Context:
Given two random variables $x$ and $y$, consider the inequality:

$$-g(x) < f(x, y) < g(x)$$

where $g(x) > 0$. The probability of each part of the inequality is the same, denoted as $\alpha$:

$$P[-g(x) < f(x, y)] = P[f(x, y) < g(x)] = \alpha$$

Determine the probability that the inequality holds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Vera has a set of weights, each of which has a distinct mass and weighs an integer number of grams. It is known that the lightest weight in the set weighs 71 times less than the sum of the weights of all other weights in the set. It is also known that the two lightest weights together weigh 34 times less than the sum of the weights of all other weights in the set. What is the minimum number of grams the lightest weight can weigh?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a set $B \subseteq \mathbb{R}$ such that for every closed set $F$ containing $B$, the intersection $|B \cap F|$ is infinite.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Lil writes one of the letters P, Q, R, S in each cell of a 2x4 table. She does this in such a way that, in each row and in each 2x2 square, all four letters appear. How many distinct ways can she fill this table? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Denote by $ S$ the set of all positive integers. Find all functions $ f: S \rightarrow S$ such that
\[ f (f^2(m) + 2f^2(n)) = m^2 + 2 n^2\]
for all $ m,n \in S$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \(\lim_{x\to y}\frac{f_1'(y)-f_3'(y)}{f_2'(y)}\) where \(f_1(y)=x^y\), \(f_2(y)=y^x\), and \(f_3(y)=x^x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Nancy now drinks the equivalent of 75% of her body weight (W) in water, in addition to consuming 2 pounds of water from her food intake. Furthermore, some of that water evaporates during the day, and the evaporated water (E) makes up 5% of her total daily water intake. If Nancy's total remaining daily water intake (R), excluding the evaporated water, is 64 pounds, calculate her weight (W) using this system of equations:

W = R - 2 (amount of water excluding food)
0.05(W+2) = E (amount of water evaporated)
R - E = 64 (remaining daily water intake)

Use these equations to find Nancy's weight (W).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A firm is about to hire one of three candidates who take a test with scores that are continuous random variables distributed as \( \text{Unif}[0,100] \). Candidates apply sequentially, and the firm must decide to either select a candidate immediately or reject them permanently. If the first two candidates are rejected, the third candidate must be selected. What is the score threshold below which the first candidate will be rejected to maximize the expected score of the admitted candidate?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A boulevard has 25 houses on each side, for a total of 50 houses. The addresses on the east side of the boulevard follow an arithmetic sequence, as do the addresses on the west side. On the east side, the addresses start at 5 and increase by 7 (i.e., 5, 12, 19, etc.), while on the west side, they start at 2 and increase by 5 (i.e., 2, 7, 12, etc.). A sign painter charges $\$1$ per digit to paint house numbers. If he paints the house number on each of the 50 houses, how much will he earn?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let us call a number \( \mathrm{X} \) "50-podpyirayushchim" if for any 50 real numbers \( a_{1}, \ldots, a_{50} \) whose sum is an integer, there exists at least one \( a_i \) such that \( \left|a_{i}-\frac{1}{2}\right| \geq X \).

Find the greatest 50-podpyirayushchee \( X \), rounded to the nearest hundredth according to standard mathematical rules.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Laura's House is a 20-mile round trip from her school. The supermarket is an additional 10 miles farther away from the school. Laura also visits her local gym 3 times a week, which is 5 miles from her house, and visits a friend's house every Saturday, located 12 miles away from her house. If Laura drives to school every morning, to the supermarket two afternoons a week, and includes her trips to the gym and her friend's house, how many miles does Laura drive per week?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest \( k \) for which there exists a permutation
 \( \left( {{a}_{1},{a}_{2},\ldots ,{a}_{2022}}\right) \) of integers from 1 to 2022 such that
 for at least \( k \) distinct \( i \) with \( 1 \leq  i \leq  {2022} \) the number
 \( \frac{{a}_{1} + {a}_{2} + \ldots  + {a}_{i}}{1 + 2 + \ldots  + i} \) is an integer larger than 1 .
Please reason step by step, and put your final answer within \boxed{}.


Context:
For $k \in \{0, 1, \ldots, 9\}$, let $\epsilon_k \in \{-1, 1\}$. If the minimum possible value of $\sum_{i=1}^9 \sum_{j=0}^{i-1} \epsilon_i \epsilon_j 2^{i+j}$ is $m$, find $|m|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A number, when divided by the sum of a certain number and 445, gives 2 times their difference as quotient and 50 as remainder. The number is 220050. What is the first number?
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Context:
Use Plancharel's theorem to evaluate the integral \( \int_{-\infty}^{\infty}\frac{\cos^2 x}{x^2}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A positive integer $n$ is called \textit{good} if $2 \mid \tau(n)$ and if its divisors are $1 = d_1 < d_2 < \ldots < d_{2k-1} < d_{2k} = n$, then $d_{k+1} - d_k = 2$ and $d_{k+2} - d_{k-1} = 65$. Find the smallest \textit{good} number.
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Context:
Determine the number of subgroups isomorphic to \(\mathbb{Z}_{2} \times \mathbb{Z}_{2}\) in the dihedral group \(D_8 = \langle \sigma, \rho \; | \; \rho^8 = \sigma^2 = e \text{ and } \sigma\rho\sigma = \rho^{-1} \rangle\).
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Context:
Moor has $3$ different shirts, labeled $T$, $E$, and $A$. Across $5$ days, the only days Moor can wear shirt $T$ are days $2$ and $5$. How many different sequences of shirts can Moor wear across these $5$ days?
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Context:
Determine the splitting field of the polynomial \((x^3+x^2+1)(x^3+x+1)(x^2+x+1)\) over the finite field \(\mathbb{F}_2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that $a$ and $b$ are natural numbers such that \[ p = \frac{b}{4} \sqrt{\frac{2a-b}{2a+b}} \] is a prime number. What is the maximum possible value of $p$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $a$ in the inequality $ax^2 + 2x + 20$ with the solution set $(\-\frac{1}{2}, \frac{1}{3})$.
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Context:
Evaluate the double integral \( \iint_D (x + y)\, dx \, dy \) where \(D\) is the trapezoidal region with vertices \((0,0), (5,0), (5/2, 5/2), (5/2, -5/2)\), using the transformations \( x = 2u + 3v \) and \( y = 2u - 3v\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of distinct elements in the group defined by the presentation \( \langle x,y \mid x^3=y^2=(xy)^3=e \rangle \) by determining a normal form for its elements.
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Context:
Let $R$  be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$?
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Context:
Suppose you have $n$ red balls, $n$ green balls, and an arbitrary number of white balls. Each ball is selected into a set with a probability of $p = \frac{1}{n}$ and excluded with a probability of $1 - p = 1 - \frac{1}{n}$, independently of one another. Determine a constant lower bound (independent of $n$) on the probability that the set will contain at least one green ball but no red balls.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the infinite series: \[ \sum_{n=0}^\infty \frac{2n^7 + n^6 + n^5 + 2n^2}{n!} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $p$ for which the integral \( \int_0^{\infty} \frac{1-\cos x}{x^p} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the infinite product: \[ \prod_{p \in \mathbb{P}} \frac{p^4+1}{p^4-1} \] where \( \mathbb{P} \) is the set of prime numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The base of the tetrahedron $S-ABCD$ is a square $ABCD$ with side length 1, and $SO$ is the height of the pyramid with $SO = \frac{1}{2}$. If $SA = SD$ and $SB = SC$, what is the dihedral angle between the planes $ASB$ and $CSD$?
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Context:
Find the value of \( \beta \) such that the function \( u(x_1, x_2, \ldots, x_n) = \left(\sum_{i=1}^{n}x_{i}^2\right)^{\beta/2} \) satisfies \( \nabla^2 u = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), it is known that \(\angle BAC = \alpha\) and \(AC = b\). The incircle touches the sides \(AB\) and \(BC\) at points \(M\) and \(N\), respectively. The angle bisector of \(\angle BAC\) intersects the line \(MN\) at point \(K\). Find the distance from point \(K\) to the line \(AC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a group of some students, each student is registered for at least one of 3 classes – history, math, and English. 20 - one students are registered for history, 20 - 6 students are registered for math, and thirty - 4 students are registered for English. If only 3 students are registered for all 3 classes, and 7 students are registered for exactly two classes, how many students are there in the group?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The ship decided to determine the depth of the ocean at its location. The signal sent by the echo sounder was received on the ship 8 seconds later. The speed of sound in water is 1.5 km/s. Determine the depth of the ocean.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two positive integers \( m \) and \( n \) satisfying \( 1 \leq m \leq n \leq 40 \), find the number of pairs \( (m, n) \) such that their product \( mn \) is divisible by 33.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the last two digits of the decimal representation of $9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}$. Express your answer as a two-digit number.
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Context:
a wire in the form of a circle of radius 3.5 m is bent in the form of a rectangule , whose length and breadth are in the ratio of 6 : 5 . what is the area of the rectangle ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the mass of the solid $E$ enclosed by the planes $z = 0$, $z = x + y + 5$, and the cylinders $x^2 + y^2 = 4$, $x^2 + y^2 = 9$. The density at any point is equal to its distance from the $yz$-plane.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The speed of a boat in still water is 65 km/hr and the rate of the current is 15 km/hr. The boat travelled downstream for a certain amount of time and covered a distance of 33.33 km. How long did the boat travel downstream?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $z$ be a complex number satisfying $(z+\frac{1}{z})(z+\frac{1}{z}+1)=1$. Evaluate $(3z^{100}+\frac{2}{z^{100}}+1)(z^{100}+\frac{2}{z^{100}}+3)$.
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Context:
Rob and Mark plan to meet at the national park. It takes 1 hour for Rob to get to the park and it takes three times as much time for Mark to get there. Rob lives in a time zone 2 hours behind Mark's time zone. If Rob leaves his home at 11 a.m. local time, at what local time should Mark leave his home so that they both arrive at the national park at the same time?
Please reason step by step, and put your final answer within \boxed{}.


Context:
4. Among the 21 numbers $8,9, \cdots, 28$, the probability that the sum of three randomly selected different numbers is exactly divisible by 9 is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jerry starts at $0$ on the real number line. He tosses a fair coin $8$ times. When he gets heads, he moves $1$ unit in the positive direction; when he gets tails, he moves $1$ unit in the negative direction. The probability that he reaches $4$ at some time during this process is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is the value of $a + b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH.$)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\triangle ABC$ have side lengths $AB=13$, $AC=14$, and $BC=15$.  There are two circles located inside $\angle BAC$ which are tangent to rays $\overline{AB}$, $\overline{AC}$, and segment $\overline{BC}$.  Compute the distance between the centers of these two circles.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Among the 99 positive integers from 1, 2, ..., 99, determine the minimum value of k such that any selection of k numbers must include at least two numbers a and b (a != b) satisfying 1/2 <= b/a <= 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the volume of a cuboctahedron with a side length of 1?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the smallest value of $n$, the following condition is met: if $n$ crosses are placed in some cells of a $6 \times 6$ table in any order (no more than one cross per cell), three cells will definitely be found forming a strip of length 3 (vertical or horizontal) in each of which a cross is placed.
Please reason step by step, and put your final answer within \boxed{}.


Context:
7. Point $P$ is located inside a square $A B C D$ of side length 10 . Let $O_{1}, O_{2}, O_{3}, O_{4}$ be the circumcenters of $P A B, P B C, P C D$, and $P D A$, respectively. Given that $P A+P B+P C+P D=23 \sqrt{2}$ and the area of $O_{1} O_{2} O_{3} O_{4}$ is 50 , the second largest of the lengths $O_{1} O_{2}, O_{2} O_{3}, O_{3} O_{4}, O_{4} O_{1}$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{\partial B_1(0)} P(x, y) \, dA_y \) for the Poisson kernel \( P(x, y) \) defined as \[ P(x,y)=\frac{1}{\omega_n R} \frac{R^2-|x|^2}{|x-y|^n}, \] where \( x \) is in the open ball \( B_R(0) \) in \( \mathbb{R}^n \) with \( |x|<R \), \( y \) lies on the boundary of \( B_R(0) \) with \( |y|=R \), and \( \omega_n \) is the surface area of an \( n \)-dimensional unit sphere.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a $5 \times 5$ grid, cover the area with several S-Tetrominos (also known as Z-Tetrominos) such that in every square, there are two or fewer tiles covering it (tiles can overlap). Determine the maximum possible number of squares that can be covered by at least one tile.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In trapezoid $KLMN$, base $KL$ is $40 \text{ cm}$ and base $MN$ is $16 \text{ cm}$. Point $P$ lies on segment $KL$ such that segment $NP$ divides the trapezoid into two parts with equal areas.

Determine the length of segment $KP$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
John drives 1,000 miles a month, totaling 12,000 miles a year. He needs to get an oil change every 3,000 miles, and the price of an oil change fluctuates throughout the year. In the first quarter, the price is $55, in the second quarter, it is $45, in the third quarter, it is $50, and in the fourth quarter, it is $40. John receives one free oil change per year through his auto shop's loyalty program.

Additionally, he needs to get his tires rotated every 6,000 miles, which costs $40 per rotation. John is also eligible for a 10% discount on tire rotations, as he holds a membership with a local automobile association.

Furthermore, he needs to replace his brake pads every 24,000 miles, which costs $200. The local auto shop offers a biennial membership for $60, which grants a 20% discount on brake pad replacements.

Calculate John's total annual car maintenance expenses for oil changes, tire rotations, and brake pad replacements, taking into account fluctuating oil change prices, the discount on tire rotations, and the biennial membership for brake pad replacement discounts.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim _{x\rightarrow0}\frac{1-\cos\left(\frac{1-\cos x \cos 2x}{x^2}-\frac {5}{2}\right)\cos2x}{x^2} \]
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Context:
How many four-digit integers $abcd$, with $a \not\equiv 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the probability of obtaining an infinite sequence of the same number when rolling a fair six-sided die infinitely many times.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an equilateral triangle with side length \( a \), find the segment connecting a vertex of the triangle to a point that divides the opposite side in the ratio \( 2:1 \).
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Context:
A frog starts jumping from the point (1,2) and makes a series of jumps, each of which is parallel to one of the coordinate axes and is 1 unit in length. The direction of each jump (up, down, right, or left) is chosen at random. The frog stops jumping when it reaches the boundary of the square with vertices (0,0), (0,4), (4,4), and (4,0). What is the probability that the frog ends its jumps on a vertical side of the square? Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
双曲线 $x^{2}-y^{2}=1$ 的右半支与直线 $x=100$ 围成的区域内部（不含边界）整点 （纵横坐标均为整数的点）的个数是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Square $EFGH$ has a side length of $40$. Point $Q$ lies inside the square such that $EQ = 16$ and $FQ = 34$. The centroids of $\triangle{EFQ}$, $\triangle{FGQ}$, $\triangle{GHQ}$, and $\triangle{HEQ}$ are the vertices of a convex quadrilateral. Calculate the area of this quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the set of points S with Cartesian coordinates (x, y) that satisfy all the following five conditions:
(i) a/2 ≤ x ≤ 2a.
(ii) a/2 ≤ y ≤ 2a.
(iii) x + y ≥ a.
(iv) x + a ≥ y.
(v) y + a ≥ x.

Determine the number of edges of the polygon that forms the boundary of the set S. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The average age of 6 boys is 18 years, and their ages are in proportion x : x+2 : x+4 : 2x-3 : 2x-1 : 2x+1. What is the age in years of the eldest boy, given that the youngest boy is twice as old as his sister who is currently 5 years old?
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Context:
A man buys three items: a cycle for Rs. 2300, a scooter for Rs. 12,000, and a motorbike for Rs. 25,000. He sells the cycle at a loss of 30%, the scooter at a profit of 25%, and the motorbike at a profit of 15%. What is the combined selling price of these three items?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设函数 $f_{0}(x)=|x|, f_{1}(x)=\left|f_{0}(x)-1\right|, f_{2}(x)=\left|f_{1}(x)-2\right|$, 则函数 $y=f_{2}(x)$ 的图象与 $x$ 轴所围成图形中的封闭部分的面积是 $\qquad$.
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Context:
Let the set \( A = \{1, 2, \cdots, 2016\} \). For any 1008-element subset \( X \) of \( A \), if there exist \( x \) and \( y \in X \) such that \( x < y \) and \( x \mid y \), then \( X \) is called a "good set". Find the largest positive integer \( a \) (where \( a \in A \)) such that any 1008-element subset containing \( a \) is a good set.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three numbers,  $a_1$ ,  $a_2$ ,  $a_3$ , are drawn randomly and without replacement from the set  $\{1, 2, 3, \dots, 1000\}$ .  Three other numbers,  $b_1$ ,  $b_2$ ,  $b_3$ , are then drawn randomly and without replacement from the remaining set of 997 numbers.  Let  $p$  be the probability that, after a suitable rotation, a brick of dimensions  $a_1 \times a_2 \times a_3$  can be enclosed in a box of dimensions  $b_1 \times b_2 \times b_3$ , with the sides of the brick parallel to the sides of the box.  If  $p$  is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a circle O: x^2 + y^2 = 1 and a point P(-1, sqrt(3)), where A and B are two moving points on circle O, find the maximum value of angle APB in radians.
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Context:
Eva draws an equilateral triangle and its altitudes. In the first step, she draws the center triangle of the equilateral triangle. In the second step, she draws the center triangle of this center triangle, and so on. After each step, Eva counts all triangles whose sides lie completely on the drawn lines. What is the minimum number of center triangles she must have drawn so that the figure contains more than 2022 such triangles?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each face of a $6 \times 6 \times 6$ cube is divided into $1 \times 1$ cells. The cube is covered with $2 \times 2$ squares such that each square covers exactly four cells, no squares overlap, and each cell is covered by an equal number of squares. What is the maximum value of this equal number? (A square can be folded over an edge.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
设整数 $n \geq 2$. 某天 $n$ 个人组成了一个社团，每人向政府交 1 元注册费。接下来每天都发生如下事情：每一个前一天组建的社团都分裂成两个团体，其中含至少两人的团体组建成一个新社团，每个新社团的成员向政府交 1 元注册费，只有一名成员的团体不再组建新社团。上述过程直到不能继续进行下去为止。当$n=6$时，求政府总共收到的注册费的最大值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle is circumscribed around a right triangle \(ABC\) with legs \(AC = 5\) and \(BC = 12\). Points \(E\) and \(G\) are the midpoints of the smaller arcs \(AC\) and \(BC\) of this circle, respectively. Point \(F\) is the midpoint of the arc \(AB\) that does not contain point \(C\).

Find the area of quadrilateral \(AEGF\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
求 \(\sin ^{3} 6^{\circ}-\sin ^{3} 114^{\circ}+\sin ^{3} 126^{\circ}\)。题目的原始答案是\(\frac{a - b \sqrt{c}}{d}\)的格式，请给出a+b+c+d的值。
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Context:
Let $f:\mathbb{N}\to\mathbb{N}$ be a function. Derive an upper bound for $$\frac{a}{b}+\frac{b}{a},$$ given that $$a < b$$ and $$\frac{f(b)/b}{f(a)/a}<2.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
# Problem 3.

In triangle $A B C$, the bisector $B E$ and the median $A D$ are equal and perpendicular. Find the area of triangle $A B C$, if $A B=\sqrt{13}$.
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Context:
Evaluate the integral \( I = \int_{-\frac{\pi }{2}}^{\frac{\pi }{2}} \frac{\cos (x)}{\gamma ^{H_x-H_{-x}}+1} \, dx \), where \( \gamma \) is Euler's gamma and \( H_x \) is the harmonic number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An even function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x+2) = -f(x)$ and is monotonic in the interval $[0,2]$. If there exists $x \in (0,2)$ such that $f(x) = 0$, then the sum of all roots of the equation $f(x) = 0$ in the interval $[2002, 2010]$ is ______.
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Context:
The New York Public Library requires patrons to choose a 4-digit Personal Identification Number (PIN) to access its online system. (Leading zeros are allowed.) The PIN is not allowed to contain either of the following two forbidden patterns:

1. A digit that is repeated 3 or more times in a row. For example, $0001$ and $5555$ are not PINs, but $0010$ is a PIN.
2. A pair of digits that is duplicated. For example, $1212$ and $6363$ are not PINs, but $1221$ and $6633$ are PINs.

How many distinct possible PINs are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( 0 \leqslant a_{k} \leqslant 1 \) for \( k = 1, 2, \cdots, 2002 \), and let \( a_{2003} = a_{1} \) and \( a_{2004} = a_{2} \), find the maximum value of \( \sum_{k=1}^{20002} \left( a_{k} - a_{k+1} a_{k+2} \right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a regular quadrilateral pyramid $P-A B C D$, the dihedral angle between a lateral face and the base is $\alpha$, and the dihedral angle between two adjacent lateral faces is $\beta$. Find the value of $2 \cos \beta + \cos 2 \alpha$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the largest integer $n$ for which

$$
\frac{\sqrt{7}+2 \sqrt{n}}{2 \sqrt{7}-\sqrt{n}}
$$

is an integer?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mom made 32 chocolate chip cookies. It took some time to mix the ingredients and 16 minutes to bake. Julie and Matt ate 9 cookies after dinner. There were 23 cookies left. How many minutes did it take to mix the ingredients?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sum of positive numbers \(a, b, c,\) and \(d\) does not exceed 4. Find the maximum value of the expression

\[
\sqrt[4]{a^{2}(a+b)}+\sqrt[4]{b^{2}(b+c)}+\sqrt[4]{c^{2}(c+d)}+\sqrt[4]{d^{2}(d+a)}
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer \( n > 4 \) such that there exists a graph with \( n \) vertices, no triangles, and for every pair of non-adjacent vertices, there are exactly two vertices adjacent to both.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine how the prime number 2 splits in the splitting field of the polynomial \(X^3 - 3X + 1 = 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A straight one-way city street has 8 consecutive traffic lights. Every light remains green for 1.5 minutes, yellow for 3 seconds, and red for 1.5 minutes. The lights are synchronized so that each light turns red 10 seconds after the preceding one turns red. Calculate the longest interval of time, in seconds, during which all 8 lights are green.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the diagram, points R and S lie on PT and PQ, respectively. If ∠PQR = 90°, ∠QRT = 158°, and ∠PRS = ∠QRS, what is the measure of ∠QSR? Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
有限项等差数列的公差为 4 ，第二项起各项之和加上首项的平方小于 100 ，则该数列可能有几项？请提供可能项数的最小值与最大值之和。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three couples sit for a photograph in  $2$  rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A B C$ be a triangle, and let $D, E$, and $F$ be the midpoints of sides $B C, C A$, and $A B$, respectively. Let the angle bisectors of $\angle F D E$ and $\angle F B D$ meet at $P$. Given that $\angle B A C=37^{\circ}$ and $\angle C B A=85^{\circ}$, determine the degree measure of $\angle B P D$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the principle on a certain sum of money at 5% per annum for 6 years if the amount being Rs. 1120. What is the principle amount?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Provide an example of a function \( f: \mathbb{R} \to \mathbb{R} \) such that \( \rho(Af(x), Af(y)) < \rho(f(x), f(y)) \) for all \( x \neq y \), but \( f \) does not have a fixed point.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equations \(x = 3\sqrt{y\sqrt{3\sqrt{y\cdots}}}\) and \(y = 3\sqrt{x\sqrt{3\sqrt{x\cdots}}}\), find the value of \(x + y\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that $u_1$ and $u_2$ are vectors such that $|u_1| = 2$, $|u_2| = 3$, and $u_1 \cdot u_2 = 5$. Define the vectors $u_3, u_4, u_5, u_6, \ldots$ by $u_3 = \text{proj}_{u_1} u_2$, $u_4 = \text{proj}_{u_2} u_3$, $u_5 = \text{proj}_{u_3} u_4$, and so on. Find the value of the infinite sum $$\sum_{n=1}^\infty |u_n|.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $p$ for which the $p$-norm of a matrix is invariant under orthogonal transformations. Specifically, find the values of $p$ such that for any orthogonal matrix $A$ and any matrix $X$, the equality $\|AX\|_p = \|X\|_p$ holds.
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Context:
Determine the asymptotic complexity of the recurrence relation \( T(n) = T(\sqrt{n}) + 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the series \(1 + \frac{1}{2} -\frac{1}{3} + \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \dots\) converges. If it diverges, specify whether it diverges to \(\infty\) or has no limit. The generic term of the series is given by:
\[a_n = \begin{cases} -\frac{1}{n}, & \text{if } 3 \mid n \\ \frac{1}{n}, & \text{otherwise} \end{cases}\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the matrix \( A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \), find the sum of all elements in the matrix \( M = \sum_{i=1}^{20} A^i \).
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Context:
For some positive integers \( p \) , there is a quadrilateral \( {ABCD} \) with positive integer side lengths, perimeter \( p \) , right angles at \( E \)
 and \( C,{AB} = 2 \) , and \( {CD} = {AD} \) . How many different values of \( p < {2015} \) are possible?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected stopping time $E[S]$ for the first occurrence of the sequence HTH in a series of fair coin tosses using the optional stopping theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the recursive sequence defined by \(x_0 = 0\) and \(2x_{n+1} = x_n^2 + a\), find the range of values of \(a\) for which the sequence converges to a finite limit, considering only real-valued sequences.
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Context:
You are playing a game with 3 envelopes, each containing a uniformly random amount of money between $0$ and $1000$. At each step, you can choose an envelope, look at its contents, and decide to either keep it or discard it and choose from the remaining envelopes. If you play to maximize your expected winnings, what is \( \lfloor E \rfloor \), the greatest integer less than or equal to your expected winnings?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements of order 6 in the symmetric group $S_7$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine for which values of $n$ the ring $\mathbb{C}[x_1,\dots,x_n]/(\sum_{i=1}^n x_i^2)$ contains no nilpotent elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using Rouche's theorem, determine the number of roots of the polynomial \( z^9 + z^5 - 8z^3 + 2z + 1 \) that lie in the annular region between the circles \(|z|=1\) and \(|z|=2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $\{a_1,a_2,a_3,\ldots,a_n\}$ is a set of real numbers, indexed so that $a_1<a_2<a_3<\cdots<a_n,$ its complex power sum is defined to be $a_1i+a_2i^2+a_3i^3+\cdots+a_ni^n,$ where $i^2=-1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1,2,\ldots,n\}.$ Given that $S_8=-176-64i$ and $S_9=p+qi,$ were $p$ and $q$ are integers, find $|p|+|q|.$
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Context:
A rope of length 10 m is tied tautly from the top of a flagpole to the ground 6 m away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 cm/min. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer so that the sun is directly overhead.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the outward flux of the vector field \( F(x,y,z) = (xy+2xz, x^2+y^2, xy-z) \) through the surface of the cylinder \( x^2+y^2=4 \) bounded by the planes \( z=y-2 \) and \( z=0 \), without using the Divergence Theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum number of isolated local maxima of the function \(|p(x)|\), where \(p: \Bbb R^n \to \Bbb R\) is a quadratic polynomial?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of generators in the group \((\mathbb{Z}_{100}^*, \cdot_{100})\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the integral \( \int_{|z|=1} \frac{z^7}{10z^8-5z^4+2z^2+iz+1} \, dz \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value of the maximum between a random variable $T$ and a constant $5$, given that $T$ follows an exponential distribution with a rate parameter of $\frac{1}{30}$, denoted as $T\sim\exp(\frac{1}{30})$. Provide your answer to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the following integral for $\sigma > 1$:
\[ \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T}\left|\zeta{(\sigma + it)}\right|^2dt .\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a simple graph with 239 vertices, where the graph is not bipartite and each vertex has a degree of at least 3, determine the smallest integer $k$ such that every odd cycle in the graph has a length of at most $k$. What is the value of $k$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the greatest integer $A$ such that in any permutation of the numbers $1, 2, \ldots, 100$, there exist ten consecutive numbers whose sum is at least $A$. What is $A$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a monic integer polynomial of degree $n$ whose Galois group over $\mathbb{Q}$ is $S_n$. Provide the polynomial.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $M$ be the set of all holomorphic functions $f$ from the unit disk to itself such that $f(0)=\frac{2}{3}$. Determine the supremum of $|f'(0)|$ over all functions in $M$.
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Context:
Two airlines each operate a flight from country A to country B. A total of 1000 passengers choose between the two airlines, each with a probability of \( \frac{1}{2} \). Each airline offers a plane with 510 seats. What is the probability that at least one passenger does not get a seat on their chosen flight?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimal number of time steps required to reach the state $x_f = \begin{bmatrix} 1 & 1 & 0 \end{bmatrix}^T$ from the initial state $x_0 = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}^T$ in the discrete-time state-space system given by:

$$x(t+1) = Ax(t) + Bu(t), \quad y(t) = Cx(t)$$

where

$$A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 1 & 1 \\ 2 & 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{1}{(e^{-x}+1)(\pi^2+x^2)}dx \) using a complex-analytic approach.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of conjugacy classes of order 3 matrices in the group \( PGL(3, \mathbb{Z}) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the remainder when \(41!\) is divided by \(83\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of degrees of freedom of a rank-1 matrix $X = ab^T$, where $a \in \mathbf{R}^m$ and $b \in \mathbf{R}^n$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bag contains 25 marbles numbered from 1 to 25. If you randomly select 12 marbles from the bag, what is the expected value of the second-largest marble?
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Context:
Find a natural number whose Collatz sequence length exceeds the logarithmic trend of $\ln(x)^C$, where $C$ is a constant.
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Context:
Find the smallest known value of $C$ such that there are infinitely many prime pairs differing by $C$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$, $b$, and $c$ be three distinct real numbers satisfying the equation \( \frac{a}{1+ab} = \frac{b}{1+bc} = \frac{c}{1+ca} \). Find the value of the product $abc$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions to the differential equation \( x'(t) = x(t) - e^{-t^2} \) that satisfy the condition \( \lim_{t \rightarrow \pm \infty} x(t) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X$ be a linear space and $T$ be a linear operator on $X$. Define a new linear operator $S$ by $S(x) = T(T(x))$. If $T^*$ is the adjoint operator of $T$, determine the adjoint operator of $S$.
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Context:
Evaluate the limit: $$\lim_{n\to\infty} \int\limits_{0}^{1/2} \cos(1-\cos(1-\cos(...(1-\cos(x))...))_{n\text{ times}} \ \mathrm{d}x$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence of the Taylor series expansion of the function \( f(x)= \begin{cases} e^{-\frac{1}{x^2}}\quad &\text{if $x\neq 0$}\\ 0\quad &\text{if $x=0$} \end{cases} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral of the differential form \( \varphi = e^x dx \wedge dy + y dz \wedge dx + x dy \wedge dz \) over the surface \( S \) of the cylinder defined by \( x^2 + y^2 = 1 \) between \( z = 1 \) and \( z = 3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p(z) = z^7 + z(z-3)^3 + 1$. Find the number of zeros (including multiplicities) of $p(z)$ in the open disk $|z-3|<1$.
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Context:
Suppose we roll a fair 6-sided die repeatedly. What is the expected number of rolls required to see 3 of the same number in succession?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Earl and Bob start their new jobs on the same day. Earl's work schedule is to work for 3 days followed by 1 day off, while Bob's work schedule is to work for 7 days followed by 3 days off. In the first 1000 days, how many days off do they have in common?
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Context:
Rupert and Ronald aced their math test, so their mother bought them a fantastic trampoline. Over the course of a week, they jumped on the trampoline every day with varying amounts of enthusiasm. On Monday, Ronald jumped 157 times while Rupert jumped 86 more times than Ronald. On Tuesday, Ronald increased his jumps to 193 times while Rupert decreased his jumps by 35 times compared to Monday. On Wednesday, both brothers doubled the number of jumps from their respective Tuesday jumps. For the remaining four days, from Thursday to Sunday, their jumps decreased by a constant rate of 20 times each day. Calculate the total number of jumps made by Rupert and Ronald throughout the week.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There is a magical tree with 123 fruits. On the first day, 1 fruit falls from the tree. From the second day onwards, the number of fruits falling each day increases by 1 compared to the previous day. However, if the number of fruits on the tree is less than the number of fruits that should fall on a given day, the falling process restarts from 1 fruit on that day and a new cycle begins. Following this pattern, on which day will all the fruits have fallen from the tree?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Marvin had a birthday on Tuesday, May 27 in the leap year 2008. In what year will his birthday next fall on a Saturday? Express your answer as a single year.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. Find the probability that the lengths of these three segments are the three side lengths of a triangle with positive area. Provide your answer in the form \(\frac{k}{m}\) where the fraction is simplified, and give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $w$ and $z$ be complex numbers such that $|w+z|=2$ and $|w^2+z^2|=18$. Find the smallest possible value of $|w^3+z^3|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the sequence:
$$a_n = \frac{1}{n} \sqrt[n]{n(n+1)(n+2)...(2n)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
9. Four black $1 \times 1 \times 1$ cubes and four white $1 \times 1 \times 1$ cubes can form $\qquad$ different $2 \times 2 \times 2$ cubes (cubes that are the same after rotation are considered the same type).
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task 3 - 331213

In a beauty contest for poodles, Asta, Benno, Cäsar, and Dolly face a jury of four members. Each jury member votes for one of the dogs by raising a card with the initial letter of the dog's name. As a rule for evaluating this voting result, it was stipulated: If two dogs clearly receive more votes than the other two dogs, they are considered qualified. However, if the voting result does not clearly show two dogs with more votes than the other two, a supplementary regulation will be made (e.g., a new vote will be scheduled).

Determine the number of all those different voting results that lead to this latter case! Voting results are considered the same only if they not only match in the number of votes for the dogs but also in which jury members voted for the respective dogs. For example, the voting results $A A B C$ and $C A B A$ are considered different.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( f(x)=a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \cdots + a_{n} x^{n} \), where \( a_{1}, a_{2}, a_{3}, \cdots, a_{n} \) form an arithmetic sequence, and \( n \) is a positive even number. Also, given \( f(1) = n^{2} \) and \( f(-1) = n \), compare the value of \( f\left(\frac{1}{2}\right) \) to 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a rectangle, a broken line is drawn where adjacent segments are perpendicular and equal to the shorter side of the rectangle. Find the ratio of the sides of the rectangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The students were fishing. It is known that \( a_{1} \) fishermen caught at least one fish, \( a_{2} \) fishermen caught more than one fish, \( a_{3} \) - more than two fish, and so on, up to \( a_{10} \) - more than nine fish. No one caught more than 10 fish. How many fish did the students catch in total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bucket A is 1/4 full of sand, and bucket B, which has one-half the capacity of bucket A, is 3/8 full of sand. Bucket C has twice the capacity of bucket A and is currently 1/3 full of sand. Jana first pours the contents of bucket C into bucket B, then she pours the contents of bucket B into bucket A. If any sand overflows from bucket A during this process, how much sand would overflow in terms of bucket A's capacity?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The base of a hill located beside a river is some distance above the seabed. This depth is a quarter of the vertical distance between the riverbed and the peak of the hill. The hill is 900m high. How far above the seabed is the base of the hill?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Cartesian coordinate system, the vertices of $\triangle ABC$ are $A(-1, 0)$, $B(3, 0)$, and $C(0, 2)$. It is known that a moving line $y = m (0 < m < 2)$ intersects segments $AC$ and $BC$ at points $D$ and $E$ respectively, and there exists a point $P$ on the x-axis such that $\triangle DEP$ is an isosceles right triangle. Find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the integral:
\[
I = \lim_{\epsilon\to 0} \frac{1}{2\pi\epsilon}\int_{-\infty}^\infty\int_{-\infty}^\infty f(x,y)\frac{y}{\epsilon}\exp{\frac{-(x^2+y^2)}{2\epsilon}}dxdy
\]
where \(f\) and \(\nabla f\) are continuous and bounded.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value, for \(a, b > 0\), of the expression

\[
\frac{|a + 3b - b(a + 9b)| + |3b - a + 3b(a - b)|}{\sqrt{a^{2} + 9b^{2}}}
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A train 120 meters long takes 26.997840172786177 seconds to cross a bridge of a certain length if the speed of the train is 36 kmph. How long is the bridge?
Please reason step by step, and put your final answer within \boxed{}.


Context:
11.175. A right parallelepiped is described around a sphere, with the diagonals of the base being $a$ and $b$. Determine the total surface area of the parallelepiped.
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Context:
Margo walks to her friend's house uphill, taking 15 minutes. She returns downhill via the same route, which takes only 10 minutes. If her walking rate uphill is 3 miles per hour and downhill is 6 miles per hour, how far is Margo's friend's house?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can \(2^{n}\) be expressed as the sum of four squares of positive integers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
若正整数 $m 、 n$ 满足 $\frac{(m+n)!}{n!}=5040$,则 $m!n$ 的值为 $\qquad$。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(X_1, \ldots, X_n) \sim \operatorname{Unif}(0,b), b>0$. Find $E\left[\sum \frac{X_i }{X_{(n)}}\right]$ where $X_{(n)} = \max_i X_i$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
# Problem 6. (3 points)

Three runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions.

At some point, the first runner meets the second. After 15 minutes, the second runner meets the third for the first time. Another 25 minutes later, the third runner meets the first for the first time.

How many minutes does it take for one runner to complete the entire track?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Will was organizing his baseball cards in a binder with 3 on each page. He wanted to place the cards in consecutive order based on the year of issue. If he had 8 new cards from 2020 and 10 old cards from various years between 2015 and 2019, with 2 duplicates, how many pages would he use to arrange the cards so that no two cards from the same year share a page?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the square root of 81. Provide both positive and negative roots, if applicable. Express your answer in the form of ±n, where n is the absolute value of the root.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $s_1, s_2, s_3$ be the three roots of the polynomial $x^3 + x^2 + \frac{9}{2}x + 9$.

Find the value of $a + b + c$ if the expression $\prod_{i=1}^{3}(4s^4_i + 81)$ can be written as $2^a3^b5^c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The original price of an article is increased by a certain percentage, and then the new price is increased by the same percentage. This is equivalent to increasing the original price by 16.64 percent. What is the percentage increase applied twice?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the following expression as x approaches 2:
$$\lim _{ x\rightarrow 2 }{ \frac { f\left( x \right) -2 }{ \sqrt { x-1 } -1 } } $$
Given that
$$\lim_{x\to2}\frac{\ xf(x)-4}{x-2}=3.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jenny leaves her house at 8:00 with some cupcakes for her daughter's birthday party at school. She jogs to the school at 15 miles per hour. Half an hour later, her wife Anna realizes that Jenny used peanut butter frosting because she forgot one of the kids in the class had a severe peanut allergy. Jenny doesn't have her cell phone, so Anna leaves the house driving at 45 miles her hour to catch Jenny and warn her. How long does Anna spend traveling in minutes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A plane is passed through the midpoint of an edge of a regular tetrahedron and is perpendicular to an adjacent edge. Find the area of the resulting cross-section if the edge of the tetrahedron is equal to \(a\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A company needs to transport at least 180 tons of goods every day. The company has 8 Type A trucks with a capacity of 6 tons each and 4 Type B trucks with a capacity of 10 tons each. Type A trucks can make 4 round trips per day, and Type B trucks can make 3 round trips per day. The daily cost for a Type A truck is 320 yuan, and for a Type B truck, it is 504 yuan. Determine the minimum daily expense for the company by reasonably allocating the vehicles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the vertices of triangle ABC as A(2,0), B(0,4), and C(4,1), where AE is the angle bisector of angle BAC, calculate the coordinates of a directional vector of line AE. Express your answer as a vector in the form (x, y).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A B C$ be a right triangle with $\angle A=90^{\circ}$. Let $D$ be the midpoint of $A B$ and let $E$ be a point on segment $A C$ such that $A D=A E$. Let $B E$ meet $C D$ at $F$. If $\angle B F C=135^{\circ}$, determine $B C / A B$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence ${a_n}$ with first term $a_1 = a$, its sum of the first $n$ terms is $S_n$, and it satisfies the equation $S_n + S_{n-1} = 3n^2$ for $n \geq 2$. If $a_n < a_{n+1}$ holds true for any $n \in \mathbb{N}^*$, find the range of $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( p \) be an odd prime. Find the number of \( p \)-element subsets \( A \) of \( \{1, 2, \ldots, 2p \} \) such that the sum of all elements of \( A \) is divisible by \( p \).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{1}{p}\left(\binom{2p}{p} - 2\right) + 2}


Context:
Unit square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$. Line $EF$ is parallel to $AB$, and line $GH$ is parallel to $BC$. The point $P$ is the intersection of $EF$ and $GH$. It is given that $BF = \frac{1}{4}$ and $BF + DH = FH$. Calculate the nearest integer to the degree measure of $\angle FAH$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
4. An infinite sequence of integers, $a_{0}, a_{1}, a_{3} \ldots$, with $a_{0}>0$, has the property that for any $n \geq 0, a_{n+1}=a_{n}-b_{n}$, where $b_{n}$ is the number having the same sign as $a_{n}$, but having the digits written in the reverse order. For example if $a_{0}=1210$, $a_{1}=1089$ and $a_{2}=-8712$, etc. Find the smallest value of $a_{0}$ so that $a_{n} \neq 0$ for all $n \geq 1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a 4x4 grid of points (equally spaced). How many rectangles, of any size, can be formed where each of its four vertices are points on this grid?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A very large number of small particles forms a spherical cloud. Initially, they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not interact with each other in any other way.

Find the time that passes until the cloud collapses fully, expressed in the form $\frac{k}{\sqrt{G \rho_0}}$. Please give the value of $10000k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $ab \gt 0$, then the value of $\frac{a}{|a|}+\frac{b}{|b|}+\frac{ab}{{|{ab}|}}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the convergence and find the limit of the sequence $b_n = \frac{n+2}{2^{n+2}}\left(\frac{2}{1}+\frac{2^2}{2}+...+\frac{2^{n+1}}{n+1}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $I$ be the incenter of $\triangle ABC$. If $BC = AC + AI$ and $\angle ABC - \angle ACB = 13^\circ$, find $\angle BAC$.
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Context:
For each of the $9$ positive integers $n, 2n, 3n, \ldots, 9n$, Alice takes the first decimal digit (from the left) and writes it onto a blackboard. She selected $n$ such that among the nine digits on the blackboard, there is the least possible number of different digits. What is this number of different digits equal to?
Please reason step by step, and put your final answer within \boxed{}.


Context:
1. Find the smallest 10-digit number, the sum of whose digits is not less than that of any smaller number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the initial value problem: $$\dfrac{d^2y}{dx^2}=\left( \dfrac{dy}{dx} \right)^2 + 1$$ with initial conditions \( y(0)=0 \) and \( \dfrac{dy}{dx}(0)=0 \) on the interval \( \left( \dfrac{\pi}{2},-\dfrac{\pi}{2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the sequence listing all positive rational numbers not exceeding \(\frac{1}{2}\), where fractions are ordered by increasing denominator and then by increasing numerator, find the sum of the numerator and denominator of the 2012th fraction. If the fraction is \(\frac{m}{n}\) with \(m\) and \(n\) relatively prime, compute \(m+n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( \lim\sup_{n \to \infty} \frac{E_n}{n!} \), where \( E_n \) is the number of permutations in \( S_n \) that can be obtained by using each transposition exactly once.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the recurrence relation using the Divide and Conquer method: \[ T(n) = T\left(\frac{n}{7}\right) + T\left(\frac{11n}{14}\right) + n \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the work done by the force field \( F(x,y,z) = e^{y+2z}(i+xj+2xk) \) as a particle moves from \((0,0,0)\) to \((1,1,1)\) along the curve defined by \(x=t\), \(y=t^2\), and \(z=t^4\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
5 级方阵 A 满足: 第一行元素均为 1, 第二行元素从左至右依次为 $1,-1,1,-1,1$,且 $\operatorname{trace}\left(A A^{\prime}\right)=28$. 记 $\operatorname{det} A$ 的最大值为 M, 则 $[M]=\qquad$.
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Context:
Let  $ABC$  be an acute-angled triangle with altitude  $AT = h$ . The line passing through its circumcenter  $O$  and incenter  $I$  meets the sides  $AB$  and  $AC$  at points  $F$  and  $N$ , respectively. It is known that  $BFNC$  is a cyclic quadrilateral. Find the sum of the distances from the orthocenter of  $ABC$  to its vertices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the linear model \( Y_i = \beta + U_i \) for \( i = 1, 2, 3 \), with conditions \( E(U_1) = E(U_2) = E(U_3) = 0 \), \( \text{Var}(U_1) = 1 \), \( \text{Var}(U_2) = 2 \), and \( \text{Var}(U_3) = 4 \), determine the best linear unbiased estimate (BLUE) of the variance of \( \beta \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $\underline{xyz}$  represent the three-digit number with hundreds digit  $x$ , tens digit  $y$ , and units digit  $z$ , and similarly let  $\underline{yz}$  represent the two-digit number with tens digit  $y$  and units digit  $z$ .  How many three-digit numbers  $\underline{abc}$ , none of whose digits are 0, are there such that  $\underline{ab}>\underline{bc}>\underline{ca}$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the function f(x) defined on R satisfies: the graph of the function y=f(x-1) is symmetrical about the point (1,0), and f(x+2)=f(x) always holds when x≥0, and f(x)=e^x-1 when x∈[0,1], find the value of f(-2017)+f(2018)=____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a Brownian Motion starting at $0$, calculate the probability that $X(2T) > 0$ and $X(T) > 0$. Express your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the sufficient and necessary condition for the inequality $|x-m| < 1$ to hold is $\frac{1}{3} < x < \frac{1}{2}$, then the range of values for the real number $m$ is        .
Please reason step by step, and put your final answer within \boxed{}.

\boxed{-\frac{1}{2} \leq m \leq \frac{4}{3}}


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?

[i]Proposed by Lewis Chen[/i]
Please reason step by step, and put your final answer within \boxed{}.


Context:
The average height of 30 students in a class was calculated as a certain value. It was later found that the height of one of the students in the class was incorrectly written as 151 cm, whereas the actual height was 136 cm. The actual average height of the students in the class was 174.5 cm. What was the initially calculated average height?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface integral \( \iint_Y G \cdot N\, dS \) where \( Y \) is the surface defined by \((x+y)^2+(y+z)^2=4+z^2\) for \( 0 \leq z \leq 2 \), \( N \) is the outward-pointing normal vector, and \( G = (x^2y, y^2z, z^2x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum of the series: \[ \sum_{m=1}^\infty \frac{\left(\frac{1}{2}\right)^\underline{m}}{m!} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Acute triangle $A B C$ has circumcenter $O$. The bisector of $\angle A B C$ and the altitude from $C$ to side $A B$ intersect at $X$. Suppose that there is a circle passing through $B, O, X$, and $C$. If $\angle B A C=n^{\circ}$, where $n$ is a positive integer, compute the largest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of $\cos 96^\circ \cos 24^\circ - \sin 96^\circ \sin 66^\circ$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sequence $\{a_n\}$ with the sum of its first $n$ terms $S_n = 6n - n^2$, find the sum of the first $20$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}}\right\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the value of the infinite nested radical expression: \[ \sqrt{-1+1\sqrt{-2+2\sqrt{-3+3\sqrt{-4+4\sqrt{\ldots}}}}} \]
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Context:
Given sets A = {2, 4, a} and B = {1, 2, 3}, if A ∪ B = {1, 2, 3, 4}, find the possible values of a. Express your answer as a list or set of values.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知正三棱锥 $P-A B C$ 的底面 $A B C$ 为正三角形，该正三棱锥的外接球的球心 $O$ 满足 $\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}=0$. 求二面角 $A-P B-C$ 的余弦值。答案的形式为\frac{k}{m}，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each vertex of a regular dodecagon ($12$-gon) is to be colored either red or blue, and thus there are $2^{12}$ possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a population of drivers, 30% belong to Class I, where the number of accidents per driver follows a geometric distribution with a parameter uniformly distributed over (0, 0.5). The remaining drivers belong to Class II, where each driver has an expected number of claims of 0.4. Calculate the probability that a randomly selected driver from this population has at most three accidents.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 5 apples, their weights are 125g, $a$, 121g, $b$, and 127g respectively. If the median and the average of this sample are both 124, then the standard deviation $s=$ ______ (Answer with a number).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ann and Bob each simultaneously toss a fair coin in each round. Let $X_n$ be the number of heads obtained in the $2n$ flips during the first $n$ rounds. For each integer $m > 0$, let $r_m$ be the probability that there exists an $n$ such that $X_n = m$. Determine the value of $\lim_{m \to \infty} r_m$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to traverse an 8-cycle graph in 10 moves, starting from a given vertex and ending at the vertex directly opposite to it.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two parallel lines \(\ell_{1}\) and \(\ell_{2}\) are tangent to a circle \(\omega_{1}\) with center \(O_{1}\) at points \(A\) and \(B\) respectively. A second circle \(\omega_{2}\) with center \(O_{2}\) is tangent to line \(\ell_{1}\) at point \(D\), intersects line \(\ell_{2}\) at points \(B\) and \(E\), and intersects circle \(\omega_{1}\) again at point \(C\) (point \(O_{2}\) lies between lines \(\ell_{1}\) and \(\ell_{2}\)). Given that the ratio of the area of quadrilateral \(B O_{1} C O_{2}\) to the area of triangle \(O_{2} B E\) is \(\frac{6}{5}\), find the ratio of the radii of circles \(\omega_{2}\) and \(\omega_{1}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the norm of the functional $l: C[0,\pi] \rightarrow \mathbb{R}$ defined by \[ l(x) = x(0) - x(\pi/4) + \int_{0}^{\pi} x(s) \sin s \, ds. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given some $4$-tuples of positive real numbers $(a_1,b_1,c_1,d_1),\dots,(a_n,b_n,c_n,d_n)$ with all $a_i,b_i,c_i,d_i\leq 1$, determine the best possible value $k$ between $1$ and $3$ such that we can always partition $\{1,2,\dots,n\}$ into two subsets $X,Y$ satisfying:
\[1+\sum_{i \in X}a_i\geq \sum_{i \in Y}a_i, \quad 1+\sum_{i \in X}b_i\geq \sum_{i \in Y}b_i,\]
\[\sum_{i \in X}c_i\leq 1+\sum_{i \in Y}c_i, \quad \sum_{i \in X}d_i\leq 1+\sum_{i \in Y}d_i.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( p \) such that the series of functions \( f(x) = \sum_{n=1}^\infty \frac{\log(1+ n^2x^2)}{n^p} \) converges and can be differentiated term-by-term for all \( x \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest common divisor (GCD) of the complex numbers \(29-3i\) and \(11+10i\) in the Gaussian integers \(\mathbb{Z}[i]\).
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Context:
A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equals the third, calculate the probability that at least one "2" is tossed. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the largest number of shells that cannot be delivered using whole boxes containing 15, 18, and 20 shells?
Please reason step by step, and put your final answer within \boxed{}.


Context:
John goes for a jog every 3 days. He went for a jog on Monday, January 5, and his next jog was on January 8. What is the date of the next Monday on which he will go for a jog? Express your answer as a date in the format 'Month Day'.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The non-isosceles triangle \(ABC\) with a perimeter of 12 is inscribed in a circle \(\omega\). Points \(P\) and \(Q\) are the midpoints of the arcs \(ABC\) and \(ACB\) respectively. The tangent drawn to the circle \(\omega\) at point \(A\) intersects the ray \(PQ\) at point \(R\). It turns out that the midpoint of the segment \(AR\) lies on the line \(BC\). Find the length of segment \(BC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the integral \( \int_M \omega \) where \( M = \{(x,y,z) : z = x^2 + y^2, z < 1\} \) is a smooth 2-manifold in \( \mathbb{R}^3 \) and \( \omega = x \, dy \wedge dz + y \, dz \wedge dx + z \, dx \wedge dy \in \Omega^2(\mathbb{R}^3) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the integral \( \int_{|z|=2}\frac{f'(z)}{f(z)}dz \), where \( f(z) = 1 + \frac{1}{z}+\frac{1}{2!z^2}+\dots+\frac{1}{n!z^n} \). Given that \( f \) has no zeros with \( |z|\geq 2 \), apply the Argument Principle to determine the value of the integral.
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Context:
Given that $\alpha$ is an obtuse angle and $\beta$ is also an obtuse angle with $\cos\alpha = -\frac{2\sqrt{5}}{5}$ and $\sin\beta = \frac{\sqrt{10}}{10}$, find the value of $\alpha + \beta$.
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Context:
The hour and minute hands on a certain 12-hour analog clock are indistinguishable. If the hands of the clock move continuously, compute the number of times strictly between noon and midnight for which the information on the clock is not sufficient to determine the time.

[i]Proposed by Lewis Chen[/i]
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many integer values of x make the polynomial p = x^4 + 6x^3 + 11x^2 + 3x + 31 a perfect square?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \int\int_S z \, dx \, dy \) over the complete surface \( S \) of a circular cylinder with radius \( a \) and axis along the \( OZ \) direction, where the ends of the cylinder are in the planes \( z = 0 \) and \( z = 2a \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of all trees planted at a five-foot distance from each other on a rectangular plot of land, the sides of which are 120 feet and 70 feet.
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Context:
Use differentials to approximate the value of \( h(1.02, 1.99) \), where \( h = f \circ \vec{g} \), \( f(u,v) = 3u + v^2 \), \( \vec{g}(1,2) = (3,6) \), and the Jacobian of \( \vec{g} \) at \( (1,2) \) is \( \begin{pmatrix} 2 & 1 \\ 3 & 5 \end{pmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the dimension of the null space of a linear transformation $T \in \wedge^{n-2}V^*$, where $V$ is a vector space with dimension $n$ and $T \neq 0$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit of the following expression as \( n \) approaches infinity, using the asymptotic expansion of the Barnes G-function:

\[\lim_{n\to\infty} \frac{G(n+\frac{11}{8})G(n+\frac{13}{8})^{2}G(n+\frac{15}{8})}{G(n+\frac{5}{2})^{2}G(n+\frac{7}{4})^{2}}\]
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Context:
Semen has 20 numbers: \(1, 2, 3, \ldots, 19, 20\). He created 10 fractions by writing ten of these numbers in some order as the numerators and the remaining ten numbers in some order as the denominators. What is the maximum number of integers that Semen could obtain after reducing all the written fractions?
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Context:
Determine the time complexity of the recurrence relation: $$T(n)=T\left(n-\frac{n}{\lg n}\right)+ O(n)$$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let acute triangle $ABC$ have circumcenter $O$, and let $M$ be the midpoint of $BC$. Let $P$ be the unique point such that $\angle BAP=\angle CAM, \angle CAP=\angle BAM$, and $\angle APO=90^{\circ}$. If $AO=53, OM=28$, and $AM=75$, compute the perimeter of $\triangle BPC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of pairwise non-isomorphic connected 3-regular graphs of order 14.
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Context:
At the round table,  $10$  people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) .  It is clear thath I have at least one knight and at least one liar.
What is the largest number of those sitting at the table can say: ''Both of my neighbors are knights '' ?
(A statement that is at least partially false is considered false.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Twelve students participated in a theater festival consisting of \( n \) different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine the largest possible value of \( n \).
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Context:
Find the nearest prime number to 2304 that is divisible by 23.
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Context:
A special clock in the museum has 20 positions on its dial. Every 7 minutes, the hand jumps once, and each jump covers 9 positions. This morning at 8:00 AM, the hand moved from position 0 to position 9. Where was the hand pointing at 8:00 PM yesterday?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value of the time at which the trajectory of the Wiener process, $W_t$, first crosses the line $y=t$.
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Context:
You have four charged batteries and four uncharged batteries, but you do not know which are which. A radio requires two charged batteries to operate. What is the minimum number of attempts needed to ensure the radio works? An attempt consists of inserting two batteries into the radio and checking if it operates.
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Context:
A point is randomly thrown onto the interval $[6, 10]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-3k-10\right)x^{2}+(3k-8)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The inclination angle of the line $x - y + 3 = 0$ is ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A variable plane intersects the coordinate planes to form a tetrahedron with a constant volume of \(96k^3\). Determine the locus of the center of gravity of this tetrahedron.
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Context:
From a large sheet of aluminum, triangular sheets (with each cell side equal to 1) are cut with vertices at marked points. What is the minimum area of the triangle that can be obtained?
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Context:
Construction Mayhem University is expanding its campus with a new construction schedule. A new project begins every two days, with each project taking exactly one more day than the previous one to complete. The first project takes 3 days, the second takes 4 days, and so on.

Suppose the new schedule starts on Day 1. On which day will there first be at least $10$ projects in place at the same time?
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Context:
Tara is saving up to buy a new clarinet. She already has $10 saved. The clarinet costs $90. She plans to sell her old books to make the money. She sells each book for a certain amount. However, when she is halfway towards her goal, she loses all her savings and has to start over. She sells a total of 25 books by the time she reaches her goal. How much does she sell each book for?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. Find the value of $\overline{AR}+\overline{BR}+\overline{CR}+\overline{DR}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
By placing addition signs between some of the digits of the number 123456789, we can obtain various sums. What is the minimum number of addition signs necessary to obtain the number 54 in this manner? Provide your answer as a whole number.
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Context:
How many Young diagrams are there with at most $p$ rows and $q$ columns, without restrictions on the weight of the diagrams?
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Context:
Pompous Vova has an iPhone XXX, and on that iPhone, he has a calculator with voice commands: "Multiply my number by two and subtract two from the result," "Multiply my number by three and then add four," and lastly, "Add seven to my number!" The iPhone knows that initially, Vova's number was 1. How many four-digit numbers could the iPhone XXX theoretically achieve by obediently following Vova's commands?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the smallest number that must be added to 956734 to make it exactly divisible by a combination of prime numbers, such as the product of 751 and 503?
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Context:
The points \( K, L, M, N \) are the centers of the circles inscribed in the faces \( S A B, S A C, S B C, \) and \( A B C \) of the tetrahedron \( S A B C \). It is known that \( A B = S C = 5 \), \( A C = S B = 7 \), \( B C = S A = 8 \). Find the volume of the tetrahedron \( K L M N \). If necessary, round your answer to two decimal places.
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Context:
Determine the coordinates of the point \( P(m, n) \) on the ellipse \( \frac{x^2}{8} + \frac{y^2}{18} = 1 \) such that the tangent line at \( P \) intersects the coordinate axes to form a triangle \( OAB \) with the origin \( O \) and the least possible area.
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Context:
Two equally strong tennis players play against each other until one player wins three consecutive games. Each game is independent, and each player has a probability of \( \frac{1}{2} \) of winning any given game. What is the expected number of games they will play?
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Context:
Determine if there exists a positive integer $k$ such that the inequality \( \sum_{i=1}^{k} \frac{y_i^2}{x_i} \leq \frac{(\sum_{i=1}^{k}y_i)^2}{\sum_{i=1}^{k}x_i} \) holds, where all variables $x_i$ and $y_i$ are positive integers.
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Context:
The price of Margit néni's favorite chocolate was increased by 30%, and at the same time her pension increased by 15%. By what percentage does Margit néni's chocolate consumption decrease if she can spend only 15% more on chocolate?
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Context:
Use Green's Theorem to evaluate the line integral \( \int_C P\,dx + Q\,dy \), where \( P = xy \) and \( Q = x^2 \). The curve \( C \) is the loop in the first quadrant described by the polar equation \( r = \sin 2\theta \).
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Context:
Find the integers \( n \) such that there exist \( x \), \( y \), and \( k \) with \( x \) and \( y \) being coprime and
\[
3^n = x^k + y^k
\]
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Context:
Let ABC be a triangle with $AB = 13, BC = 14, CA = 15$. The altitude from $A$ intersects $BC$ at $D$.
Let $\omega_1$ and $\omega_2$ be the incircles of $ABD$ and $ACD$, and let the common external tangent of $\omega_1$ and $\omega_2$ (other than $BC$) intersect $AD$ at $E$. Compute the length of $AE$.
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Context:
Let $x_1, x_2, \dots, x_n$ be a positive integer sequence such that each term is less than or equal to $2001$ and for every $i\geq 3$, $x_i = |x_{i-1}-x_{i-2}|$. What is the largest possible value of $n$?
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Context:
The frame of a painting has the form of a $105^{\\prime\\prime}$ by $105^{\\prime\\prime}$ square with a $95^{\\prime\\prime}$ by $95^{\\prime\\prime}$ square removed from its center.  The frame is built out of congruent isosceles trapezoids with angles measuring $45^\\circ$ and $135^\\circ$.  Each trapezoid has one base on the frame\u0027s outer edge and one base on the frame\u0027s inner edge.  Each outer edge of the frame contains an odd number of trapezoid bases that alternate long, short, long, short, etc.  What is the maximum possible number of trapezoids in the frame?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\ge 1,$ find the minimum possible value of $|x_1+x_2+\cdots+x_{2006}|.$
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Context:
Evaluate the integral \( \int_0^{\infty} \frac{\cos^2 x \sin 2x}{x e^{x \sqrt{2}}} \, dx \).
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Context:
Find the smallest positive integer $n$ such that an $n \times n$ grid can be colored with red, yellow, and blue, with each color appearing the same number of times, and satisfying the condition that if a row contains a yellow cell, it must also contain a blue cell and cannot contain a red cell.
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Context:
Calculate the limit as $n$ approaches infinity for the following expression, given a sequence of independent and identically distributed (i.i.d.) non-negative random variables $X_i$ with mean $\mu = 4$ and variance $\sigma^2 = 16$:
$$ \lim_{n \to \infty} E\left[\cos\left(\sqrt{S_n} - 2\sqrt{n}\right)\right],$$
where $S_n = \sum_{i=1}^n X_i$. Provide your answer as a single value.
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Context:
Evaluate the integral \( \int_0^{30} \frac{1}{48!} t^{48} e^{-t} \, dt \).
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Context:
Find an explicit bijection between the set of equivalence classes of the equivalence relation defined on \( \mathbb{R}[X] \) by \( P_1 \sim P_2 \) if \( P_1 - P_2 \) is divisible by \( X \), and the set of real numbers \( \mathbb{R} \).
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Context:
Find the spectrum of the operator $T = S + S^*$, where $S$ is the unilateral shift operator on $\mathcal{l}^2$ and $S^*$ is its adjoint, the backward shift operator.
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Context:
Find the total number of different integers the function

\[ f(x) = \left\lfloor x \right\rfloor + \left\lfloor 2 \cdot x \right\rfloor + \left\lfloor \frac{5 \cdot x}{3} \right\rfloor + \left\lfloor 3 \cdot x \right\rfloor + \left\lfloor 4 \cdot x \right\rfloor \]

takes for $0 \leq x \leq 100.$
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Context:
In a game played on a 4x4 checkerboard, two players, Cat and Mouse, each place a checker on a cell of the board, with Cat going first. The players then take turns moving their checkers to an adjacent square, either vertically or horizontally, with Cat moving first. If, after any move, the two checkers occupy the same square, Cat wins. If each checker has made 2013 moves without this occurring, Mouse wins. Determine which player has a winning strategy.
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Context:
Simplify the following expression:

$$
\sqrt{2-\sqrt{3}} \cdot \sqrt{2-\sqrt{2-\sqrt{3}}} \cdot \sqrt{2-\sqrt{2-\sqrt{2-\sqrt{3}}}} \cdot \sqrt{2+\sqrt{2-\sqrt{2-\sqrt{3}}}}
$$

Express your answer as a single numerical value.
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Context:
Using the digits 1, 2, 3, 4, and 5 without repetition, how many 5-digit wave numbers can be formed such that the tens and thousands digits are each larger than the digits adjacent to them? Express your answer as a single integer.
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Context:
Colour a $20000 \times 20000$ square grid using 2000 different colours, with one colour per square. Two squares are considered neighbours if they share a vertex. Define a path as a sequence of squares where each pair of successive squares are neighbours.

Mark $k$ squares on the grid. For each unmarked square $x$, there is exactly one marked square $y$ of the same colour such that $x$ and $y$ are connected by a path consisting of squares of the same colour.

Furthermore, for any two marked squares of the same colour, any path connecting them must pass through squares of all the different colours.

Find the maximum value of $k$.
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Context:
A square board with dimensions 30 cm by 30 cm is divided along the grid lines into 225 smaller squares of equal area. Determine the maximum possible total length of the cuts made on the board.
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Context:
A baseball team played 45 games in a season, out of which 15 were away games. They won 6 home games and 7 away games, with 3 of their wins at home being in extra innings. What is the ratio of the number of away game losses to the number of home game losses that did not go into extra innings?
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Context:
Given that the function $f(x) = ax + \sin x$ has extreme points, find the range of the real number $a$. Express your answer as an interval or inequality.
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Context:
Rewrite the expression $y = \frac{(s-r)(q-p)}{(q-r)(s-p)}$ in terms of $x = \frac{(r-p)(s-q)}{(r-q)(s-p)}$.
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Context:
The probability that a fire will occur is 0.001. If there is a fire, the amount of damage, $X$, follows a Pareto distribution with $P(X>x) = \left(\frac{2(10)^6}{2(10)^6+x} \right)^2$. An insurance policy covers the excess loss over a deductible of $100,000$. The one-time insurance premium is 110% of the expected payment. Calculate the insurance premium.
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Context:
Given the equation \(ax + by + 2c = 0\) with conditions \(c \neq 0\) and \(ab - c^2 \geq 0\), find the maximum value of \(xy\). Express your answer as a single number.
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Context:
Let $a, b, c$ be the roots of the equation $x^3 - 9x^2 + 11x - 1 = 0$, and define $s = \sqrt{a} + \sqrt{b} + \sqrt{c}$. Compute $s^4 - 18s^2 - 8s$.
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Context:
Given that
\[\sum_{x=1}^{70} \sum_{y=1}^{70} \frac{x^{y}}{y} = \frac{m}{67!}\]
for some positive integer $m$, find $m \pmod{71}$.
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Context:
将等差数列 $1,5,9,13, \cdots, 2017$ 排成一个大数 $15913 \cdots 2017$ ，求该数值被 9 除的余数。
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Context:
Given the sequence ${a_n}=C_{200}^n⋅{({\sqrt[3]{6}})^{200-n}}⋅{({\frac{1}{{\sqrt{2}}}})^n}$, where $n$ ranges from $1$ to $95$, find the number of integer terms in the sequence.
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Context:
The hands of a clock are straight 22 times in a certain period. What is the period in which the hands of a clock are straight 22 times?
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Context:
Bobby buys two packets of candy. He eats two candies every day from Monday through Friday and takes a certain number of candies during the remaining days of the week. It takes him 3 such weeks to finish the 2 packets, and there are 18 candies in a packet. How many candies does Bobby take during the remaining days of the week?
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Context:
## Task Condition

Find the derivative.

$y=5 x-\ln \left(1+\sqrt{1-e^{10 x}}\right)-e^{-5 x} \cdot \arcsin \left(e^{5 x}\right)$
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Context:
s is a set of 81 consecutive multiples of 7. If the sum of the two smallest numbers in s is 145, then find the sum of the greatest number in s and the median of set s.
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Context:
Given the sets \( A=\{(x, y) \mid y=-|x|-2\} \) and \( B=\{(x, y) \mid (x-a)^2+y^2=a^2\} \) such that \( A \cap B = \varnothing \), determine the range of the real number \( a \).
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Context:
在 $\triangle A B C$ 中, $a=B C, b=C A, c=A B$. 若 $a^{2}+b^{2}=6, \cos (A-B) \cos C=\frac{2}{3}$, 求 $\triangle A B C$ 的外接圆半径。原始答案为 $\frac{m \sqrt{n}}{p}$ 的形式，请给出 $m + n + p$ 的值。
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Context:
Given that the solution set of the inequality $x^2+ax+4<0$ is empty, find the range of values for $a$.
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Context:
Given real numbers \( x \) and \( y \) such that

\[ \frac{x^2 + y^2}{x^2 - y^2} + \frac{x^2 - y^2}{x^2 + y^2} = k, \]

find \( \frac{x^8 + y^8}{x^8 - y^8} + \frac{x^8 - y^8}{x^8 + y^8} \) in terms of \( k \).
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Context:
Alice places a coin, heads up, on a table then turns off the light and leaves the room. Bill enters the room with 2 coins and flips them onto the table and leaves. Carl enters the room, in the dark, and removes a coin at random. Alice reenters the room, turns on the light and notices that both coins are heads. What is the probability that the coin Carl removed was also heads?
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Context:
On the side \( AD \) of a cyclic quadrilateral \( ABCD \), there is the center of the circle that touches the other three sides of the quadrilateral. Find \( AD \) if \( AB = 2 \) and \( CD = 3 \).
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Context:
[Example 2.2.1] Simplify: $M=\frac{2}{\sqrt{4-3 \sqrt[4]{5}+2 \sqrt[4]{25}-\sqrt[4]{125}}}$.
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Context:
Given that \( x \) and \( y \) are integers and satisfy the equation
$$
\left(\frac{1}{x} + \frac{1}{y}\right) \left(\frac{1}{x^2} + \frac{1}{y^2}\right) = -\frac{2}{3} \left(\frac{1}{x^4} - \frac{1}{y^4}\right),
$$
how many possible values of \( x + y \) are there? Express your answer as a single integer.
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Context:
For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less than or equal to $97$ does $D(n) = 2$?
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Context:
The angles $\alpha$ that satisfy the condition $|\sin\alpha| = -\sin\alpha$ are.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{[2k\pi-\pi, 2k\pi], k \in \mathbb{Z}}


Context:
9-6. Petya wants to place 99 coins in the cells of a $2 \times 100$ board so that no two coins are in cells that share a side, and no more than one coin is in any cell. How many ways are there to place the coins?
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Context:
From a list of integers from $1$ to $2022$, inclusive, delete all numbers in which at least one of its digits is a prime. How many numbers remain without erasing?
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Context:
A heavy concrete platform anchored to the seabed in the North Sea supported an oil rig that stood 40 m above the calm water surface. During a severe storm, the rig toppled over. The catastrophe was captured from a nearby platform, and it was observed that the top of the rig disappeared into the depths 84 m from the point where the rig originally stood.

What is the depth at this location? (Neglect the height of the waves.)
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Context:
Let  $ABC$  be triangle such that  $|AB| = 5$ ,  $|BC| = 9$  and  $|AC| = 8$ . The angle bisector of  $\widehat{BCA}$  meets  $BA$  at  $X$  and the angle bisector of  $\widehat{CAB}$  meets  $BC$  at  $Y$ . Let  $Z$  be the intersection of lines  $XY$  and  $AC$ . What is  $|AZ|$ ?
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Context:
Consider polynomials $P$ of degree $2015$, all of whose coefficients are in the set \{0,1,\dots,2010\}. Call such a polynomial "good" if for every integer $m$, one of the numbers $P(m)-20$, $P(m)-15$, $P(m)-1234$ is divisible by $2011$, and there exist integers $m_{20}, m_{15}, m_{1234}$ such that $P(m_{20})-20$, $P(m_{15})-15$, $P(m_{1234})-1234$ are all multiples of $2011$. Let $N$ be the number of good polynomials. Find the remainder when $N$ is divided by $1000$.
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Context:
Three male students and three female students, a total of six students, stand in a row. If male student A does not stand at either end, and exactly two of the three female students stand next to each other, then the number of different arrangements is ______.
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Context:
Over a period of six months, the number of animal sightings and family visits to a national park fluctuates due to various factors such as seasonal changes, animal behavior, and park events. The pattern of family visits and animal sightings during this time is as follows:

- In January, 100 families visit the park and see animals 26 times.
- In February, family visits increase by 50% and they see animals three times as many as in January.
- In March, animal sightings decline due to the change in season, and families see animals only half as many times as in February. Additionally, the number of families visiting the park decreases by 20% compared to February.
- In April, the park hosts a special event, resulting in a 70% increase in family visits compared to March. During this month, families see animals 40% more often compared to March.
- In May, the number of families remains the same as in April, but animal sightings decrease by 25% due to the warmer climate.
- In June, with the onset of summer vacations, there is a 30% increase in family visits, but animal sightings remain the same as in May due to the hot weather.

Assuming that each family visits the park only once per month, calculate the total number of animal sightings by families during these six months.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A  quartic funtion  $ y \equal{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx\plus{}e\ (a\neq 0)$  touches the line  $ y \equal{} px \plus{} q$  at  $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta ).$  Find the area of the region bounded by these graphs in terms of  $ a,\ \alpha ,\ \beta$ .
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{a (\beta - \alpha)^{5}}{30}}


Context:
设 X=\{1,2,\cdots,10\}，f:X\rightarrow X 的映射满足① f\circ f=I_X，其中 f\circ f 是复合映射，I_X 为 X 上的恒等映射；② |f(i)-i|\leq 2，对任意 i\in X.求映射 f 的数目.
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Context:
Paolo has 14 coconuts, while Dante has a certain number of times more coconuts than Paolo. If Dante sold 10 of his coconuts and had 32 coconuts left, what is the ratio of Dante's coconuts to Paolo's coconuts?
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Context:
Let  $f_1(x)=x^2-1$ , and for each positive integer  $n \geq 2$  define  $f_n(x) = f_{n-1}(f_1(x))$ . How many distinct real roots does the polynomial  $f_{2004}$  have?
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Context:
A regular hexagonal paper with a side length of 1 is joined with a regular triangular paper with a side length of 1. The pieces must be aligned edge-to-edge without overlapping. Find the number of edges in the resulting new shape.
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Context:
if the function q is defined by the formula q = 7w / (6mx (z^3)), by what factor will q be multiplied if w is quadrupled, m is doubled, x is reduced to half, and z is raised to the power of 4?
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Context:
Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$
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Context:
Determine the number of integers $a$ with $1 \leq a \leq 1007$ such that both $a$ and $a+1$ are quadratic residues modulo $1009$.
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Context:
How many of the first $1000$ positive integers can be written as the sum of finitely many distinct numbers from the sequence $3^0$, $3^1$, $3^2$, $\ldots$?
Please reason step by step, and put your final answer within \boxed{}.