Context:
Find the Cramer-Rao Lower Bound for the variance of any unbiased estimator of \( \log \sigma \), where \( X_1, X_2, \ldots, X_n \) form a random sample from a normal distribution with mean 0 and unknown standard deviation \( \sigma > 0 \).
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Context:
Two knights placed on distinct square of an $8\times8$ chessboard, whose squares are unit squares, are said to attack each other if the distance between the centers of the squares on which the knights lie is $\sqrt{5}.$ In how many ways can two identical knights be placed on distinct squares of an $8\times8$ chessboard such that they do NOT attack each other?
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Context:
Evaluate the sum \( \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{\alpha}} \) and determine its limit as \( \alpha \) approaches \( 0^+ \) by grouping successive terms.
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Context:
As the order of trees tends to infinity, does the ratio of the number of trees of order \(2n\) with a perfect matching to the total number of trees of order \(2n\) converge to zero?
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Context:
Let $P(x) = x^{2013}+4x^{2012}+9x^{2011}+16x^{2010}+\cdots + 4052169x + 4056196 = \sum\limits_{j=1}^{2014}j^2x^{2014-j}.$ If $a_1, a_2, \cdots a_{2013}$ are its roots, then compute the remainder when $a_1^{997}+a_2^{997}+\cdots + a_{2013}^{997}$ is divided by 997.

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Context:
Given that
 \( {34}! = {95232799cd96041408476186096435ab000000}{}_{\left( {10}\right) }, \)
 determine the digits \( a,b,c \) , and \( d \) .
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Context:
Let \( g(x) = C \) be a constant function. Determine whether the Riemann-Stieltjes integral \( \int_a^b f \, dg \) exists for any function \( f \).
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Context:
Determine the convergence or divergence of the improper integral $\int^\infty_1 \frac{e^{1/x}}{x^2} dx$.
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Context:
A circle with radius \( r \) is inscribed in a right trapezoid. Find the sides of the trapezoid if its shorter base is equal to \( \frac{4r}{3} \).
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Context:
In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?
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Context:
Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three aligned numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$?
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Context:
The rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively.  The area of quadrilateral $ABDF$ is
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Context:
What is the difference between the compound interest on Rs. 8000 for 1 1/2 years at a certain interest rate per annum compounded yearly and half-yearly, if the difference is Rs. 3.263999999999214?
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Context:
Let $0 \le a, b, c \le 5$ be integers. For how many ordered triples $(a,b,c)$ is $a^2b+b^2c+c^2a-ab^2-bc^2-ca^2 = 0$?
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Context:
Each cell of a  $29 \times 29$  table contains one of the integers  $1, 2, 3, \ldots , 29$ , and each of these integers appears  $29$  times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.
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Context:
A manufacturer of airplane parts makes a certain engine that has a probability $p$ of failing on any given flight. There are two planes that can be made with this sort of engine, one that has 3 engines and one that has 5. A plane crashes if more than half its engines fail. For what values of $p$ do the two plane models have the same probability of crashing?
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Context:
从1,2,3,4,5,7,9这7个数字中任取两个数字分别做成对数的底数和真数，则可构成不相等的对数值的数目是.
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Context:
Two tourists started from $A$ to $B$ at the same time. The first tourist walked the first half of the time with a speed of $5 \mathrm{km/h}$ and the remaining time with a speed of $4 \mathrm{km/h}$. The second tourist walked the first half of the distance with a speed of $5 \mathrm{km/h}$ and the remaining distance with a speed of $4 \mathrm{km/h}$. Who will arrive at $B$ first?
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Context:
The sum of three positive numbers that form an arithmetic sequence is 15, and these three numbers, when increased by 2, 5, and 13 respectively, form a geometric sequence. Find these three positive numbers.
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Context:
Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \times 2$ rectangle.
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Context:
In the diagram, each of \( \triangle W X Z \) and \( \triangle X Y Z \) is an isosceles right-angled triangle. The length of \( W X \) is \( 6 \sqrt{2} \). The perimeter of quadrilateral \( W X Y Z \) is closest to
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Context:
In figure 1, \(BEC\) is a semicircle and \(F\) is a point on the diameter \(BC\). Given that \(BF : FC = 3 : 1\), \(AB = 8\), and \(AE = 4\), find the length of \(EC\).
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Context:
Through the point with coordinates $(2,2)$, lines (including two parallel to the coordinate axes) are drawn, which divide the plane into angles of $18^{\circ}$. Find the sum of the x-coordinates of the points of intersection of these lines with the line $y = 2016 - x$.
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Context:
Determine all pairs of natural integers \(a, b\) such that \(a^{b} = b^{a}\).
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Context:
平面α,β所成的锐角为40°，过空间中一个定点P作与α，β所成的角都等于80°的平面，这样的平面有且仅有多少个．
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Context:
Several students are competing in a series of three races.  A student earns $5$ points for winning a race, $3$ points for finishing second and $1$ point for finishing third.  There are no ties.  What is the smallest number of points that a student must earn in the three races to be guaranteed of earning more points than any other student?
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Context:
Given $V = \mathbb{C}[x]/(x-2) \oplus \mathbb{C}[x]/(x^2)$, find the minimal and characteristic polynomials of the linear transformation $T: V \to V$ where $T$ is multiplication by $x$.
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Context:
The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden? Provide your answer as an integer.
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Context:
A train that is initially 160 meters long is moving with a speed of 30 m/s. It starts increasing in length at the rate of 2 m/s while it keeps moving. The train crosses a man standing on the platform in certain amount of time. What is the final speed of the train as it finishes crossing the man, if the acceleration due to increase in length is considered? Assume no wind resistance or other external factors.
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Context:
求出所有满足下面要求的不小于 -1 的实数 $t$ ：对任意 $a \in[-2, t]$, 总存在 $b, c \in[-2, t]$, 使得 $a b+c=1$. 请给出满足条件的最小整数。
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Context:
Bob plays a game where, for some number $n$, he chooses a random integer between 0 and $n-1$, inclusive. If Bob plays this game for each of the first four prime numbers, what is the probability that the sum of the numbers he gets is greater than 0?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  $A$  be a subset of  $\{1, 2, 3, \ldots, 50\}$  with the property: for every  $x,y\in A$  with  $x\neq y$ , it holds that
\[\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}.\]
Determine the largest possible number of elements that the set  $A$  can have.
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Context:
The fraction \[\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\] where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$?
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Context:
空间给定9个点，其中任何4点不共面，在9点间连接若干条线段，使图中不存在四面体，问图中最多有多少个三角形？ （1994年中国国家队测验题）
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Context:
In the convex trapezoid $ABCD$, the longer parallel side is fixed. The side $CD$ moves such that neither its length nor the perimeter of the trapezoid changes. What path does the intersection point of the extensions of the non-parallel sides trace?
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Context:
Let  $S$  be the set of integers that represent the number of intersections of some four distinct lines in the plane. List the elements of  $S$  in ascending order.
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Context:
Which numbers have the last two digits that match the last two digits of their square?
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Context:
Let  $P$  be an interior point of triangle  $ABC$ . Let  $a,b,c$  be the sidelengths of triangle  $ABC$  and let  $p$  be it's semiperimeter. Find the maximum possible value of  $$  \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right) $$  taking into consideration all possible choices of triangle  $ABC$  and of point  $P$ .

by Elton Bojaxhiu, Albania
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Context:
The height of an isosceles trapezoid is \( h \). The upper base of the trapezoid is viewed from the midpoint of the lower base at an angle of \( 2\alpha \), and the lower base is viewed from the midpoint of the upper base at an angle of \( 2\beta \). Find the area of the trapezoid in general, and calculate it without tables if \( h = 2 \), \( \alpha = 15^\circ \), and \( \beta = 75^\circ \).
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Context:
In the following diagram, two sides of a square are tangent to a circle with a diameter of $8$. One corner of the square lies on the circle. There are positive integers $m$ and $n$ such that the area of the square is $m + \sqrt{n}$. Find $m + n$.
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Context:
Given a parallelogram with area  $1$  and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the  $8$  lines formed we have  a octagon inside of the parallelogram. Determine the area of this octagon
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Context:
Find all the solutions to
\[3 \sqrt[3]{3x - 2} = x^3 + 2.\]Enter all the solutions, separated by commas.
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Context:
Andrew wants to write the numbers 1, 2, 3, 4, 5, 6, and 7 in the circles in the diagram so that the sum of the three numbers joined by each straight line is the same. What number should he write in the top circle to satisfy this condition? Express your answer as a single integer.
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Context:
Find the least positive integer $n$ such that
$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$
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Context:
Find all prime numbers $p$ that satisfy the following condition: For any prime number $q < p$, if $p = k q + r$ with $0 \leq r < q$, there does not exist an integer $a > 1$ such that $a^2$ divides $r$.
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Context:
Is it possible to append two digits to the right of the number 277 so that the resulting number is divisible by any number from 2 to 12?
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Context:
An artist decided to purchase a canvas for a miniature, which should have an area of 72 cm$^{2}$. To stretch the miniature on a frame, there must be strips of blank canvas 4 cm wide on the top and bottom, and 2 cm wide on the sides.

What are the minimum dimensions of the required canvas?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{10 \, \text{cm} \times 20 \, \text{cm}}


Context:
Taking a box of matches, I found that I could use them to form any pair of the regular polygons shown in the illustration, using all the matches each time. For example, if I had 11 matches, I could form either a triangle and a pentagon, a pentagon and a hexagon, or a square and a triangle (using only 3 matches for the triangle). However, 11 matches cannot form a triangle with a hexagon, a square with a pentagon, or a square with a hexagon. Of course, each side of the shape must use an equal number of matches.

What is the smallest number of matches that could be in my box?
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Context:
(4 points) $f(x)$ is an even function defined on $\mathbb{R}$, and when $x \geq 0$, $f(x) = 2x + 1$. If $f(m) = 5$, then the value of $m$ is ___.
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Context:
Let  $x, y$  and  $z$  be consecutive integers such that
\[\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.\]
Find the maximum value of  $x + y + z$ .
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Context:
满足 $A B=1, B C=2, C D=4, D A=3$ 的凸四边形 $A B C D$ 的内切圆半径的取值范围是 $(a, \frac{k \sqrt{m}}{n}]$。请找出 a, k, m, n 的值并计算 a + k + m + n。
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Context:
The product of all the prime numbers between 1 and 100 is equal to $P$. What is the remainder when $P$ is divided by 16?
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Context:
In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, the value of $y$ is
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Context:
For positive integers $n$ and $k$, let $\mho(n,k)$ be the number of distinct prime divisors of $n$ that are at least $k$. For example, $\mho(90, 3)=2$, since the only prime factors of $90$ that are at least $3$ are $3$ and $5$. Find the closest integer to \[\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.\]
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Context:
In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. What is the area of quadrilateral $ABCD$?
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Context:
What is the Fourier transform of the function $\hat{f}(w)$ defined by $\hat{f}(w) = \int_{-\infty}^{+\infty}\max(t-1,0)e^{-i\omega t}dt$, and under what conditions does it converge?
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Context:
Misha created a homemade dartboard during the summer at his dacha. The circular board is divided into several sectors by circles, and darts can be thrown at it. Points are awarded according to the number written in the sector, as shown in the illustration.

Misha threw 8 darts three separate times. The second time, he scored 2 times more points than the first time, and the third time, he scored 1.5 times more points than the second time. How many points did he score the second time?
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Context:
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. Find the coordinates of $p_{2015}$ in the form $(x, y)$, and provide the sum $x + y$.
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Context:
In a math competition with problems $A$, $B$, and $C$, there are 39 participants, each of whom answered at least one question correctly. Among those who answered problem $A$ correctly, the number of participants who answered only problem $A$ is 5 more than those who also answered other problems. Among those who did not answer problem $A$ correctly, the number of participants who answered problem $B$ is twice the number of those who answered problem $C$. It is also given that the number of participants who answered only problem $A$ is equal to the sum of the participants who answered only problem $B$ and only problem $C$. What is the maximum number of participants who answered problem $A$?
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Context:
1、设X是N＊的子集，X的最小元为1，最大元为100，对X中任何一个大于1的数，都可表成X中两个数（可以相同）的和，求|X|的最小值.
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Context:
1. How many ways can the vertices of a cube be colored red or blue so that the color of each vertex is the color of the majority of the three vertices adjacent to it?
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Context:
Let $A$ be a set of ten distinct positive numbers (not necessarily integers). Determine the maximum possible number of arithmetic progressions consisting of three distinct numbers from the set $A$.
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Context:
In triangle  $ABC$  we have  $|AB| \ne |AC|$ . The bisectors of  $\angle ABC$  and  $\angle  ACB$  meet  $AC$  and  $AB$  at  $E$  and  $F$ , respectively, and intersect at I. If  $|EI| = |FI|$  find the measure of  $\angle  BAC$ .
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Context:
A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?
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Context:
The figure shows the map of an (imaginary) country consisting of five states. The goal is to color this map with green, blue, and yellow in such a way that no two neighboring states share the same color. Calculate the exact number of different ways the map can be painted.
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Context:
An orchard has 50 apple trees and 30 orange trees. Each apple tree can fill 25 baskets with 18 apples each, while each orange tree can fill 15 baskets with 12 oranges each. How many apples and oranges can you get from the entire orchard?
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Context:
Let $\{\omega_1,\omega_2,\cdots,\omega_{100}\}$ be the roots of $\frac{x^{101}-1}{x-1}$ (in some order). Consider the set $$S=\{\omega_1^1,\omega_2^2,\omega_3^3,\cdots,\omega_{100}^{100}\}.$$ Let $M$ be the maximum possible number of unique values in $S,$ and let $N$ be the minimum possible number of unique values in $S.$ Find $M-N.$
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Context:
Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N = 749$, Bernardo writes the numbers $10,444$ and $3,245$, and LeRoy obtains the sum $S = 13,689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$?
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Context:
Find the smallest positive $\alpha$ (in degrees) for which all the numbers $\cos{\alpha}, \cos{2\alpha}, \ldots, \cos{2^n\alpha}, \ldots$ are negative.
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Context:
Let $a, b, c, d$ three strictly positive real numbers such that \[a^{2}+b^{2}+c^{2}=d^{2}+e^{2},\]  \[a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.\] Compare \[a^{3}+b^{3}+c^{3}\] with \[d^{3}+e^{3},\]
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Context:
设S是由一些不大于15的正整数组成的集合，且S中任何两个不相交的子集，各子集的元素和互不相等，求具有这个性质的S中的元素和的最大值.
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Context:
Fifteen square tiles with side 10 units long are arranged as shown. An ant walks along the edges of the tiles, always keeping a black tile on its left. Find the shortest distance that the ant would walk in going from point \( P \) to point \( Q \).
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Context:
若等比数列1，a+bi,b+ai,⋯,(a,b∈R且a<0)则其前12项的和与积＝__
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Context:
$A B C D$ is a regular tetrahedron of volume 1. Maria glues regular tetrahedra $A^{\prime} B C D, A B^{\prime} C D$, $A B C^{\prime} D$, and $A B C D^{\prime}$ to the faces of $A B C D$. What is the volume of the tetrahedron $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$?
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Context:
A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles?
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Context:
The product of three natural numbers is 600. If one of the factors is decreased by 10, the product decreases by 400. If instead that factor is increased by 5, the product doubles. Which three natural numbers have this property?
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Context:
Let $f(x) = x^2-3x$. For what values of $x$ is $f(f(x)) = f(x)$?  Enter all the solutions, separated by commas.
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Context:
Let  $P_1, P_2, \ldots, P_6$  be points in the complex plane, which are also roots of the equation  $x^6+6x^3-216=0$ . Given that  $P_1P_2P_3P_4P_5P_6$  is a convex hexagon, determine the area of this hexagon.
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Context:
Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and
\[
a_{n} = 2 a_{n-1}^2 - 1
\]
for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality
\[
a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}.
\]
What is the value of $100c$, rounded to the nearest integer?
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Context:
Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$. Determine the maximum possible number of real roots of $ P(x)$.
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Context:
What is the probability of having  $2$  adjacent white balls or  $2$  adjacent blue balls in a random arrangement of  $3$  red,  $2$  white and  $2$  blue balls?
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Context:
For any finite set $X$, let $| X |$ denote the number of elements in $X$. Define \[S_n = \sum | A \cap B | ,\] where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\left\{ 1 , 2 , 3, \cdots , n \right\}$ with $|A| = |B|$. For example, $S_2 = 4$ because the sum is taken over the pairs of subsets \[(A, B) \in \left\{ (\emptyset, \emptyset) , ( \{1\} , \{1\} ), ( \{1\} , \{2\} ) , ( \{2\} , \{1\} ) , ( \{2\} , \{2\} ) , ( \{1 , 2\} , \{1 , 2\} ) \right\} ,\] giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4$. Let $\frac{S_{2022}}{S_{2021}} = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by 1000.
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Context:
Doug and Ryan are competing in the 2005 Wiffle Ball Home Run Derby. In each round, each player takes a series of swings. Each swing results in either a home run or an out, and an out ends the series. When Doug swings, the probability that he will hit a home run is $1 / 3$. When Ryan swings, the probability that he will hit a home run is $1 / 2$. In one round, what is the probability that Doug will hit more home runs than Ryan hits?
Please reason step by step, and put your final answer within \boxed{}.


Context:
将4个数1,9,8,8写成一行并进行以下操作：对每一对相邻的数都作一次减法，即用右边的数减左边的数，然后将所得的数写在两数之间，算是完成了一次操作.然后再对这7个数所排成的一行进行同样的操作，如此继续下去，共操作100次，求最后所得到的一行数的和. （第51届莫斯科市奥林匹克试题）
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \[ \text{lcm}(m,n) = 3m \times \gcd(m,n)? \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of real solutions to the equation
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{50}{x - 50} = x + 5.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest whole number that is larger than the sum \[2\dfrac{1}{2}+3\dfrac{1}{3}+4\dfrac{1}{4}+5\dfrac{1}{5}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular hexagon \( K L M N O P \) is inscribed in an equilateral triangle \( A B C \) such that the points \( K, M, O \) lie at the midpoints of the sides \( A B, B C, \) and \( A C \), respectively. Calculate the area of the hexagon \( K L M N O P \) given that the area of triangle \( A B C \) is \( 60 \text{ cm}^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Janet lives in a city built on a grid system.  She walks 3 blocks north, then seven times as many blocks west. Then she turns around and walks 8 blocks south and twice as many blocks east in the direction of her home. If Janet can walk 2 blocks/minute, how long will it take her to get home?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $a,b,c,d$  be positive integers such that the number of pairs  $(x,y) \in (0,1)^2$  such that both  $ax+by$  and  $cx+dy$  are integers is equal with 2004. If  $\gcd (a,c)=6$  find  $\gcd (b,d)$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A triple of integers \((a, b, c)\) satisfies \(a+b c=2017\) and \(b+c a=8\). Find all possible values of \(c\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A two-player game is played on a grid of varying sizes (6x7, 6x8, 7x7, 7x8, and 8x8). The game starts with a piece at the bottom-left corner, and players take turns moving the piece up, right, or diagonally up-right. The game ends when a player cannot make a move. How many of these grid sizes allow the first player to guarantee a win? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all integers $n$, not necessarily positive, for which there exist positive integers $a, b, c$ satisfying $a^{n}+b^{n}=c^{n}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sequence consists of the digits $122333444455555\ldots$ such that the each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the $4501$st and $4052$nd digits of this sequence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all real numbers \( x, y, z \) satisfying \( x + y + z = 2 \), \( x^{2} + y^{2} + z^{2} = 6 \), and \( x^{3} + y^{3} + z^{3} = 8 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Circle \(C_1\) and \(C_2\) each have radius \(1\), and the distance between their centers is \(\frac{1}{2}\). Circle \(C_3\) is the largest circle internally tangent to both \(C_1\) and \(C_2\). Circle \(C_4\) is internally tangent to both \(C_1\) and \(C_2\) and externally tangent to \(C_3\). The radius of \(C_4\) is given in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $C$  be a unit cube and let  $p$  denote the orthogonal projection onto the plane. Find the maximum area of  $p(C)$ .
Please reason step by step, and put your final answer within \boxed{}.

\boxed{The maximum area of p(C) is \(\sqrt{3}\).}


Context:
The Red Sox play the Yankees in a best-of-seven series that ends as soon as one team wins four games. Suppose that the probability that the Red Sox win Game $n$ is $\frac{n-1}{6}$. What is the probability that the Red Sox will win the series?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sequence of positive integers  $a_1,a_2,\ldots $  is such that for each  $m$  and  $n$  the following holds: if  $m$  is a divisor of  $n$  and  $m<n$ , then  $a_m$  is a divisor of  $a_n$  and  $a_m<a_n$ . Find the least possible value of  $a_{2000}$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
We have a calculator with two buttons that displays an integer $x$. Pressing the first button replaces $x$ by $\lfloor \frac{x}{2} \rfloor$, and pressing the second button replaces $x$ by $4x+1$. Initially, the calculator displays $0$. How many integers less than or equal to $2014$ can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y \rfloor$ denotes the greatest integer less than or equal to the real number $y$).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation:
\[
[x+0.1]+[x+0.2]+[x+0.3]+[x+0.4]+[x+0.5]+[x+0.6]+[x+0.7]+[x+0.8]+[x+0.9]=104
\]
where \([x]\) denotes the greatest integer less than or equal to \(x\), find the smallest value of \(x\) that satisfies this equation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Is there a pair of rational numbers \(a\) and \(b\) that satisfies the following equality?

$$
\frac{a+b}{a}+\frac{a}{a+b}=b
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the parallelogram $ABCD$ ($AB \parallel CD$), diagonal $BD = a$, and $O$ is the point where the diagonals intersect. Find the area of the parallelogram, given that $\angle DBA = 45^\circ$ and $\angle AOB = 105^\circ$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A secret facility is in the shape of a rectangle measuring $200 \times 300$ meters. There is a guard at each of the four corners outside the facility. An intruder approached the perimeter of the secret facility from the outside, and all the guards ran towards the intruder by the shortest paths along the external perimeter (while the intruder remained in place). Three guards ran a total of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. Give the answer in the form $r+s\sqrt{t}$ with $r$, $s$, and $t$ positive integers，writing r.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle $AHI$ is equilateral. We know $\overline{BC}$, $\overline{DE}$ and $\overline{FG}$ are all parallel to $\overline{HI}$ and $AB = BD = DF = FH$. What is the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$? Express your answer as a common fraction.

[asy]
unitsize(0.2inch);
defaultpen(linewidth(0.7));
real f(real y)
{
  return (5*sqrt(3)-y)/sqrt(3);
}
draw((-5,0)--(5,0)--(0,5*sqrt(3))--cycle);
draw((-f(5*sqrt(3)/4),5*sqrt(3)/4)--(f(5*sqrt(3)/4),5*sqrt(3)/4));
draw((-f(5*sqrt(3)/2),5*sqrt(3)/2)--(f(5*sqrt(3)/2),5*sqrt(3)/2));
draw((-f(15*sqrt(3)/4),15*sqrt(3)/4)--(f(15*sqrt(3)/4),15*sqrt(3)/4));
label("$A$",(0,5*sqrt(3)),N);
label("$B$",(-f(15*sqrt(3)/4),15*sqrt(3)/4),WNW);
label("$C$",(f(15*sqrt(3)/4),15*sqrt(3)/4),ENE);
label("$D$",(-f(5*sqrt(3)/2),5*sqrt(3)/2),WNW);
label("$E$",(f(5*sqrt(3)/2),5*sqrt(3)/2),ENE);
label("$F$",(-f(5*sqrt(3)/4),5*sqrt(3)/4),WNW);
label("$G$",(f(5*sqrt(3)/4),5*sqrt(3)/4),ENE);
label("$H$",(-5,0),W);
label("$I$",(5,0),E);[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
$S$ is the sum of the first 14 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{9} a_{17} > S + 12$ and $a_{11} a_{15} < S + 47$. Indicate all possible values of $a_{1}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a natural number  $n \ge 3$ , we draw  $n - 3$  internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the  $n$ -gon into  $n - 2$  triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of  $n$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sequence  $ (a_n)$  satisfies  $ a_0 \equal{} 0$  and  $ \displaystyle a_{n \plus{} 1} \equal{} \frac85a_n \plus{} \frac65\sqrt {4^n \minus{} a_n^2}$  for  $ n\ge0$ . Find the greatest integer less than or equal to  $ a_{10}$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
The numbers \(a, b, c\) satisfy the equations

\[
a b + a + b = c, \quad b c + b + c = a, \quad c a + c + a = b
\]

Find all possible values of \(a\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Determine the age that is not one of Mr. Jones's children. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are  $n\leq 99$  people around a circular table. At every moment everyone can either be truthful (always says the truth) or a liar (always lies). Initially some of people (possibly none) are truthful and the rest are liars. At every minute everyone answers at the same time the question "Is your left neighbour truthful or a liar?" and then becomes the same type of person as his answer. Determine the largest  $n$  for which, no matter who are the truthful people in the beginning, at some point everyone will become truthful forever.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Carl only eats food in the shape of equilateral pentagons. Unfortunately, for dinner he receives a piece of steak in the shape of an equilateral triangle. So that he can eat it, he cuts off two corners with straight cuts to form an equilateral pentagon. The set of possible perimeters of the pentagon he obtains is exactly the interval $[a, b)$, where $a$ and $b$ are positive real numbers. Compute $\frac{a}{b}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The figure below was created by joining together 5 non-overlapping congruent squares. The total area of the figure is 45. Determine the perimeter of the figure.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$Q$ is the point of intersection of the diagonals of one face of a cube whose edges have length 2 units. The length of $Q R$ is
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given points $P = (1: -2: 3), Q = (2: 2: -1)$ and $R = (3: 0: 2)$ on a line $g$ in $\mathbb{P}^{2}\mathbb{R}$, find a coordinate for $g$ such that $(P, Q, R)$ has coordinates $(0, 1, ∞)$. How many possibilities are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In three out of six circles on the diagram, the numbers 4, 14, and 6 are written. In how many ways can the remaining three circles be filled with natural numbers such that the product of the three numbers along each side of the triangular diagram is the same?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Al told Bob that he was thinking of $2011$ distinct positive integers. He also told Bob the sum of those $2011$ distinct positive integers. From this information, Bob was able to determine all $2011$ integers. How many possible sums could Al have told Bob?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two bored millionaires, Bilion and Trilion, decide to play a game. They each have a sufficient supply of  $\$  1, \ $ 2,\$  5 $, and $ \ $ 10$  bills. Starting with Bilion, they take turns putting one of the bills they have into a pile. The game ends when the bills in the pile total exactly  $\$ 1{,}000{,}000 $, and whoever makes the last move wins the $ \ $1{,}000{,}000$  in the pile (if the pile is worth more than  $\$ 1{,}000{,}000$ after a move, then the person who made the last move loses instead, and the other person wins the amount of cash in the pile). Assuming optimal play, how many dollars will the winning player gain?

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


Context:
At a school, after classes, someone broke a mirror in the vestibule. The last to leave were 9 students. The school principal asked each of them, "Who broke the mirror?" The answers were as follows:

- Roland: "It was Percy."
- Alex: "No, it wasn't Percy."
- Ted: "I broke the mirror."
- Fred: "It was either Ted or Walter."
- Percy: "Alex is lying."
- Tom: "Ted is guilty."
- John: "Ted is not guilty."
- Walter: "Neither I nor Ted broke the mirror."
- Charlie: "Walter is telling the truth. Percy is also not guilty."

The students later admitted that only three of their statements were true. Who broke the mirror?
Please reason step by step, and put your final answer within \boxed{}.


Context:
6. A uniformly growing uniform grassland is divided into two unequal parts, $\mathrm{A}$ and $\mathrm{B}$. A herd of cows first eats the grass on plot A, finishing it in exactly 7 days. Then the herd continues to eat the grass on plot B at the same speed, finishing it in 4 days, at which point plot A has just recovered to the state it was in when the cows started eating. What is the simplest integer ratio of the area of plot A to plot B? $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
An ancient British merchant had a 15-pound weight, which shattered into 4 pieces after falling to the ground. Each piece weighed an integer number of pounds, and it was possible to use these 4 pieces to measure any integer weight from 1 to 15 pounds (weights could only be placed on one side of the balance). What are the weights of the 4 pieces?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways every unit square of a  $2018$  x  $2018$  board can be colored in red or white such that number of red unit squares in any two rows are distinct and number of red squares in any two columns are distinct.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $i$ is the imaginary unit, determine in which quadrant the complex conjugate of the complex number $i(1+i)$ is located in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle ABC of side length $3$. The perimeter of the remaining quadrilateral is
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the picture, there is a grid consisting of 25 small equilateral triangles. How many rhombuses can be made from two adjacent small triangles?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all real numbers \(a\) for which there exist three distinct real numbers \(x, y, z\) such that \(a = x + \frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
the length of a rectangle is increased by 40 % while its width is halved . what is the % change in area ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three equal circles of radius \( R \) touch each other externally. Find the sides and angles of the triangle whose vertices are the points of tangency.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x$ be a complex number such that $x+x^{-1}$ is a root of the polynomial $p(t)=t^{3}+t^{2}-2 t-1$. Find all possible values of $x^{7}+x^{-7}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the generalized Monty Hall problem, there are $n$ doors with $t$ cars hidden behind them ($t < n$). A contestant initially selects a door. The host, who knows the locations of the cars, opens $s$ doors revealing goats. Under what condition should the contestant switch their choice to maximize the chance of winning a car? Specifically, find the minimum $s$ such that switching always increases the probability of winning.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of squares in the sequence given by  $ a_0\equal{}91$  and  $ a_{n\plus{}1}\equal{}10a_n\plus{}(\minus{}1)^n$  for  $ n \ge 0.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all solutions to the equation $$(z^2 - 3z + 1)^2 - 3(z^2 - 3z + 1) + 1 = z.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Line segments PQ and RS are parallel. Points T, U, and V are placed so that ∠QTV = 30°, ∠SUV = 40°, and ∠TVU = x°, as shown. Find the value of x. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In convex quadrilateral $ABCD$, $\angle BAD = \angle BCD = 90^o$, and $BC = CD$. Let $E$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Given that $\angle AED = 123^o$, find the degree measure of $\angle ABD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the diagram, \( KL \) is parallel to \( MN \), \( AB = BC \), and \(\angle KAC = 50^\circ\). Find the value of \( x \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. She starts with a triangle $A_{0} A_{1} A_{2}$ where angle $A_{0}$ is $90^{\circ}$, angle $A_{1}$ is $60^{\circ}$, and $A_{0} A_{1}$ is 1. She then extends the pasture. First, she extends $A_{2} A_{0}$ to $A_{3}$ such that $A_{3} A_{0}=\frac{1}{2} A_{2} A_{0}$ and the new pasture is triangle $A_{1} A_{2} A_{3}$. Next, she extends $A_{3} A_{1}$ to $A_{4}$ such that $A_{4} A_{1}=\frac{1}{6} A_{3} A_{1}$. She continues, each time extending $A_{n} A_{n-2}$ to $A_{n+1}$ such that $A_{n+1} A_{n-2}=\frac{1}{2^{n}-2} A_{n} A_{n-2}$. What is the smallest $K$ such that her pasture never exceeds an area of $K$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle $ABC$ has side lengths $AB=18$, $BC=36$, and $CA=24$. The circle $\Gamma$ passes through point $C$ and is tangent to segment $AB$ at point $A$.

Let $X$, distinct from $C$, be the second intersection of $\Gamma$ with $BC$. Moreover, let $Y$ be the point on $\Gamma$ such that segment $AY$ is an angle bisector of $\angle XAC$.

Suppose the length of segment $AY$ can be written in the form $AY=\frac{p\sqrt{r}}{q}$ where $p$, $q$, and $r$ are positive integers such that $\gcd(p, q)=1$ and $r$ is square free.

Find the value of $p+q+r$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在平面直角坐标系 $x O y$ 中, 圆 $C_{1}: x^{2}+y^{2}-a=0$ 关于直线 $l$ 对称的圆为 $C_{2}: x^{2}+y^{2}+2 x-2 a y+3=0$, 请给出直线 $l$ 的方程中系数的和。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an $8 \times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e. it cannot stay in the same square). What is the expected number of minutes until the rook reaches the upper right corner?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation of the circle $(x-1)^2 + y^2 = 1$, identify the center $(a, b)$ and radius $r$. Express your answer as an ordered pair for the center and a single value for the radius.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the diagram, each of the two circles has center \(O\). Also, \(O P: P Q = 1:2\). If the radius of the larger circle is 9, what is the area of the shaded region?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A grasshopper is sitting on the number line at the number 0. Each second, it jumps one unit to the left or to the right, with equal probability. The directions of the jumps are chosen independently of each other. Let \( p \) denote the probability that, after 2022 jumps, the grasshopper is sitting at a number divisible by 5. Determine whether \( p < \frac{1}{5} \), \( p = \frac{1}{5} \), or \( p > \frac{1}{5} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Which East Asian country was supposed to host the 1980 International Mathematical Olympiad (IMO) but was canceled due to internal strife in its host country?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two lines $l_{1}: ax+(a+2)y+1=0$ and $l_{2}: x+ay+2=0$. If $l_{1} \perpendicular l_{2}$, then find the value(s) of the real number $a$. Express your answer as a numerical value or a set of numerical values.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the real and imaginary parts of $(1 + i)^{19}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the diagram, semicircles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, such that they are mutually tangent. Point $C$ is at one third the distance from $A$ to $B$, so $\overline{AC} = \frac{1}{3} \overline{AB}$ and $\overline{CB} = \frac{2}{3} \overline{AB}$. If $\overline{CD} \perp \overline{AB}$, find the ratio of the shaded area to the area of a circle with $\overline{CD}$ as radius.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest  $\ell > 0$  such that there exists a square of side  $\ell$  contained in a cube with edge  $1$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T_1$ be a triangle with side lengths $2011$, $2012$, 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:
Three members of the Euclid Middle School girls' softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all $2$-digit primes.
Bethany : And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.
Ashley: And the sum of your two uniform numbers is today's date.
What number does Caitlin wear?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A plane contains points $A$ and $B$ with $AB = 1$.  Let $S$ be the union of all disks of radius 1 in the plane that cover $\overline{AB}$. What is the area of $S$? Express your answer in terms of $\pi$ and in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Number  $125$  is written as the sum of several pairwise distinct and relatively prime numbers, greater than  $1$ . What is the maximal possible number of terms in this sum?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The diagram shows two semicircles with a common center \( O \) and a rectangle \( OABC \). The line through \( O \) and \( C \) meets the small semicircle at \( D \) and the large semicircle at \( E \). The lengths \( CD \) and \( CE \) are equal. What fraction of the large semicircle is shaded?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $O$  and  $I$  be the circumcenter and incenter of triangle  $ABC$ . The perpendicular from  $I$  to  $OI$  meets  $AB$  and the external bisector of angle  $C$  at points  $X$  and  $Y$  respectively. In what ratio does  $I$  divide the segment  $XY$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $ a $  ,  $ b $  ,  $ c $  ,  $ d $  ,  $ (a + b + c + 18 + d) $  ,  $ (a + b + c + 18 - d) $  ,  $ (b + c) $  , and  $ (c + d) $  be distinct prime numbers such that  $ a + b + c = 2010 $ ,  $ a $ ,  $ b $ ,  $  c $ ,  $ d \neq 3 $  , and  $ d \le 50 $ . Find the maximum value of the diﬀerence between two of these prime numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Complete the conversion between the following number systems: $101101_{(2)} = \_\_\_\_\_\_\_\_\_\_\_\_{(10)}\_\_\_\_\_\_\_\_\_\_\_\_{(7)}$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle  $ABC$ ,  $\angle ABC$  is obtuse.  Point  $D$  lies on side  $AC$  such that  $\angle ABD$  is right, and point  $E$  lies on side  $AC$  between  $A$  and  $D$  such that  $BD$  bisects  $\angle EBC$ .  Find  $CE$  given that  $AC=35$ ,  $BC=7$ , and  $BE=5$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$, where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does not count as positive.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In chemical laboratories, the floors are covered with tiles made of acid-resistant ceramics, which have the shape of regular polygons. Which polygons can be used to completely cover the floor without gaps?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each cell of a $29 \times 29$ table contains one of the integers $1, 2, 3, \ldots, 29$, and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A B C$ be a triangle with $A B=13, B C=14$, and $C A=15$. We construct isosceles right triangle $A C D$ with $\angle A D C=90^{\circ}$, where $D, B$ are on the same side of line $A C$, and let lines $A D$ and $C B$ meet at $F$. Similarly, we construct isosceles right triangle $B C E$ with $\angle B E C=90^{\circ}$, where $E, A$ are on the same side of line $B C$, and let lines $B E$ and $C A$ meet at $G$. Find $\cos \angle A G F$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are eleven positive integers $n$ such that there exists a convex polygon with $n$ sides whose angles, in degrees, are unequal integers that are in arithmetic progression. Find the sum of these values of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\theta=\frac{2\pi}{2015}$, and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form $\frac{b}{2^a}$, where $a$ is a non-negative integer and $b$ is an odd integer (not necessarily positive). Find $a+b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a $6 \times 6$ grid of squares. Edmond chooses four of these squares uniformly at random. What is the probability that the centers of these four squares form a square?
Please reason step by step, and put your final answer within \boxed{}.