Context:
Rachel has two indistinguishable tokens, and places them on the first and second square of a $1 \times 6$ grid of squares. She can move the pieces in two ways: If a token has a free square in front of it, then she can move this token one square to the right. If the square immediately to the right of a token is occupied by the other token, then she can 'leapfrog' the first token; she moves the first token two squares to the right, over the other token, so that it is on the square immediately to the right of the other token. If a token reaches the 6th square, then it cannot move forward any more, and Rachel must move the other one until it reaches the 5th square. How many different sequences of moves for the tokens can Rachel make so that the two tokens end up on the 5th square and the 6th square?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Richard has an infinite row of empty boxes labeled $1, 2, 3, \ldots$ and an infinite supply of balls. Each minute, Richard finds the smallest positive integer $k$ such that box $k$ is empty. Then, Richard puts a ball into box $k$, and if $k \geq 3$, he removes one ball from each of boxes $1,2,\ldots,k-2$. Find the smallest positive integer $n$ such that after $n$ minutes, both boxes $9$ and $10$ have at least one ball in them.

[i]Proposed by [b]vvluo[/b] & [b]richy[/b][/i]
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Context:
Determine the number of integers  $a$  with  $1\leq a\leq 1007$  and the property that both  $a$  and  $a+1$  are quadratic residues mod  $1009$ .
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Context:
Find the greatest positive real number $M$ such that for all positive real sequence $(a_n)$ and for all real number $m < M$, it is possible to find some index $n \ge 1$ that satisfies the inequality $a_1 + a_2 + a_3 + ...+ a_n +a_{n+1} > m a_n$.
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Context:
Let $\zeta_7= e^{i2\pi/7}$ be the $7$th root of unity. Find the quadratic subfield of $\mathbb Q(\zeta_7)$ which can be expressed in the form $\mathbb Q(\sqrt D)$, where $D$ is an integer.
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Context:
An engineer designed a ball so that when it was dropped, it rose with each bounce exactly one-half as high as it had fallen. The engineer dropped the ball from a certain height and caught it after it had traveled 45 meters. The ball bounced 4 times. From what height did the engineer drop the ball?
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Context:
30 volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called **unusual** if the winner of the match has fewer wins than the loser. A team is called **astonishing** if all its matches are **unusual** matches. Find the maximum number of **astonishing** teams.
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Context:
Given the recursive relationship $(2n+3)a_{n+1}-(2n+5)a_n=(2n+3)(2n+5)\lg (1+ \frac{1}{n} )$ and the initial condition $a_1=5$, find the 2017th term of the sequence $\left\{ \frac{a_n}{2n+3} \right\}$. Express your answer as a single value, including any necessary logarithmic terms.
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Context:
On sides \( BC \) and \( AC \) of triangle \( ABC \), points \( D \) and \( E \) are chosen respectively such that \( \angle BAD = 50^\circ \) and \( \angle ABE = 30^\circ \). Find \( \angle BED \) if \( \angle ABC = \angle ACB = 50^\circ \).
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Context:
A lighthouse is located 38° north by east of a ship. Determine the direction of the ship from the lighthouse, including the degree value. Express your answer in the format "Direction by Direction X°" (e.g. South by west X°).
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Context:
Given the set M = {-1, 0, 1} and N as the set of natural numbers, find the intersection M ∩ N. Express your answer as a set, with elements separated by commas.
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Context:
Evaluate the following limit:
$$L=\lim_{n\to\infty}\int_0^1\frac{\int_0^t(\frac{x^{n+1}-1}{x-1})\mathrm dx}{t}\mathrm dt $$
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Context:
Suppose rectangle $F O L K$ and square $L O R E$ are on the plane such that $R L=12$ and $R K=11$. Compute the product of all possible areas of triangle $R K L$.
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Context:
Given four distinct real numbers \( a, b, c, d \) such that \(\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4\) and \( ac = bd \), find the maximum value of \(\frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \).
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Context:
If an integer $n > 8$ is a solution of the equation $x^2 - ax + b = 0$ and the representation of $a$ in the base-$n$ number system is $18$, what is the base-$n$ representation of $b$?
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Context:
Given the function $f(x)=2x-\sin x$, if the positive real numbers $a$ and $b$ satisfy $f(a)+f(2b-1)=0$, then the minimum value of $\dfrac {1}{a}+ \dfrac {4}{b}$ is ______.
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Context:
Determine \( q(x) \) if the graph of \( \frac{x^3-2x^2-5x+6}{q(x)} \) has vertical asymptotes at \( 3 \) and \( -3 \), no horizontal asymptote, and \( q(2) = 18 \).
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Context:
If I choose four cards from a standard $52$-card deck, without replacement, what is the probability that I will end up with one card from each suit, in a sequential order (e.g., clubs, diamonds, hearts, spades)?
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Context:
Given that in a convex quadrilateral $ABCD$, $\angle BAC=30^\circ, \angle ACD=40^\circ,\angle ADB=50^\circ,\angle CBD=60^\circ$ and $\angle ABC+\angle ADC>180^\circ.$, if $\angle A=\angle C=a^\circ, \angle B=b^\circ, \angle D=c^\circ.$, find the value of $a+b+c$.
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Context:
On a plane, there are a finite number of polygons. If any two of these polygons have a line passing through the origin that intersects both of them, then these polygons are said to be properly placed. Find the smallest natural number \( m \) such that for any set of properly placed polygons, we can draw \( m \) lines passing through the origin such that each of these polygons intersects with at least one of these \( m \) lines.
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Context:
A marketing firm determined that, of 160 households surveyed, 80 used neither brand A nor brand B soap. 60 used only brand A soap and for every household that used both brands of soap, some used only brand B soap. 5 of the 200 households surveyed used both brands of soap. What is the ratio of households that used only brand B soap to those that used both brands?
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Context:
The graph of the function $f(x)=\sin(\omega x+\varphi)$, where $(\omega>0, |\varphi|<\frac{\pi}{2})$, passes through the point $(0,-\frac{1}{2})$. Find the minimum value of $\omega$ if the graph of this function is shifted to the right by $\frac{\pi}{3}$ units and becomes symmetric about the origin.
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Context:
Given the line $y=x+\sqrt{6}$, the circle $(O)$: $x^2+y^2=5$, and the ellipse $(E)$: $\frac{y^2}{a^2}+\frac{x^2}{b^2}=1$ $(b > 0)$ with an eccentricity of $e=\frac{\sqrt{3}}{3}$. The length of the chord intercepted by line $(l)$ on circle $(O)$ is equal to the length of the major axis of the ellipse. Find the product of the slopes of the two tangent lines to ellipse $(E)$ passing through any point $P$ on circle $(O)$, if the tangent lines exist.
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Context:
Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$ and $BC > AD$ such that the distance between the incenters of $\triangle ABC$ and $\triangle DBC$ is $16$. If the perimeters of $ABCD$ and $ABC$ are $120$ and $114$ respectively, then the area of $ABCD$ can be written as $m\sqrt{n}$, where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $100m+n$.
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Context:
A farmer bought $749$ sheep. He sold $700$ of them for the price paid for the $749$ sheep. The remaining $49$ sheep were sold at the same price per head as the other $700$. Based on the cost, the percent gain on the entire transaction is:
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Context:
Let $E\subseteq\mathbb{R}$ be a Borel measurable set with $m(E)=0$ and $f(x)=x^{2}$. Determine if $m(f(E))=0$.
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Context:
5. In a regular $1000-$gon, all diagonals are drawn. What is the maximum number of diagonals that can be selected such that among any three of the selected diagonals, at least two have the same length?
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Context:
Calculate the line integral of the vector field \( \mathbf{F}(x,y,z) = \left(\frac{-y}{x^2+z^2}, \frac{x}{x^2+z^2}, \frac{z}{x^2+z^2}\right) \) over the curve \( \gamma \).
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Context:
Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible integral values for $BC$.
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Context:
4. For any integer $n(n \geqslant 4)$, consider $m$ subsets $A_{1}, A_{2}, \cdots, A_{m}$ of the set $\{1,2, \cdots, n\}$, where $A_{1}$ has one element, $A_{2}$ has two elements, $\cdots \cdots A_{m}$ has $m$ elements, and no subset is contained in another subset. Find the maximum value of $m$.

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Context:
$A,B,C$ are the three angles in a triangle such that
$2\sin B\sin (A+B)-\cos A=1$,
$2\sin C\sin (B+C)-\cos B=0$
find the three angles.
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Context:
In the game of Twister, a spinner randomly selects either an arm or a leg, and also selects one of four colors, one of which is red, each with equal probability, and players have to move the appropriate body part to the appropriately colored spot on the ground. There are four players. Each player spins once, and makes the move the spinner selects. What is the probability that in these four spins, there will be exactly two moves to a red spot, and the body part selected to move will be an arm exactly $3$ times?
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Context:
 Ralph has a cylinder with height $15$ and volume $\frac{960}{\pi}$ . What is the longest distance (staying on the surface) between two points of the cylinder?

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Context:
Let $G$ be a $30$-regular graph. Determine whether $G$ must contain a path of length at least $20$. Answer with 'yes' or 'no'.
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Context:
Determine whether the series \(\sum_{k=1}^\infty \frac{1}{4^k(2k)}\binom{2k}{k}\) converges to \(\ln 2\).
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Context:
 A frog starts at $(1, 1)$. Every second, if the frog is at point $(x, y)$, it moves to $(x + 1, y)$ with probability $\frac{x}{x+y}$ and moves to $(x, y + 1)$ with probability $\frac{y}{x+y}$ . The frog stops moving when its $y$ coordinate is $10$. Suppose the probability that when the frog stops its $x$-coordinate is strictly less than $16$, is given by $m/n$ where $m, n$ are positive integers that are relatively prime. Find $m + n.$

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Context:
There are 144 bottle caps in a box. Marvin takes 63 bottle caps and gives half of them to his friend John. Then, Marvin's cousin adds 56 more bottle caps to the box. How many bottle caps are in the box now?
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Context:
6. Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7 . Play continues until either player wins, or else indefinitely. If Nathaniel goes first, determine the probability that he ends up winning.
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Context:
Find all pairs $(x,y)$ of real numbers such that $ |x|+ |y|=1340$ and $x^{3}+y^{3}+2010xy= 670^{3}$ .
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Context:
Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?
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Context:
Consider a Markov process on the state space $S = \{1,2\}$ with a positive transition matrix, meaning all entries are strictly positive. Let $S_\infty = \times_{t=1}^\infty S$ be the underlying probability space, and let $\mathbb{P}$ be the probability measure on $S_\infty$ induced by the Markov transition matrix. For each $n$, define $F_n = \{ s \in S_\infty : \text{there are at most $n$ consecutive occurrences of $1$ or $2$} \}$. Determine if $\lim_{n \to \infty} \mathbb{P}(F_n) = 1$.
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Context:
Given a matrix $A=(a_{ij})_{n\times n}$, let $C_{i,j}$ be the cofactor in position $(i,j)$. By the determinant formula, we have \(\det A=\sum_{i=1}^n a_{i,1}C_{i,1}.\) Evaluate the expression \(\sum_{i=1}^n a_{i,1}C_{i,2}.\) Must this expression evaluate to zero?
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Context:
Given a square matrix $A = (a_{ij})_{n \times n}$, evaluate whether the expression $\sum_{i=1}^n a_{i,1}C_{i,2}$ equals zero, where $C_{i,2}$ are the cofactors of the elements in the second column.
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Context:
For each integer $n\geq 2$, let $S_n$ be the sum of all products $jk$, where $j$ and $k$ are integers and $1\leq j<k\leq n$. What is the sum of the 10 least values of $n$ such that $S_n$ is divisible by $3$?
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Context:
Determine the boundary of the set \(\mathbb{Q}^n\) when considered as a subset of \(\mathbb{R}^n\) with the standard topology. Is it true that \(\partial \mathbb{Q}^n = \emptyset\)?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\partial \mathbb{Q}^n = \mathbb{R}^n}


Context:
Subsets $S$ of the first 3$5$ positive integers $\{1, 2, 3, ..., 35\}$ are called [i]contrived [/i] if $S$ has size $4$ and the sum of the squares of the elements of $S$ is divisible by $7$. Find the number of contrived sets.
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Context:
Two congruent equilateral triangles \(\triangle ABC\) and \(\triangle DEF\) lie on the same side of line \(BC\) such that \(B\), \(C\), \(E\), and \(F\) are collinear. A line intersects \(\overline{AB}\), \(\overline{AC}\), \(\overline{DE}\), and \(\overline{EF}\) at points \(W\), \(X\), \(Y\), and \(Z\), respectively, with the conditions \(\frac{AW}{BW} = \frac{2}{9}\), \(\frac{AX}{CX} = \frac{5}{6}\), and \(\frac{DY}{EY} = \frac{9}{2}\). Determine the ratio \(\frac{EZ}{FZ}\) in the form \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers, and find \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The surface area of a sphere inscribed in a regular truncated triangular pyramid is related to the total surface area of the pyramid as \(\pi : 6 \sqrt{3}\). Find the angle between the lateral face and the base plane of the pyramid.
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Context:
In how many ways can we change the sign $\ast$ with $+$ or $-$, such that the following equation is true?

\[1 \ast 2 \ast 3 \ast 4 \ast 5 \ast 6 \ast 7 \ast 8 \ast 9 \ast 10 = 29\]

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Context:
Determine whether the limit \( \lim_{x\to-\infty}\sum_{n=1}^\infty\lambda(n)\frac{x^n}{\Gamma(n)} \) is finite.
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Context:
Determine the behavior of the function \( f(a) = (\sqrt{a})^{f(\sqrt{a})} \) at \( a = 2 \). Does it converge to a specific value, and if so, what is that value?
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Context:
What is the average of all positive integers that have three digits when written in base $5$, but two digits when written in base $8$? Write your answer in base $10$.
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Context:
Determine whether the condition $f^{(a)}(0) = 0$ for all $0 \leq a \leq n$ implies that $\lim_{x \to 0} \frac{f(x)}{x^n} = 0$, where $f^{(n)}$ denotes the $n$-th derivative and $f^{(0)} = f$. Assume that all $f^{(a)}$ are defined in an interval around $0$.
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Context:
Let $d(n)$ be the number of all positive divisors of a natural number $n \ge 2$.
Determine all natural numbers $n \ge 3$ such that $d(n -1) + d(n) + d(n + 1) \le  8$.
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Context:
Let  $\Gamma_1, \Gamma_2, \Gamma_3$  be three pairwise externally tangent circles with radii  $1,2,3,$  respectively. A circle passes through the centers of  $\Gamma_2$  and  $\Gamma_3$  and is externally tangent to  $\Gamma_1$  at a point  $P.$  Suppose  $A$  and  $B$  are the centers of  $\Gamma_2$  and  $\Gamma_3,$  respectively. What is the value of  $\frac{{PA}^2}{{PB}^2}?$ *Proposed by Kyle Lee*
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Context:
Determine whether the sequence \((y_n)\) converges in the space \(L^1(0,1)\), where \(y_n(t) = \frac{t^n}{n^2(1+t^2)}\).
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Context:
 Starting on a triangular face of a right triangular prism and allowing moves to only adjacent faces, how many ways can you pass through each of the other four faces and return to the first face in five moves?

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Context:
Determine for which \( n \) positive integer the equation: \( a + b + c + d = n\sqrt{abcd} \) has positive integer solutions.
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Context:
Determine whether the Weyl tensor is always zero or undefined in two dimensions.
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Context:
Determine whether the field extension \( \mathbb{Q}(\cos(\theta/2)) \) over \( \mathbb{Q}(\cos(\theta)) \) is finite for \( \theta \in \mathbb{R} \). If it is finite, find its degree.
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Context:
Determine whether the infinite product \( \prod_{i=1}^{\infty}\left(\frac{p_{i}-1}{p_{i}}\right) \) converges.
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Context:
We define $N$ as the set of natural numbers $n<10^6$ with the following property:

There exists an integer exponent $k$ with $1\le k \le 43$, such that $2012|n^k-1$.

Find $|N|$.
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Context:
Determine whether the sequence \( y_n = \frac{\sqrt[n]{\prod\limits ^{n}_{k=1}( 2n+k)}}{2n} \) converges, using the logarithmic version of the Cesàro-Stolz theorem.
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Context:
Determine if there exists a number \(a\) such that all derivatives of \(x^x\) are positive on the interval \((a, \infty)\). If such a number exists, find the smallest possible value of \(a\).
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Context:
Determine the values of \( k \) for which the matrix \( \begin{bmatrix}0&1&0\\0&0&1\\0&-k-7&k+8\end{bmatrix} \) is not diagonalizable over the complex numbers \( \Bbb C \).
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Context:
Determine the rate of convergence of the sequence $e^{-n}$ as $n \to \infty$. Find a sequence $\beta_n$ that converges to 0 such that $|e^{-n} - 0| \leq K|\beta_n|$ for some positive constant $K$. Express the rate of convergence using $O(\cdot)$ notation.
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Context:
Let $X$ be a non-empty set. Consider the power set $\mathcal{P}(X)$. Define an edge between $a, b \in \mathcal{P}(X)$ if their symmetric difference is a singleton, i.e., $\textrm{card}((a\setminus b) \cup (b\setminus a)) = 1$. For a finite set $X$, the graph has chromatic number 2. Determine if the graph still has chromatic number 2 when $X$ is infinite.
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Context:
Let $f:\mathbb{R} \rightarrow\mathbb{R}$ be a continuous function at $x_0\in\mathbb{R}$. Additionally, $f'(x)$ is defined for every $x\in(x_0-\delta,x_0+\delta)$, where $x_0\notin(x_0-\delta,x_0+\delta)$. Given that $\lim_{x \to x_0} f'(x)=L$, determine if $f'(x)$ is continuous at every $x\in(x_0-\delta,x_0+\delta)$, where $x_0\notin(x_0-\delta,x_0+\delta)$. Answer with 'yes' or 'no'.
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Context:
Given that Maria scored $5, 2, 4, 3, 6, 2, 7, 4, 1,$ and $3$ goals in her first $10$ soccer matches, her total score after $11$ matches is divisible by $11$, and her total score after $12$ matches is divisible by $12$. Find the product of the number of goals she scored in the eleventh and twelfth matches.
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Context:
Determine the maximum order of a permutation in the symmetric group $S_{10}$. Additionally, find the smallest positive integer $n$ such that a bijection $f: \{1, 2, 3, \ldots, 10\} \rightarrow \{1, 2, 3, \ldots, 10\}$ satisfies $f^n = f$ but $f^k \neq f$ for all $1 \leq k < n$. Provide your answer as a single integer.
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Context:
Determine whether the integral \( \int_{-1}^1 e^{-x^{4}}\cdot(1+\ln(x+\sqrt{x^2+1})+5x^3-4x^4)\,dx \) can be evaluated.
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Context:
Let  $x_1$ ,  $x_2$ , and  $x_3$  be the roots of the polynomial  $x^3+3x+1$ .  There are relatively prime positive integers  $m$  and  $n$  such that  $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$ .  Find  $m+n$ .
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Context:
The complete graph of \( y=g(x) \), which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is \( 1 \)).

What is the sum of the \( x \)-coordinates of all points where \( g(x) = x+2 \)?
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Context:
Let \( H \) be the space of all absolutely continuous functions \( f:[0,1] \rightarrow \mathbb{C} \) such that \( f(0)=0 \) and \( f' \in L^2(0,1) \), with the inner product defined by \( \langle f, g \rangle = \int_0^1 f'(t) \overline{g'(t)} \, dt \) for \( f, g \in H \). For a fixed \( 0 < t \le 1 \), define the linear functional \( L: H \rightarrow \mathbb{C} \) by \( L(h) = h(t) \). Determine the norm \( \|L\| \) of this functional and find the vector \( h_0 \in H \) such that \( L(h) = \langle h, h_0 \rangle \) for all \( h \in H \).
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Context:
Find the set of real numbers \( \lambda \) for which the rank of the matrix \( \begin{bmatrix} -3 & 5 & -5 \\ -4 & 6 & -5 \\ -4 & 4 & -3 \end{bmatrix} - \lambda I_3 \) is less than 3.
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Context:
The distance between the $5^\text{th}$ and $26^\text{th}$ exits on an interstate highway is $118$ miles. If any two exits are at least $5$ miles apart, then what is the largest number of miles there can be between two consecutive exits that are between the $5^\text{th}$ and $26^\text{th}$ exits?

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Context:
The rectangle $ABCD^{}_{}$ below has dimensions $AB^{}_{} = 12 \sqrt{3}$ and $BC^{}_{} = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P^{}_{}$. If triangle $ABP^{}_{}$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.
[asy] pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6); draw(B--C--P--D--C^^D--A); filldraw(A--P--B--cycle, gray, black); label("$A$", A, SE); label("$B$", B, NE); label("$C$", C, NW); label("$D$", D, SW); label("$P$", P, N); label("$13\sqrt{3}$", A--D, S); label("$12\sqrt{3}$", A--B, E);[/asy]
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Context:
Let \(X_t\) be a Brownian motion and \(a < 0\). Define the stopping time \(T_x = \inf\{t \geq 0: X_t + at \geq x\}\). If \(Y = \sup_{t \geq 0}(X_t + at) = x\), determine whether \(T_x < \infty\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many natural numbers $n$ less than $10^{6}$ are there such that the sum of the digits of $n$ is even, and the sum of the digits of $(n+1)$ is also even

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


Context:
Let $H$ be a subgroup of an abelian group $A$. Suppose $a_0 \in A$ satisfies $2a_0 = h_1 + h_2$ for some $h_1, h_2 \in H$. Does this imply that $H - a_0 = a_0 - H$? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(X_n)_n$ be a sequence of random variables in $L^2$. Suppose there exists a constant $M$ such that for all $n$, $$P(X^2_n \ge M) < P(X^2_n \ge E(X^2_n)).$$ Determine whether it follows that $E(X_n^2) \le M$ for all $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( p,q,r \) be 3 prime numbers such that \( 5 \leq  p < q < r \) .
 Knowing that \( 2{p}^{2} - {r}^{2} \geq  {49} \) and \( 2{q}^{2} - {r}^{2} \leq  {193} \) ,find
 \( p,q,r \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $P$  be a  $2019-$ gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of  $P$  a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must be on the same diagonal and on every diagonal there are at most two knots from a cycle.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = x^2/16$.  The vertices of the six parabolas are evenly spaced around a circle.  The parabolas open outward with their axes being extensions of six of the circle's radii.  Parabola $P_1$ is tangent to $P_2$, which is tangent to $P_3$, which is tangent to $P_4$, which is tangent to $P_5$, which is tangent to $P_6$, which is tangent to $P_1$.  What is the diameter of the circle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the radius of the incircle of an equilateral triangle \( ABC \) be 2 with center \( I \). If point \( P \) satisfies \( PI = 1 \), what is the maximum value of the ratio of the areas of \( \triangle APB \) to \( \triangle APC \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Derive the expression for \(xz_x + yz_y\) in terms of polar coordinates \((r, \phi)\), given that \(z(x,y)\) is a function of \(x\) and \(y\), where \(x = r\cos\phi\) and \(y = r\sin\phi\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let us consider all rectangles with sides of length $a,b$ both of which are whole numbers. Do more of these rectangles have perimeter $2000$ or perimeter $2002$?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{
More rectangles have perimeter 2002.

}


Context:
Masha has three identical dice, each face of which has one of six different prime numbers with a total sum of 87.

Masha rolled all three dice twice. The first time, the sum of the numbers rolled was 10, and the second time, the sum of the numbers rolled was 62.

Exactly one of the six numbers never appeared. What number is it?
Please reason step by step, and put your final answer within \boxed{}.


Context:
You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points $(x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3})$. If $x_{1} < x_{2} < x_{3}$ and $x_{3} - x_{2} = x_{2} - x_{1}$, which of the following is necessarily the slope of the line which best fits the data? "Best fits" means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line.

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


Context:
If there exists $x\in \left[1,+\infty \right)$ such that the inequality ${(1+\frac{1}{x})}^{x+a}≥e$ holds for $x$, then the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left[\frac{1}{\ln(2)} - 1, +\infty\right)}


Context:
The ratio of the number of students from grades 1 to 6 in Wuyang Primary School is 30:29:28:27:26:25. After the current sixth graders graduate, 372 new first graders are admitted, and the remaining students are promoted to the next grade. This results in the ratio of the number of students in the lower grades (grades 1, 2, 3) to the higher grades (grades 4, 5, 6) being 10:9 at the start of the new term. What is the total number of students in Wuyang Primary School at that time?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.

"Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes.

"Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes.

"Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes.

How many pieces of candy in all did the principal give to the children who always tell the truth?

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


Context:
What is the greatest number of two-digit numbers that can be written in a sequence such that any two adjacent numbers are not coprime, while any two non-adjacent numbers are coprime?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A borrower takes out a loan of $2000 for two years. Construct a sinking fund schedule if the lender receives 10% effective annual interest on the loan, and the borrower makes semiannual deposits into a sinking fund earning 8% interest convertible quarterly. Calculate the amount of each deposit needed to replace the loan amount.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three distinct integers are selected at random between $1$ and $2016$, inclusive. Which of the following is a correct statement about the probability $p$ that the product of the three integers is odd?

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


Context:
Determine whether the initial value problem \( \frac{dy}{dx} = 2y^{1/2} \) with \( y(0) = a \) admits infinitely many solutions for \( a = 0 \) or for \( a \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A line passing through the origin $O$ intersects the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ at points $A$ and $B$. The line passing through $A$ perpendicular to $AB$ intersects $C$ at point $D$. Line $DB$ intersects the $y$-axis at point $E$. If $\overrightarrow{OB} \cdot \overrightarrow{OE} = 2|\overrightarrow{OE}|^{2}$, then the eccentricity of $C$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$.  To the nearest integer the area of triangle $ABC$ is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest natural number n such that for all integers $m > n$ there are positive integers $x$ and $y$ for which the equality 1$7x + 23y = m$ holds
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the determinant of the matrix expression $ACB + BCA$, where $A$ and $B$ are $n \times n$ diagonal matrices, and $C = cc^T$ is another $n \times n$ matrix. Can this determinant be expressed solely in terms of the determinants of $A$, $B$, and $C$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
6. When four dice are rolled randomly, the probability that the product of the four numbers obtained is a perfect square is $\qquad$ .

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


Context:
Determine whether the number of conjugacy classes of the group \( SL_2(\mathbb{F}_p) \) is equal to \( p+4 \) for an odd prime \( p \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A factory produced $n$ robots. Each robot is defective with probability $q$. If a robot is defective, a tester detects it with probability $p$. We assume that the tester detected $z$ defective robots. Let $X$ be the number of defective robots. Compute $E(X | Z = z)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose we have an infinite double random array $\{X_{ij}:i\geq 1,j\geq 1\}$, where each element is independent and identically distributed with mean zero and variance one. Define $$Y_{i,n}=\frac{1}{n}\sum_{j=1}^nX_{ij}$$ and note that by the strong law of large numbers, for any $i\geq 1$, $$Y_{i,n}\to 0,\quad\text{a.s.}\qquad Y_{i,n}^2\to 0,\quad\text{a.s.}\qquad(n\to\infty).$$ Determine if it is true that $$\frac{1}{n}\sum_{i=1}^nY_{i,n}^2\to 0\quad\text{a.s.}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$, and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\sqrt{3}$. The perimeter of hexagon $ABCDEF$ can be expressed as $m +n\sqrt{p}$, where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the function $f(x)= \frac{1}{2}(m-2)x^{2}+(n-8)x+1$ is monotonically decreasing in the interval $\left[ \frac{1}{2},2\right]$ where $m\geqslant 0$ and $n\geqslant 0$, determine the maximum value of $mn$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be a positive integer and let $f_{1}(x), \ldots, f_{n}(x)$ be affine functions from $\mathbb{R}$ to $\mathbb{R}$ such that, amongst the $n$ graphs of these functions, no two are parallel and no three are concurrent. Let $S$ be the set of all convex functions $g(x)$ from $\mathbb{R}$ to $\mathbb{R}$ such that for each $x \in \mathbb{R}$, there exists $i$ such that $g(x)=f_{i}(x)$.

Determine the largest and smallest possible values of $|S|$ in terms of $n$.
(A function $f(x)$ is affine if it is of the form $f(x)=a x+b$ for some $a, b \in \mathbb{R}$. A function $g(x)$ is convex if $g(\lambda x+(1-\lambda) y) \leq \lambda g(x)+(1-\lambda) g(y)$ for all $x, y \in \mathbb{R}$ and $0 \leq \lambda \leq 1$.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
A zoo has some pandas, paired into mates (in the same zoo). Only 25% of the panda couples get pregnant after mating. They each have one baby, and 2 panda babies are born. How many pandas are there in the zoo?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Kid and Karlsson play a game. Initially they have a square piece of chocolate $2019\times 2019$ grid with $1\times 1$ cells . On every turn Kid divides an arbitrary piece of chololate  into three rectanglular pieces by cells, and then Karlsson chooses one of them and eats it. The game finishes when it‘s impossible to make a legal move. Kid wins if there was made an even number of moves, Karlsson wins if there was made an odd number of moves.Who has the winning strategy?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a six-digit number that is both a perfect square and a perfect cube. If 6 is subtracted from this number, the result is a prime number. What is the original number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the relative homology group $H_n(M, \partial M)$ is always isomorphic to $\mathbb{Z}$ for an oriented, compact, connected $n$-dimensional smooth manifold $M$ with boundary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Baker Ram is storing cake boxes in cartons that measure 25 inches by some inches by 60 inches. If the measurement of each cake box is 8 inches by 7 inches by 5 inches, and the maximum number of cake boxes that can be placed in each carton is 210, what is the width of the carton?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  ${(a_n)_{n\ge1}} $  be a sequence with   ${a_1 = 1} $   and   ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$  for all  ${n \ge 1}$ , where  ${\lfloor x \rfloor}$  denotes the greatest integer less than or equal to  ${x}$ . Find all  ${n \le 2013}$  such that  ${a_n}$  is a perfect square
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $G$ be a Lie group acting smoothly and freely on a smooth manifold $M$. If the action is not necessarily proper, but $M/G$ is a topological manifold, is the dimension of $M/G$ equal to $\dim M - \dim G$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given four points $x_1, x_2, x_3, x_4$ on the unit circle $\mathbb{S}^1$ in the plane $\mathbb{R}^2$ such that $\sum_{i=1}^4 x_i = 0$, does this imply that the points form a rectangle or two pairs of antipodal points? Answer "yes" or "no."
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $R$ be a commutative ring and $J_1, J_2$ be two non-zero proper ideals of $R$. Suppose there exist $r_1, r_2 \in R$ such that for any $a \in R$, the conditions $r - r_1 \in J_1$ and $r - r_2 \in J_2$ are equivalent to $r \in J_2$ and $r - a \in J_1$. Does this imply that $J_1 + J_2 = R$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $N$  be the smallest positive integer  $N$  such that  $2008N$  is a perfect square and  $2007N$  is a perfect cube.  Find the remainder when  $N$  is divided by  $25$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the implicit function theorem, determine if the equation $x\cos(x-y) = 0$ implicitly defines $y$ as a function of $x$ near the point $(\pi/2, \pi/2)$. If it does, calculate the derivative $\frac{dy}{dx}$ at $x=\pi/2$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all positive integers $n$ such that $n^2+3$ divides evenly (without remainder) into $n^4-3n^2+10$ ?

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


Context:
Compute the 100$^{th}$ smallest positive integer $ n $ that satisfies the three congruences

$$
\left\lfloor \frac{n}{8} \right\rfloor \equiv 3 \pmod{4},
$$

$$
\left\lfloor \frac{n}{32} \right\rfloor \equiv 2 \pmod{4},
$$

$$
\left\lfloor \frac{n}{256} \right\rfloor \equiv 1 \pmod{4}.
$$

Here $\left\lfloor \cdot \right\rfloor$ denotes the greatest integer function.

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


Context:
Determine whether the sequence of functions \( g_n(\alpha) = \int_a^b \alpha(x) \sin(nx) \cos(nx) \, dx \) converges uniformly for all functions \( \alpha \) in the set \( X = \{ \alpha : [a,b] \to \mathbb{R} \} \), where \( \alpha'' \) exists and is continuous, and there exists a constant \( K > 0 \) such that \( |\alpha(x)|, |\alpha''(x)| \le K \) for all \( x \in [a,b] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
在一条公路上每隔100千米有一个仓库，共有五个仓库，一号仓库存有10吨货物，二号仓库有20吨货物，五号仓库存有40吨货物，其余两个仓库是空的。现在想把所有的货物集中存放在一个仓库里，如果每吨货物运输一千米需要一元的运费，那么最少要花多少运费？
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T_1$ be a triangle with sides $2011$, $2012$, and $2013$. For $n \geq 1$, if $T_n = \Delta ABC$ and $D, E$, and $F$ are the points of tangency of the incircle of $\Delta ABC$ to the sides $AB$, $BC$, and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE$, and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $\left(T_n\right)$?

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


Context:
Determine whether the quantity \( \frac{\log(\|A^{-1}\|)}{\log(\|A\|)} \) is bounded by a constant that depends only on the size \( n \) of the matrix, as \( A \) ranges over \( SL(n,\mathbb{R}) \) (the determinant one matrices).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A = \{1, 2, \cdots, 2002\}$ and $M = \{1001, 2003, 3005\}$. For any non-empty subset $B$ of $A$, if the sum of any two elements in $B$ is not in $M$, then $B$ is called an $M$-free set. If $A = A_{1} \cup A_{2}$, $A_{1} \cap A_{2} = \emptyset$, and both $A_{1}$ and $A_{2}$ are $M$-free sets, then the ordered pair $\left(A_{1}, A_{2}\right)$ is called an $M$-partition of $A$. Find the number of all $M$-partitions of $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the system of partial differential equations in cylindrical coordinates:
\[ \partial_r b_2 - \partial_{\theta}b_1 = 0, \\
\partial_{\theta} b_3 - \partial_{z}b_2 = 0, \\
\partial_rb_3 - \partial_{z}b_1 = \xi(r, z), \]
where $\xi(r, z)$ is a given function of $r$ and $z$, and $b_1, b_2, b_3$ are unknown functions of $(r, \theta, z)$. Assume that this system has a unique solution. Determine whether $b_2 = 0$ for this solution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the convergence of the integral \( J_n=\int^\infty_0 e^{-t}\big|\cos(nt+c_n)\big|^n\, dt \) as \( n\rightarrow\infty \), where \( c_n \) is an arbitrary numerical sequence. For which \( p>0 \) does \( n^pJ_n \) converge?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\Lambda_A$ be a lattice with a basis $A= \{(1,1,1)^T, (1,0,2)^T, (0,1,1)^T\}$. Determine if $\Lambda_B$, with a basis $B= \{(1,2,6)^T, (3,1,3)^T, (3,3,7)^T\}$, is a sublattice of $\Lambda_A$. If it is, calculate the index $[\Lambda_A : \Lambda_B]$. Provide your answer as a single number or state that $\Lambda_B$ is not a sublattice.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest integer \( n \) such that the remainder estimate \( |R_n| \leq \frac{M}{(n+1)!} (x-a)^{n+1} \) is satisfied, where \( M \) is the maximum value of \( |f^{(n+1)}(z)| \) on the interval \([-1,1]\) for the function \( f(x) = e^{-2x} \) expanded around \( a = 0 \). Determine \( M \) and ensure that \( |R_n| \leq \frac{1}{1000} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $xf(x)$ approaches a limit as $x$ approaches infinity, does it necessarily follow that $f(x)$ approaches 0? Specifically, if $f: \mathbb{R} \to \mathbb{R}$ satisfies $\lim_{x \to \infty} xf(x) = \lambda$ for some $\lambda \in \mathbb{R}$, does it imply that $\lim_{x \to \infty} f(x) = 0$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
$ C_{1}$ and $ C_{2}$ be two externally tangent circles with diameter $ \left[AB\right]$ and $ \left[BC\right]$, with center $ D$ and $ E$, respectively. Let $ F$ be the intersection point of tangent line from A to $ C_{2}$ and tangent line from $ C$ to $ C_{1}$ (both tangents line on the same side of $ AC$). If $ \left|DB\right| ={}\left|BE\right|={}\sqrt{2}$, find the area of triangle $ AFC$.

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


Context:
Determine if the limit \( \lim_{x\rightarrow\infty}\prod_{p\leqslant x}p^{1/(2x)} \) is finite.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In 2009, 108 U.S. skydivers fell headfirst from a height of 18,000 ft. How far had they fallen 11 seconds after having jumped from the plane?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Circles $\omega_1$, $\omega_2$, and $\omega_3$ are centered at $M$, $N$, and $O$, respectively. The points of tangency between $\omega_2$ and $\omega_3$, $\omega_3$ and $\omega_1$, and $\omega_1$ and $\omega_2$ are tangent at $A$, $B$, and $C$, respectively. Line $MO$ intersects $\omega_3$ and $\omega_1$ again at $P$ and $Q$ respectively, and line $AP$ intersects $\omega_2$ again at $R$. Given that $ABC$ is an equilateral triangle of side length $1$, compute the area of $PQR$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the Nyquist frequency for the signal \( x(t) = \sum_{n=1}^{3} \sin\left(2\pi \frac{n}{8}\frac{t}{T}\right) \), where \( T > 0 \) is a positive number. Determine the smallest frequency \( u_0 \) such that for all \( u > u_0 \), the Fourier transform \( \hat{x}(u) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given four positive integers $a < b < c < d$ that satisfy $a + d = b + c = n$, where $n$ is a constant, determine if there exist integers $p, q, r, s$ such that $a = p + q$, $b = p + r$, $c = q + s$, and $d = r + s$. Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limit of the characteristic function $\chi_{B(x,r)}$ of the ball $B(x,r)$ as the radius $r$ approaches 0.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\pi}^{\pi} \frac{\cos(x)}{\sqrt{4+3\sin(x)}} \, dx \). Determine if the integral equals zero due to symmetry.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the first continuous cohomology group \(H^1_{cont}(\mathrm{Gal}(L/K), L^{\times})\) vanishes for algebraic extensions \(L/K\) of \(\mathbb{Q}_p\) with \(L^{\times}\) having the \(p\)-adic topology.
Please reason step by step, and put your final answer within \boxed{}.


Context:
[ Sphere inscribed in a pyramid ]
[Volume helps solve the problem.]

In a triangular pyramid, two opposite edges are equal to 12 and 4, and the other edges are equal to 7. A sphere is inscribed in the pyramid. Find the distance from the center of the sphere to the edge equal to 12.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Inside the right triangle \(ABC\) (with angle \(B\) being the right angle), a point \(D\) is placed such that the areas of triangles \(ABD\) and \(BCD\) are, respectively, three times and four times smaller than the area of triangle \(ABC\). The segments \(AD\) and \(DC\) have lengths \(a\) and \(c\), respectively. Find \(BD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
19 In a carriage, any $m(m \geqslant 3)$ passengers have a unique common friend (if A is a friend of B, then B is also a friend of A, and no one is a friend of themselves), how many friends does the person with the most friends have in this carriage?

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


Context:
Determine whether the integral \( \int_0^\infty \frac{xe^{-bx}}{\sqrt{x^2 +a^2}}dx \) can be solved analytically, where \(a\) and \(b\) are constants.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of each square is 24, and the perimeter of each small rectangle is 16. What is the perimeter of the large rectangle?

The perimeter of a figure is the sum of the lengths of all its sides.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a set of linearly independent $d \times d$ complex unitary matrices $\{U_i\}_{i=1}^n \subseteq M_d$ with $n \geq d$, determine if there exists a vector $v \in \mathbb{C}^d$ such that the span of $\{U_1v, U_2v, \ldots , U_nv\}$ is $\mathbb{C}^d$. Answer with 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If each of the three operation signs, $+$, $-$, $\times $, is used exactly ONCE in one of the blanks in the expression

\[5\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}4\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}6\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}3\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alice is thinking of a number. She gives the following 3 clues. "My number has 243 as a factor. My number is a multiple of 36. My number is between 1000 and 3000." What is Alice's number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximal number of solutions can the equation have   $$ \max  \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0 $$  where  $a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10}$  are real numbers, all  $a_i$  not equal to  $0$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a linear operator $T$ on $\mathbb{R}^5$ where $T^4 \neq 0$ and $T^5 = 0$, determine the dimensions of the range of $T^i$ for $i = 1, 2, 3, 4$. Provide your answers as a list of dimensions for each power.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the probability of being in state $3$ after $500,000$ steps in a Markov chain, given that the chain starts in state $1$. Assume the transition probabilities are defined as in the original problem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the zeros of a holomorphic function can have an accumulation point on the boundary of its domain. Answer 'yes' or 'no'.
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Context:
In the Cartesian coordinate system \( xOy \), the area of the region corresponding to the set of points \( K = \{(x, y) \mid (|x| + |3y| - 6)(|3x| + |y| - 6) \leq 0 \} \) is ________.
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Context:
Given a positive integer \( k \), find the largest positive integer \( m \) such that if no more than \( m \) of any set of \( 3k \) points in the plane are collinear, then the points can be divided into \( k \) sets of three points each, where the points in each set form the vertices of a triangle.
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Context:
How many numbers not more than 100 whose number at tens place and ones place is different have 0, 1, 7, and 9?
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Context:
Determine whether the function \( f(x, y) = \frac{xy}{\sqrt{x} + y^2} \) for \( x, y \neq 0 \) and \( f(0, 0) = 0 \) has a continuous limit at the origin \((0,0)\).
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Context:
Charles has $5q + 1$ quarters and Richard has $q + 5$ quarters. The difference in their money in dimes is:

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Context:
The parabola $y = x^2$ is tangent to the graph of $y = x^4 + ax^3 + x^2 + bx + 1$ at two points.  Find the positive difference between the $x$-coordinates of the points of tangency.
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Context:
Determine the exact value of $a > 1$ for which the periodic function $g$ defined by $g(x) = f(x)$ for $0 \le x < a$ and $g(x) = g(x+a)$ for all $x \in \mathbb{R}$ is continuous.
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Context:
For a given positive integer \( k \) denote the square of the sum of its digits by
 \( {f}_{1}\left( k\right) \) and let \( {f}_{n + 1}\left( k\right)  = {f}_{1}\left( {{f}_{n}\left( k\right) }\right) \) . Determine the value of \( {f}_{1991}\left( {2}^{1990}\right) \)
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Context:
Find the splitting field $K$ of the polynomial $x^{12} - 9$ over the field of rational numbers $\mathbb{Q}$, and determine the degree of the extension $[K: \mathbb{Q}]$. Provide your answer as a single number.
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Context:
5. Let $A B C D E F$ be a regular hexagon. A frog starts at vertex $A$ and can randomly jump to one of the two adjacent vertices each time. If it reaches point $D$ within 5 jumps, it stops jumping; if it does not reach point $D$ within 5 jumps, it stops after 5 jumps. How many different possible jumping sequences are there from the start to the stop? $\qquad$
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Context:
If $b \gt a \gt 1$, and $3\log _{a}b+2\log _{b}a=7$, then the minimum value of ${a^2}+\frac{3}{{b-1}}$ is ____.
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Context:
Decrypt the ElGamal encrypted message given the parameters: \( p = 29 \), \( g = 2 \), and a public key of 28. The intercepted encrypted message is: (28, 28), (23, 1), (18, 1), (16, 6), (22, 13). Provide the decrypted message.
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Context:
The Pell numbers $P_n$ satisfy $P_0 = 0$, $P_1 = 1$, and $P_n=2P_{n-1}+P_{n-2}$ for $n\geq 2$. Find $$\sum \limits_{n=1}^{\infty} \left (\tan^{-1}\frac{1}{P_{2n}}+\tan^{-1}\frac{1}{P_{2n+2}}\right )\tan^{-1}\frac{2}{P_{2n+1}}$$
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Context:
Leon has cards with digits from 1 to 7. How many ways are there to glue two three-digit numbers (one card will not be used) so that their product is divisible by 81, and their sum is divisible by 9?
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Context:
If $\cos(α + \frac{π}{3}) = -\frac{\sqrt{3}}{3}$, find the value of $\sin α$.
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Context:
Compute the Galois group of the polynomial $x^4+12x^3+44x^2+48x+40$ over $\mathbb{Q}$, given that its discriminant is $2^{14}3^27^2$.
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Context:
Find the smallest positive integer $k$ such that $z^{10}+z^{9}+z^{6}+z^{5}+z^{4}+z+1$ divides $z^{k}-1$.
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Context:
(6?3) + 4 - (2 - 1) = 5. To make this statement true, the question mark between the 6 and the 3 should be replaced by
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Context:
Given a stochastic variable $x(t)$ with a steady-state Fokker-Planck distribution $P(x)$, where the expectation value for any function $f(x)$ is given by
$$\overline{(f(x))} = \int dx f(x) P(x),$$
and the stochastic differential equation is $\dot{x}(t) = -ax(t) + b\eta(t)$ with $\overline{\eta(t)} = 0$ and $\overline{\eta(t)\eta(t')} = \delta(t-t')$, calculate $\overline{\dot{x}(t)}$ and determine its relationship to $\overline{x(t)}$. What is the value of $\overline{\dot{x}(t)}$?
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Context:
Given that vehicles are not allowed to turn back at a crossroads, calculate the total number of driving routes.
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Context:
Determine whether the coefficient \( m_{a,b,c} \) of the monomial \( x_1^{a+c+1}x_2^{a+b+1}x_3^{b+c+1} \) in the expansion of the polynomial \( f_{a,b,c}(x_1,x_2,x_3) = (x_1-x_2)^{2a+1}(x_2-x_3)^{2b+1}(x_3-x_1)^{2c+1} \) is zero when \( a, b, c \) are pairwise unequal nonnegative integers.
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Context:
Given the set $X=\{1,2,5,7,11,13,16,17\}$, let $x_{i}$, $x_{j}\in X$. If the equation $x_{i}-x_{j}=k\left(k \gt 0\right)$ has at least three different solutions, write down all possible values of $k$ as ______.
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Context:
In the figure, points $A$, $B$, $C$, and $D$ are on the same line and are the centers of four tangent circles at the same point. Segment $AB$ measures $8$, and segment $CD$ measures $4$. The circumferences with centers $A$ and $C$ are of equal size. We know that the sum of the areas of the two medium circles is equivalent to the sum of the areas of the small and large circles. What is the length of segment $AD$?
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Context:
A rectangle with perimeter $176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles?
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Context:
Among the $n$ angles of a convex $n$-gon, $n-1$ angles are equal to $150^\circ$, and the remaining angle is less than $150^\circ$. For which $n$ is this possible? List all possible answers.
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Context:
5. Given the number $800 \ldots 008$ (80 zeros). It is required to replace some two zeros with non-zero digits so that after the replacement, the resulting number is divisible by 198. In how many ways can this be done?
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Context:
An ideal gas is used as the working substance of a heat engine operating cyclically. The cycle consists of three stages: isochoric pressure reduction from $3 P_{0}$ to $P_{0}$, isobaric density increase from $\rho_{0}$ to $3 \rho_{0}$, and a return to the initial state, represented as a quarter circle in the $P / P_{0}, \rho / \rho_{0}$ coordinates with the center at point $(1,1)$. Determine the efficiency of this cycle, knowing that it is 8 times less than the maximum possible efficiency for the same minimum and maximum gas temperatures as in the given cycle.
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Context:
There are four boxes of apples. By weighing three of the boxes at a time, the following four different weights are obtained: $70 \mathrm{~kg}, 80 \mathrm{~kg}, 73 \mathrm{~kg}, 77 \mathrm{~kg}$. The weights of the four boxes of apples are $\qquad$ kg, $\qquad$ kg, $\qquad$ kg, $\qquad$ kg.
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Context:
平面上的n个圆满足：存在平面上的6个点，任一点都至少在这n个圆中的3个圆上．求n的最小值
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Context:
Given an isosceles triangle with an internal angle of 40°, determine the possible degrees of its apex angle. Express your answer in degrees (°), including all possible angles.
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Context:
A prime number is divided by 21 with a remainder. Find all values of the remainder that are composite numbers.
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Context:
In five pots arranged in a row, Rabbit poured three kilograms of honey (not necessarily into each pot and not necessarily evenly). Winnie-the-Pooh can take any two pots that are next to each other. What is the maximum amount of honey that Winnie-the-Pooh can be guaranteed to eat?
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Context:
The original price of a saree is Rs. 600 after successive discounts of 20%, 15%, and 10% are applied. If a new saree with the same original price goes through successive discounts of 20%, 15%, 10%, and an additional discount of 'x%', what is the final sale price of the new saree in terms of 'x'?
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Context:
空间中有9个点，其中任意4个点不共面．在这9点之间连线，要求不产生四面体.最多有几个三角形？
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Context:
When conducting market research, what statistical measure of shoe models should a shoe store focus on in order to increase sales? Provide your answer as a single word or brief phrase.
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Context:
For how many positive integers $n \le 1000$ is
$\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$
not divisible by $3$?
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Context:
$2019$  circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours?
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Context:
What is the value of \[3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}?\] The original answer is in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\).
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Context:
29 函数f定义在有序正整数对的集合上，且满足下列性质：

(1)f(x,x)=x,

(2)f(x,y)=f(y,x)-

(3)(x+y)f(x,y)=yf(x,x+y).

求f(14,52)
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Context:
Determine the greatest positive integer  $ n$  such that in three-dimensional space, there exist n points  $ P_{1},P_{2},\cdots,P_{n},$  among  $ n$  points no three points are collinear, and for arbitary  $ 1\leq i < j < k\leq n$ ,  $ P_{i}P_{j}P_{k}$  isn't obtuse triangle.
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Context:
A paper cup has a base that is a circle with radius $r$, a top that is a circle with radius $2r$, and sides that connect the two circles with straight line segments. This cup has height $h$ and volume $V$. A second cup, exactly the same shape as the first, is held upright inside the first cup so that its base is a distance of $\frac{h}{2}$ from the base of the first cup. The volume of liquid that will fit inside the first cup and outside the second cup can be written as $\frac{m}{n} \cdot V$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
Using equal-length toothpicks to form a rectangular diagram as shown, if the length of the rectangle is 20 toothpicks long and the width is 10 toothpicks long, how many toothpicks are used?
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Context:
一个不均匀的骰子，掷出 $1,2,3,4,5,6$ 点的概率依次成等差数列.独立地先后掷该骰子两次，所得的点数分别记为 $a, b$ .若事件 " $a+b=7$ " 发生的概率为 $\frac{1}{7}$ ，则事件 " $a=b$ "发生的概率为 $\qquad$.  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:
In the diagram, \(P Q R S\) is a rectangle. Also, \(\triangle S T U\), \(\triangle U V W\), and \(\triangle W X R\) are congruent. Calculate the fraction of the area of rectangle \(P Q R S\) that is shaded. Express your answer as a simplified fraction.
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Context:
The polar curve $r = 1 + \cos \theta$ is rotated once around the point with polar coordinates $(2,0).$  What is the area of the region it sweeps over?
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Context:
A social survey agency conducted a monthly income survey on 10,000 residents of a certain area and drew a frequency distribution histogram of the sample based on the collected data (as shown in the diagram). To analyze the relationship between residents' income and various factors such as age, education, and occupation, 100 people need to be further surveyed using the stratified sampling method from these 10,000 individuals. The question is: how many people should be sampled from the monthly income bracket of [¥2500, ¥3000)?
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Context:
Find the dihedral angles of a trihedral angle, whose plane angles are $90^{\circ}, 90^{\circ}$, and $\alpha$.
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Context:
Five square sheets with a side length of 10 centimeters are placed in such a manner that each small square's covered part is a smaller square whose side length is half of the original square's side length. Determine the perimeter of the outer contour (thick lines) in the figure in centimeters.
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Context:
Given the set M={α, β, γ}, where α, β, and γ are complex numbers, if the product of any two numbers in set M and the square of any one number in set M are still elements in M, then set M is equal to \_\_\_\_\_\_.
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Context:
Find all ordered triples of primes  $(p, q, r)$  such that \[ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1.  \] *Reid Barton*
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Context:
In the diagram, the side lengths of four squares are shown. The area of the fifth square is denoted by k. Using the given relationships between the heights of the squares on the left and right sides, find the value of k. Express your answer as a whole number.
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Context:
Given an inscribed quadrilateral $A B C D$. The rays $A B$ and $D C$ intersect at point $K$. It turns out that points $B, D$, as well as the midpoints of segments $A C$ and $K C$, lie on the same circle. What values can the angle $A D C$ take?
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Context:
A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$?
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Context:
The graph of $y=f(x)$ is obtained by shifting the graph of $y=\cos \left(2x+\frac{\pi}{6}\right)$ to the left $\frac{\pi}{6}$ units. How many intersection points does the graph of $y=f(x)$ have with line $y=\frac{1}{2}x-\frac{1}{2}$?
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Context:
Given four positive numbers, the first three form an arithmetic sequence with a sum of 48. The last three form a geometric sequence, with the last number being the maximum value of the function $y=21-4x-x^2$. Find these four numbers.
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Context:
The odd numbers from 5 to 21 are used to build a 3 by 3 magic square. If 5, 9 and 17 are placed as shown, what is the value of $x$?
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Context:
Given a positive integer  $n$ , determine the maximum number of lattice points in the plane a square of side length  $n +\frac{1}{2n+1}$  may cover.
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Context:
Two bicycle enthusiasts are leaving Cincinnati at the same time. One is biking 840 miles to Boston, and the other is biking 440 miles to Atlanta. They plan to travel for seven days, covering equal distances each day. However, the terrain and weather conditions vary along their routes. The biker going to Boston will face 2 days of hilly terrain (decreasing daily mileage by 20%), 2 days of rain (decreasing daily mileage by 10%), and 1 rest day (no mileage). The biker going to Atlanta will face 1 day of hilly terrain (decreasing daily mileage by 20%), 1 day of rain (decreasing daily mileage by 10%), and 1 rest day (no mileage). What is the greatest number of miles a day each can bike considering these factors, if they want to cover equal distances each day?
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Context:
Of the students attending a school athletic event, $80\%$ of the boys were dressed in the school colors, $60\%$ of the girls were dressed in the school colors, and $45\% $ of the students were girls. Find the percentage of students attending the event who were wearing the school colors.
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Context:
Let $n>3$ be a positive integer. Equilateral triangle $ABC$ is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$. Please provide the value of $m-d$ when $n = 5$. The answer should be an integer.
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Context:
In the diagram, \(A B C D\) is a rectangle, \(P\) is on \(B C\), \(Q\) is on \(C D\), and \(R\) is inside \(A B C D\). Also, \(\angle P R Q = 30^\circ\), \(\angle R Q D = w^\circ\), \(\angle P Q C = x^\circ\), \(\angle C P Q = y^\circ\), and \(\angle B P R = z^\circ\). What is the value of \(w + x + y + z\)?
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Context:
Find all numbers between  $100$  and  $999$  which equal the sum of the cubes of their digits.
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Context:
求出具有下述性质的正整数 $n$ : 它被 $\leqslant \sqrt{n}$ 的所有正整数整除.
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Context:
Find all solutions in integers for the equation: \( x^{3} + y^{3} = 2^{30} \).
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Context:
the cricket team of 11 members is 29 yrs old & the wicket keeper is 3 yrs older . if the ages ofthese 2 are excluded , the average age of theremaining players is 1 year less than the average age of the whole team . what is the average age of the team ?
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Context:
In a 4 by 4 grid, each of the 16 small squares measures 3 cm by 3 cm and is shaded. Four unshaded circles are then placed on top of the grid as shown. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value $A+B$?
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Context:
Find the amount of Water formed on combining some moles of Sodium hydroxide and 1 mole of Perchloric acid if 18 grams of Water are formed.
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Context:
Four identical isosceles trapezoids are placed such that their lower bases form the two diagonals of a square. If $PX = 3XQ$, find the ratio of the area of the shaded regions to the area of the entire square.
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Context:
The gardener Fedya has a miracle tree with seven branches in his garden. On each branch, there can either grow 6 apples, 5 pears, or 3 oranges. Fedya discovered that the tree has fruit of all types, with the most pears and the fewest apples.

How many fruits in total grew on the miracle tree?
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Context:
Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS}$?
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Context:
Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{1} + \frac{1}{1} + \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{1} + \frac{1}{3}$.

Find the least positive integer $k$ making the number $k! \cdot S_{2016}$ an integer.
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Context:
Determine whether the set $$ D:= \bigcap_{n \in \mathbb{N}} ]\frac{1}{n},1]$$ has a minimum or infimum.
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Context:
Let $f(x) = (x - 5)(x - 12)$ and $g(x) = (x - 6)(x - 10)$. Find the sum of all integers $n$ such that $\frac{f(g(n))}{f(n)^2}$ is defined and an integer.
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Context:
Charles has $5q + 1$ quarters and Richard has $q + 5$ quarters. The difference in their money in dimes is:
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Context:
Let $\tau = \frac{1 + \sqrt{5}}{2}.$  Find
\[\sum_{n = 0}^\infty \frac{\lfloor \tau^n \rceil}{2^n}.\]Note: For a real number $x,$ $\lfloor x \rceil$ denotes the integer closest to $x.$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 $P(x)=x^{2}-1$. How many distinct real solutions does the following equation have:
$$
\underbrace{P(P(\ldots(P}_{2004}(x)) \ldots))=0?
$$
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Context:
In a school tennis tournament with $m \ge 2$ participants, each match consists of 4 sets. A player who wins more than half of all sets during a match gets 2 points for this match. A player who wins exactly half of all sets during the match gets 1 point, and a player who wins less than half of all sets gets 0 points. During the tournament, each participant plays exactly one match against each remaining player. Find the least number of participants $m$ for which it is possible that some participant wins more sets than any other participant but obtains less points than any other participant.
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Context:
A person walks on a 400-meter circular track at a speed of 6 kilometers per hour. They start from point A and walk clockwise for 1 minute, then counterclockwise for 3 minutes, and clockwise again for 5 minutes. What is the minimum time they need to return to the starting point A? Express your answer in minutes as a whole number.
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Context:
If Bill can buy 3 pairs of jeans, 2 shirts, and 1 hat for $89, or 2 pairs of jeans, 3 shirts, and 2 hats for $102, or 4 pairs of jeans, 1 shirt, and 3 hats for $125, how much does one shirt cost?
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Context:
Vasya cut a triangle out of cardboard and numbered its vertices with the numbers 1, 2, and 3. It turns out that if Vasya rotates his triangle 15 times clockwise around the vertex numbered 1 by an angle equal to the angle at this vertex, the triangle returns to its original position. If Vasya rotates his triangle 6 times clockwise around the vertex numbered 2 by an angle equal to the angle at this vertex, the triangle returns to its original position. Vasya claims that if he rotates his triangle \( n \) times around the vertex numbered 3 by an angle equal to the angle at this vertex, the triangle will return to its original position. What is the minimal \( n \) that Vasya could name so that his statement is true for at least some cardboard triangle?
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Context:
394. The random variable $X$ is given by the probability density function $f(x)=(1 / 2) \sin x$ in the interval $(0, \pi)$; outside this interval, $f(x)=0$. Find the variance of the function $Y=\varphi(X)=X^{2}$, using the density function $g(y)$.
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Context:
一个三位数等于它的个位数字的阶乘之和，则此三位数的个位数字之和为多少？
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Context:
Among the $n$ angles of a convex $n$-gon, $n-1$ angles are equal to $150^{\circ}$, and the remaining angle is less than $150^{\circ}$. For which values of $n$ is this possible? List all possible answers.
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Context:
在平面直角坐标系中, 点 $A, B, C$ 在双曲线 $x y=1$ 上, 满足 $\triangle A B C$ 为等腰直角三角形。求 $\triangle A B C$ 的面积的最小值.
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  $6$ -inch-wide rectangle is rotated  $90$  degrees about one of its corners, sweeping out an area of  $45\pi$  square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches.
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Context:
Given that ${F_1}$ and ${F_2}$ are the left and right foci of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, $P$ is a point on the line $x = \frac{3a}{2}$, and $\Delta F_2 P F_1$ is an isosceles triangle with a base angle of $30^\circ$, calculate the eccentricity of $E$. Express your answer as a single fraction.
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Context:
给定整数 $n \geqslant 2$. 求最小的正实数 $c$, 使得对任意复数 $z_{1}, z_{2}, \cdots, z_{n}$, 均有
$$
\left|\sum_{i=1}^{n} z_{i}\right|+c \sum_{1 \leq i<j \leqslant n}\left|z_{i}-z_{j}\right| \geqslant \sum_{i=1}^{n}\left|z_{i}\right|
$$
若$n=5$，答案写为$\frac{a}{b}$，其中$a,b$互质，求$a+b$.
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Context:
The marked price of a book was 30% less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay?
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Context:
Find the mass percentage of O in Chromic acid.
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Context:
Let  $ f : \mathbf {R} \to \mathbf {R} $  be a smooth function such that  $ f'(x)^2 = f(x) f''(x) $  for all  $x$ . Suppose  $f(0)=1$  and  $f^{(4)} (0) = 9$ . Find all possible values of  $f'(0)$ .
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Context:
Jason and Jared take turns placing blocks within a game board with dimensions  $3 \times 300$ , with Jason going first, such that no two blocks can overlap. The player who cannot place a block within the boundaries loses. Jason can only place  $2 \times 100$  blocks, and Jared can only place  $2 \times n$  blocks where  $n$  is some positive integer greater than 3. Find the smallest integer value of  $n$  that still guarantees Jason a win (given both players are playing optimally).
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Context:
If there is a positive integer \( m \) such that \( m! \) (m factorial) ends with exactly \( n \) zeros, then the positive integer \( n \) is called a "factorial tail number." How many numbers less than 1992 are not "factorial tail numbers"?
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Context:
Let $x_1=1$ and $x_{n+1} =x_n+\left\lfloor \frac{x_n}{n}\right\rfloor +2$, for $n=1,2,3,\ldots $ where $x$ denotes the largest integer not greater than $x$. Determine $x_{1997}$.
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Context:
Determine the smallest natural number that can be represented both as the sum of $2002$ positive integers with the same sum of decimal digits, and as the sum of $2003$ integers with the same sum of decimal digits.
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Context:
In an old script found in the ruins of Persepolis, it is written:

"This script has been finished in a year whose 13th power is 258145266804692077858261512663. If you are skilled in Arithmetic, you will know the year this script is finished easily."

Find the year the script is finished. Provide a reason for your answer.
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Context:
Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?
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Context:
The diagram shows an equilateral triangle inside a rectangle. Calculate the sum of angles x and y. Express your answer in degrees.
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Context:
The cube below has sides of length 5 feet. If a cylindrical section of radius 1 foot is removed from the solid at an angle of $45^\circ$ to the top face, what is the total remaining volume of the cube? Express your answer in cubic feet in terms of $\pi$.
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Context:
Anya, Vanya, Danya, and Tanya collected apples. Each of them collected a whole number percentage from the total number of apples, and all these numbers are distinct and more than zero. Then Tanya, who collected the most apples, ate her apples. After that, each of the remaining kids still had a whole percentage of the remaining apples. What is the minimum number of apples that could have been collected?
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Context:
Given a finite set $S \subset \mathbb{R}^3$, define $f(S)$ to be the minimum integer $k$ such that there exist $k$ planes that divide $\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that
\[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\]
Evaluate $M(200) \cdot m(200)$.
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Context:
It is known that the sum of the digits of number A is 59, and the sum of the digits of number B is 77. What is the minimum sum of the digits that the number A+B can have?
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Context:
Given an acute triangle \(ABC\). The altitude \(AA_1\) is extended through vertex \(A\) by a segment \(AA_2 = BC\). The altitude \(CC_1\) is extended through vertex \(C\) by a segment \(CC_2 = AB\). Find the angles of triangle \(A_2BC_2\).
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Context:
Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$?
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Context:
If one can find a student with at least $k$ friends in any class which has 21 students such that at least two of any three of these students are friends, what is the largest possible value of $k$?
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Context:
There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins that is equal to the difference between the number of stones in the box the stone was put in, and that in the box the stone was taken from (the moved stone does not count). If this difference is negative, then Sisyphus returns the corresponding amount to Zeus (if Sisyphus cannot pay, generous Zeus allows him to make the move and pay later).
After some time all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment?
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Context:
One day, a group of young people came to the Platonic Academy located in the outskirts of Athens. The academy's gate was closed, and above the gate a sign read: "No one ignorant of geometry may enter!" Next to the sign was a diagram with four small rectangles of areas $20, 40, 48, \text{and } 42$ forming a larger rectangle. Find the shaded area to gain entry. Euclid confidently walked up to the gatekeeper, gave a number, and the gatekeeper nodded and opened the academy gate.
What was the answer given by Euclid?
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Context:
Three players play tic-tac-toe together. In other words, the three players take turns placing an "A", "B", and "C", respectively, in one of the free spots of a $3 \times 3$ grid, and the first player to have three of their label in a row, column, or diagonal wins. How many possible final boards are there where the player who goes third wins the game? (Rotations and reflections are considered different boards, but the order of placement does not matter.)
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Context:
Determine the number of cyclic subgroups of order 12 in the group \( \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z} \).
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Context:
In the diagram below, three squares are inscribed in right triangles. Their areas are $A$, $M$, and $N$, as indicated in the diagram. If $M = 5$ and $N = 12$, then $A$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$.
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Context:
A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("flipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it). Thus, Student 3 will close the third locker, open the sixth, close the ninth. . . . Student 5 then goes through and "flips"every 5th locker. This process continues with all students with odd numbers  $n<100$  going through and "flipping" every  $n$ th locker.

How many lockers are open after this process?
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Context:
In the diagram, each of the numbers 0, 1, 2, 3, 4, 5, 6, and 7 is to be used to label a vertex of the cube. The numbers 0, 2, and 3 are placed as shown. The sum of the numbers at the ends of each edge must be a prime number. Find the sum of the values of M, N, P, and Q.
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Context:
In a country there are  $15$  cities, some pairs of which are connected by a single two-way airline of a company. There are  $3$  companies and if any of them cancels all its flights, then it would still be possible to reach every city from every other city using the other two companies. At least how many two-way airlines are there?
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Context:
The integer  $n$  has exactly six positive divisors, and they are:  $1<a<b<c<d<n$ . Let  $k=a-1$ . If the  $k$ -th divisor (according to above ordering) of  $n$  is equal to  $(1+a+b)b$ , find the highest possible value of  $n$ .
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Context:
For a gift to his son, a father bought 4 books. All the books except the first one cost 84 kopecks, without the second one they cost 80 kopecks, without the third one they cost 76 kopecks, and without the fourth one they cost 72 kopecks. What is the cost of each book?
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Context:
In the diagram, \( S \) lies on \( R T \), \( \angle Q T S = 40^{\circ} \), \( Q S = Q T \), and \( \triangle P R S \) is equilateral. The value of \( x \) is
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Context:
Evaluate the limit: $$\lim_{\delta\to 0}\int_{\frac{\delta}{1-\delta}}^{\delta}\frac{e^{-x^2}}{x^2}dx$$
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Context:
Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,150 \pi).$
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Context:
Given subsets $A$ and $B$ of a set $M$, let $S_0 = \{A, B\}$. For $i \geq 0$, define $S_{i+1}$ recursively as the collection of subsets $X$ of $M$ formed by the union, intersection, or complement of pairs $C, D \in S_i$. Denote $S = \cup^\infty_{i=0} S_i$. What is the maximum possible number of elements in $S$?
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Context:
Consider  $13$  marbles that are labeled with positive integers such that the product of all  $13$  integers is  $360$ . Moor randomly picks up  $5$  marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain?
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Context:
Determine the number of homomorphisms from the dihedral group $D_6$ to the dihedral group $D_5$. Provide your answer as a single integer.
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Context:
If there are several heaps of stones on the table, it is said that there are  $\textit{many}$  stones on the table, if we can find  $50$  piles and number them with the numbers from  $1$  to  $50$  so that the first pile contains at least one stone, the second - at least two stones,..., the  $50$ -th has at least  $50$  stones. Let the table be initially contain  $100$  piles of  $100$  stones each. Find the largest  $n \leq 10 000$  such that after removing any  $n$  stones, there will still be  $\textit{many}$  stones left on the table.
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Context:
The measurements of all the edges of a cuboid are whole numbers. The sum of the cuboid's volume, half of its surface area, and the lengths of the edges meeting at one vertex totals 2000. What are the lengths of the edges of the cuboid?
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Context:
Determine the number of ring homomorphisms from the ring of real numbers \( \mathbb{R} \) to itself.
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Context:
Construct the projections of a cylinder that is tangent to two given planes and whose base (circle) is tangent to the projection axis. (How many solutions are possible?)
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Context:
Find the x-coordinate of the center of mass of a half cylinder with radius \( r \).
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Context:
In the figure, a semicircle is folded along the $AN$ string and intersects the $MN$ diameter at $B$. It is given that $MB:BN = 2:3$ and $MN = 10$. If $AN = x$, what is the value of $x^2$?
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Context:
Suppose that 4 letters are chosen at random without replacement from the phrase “MAZAMORRA MORADA.” What is the probability that the four letters chosen can be arranged to spell AMOR? Express your answer as a common fraction.
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Context:
Shelly writes down a vector $v=(a, b, c, d)$, where $0<a<b<c<d$ are integers. Let $\sigma(v)$ denote the set of 24 vectors whose coordinates are $a, b, c$, and $d$ in some order. For instance, $\sigma(v)$ contains $(b, c, d, a)$. Shelly notes that there are 3 vectors in $\sigma(v)$ whose sum is of the form $(s, s, s, s)$ for some $s$. What is the smallest possible value of $d$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The top view of a certain solid is shown as a rectangle in the right figure, the front view (or main view) is an isosceles triangle with a base length of 8 and a height of 5, and the side view (or left view) is an isosceles triangle with a base length of 6 and a height of 5. Then, the volume of this solid is ▲.
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Context:
Calculate the increase in size of a 10mm urchin over 60 days using the Johnson growth model. The differential equation is given by \( \frac{dS}{dt} = k \cdot S_t \cdot dT \cdot (\ln(S_\infty) - \ln(S_t))^2 \), where \( S_t \) is the size on day \( t \), \( k = 2.92 \), and \( S_\infty = 90 \).
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Context:
There are $2022$ equally spaced points on a circular track $\gamma$ of circumference $2022$. The points are labeled $A_1, A_2, \ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$, until she reaches $A_{2022}$, after which she hops back to $A_1$. When hopping from $P$ to $Q$, she always hops along the shorter of the two arcs $\widehat{PQ}$ of $\gamma$; if $\overline{PQ}$ is a diameter of $\gamma$, she moves along either semicircle.

Determine the maximal possible sum of the lengths of the $2022$ arcs which Bunbun traveled, over all possible labellings of the $2022$ points.

[i]Kevin Cong[/i]
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Context:
Find the set of integers \(a\) such that 35 divides \(a^3 - 1\).
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Context:
Given mobile points  $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$  on the  $x$ - $y$  plane.
Denote by  $D$  the part in which line segment  $PQ$  sweeps. Find the volume  $V$  generated by a rotation of  $D$  around the  $x$ -axis.
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Context:
Emma is 7 years old.  If her sister is 9 years older than her, how old will Emma be when her sister is 56?
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Context:
Tam created the mosaic shown using a regular hexagon, squares, and equilateral triangles. If the side length of the hexagon is \( 20 \text{ cm} \), what is the outside perimeter of the mosaic?
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Context:
The radius of the circumscribed circle around a right triangle is \( 15 \) cm, and the radius of the inscribed circle inside it is \( 6 \) cm. Find the sides of the triangle.
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Context:
A mosquito is walking at random on the nonnegative number line. She starts at position 1. When she is at position 0, she always takes a step 1 unit to the right. From any positive position on the line, she randomly moves left or right 1 unit with equal probability. What is the expected number of times the mosquito will visit position 0 before she visits position 4 for the first time?
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Context:
A computer manufacturer produces two electronic components, A and B. The production cost for component A is $80 per unit, and the shipping cost is $2 per unit. Component B has a production cost of $60 per unit and a shipping cost of $3 per unit. The manufacturer has fixed costs of $16,200 per month related to both components, regardless of how many are produced.

If the manufacturer produces and sells 200 components A and 300 components B every month, what is the lowest price it can sell each component for such that the overall costs do not exceed the total revenues?
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Context:
Alison is eating 2401 grains of rice for lunch. She eats the rice in a very peculiar manner: every step, if she has only one grain of rice remaining, she eats it. Otherwise, she finds the smallest positive integer $d>1$ for which she can group the rice into equal groups of size $d$ with none left over. She then groups the rice into groups of size $d$, eats one grain from each group, and puts the rice back into a single pile. How many steps does it take her to finish all her rice?
Please reason step by step, and put your final answer within \boxed{}.