Context:
Let $N$ be the largest positive integer that can be expressed as a 2013-digit base -4 number. What is the remainder when $N$ is divided by 210?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Jordan Normal Form of a nilpotent matrix $N \in \text{Mat}(10 \times 10, \mathbb{C})$ given that $\dim \ker N = 3$, $\dim \ker N^2 = 6$, and $\dim \ker N^3 = 7$. Provide the sizes of the Jordan blocks in the form of a list.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of subgroups in an abelian group of order 1,000.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x \to 0}\left(\Gamma\left(\frac{x}{3}\right) - \frac{3}{x} + 3\gamma \right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\rightarrow\infty}\sum_{k=2}^n\frac{1-e^{-(x_k-x_{k-1})}}{1+e^{-(x_k-x_{k-1})}}$$ where the points $\{x_1,..,x_n\} \in [0,1]$ are ordered $x_1 < x_2 < ... < x_n$ and form a partition of $[0,1]$. Moreover, the maximum difference $x_j - x_{j-1}$ tends to zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all weak limit points of the sequence of measures \( \{\mu_n\} \) on \((\mathbb{R}, B)\), where \( \mu_n = U_{\{\frac{1}{n}, \frac{2}{n}, \dots, \frac{n-1}{n}, 1\}} \) and \( U \) denotes the uniform distribution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Twenty circles are drawn in the plane, all passing through the origin, with no two circles tangent at the origin. Additionally, except for the origin, no three circles pass through a common point. Determine the number of regions created in the plane by these circles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the least number which when divided by 38 and another number leaves a remainder of 1 in each case. The least number is 115. What is the other divisor?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four lighthouses are located at points $A$, $B$, $C$, and $D$. The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B$, the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C$, and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C$. To an observer at $A$, the angle determined by the lights at $B$ and $D$ and the angle determined by the lights at $C$ and $D$ are equal. To an observer at $C$, the angle determined by the lights at $A$ and $B$ and the angle determined by the lights at $D$ and $B$ are equal. The number of kilometers from $A$ to $D$ is given by $\frac {p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are relatively prime positive integers, and $r$ is not divisible by the square of any prime. Find $p$ + $q$ + $r$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the integral $$\lim_{n \to \infty} \int_0^n \frac{dx}{1+n^2\cos^2x}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the kernel of the homomorphism \( k: R[x,y] \to R[x,y] \) defined by \( f(x,y) \mapsto \frac{\partial^2 f}{\partial x \partial y} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How can 50 cities be connected with the smallest number of airline routes so that it is possible to travel from any city to any other city with no more than two layovers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the optimal reserve price for a second-price sealed-bid auction where a seller offers a single private value good to two potential buyers. The buyers' valuations are uniformly distributed between 0 and 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2027\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three cards are dealt at random from a standard deck of 52 cards.  What is the probability that the first card is a 4, the second card is a $\clubsuit$, and the third card is a 2?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum for positive integers $n$ and $m$:
$$S(n, m) = \sum_{i=0}^n \frac{1}{2^{m+i+1}}\binom{m+i}{i} + \sum_{i=0}^m \frac{1}{2^{n+i+1}}\binom{n+i}{i}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the number of positive integers $n$ between $2017$ and $2017^2$ such that $n^n \equiv 1 \pmod{2017}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $2^{a}=3$ and $\log_{3}5=b$, then $\log_{15}20=$ \_\_\_\_\_\_ (express in terms of $a$ and $b$).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a convex quadrilateral with $AC \perp BD$, and let $P$ be the intersection of $AC$ and $BD$. Suppose that the distance from $P$ to $AB$ is $99$, the distance from $P$ to $BC$ is $63$, and the distance from $P$ to $CD$ is $77$. What is the distance from $P$ to $AD$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the probability of germination for each seed be 0.9, and 1000 seeds have been planted. For each seed that does not germinate, 2 more seeds need to be replanted. Let X be the number of replanted seeds. Calculate the expected value of X.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jill and her brother Jack are going apple picking. Jill's basket can hold twice as much as Jack's basket when both are full. Jack's basket is full when it has 12 apples and currently has space for 4 more. Each apple has an average weight of 150 grams. The empty baskets themselves also have weight; Jill's basket weighs 300 grams and Jack's basket weighs 200 grams. What is the total weight of apples in Jill's basket when it is full?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A standard six-sided die is rolled repeatedly until a 6 appears. Calculate the probability that the sum of all rolls, including the roll that results in a 6, is even.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What percent of the first $20$ positive integers are divisible by $3$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a day, the hands of a clock are at right angles a certain number of times. How many times are they at right angles in a day?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A fair coin is flipped 30 times. Let \( X \) denote the number of heads among the first 20 coin flips, and \( Y \) denote the number of heads among the last 20 coin flips. Compute the correlation coefficient of \( X \) and \( Y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a 4 × 4 grid of squares, where each square is colored either black or white. Two colorings are considered identical if one can be transformed into the other by any combination of flipping, rotating, or swapping the two colors (changing all black squares to white and vice versa). Determine the number of distinct colorings of the grid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that a group of students is sitting evenly spaced around a circular table and a bag containing 120 pieces of candy is circulated among them, determine the possible number of students if Sam picks both the first and a final piece after the bag has completed exactly two full rounds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ \mathcal{C}$ be the hyperbola $ y^2 \minus{} x^2 \equal{} 1$.  Given a point $ P_0$ on the $ x$-axis, we construct a sequence of points $ (P_n)$ on the $ x$-axis in the following manner: let $ \ell_n$ be the line with slope $ 1$ passing passing through $ P_n$, then $ P_{n\plus{}1}$ is the orthogonal projection of the point of intersection of $ \ell_n$ and $ \mathcal C$ onto the $ x$-axis. (If $ P_n \equal{} 0$, then the sequence simply terminates.)

Let $ N$ be the number of starting positions $ P_0$ on the $ x$-axis such that $ P_0 \equal{} P_{2008}$. Determine the remainder of $ N$ when divided by $ 2008$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Fill the numbers $1$ to $9$ into a $3 \times 3$ grid, such that in each row, the number to the right is greater than the number to the left, and in each column, the number below is greater than the number above. Given that $1$, $4$, and $9$ are already placed in the grid, determine the number of different ways to fill the remaining 6 integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each triangle in this figure is an isosceles right triangle. The length of $\overline{BC}$ is 2 units. What is the number of units in the perimeter of quadrilateral $ABCD$? Express your answer in simplest radical form.

[asy]
draw((0,0)--(10,0)--(0,10)--cycle);
draw((10,0)--(0,10)--(10,20)--cycle);
draw((1,0)--(1,1)--(0,1));
draw((1,11)--(2,10)--(1,9));
label("$A$",(0,0),S);
label("$B$",(10,0),S);
label("$C$",(10,20),N);
label("$D$",(0,10),W);

[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider two distinct positive integers $a$ and $b$ that have integer arithmetic, geometric, and harmonic means. Find the minimal value of $|a-b|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider $m$ segments on the real line. Each segment has its two endpoints in the set of integers $\{1, 2, \ldots, 2024\}$, and no two segments have the same length. No segment is entirely contained in another segment, but two segments may partially overlap each other.

What is the maximum value of $m$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the distance from the ellipsoid \(x^2+y^2+4z^2=4\) to the plane \(x+y+z=6\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知正实数 $x, y$ 满足: $\frac{1}{x+3 y}+\frac{1}{2 x+y}=1$, 则 $x+y$ 的最小值是$\frac{a}{b}(c+d \sqrt{d})$，求$a+b+c+d$的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit: $$\lim_{{n \rightarrow \infty}}{\prod_{j=1}^{n}\Big(1+\frac{2j-1}{2n}\Big)^{\frac{1}{2n}}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest number \( m \) such that for any positive numbers \( a, b, \) and \( c \) whose sum equals 1, the following inequality holds:

$$
\sqrt{\frac{a b}{c+a b}}+\sqrt{\frac{b c}{a+b c}}+\sqrt{\frac{c a}{b+c a}} \geqslant m
$$

(E. Emelyanov)
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the ratio of the legs in a right triangle, if the triangle formed by its altitudes as sides is also a right triangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum possible order of a group $G$ in which every element $g$ satisfies $g^3 = e$, the identity element?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( OP \) be the diameter of the circle \( \Omega \), and let \( \omega \) be a circle with its center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of circle \( \Omega \) intersects the second circle at point \( A \). Find the length of the segment \( AB \) if \( BD \cdot BC = 5 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a, b, c, x, y,$ and $z$ be complex numbers such that \[ a = \frac{b+c}{x-2}, \ b = \frac{c+a}{y-2}, \ c = \frac{a+b}{z-2}. \] If $xy + yz + xz = 67$ and $x + y + z = 2010$, find the value of $xyz$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For $\pi \le \theta < 2\pi$, let
\begin{align*} P &= \frac12\cos\theta - \frac14\sin 2\theta - \frac18\cos 3\theta + \frac{1}{16}\sin 4\theta + \frac{1}{32} \cos 5\theta - \frac{1}{64} \sin 6\theta - \frac{1}{128} \cos 7\theta + \cdots \end{align*}
 and
\begin{align*} Q &= 1 - \frac12\sin\theta -\frac14\cos 2\theta + \frac18 \sin 3\theta + \frac{1}{16}\cos 4\theta - \frac{1}{32}\sin 5\theta - \frac{1}{64}\cos 6\theta +\frac{1}{128}\sin 7\theta + \cdots \end{align*}
 so that $\frac{P}{Q} = \frac{2\sqrt2}{7}$. Then $\sin\theta = -\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( N \) denote the number of subsets of \(\{1,2,3, \ldots, 100\}\) that contain more prime numbers than multiples of 4. Compute the largest integer \( k \) such that \( 2^{k} \) divides \( N \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that a^2 - b^2 = 10 and a * b = some value, find the value if a^4 + b^4 = 228.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A motorist embarks on a complex journey lasting 8 hours. He begins by driving for 2 hours at a speed of 60 kmph, followed by 3 hours of travel on a train at a constant speed of 100 kmph. He then takes a 45-minute break before driving for the remaining portion of the trip at varying speeds: for the first 75 minutes, he drives at 72 kmph, then slows down to 50 kmph for 95 minutes. Calculate the total distance covered by the motorist throughout the journey.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let's find such an $n$ and such numbers $a_{i}, b_{i}, c_{i},(i=1,2, \ldots, n)$, for which the points $(x, y)$ satisfying the condition

$$
\sum_{i=1}^{n}\left|a_{i} x+b_{i} y+c_{i}\right|=10
$$

when plotted on the coordinate system, form the vertices of a regular octagon!
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T: \mathcal P_2(\Bbb R) \rightarrow \mathcal P_2(\Bbb R)$ be a linear transformation defined by $T(p(x)) = p(1)(x^2 + x) + p(0)$. Consider the standard ordered basis $\alpha = \{x^2, x, 1\}$ for $\mathcal P_2(\Bbb R)$. Determine the matrix $[T]_\alpha$ that represents $T$ with respect to basis $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
a 180 meter long train crosses a man standing on the platform in 9 sec . what is the speed of the train ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( x \in \mathbf{R} \), the sum of the maximum and minimum values of the function \( f(x)=\max \left\{\sin x, \cos x, \frac{\sin x+\cos x}{\sqrt{2}}\right\} \) is equal to?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The function \( f(x) = -\log_{\frac{1}{2}}\left(x^{2} - ax - a\right) \) is a decreasing function on the interval \( (-\infty, 1 - \sqrt{3}) \). Find the range of values for \( a \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $S$  be the set of positive integers not divisible by  $p^4$  for all primes  $p$ . Anastasia and Bananastasia play a game.

At the beginning, Anastasia writes down the positive integer  $N$  on the board. Then the players take moves in turn; Bananastasia moves first. On any move of his, Bananastasia replaces the number  $n$  on the blackboard with a number of the form  $n-a$ , where  $a\in S$  is a positive integer. On any move of hers, Anastasia replaces the number  $n$  on the blackboard with a number of the form  $n^k$ , where  $k$  is a positive integer. Bananastasia wins if the number on the board becomes zero.

Compute the second-smallest possible value of  $N$  for which Anastasia can prevent Bananastasia from winning.
Please reason step by step, and put your final answer within \boxed{}.


Context:
One end of a bus route is at Station $A$ and the other end is at Station $B$. The bus company has the following rules:
(1) Each bus must complete a one-way trip within 50 minutes (including the stopping time at intermediate stations), and it stops for 10 minutes when reaching either end.
(2) A bus departs from both Station $A$ and Station $B$ every 6 minutes. Determine the minimum number of buses required for this bus route.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Michael writes down all the integers between $1$ and $N$ inclusive on a piece of paper and discovers that exactly $40\%$ of them have leftmost digit $1$. Given that $N > 2017$, find the smallest possible value of $N$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the integral $$\begin{align}I=\int_{0}^{1}\frac{\ln(x)\,dx}{1-x^2}\end{align}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute
\[\frac{\lfloor \sqrt[4]{1} \rfloor \cdot \lfloor \sqrt[4]{3} \rfloor \cdot \lfloor \sqrt[4]{5} \rfloor \dotsm \lfloor \sqrt[4]{2015} \rfloor}{\lfloor \sqrt[4]{2} \rfloor \cdot \lfloor \sqrt[4]{4} \rfloor \cdot \lfloor \sqrt[4]{6} \rfloor \dotsm \lfloor \sqrt[4]{2016} \rfloor}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
6. Given the set
$$
A=\left\{n \in \mathbf{Z} \left\lvert\,\left(1+\frac{1}{n}\right)^{n+1}=\left(1+\frac{1}{2020}\right)^{2000}\right.\right\} \text {. }
$$

Then the number of elements in set $A$ is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
5th Putnam 1942 Problem B4 A particle moves in a circle through the origin under the influence of a force a/r k towards the origin (where r is its distance from the origin). Find k.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If a taxpayer has an annual income before tax of $42,400 from three different sources - salary ($28,000), freelance work ($10,000), and investments ($4,400) - with a standard deduction of $12,000, a personal exemption of $4,000, and additional deductible expenses (such as medical and child care) of $6,500, and they are filing as a single individual, what is the differential savings in their tax liability if the income tax rates change in the following manner for each tax bracket:

Current Tax Rates:
- 10% on income up to $9,850
- 15% on income between $9,851 and $23,250
- 25% on income between $23,251 and 37,000
- 42% on income over $37,001

New Tax Rates:
- 10% on income up to $13,500
- 15% on income between $13,501 and $25,000
- 25% on income between $25,001 and $39,000
- 32% on income over $39,001

Calculate the differential savings for the tax payer considering these additional factors.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The variables x and y have a linear correlation. When x takes the values of 16, 14, 12, and 8, the observed values of y are 11, 9, 8, and 5 respectively. Using the reference formulas for linear regression, predict the approximate value of x when y = 10. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Insert a digit in the middle of a two-digit number to form a three-digit number. For some two-digit numbers, the resulting three-digit number can be $k$ times the original two-digit number (where $k$ is a positive integer). What is the maximum value of $k$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If "+" or "-" is added in front of the numbers $\frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}$, making their algebraic sum equal to $1$, then there are a total of _______ ways to do this.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Clever Cat sleeps under the oak tree from noon till midnight and is awake telling stories from midnight till noon. A poster on the tree above him says "Two hours ago, Clever Cat was doing the same thing as he will be doing in one hour's time". Calculate the number of hours a day the poster tells the truth.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If both \( 7n+1 \) and \( 8n+1 \) can be represented as the sum of three distinct positive integers in a geometric progression, what is the smallest value of the positive integer \( n \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all odd integers \( n > 1 \) for which there is at least one permutation \( a_1, a_2, \ldots, a_n \) of \( 1, 2, 3, \ldots, n \) such that the sums \( a_1 - a_2 + a_3 - \ldots + a_n \), \( a_2 - a_3 + a_4 - \ldots - a_n + a_1 \), \( a_3 - a_4 + \ldots + a_n - a_1 + a_2 \), \( \ldots \), \( a_n - a_1 + a_2 - \ldots + a_{n-1} \) are all positive.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circular disk is divided by 2n equally spaced radii (n > 0) and one chord. Calculate the maximum number of non-overlapping regions into which the disk can be divided. Express your answer as a formula in terms of n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sum of the first four terms of an arithmetic progression, as well as the sum of the first seven terms, are natural numbers. Furthermore, its first term \(a_1\) satisfies the inequality \(a_1 \leq \frac{2}{3}\). What is the greatest value that \(a_1\) can take?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the set \( S \) contain 2012 elements, where the ratio of any two elements is not an integer. An element \( x \) in \( S \) is called a "good element" if there exist distinct elements \( y \) and \( z \) in \( S \) such that \( x^2 \) divides \( y \cdot z \). Find the maximum possible number of good elements in \( S \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In equilateral triangle $ABC$ with side length 2, let the parabola with focus $A$ and directrix $BC$ intersect sides $AB$ and $AC$ at $A_{1}$ and $A_{2}$, respectively. Similarly, let the parabola with focus $B$ and directrix $CA$ intersect sides $BC$ and $BA$ at $B_{1}$ and $B_{2}$, respectively. Finally, let the parabola with focus $C$ and directrix $AB$ intersect sides $CA$ and $CB$ at $C_{1}$ and $C_{2}$, respectively. Find the perimeter of the triangle formed by lines $A_{1}A_{2}, B_{1}B_{2}, C_{1}C_{2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Task 4 - 250914 Three circles with the given radius $r$ may lie in a plane such that each touches the other two. A common tangent is drawn to each pair of these three circles, which does not have any point in common with the third circle. With these three tangents, an equilateral triangle is constructed.

Calculate the area of this triangle in dependence on $r$!

Hint: For numerical values that occur in the area specification, the use of approximate values is permitted (but not required); if so, however, with an indication - and justification - of how many decimal places the approximate value is accurate to.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum length of a closed self-avoiding polygon that can travel along the grid lines of an $8 \times 8$ square grid?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The dollar is now worth $\frac{1}{980}$ ounce of gold. After the $n^{th}$ 7001 billion dollars bailout package passed by congress, the dollar gains $\frac{1}{2^{2^{n-1}}}$ of its $(n-1)^{th}$ value in gold. After four bank bailouts, the dollar is worth $\frac{1}{b}(1-\frac{1}{2^c})$ in gold, where $b, c$ are positive integers. Find $b + c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the last two digits of the decimal representation of  $9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alice is once again very bored in class. On a whim, she chooses three primes $p, q, r$ independently and uniformly at random from the set of primes of at most 30. She then calculates the roots of $p x^{2}+q x+r$. What is the probability that at least one of her roots is an integer?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A B C D$ be a rectangle with $A B=8$ and $A D=20$. Two circles of radius 5 are drawn with centers in the interior of the rectangle - one tangent to $A B$ and $A D$, and the other passing through both $C$ and $D$. What is the area inside the rectangle and outside of both circles?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Gus has to make a list of  $250$  positive integers, not necessarily distinct, such that each number is equal to the number of numbers in the list that are different from it. For example, if  $15$  is a number from the list so the list contains  $15$  numbers other than  $15$ . Determine the maximum number of distinct numbers the Gus list can contain.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An archipelago consisting of an infinite number of islands is spread out along the southern shore of an endless sea in a chain. The islands are connected by an infinite chain of bridges, and each island is also connected to the shore by a bridge. In the event of a strong earthquake, each bridge independently has a probability $p=0.5$ of being destroyed. What is the probability that, after a strong earthquake, it will be possible to travel from the first island to the shore using the remaining bridges?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of positive integers less than or equal to  $2017$  that have at least one pair of adjacent digits that are both even. For example, count the numbers  $24$ ,  $1862$ , and  $2012$ , but not  $4$ ,  $58$ , or  $1276$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\triangle ABC$ have $AB=9$ and $AC=10$. A semicircle is inscribed in $\triangle ABC$ with its center on segment $BC$ such that it is tangent to $AB$ at point $D$ and $AC$ at point $E$. If $AD=2DB$ and $r$ is the radius of the semicircle, $r^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Edward starts at his house located at \((0,0)\) and needs to reach the point \((6,4)\), which is the coordinate for his school. However, there is a park shaped as a square with coordinates \((2,1), (2,3), (4,1), (4,3)\). Edward cannot walk inside the park, but there is a road along the perimeter of the square. Determine how many different shortest routes exist from Edward's house to his school.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four cubes with edge lengths $2$, $3$, $4$, and $5$ are stacked with their bottom faces on the $xy$-plane, and one vertex at the origin $(0,0,0)$. The stack sequence follows the increasing order of cube sizes from the bottom. If point $X$ is at $(0,0,0)$ and point $Y$ is at the top vertex of the uppermost cube, determine the length of the portion of $\overline{XY}$ contained in the cube with edge length $4$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four points, $A, B, C$, and $D$, are chosen randomly on the circumference of a circle with independent uniform probability. What is the expected number of sides of triangle $A B C$ for which the projection of $D$ onto the line containing the side lies between the two vertices?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $M$ is a point on the parabola $x^2 = 4y$, and $F$ is its focus. Point $A$ is on the circle $C: (x+1)^2 + (y-5)^2 = 1$. Find the minimum value of $|MA| + |MF|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sample set with a capacity of 20 that is divided into groups, the group intervals and their respective frequencies are as follows: (10, 20), 2; (20, 30), 3; (30, 40), 4; (40, 50), 5; (50, 60), 4; (60, 70), 2. Calculate the frequency of the sample in the interval $(-∞, 50)$. Express your answer as a decimal value, rounded to 2 decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The coordinates of the point $P(2,3)$ with respect to the $x$-axis are ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The solution set of the inequality $(x-2)\sqrt{x+3} \geq 0$ is __________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f$ be a function defined on positive integers such that $f(1)=4$, $f(2n)=f(n)$ and $f(2n+1)=f(n)+2$ for every positive integer $n$. Find the number of positive integers $k$ less than $2014$ for which $f(k)=8$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are relatively prime positive integers $p$ and $q$ such that $\dfrac{p}{q}=\displaystyle\sum_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$.  Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The 12 numbers from 1 to 12 on a clock face divide the circumference into 12 equal parts. Using any 4 of these division points as vertices to form a quadrilateral, find the total number of rectangles that can be formed.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The problem of transporting goods from a riverside city A to another location B. The shortest distance from B to the river bank is BC = b kilometers, and the distance of AC along the river is a kilometers. If the water transportation cost is \(\frac{1}{n}\) times the road transportation cost (where \(n\) is a number greater than 1), to minimize the total transportation cost from A to B, what is \(DC = \qquad\) kilometers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two lines $l_{1}$: $(a+2)x+(a+3)y-5=0$ and $l_{2}$: $6x+(2a-1)y-5=0$ are parallel, then $a=$        .
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task 11/78

A mathematician writes to another: "This year I will be 100 years old, next year 200."

Obviously, the age references are not in the decimal system. How old is the mathematician?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tanika is selling boxes of crackers for her scout troop's fund-raiser. On Thursday, she sold 60 boxes at a 10% discount. On Friday, she sold 50% more boxes than on Thursday at a 15% discount. On Saturday, she sold 80% more boxes than on Friday at a 20% discount. On Sunday, she sold 30% less boxes than on Saturday at a 25% discount. How many boxes did she sell, in total, over the four days?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the flux of the vector field $\mathbf{F} = 5x\hat{i} + y\hat{j} + z\hat{k}$ through the surface $S_1$ of the paraboloid $x = y^2 + z^2$ oriented in the negative $x$ direction, bounded by the plane $x = 4$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of strictly increasing sequences of nonnegative integers with the following properties:
- The first term is 0 and the last term is 12. The sequence has at least two terms.
- Among any two consecutive terms, exactly one of them is even.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Write down a set of Pythagorean triples (three positive integers that can form the three sides of a right triangle): $\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(b_1,b_2,b_3,\ldots,b_{10})$ be a permutation of $(1,2,3,\ldots,10)$ for which
$b_1>b_2>b_3>b_4 \mathrm{\ and \ } b_4<b_5<b_6<b_7<b_8<b_9<b_{10}.$
Find the number of such permutations.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $S$ cans of soda can be purchased from a vending machine for $Q$ quarters. Which of the following expressions describes the number of cans of soda that can be purchased for $D$ dollars, where $1$ dollar is worth $4$ quarters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose  $a_0,a_1,\ldots, a_{2018}$  are integers such that \[(x^2-3x+1)^{1009} = \sum_{k=0}^{2018}a_kx^k\] for all real numbers  $x$ .  Compute the remainder when  $a_0^2 + a_1^2 + \cdots + a_{2018}^2$  is divided by  $2017$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
$2 \cdot 107$ Let the natural number $n \geqslant 5, n$ different natural numbers $a_{1}, a_{2}, \cdots, a_{n}$ have the following property: for any two different non-empty subsets $A$ and $B$ of the set
$$
S=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}
$$

the sum of all numbers in $A$ and the sum of all numbers in $B$ will not be equal. Under the above conditions, find
$$
\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}
$$

the maximum value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the surface integral $$\int_S yds$$ where $S$ is the part of the plane $$z=1+y$$ that lies inside the cone $$z=\sqrt{2(x^2+y^2)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the pyramid \( PABCD \), which has a quadrilateral \( ABCD \) with sides 5, 5, 10, and 10 as the base, its shorter diagonal being \( 4\sqrt{5} \), and all the lateral faces inclined at an angle of \( 45^\circ \) to the base.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A particular novel contains $40,\!000$ words. If the author completed the novel in $80$ hours, on average how many words per hour did she write?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that the probabilities of person A and person B hitting the target in each shot are $\frac{3}{4}$ and $\frac{4}{5}$, respectively, and that each shot is independent. If they take shots in the order of A, B, A, B, until one of them hits the target and then stop shooting, what is the probability that A has taken two shots when they stop shooting?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are  $20$  geese numbered  $1-20$  standing in a line. The even numbered geese are standing at the front in the order  $2,4,\dots,20,$  where  $2$  is at the front of the line. Then the odd numbered geese are standing behind them in the order,  $1,3,5,\dots ,19,$  where  $19$  is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered  $1,2,\dots,20$  (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an infinite sequence $\{a_n\}$, where $a_1=1$, $a_2=2$, and for any $n\in N^*$, $a_{n+2}=a_n$. The sequence $\{b_n\}$ satisfies $b_{n+1}-b_n=a_n$ ($n\in N^*$). If any term in the sequence $\left\{ \frac{b_{2n}}{a_n} \right\}$ repeats infinitely many times within the sequence, then the value of $b_1$ that meets the requirement is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If three angles \(x, y, z\) form an arithmetic sequence with a common difference of \(\frac{\pi}{2}\), then \(\tan x \tan y + \tan y \tan z + \tan z \tan x = \) ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two equal circles intersect at point \( C \). Through point \( C \), two lines are drawn, intersecting the circles at points \( A, B \) and \( M, N \) respectively. The line \( AB \) is parallel to the line of centers, and the line \( MN \) forms an angle \( \alpha \) with the line of centers. Given that \( AB = a \), find \( NM \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The opening of a pit is a circle with a diameter of $100 \mathrm{~m}$, and its cross-sections passing through the center of the circle are equilateral triangles. Let $A$ and $B$ be two diametrically opposite points on the opening, $C$ the deepest point of the pit, and $H$ the midpoint of $BC$. What is the length of the shortest path connecting $A$ with $H$?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{50\sqrt{5} \approx 111.8 \text{ m}}


Context:
17 Find the number of all positive integers $a$ that satisfy the following condition: there exist non-negative integers $x_{0}, x_{1}, x_{2}, \cdots$, $x_{2001}$, such that $a^{x_{0}}=a^{x_{1}}+a^{x_{2}}+\cdots+a^{x_{2001}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A divisor of a natural number is called proper if it is different from 1 and the number itself. A number is called interesting if it has two proper divisors, one of which is a prime number, and the other is a perfect square, and their sum is also a perfect square. How many interesting numbers are there that do not exceed 1000?
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom{9}{2}=36$ lines. Assume that all 36 lines drawn are distinct. What is the maximum possible number of points which lie on at least two of the drawn lines?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle is circumscribed around $ABCD$ as follows: [asy]
pair pA, pB, pC, pD, pO;
pO = (0, 0);
pA = pO + dir(-40);
pB = pO + dir(40);
pC = pO + dir(130);
pD = pO + dir(190);
draw(pA--pB--pC--pA);
draw(pA--pD--pC--pA);
label("$A$", pA, SE);
label("$B$", pB, NE);
label("$C$", pC, NW);
label("$D$", pD, SW);
draw(circle(pO, 1));
label("$30^\circ$", pA + dir(150) * .45);
label("$40^\circ$", pC + dir(-20) * .35);
[/asy] How many degrees are in $\angle CAB + \angle ACD$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
15 points $P_i$ are placed within the unit sphere. Find the maximum possible value of \(\sum_{1\leq i < j \leq 15} |P_i P_j|^2\) given that \(x_i^2+y_i^2+z_i^2 \leq 1\) for each point \(P_i = (x_i, y_i, z_i)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(ABCD\) be a square of side length 2. Let points \(X, Y\), and \(Z\) be constructed inside \(ABCD\) such that \(ABX, BCY\), and \(CDZ\) are equilateral triangles. Let point \(W\) be outside \(ABCD\) such that triangle \(DAW\) is equilateral. Let the area of quadrilateral \(WXYZ\) be \(a+\sqrt{b}\), where \(a\) and \(b\) are integers. Find \(a+b\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a track for remote-controlled racing cars, racing car A completes the track in some seconds, while racing car B completes it in 24 seconds. If they both start at the same time, they will be side by side again after 168 seconds. How many seconds does it take for racing car A to complete the track?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $N=10^6$ . For which integer  $a$  with  $0 \leq a \leq N-1$  is the value of \[\binom{N}{a+1}-\binom{N}{a}\] maximized?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum number of rooks that can be placed in an $8 \times 8 \times 8$ cube so that no two rooks attack each other?
Please reason step by step, and put your final answer within \boxed{}.


Context:
During a town winter carnival, a group of friends - Steven, Stephanie, Sonya, Sam, Sue, and Sophie - decided to go ice skating. Steven, who is 20 years old and has been skating since he was 6, only fell down 3 times, but Stephanie, who is 24 and has only been skating for a year, had 13 more falls than Steven. Sonya, on the other hand, fell down 2 times less than half the number of times Stephanie fell. Sam, who is the same age as Stephanie but has been skating for 10 years, fell only once, whereas Sophie, who is two years younger than Sonya and has been skating for half as long as Sonya, fell four times more than Sam. Sue, the oldest in the group at 26 years old, has 15 years of skating experience. She fell one time more per year of her age difference with the youngest member of the group. Taking into account the complex relationships between the friends' ages, experience levels, and fall patterns, how many times did Sue fall?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $\sin(x-y)\cos x - \cos(x-y)\sin x = \frac{3}{5}$, then the value of $\tan 2y$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p(z) = az^2+bz+c$ be a quadratic polynomial with real coefficients. If $a > 0$, $|c| > 1$, and $b^2-2ac > 0$, find the number of roots of the function $g(z) = p(z) - e^{z}$ in the first quadrant of the complex plane.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangular piece of paper is divided into polygonal areas in the following way: at each step, one of the existing surfaces is cut by a straight line, resulting in two new areas. What is the minimum number of cuts needed such that among the obtained polygons, there are $251$ polygons with $11$ sides?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find \( y'' \) if \( x = \ln t \) and \( y = \sin 2t \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $a_0 + a_1 + a_2 + \cdots + a_6$ given that $(2-x)^7 = a_0 + a_1(1+x)^2 + \cdots + a_7(1+x)^7$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Right $\triangle ABC$ has $AB=3, BC=4,$ and $AC=5.$ Square $XYZW$ is inscribed in $\triangle ABC$ with $X$ and $Y$ on $\overline{AC}, W$ on $\overline{AB},$ and $Z$ on $\overline{BC}.$ What is the side length of the square?
Please reason step by step, and put your final answer within \boxed{}.


Context:
小明同学任选一个各项均为非零整数的等差数列 $\left\{a_{n}\right\}$ ，随后他列出 2020 个关于 $x$ 的一元二次方程:
$$
E_{i}: a_{i} x^{2}+a_{i+1} x+a_{i+2}=0(i=1,2, \cdots, 2020),
$$
再将上述每个方程的根都写在黑板上.求此时黑板上两两不同的实根的个数的最大可能值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit: $$\lim_{n\to\infty}\left(n-(\arccos(1/n)+\cdots+\arccos(n/n))\right).$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $A B C$ is a triangle with circumcenter $O$ and orthocenter $H$ such that $A, B, C, O$, and $H$ are all on distinct points with integer coordinates. What is the second smallest possible value of the circumradius of $A B C$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence $\{a_n\}$ where $a_n = n$, for each positive integer $k$, in between $a_k$ and $a_{k+1}$, insert $3^{k-1}$ twos (for example, between $a_1$ and $a_2$, insert three twos, between $a_2$ and $a_3$, insert $3^1$ twos, between $a_3$ and $a_4$, insert $3^2$ twos, etc.), to form a new sequence $\{d_n\}$. Let $S_n$ denote the sum of the first $n$ terms of the sequence $\{d_n\}$. Find the value of $S_{120}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation \(f(x-1) + f(x+3) = f(x+1) + f(x+5)\), determine the period of the function \(f(x)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the quadratic function $y=2mx^{2}+\left(8m+1\right)x+8m$ with respect to $x$, the graph intersects the $x$-axis. Find the range of $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \large \int_{0}^{\pi/2}\cos(2x)\ln^2(\tan(x/2))\mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=\frac{1}{2}ax^{2}-(x-1)e^{x}$, if for any real numbers $(x_{1}, x_{2}, x_{3})$ in the interval $[0,1]$, we have $f(x_{1})+f(x_{2})\geqslant f(x_{3})$, then the range of values for the real number $a$ is ________
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set of all injective functions from an uncountable set to the set of bounded functions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The organizers of a fair projected a 25 percent increase in attendance this year over that of last year. Last year, the fair saw A attendees. However, due to an increase in ticket prices by $5 and unfavorable weather conditions, attendance this year actually decreased by 20 percent. The weather affected V percentage of the days of the fair, causing a reduction by W visitors per day due to the harsh conditions. Taking into account these factors, calculate the percent of the projected attendance that was the actual attendance this year.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an equilateral triangle \(ABC\) with side length 1, points \(D\) and \(E\) are located on sides \(AB\) and \(AC\) respectively. When the triangle is folded along the line segment \(DE\), vertex \(A\) lands exactly on side \(BC\). Determine the minimum length of \(AD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The numbers $3, 6, 10, 15, \cdots$ are called triangular numbers because they can be arranged in the form of an equilateral triangle. Find the 19th triangular number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $\cos 78^{\circ}$ is approximately $\frac{1}{5}$, what is the approximate value of $\sin 66^{\circ}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $Q(1)$ if $Q(x)$ is a polynomial satisfying $Q(x)-Q'(x) = (x-1)^n$, where $n$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a regular tetrahedron $S-ABCD$, $O$ is the projection of the apex $S$ onto the base, $P$ is the midpoint of the lateral edge $SD$, and $SO=OD$. The angle formed by line $BC$ and plane $PAC$ is equal to _____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There is a jar containing $m$ white balls and $n$ black balls $(m > n)$. Balls are drawn one by one without replacement. If at some point during the drawing process, the number of white balls drawn equals the number of black balls drawn, this is called a coincidence. Find the probability of having at least one coincidence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose Sachin borrows an amount X from a lender at an annual interest rate of R1% for T1 years. He immediately lends money to Rahul and Ravi at the annual interest rates of R2% and R3% for T2 and T3 years, respectively. Calculate the gain of Sachin in one year if:

X = Rs. 5000
R1 = 4% p.a.
T1 = 2 years
R2 = 6.25% p.a. (25/4%)
T2 = 2 years
R3 = 7.5% p.a.
T3 = 3 years

(Note that only Rahul repays the loan within the same period as the original borrowed amount, which is 2 years. Ravi repays the loan after 3 years.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 6.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given four numbers, the sum of every three numbers among them is 17, 21, 25, and 30, respectively. Find these four numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For some integer values of n, the value of the expression 4n + 7 is an integer greater than a certain number and less than 40. There are 10 such integer values of n. What is the lower bound if the expression is greater than 1?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute $$2 \sqrt{2 \sqrt[3]{2 \sqrt[4]{2 \sqrt[5]{2 \cdots}}}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T$ be the set of ordered triples $(x,y,z)$, where $x,y,z$ are integers with $0\leq x,y,z\leq9$. Players $A$ and $B$ play the following guessing game. Player $A$ chooses a triple $(x,y,z)$ in $T$, and Player $B$ has to discover $A$[i]'s[/i] triple in as few moves as possible.  A [i]move[/i] consists of the following: $B$ gives $A$ a triple $(a,b,c)$ in $T$, and $A$ replies by giving $B$ the number $\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|$. Find the minimum number of moves that $B$ needs to be sure of determining $A$[i]'s[/i]  triple.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a large bag of decorative ribbons, $\frac{1}{4}$ are yellow, $\frac{1}{3}$ are purple, $\frac{1}{8}$ are orange, and the remaining 45 ribbons are silver. How many of the ribbons are orange?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A chess king is placed in the bottom-left corner of a chessboard. Two players take turns moving the king. On each turn, the king can be moved one square to the right, one square up, or one square diagonally "up-right." The player who places the king in the top-right corner of the board wins the game. Assuming both players play optimally, which player will win?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The numbers from 1 to 1996 are written sequentially as follows: 12345678910111213... up to 1996. Find how many zeros appear in this sequence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a positive integer \(a\), there exists a quadratic trinomial with integer coefficients, where \(a\) is the coefficient of the quadratic term, and it has two distinct positive roots both less than 1. Find the smallest value of \(a\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jane is considering buying a sweater priced at $50. A store is offering a 10% discount on the sweater. After applying the discount, Jane needs to pay a state sales tax of 7.5%, and a local sales tax of 7%. Calculate the difference between the state and local sales taxes that Jane has to pay.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A vertical wooden pillar has a rope tied to its top. The part of the rope hanging down from the top of the pillar to the ground measures 4 feet. When pulling the rope, it runs out when 8 feet away from the base of the pillar. Calculate the total length of the rope. Express your answer in feet.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sum was put at simple interest at a certain rate for 9 years. Had it been put at a higher rate, it would have fetched $1,350 more. The sum is $3,000. What is the difference in percentage between the original rate and the higher rate?
Please reason step by step, and put your final answer within \boxed{}.


Context:
20. Suppose that $a, b, c$ are real numbers such that
$$
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=4\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)=\frac{c}{a+b}+\frac{a}{b+c}+\frac{b}{c+a}=4 .
$$

Determine the value of $a b c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any natural values \( m \) and \( n \) (with \( n > 1 \)), a function is defined as \( f\left(\frac{m}{n}\right)=\frac{\sqrt[n]{3^{m}}}{\sqrt[n]{3^{m}}+3} \). Compute the sum

\[ f\left(\frac{1}{2020}\right)+f\left(\frac{2}{2020}\right)+f\left(\frac{3}{2020}\right)+\cdots+f\left(\frac{4039}{2020}\right)+f\left(\frac{4040}{2020}\right) \].
Please reason step by step, and put your final answer within \boxed{}.


Context:
We can label the squares of an 8 x 8 chessboard from 1 to 64 in 64! different ways. For each way, we find \( D \), the largest difference between the labels of two squares that are adjacent (orthogonally or diagonally). What is the smallest possible \( D \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, point $F$ is on segment $AB$ such that $CF$ bisects $\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD$ and $BE$ are perpendicular to $CF$. Let $M$ be the midpoint of $AB$. If $ME = 13$, $AD = 15$, and $BE = 25$, find $AC + CB$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
1. Given that $A D$ and $B E$ are the angle bisectors of $\triangle A B C$, and they intersect at point $I$, with $I D=I E$. If $\angle C A B=80^{\circ}$, then $\angle A B C=$ $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $A$ is a square matrix satisfying $$A^{T}A = I,$$ write the value of $|A|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Gloria wants to buy the $129,000 mountain cabin that her friend Alfonso is selling. She only has $150 in cash. She intends to raise the remaining amount by selling her mature trees for lumber. She has 20 cypress trees, 600 pine trees, and 24 maple trees. She will get $100 for each cypress tree, a certain amount for a maple tree, and $200 per pine tree. After paying Alfonso for the cabin, Gloria has $350 left. How much does Gloria get for each maple tree?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many 9-digit numbers, divisible by 2, can be formed by permuting the digits of the number 231157152?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each of the twelve letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "MIST"? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the covariance of the process $X(t) = \int_0^t B(u)\,du$ where $B$ is a standard Brownian motion? i.e., I wish to find $E[X(t)X(s)]$, for $0<s<t<\infty$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Richard starts with the string $HHMMMMTT$. A move consists of replacing an instance of $HM$ with $MH$, replacing an instance of $MT$ with $TM$, or replacing an instance of $TH$ with $HT$. Compute the number of possible strings he can end up with after performing zero or more moves.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a, b, c, d, e, f \) be integers selected from the set \( \{1,2, \ldots, 100\} \), uniformly and at random with replacement. Set
\[ M = a + 2b + 4c + 8d + 16e + 32f. \]
What is the expected value of the remainder when \( M \) is divided by 64?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit:

$$\lim_{n \to \infty} \int_{\epsilon}^1\frac {n+1}{2^{n+1}}\frac{(1+t)^{n+1}-(1-t)^{n+1}}t\,dt$$

where $\epsilon \in (0, 1)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
2. 8 girls and 25 boys stand in a circle, with at least two boys standing between any two girls. How many different arrangements are there (arrangements that coincide by rotating the circle are considered the same)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three circles with radii \(\frac{1}{2}\), \(\frac{1}{2}\), and \(\sqrt{\frac{13-6 \sqrt{3}}{13}}\) are positioned such that the triangle formed by the centers of these circles is equilateral with a side length of \(\frac{\sqrt{3}}{2}\). Find the radius of the circumcircle around the triangle, each vertex of which is the intersection point of two of these circles, and is farther from the center of the third circle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sum of the absolute values of the terms of a finite arithmetic progression is 100. If all its terms are increased by 1 or all its terms are increased by 2, the sum of the absolute values of the terms of the resulting progression will also be 100. What values can the quantity $n^{2} d$ take under these conditions, where $d$ is the common difference of the progression and $n$ is the number of its terms?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( I \) over the region \( Q \) in \( \mathbb{R}^3 \) defined by:

\[ I = \int_{Q} e^{-x-y-z} \, dV \]

where \( Q = \{ (x, y, z) : x + y > 0, \ y + z > 0, \ x + z > 0 \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The monotonic decreasing interval of the function $y=2^{x^2-2x+1}$ is ______________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \(\displaystyle\lim_{n\to\infty}\sqrt[n]{|1-|z|^n|}\) for \(z\in\mathbb{C}\) with \(|z|\ne 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside the region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $L = \mathbb{Q}(\sqrt[4]{2}, \omega)$ where $\omega = e^{\frac{2\pi i}{8}}$. Determine the degree of the field extension $[L: \mathbb{Q}(\sqrt[4]{2})]$. \( \boxed{\text{Your answer here}} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a tight asymptotic upper bound for the recurrence relation \( p(n) = \sqrt{n}p(\sqrt{n}) + \sqrt{n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), points \(M\) and \(N\) are marked on the sides \(AB\) and \(BC\) respectively, such that \(BM = BN\). A line is drawn through point \(M\) perpendicular to \(BC\), and another line is drawn through point \(N\) perpendicular to \(AB\). These lines intersect at point \(O\). The extension of segment \(BO\) intersects side \(AC\) at point \(P\) and divides it into segments \(AP = 5\) and \(PC = 4\). Find the length of segment \(BP\), given that \(BC = 6\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A man saves 10% of his monthly salary. If on account of dearness of things he is to increase his monthly expenses by 5%, he is only able to save Rs. 400 per month. What is his monthly salary?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can we change the sign $ *$ with $ +$ or $ -$, such that the following equation is true?
\[ 1 *2*3*4*5*6*7*8*9*10=29\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sequence \( a_1, a_2, a_3, \ldots \) is defined as follows: \( a_1 = 1 \). For \( n \geq 2 \), \( a_n \) is the smallest integer greater than \( a_{n-1} \) such that there do not exist \( 1 \leq i, j, k \leq n \) (not necessarily distinct) such that \( a_i + a_j = 3a_k \). Find \( a_{1998} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all values of $p$ for which the divergence of the vector field $\vec{F} = r^p\vec{r}$ is zero, where $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$ and $r = \|\vec{r}\|$. Provide your answer in terms of $p$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jamie, a history enthusiast with a passion for social and educational movements, is organizing a series of educational talks about influential figures in history. Jamie has invited 5 speakers, each specializing in a different movement: civil rights, feminism, environmentalism, educational reform, and labor rights.

Each speaker will give two talks, one in the morning and one in the afternoon. Jamie plans to offer 3 history-themed snacks for each talk: one representing the era, one representing the movement, and one representing the speaker's favorite historical figure.

If each snack costs $2 and Jamie expects 20 people to attend each talk, how much will Jamie spend on snacks for all the talks combined?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation \(\frac{x^3+2x}{x^2-1}=\sqrt{x^2-\frac{1}{x}}\) for \(x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume enclosed by the paraboloids \(z = x^2 + (y-1)^2\) and \(z = \frac{5}{2} - x^2 - y^2\) for \(x \geq 0\). Use a triple integral to find the volume.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint_{R} f \cdot g \, dx \, dy \) over the circular disk \( R \) with boundary \( x^2 + y^2 = 1 \), where the vector fields \( f \) and \( g \) are defined as follows:

\[ f(x,y) = v(x,y) \mathbf{i} + u(x,y) \mathbf{j} \]
\[ g(x,y) = \left(\frac{\partial u}{\partial x} - \frac{\partial u}{\partial y}\right) \mathbf{i} + \left(\frac{\partial v}{\partial x} - \frac{\partial v}{\partial y}\right) \mathbf{j} \]

Given that on the boundary of \( R \), \( u(x,y) = 1 \) and \( v(x,y) = y \), find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知离心率为$\frac{2}{3}$的椭圆$C:\frac{{{x}^{2}}}{{{a2}}}+\frac{{{y}^{2}}}{{{b2}}}=1\left( a>b>0 \right)$的左、右焦点分别为${{F}_{1}}$、${{F2}}$，点$P$在椭圆$C$上，点$M$为$\Delta P{{F}_{1}}{{F2}}$的内心，且$\Delta MP{{F}_{1}}$、$\Delta MP{{F2}}$、$\Delta M{{F}_{1}}{{F2}}$的面积分别为${{S}_{\Delta MP{{F1}}}}$、${{S}_{\Delta MP{{F2}}}}$、${{S}_{\Delta M{{F1}}{{F2}}}}$，若${{S}_{\Delta MP{{F1}}}}+3{{S}_{\Delta MP{{F2}}}}=2{{S}_{\Delta M{{F1}}{{F2}}}}$，则$\frac{{{S}_{\Delta MP{{F1}}}}}{{{S}_{\Delta MP{{F2}}}}}$的值为__________．
Please reason step by step, and put your final answer within \boxed{}.


Context:
Fedja used matches to form a parallelogram, ensuring that the sides are of equal length and the vertices do not lie on a single line. It is observed that exactly 7 or 9 matches fit into the diagonals. Determine the number of matches that make up the perimeter of the parallelogram.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$. What is the smallest positive integer $n$ such that \[ \sum_{t \mid n} d(t)^3 \] is divisible by $35$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知正整数 $n$ 满足 $n \neq 2017$ ，且 $n^{n}$ 与 $2017^{2017}$ 有相同的个位数字，求 $|2017-n|$ 的最小值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $A$ 为 20 ! 的所有因数构成的集合, $x_{i} \in A(1 \leq i \leq n)$ .若对任意的正整数 $y_{i}(1 \leq i \leq n)$, 存在正整数 $z$, 使得 $\frac{y_{i}+z}{x_{i}}$ 为正整数.
求 $n$ 的最大值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$$
\begin{aligned}
& \text { [Isosceles, inscribed, and circumscribed trapezoids] } \\
& \text { [ Right triangle with an angle of \$30^\circ\$ ] } \\
& \text { [ Pythagorean theorem (direct and inverse). ] } \\
& \text { [ Equilateral (equiangular) triangle ] }
\end{aligned}
$$

Quadrilateral $A B C D$ is inscribed in a circle with center $O, \angle B O A=\angle C O D=60^{\circ}$. The perpendicular $B K$, dropped to side $A D$, is equal to $6 ; A D=3 B C$.

Find the area of triangle $C O D$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The product of the two $102$-digit numbers $404,040,404,...,040,404$ and $707,070,707,...,070,707$ has thousands digit $A$ and units digit $B$. Calculate the sum of $A$ and $B$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the solution set of the inequality $x^{2} < |x-1|+a$ is a subset of the interval $(-3,3)$, then the range of the real number $a$ is ________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $x \gt 0$, $y \gt 0$, and $\left(x+1\right)\left(y+1\right)=2x+2y+4$, find the minimum value of $xy$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\frac14$ of the way to his destination, and in $45$ minutes he will be exactly $\frac13$ of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from the point where he passes his school.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设正三棱雉 $P-A B C$ 的底面边长为 1, 高为 $\sqrt{2}$, 过底边 $B C$ 作此三棱雉的截面，则截面面积的最小值为. 原始答案为\frac{m\sqrt{n}}{k}的形式，请给出m + n + k的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m + n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the country of Anchuria, a unified state exam takes place. The probability of guessing the correct answer to each question on the exam is 0.25. In 2011, to receive a certificate, one needed to answer correctly 3 questions out of 20. In 2012, the School Management of Anchuria decided that 3 questions were too few. Now, one needs to correctly answer 6 questions out of 40. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of receiving an Anchurian certificate higher - in 2011 or in 2012?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate: $u =\displaystyle\int_0^\infty\dfrac{dx}{x^4 +7x^2+1}$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an arithmetic-geometric sequence $\{a\_n\}$, the sum of its first $n$ terms is denoted as $S\_n$. If $a\_3 = 2S\_2 + 1$ and $a\_4 = 2S\_3 + 1$, find the common ratio $q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a finite sequence $\{a_n\}$ with $k$ terms, and it satisfies $a_{n+1}=a_{n-1}- \frac {n}{a_{n}}$ for $n=2,…,k-1$. If $a_{1}=24$, $a_{2}=51$, and $a_{k}=0$, then $k=$ ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
数列 \{a_n\} 定义如下：设 \df{(2n)!}{n!(n+2024)!} 写成既约分数后的分母为 A(n)，a_n 等于 2A(n) 的最大质因数，则 a_n 的最大值等于__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a scale, there are 20 carrots, 10 apples, and 5 oranges weighing a total of 8.70 kg. When 4 carrots and 2 apples are removed from the scale, the average weight of the remaining 16 carrots and 8 apples is 206 grams. If the average weight of an apple is 210 grams, what is the average weight (in grams) of the 4 carrots which were removed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two.

[color=#BF0000]Rewording of the last line for clarification:[/color]
Find the smallest value of $k$ such that there (always) exists $3$ mathematicians $X,Y,Z$ such that $X$ and $Y$ know each other, $X$ and $Z$ know each other and $Y$ and $Z$ know each other.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the series representation $$\sum_{k=0}^n \frac{A_k}{x+k}=\frac{n!}{x(x+1)(x+2)(x+3)\cdots(x+n)}$$ find the value of $A_7$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
During the holidays, Jordan read 130 French novels, while his brother Alexandre read 1/10 of what Jordan read. Their sister Camille read 2 times as many novels as Alexandre, and their friend Maxime read 5 less than half the total number of novels read by Jordan, Alexandre, and Camille combined. How many more novels did Jordan read than Maxime?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Among all triangles formed by selecting any 3 vertices from the 8 vertices of a rectangular prism, calculate the number of acute-angled triangles. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
12.074. A sphere is inscribed in a cone. The ratio of the radius of the circle of contact between the spherical and conical surfaces to the radius of the base of the cone is $k$. Find the cosine of the angle between the slant height of the cone and the plane of the base.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Fill the natural numbers \(1 \sim 22\) in the boxes below (each box can hold only one number). Among the 11 fractions formed, the maximum number of fractions with integer values is \(\qquad\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest composite number that has no prime factors less than 20.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In rectangle $ABCD$, $AB = 4$ cm, $BC = 10$ cm, and $DE = DF$. The area of triangle $DEF$ is one-fourth the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two swimmers start at opposite sides of a 90-foot swimming pool. One swims at 3 feet per second, while the other swims at 2 feet per second. They swim back and forth for 12 minutes without taking any time to turn around. How many times do they meet?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c, a \neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\{a,b,c\}=\{r,s\}$.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
8. Given that the three sides of $\triangle A B C$ are $10$, $12$, and $14$, respectively, and that $\triangle A B C$ is folded into a tetrahedron along its three midlines. Then the diameter of the circumscribed sphere of the tetrahedron is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sets $P=\{0, m\}$ and $Q=\{x|2x^2-5x<0, x\in \mathbb{Z}\}$, if $P\cap Q \neq \emptyset$, then find all possible values of $m$. Express your answer as a list of integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a number that is both the sum of two perfect squares and the difference of two perfect cubes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A pool contains one-eighteenth of its capacity in water. Two water pipes begin to fill the pool simultaneously. When the pool reaches two-ninths of its capacity, the first pipe starts to fill the pool alone and takes 81 minutes to fill an amount equal to what the second pipe had already added. Then, the second pipe fills the pool alone for 49 minutes, and at this point, the total amount of water added by both pipes is the same. After that, both pipes continue to fill the pool. How many additional minutes are needed for both pipes to completely fill the pool?
Please reason step by step, and put your final answer within \boxed{}.


Context:
在某片树林中有n(≥3)个鸟巢，而且它们之间的距离不相等.每一个巢中各有一只鸟，在某一时刻某些鸟离开自己的巢而飞落到其他的巢中.如果某一对鸟之间的距离小于另外一对鸟之间的距离（一只鸟可以与其他任何已知鸟结成“一对”），那么在飞落之后，第一对鸟之间的距离就大于第二对鸟之间的距离.问：当n取何值时能做到这一点？
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many integers $N$ less than $1000$ can be written as the sum of $j$ consecutive positive odd integers from exactly $5$ values of $j \ge 1$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Select 3 out of 5 members from the class committee to serve as the class's Academic Affairs, Entertainment, and Sports commissioners. Given that members A and B cannot serve as the Entertainment commissioner, determine the number of different methods for this selection.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of 6-tuples \((a_1,a_2,a_3,a_4,a_5,a_6)\) such that each element \(a_i\) is chosen from the set \(\{1,2,3,4\}\) and the expressions \[ a_j^2 - a_j a_{j+1} + a_{j+1}^2 \] for \(j = 1, 2, 3, 4, 5, 6\) (with \(a_7 = a_1\)) are all equal to each other.
Please reason step by step, and put your final answer within \boxed{}.


Context:
From the center \( O \) of the inscribed circle of a right triangle, the half of the hypotenuse that is closer to \( O \) appears at a right angle. What is the ratio of the sides of the triangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $y=\log_{a}(x-3)+2$ $(a > 0, a \neq 1)$, its graph passes through a fixed point $A$. If point $A$ is also on the graph of the power function $f(x)$, then $f(2)=$ ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Carla is taking attendance for her sixth-grade class. She knows that 2 students are in the restroom, and a certain multiple of that number of students minus one are absent. There are four rows with six desks each that are all 2/3 full. Carla teaches 23 students, including the ones in and out of the classroom. What is the multiple of the number of students in the restroom to find the number of absent students?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If
\[ 1 \cdot 1995 + 2 \cdot 1994 + 3 \cdot 1993 + \dots + 1994 \cdot 2 + 1995 \cdot 1 = 1995 \cdot 997 \cdot y, \]
compute the integer \( y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the bisectors of the exterior angles at $B$ and $C$ of triangle $ABC$ meet at $D$. Then, if all measurements are in degrees, angle $BDC$ equals:
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the incircle of triangle ABC, the line AA1 intersects the incircle again at Q. The lines A1C1 and A1B1 intersect the line passing through A and parallel to BC at P and R respectively. Given that angle PQC1 = 45 degrees and angle RQB1 = 65 degrees, calculate the angle PQR. Express your answer in degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. A $20 \times 19$ rectangle is plotted on the Cartesian plane with one corner at the origin and with sides parallel to the coordinate axes. How many unit squares do the two diagonals of this rectangle pass through?
Please reason step by step, and put your final answer within \boxed{}.


Context:
We will say that a list of positive integers is *admissible* if all its numbers are less than or equal to $100$ and their sum is greater than $1810$. Find the smallest positive integer $d$ such that from each admissible list, some numbers can be crossed out such that the sum of the numbers left uncrossed is greater than or equal to $1810-d$ and less than or equal to $1810+d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A triangle has sides of length $48$, $55$, and $73$. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ and $d$ are relatively prime positive integers such that $\frac{c}{d}$ is the length of a side of the square, find the value of $c+d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all real values of $p$ for which the function $d(x, y) = |x-y|^p$ is a metric on $\mathbb{R}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A math class has fewer than 50 students. When the students try to sit in rows of 8, 5 students are left in the last row. When the students try to sit in rows of 6, 3 students remain in the last row. How many students are in this class?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a $4 \times 4$ grid, place 4 pieces such that there is exactly one piece in each row, each column, and each diagonal. How many ways are there to do this?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can the numbers \(1, 2, \ldots, 2002\) be placed at the vertices of a regular 2002-gon so that no two adjacent numbers differ by more than 2? (Rotations and reflections are considered distinct.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
对正整数 n, 考虑含有 n 阶元的有限非交换群, 记 $a_{n}$ 为这些群的阶的最小可能值.则 $a_{1}+a_{2}+a_{3}+a_{4}=\qquad.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of $p$ for which the series $\sum_{n=1}^{\infty} (-1)^n \cdot \arctan \left(\frac{1}{\sqrt{n^p}}\right) \cdot \log \left(1+\frac{1}{n^{2p}}\right)$ converges conditionally.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of  $50$  meters per second, while each of the spiders has a speed of  $r$  meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum  $c$  so that for any  $r<c,$   the fly can always avoid being caught?
Please reason step by step, and put your final answer within \boxed{}.


Context:
从一个正九边形的 9 个顶点中选 3 个使得它们是一个等腰三角形的三个顶点的方法数是多少？
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 300 integers between a certain number and 1000 such that their unit digit is odd. What is the starting number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer value of n such that the expansion of (3x + 1/x sqrt(x))^n contains a constant term. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the graph of the power function $y=f(x)$ passes through the point $\left( -2,-\frac{1}{8} \right)$, find the value(s) of $x$ that satisfy $f(x)=27$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer \( n \) such that there exists a complex number \( z \), with positive real and imaginary part, satisfying \( z^{n} = (\bar{z})^{n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $ a,b$  be integers greater than  $ 1$ . What is the largest  $ n$  which cannot be written in the form  $ n \equal{} 7a \plus{} 5b$  ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A top hat contains $3$ red chips and $2$ green chips. Chips are drawn randomly, one at a time without replacement, until all $3$ of the reds are drawn or until both green chips are drawn. What is the probability that the $3$ reds are drawn? Express your answer in the form of \(\frac{k}{m}\) as a common fraction, and give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Eight-sided polygon ABCDEFGH has integer side lengths. It can be divided into a rectangle and a square. The area of the square is greater than the area of the rectangle. The product of the two areas is equal to 98. What could be the perimeter of ABCDEFGH?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The length of edge PQ of a tetrahedron PQRS measures 51 units, and the lengths of the other edges are 12, 19, 24, 33, and 42 units. Determine the length of edge RS.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any integer $n \ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ whenever $m$ divides the digit sum of $N$ written in base $b_n$, for $2 \le  m \le n$. Find the integer nearest to $b_{36}/b_{25}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=2$, $PD=6$, and $DE=2\sqrt{10}$. Determine the area of quadrilateral $AEDC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Point  $P$  lies outside a circle, and two rays are drawn from  $P$  that intersect the circle as shown. One ray intersects the circle at points  $A$  and  $B$  while the other ray intersects the circle at  $M$  and  $N$ .  $AN$  and  $MB$  intersect at  $X$ . Given that  $\angle AXB$  measures  $127^{\circ}$  and the minor arc  $AM$  measures  $14^{\circ}$ , compute the measure of the angle at  $P$ .

[asy]
size(200);
defaultpen(fontsize(10pt));
pair P=(40,10),C=(-20,10),K=(-20,-10);
path CC=circle((0,0),20), PC=P--C, PK=P--K;
pair A=intersectionpoints(CC,PC)[0],
B=intersectionpoints(CC,PC)[1],
M=intersectionpoints(CC,PK)[0],
N=intersectionpoints(CC,PK)[1],
X=intersectionpoint(A--N,B--M);
draw(CC);draw(PC);draw(PK);draw(A--N);draw(B--M);
label(" $A$ ",A,plain.NE);label(" $B$ ",B,plain.NW);label(" $M$ ",M,SE);
label(" $P$ ",P,E);label(" $N$ ",N,dir(250));label(" $X$ ",X,plain.N);[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the remainder when $712!$ is divided by $719$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $1000 \leq n = \text{ABCD}_{10} \leq 9999$ be a positive integer whose digits $\text{ABCD}$ satisfy the divisibility condition: $$1111 \mid (\text{ABCD} + \text{AB} \times \text{CD}).$$ Determine the smallest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a binary search tree, when searching for the key value 30, nodes containing the key values 10, 20, 40, 50, 70, 80, and 90 are traversed, not necessarily in the order given. How many different orders are possible in which these key values can occur on the search path from the root to the node containing the value 30?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit of the function:

$$\lim_{x \rightarrow \frac{1}{3}} \frac{\sqrt[3]{\frac{x}{9}}-\frac{1}{3}}{\sqrt{\frac{1}{3}+x}-\sqrt{2x}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $M$ be the set of all integers from $1$ to $2013$. Each subset of $M$ is assigned one of $k$ available colors. The only condition is that if the union of two different subsets $A$ and $B$ equals $M$, then $A$ and $B$ must be given different colors. What is the least possible value of $k$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Sum up those natural numbers less than 1980, in which the sum of the digits is even!
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sphere with a radius of 2017 is placed at the base of a tower. On top of this sphere, three hemispheres are stacked, each with the same radius of 2017. The centers of the sphere and the hemispheres are collinear. Calculate the maximum height of this tower.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Archit and Ayush are walking around on the set of points $(x,y)$ for all integers $-1 \leq x,y \leq 1$. Archit starts at $(1,1)$ and Ayush starts at $(1,0)$. Each second, they move to another point in the set chosen uniformly at random among the points with distance $1$ away from them. If the probability that Archit goes to the point $(0,0)$ strictly before Ayush does can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m,n$, compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The numbers \(1,2,\cdots,9\) are randomly placed into the \(9\) squares of a \(3 \times 3\) grid. Each square gets one number, and each of the numbers is used once. Find the probability that the sum of the numbers in each row and each column is odd. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a triangle $\Delta ABC$, it is given that \( \frac{2\cos A}{a} +\frac{\cos B}{b} +\frac{2\cos C}{c}=\frac{a}{bc} +\frac{b}{ca} \). Find the angle $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Provide an example of a simple equation with rational coefficients whose graph has no rational coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Wei decides to modify the design of his logo by using a larger square and three tangent circles instead. Each circle remains tangent to two sides of the square and to one adjacent circle. If each side of the square is now 24 inches, calculate the number of square inches that will be shaded.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the function \( f(n) = \sum_{j \ge 0, p_1\ldots p_j |2n, p_i >p_{i+1}} (-1)^j \left\lfloor \frac{n}{p_1\ldots p_j}\right\rfloor \) at \( n = 12 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $r_{1}, r_{2}, \ldots, r_{7}$ be the distinct complex roots of the polynomial $P(x)=x^{7}-7$. Let $$K=\prod_{1 \leq i<j \leq 7}\left(r_{i}+r_{j}\right)$$ that is, the product of all numbers of the form $r_{i}+r_{j}$, where $i$ and $j$ are integers for which $1 \leq i<j \leq 7$. Determine the value of $K^{2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) \) be a monic cubic polynomial. The solutions of the equation \( f\left(\frac{1}{x}\right)=0 \) are \( \alpha, \beta, \gamma \). Given that \( \alpha + \beta + \gamma = 10 \) and \( \alpha \beta \gamma = 15 \), find \( \left \lceil{f(10)-f(-10)}\right \rceil \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many right trapezoids \(ABCD\) exist, where angles \(A\) and \(B\) are right angles, \(AD = 2\), \(CD = BC\), the sides have integer lengths, and the perimeter is less than 100?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the domain of validity for the Laurent series of the function \( f(z) = \frac{e^{-z}}{(z-1)^2} \) around the circle \( |z| = 3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Eliminate every second number in a clockwise direction from numbers $1, 2, 3, \cdots, 2001$ that have been placed on a circle, starting with the number 2, until only one number remains. What is the last remaining number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the square root of a positive number is $2a-3$ and $3a-22$. Find this positive number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On Monday, Knight Milivoj traveled 25 miles and spent the night in Zubín. The next day, Tuesday, he reached Veselín. On the way back, he traveled 6 miles more on Thursday than on Monday and spent the night in Kostín. On Friday, he traveled the remaining 11 miles to Rytířov. Determine the distance between Zubín and Veselín.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A_0=(0,0)$. Distinct points $A_1,A_2,\cdots$ lie on the $x$-axis, and distinct points $B_1,B_2,\cdots$ lie on the graph of $y=\sqrt{x}$. For every positive integer $n$, $A_{n-1}B_nA_n$ is an equilateral triangle. Find the least $n$ for which the length $A_0A_n\geq100$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the recurrence relation \( 2y_n = 11 y_{n-1} - 20y_{n-2} + 12y_{n-3} + 2 \) with the initial conditions \( y_0 = \frac{9}{8} \), \( y_1 = \frac{-5}{4} \), and \( y_2 = \frac{-1}{2} \). Find the next term \( y_3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a certain university, some percentage of the professors are women, and 70% of the professors are tenured. If 90% of the professors are women, tenured, or both, and 52% of the men are tenured, what percent of the professors are women?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate $h(6000)$ for the function \( h(n) = \sum_{d|n}g(d) \) where \( g(n) \) is defined as \( g(n) = \sum_{d|n}f(d) \) and \( f(n) \) is the greatest power of 3 that divides \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a gambler's ruin Markov chain with equal probability of winning and losing each round, consider a condition where the game stops if the player earns N dollars, making x=N an absorbing state along with x=0. Compute the probability $P_x(N(0)<\infty)$, where $N(0)$ is the number of visits to state 0, starting from state x.
Please reason step by step, and put your final answer within \boxed{}.


Context:
During 200 test demands, zero failures were discovered. A business software system is required to achieve at most 1 failure in 1000 demands, equivalent to a probability on demand: \( pfd = 10^{-3} \). The business manager believes there is a 70% chance the system meets this requirement and a 30% chance it is an order of magnitude worse (\( pfd = 10^{-2} \)).

Given the prior probabilities:
\[ P(pfd = 10^{-3}) = 0.7 \]
\[ P(pfd = 10^{-2}) = 0.3 \]

The likelihood of observing \( f \) failures in \( d \) tests is defined by the binomial distribution:
\[ P(f|d,pfd) = \frac{d!}{f!(d-f)!}(pfd)^{f}(1-pfd)^{d-f} \]

Using Bayes' theorem, the posterior distribution for \( pfd \) is:
\[ P(pfd|d,f) = \frac{P(f|pfd,d)P(pfd)}{P(f)} \]

Calculate the posterior probability that the software meets the business requirement after observing zero failures in 200 test demands.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the differential equation \( \frac{d^2y(t)}{dt^2} + 2\frac{dy(t)}{dt} + 10y(t) = \frac{dx(t)}{dt} + x(t) \), where \( x(t) \) is the input and \( y(t) \) is the output, and the input is \( \frac{3u(t)}{e^t} \), find the zero-state output of the system.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \frac{1}{\pi^2}\int_{x=0}^1\int_{y=\sin^{-1} x}^{\dfrac{\pi}{2}}f(x,y)\mathop{dy}\mathop{dx} \) where \( f(x,y)=\begin{cases} \frac{y}{\sin y} & \text{if } y\neq 0 \\ 1 & \text{if } y=0 \end{cases} \). Provide the answer correct to three decimal places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int \int_E f(x,y) \, dx \, dy \) where \( f(x,y) = \begin{cases} (x-1/2)^{-3} & \text{if } 0 < y < |x-1/2| \\ 0 & \text{else} \end{cases} \) and \( E = [0,1] \times [0,1] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct homeomorphism classes of topologies on an infinite set $X$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of the pole of the function \( f(z) = \frac{z}{(1-e^z) \sin z} \) at \( z = 2\pi ik \) where \( k \in \mathbb{Z} \) and \( k \neq 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Through the vertices \( A \) and \( C \) of triangle \( ABC \), lines are drawn perpendicular to the bisector of angle \( ABC \), intersecting lines \( CB \) and \( BA \) at points \( K \) and \( M \) respectively. Find \( AB \) if \( BM = 10 \) and \( KC = 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
James decides to sell 80% of his toys. He bought them for $20 each and sells them for a certain amount. He had 200 toys and made $800 more compared to before he bought them. How much did he sell each toy for?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Example 7 Given the set $A=\left\{z \mid z^{2 n-1}=\bar{z}, z \in \mathbf{C}, n \in \mathbf{N}, n \geqslant 2\right\}$, on the complex plane, how many right triangles can be formed with the points corresponding to the complex numbers in $A$ as vertices?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the measure of the set of points in the interval \((0,1]\) for which a fixed binary string appears finitely many times in their binary expansion.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \int_S{\bf F}\cdot{\bf\hat n}\,dS \) for the vector field \( {\bf F}=2y{\bf i}-z{\bf j}+x^2{\bf k} \), where \( S \) is the surface of the parabolic cylinder \( y^2=8x \) in the first octant, bounded by the planes \( y=4 \) and \( z=6 \). The unit normal vector \( \bf\hat n \) points in the direction of increasing \( x \). Project the integral onto the plane \( x=0 \) to find the result.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ l = \lim_{x\to 0}\frac{e^{\sin x}-e^{\tan x}}{e^{\sin 2x}-e^{\tan 2x}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008 \cdot 2 + 0 \cdot 3$ and $402 \cdot 2 + 404 \cdot 3$ are two such ways.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the linear function $y = -3x + 2$, its graph does not pass through the ______ quadrant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \iiint z(x^2+y^2+z^2)^{-3/2} \, dx \, dy \, dz \) over the region defined by the part of the ball \( x^2+y^2+z^2 \le 81 \) where \( z \ge 4.5 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T_1, T_2: V \rightarrow W$ be linear transformations, where $V$ and $W$ are vector spaces over a field $\mathbb{F}$. Given that $\dim(V) = n$, $\ker(T_1) \neq \ker(T_2)$, and $\dim(\operatorname{Image}(T_1)) + \dim(\operatorname{Image}(T_2)) = \dim(W)$, determine $\dim(\ker(T_1) \cap \ker(T_2))$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the function \( f = \sin\theta_1 + \sin\theta_2 \), given that \( \theta_1 + \theta_2 + \phi = \pi \) and \( 0 < \phi < \pi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \(\int_{-\pi}^{\pi}e^{\sin(\theta)}\cos(\theta+\cos(\theta))d\theta\) using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{\infty} e^{ix} \, dx \) and determine the correct result.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area enclosed by the polar curve \( r = 2 + 2\sin(\theta) \) but outside the curve \( r = 4\sin(\theta) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alf, the alien from the $1980$s TV show, has a big appetite for the mineral apatite. However, he’s currently on a diet, so for each integer $k \ge 1,$ he can eat exactly $k$ pieces of apatite on day $k.$ Additionally, if he eats apatite on day $k,$ he cannot eat on any of days $k + 1, k + 2, \ldots, 2k - 1.$ Compute the maximum total number of pieces of apatite Alf could eat over days $1,2, \ldots,99,100.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area between the function \( y = x + \sin x \) and its inverse function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( L \) be the field generated by \( \omega \) and \( 2^{1/3} \) over \( \mathbb{Q} \), where \( \omega \neq 1 \) is a complex number such that \( \omega^3 = 1 \). Determine the number of subfields \( K \) of \( L \) such that \( \mathbb{Q} \subsetneq K \subsetneq L \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of the triple factorial of -2, denoted as \((-2)!!!\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the ring $\mathbb{Z}_5[x]/(x^3 + 3x + 2)$, determine the inverse of the element $(x+2) + (x^3 + 3x + 2)$, if it exists.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Provide an example of a continuous function \( f(x) \) such that \( \int_{1}^{\infty} f(x) \, dx \) converges, but \( \lim_{x \to \infty} f(x) \) does not exist.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface integral \( \int_S dx \wedge dy + dy \wedge dz \), where \( S \) is the surface defined by \( S = \{(x,y,z) : x = z^2 + y^2 - 1, x < 0\} \).
Please reason step by step, and put your final answer within \boxed{}.