Context:
Let $ABC$ be a triangle with incenter $I$. Let $M_b$ and $M_a$ be the midpoints of $AC$ and $BC$, respectively. Let $B'$ be the point of intersection of lines $M_bI$ and $BC$, and let $A'$ be the point of intersection of lines $M_aI$ and $AC$.

If triangles $ABC$ and $A'B'C$ have the same area, what are the possible values of $\angle ACB$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $f(x)=e^{ax+1}-x(\ln x-2)$ exists a monotonically decreasing interval, then the range of positive number $a$ is ______.
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Context:
Calculate the integral \( \int_{-\infty}^{\infty} \frac{x \sin(\pi x)}{(x-3)(x-2)} \, dx \) using the Residue Theorem.
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Context:
Determine whether the sequence $P_n$ is equal to the sequence $G_n$, where $G_0=1$ and $G_{n+1}=F_0G_n+F_1G_{n-1}+\cdots+F_nG_0$, with $F_n$ being the $n$th term of the Fibonacci sequence defined by $F_0=F_1=1$ and $F_{n+1}=F_n+F_{n-1}$. The sequence $P_n$ is defined by $P_0=P_1=1$, $P_2=2$, and $P_{n+1}=2P_n+P_{n-1}$ for $n>1$. Is $P_n=G_n$?
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Context:
A question and answer forum has 200 members. The average number of answers posted by each member on the forum is three times as many as the number of questions asked. Each user posts an average of 3 questions per hour. How many questions and answers are posted on the forum by its users in a day?
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Context:
Find the derivative with respect to \( a \) of the solution to the initial value problem at \( a = 0 \):

\[ x' = x^2 - x\cos t, \quad x(0) = a \]
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Context:
4. Find the number of all 5-digit numbers (in base 10) each of which contains the block 15 and is divisible by 15 . (For example, 31545,34155 are two such numbers.)

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Context:
Suppose $f$ is an analytic function on the unit disk $D = \{ z \in \mathbb{C} : |z| < 1 \}$ and continuous up to the closure $\bar{D}$. If $f$ vanishes on a non-null segment of the boundary, such as $\{ e^{it}, 0 \leq t \leq \pi/2 \}$, is $f$ necessarily zero everywhere on the boundary? Answer "yes" or "no".
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Context:
Let $f$ be a differentiable function defined over a convex set $X$. Suppose that for all $x, \overline{x} \in X$ with $x \neq \overline{x}$, the following strict inequality holds: $\langle \nabla f(\overline{x}), x - \overline{x}\rangle > 0$. Does this imply that $\overline{x}$ is a local minimum? Answer "yes" or "no."
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Context:
Find the equation of the tangent line to the curve $y = g(u(x), v(x))$ at $x = 1$, given that $(u'(1), v'(1)) = (3, 1)$ and the tangent plane to the surface $z = g(u, v)$ at the point $(1, 2)$ is $4 - z - 2u + 5v = 0$. Assume $g(u, v)$ and $g(u(x), v(x))$ are related functions.
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Context:
Let \( a,b,c,d,e, \) and \( f \) be real numbers. Define the polynomials
 \( P\left( x\right)  = 2{x}^{4} - {26}{x}^{3} + a{x}^{2} + {bx} + c\; \) and \( \;Q\left( x\right)  = 5{x}^{4} - {80}{x}^{3} + d{x}^{2} + {ex} + f. \)
  Let \( S \) be the set of all complex numbers which are a root of either \( P \) or \( Q \) (or both).
 Given that \( S = \{ 1,2,3,4,5\} \) ,compute \( P\left( 6\right)  \cdot  Q\left( 6\right) \) .
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Context:
Given that \(a^2 + b^2 + c^2 = 1\) where \(a, b, c\) are real numbers, find the maximum value of the expression \((4a-3b)^2 + (5b-4c)^2 + (3c-5a)^2\).
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Context:
Determine the range of values of \( p \) for which the improper integral \( \int_{0}^{1} \frac{\ln x}{x^p(1-x)^{1-p}} \, dx \) converges.
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Context:
Ricsi, Dénes, and Attila often play ping pong against each other, with two of them standing on one side. Dénes and Attila win against Ricsi three times as often as they lose; Dénes wins against Ricsi and Attila as often as he loses; finally, Attila wins against Ricsi and Dénes twice as often as he loses. Recently, they spent an entire afternoon playing, during which they played six matches, with two players on each side. What is the probability that Ricsi won at least once, either alone or as part of a pair?
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Context:
There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that

($1$) for each $k,$ the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i + n$ is seat $i$);

($2$) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break.

Find the number of possible values of $n$ with $1 < n < 1000.$
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Context:
6. 22 For a given positive integer $k$, define $f_{1}(k)$ as the square of the sum of the digits of $k$, and let
$$f_{n+1}(k)=f_{1}\left(f_{n}(k)\right),$$

Find: $f_{1991}\left(2^{1990}\right)$.

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Context:
For the curve defined by \(x^3 + y^3 = c^3\), the tangent at the point \((a, b)\) passes through the point \((a_1, b_1)\). Given that \(\frac{a_1}{a} + \frac{b_1}{b} = -k\), find the value of \(k\). Assume that \((a_1, b_1)\) also lies on the curve.
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Context:
Four siblings BRYAN, BARRY, SARAH and SHANA are having their names monogrammed on their towels. Different letters may cost different amounts to monogram. If it costs $21 to monogram BRYAN,$25 to monogram BARRY and $18 to monogram SARAH, how much does it cost to monogram SHANA?

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Context:
What is the largest integer that is a divisor of \[
(n+1)(n+3)(n+5)(n+7)(n+9)
\]for all positive even integers $n$?
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Context:
Determine the values of $x$ for which the series $$\sum_{n=2}^{\infty} \frac{x^{n^2}}{n \log(n)}$$ converges.
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Context:
A rectangular box has integer side lengths in the ratio $1: 3: 4$. Which of the following could be the volume of the box?
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Context:
99.2. Consider 7-gons inscribed in a circle such that all sides of the 7-gon are of different length. Determine the maximal number of $120^{\circ}$ angles in this kind of a 7-gon.

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Context:
Find the degree of the splitting field of the polynomial \( f = X^4 + 2X^2 + 2 \) over \( \mathbf{Q} \).
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Context:
One rainy afternoon you write the number  $1$  once, the number  $2$  twice, the number  $3$  three times, and so forth until you have written the number  $99$  ninety-nine times. What is the  $2005$  th digit that you write?
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Context:
Consider the inequality
\[(a_1+a_2+\dots+a_n)^2\ge 4(a_1a_2+a_2a_3+\cdots+a_na_1).\]
 Find all $n\ge 3$ such that the inequality is true for positive reals.

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Context:
A biased coin has a $ \dfrac{6 + 2\sqrt{3}}{12} $ chance of landing heads, and a $ \dfrac{6 - 2\sqrt{3}}{12} $ chance of landing tails. What is the probability that the number of times the coin lands heads after being flipped 100 times is a multiple of 4? The answer can be expressed as $ \dfrac{1}{4} + \dfrac{1 + a^b}{c \cdot d^e} $ where $ a, b, c, d, e $ are positive integers. Find the minimal possible value of $ a + b + c + d + e $.
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Context:
8、A unit square has its center coinciding with the center of a circle and is located inside the circle. If a point is randomly selected on the circle, and the probability that from this point one can see two complete sides of the square is $\mathbf{1 / 2}$, then the radius of the circle is $\qquad$
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Context:
Use Parseval's identity to find the length \(|f|\) of the function \(f(x) = 2 \cos(14x) + 4 \cos(11x) + 2 \sin(27x) - \cos(19x) + 5 \cos(140x)\).
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Context:
Identify the space obtained as the $L^2$-closure of polynomials $p(z)$ in the $z$-variable, where $z \in \Omega \subset \mathbb{C}$ for a given open and bounded set $\Omega$. Does this space have a classical interpretation in complex analysis?
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Context:
Let  $ABC$  be a triangle with circumcenter  $O$  such that  $AC = 7$ . Suppose that the circumcircle of  $AOC$  is tangent to  $BC$  at  $C$  and intersects the line  $AB$  at  $A$  and  $F$ . Let  $FO$  intersect  $BC$  at  $E$ . Compute  $BE$ .
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Context:
Parkway Elementary School has 800 seventh-grade students. Out of these, 420 are boys and 380 are girls. These students can participate in a range of extracurricular activities including soccer, basketball, chess club, math club, and glee club. The school allows each student to participate in only one extracurricular activity. There are 320 students participating in soccer, out of which 70% are boys, 280 students participating in basketball, of which 65% are girls, 70 students participating in the chess club that's majorly boys (80%), and the math club has a total of 50 members with an equal gender representation. The remaining students participate in the glee club.

The school also has an absentee record where 5% of total boys and 8% of total girls were absent throughout the year and were not able to participate in any of the clubs. How many girl students in seventh grade at Parkway Elementary School are not participating in any extracurricular activities, considering the absenteeism as well?
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Context:
Given the XOR result X = 0000 0011, determine if it is possible to uniquely identify the original values A and B, where A and B are 7-bit ASCII characters ranging from 32 to 126. Calculate the number of combinations a computer would need to try to find the correct values of A and B.
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Context:
Identify the group \((D, \circ)\) of the symmetries of the circle.
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Context:
Angelo wrote down the word TEAM. He then swapped two adjacent letters around and wrote down the new order of the letters. He proceeded in this way until he obtained the word MATE. What is the minimum number of swaps that Angelo could have used to transform the word "TEAM" into "MATE"? Express your answer as a single integer.
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Context:
How many ordered triples $(A,B,C)$ of subsets of $\{1,2,...,n\}$ exist such that $A\cap B\cap C = \emptyset$, $A\cap B \neq \emptyset$, and $A\cap C \neq \emptyset$?
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Context:
Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?

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Context:
## C2

In a country with $n$ cities, all direct airlines are two-way. There are $r>2014$ routes between pairs of different cities that include no more than one intermediate stop (the direction of each route matters). Find the least possible $n$ and the least possible $r$ for that value of $n$.

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Context:
Given a continuous function $f: \mathbb{R} \to \mathbb{C}$, if $\int_{\mathbb{R}} |f(x)| \, dm(x) < \infty$, is it true that $\text{ess}\sup |f| < \infty$? Answer "yes" or "no."
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the superstructure $\mathbb V$ over $\mathbb R$ and its subset $\mathcal{P}_{fin}(\mathbb V)$, where $\mathbb V=\cup_{n<\omega}V_n$ with $V_0=\mathbb R$ and $V_{n+1}=V_n\cup \mathcal{P}(V_n)$. What is the cardinality of these sets?
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Context:
Let \( S \) be the set of all points(x,y)in the coordinate plane such that \( 0 \leq  x \leq  \frac{\pi }{2} \) and \( 0 \leq  y \leq  \frac{\pi }{2} \) . What is the area of the
 subset of \( S \) for which
 \( {\sin }^{2}x - \sin x\sin y + {\sin }^{2}y \leq  \frac{3}{4}? \)

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Context:
8,9

Point $K$ is located on edge $AD$ of tetrahedron $ABCD$, point $N$ is on the extension of edge $AB$ beyond point $B$, and point $M$ is on the extension of edge $AC$ beyond point $C$, such that $AK:KD=3:1$, $BN=AB$, and $CM:AC=1:3$. Construct the section of the tetrahedron by the plane passing through points $K, M, N$. In what ratio does this plane divide the volume of the tetrahedron?
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Context:
Determine the rank of the field of rational numbers \(\mathbb{Q}\) over the ring of integers \(\mathbb{Z}\).
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Context:
There are 1996 points on a circle, each painted in one of several different colors. The number of points of each color is different. Determine the number of colors and the distribution of points per color that maximizes the number of inscribed polygons that can be formed by selecting one point from each color, ensuring that all vertices of the polygon are of different colors.
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Context:
Let \( K = \mathbb{Q} \) and consider the polynomial \( f = X^3 - 3 \). Determine the basis and the degree of the splitting field of \( f \) over \( K \).
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Context:
Determine whether the set $X=\{A \in P : 1 \in A\}$, where $P=\{A \subseteq \mathbb{N} : A \text{ is non-empty, finite, and has an even number of elements}\}$, has a lower bound in the partially ordered set $(P,\subseteq)$. Provide your answer as "yes" or "no".
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Context:
Determine whether the growth in the number of decimal digits of the factorial function is exponential or sub-exponential. For example, 7! = 5040, which has 4 digits.
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Context:
The equation \( x^{2} + mx + 1 + 2i = 0 \) has real roots. Find the minimum value of the modulus of the complex number \( m \).
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Context:
Among all the triangles inscribed in a given circle, find the one for which the sum of the squares of the lengths of the sides is maximized.
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Context:
Given the set $M=\{x|x^2-5x-6\leq0, x\in \mathbb{N}\}$, if two numbers are randomly selected from $M$ and added together, the sum will be an element of set $N$. How many non-empty proper subsets does $N$ have?
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Context:
Ana, Bob, and CAO bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point $D$ on the south edge of the field. Cao arrives at point $D$ at the same time that Ana and Bob arrive at $D$ for the first time. The ratio of the field's length to the field's width to the distance from point $D$ to the southeast corner of the field can be represented as $p : q : r$, where $p$, $q$, and $r$ are positive integers with $p$ and $q$ relatively prime. Find $p+q+r$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$2006$ points are evenly spaced on a circle. Given one point, find the maximum number of points that are less than one radius distance away from that point.

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Context:
In a pentagon, all diagonals are drawn. Identify seven angles between two diagonals or between diagonals and sides such that if these angles are equal, the pentagon is regular.
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Context:
Determine whether the functional equation \[3=\sqrt{1+f(2)\sqrt{1+f(3)\sqrt{1+\cdots}}}\] has a unique solution \(f(x)=x\) for any positive, continuous, and differentiable function \(f(x)\) on \((0,\infty)\) that is not constant.
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Context:
Determine if the integral \( \int_0^\infty \frac{1-\cos x}{x^2(x^2+1)}\,dx \) can be evaluated without using complex analysis techniques such as complex integration, residues, and Cauchy's Theorem.
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Context:
 Lucy has some spare change. If she were to convert it into quarters and pennies, the minimum number of coins she would need is $66$. If she were to convert it into dimes and pennies, the minimum number of coins she would need is $147$. How much money, in cents, does Lucy have?

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Context:
Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let $m$ be the maximum number of red marbles for which such an arrangement is possible, and let $N$ be the number of ways he can arrange the $m+5$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
1. A bear walks one mile south, one mile east, and one mile north, only to find itself where it started. Another bear, more energetic than the first, walks two miles south, two miles east, and two miles north, only to find itself where it started. However, the bears are not white and did not start at the north pole. At most how many miles apart, to the nearest .001 mile, are the two bears' starting points?
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Context:
Suppose that \( \sum_{j=1}^{\infty} a_j \) is a convergent series with \( a_j \geq 0 \). Determine whether it is true that \( \lim_{n \to \infty} S_n = 0 \), where \( S_n = \sum_{j=1}^{n-1} \frac{a_j}{n-j} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a square matrix \(A\) of dimension \(n \times n\) where \(n = 2, 3, 4, \ldots\). The matrix is hollow (all elements on the principal diagonal are zero), symmetric, and all off-diagonal elements are equal to \(a\). Determine if the spectral radius (maximum eigenvalue) of this matrix is equal to \((n-1) \times a\).
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Context:
Determine whether \( \biggl\|\frac{vv^T}{v^Tv}\biggr\| = 1 \) for any vector \( v \in \mathbb{R}^n \), where \( \|.\| \) denotes a matrix norm.
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Context:
Determine the cardinality of the set \(X = \bigcup_{n \in \mathbb{N}} X_n\), where each set \(X_n\) has infinite cardinality \(\alpha\). Is the cardinality of \(X\) equal to \(\aleph_0 \alpha\)?
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Context:
The first AMC $8$ was given in $1985$ and it has been given annually since that time. Samantha turned $12$ years old the year that she took the seventh AMC $8$. In what year was Samantha born?
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Context:
Determine whether the limit \( \lim_{x \to \infty} \sum_{n=1}^{x-1} \frac{1}{x-n} \) exists.
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Context:
Determine if the function \( f: \mathbb{R}^2 \to \mathbb{R} \) is differentiable at the point \((0,0)\):
\[
f(x,y) = \begin{cases} \cfrac{x^n y^m}{x^2 - xy + y^2} & \text{for } (x,y) \ne (0,0) \\ 0 & \text{for } (x,y) = (0,0) \end{cases}
\]
where \( m, n \in \mathbb{N}^* \).
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Context:
4. Consider a 20 -sided convex polygon $K$, with vertices $A_{1}, A_{2}, \ldots, A_{20}$ in that order. Find the number of ways in which three sides of $K$ can be chosen so that every pair among them has at least two sides of $K$ between them. (For example $\left(A_{1} A_{2}, A_{4} A_{5}, A_{11} A_{12}\right)$ is an admissible triple while $\left(A_{1} A_{2}, A_{4} A_{5}, A_{19} A_{20}\right)$ is not.)

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Context:
Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10?$
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Context:
Determine whether the following limit is true: \[ \lim_{x\rightarrow\infty}\frac{x}{\pi(x)}-\ln(x) =-1, \] where \( \pi(x) \) is the number of primes not exceeding \( x \).
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Context:
Find all positive integer values of $c$ such that the equation $x^2-7x+c=0$ only has roots that are real and rational. Express them in decreasing order, separated by commas.
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Context:
Given two sequences of vectors \(\{a_n\}\) and \(\{b_n\}\) that converge to vectors \(a\) and \(b\), respectively, determine if the sequence of cross products \(\{a_n \times b_n\}\) converges to \(a \times b\).
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Context:
In Heidi's history class, the only grades that count toward the semester average are the $6$ tests she has already taken and the upcoming final exam. The final exam counts as two tests. Heidi has determined that if she earns $99$ points on the final she will have exactly a $90$-point average for the semester. On average, how many points has Heidi scored on each test prior to the final exam?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the following identity holds for all integers \( n \):
\[ \frac{\sum_{k=0}^{\infty}\frac{F_{n+k}}{k!}}{\sum_{k=0}^{\infty}\frac{F_{n-k}}{k!}}=e, \]
where \( F_n \) denotes the Fibonacci sequence.
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Context:
In the plane, $2022$ points are chosen such that no three points lie on the same line. Each of the points is coloured red or blue such that each triangle formed by three distinct red points contains at least one blue point.
What is the largest possible number of red points?

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Context:
Determine whether the sum \( \sum_{k=1}^N \frac{(k+2)^3 c_k}{(N+3)^4} \) converges, given that \( \{c_k\} \) is a convergent sequence with limit \( c \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that a polynomial function $p(x)$ crosses the horizontal line $y = 1$ at exactly one point, crosses the horizontal line $y = 3$ at exactly three points, and crosses the horizontal line $y = 5$ at exactly five points ($p$ is not tangent at any point to any of those three horizontal lines). What is the least possible degree of $p$?
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Context:
Determine if the function \( f(x) = \int_{0}^{\infty} t^{x^2+1} e^{-2t} \, dt \) can be expressed as a power series in a neighborhood of zero. If it can, find the radius of convergence of this series.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the convergence radius and convergence domain of the complex power series $\sum\limits_{n=0}^\infty \frac{1}{2^{n}} z^{2^n}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked as shown. Find the length of the portion of $\overline{XY}$ contained in the cube with edge length $3$. The original answer is in the format \(\frac{k\sqrt{n}}{m}\). Please calculate the value of k + m + n.
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Context:
Evaluate the limit: $$\lim_{x\to \infty} \left(\cosh^{-1}(x^{3}) + \coth^{-1}(\sqrt{x^{2}+1}) - 3\sinh^{-1}(x)\right)$$
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Context:
In the $4 \times 5$ grid shown, six of the $1 \times 1$ squares are not intersected by either diagonal. When the two diagonals of an $8 \times 10$ grid are drawn, how many of the $1 \times 1$ squares are not intersected by either diagonal?
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Context:
Determine the number of continuous functions \( f(x) \) defined on the interval \([0,1]\) that satisfy the following conditions: \( \int_0^1 f(x) \, dx = 1 \), \( \int_0^1 x f(x) \, dx = a \), and \( \int_0^1 x^2 f(x) \, dx = a^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three balls are drawn simultaneously from the urn (as described in Problem 4). Find the probability that all the drawn balls are blue (event $B$).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a fluid dynamics context, suppose you have a piece of wood with a mass of 0.5 kg and densities of different materials as follows: $\rho_{\text{wood}} = 600 \ \mathrm{kg/m^3}$, $\rho_{\text{copper}} = 8900\ \mathrm{kg/m^3}$, and $\rho_{\text{water}} = 1000 \ \mathrm{kg/m^3}$. What is the minimum mass of copper, in kilograms, that must be attached to the wood to make the combined object submerged in water?
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Context:
In the diagram, \( \triangle QRS \) is an isosceles right-angled triangle with \( QR = SR \) and \( \angle QRS = 90^{\circ} \). Line segment \( PT \) intersects \( SQ \) at \( U \) and \( SR \) at \( V \). If \( \angle PUQ = \angle RVT = y^{\circ} \), calculate the value of \( y \) in degrees.
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Context:
Find all integer triplets \((x, y, z)\) such that:
\[ x^{3} + y^{3} + z^{3} - 3xyz = 2003. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
It's pouring down rain, and the amount of rain hitting point  $(x,y)$  is given by   $$ f(x,y)=|x^3+2x^2y-5xy^2-6y^3|. $$  If you start at the origin  $(0,0)$ , find all the possibilities for  $m$  such that  $y=mx$  is a straight line along which you could walk without any rain falling on you.
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Context:
In the diagram, square PQRS has side length 40. Points J, K, L, and M are on the sides of PQRS, so that JQ = KR = LS = MP = 10. Line segments JZ, KW, LX, and MY are drawn parallel to the diagonals of the square so that W is on JZ, X is on KW, Y is on LX, and Z is on MY. Find the area of quadrilateral WXYZ. Express your answer in square units as a whole number.
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Context:
Find the limit of the infinite series \( \lim_{n\rightarrow \infty} e^{-n}\displaystyle\sum_{k=0}^{n-1}\frac{n^{k-1}}{k!} \). What is the value of this limit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all integers \( x \) such that \( 2x^2 + x - 6 \) is a positive integral power of a prime positive integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
给定正整数 $m(m \geq 3)$. 设正项等差数列 $\left\{a_{n}\right\}$ 与正项等比数列 $\left\{b_{n}\right\}$ 满足: $\left\{a_{n}\right\}$ 的首项等于 $\left\{b_{n}\right\}$ 的公比, $\left\{b_{n}\right\}$ 的首项等于 $\left\{a_{n}\right\}$ 的公差, 且 $a_{m}=b_{m}$ 。求 $a_{m}$ 的最小值, 并确定当 $a_{m}$ 取到最小值时 $a_{1}$ 与 $b_{1}$ 的比值。请给出当 $m = 3$ 时 $a_{1}$ 与 $b_{1}$ 的比值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Pinocchio correctly solved the example but smudged his notebook.

Behind each blot is hidden the same digit different from zero. Find this digit.
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Context:
In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square $\sqrt{5}$ away as long as it isn't occupied.

One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability $p$. Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors.

The value of $p$ that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.
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Context:
In the future, MIT has attracted so many students that its buildings have become skyscrapers. Ben and Jerry decide to go ziplining together. Ben starts at the top of the Green Building, and ziplines to the bottom of the Stata Center. After waiting $a$ seconds, Jerry starts at the top of the Stata Center, and ziplines to the bottom of the Green Building. The Green Building is 160 meters tall, the Stata Center is 90 meters tall, and the two buildings are 120 meters apart. Furthermore, both zipline at 10 meters per second. Given that Ben and Jerry meet at the point where the two ziplines cross, compute $100 a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 1997 odd numbers whose sum equals their product. Among them, three numbers are not 1 but are three different prime numbers. What are these three prime numbers?
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Context:
The graph of the function \( f(x) = x^4 + 4x^3 - 16x^2 + 6x - 5 \) has a common tangent line at \( x = p \) and \( x = q \). Find the product \( pq \).
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Context:
In Zuminglish, words consist only of the letters $M, O,$ and $P$. $O$ is a vowel, while $M$ and $P$ are consonants. A valid Zuminglish word must have at least two consonants between any two $O's$. Determine the number of valid 7-letter words in Zuminglish and find the remainder when this number is divided by $1000$.
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Context:
已知平面上有 $100$ 条直线，用 $T$ 表示由这些直线中的某三条直线围成的直角三角形的集合。求 $|T|$ 的最大值。
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Context:
Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.
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Context:
At a school, all 60 students play on at least one of three teams: Basketball, Soccer, and Mathletics. 8 students play all three sports, half the students play basketball, and the ratio of the size of the math team to the size of the basketball team to the size of the soccer team is $4:3:2$. How many students at the school play on exactly two teams?
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Context:
设a,b,c是△ABC的三边,P: ${{a}^{2}}={{b2}}+{{c}^{2}}$, Q:方程x$^{2}$ +2ax+b$^{2}$ = 0与方程x$^{2}$ +2cx－b$^{2}$ = 0有公共根. 则P是Q的_____.(填：充分不必要条件,必要而不充分条件,充要条件,既不充分也不必要条件)
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Context:
A car and a bus started out from the same place, traveling in opposite directions. After 4 hours, they were 384 miles apart. At the beginning, the car was going 8 miles per hour faster than the bus. However, after 2 hours, the car's speed decreased by 10 miles per hour due to a technical problem. How fast were the car and the bus going initially?
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Context:
17. In $\triangle A B C, \angle C=90^{\circ}, \angle B A C<45^{\circ}$ and $A B=7$. $P$ is a point on $A B$ such that $\angle A P C$ $=2 \angle A C P$ and $C P=1$. Find $\frac{A P}{B P}$.
(2 marks)
在 $\triangle A B C$ 中, $\angle C=90^{\circ}, \angle B A C<45^{\circ}$ 且 $A B=7 \circ P$ 是 $A B$ 上的一點, 使得 $\angle A P C=2 \angle A C P$ 及 $C P=1$ 。求 $\frac{A P}{B P}$ 。
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Context:
Suppose that $10$ mathematics teachers gather at a circular table with $25$ seats to discuss the upcoming mathematics competition. Each teacher is assigned a unique integer ID number between $1$ and $10$, and the teachers arrange themselves in such a way that teachers with consecutive ID numbers are not separated by any other teacher (IDs $1$ and $10$ are considered consecutive). In addition, each pair of teachers is separated by at least one empty seat. Given that seating arrangements obtained by rotation are considered identical, how many ways are there for the teachers to sit at the table?
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Context:
The slope of the tangent line to the curve $y=\frac{1}{3}{x^3}-\frac{2}{x}$ at $x=1$ is $\alpha$. Find $\frac{{sin\alpha cos2\alpha}}{{sin\alpha+cos\alpha}}$.
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Context:
3.365. $\cos \frac{\alpha-\beta}{2}$, if $\sin \alpha+\sin \beta=-\frac{27}{65} ; \operatorname{tg} \frac{\alpha+\beta}{2}=\frac{7}{9} ; \frac{5}{2} \pi<\alpha<3 \pi$ and $-\frac{\pi}{2}<\beta<0$.
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Context:
Two circles \( C_{1} \) and \( C_{2} \) touch each other externally and the line \( l \) is a common tangent. The line \( m \) is parallel to \( l \) and touches the two circles \( C_{1} \) and \( C_{3} \). The three circles are mutually tangent. If the radius of \( C_{2} \) is 9 and the radius of \( C_{3} \) is 4, what is the radius of \( C_{1} \)?
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Context:
Suppose \( A, B, C \) are three angles such that \( A \geq B \geq C \geq \frac{\pi}{8} \) and \( A + B + C = \frac{\pi}{2} \). Find the largest possible value of the product \( 720 \times (\sin A) \times (\cos B) \times (\sin C) \).
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Context:
There are $n$ participants at a conference, each knowing at most $k$ languages. For every group of three participants, there are at least two who speak the same language. Find the smallest value of $n$ such that for any distribution of languages satisfying these properties, one can find a language spoken by at least three delegates.
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Context:
How many ordered integer pairs (x, y) are there satisfying (x^2-18)^2 ≡ y^2 (mod 31), where 0 ≤ x, y < 31? Express your answer as a single integer.
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Context:
Determine the number of quadratic polynomials $P(x)=p_{1} x^{2}+p_{2} x-p_{3}$, where $p_{1}, p_{2}, p_{3}$ are not necessarily distinct (positive) prime numbers less than 50, whose roots are distinct rational numbers.
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Context:
After marking 12 balls with the numbers $1, 2, 3, \cdots, 12$ and placing them in a box, three people (A, B, and C) each take out 4 balls. It is known that the sum of the numbers on the balls each person took is equal. A has two balls with the numbers 6 and 11, B has two balls with the numbers 4 and 8, and C has one ball with the number 1. What are the numbers on C's other three balls?
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Context:
给定正整数 $n$ ，设 $a_{1}, a_{2}, \cdots, a_{n}$ 为非负整数序列，若其中连续若干项 (可以只有一项) 的算术平均值不小于 1 ，则称这些项组成一条 "龙"，其中第一项称为"龙头"，最后一项称为"龙尾"。已知 $a_{1}, a_{2}, \cdots, a_{n}$ 中每一项均为龙头或者龙尾。求 $\sum_{i=1}^{n} a_{i}$的最小值.若$n=5$，
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Context:
The café "Buratino" operates 6 days a week with Mondays off. Kolya said that from April 1 to April 20, the café was open for 17 days, and from April 10 to April 30, it was open for 18 days. It is known that he made a mistake once. What was the date of the last Tuesday in April?
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Context:
Given that $a > 0$, $b > 0$, and $2 - \log_2 a = 3 - \log_3 b = \log_6 \frac{1}{a + b}$, find the value of $\frac{1}{a} + \frac{1}{b}$.
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Context:
已知数列$\left\{ {{a}_{n}} \right\}$中，${{a1}}=1$，${{a}_{n}}=n-{{a2n}}$，${{a}_{2n+1}}={{an}}+1$，则${{a}_{1}}+{{a2}}+\cdots +{{a}_{99}}$的值为 _____．
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Context:
Call a day a *perfect* day if the sum of the digits of the month plus the sum of the digits of the day equals the sum of the digits of the year. For example, February $28$, $2028$ is a perfect day because $2+2+8=2+0+2+8$. Find the number of perfect days in $2018$.
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Context:
In the following right-angled triangle ABC, ∠C = BC = 1 and DEF is an arc of a circle with center A. Suppose the shaded areas BDE and CEF are equal and AD = x/√π. Find the value of x. Express your answer as a single integer.
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Context:
Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$
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Context:
In the fraction (x+y)/(x^2+y^2), if the values of x and y are both enlarged by a factor of 3, describe the resulting change in the fraction's value. Specify if the value is enlarged or reduced and by what factor. Express your answer in the format "enlarged/reduced by a factor of [factor]".
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Context:
In order to travel from the first to the second university building, Sasha took a car from a car-sharing service, and Zhenya rented a scooter. Sasha and Zhenya left the first building for the second one simultaneously, and at the same time, Professor Vladimir Sergeevich left the second building for the first in a car. The cars met after 3 minutes, and the scooter and the professor's car met at a distance of 1 km from the first building. Find the speeds of the cars and the scooter, given that the scooter covers a distance of 30 km in 1.25 hours longer than the car, and its speed is 4 times less than the speed of Vladimir Sergeevich's car. Provide the speeds in $\frac{\kappa M}{4}$ in ascending order.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $C=\{4,6,8,9,10, \ldots\}$ be the set of composite positive integers. For each $n \in C$ let $a_{n}$ be the smallest positive integer $k$ such that $k$ ! is divisible by $n$. Determine whether the following series converges: $$\sum_{n \in C}\left(\frac{a_{n}}{n}\right)^{n}$$
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Context:
Determine the elementary divisors and invariant factors of the abelian group: $$ \mathbb Z_4\oplus\mathbb Z_{10}\oplus\mathbb Z_{25}$$
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Context:
Each face of a die is arranged so that the sum of the numbers on opposite faces is 7. In the arrangement shown with three dice, only seven faces are visible. What is the sum of the numbers on the faces that are not visible in the given image?
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Context:
A new website registered $2000$ people. Each of them invited $1000$ other registered people to be their friends. Two people are considered to be friends if and only if they have invited each other. What is the minimum number of pairs of friends on this website?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence
\[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \]
for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S$ be the smallest subset of the integers with the property that $0 \in S$ and for any $x \in S$, we have $3 x \in S$ and $3 x+1 \in S$. Determine the number of non-negative integers in $S$ less than 2008.
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Context:
For a $5 \times 5$ chessboard colored as shown below, place 5 different rooks on black squares such that no two rooks can attack each other (rooks attack if they are in the same row or column). How many different ways are there to do this?
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Context:
某种比赛中每个队恰好与其他队各赛一场，每场比赛中胜者得2分，负者得0分，平局各得1分，比赛结束后发现，每队所得分数中恰有一半是该队同十个得分最低的队的比赛中得到的（十个得分最低的队所得分数中一半是他们彼此比赛中得到的），问共有几个队参加比赛？
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Context:
There are  $2017$  turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after  $2015$  seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form  $\tfrac{p}{q}$  where  $p$  and  $q$  are relatively prime positive integers. Find  $p + q$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
2000 people registered on a new website. Each person invited 1000 others to be friends. Two people are considered friends if and only if both have invited each other to be friends. What is the minimum number of friend pairs that could have been formed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, $BC=1$, $\angle B= \frac{\pi}{3}$, and the area of $\triangle ABC$ is $S=\sqrt{3}$. Find the value of $\sin C$.
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Context:
How many non-congruent  triangles can be drawn such that their vertices lie on the corners of a unit cube?
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Context:
Unconventional dice are to be designed such that the six faces are marked with numbers from  $1$  to  $6$  with  $1$  and  $2$  appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.
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Context:
Find the limit: \[ \lim_{x \to \infty} \left(\sum_{n=1}^{\infty} \left(\frac{x}{n}\right)^{n}\right)^{\frac{1}{x}} \]
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Context:
Given an ellipse with equation x^2/a^2 + y^2/b^2 = 1 (a > b > 0), if there exist three points on the ellipse such that these three points and the center of the ellipse exactly form the four vertices of a square, calculate the eccentricity of the ellipse. Express your answer as a fraction or decimal, including necessary mathematical notation (e.g., square root).
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Context:
Given the function $f(x) = mx^2 - mx - 1$. If for all $x \in \mathbb{R}$, $f(x) < 0$ always holds, then the range of the real number $m$ is.
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Context:
Mary has a mystery book library. She started with 200 mystery books at the beginning of the year. She joined a book club that sent her 2 books each month of the year. After the first 3 months, she lent 10 books to her friend Jane and never got them back. In the 4th month, her other friend, Alice, returned 5 books that she had borrowed last year. She went to a bookstore in the 5th month and bought 15 more books, then bought 8 books at different yard sales throughout the next 2 months. For her birthday in the 7th month, her daughter gave her a book, and her mother gave her a series of 8 books. In the 8th month, she borrowed 4 books from a friend, Mark, and returned them after 2 months. She also received 6 books from her sister in the 9th month. In the 10th month, she got 4 books on a buy-one-get-one-free deal at a bookstore. She donated 30 books she didn't want to read again to a charity shop in the 11th month. Her neighbor borrowed 5 books from her in the last month of the year and had not returned them by the year end. Finally, she sold 7 books to a used book store in the 12th month. How many books did she have at the end of the year?
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Context:
Given that $f(z)$ satisfies the condition $\Im (f'(z))=6x(2y-1)$, with $f(0) = 3-2i$ and $f(1) = 6-5i$, determine the value of $f(1+i)$. \(z = x + yi\) is a complex number.
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Context:
Given vectors $\overrightarrow{a}=(1,-1)$ and $\overrightarrow{b}=(2,-1)$, the projection of $\overrightarrow{a}+\overrightarrow{b}$ in the direction of $\overrightarrow{a}$ is ______.
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Context:
Find the points on the cubic curve $y^2z = x^3 -xz^2$ where the tangent line is vertical, given that $O = (0:1:0)$ is an inflection point.
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Context:
What is the smallest number of additional squares that must be shaded in the given figure so that it has at least one line of symmetry and rotational symmetry of order 2? Provide your answer as a whole number.
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Context:
Let $X_i \sim N(0,1)$ for $0 \leq i \leq n$ be i.i.d. standard normal distributions. Define $Y_n := \frac{\sqrt{n} X_0}{\sqrt{(\sum_{i=1}^n X_i^2)}}$. Determine the limiting distribution function for $Y_n$ as $n \to \infty$. Provide your answer in terms of a known distribution.
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Context:
John now has 15 marbles of different colors, including two red, two green, and one blue marble. In how many ways can he choose 5 marbles, if exactly two of the chosen marbles are red and one is green?
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Context:
Let \( t = 2016 \) and \( p = \ln 2 \). Evaluate in closed form the sum
\[
\sum_{k=1}^{\infty}\left(1 - \sum_{n=0}^{k-1} \frac{e^{-t} t^n}{n!}\right)(1-p)^{k-1} p.
\]
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Context:
Let $P$ be the convex polygon in the complex plane whose vertices are the roots of
\[z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 = 0.\]The area of $P$ can be expressed in the form $\frac{a + b \sqrt{c}}{d},$ where $a,$ $b,$ $c,$ $d$ are positive integers, in simplest form.  Find $a + b + c + d.$
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Context:
A regular three-sided pyramid (its base is an equilateral triangle) is inscribed in a sphere, the diameter of which is nine times the height of the pyramid. What angle do two side faces enclose?
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Context:
Let $N$ denote the number of ordered pairs of sets $(A, B)$ such that $A \cup B$ is a size-$999$ subset of $\{1,2,\dots,1997\}$ and $(A \cap B) \cap \{1,2\} = \{1\}$. If $m$ and $k$ are integers such that $3^m5^k$ divides $N$, find the largest possible value of $m+k$. \(\text{Proposed by Michael Tang}\)
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Context:
 Ben and Jerry are walking together inside a train tunnel when they hear a train approaching. They decide to run in opposite directions, with Ben heading towards the train and Jerry heading away from the train. As soon as Ben finishes his $1200$ meter dash to the outside, the front of the train enters the tunnel. Coincidentally, Jerry also barely survives, with the front of the train exiting the tunnel as soon as he does. Given that Ben and Jerry both run at $1/9$ of the train’s speed, how long is the tunnel in meters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Five students $A, B, C, D, E$ participate in a competition. Two people make predictions about the results of the competition:

- The first person predicts the order to be $A, B, C, D, E$. This prediction is completely wrong: not only are none of the students' positions correct, but not even any adjacent pairs of students are correctly predicted.
- The second person predicts the order to be $D, A, E, C, B$. This prediction has two students in the correct positions and two pairs of students whose positions are adjacent correctly.

What are the actual results of the competition?
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Context:
Determine the number of distinct points in the $xy$-plane that are common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$.
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Context:
In a coconut grove, (x + 2) trees yield 30 nuts per year, x trees yield a certain number of nuts per year and (x – 2) trees yield 180 nuts per year. If the average yield per year per tree is 100, and x is 10, how many nuts per year do x trees yield?
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Context:
If $A=\{1,2,3,4,5\}$, find the number of functions $f:A\to A$ such that $f(f(f(x)))=f(f(x))$ for all $x\in A$.
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Context:
Given the function $f(x)=\sin^{2}ωx+\sqrt{3}\sin ωx\cos ωx$, where $x\in R$, $f(α)=-\frac{1}{2}$, $f(β)=\frac{1}{2}$, and the minimum value of $|α-β|$ is $\frac{3}{4}π$, calculate the value of the positive number $ω$. Express your answer as a single value.
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Context:
There exist two distinct positive integers, both of which are divisors of  $10^{10}$ , with sum equal to  $157$ .  What are they?
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Context:
A sequence $\left\{a_{n}\right\}_{n \geq 0}$ of real numbers satisfies the recursion $a_{n+1}=a_{n}^{3}-3 a_{n}^{2}+3$ for all positive integers $n$. For how many values of $a_{0}$ does $a_{2007}=a_{0}$ ?
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Context:
Find the condition for the equation $|z^2| - \text{Re}(bz) + a = 0$ to have a solution, where $a \in \mathbb{R}$ and $b \in \mathbb{C}$. Determine the nature of the solution set when solutions exist.
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Context:
Solve the equation \(\frac{x}{3}+\frac{x}{\sqrt{x^{2}-9}}=\frac{35}{12}\). In the answer, write the sum of all obtained solutions.
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Context:
Let the sample space be $\Omega =\{1,2,3,4,5,6,7,8\}$ with equally likely sample points, and events $A=\{1,2,3,4\}$, $B=\{1,2,3,5\}$, $C=\{1,m,n,8\}$, such that $p\left(ABC\right)=p\left(A\right)p\left(B\right)p\left(C\right)$, and satisfying that events $A$, $B$, and $C$ are not pairwise independent. Find $m+n$.
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Context:
A point is randomly thrown onto the interval \( [11; 18] \) and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^2 + 2k - 99)x^2 + (3k - 7)x + 2 = 0\) satisfy the condition \( x_1 \leq 2x_2 \).
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Context:
To build a wall of 8 m x 6 m x some height, 1600 bricks, each measuring 100 cm x 11.25 cm x 6 cm, will be needed. What is the height of the wall?
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Context:
On January 15 in the stormy town of Stormville, there is a  $50\%$  chance of rain. Every day, the probability of it raining has a  $50\%$  chance of being  $\frac{2017}{2016}$  times that of the previous day (or  $100\%$  if this new quantity is over  $100\%$ ) and a  $50\%$  chance of being  $\frac{1007}{2016}$  times that of the previous day. What is the probability that it rains on January 20?

*2018 CCA Math Bonanza Lightning Round #3.3*
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Context:
We say that some positive integer $m$ covers the number $1998$, if $1,9,9,8$ appear in this order as digits of $m$. (For instance $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$.) Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n\ge 5$), all different from $0$. What is the remainder of $k(n)$ on division by $8$?
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Context:
There are three cups of solution with equal weight, and their concentrations are $10 \%, 20 \%, 45 \%$ respectively. If $\frac{1}{2}$, $\frac{1}{4}$, and $\frac{1}{5}$ of the weights from the three cups are poured into a fourth empty cup in order, what is the concentration of the solution in the fourth cup? $\qquad$ $\%$.
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Context:
在直三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, $A B=1$, $B C=C C_{1}=\sqrt{3}, \angle A B C=90^{\circ}, P$ 是平面 $A B C$ 上一动点. 则 $A_{1} P+\frac{1}{2} P C$ 的最小值是 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
\[
\left(\left(\sqrt{mn} - \frac{mn}{m+\sqrt{mn}}\right) \div \frac{\sqrt[4]{mn} - \sqrt{n}}{m-n} - m \sqrt{n}\right)^{2} \div \sqrt[3]{mn \sqrt{mn}} - \left(\frac{m}{\sqrt{m^{4}-1}}\right)^{-2}.
\]
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Context:
A quadrilateral \(ABCD\) is inscribed in a circle with a diameter of 1, where \(\angle D\) is a right angle and \(AB = BC\).

Find the area of quadrilateral \(ABCD\) if its perimeter is \(\frac{9\sqrt{2}}{5}\).
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Context:
The numbers 1 to 9 are randomly placed into a 3x3 grid such that each cell contains exactly one number and each number is used exactly once. Calculate the probability that the sum of the numbers in each row and each column is odd. Express your answer as a simplified fraction.
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Context:
Identify a ring $R$ where the element $2$ is invertible and the equation $x^2 - 1 = 0$ has exactly four distinct solutions.
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Context:
At a temperature of $13^{\circ}$, a 41% sugar solution in water becomes saturated. How many grams of sugar will not dissolve in water (and will precipitate) if (at a temperature of $15^{\circ}$) you thoroughly mix a glass of water (220 g) with 280 g of sugar?
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Context:
In the plane Cartesian coordinate system \( xOy \), a moving line \( l \) is tangent to the parabola \( \Gamma: y^{2} = 4x \), and intersects the hyperbola \( \Omega: x^{2} - y^{2} = 1 \) at one point on each of its branches, left and right, labeled \( A \) and \( B \). Find the minimum area of \(\triangle AOB\).
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Context:
Find the limit of the integral \( \int_0^{+\infty}\frac{\mathrm{d}x}{(1+x^2)^{1/n}} \) as \( n \) approaches infinity.
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Context:
Given the sets $A=\{x|(x-3)(x+1)\geqslant 0\}$ and $B=\left\{ x \mid x < - \frac{4}{5}\right\}$, calculate the intersection $A\cap B$. Express your answer as a set definition.
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Context:
Around the vertices of an equilateral triangle with a side length of 4 units, arcs of circles with radii of 1 and 5 units were drawn as shown in the figure.

What is the width of the narrowest strip between two parallel lines that can contain the resulting plane figure?
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Context:
Find the annual yield on a 4-year zero-coupon bond given the following information: You have two 4-year annual-coupon bonds, each with a face value and redemption value of 8000. The first bond has a coupon rate of 7% and a price of 7908.57, while the second bond has a coupon rate of 8% and a price of 8101.55.
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Context:
Let $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$ . Compute the following expression in terms of $k$ : \[E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}.\]
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Context:
For  $ 0<a<1$ , let  $ S(a)$  is the area of the figure bounded by three curves  $ y\equal{}e^x,\ y\equal{}e^{\frac{1\plus{}a}{1\minus{}a}x}$  and  $ y\equal{}e^{2\minus{}x}$ .
Find  $ \lim_{a\rightarrow 0} \frac{S(a)}{a}$ .
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Context:
Evaluate the integral \( \oint_{|z| = 1} \frac{z+\bar{z}}{(2z-i)^2} \, dz \) using the Residue theorem.
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Context:
将一个四棱锥的每一个顶点染色，要求同一条棱上的两个顶点异色．如果只有5种颜色可供选用，一共有多少种不同的染色方法？
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Context:
What is the units (i.e., rightmost) digit of $\left[\frac{10^{20000}}{10^{100}+3}\right]$ ? Here $[x]$ is the greatest integer $\leqslant x$.
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Context:
Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches, including the top, bottom, and sides.
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Context:
Ranu starts with one standard die on a table.  At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step.  After 2013 such steps, let  $D$  be the number of dice on the table.  What is the expected value (average value) of  $6^D$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer $k$ such that
\[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\]
for some ordered triple of positive integers $(a,b,c)$.
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Context:
What is the molecular weight of 3 moles of Na2SO4·10H2O (sodium sulfate decahydrate)?
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Context:
What is the cube root of a certain number if the result is 0.06000000000000001?
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Context:
A train passes a platform in 40 seconds. The same train passes a man standing on the platform in 20 seconds. The speed of the train is some km/hr, and the length of the platform is 300.024 meters. What is the speed of the train?
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Context:
A cake has a shape of triangle with sides  $19,20$  and  $21$ . It is allowed to cut it it with a line into two pieces and put them on a round plate such that pieces don't overlap each other and don't  stick out of the plate. What is the minimal diameter of the plate?
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Context:
A robot is standing on the bottom left vertex  $(0,0)$  of a  $5\times5$  grid, and wants to go to  $(5,5)$ , only moving to the right  $(a,b)\mapsto(a+1,b)$  or upward  $(a,b)\mapsto(a,b+1)$ . However this robot is not programmed perfectly, and sometimes takes the upper-left diagonal path  $(a,b)\mapsto(a-1,b+1)$ . As the grid is surrounded by walls, the robot cannot go outside the region  $0\leq a,b\leq5$ . Supposing that the robot takes the diagonal path exactly once, compute the number of different routes the robot can take.
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Context:
Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1.  Length $r$ is found to be $32$ inches.  After rearranging the blocks as in Figure 2, length $s$ is found to be $28$ inches.  How high is the table?
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Context:
$P$ 是 $\triangle A B C$ 内一点, 引线段 $A P D, B P E$ 和 $C P F$, 使 $D$ 在 $B C$ 上, $E$ 在 $A C$ 上, $F$ 在 $A B$上. 已知 $A P=6, B P=9, P D=6, P E=3, C F=20$, 求 $\triangle A B C$ 的面积.
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Context:
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?
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Context:
A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds \(T_{max}\). One end of the cord is attached to a fixed point, the other is attached to an object of mass \(3m\). If a second, smaller object of mass \(m\) moving at an initial speed \(v_0\) strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of \(v_f\). All motion occurs on a horizontal, frictionless surface.

Find \(\frac{v_f}{v_0}\). The answer should be in the form \(\frac{k}{\sqrt{m}}\). Please calculate the value of k + m.
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Context:
In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. What is the perimeter of the hexagon? Express your answer in the form $k\sqrt{m}$, and find the value of $k + m$.
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Context:
Let O be the origin (0, 0) and C the line segment { (x, y) : x ∈ [1, 3], y = 1 }. Let K be the curve { P : for some Q ∈ C, P lies on OQ and PQ = 0.01 }. Let k be the length of the curve K. Is k greater or less than 2?
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Context:
You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open all the doors and disarm all the mines. In the room is a panel with  $3$  buttons, which conveniently contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially  $3$  doors are closed and  $3$  mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?
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Context:
Two congruent pentagons are each formed by removing a right-angled isosceles triangle from a square of side-length 1. The two pentagons are then fitted together as shown. Calculate the length of the perimeter of the octagon formed.
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Context:
Find the area of the smallest region bounded by the graphs of $y=|x|$ and $x^2+y^2=4$.
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Context:
Points $A, B, C,$ and $D$ are sequentially located on a circle. It is known that the degree measures of the smaller arcs $AB, BC, CD,$ and $AD$ have a ratio of $1: 3: 5: 6$. Find the angles of the quadrilateral $ABCD$.
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Context:
Determine the length of  $BC$  in an acute triangle  $ABC$  with  $\angle ABC = 45^{\circ}$ ,  $OG = 1$  and  $OG \parallel BC$ . (As usual  $O$  is the circumcenter and  $G$  is the centroid.)
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Context:
Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)
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Context:
In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}.$  Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB.$  Find the greatest integer that does not exceed $1000r.$
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Context:
Find the angles of a triangle, given that the distance between the center of the circumscribed circle and the orthocenter is half of the longest side and equal to the shortest side.
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Context:
对于四面体 $A B C D, A B \perp B C, C D \perp$ $B C, B C=2$ ，且异面直线 $A B$ 与 $C D$ 所成的角为 $60^{\circ}$. 若四面体 $A B C D$ 的外接球半径为 $\sqrt{5}$,则四面体 $A B C D$ 的体积的最大值为         .原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer layer of unit cubes are removed from the block, more than half the original unit cubes will still remain?
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Context:
Point $P$ is selected at random from the interior of the pentagon with vertices $A=(0,2)$, $B= (4,0)$, $C = (2\pi +1, 0)$, $D=(2\pi
+1,4)$, and $E=(0,4)$. What is the probability that $\angle APB$ is obtuse? Express your answer as a common fraction.

[asy]
pair A,B,C,D,I;
A=(0,2);
B=(4,0);
C=(7.3,0);
D=(7.3,4);
I=(0,4);
draw(A--B--C--D--I--cycle);
label("$A$",A,W);
label("$B$",B,S);
label("$C$",C,E);
label("$D$",D,E);
label("$E$",I,W);
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximal possible finite number of roots of the equation   $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$ , where  $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$  are distinct reals.
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Context:
A train is moving at a speed of 132 km/hour. The length of the train is 110 meters. How long will it take to cross a railway platform of a certain length if it takes 7.499400047996161 seconds to cross it?
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Context:
A circle centered at \(A\) with a radius of 1 and a circle centered at \(B\) with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is in the simplest fractional form \(\frac{k}{m}\). Please find the value of \(k + m\).
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Context:
How can we calculate the following sum efficiently?
$$\sum_{k=0}^{38\,204\,629\,939\,869} \frac{\binom{38\,204\,629\,939\,869}{k}}{\binom{76\,409\,259\,879\,737}{k}}\,.$$
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Context:
Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\angle A = \frac{5}{2} \angle B$. What is the degree measure of $\angle D$?
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Context:
When a polynomial is divided by \(3x^3 - 4x^2 + 5x - 6\), what are the possible degrees of the remainder? Enter all the possible values, separated by commas.
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Context:
Let the conjugate of complex number $z$ be $\overline{z}$. If $z+\overline{z}=4$ and $z\cdot \overline{z}=8$, find the value of $\dfrac{\overline{z}}{z}$.
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Context:
The polynomial  $P$  is a quadratic with integer coefficients.  For every positive integer  $n$ , the integers  $P(n)$  and  $P(P(n))$  are relatively prime to  $n$ .  If  $P(3) = 89$ , what is the value of  $P(10)$ ?
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Context:
The set Φ consists of a finite number of points on the plane. The distance between any two points from Φ is at least  $\sqrt{2}$ . It is known that a regular triangle with side lenght  $3$  cut out of paper can cover all points of Φ. What is the greatest number of points that Φ can consist of?
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Context:
Determine all triangles such that the lengths of the three sides and its area are given by four consecutive natural numbers.
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Context:
How can we determine the number of distinct 3-cycles in $A_4$ that can be formed from the product of two non-disjoint transpositions?
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Context:
设函数 $\varphi_{1}(x), \varphi_{2}(x)$ 并足 $\varphi_{i}^{\prime \prime}(x)-\ln (x+1) \varphi_{i}^{\prime}(x)+e^{x} \varphi_{i}(x)=0, i=1,2, x>-1$. 定义 $w(x)=\varphi_{1}^{\prime}(x) \varphi_{2}(x)-\varphi_{1}(x) \varphi_{2}^{\prime}(x)$. 若 $w(0)=1$, 则 $[w(2)]=$ $\qquad$  .
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Context:
三位数（100,101，···，999）共900个．在卡片上打印这些三位数，每张卡片打印一个三位数，有的卡片所印的，倒过来看仍为三位数，如198倒过来看是861（1倒过来看仍视为1）；有的卡片则不然，如531倒过来看是189．因此有些卡片可以一卡两用，有的不能.于是，至多可少打印＿张卡片.
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Context:
Determine $x^2+y^2+z^2+w^2$ if
$\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$
$\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$
$\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$
$\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1$
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Context:
$ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$   satisfy the given conditions

 $ a)$   $ f(0,0)\equal{}1$  ,  $ f(0,1)\equal{}1$  ,

 $ b)$   $ \forall k \notin \left\{0,1\right\}$    $ f(0,k)\equal{}0$  and

 $ c)$   $ \forall n \geq 1$  and  $ k$  ,  $ f(n,k)\equal{}f(n\minus{}1,k)\plus{}f(n\minus{}1,k\minus{}2n)$

find the sum  $ \displaystyle\sum_{k\equal{}0}^{\binom{2009}{2}}f(2008,k)$
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Context:
Find the largest positive  $p$  ( $p>1$ ) such, that  $\forall a,b,c\in[\frac1p,p]$  the following inequality takes place
\[9(ab+bc+ca)(a^2+b^2+c^2)\geq(a+b+c)^4\]
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Context:
What is the molecular weight of 9 moles of Carbon tetrachloride?
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Context:
Each vertex of a tetrahedron has edges of lengths $5$, $\sqrt{41}$, and $\sqrt{34}$ meeting. What are the lengths of the segments connecting the midpoints of the opposite edges of the tetrahedron?
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Context:
A frame for three square photos has an equal width all around (see the diagram). The perimeter of one photo opening is 60 cm, and the total perimeter of the entire frame is 180 cm. What is the width of the frame?
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Context:
Determine all positive integers \( k \) for which there exist positive integers \( r \) and \( s \) that satisfy the equation
$$
\left(k^{2}-6k+11\right)^{r-1}=(2k-7)^{s}
$$
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Context:
What is the minimum number of planes determined by  $6$  points in space which are not all coplanar, and among which no three are collinear?
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Context:
Does there exist an integer \( n \) such that \( 21n \equiv 1 \mod 74 \)?
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Context:
Two identical cylindrical containers are connected at the bottom by a small tube with a tap. While the tap was closed, water was poured into the first container, and oil was poured into the second one, so that the liquid levels were the same and equal to $h = 40$ cm. At what level will the water in the first container settle if the tap is opened? The density of water is 1000 kg/$\mathrm{m}^3$, and the density of oil is 700 kg/$\mathrm{m}^3$. Neglect the volume of the connecting tube. Give the answer in centimeters.
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Context:
A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out.  What is the probability that all beads in the bag are red after three such replacements? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Let $n$ be the answer to this problem. Given $n>0$, find the number of distinct (i.e. non-congruent), non-degenerate triangles with integer side lengths and perimeter $n$.
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Context:
Let  $\alpha$  be a root of  $x^6-x-1$ , and call two polynomials  $p$  and  $q$  with integer coefficients  $\textit{equivalent}$  if  $p(\alpha)\equiv q(\alpha)\pmod3$ .  It is known that every such polynomial is equivalent to exactly one of  $0,1,x,x^2,\ldots,x^{727}$ .  Find the largest integer  $n<728$  for which there exists a polynomial  $p$  such that  $p^3-p-x^n$  is equivalent to  $0$ .
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Context:
最多能取多少个两两不等的正整数，使得其中任意三个数之和都为素数.
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Context:
For an integer $n \geq 3$, define the sequence $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}$ by the prime factorization of $n!$ as $n! = p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \ldots p_{k}^{\alpha_{k}}$, where the primes $p_{1}<p_{2}<\ldots <p_{k}$ correspond to the respective exponents $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}$. Determine all integers $n \geq 3$ for which $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}$ form a geometric sequence.
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Context:
Five integers form an arithmetic sequence. Whether we take the sum of the cubes of the first four terms, or the sum of the cubes of the last four terms, in both cases, we obtain 16 times the square of the sum of the considered terms. Determine the numbers.
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Context:
Suppose that in the previous problem, the person looking at the portrait answered the question like this: "In my family, I grew up alone, completely alone. Nonetheless, the son of the one in the portrait is the son of my father (you heard correctly, it's the son!)."

Whose portrait is this person looking at?
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Context:
Some friends went hiking for 3.5 hours. They traveled 21 kilometers in that time. Birgit was faster than the average time by a certain amount. If Birgit kept the same pace, it would take her 48 minutes to go 8 kilometers. How much faster was Birgit than the average time?
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Context:
Compute the limit of the function

$$
\lim _{x \rightarrow 1}\left(\frac{2 x-1}{x}\right)^{\ln (3+2 x) / \ln (2-x)}
$$
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Context:
设复数 $z, w$ 满足 $z+w=2$, 求 $S=\left|z^{2}-2 w\right|+\left|w^{2}-2 z\right|$的最小可能值。原始答案为$a\sqrt{b}-c$的格式，请给出$a+b+c$的值。
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Context:
Let $Q=[b_{ij}] $ be a $5\times 5$ matrix over $\mathbb C$ such that $b_{ij}=2$ for $i\ne j$ and otherwise $b_{ii}=c+2$ where $c>0$. Let $\gamma,\delta$ be the largest and the smallest eigenvalue of $Q$. If $\gamma-\delta=6$, then what is the trace of $Q$?
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Context:
If \(\frac{a}{b} = 5\), \(\frac{b}{c} = \frac{1}{4}\), and \(\frac{c^2}{d} = 16\), then what is \(\frac{d}{a}\)?
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Context:
A circle is tangent to the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $AB$ is 4. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is four times the other.
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Context:
Consider the sequence \(\left\{a_{n}\right\}\) defined by \(a_{0}=\frac{1}{2}\) and \(a_{n+1}=a_{n}+\frac{a_{n}^{2}}{2023}\) for \(n = 0, 1, \ldots\). Find the integer \(k\) such that \(a_{k} < 1 < a_{k+1}\).
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Context:
A car travels 200 kilometers in 4 hours, and a train travels 210 kilometers in 3 hours. Which one has a higher speed?
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Context:
If, in the expression $x^2 - 3$, $x$ increases or decreases by a positive amount of $a$, the expression changes by an amount:
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Context:
sophia finished 23 of a book . she calculated that she finished 90 more pages than she has yet to read . how long is her book ?
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Context:
Evaluate the improper integral \( \int_1^\infty \frac{\ln x}{x} \, dx \) by determining its convergence or divergence as the upper limit approaches infinity.
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Context:
$PS$  is a line segment of length  $4$  and  $O$  is the midpoint of  $PS$ . A semicircular arc is drawn with  $PS$  as diameter. Let  $X$  be the midpoint of this arc.  $Q$  and  $R$  are points on the arc  $PXS$  such that  $QR$  is parallel to  $PS$  and the semicircular arc drawn with  $QR$  as diameter is tangent to  $PS$ . What is the area of the region  $QXROQ$  bounded by the two semicircular arcs?
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Context:
Let \( A = \{1, 2, 3, \ldots, 9\} \). Determine the number of bijective functions \( f: A \rightarrow A \) such that there exists at least one \( i \in A \) for which \( \left| f(i) - f^{-1}(i) \right| > 1 \).
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Context:
How to find the sum of the series $\sin(x)-\dfrac{1}{2}\sin(2x)+\dfrac{1}{3}\sin(3x)-\dfrac{1}{4}\sin(4x)+...$?
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Context:
Let  $n \ge 3$  be an integer. What is the largest possible number of interior angles greater than  $180^\circ$  in an  $n$ -gon in the plane, given that the  $n$ -gon does not intersect itself and all its sides have the same length?
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Context:
A cyclist climbs a hill with a length of 400 m with a certain speed. When descending, the speed is two times greater. The cyclist takes 30 minutes to climb and descend the hill. What is the speed of the cyclist when climbing the hill?
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Context:
6. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C: \frac{x^{2}}{4}-\frac{y^{2}}{5}=1$, respectively. Point $P$ is on the right branch of the hyperbola $C$, and the excenter of $\triangle P F_{1} F_{2}$ opposite to $\angle P F_{1} F_{2}$ is $I$. The line $P I$ intersects the $x$-axis at point $Q$. Then
$$
\frac{|P Q|}{|P I|}+\frac{\left|F_{1} Q\right|}{\left|F_{1} P\right|}=
$$
$\qquad$ .
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Context:
How can we count the number of homomorphisms from $S_5$ to $A_6$ with a kernel of $A_5$?
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Context:
Let $G = \Bbb Z \times \Bbb Z$ with group law given by addition. Let $H$ be the subgroup generated by $(2,3)$, so $H$ consists of all elements of the form $(2a,3a)$ for some $a \in \Bbb Z$. To which group is $G/H$ isomorphic?
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Context:
Consider the sequence \( \{a_{n}\} \) defined by \( a_{1} = 1 \), \( a_{2} = 2 \), \( a_{3} = 3 \), \( a_{4} = 4 \), \( a_{5} = 5 \), and for \( n \geq 5 \), \( a_{n+1} = a_{1} a_{2} \cdots a_{n} - 1 \). Determine the number of positive integers \( m \) such that \( a_{1}a_{2}\cdots a_{m} = a_{1}^{2} + a_{2}^{2} + \cdots + a_{m}^{2} \).
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Context:
Determine the spectrum of the operator $B: D \subset Y \rightarrow Y$, where $D=\{g \in \mathbb{L}^2(\mathbb{R}) : \int |yg(y)|^2 < \infty \}$, $Y=\mathbb{L}^2(\mathbb{R})$, and $Bg(y):=yg(y)$. Provide your answer in terms of the spectrum of $B$.
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Context:
Evaluate the limit: $$\lim_{N\to \infty} \frac{2}{N} \sum_{j=0}^{N-1} \left(\frac{3}{2}\left[\frac{2j}{N} + 1\right]\right)$$
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Context:
Given matrices $A$, $B$, and $C$ with sizes $m \times m$, $n \times n$, and $n \times m$, respectively, where $\det(A) = 2$ and $\det(B) = 3$, find the determinant of the block matrix:
\[ \det \begin{pmatrix} 0 & A \\ B & C \end{pmatrix} \]
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Context:
If $X$ is a random variable that follows a Poisson distribution with parameter $u$, what is the variance of $X^2$?
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Context:
Evaluate the expression \( \log_{2}\left(\prod_{a=1}^{2015}\prod_{b=1}^{2015}\left(1+e^{\frac{2\pi iab}{2015}}\right)\right) \).
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Context:
Determine the signature of the symmetric bilinear form on \( \mathbb{R}^n \) defined by \( e_i \cdot e_j = 1 - \delta_{ij} \), where \( e_1 \cdot e_1 = e_2 \cdot e_2 = e_3 \cdot e_3 = \cdots = 0 \) and \( e_1 \cdot e_2 = e_1 \cdot e_3 = e_2 \cdot e_3 = \cdots = 1 \). The signature is the number of vectors in an orthogonal basis squaring to \(+1, -1, 0\). What is the signature for \( n > 1 \)?
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Context:
Compute the derivative of the function \( \frac{d(x^TWW^Tx)}{dW} \), where \( x \) is an \( n \times 1 \) vector and \( W \) is an \( n \times m \) matrix, using the fact that \( \frac{d(x^TWx)}{dW} = x^Tx \).
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Context:
Evaluate the limit: $$\lim_{n\to\infty} \frac{1+\frac{n}{1!}+\cdots+\frac{n^n}{n!}}{e^n}$$
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Context:
Solve for \(x\) in the equation: \[ 7000 = \sum_{n=1}^{5} \frac{1000}{(1 + x)^n} + \sum_{n=6}^{10} \frac{2000}{(1 + x)^n}. \]
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Context:
Find the Fredholm index of the Toeplitz operator $T_{e^{2ix}}$ on the Hilbert space $L_2^+(S^1)$. Provide your answer as a single integer.
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Context:
Determine the cardinality of the set of all real-valued functions that have a countable number of discontinuities.
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Context:
3. Through a point lying inside a triangle, three lines parallel to its sides are drawn, which divide the triangle into six parts: three triangles and three quadrilaterals. The areas of all three inner triangles are equal. Determine the range in which the ratio of the area of each inner triangle to the area of the original triangle can lie.

#
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Context:
Consider the Collatz Conjecture, where for a given non-zero natural number, if it is even, you divide it by 2, and if it is odd, you multiply it by 3 and add 1. This process is repeated, and the number of transformations needed for a number to first reach 1 is called its path length. Calculate the sum of all natural numbers that have a path length of 10.
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Context:
Find the expected value of \( Z = 2Y_1 + Y_2 \), where \( Y_1 = \min(X_1, X_2) \) and \( Y_2 = \max(X_1, X_2) \). Each \( X_i \) is an independent exponential random variable with parameter \( \theta = 2 \).
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Context:
Calculate the Hilbert polynomial of the twisted cubic $C$ in $\mathbb{P}^3$, defined by the ideal $V(XZ-Y^2, YW-Z^2, XW-YZ)$. Denote this polynomial as $P_C(n)$. What is $P_C(n)$?
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Context:
Find the kernel of the ring homomorphism \( \phi: \mathbb{Z}[a, b, c, d] \to \mathbb{Z}[p, q, r, s] \) defined by \( \phi(a) = pq, \phi(b) = pr, \phi(c) = qs, \phi(d) = rs \).
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Context:
How many integers between 30,000 and 100,000 have a sum of digits that is 15 or less?
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Context:
Find the residue of the function \( \frac{1}{e^{e^{\frac{1}{z}}}} \) at \( z=0 \).
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Context:
Find the limit of the power series \( \lim_{{x\rightarrow +\infty}} \sum_{k=1}^\infty (-1)^k \frac{x^k}{k^{k-m}} \), where \( m>1 \).
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Context:
Find a non-constant polynomial in \( \mathbb{Z}/8\mathbb{Z}[x] \) that is a unit.
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Context:
Let $\mathcal{H} = l^2(\mathbb{N})$. For $N \geq 1$, define the linear functional $L: \mathcal{H} \rightarrow \mathbb{F}$ by $L(\{\alpha_n\}) = \alpha_N$. Find the vector $h_0$ in $\mathcal{H}$ such that $L(h) = \langle h, h_0 \rangle$ for every $h$ in $\mathcal{H}$.
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Context:
Given that $\Phi(x)$ is the fundamental solution matrix of the system $\dot{x}=Ax$, where $A$ is a constant matrix, and $\Phi(0)=E$ (the identity matrix), find an expression for $\Phi^{-1}(x)$. Provide your answer in terms of $\Phi(x)$.
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Context:
Evaluate the limit: $$\lim_{n\to\infty}\frac{\displaystyle\sum_{k=1}^{n}\frac{2n}{k^2+kn+n^2}-\int_0^1\frac{1}{x^2-x+1}dx}{\displaystyle\sum_{k=1}^{n}\frac{n}{k^2+kn+n^2}-\int_0^1\frac{1}{x^2+x+1}dx}$$
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Context:
Angela has $128$ coins. $127$ of them have the same weight, but the one remaining coin is heavier than the others. Angela has a balance that she can use to compare the weight of two collections of coins against each other (that is, the balance will not tell Angela the weight of a collection of coins, but it will say which of two collections is heavier). What is the minumum number of weighings Angela must perform to guarantee she can determine which coin is heavier?
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Context:
Find the set of values for $n$ such that the integral \( \int_0^\infty \frac{\sin(x+x^2)}{x^n} \, \mathrm{d}x \) converges.
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Context:
A plane has 144 seats, and it is known that 15% of ticket holders do not show up for the flight, with each person's decision being independent. How many tickets can you sell to be 99% sure that not more than 144 people will show up for the flight?
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Context:
Identify the space for which the set \( \{1, \sin(x), \cos(x), \sin(2x), \cos(2x), \ldots\} \) forms a basis.
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Context:
Find the residue of the function \( f(z) = \frac{e^z\sin(z)}{z(1-\cos(z))} \) at \( z = 0 \) using its Laurent series expansion.
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Context:
In a triangulation of the $k$-simplex \( \sigma_k = \{(x_1, \ldots, x_{k+1}) : x_1 + \cdots + x_{k+1} = 1, x_i \ge 0 \} \), determine the number of $k$-simplices that contain a given non-boundary $(k-1)$-simplex.
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Context:
Consider the positive integers less than or equal to one billion, i.e., $1, 2, 3, \dots, 10^{9}-1, 10^{9}$. What fraction of these integers can be expressed in the form $x^2 + y^3 + z^5$, where $x, y$, and $z$ are positive integers?
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Context:
Let \(X_1, \ldots, X_n\) be iid Bernoulli\((p)\) random variables with \(0 < p < 1\). Given a sample where \(x_1 = \ldots = x_n = 1\), and assuming the prior distribution of \(p\) is Uniform\((0,1)\), consider a Bayesian test for \(H_0: p \geq 0.9\) versus \(H_1: p < 0.9\). Find the necessary and sufficient condition on \(n\) such that \(H_0\) is rejected.
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Context:
Determine the maximum number of rooks that can be placed on an \( n \times n \) chessboard such that each rook is threatened by at most \( 2k \) other rooks.
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Context:
What is the maximum number of smaller balls that can be placed inside a larger ball without intersecting each other?
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Context:
Determine the number of subgroups of index 4 in the group \((\mathbb{Z}/4\mathbb{Z})^2\).
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Context:
Calculate \( F''(\pi) \) for the function \( F(x) = \int_x^{\sin x}\left( \int_0^{\sin t}\sqrt{1+u^4} \, du \right) dt \) using the fundamental theorem of calculus.
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Context:
In a pirate coin distribution game, determine the minimum number of coins the third youngest pirate must be offered to vote in favor of a proposal.
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Context:
Determine the range of values for $r$ such that the series \(-\sum\limits_{n=1}^{\infty} \frac{1}{n^{r-1}}\sin(nx)\) does not converge uniformly for \(x \in \mathbb{R}\).
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Context:
Frank mistakenly believes that the number $1011$ is prime and for some integer $x$ writes down $(x+1)^{1011}\equiv x^{1011}+1\pmod{1011}$. However, it turns out that for Frank's choice of $x$, this statement is actually true. If $x$ is positive and less than $1011$, what is the sum of the possible values of $x$?
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Context:
The sum of the three largest natural divisors of a natural number \( N \) is 10 times the sum of its three smallest natural divisors. Find all possible values of \( N \).
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Context:
Determine the limit superior of the sequence \(|\sin n|^n\) as \(n\) approaches infinity.
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Context:
Consider the set \( A = \{1, 2, \cdots, 6\} \). A mapping \( f: A \rightarrow A \) is a bijection. Define \( f^{(2)}(x) = f(f(x)) \) and \( f^{(n+1)}(x) = f(f^{(n)}(x)) \) for \( n \geq 1 \). Determine the number of bijections \( f \) such that \( f^{(12)}(x) = x \) for all \( x \in A \).
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Context:
Determine the values of the constant $c$ for which the series \( S=\sum_{n=1}^{\infty}\frac{n^{cn+d}}{(cn+d)^n} \) converges, given that $c \neq 1$ and $c$ and $d$ are positive constants.
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Context:
A function $f$ is defined by \(f(z)=(4+i)z^2+a z+b\) for all complex numbers $z$, where $a$ and $b$ are complex numbers. Given that $f(1)$ and $f(i)$ are both purely real, find the minimum value of $|a|+|b|$. \(i=\sqrt{-1}\)
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Context:
Let $L$ be the line of intersection of the planes $ax+by+cz=1$ and $a'x+b'y+c'z=1$. Given that $$(a-a')^2+(b-b')^2+(c-c')^2=16$$ and $$(ab'-a'b)^2+(bc'-b'c)^2+(ac'-a'c)^2=9,$$ find the perpendicular distance of $L$ from the origin.
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Context:
Determine the greatest possible value of \( \frac{P}{20!} \), where \( P \) is the product of the numbers in a list of 20 numbers. For each positive integer \( n \) from 1 to 20, the \( n \)th number in the list is either \( \frac{1}{2}n \) or \( 2n \). The mean of the numbers in the list is exactly 12.
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Context:
Compute \( \int_{S^1} i^* \omega \) where \( \omega \in \Omega^1(\mathbb{R}^2) \) is a 1-form such that \( d\omega = dx \wedge dy \). Let \( i: S^1 \to \mathbb{R}^2 \) be the inclusion map and \( S^1 \) be the unit circle. Use Stokes' theorem to evaluate the integral.
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Context:
Find all positive integers \( n \) such that \( n \mid a^{25} - a \) for all positive integers \( a \).
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Context:
Given a solution $u \in H^2(\mathbb{R}^3)$ to the Hartree equation with an odd real-valued potential $V$, find the time derivative of the modified Hartree energy.
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Context:
Given independent random variables \( X_1 \sim N(\mu = 2, \sigma = 2) \), \( X_2 \sim N(\mu = 1, \sigma = 4) \), and \( X_3 \sim N(\mu = -4, \sigma = 3) \), consider \( Y = (X_1 + 2X_2 + X_3)^2 \). If \( E(Y) = 53 \), determine the probability \( P(Y > E(Y)) \).
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Context:
In how many ways can 15 different books be distributed to 15 children such that each child receives exactly one book, and then the books are collected and redistributed so that no child receives the same book they initially received?
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Context:
Let the function \( f(x) = 1 - |1 - 2x| \) and \( g(x) = x^2 - 2x + 1 \) for \( x \in [0,1] \). Define the function \( F(x) \) as follows:

\[
F(x) =
\begin{cases}
f(x) & \text{if } f(x) \geq g(x), \\
g(x) & \text{if } f(x) < g(x).
\end{cases}
 \]

Determine the number of real roots of the equation \( F(x) \cdot 2^x = 1 \).
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Context:
Let $ABC$ be a triangle with incenter $I$ such that $AB=20$ and $AC=19$. Point $P \neq A$ lies on line $AB$ and point $Q \neq A$ lies on line $AC$. Suppose that $IA=IP=IQ$ and that line $PQ$ passes through the midpoint of side $BC$. Suppose that $BC=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
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