Context:
Let $\{N(t): t \geq 0\}$ be a Poisson process with rate $1$, and let $T_1 < T_2 < \cdots$ denote the arrival times. Determine the probability density function of the random variable $Y = \frac{T_1}{T_3}$. Provide your answer in terms of $y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest integer $n$ such that the group of integers modulo $n$ has a proper subgroup of order $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( f(x) = \frac{\sqrt{2-e^{2x}}\sqrt[4]{2-e^{4x}}\cdot\ldots\cdot\sqrt[50]{2-e^{50x}}}{(2-e^x)\sqrt[3]{2-e^{3x}}\cdot\ldots\cdot\sqrt[99]{2-e^{99x}}} \), find the derivative \( f'(0) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the complex matrix \( z = \left| \begin{array}{ccc} 3 + 2i & 1 & i \\ 2 & 3 - 2i & 1 + i \\ 1 - i & -i & 3 \end{array} \right| \), and the condition \( |z + \overline{z}| = k|z| \), find the value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of triples of natural numbers \((a, b, c)\) that satisfy the following system of equations:

\[
\left\{\begin{array}{l}
\gcd(a, b, c) = 33, \\
\operatorname{lcm}(a, b, c) = 3^{19} \cdot 11^{15}.
\end{array}\right.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of \( \sqrt[n^2]{a_n} \) given that \( a_2>0,\;a_3>0 ,\; a_{n+2}a_n=(1+1/n)^n a_{n+1}^2, n\ge2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a chess marathon with 2000 players, each pair of players plays exactly one game against each other. For every pair of players, at least one of them plays no more than 30 games during the marathon. What is the maximum number of games that can be played in this marathon?
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Context:
If \( \lim_{x\to 0}\frac{1}{x^3}\left(\frac{1}{\sqrt{1+x}}-\frac{1+ax}{1+bx}\right)=l \), find the value of \( \frac{1}{a}-\frac{2}{l}+\frac{3}{b} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any set  $A = \{x_1, x_2, x_3, x_4, x_5\}$  of five distinct positive integers denote by  $S_A$  the sum of its elements, and denote by  $T_A$  the number of triples  $(i, j, k)$  with  $1 \le i < j < k \le 5$  for which  $x_i + x_j + x_k$  divides  $S_A$ .
Find the largest possible value of  $T_A$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $p\ge 3$  be a prime number. Each side of a triangle is divided into  $p$  equal parts, and we draw a line from each division point to the opposite vertex. Find the maximum number of regions, every two of them disjoint, that are formed inside the triangle.
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Context:
设整数 $n>4,(x+2 \sqrt{y}-1)^{n}$ 的展开式中 $x^{n-4}$ 与 $x y$ 两项的系数相等, 则 $n$ 的值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. (25 points) Six resistors with resistances $R_{1}=1$ Ohm, $R_{2}=2$ Ohm, $R_{3}=3$ Ohm, $R_{4}=4$ Ohm, $R_{5}=5$ Ohm, and $R_{6}=6$ Ohm are connected in series and form a loop. A source of constant voltage is connected to the resulting circuit such that the resistance between its terminals is maximized. The voltage of the source is $U=12 \mathrm{~V}$. Find the power $P_{3}$ dissipated on the resistor $R_{3}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The square of a three-digit number ends with three identical digits different from zero. Write the smallest such three-digit number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the line $l$: $x-2y+2=0$ passes through the left focus $F\_1$ and one vertex $B$ of an ellipse. Determine the eccentricity of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the shopper saves $\textdollar{3.75}$ when buying a coat on sale and the final price paid for the coat was $\textdollar{50}$, determine the percentage that the shopper saved.
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Context:
Find the number of ordered pairs of integers $(a,b) \in \{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a quadratic residue modulo $x^2+1$ and $35$. This means there exists a polynomial $f(x)$ with integer coefficients such that either of the following equivalent conditions holds:

- There exist polynomials $P(x)$, $Q(x)$ with integer coefficients such that $f(x)^2 - (ax+b) = (x^2+1)P(x) + 35Q(x)$.
- The remainder when the polynomial $f(x)^2 - (ax+b)$ is divided by the polynomial $x^2+1$ is a polynomial with integer coefficients, all divisible by $35$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A student is rearranging all his stamps into a new album. If he pastes 20 stamps per page, the album will be insufficient. If he pastes 23 stamps per page, at least one page will remain empty. If he receives an additional identical album, each page containing 21 stamps, he will have a total of 500 stamps. How many pages are in the album?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a certain university, there are 5 indicators to be allocated to 3 senior classes, with each class receiving at least one indicator. How many different allocation methods are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the power function $f(x) = (m^2 - m - 1)x^{1-m}$ is symmetric about the y-axis, find the value of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There is a greatest number which, when dividing 1642 and another number, leaves remainders 6 and 4 respectively. The greatest number is 4. What is the second number?
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Context:
Given a function $f(x)$ defined on $\mathbb{R}$ that satisfies $f(1)=1$, and for any $x$, $f'(x) < \frac{1}{2}$ always holds, then the solution set of the inequality $f(\lg^2x) < \frac{\lg^2x}{2}+ \frac{1}{2}$ is __________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two functions $f(x) = e^{2x-3}$ and $g(x) = \frac{1}{4} + \ln \frac{x}{2}$, if $f(m) = g(n)$ holds, calculate the minimum value of $n-m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Berlov S.L.

A convex 2011-gon is drawn on the board. Petya sequentially draws diagonals in it such that each newly drawn diagonal intersects no more than one of the previously drawn diagonals at an internal point. What is the maximum number of diagonals Petya can draw?
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Context:
What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?
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Context:
Given the function $f(x)=(ax-1)(x-b)$, if the solution set of the inequality $f(x) > 0$ is $(-1,3)$, then the solution set of the inequality $f(-x) < 0$ is _______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any positive integer  $n$  , let  $s(n)$  denote the number of ordered pairs  $(x,y)$  of positive integers for which  $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$  . Determine the set of positive integers for which  $s(n) = 5$
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\{p^2 \mid p \text{ is a prime number}\}}


Context:
In how many ways can 10 people, including 3 sets of clones (2 clones in each set), be arranged in a row such that no clone is adjacent to another clone from the same set?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let a, b, c, d be rational numbers with a > 0. If for every integer n >= 0, the number an^3 + bn^2 + cn + d is also an integer, then find the minimal value of a. Express your answer as a rational number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim _{x\to \infty }\frac{\left(x!\right)^3\left(3\left(x+1\right)\right)!}{\left(\left(x+1\right)!\right)^3\left(3x\right)!}$$
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Context:
The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence of the series:

$$\sum_{n=1}^\infty \left[\frac {z(z+n)}{n}\right]^{n}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Positive integers  $ a$ ,  $ b$ ,  $ c$ , and  $ d$  satisfy  $ a > b > c > d$ ,  $ a \plus{} b \plus{} c \plus{} d \equal{} 2010$ , and  $ a^2 \minus{} b^2 \plus{} c^2 \minus{} d^2 \equal{} 2010$ . Find the number of possible values of  $ a$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square $S$ with side length $2n$ units is divided into $4n^2$ unit squares by drawing $2n-1$ horizontal and $2n-1$ vertical lines, each one unit apart. A circle with diameter $2n-1$ is inscribed in $S$, with its center at the intersection of the diagonals. Determine the number of unit squares that have at least a portion of their boundary intersected by the circle's circumference.
Please reason step by step, and put your final answer within \boxed{}.


Context:
All of the square-shaped tiles are lined up to make a larger square. If the number of tiles on the perimeter is 52, find the number of tiles on the outermost of one side.
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Context:
Given any $n$ points on a plane $p_{1}, p_{2}, \cdots, p_{n}$, where no three points are collinear, and each point $p_{i}(i=1,2,\cdots,n)$ is colored either red or blue. Let $S$ be a set of some triangles with vertex sets $\left\{p_{1}, p_{2}, \cdots, p_{n}\right\}$, having the property that for any two line segments $p_{i} p_{j}$ and $p_{h} p_{k}$, the number of triangles in $S$ with $p_{i} p_{j}$ as a side is equal to the number of triangles with $p_{h} p_{k}$ as a side. Find the smallest positive integer $n$ such that there are always two triangles in the set $S$ where each triangle's three vertices are the same color.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a,$ $b,$ and $c$ be positive real numbers such that $a > b$ and $a + b + c = 4.$  Find the minimum value of
\[4a + 3b + \frac{c^3}{(a - b)b}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest possible number of digits in a natural number \( N \) such that its decimal representation consists only of the digits 1 and 2, and by removing digits from this number, any of the 10,000 numbers consisting of 9999 ones and one two can be obtained.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A  $13$ -foot tall extraterrestrial is standing on a very small spherical planet with radius  $156$  feet. It sees an ant crawling along the horizon. If the ant circles the extraterrestrial once, always staying on the horizon, how far will it travel (in feet)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The negation of the proposition "The cube of every even number is even" is __________.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{there exists an even number whose cube is odd}


Context:
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. Find the probability that the dart lands within the center square. The original answer is in the form \(\frac{\sqrt{k} - 1}{m}\). Please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all positive integers  $a$  for which there exist exactly  $2014$  positive integers  $b$  such that  $\displaystyle2\leq\frac{a}{b}\leq5$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cylindrical drinking cup is made from a rectangular piece of paper with length \( L \) and width \( W \) by rolling the paper and joining the edges. Determine the ratio \( \frac{L}{W} \) that maximizes the volume of the resulting cup.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine $x^2+y^2+z^2+w^2$ if $\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$ $\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$ $\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$ $\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A fair coin is flipped indefinitely. What is the probability that a run of three consecutive tails occurs before a run of four consecutive heads?
Please reason step by step, and put your final answer within \boxed{}.


Context:
将1、2、3、4、5、6、7、8分别放在正方体的八个顶点上，使得每一个面上的任意三个数之和均不小于10．求每一面上四个数之和的最小值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of possible values of \( x \) that satisfy the equation \( \frac{x}{100} = \sin(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a certain DNA fragment consists of 500 base pairs, with A+T making up 34% of the total number of bases, calculate the total number of free cytosine deoxyribonucleotide molecules required when this DNA fragment is replicated twice.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangle of paper of $3$ cm by $9$ cm is folded along a straight line, making two opposite vertices coincide. In this way a pentagon is formed. Calculate it's area.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A best-of-9 series is to be played between two teams; that is, the first team to win 5 games is the winner. The Mathletes have a chance of  $\tfrac{2}{3}$  of winning any given game. What is the probability that exactly 7 games will need to be played to determine a winner?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $-1$, $0$, or $1$. How many distinct lines pass through at least two members of $S$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive real number $r$, let $M_r$ be the regular surface $M_r = \{(x, y, z) | x^2 + y^2 = z < r^2, x > 0, y > 0\}$. Let $K$ denote the Gaussian curvature of $M_r$. Determine $\int_{M_r} K dA$ and $\lim_{r \rightarrow \infty} \int_{M_r} K dA$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $Q$  be a  $(2n+1) \times (2n+1)$  board. Some of its cells are colored black in such a way that every  $2 \times 2$  board of  $Q$  has at most  $2$  black cells. Find the maximum amount of black cells that the board may have.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate:
$$
\left(10^{4}-9^{4}+8^{4}-7^{4}+\cdots+2^{4}-1^{4}\right)+\left(10^{2}+9^{2}+5 \times 8^{2}+5 \times 7^{2}+9 \times 6^{2}+9 \times 5^{2}+13 \times 4^{2}+13 \times 3^{2}\right) =
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
As shown in the figure, the line segment $AB$ of length 9 is divided into nine equal parts. What is the total length of all the line segments in the figure?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that $P(z), Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z)=R(z)$. What is the minimum possible value of $N$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many distinct sets of  $5$  distinct positive integers  $A$  satisfy the property that for any positive integer  $x\le 29$ , a subset of  $A$  sums to  $x$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum

$$
\sum_{k=1}^{\infty}\left(\prod_{i=1}^{k} \frac{P_{i}-1}{P_{i+1}}\right)=\frac{1}{3}+\frac{1}{3} \cdot \frac{2}{5}+\frac{1}{3} \cdot \frac{2}{5} \cdot \frac{4}{7}+\frac{1}{3} \cdot \frac{2}{5} \cdot \frac{4}{7} \cdot \frac{6}{11}+\ldots
$$

where $P_{n}$ denotes the $n^{\text {th }}$ prime number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the molecular weight of some moles of N2O3 if the molecular weight of the given moles is 304?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of all "auspicious numbers" with up to 3 digits, where a positive integer \( n \) with \( n \) digits is called an "auspicious number" if, when appended to the end of any two positive integers, the product of these two new numbers ends in \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The side lengths  $a,b,c$  of a triangle  $ABC$  are positive integers. Let:

\[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\]

for any positive integer  $n$ . If  $\frac{T_{2}}{2T_{1}}=2023$  and  $a>b>c$  , determine all possible perimeters of the triangle  $ABC$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
To fold a paper airplane, Austin starts with a square paper  $F OLD$  with side length  $2$ . First, he  folds corners  $L$  and  $D$  to the square’s center. Then, he folds corner  $F$  to corner  $O$ . What is the longest distance between two corners of the resulting figure?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_n$ be a sequence of random variables defined as $X_n=\sum_{i=1}^{n}U_i$, where $U_i$ are independent random variables such that $P(U_i=1)=P(U_i=-1)=\frac{1}{2}$. For $m\geq n$, find the variance of $E(U_n|X_m)$. \( \text{Express your answer as a single number.} \)
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Context:
A convex pentagon  $ ABCDE$  is inscribed in a circle. The distances of  $ A$  from the lines  $ BC,CD,DE$  are  $ a,b,c,$  respectively. Compute the distance of  $ A$  from the line  $ BE$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
The one hundred U.S. Senators are standing in a line in alphabetical order. Each senator either always tells the truth or always lies. The $i$th person in line says:

"Of the $101-i$ people who are not ahead of me in line (including myself), more than half of them are truth-tellers."

How many possibilities are there for the set of truth-tellers on the U.S. Senate?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the value of \[2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?\]
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Context:
Let  $ABCDEF$  be a regular hexagon with side length 10 inscribed in a circle  $\omega$ .  $X$ ,  $Y$ , and  $Z$  are points on  $\omega$  such that  $X$  is on minor arc  $AB$ ,  $Y$  is on minor arc  $CD$ , and  $Z$  is on minor arc  $EF$ , where  $X$  may coincide with  $A$  or  $B$  (and similarly for  $Y$  and  $Z$ ). Compute the square of the smallest possible area of  $XYZ$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A trapezoid \(ABCD\) (\(AD \parallel BC\)) and a rectangle \(A_{1}B_{1}C_{1}D_{1}\) are inscribed in a circle \(\Omega\) with a radius of 13 such that \(AC \parallel B_{1}D_{1}\) and \(BD \parallel A_{1}C_{1}\). Find the ratio of the areas of \(ABCD\) and \(A_{1}B_{1}C_{1}D_{1}\), given that \(AD = 24\) and \(BC = 10\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the behavior of the sequence \( S_n = \frac{p_n-1}{\phi(p_n-1)} \) as \( n \) approaches infinity, where \( p_n \) is the \( n \)th prime number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x\]?
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Context:
设A={1,2,3,4,5,6},B={7,8,9,⋯,n}，在A中任取3个数，在B中取2个数，组成含有5个元素的集合$A _ { i } ( i = 1 , 2 , 3 , \cdots , 2 0 )$，使得$\vert A _ { i } \cap A _ { j } \vert \leq 2 , 1 \leq i < j \leq 2 0 .$求n的最小值. （2002年IMO中国国家集训队选拨考试试题）
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Context:
One country has  $n$  cities and every two of them are linked by a railroad. A railway worker should travel by train exactly once through the entire railroad system (reaching each city exactly once). If it is impossible for worker to travel by train between two cities, he can travel by plane. What is the minimal number of flights that the worker will have to use?
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Context:
There is a  $2n\times 2n$  rectangular grid and a chair in each cell of the grid. Now, there are  $2n^2$  pairs of couple are going to take seats. Define the distance of a pair of couple to be the sum of column difference and row difference between them. For example, if a pair of couple seating at  $(3,3)$  and  $(2,5)$  respectively, then the distance between them is  $|3-2|+|3-5|=3$ . Moreover, define the total distance to be the sum of the distance in each pair. Find the maximal total distance among all possibilities.
Please reason step by step, and put your final answer within \boxed{}.


Context:
学校水果店里有苹果、香蕉、石榴、橘子、葡萄、西梅6种水果，西梅数量不多，只够一人购买.甲、乙、丙、丁4位同学前去购买，每人只选择其中一种，这4位同学购买后，恰好买了其中3种水果，则他们购买水果的可能情况有________种.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the transition probability matrix \( P = \begin{bmatrix} 1 & 0.1 & 0.8 \\ 0 & 0.7 & 0.2 \\ 0 & 0.2 & 0 \end{bmatrix} \) and the initial state \( s_0 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \), calculate the expected number of steps needed for absorption.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be an annulus defined as $A=\{z:r<|z|<R\}$, where $r<1<R$. If $f \in H(A)$ and its Laurent series is $f(z) = \sum_{n=-\infty}^\infty a_n z^n$, find the condition on the coefficients $a_n$ such that there exists a function $g \in H(A)$ with $f=g'$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $|x|<1$, then what is the coefficient of $x^3$ in the expansion of $\dfrac{1}{e^x\cdot (1+x)}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the remainder when the number \( S = \sum_{i=0}^{99} (n+i)^6 + 2^{2^{2558}} + 1 \) is divided by 100, where \( n \) is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(ABCD\) be a cyclic quadrilateral. The side lengths of \(ABCD\) are distinct integers less than \(15\) such that \(BC\cdot CD=AB\cdot DA\). Find the largest possible value of \(BD\) in the form \(\sqrt{\frac{x}{y}}\). Please provide the value of \(x + y\).
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Context:
Given that the solution set of the inequality $\frac{x+1}{x-3}\leqslant 0$ is $A$, and the solution set of the inequality ${x^{2}}-2ax-3{{a}^{2}}\leqslant 0$ is $B$, and $B$ is a subset of $A$, find the range of values for $a$.
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Context:
How many different five-colored flags with five vertical stripes of equal width can be made if 11 different colors of fabric are available? (The flag here is considered simply as a piece of cloth, not attached to a pole or anything else.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. Find the number of pairs of integers $(x ; y)$ that satisfy the condition $5 x^{2}-6 x y+y^{2}=6^{100}$.
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Context:
Evaluate the limit: $$\lim_{x\to0}\frac{\sin^2(\sin{x})-x^2\sin{x}}{x^4}$$
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Context:
Given that line $l\_1$ passes through points $A(m,1)$ and $B(-3,4)$, and line $l\_2$ passes through points $C(1,m)$ and $D(-1,m+1)$, find the values of the real number $m$ when $l\_1$ is parallel to $l\_2$ or $l\_1$ is perpendicular to $l\_2$.
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Context:
If $\triangle PQR$ is right-angled at $P$ with $PR=12$, $SQ=11$, and $SR=13$, what is the perimeter of $\triangle QRS$?
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Context:
Let $x, y, z, w \in [0,1]$. Find the maximum value of $S=x^{2} y+y^{2} z+z^{2} w+w^{2} x-x y^{2}-$ $y z^{2}-z w^{2}-w x^{2}$.
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Context:
A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and turns yellow when in the sun. Initially, the ratio of green to yellow frogs was $3 : 1$. Then $3$ green frogs moved to the sunny side and $5$ yellow frogs moved to the shady side. Now the ratio is $4 : 1$. What is the difference between the number of green frogs and the number of yellow frogs now?
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Context:
How many roots does the equation \(\sin x - \log_{10} x = 0\) have?
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Context:
Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.
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Context:
Determine the minimum girth of a $k$-regular graph with $v_G > 2k$ vertices that ensures the graph is not Hamiltonian.
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Context:
4.4. Two identical cylindrical vessels are connected at the bottom by a small-bore pipe with a valve. While the valve was closed, water was poured into the first vessel, and oil into the second, so that the level of the liquids was the same and equal to \( h = 40 \) cm. At what level will the water in the first vessel stabilize if the valve is opened? The density of water is 1000 kg \(/ \mathrm{m}^{3}\), and the density of oil is 700 kg \(/ \mathrm{m}^{3}\). Neglect the volume of the connecting pipe. Provide the answer in centimeters.
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Context:
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\]
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Context:
Jordan has a new hit song on Spotify. Some months are left in the year, and she currently has 60,000 listens. If the number of listens per month doubles, the song will have 900,000 listens by the end of the year. How many months are left in the year?
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Context:
A chord of a circle is at a distance $h$ from the center. In each of the segments subtended by the chord, a square is inscribed such that two adjacent vertices of the square lie on the arc and the other two on the chord or its extension. What is the difference in the side lengths of these squares?
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Context:
In base- $2$  notation, digits are  $0$  and  $1$  only and the places go up in powers of  $-2$ . For example,  $11011$  stands for  $(-2)^4+(-2)^3+(-2)^1+(-2)^0$  and equals number  $7$  in base  $10$ . If the decimal number  $2019$  is expressed in base  $-2$  how many non-zero digits does it contain ?
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Context:
Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select $4$ points on the segment $P_0P_5$, and immediately after selecting those points, we label those $4$ selected points $P_1, P_2, P_3, P_4$ from left to right. Similarly, we randomly select $4$ points on the segment $Q_0Q_5$, and immediately after selecting those points, we label those $4$ points $Q_1, Q_2, Q_3, Q_4$ from left to right. The segments $P_1Q_1, P_2Q_2, P_3Q_3, P_4Q_4$ divide the rectangular chocolate bar into $5$ smaller trapezoidal pieces of chocolate. The probability that exactly $3$ pieces of chocolate contain both dark and white chocolate can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[Diagram in the individuals file for this exam on the Chmmc website]
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Context:
If the solution set of the inequality $ax^2+bx+a^2-1\leq0$ with respect to $x$ is $[-1,+\infty)$, then the values of the real numbers $a$ and $b$ are respectively.
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Context:
Consider the set \( \left\{ 1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{64} \right\} \). Define an operation that replaces two elements \( a \) and \( b \) with \( \frac{2ab}{a+b} \). If, after a number of steps, all elements in the set become equal, what is the value of each element?
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Context:
Determine the condition on the value of 'a' such that the characteristic polynomial \(x^3 - 3x + a\) has three distinct real roots.
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Context:
Find the remainder when \( \sum_{n=1}^{2024^{2024}} f(n) \) is divided by 1000, where \( f(n) \) denotes the number of divisors of \( 2024^{2024} \) that are either less than \( n \) or share at least one prime factor with \( n \).
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Context:
A net of a cube is shown with one integer on each face. A larger cube is constructed using 27 copies of this cube. What is the minimum possible sum of all of the integers showing on the six faces of the larger cube?
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Context:
Tim is participating in the following three math contests.  On each contest his score is the number of correct answers.
     $\bullet$  The Local Area Inspirational Math Exam consists of 15 problems.
     $\bullet$  The Further Away Regional Math League has 10 problems.
     $\bullet$  The Distance-Optimized Math Open has 50 problems.
For every positive integer  $n$ , Tim knows the answer to the  $n$ th problems on each contest (which are pairwise distinct), if they exist; however, these answers have been randomly permuted so that he does not know which answer corresponds to which contest.  Unaware of the shuffling, he competes with his modified answers.  Compute the expected value of the sum of his scores on all three contests.
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Context:
For each permutation $(a_1,a_2,\dots,a_{11})$ of the numbers $1,2,3,4,5,6,7,8,9,10,11$, we can determine at least $k$ of $a_i$s when we get $(a_1+a_3, a_2+a_4,a_3+a_5,\dots,a_8+a_{10},a_9+a_{11})$. Determine the maximum possible value of $k$. Express your answer as a single integer.
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Context:
Determine the interval of convergence for the power series \( \sum_{n=0}^{\infty}\left(\frac{1}{\sqrt{3n}}-\frac{1}{n!}\right)(x+2)^n \).
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Context:
Approximate the probability that a simple symmetric random walk \( \{X_n\} \) with \( p = 1/2 \) satisfies \( X_n \in [-3\sqrt{n}, 2\sqrt{n}] \) for large \( n \). Use the central limit theorem to express this probability in terms of a standard normal distribution.
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Context:
Points  $A$ ,  $B$ ,  $C$ , and  $D$  lie on a circle such that chords  $\overline{AC}$  and  $\overline{BD}$  intersect at a point  $E$  inside the circle.  Suppose that  $\angle ADE =\angle CBE = 75^\circ$ ,  $BE=4$ , and  $DE=8$ .  The value of  $AB^2$  can be written in the form  $a+b\sqrt{c}$  for positive integers  $a$ ,  $b$ , and  $c$  such that  $c$  is not divisible by the square of any prime. Find  $a+b+c$ .

*Proposed by Tony Kim*
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Context:
A rectangular plot $ABCD$ is divided into two rectangles as shown in the diagram and is contracted to two households, Jia and Yi. Jia's vegetable greenhouse has the same area as Yi’s chicken farm, while the remaining part of Jia's area is 96 acres more than Yi's. Given that $BF = 3 CF$, what is the total area of the rectangular plot $ABCD$ in acres?
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Context:
From a $6 \times 6$ squared grid, gray triangles were cut out. What is the area of the remaining figure? The side length of each cell is 1 cm. Provide the answer in square centimeters.
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Context:
Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then compute the polynomial $f(x)=x^{3}+p x^{2}+q x+r$ with roots $a, b, c$. If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\frac{m}{n}$ for relatively prime positive integers $m, n$, find the remainder when $m+n$ is divided by 1000 .
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Context:
Suppose that the series \(\sum a_n x^n\) and \(\sum b_n x^n\) have radii of convergence of 1 and 2, respectively. Determine the radius of convergence of the series \(\sum a_n b_n x^n\).
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Context:
Determine the value of \( \mu \) that minimizes the expected cost of fines and cleanup for a bottle filling process. A bottle is considered underfilled if it contains less than 500 ml and overfilled if it contains more than 560 ml. The purchaser fines the supplier \( \$1 \) per underfilled bottle, and it costs the manufacturer \( \$2 \) to clean up an overfilled bottle. Given that the variance \( \sigma^2 = 20^2 \), the expected cost \( E[X] \) is given by:
\[
E[X] = \phi\left(\frac{500-\mu}{20}\right) + 2\left[1-\phi\left(\frac{560-\mu}{20}\right)\right]
\]
Find \( \mu \) such that the derivative of \( E[X] \) with respect to \( \mu \) is zero:
\[
0 = -\frac{1}{20} \phi\left(\frac{500-\mu}{20}\right) + \frac{2}{20}\phi\left(\frac{560-\mu}{20}\right)
\]
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Context:
On a large one-way street, there are two traffic lights one after the other; each traffic light is set up so that the time interval when it shows a green light is two-thirds of the total operating time of the traffic light. A motorist noticed that when he drives through the first traffic light during the green light at his usual speed, in 3 out of 4 cases, the second traffic light does not stop him. Suppose now that the motorist passes the first traffic light during the red light. What is the probability that the second traffic light will also be red?
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Context:
Quadrilateral  $ABCD$  has both an inscribed and a circumscribed circle and sidelengths  $BC = 4, CD = 5, DA = 6$ . Find the area of  $ABCD$ .
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Context:
Determine the cardinality of the set \( E = \{a: \exists m \text{ such that } a_n = s_n \forall n \ge m \} \), where \( s = (s_1, s_2, \ldots) \) is an infinite binary word with \( s_p = 1 \) if \( p \) is a prime number.
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Context:
Seven students count from 1 to 1000 as follows: Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000. Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers. Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers. Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers. Finally, George says the only number that no one else says. What number does George say?
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Context:
In the triangle \( \triangle ABC \), \( \angle BAC = 50^\circ \), and \( BE = BD \), \( CF = CD \). Find the measure of \( \angle EDF \) in degrees.
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Context:
The center of the circle that touches the side \(BC\) of triangle \(ABC\) at point \(B\) and passes through point \(A\) lies on segment \(AC\). Find the area of triangle \(ABC\), given that \(BC = 6\) and \(AC = 9\).
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Context:
Write either  $1$  or  $-1$  in each of the cells of a  $(2n) \times (2n)$ -table, in such a way that there are exactly  $2n^2$  entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be  $M$ . Determine the largest possible value of  $M$ .
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Context:
A rectangular piece of paper $A B C D$ is folded and flattened as shown in the diagram, so that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ precisely landing on side $A B$. Given $\angle 1 = 22^\circ$, find the measure of $\angle 2$.
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Context:
Let  $a$  and  $b$  be two positive reals such that the following inequality
    \[ ax^3 + by^2 \geq xy - 1 \] is satisfied for any positive reals  $x, y \geq 1$ . Determine the smallest possible value of  $a^2 + b$ .
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Context:
\[
\frac{\sin ^{2}\left(135^{\circ}-\alpha\right)-\sin ^{2}\left(210^{\circ}-\alpha\right)-\sin 195^{\circ} \cos \left(165^{\circ}-2 \alpha\right)}{\cos ^{2}\left(225^{\circ}+\alpha\right)-\cos ^{2}\left(210^{\circ}-\alpha\right)+\sin 15^{\circ} \sin \left(75^{\circ}-2 \alpha\right)}=-1
\]
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Context:
Calculate the surface area of the region of the plane given by \( z = 1 + x + 2y \) that lies inside the cone defined by \( z = \sqrt{2x^2 + 10y^2} \).
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Context:
Four coplanar regular polygons share a common vertex but have no interior points in common. Each polygon is adjacent to two of the other polygons, and each pair of adjacent polygons has a common side length of $1$. How many possible perimeters are there for all such configurations? Provide your answer as an integer.
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Context:
in measuring the sides of a rectangle , one side is taken 6 % in excess , and the other 5 % in deficit . find the error percent in the area calculated from these measurements .
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Context:
Consider the sequence $\{a_n\}$ defined by $a_1 = 1$, $a_2 = \frac{1}{2}$, and for any positive integer $n$, \(a_{n+2} = a_{n+1} - \frac{a_n a_{n+1}}{2}.\) Determine whether the limit $\lim_{n\to\infty} na_n$ exists, and if it does, find its value.
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Context:
Let $R$ be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$?
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Context:
Find 8 positive integers \( n_1, n_2, \ldots, n_8 \) such that we can express every integer \( n \) with \( |n| < 1986 \) as \( a_1 n_1 + \cdots + a_8 n_8 \) with each \( a_i = 0, \pm 1 \).
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Context:
Flat Albert and his buddy Mike are watching the game on Sunday afternoon. Albert is drinking lemonade from a two-dimensional cup which is an isosceles triangle whose height and base measure 9 cm and 6 cm; the opening of the cup corresponds to the base, which points upwards. Every minute after the game begins, the following takes place: if $n$ minutes have elapsed, Albert stirs his drink vigorously and takes a sip of height $\frac{1}{n^{2}} \mathrm{~cm}$. Shortly afterwards, while Albert is busy watching the game, Mike adds cranberry juice to the cup until it's once again full in an attempt to create Mike's cranberry lemonade. Albert takes sips precisely every minute, and his first sip is exactly one minute after the game begins. After an infinite amount of time, let $A$ denote the amount of cranberry juice that has been poured (in square centimeters). Find the integer nearest $\frac{27}{\pi^{2}} A$.
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Context:
Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial  $f(x)$  with the following properties:

(a) for any integer  $n$ ,  $f(n)$  is an integer;
(b) the degree of  $f(x)$  is less than  $187$ .

Alice knows that  $f(x)$  satisfies (a) and (b), but she does not know  $f(x)$ . In every turn, Alice picks a number  $k$  from the set  $\{1,2,\ldots,187\}$ , and Bob will tell Alice the value of  $f(k)$ . Find the smallest positive integer  $N$  so that Alice always knows for sure the parity of  $f(0)$  within  $N$  turns.

*Proposed by YaWNeeT*
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Context:
Juan rolls a fair regular octahedral die marked with the numbers 1 through 8. Amal now rolls a fair twelve-sided die marked with the numbers 1 through 12. What is the probability that the product of the two rolls is a multiple of 4?
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Context:
Find the values of $p$ for which the limit \( \lim_{(x, y) \to (0, 0)} \frac{|x| \cdot |y|^p}{x^2 + y^2} \) exists.
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Context:
Given $\displaystyle Q_r(\theta) = \frac{r \sin \theta}{1 - 2r\cos \theta + r^2}$, which is the imaginary part of $\displaystyle\frac{1}{1-re^{i\theta}}$, evaluate the integral $\frac{1}{2\pi}\int_{-\pi}^{\pi}Q_r^2(\theta) \, d\theta$.
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Context:
2018 little ducklings numbered 1 through 2018 are standing in a line, with each holding a slip of paper with a nonnegative number on it; it is given that ducklings 1 and 2018 have the number zero.  At some point, ducklings 2 through 2017 change their number to equal the average of the numbers of the ducklings to their left and right.  Suppose the new numbers on the ducklings sum to 1000.  What is the maximum possible sum of the original numbers on all 2018 slips?
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Context:
In the expansion of the binomial ${(\sqrt{x}-\frac{1}{{2x}}})^n$, only the coefficient of the 4th term is the largest. The constant term in the expansion is ______.
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Context:
Consider all acute-angled triangles with a given side \( a \) and angle \( \alpha \).

What is the maximum of the sum of the squares of the lengths of sides \( b \) and \( c \)?
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Context:
Find a multiple of 23, which when divided by 1821 and 24 leaves remainders 710 and 13 respectively. The least such multiple is 3024. What is the divisor we are finding the multiple of?
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Context:
Evaluate the expression \(\frac{29\int_0^1 (1-x^4)^7\,dx}{4\int_0^1 (1-x^4)^6\,dx}\) without expanding the integrals.
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Context:
Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors. Suppose $M$ can be written as $\prod_{k=1}^{n}p_k^{a_k}$, where the $p_k$'s are distinct primes and the $a_k$'s are positive integers. Find $\sum_{k=1}^{n}(p_k + a_k)$.
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Context:
Compute the limit \( \lim\limits_{s\rightarrow0^{+}}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{s}} \).
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Context:
A square has coordinates at $(0, 0)$, $(4, 0)$, $(0, 4)$, and $(4, 4)$. Rohith is interested in circles of radius $ r$ centered at the point $(1, 2)$. There is a range of radii $a < r < b$ where Rohith’s circle intersects the square at exactly $6$ points, where $a$ and $b$ are positive real numbers. Then $b - a$ can be written in the form $m +\sqrt{n}$, where $m$ and $n$ are integers. Compute $m + n$.
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Context:
What is the smallest value of \( n \) such that the symmetric group \( S_n \) contains an even permutation of order 312? Given that there exists an 8-cycle, a 3-cycle, and a 13-cycle in the group.
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Context:
Given that $θ$ is an angle in the second quadrant, and $\sin θ$ and $\cos θ$ are the two roots of the equation $2x^2 + (\sqrt{3} - 1)x + m = 0$ (where $m \in \mathbb{R}$), find the value of $\sin θ - \cos θ$.
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Context:
Determine the values of $p$ for which the series \( \sum\limits_{n=1}^\infty\left(\dfrac1{n^p}\sum\limits_{k=1}^nk^a\right) \) converges, given that \( a > 0 \).
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Context:
Determine whether the sequence \(\sqrt[n]{n!}\) converges or diverges.
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Context:
Find an integer $n$ such that the decimal representation of the number $5^{n}$ contains at least 1968 consecutive zeros.
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Context:
Find a polynomial in \(\Bbb{Q}[x]\) that is irreducible over \(\Bbb{Q}\), has at least one linear factor over \(\Bbb{R}\), and at least one irreducible quadratic factor over \(\Bbb{R}\).
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Context:
Determine the values of \( p \in \mathbb{R}_{\geq 0} \) for which the integral \( \int_0^\infty x\sin(x^p) \, dx \) converges.
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Context:
The trapezoid $ABCD$ satisfies $AB \parallel CD$, $AB = 70$, $AD = 32$ and $BC = 49$. We also know that $\angle ABC = 3 \angle ADC$. How long is the base $CD$?
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Context:
Let \( r=\sqrt{\frac{\sqrt{53}}{2}+\frac{3}{2}} \). There is a unique triple of positive integers \((a, b, c)\) such that
\[
r^{100}=2 r^{98}+14 r^{96}+11 r^{94}-r^{50}+a r^{46}+b r^{44}+c r^{40}
\]
What is the value of \( a^{2}+b^{2}+c^{2} \)?
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Context:
Last year, Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?
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Context:
The number $2017$ is prime. Let $S=\sum\limits_{k=0}^{62}\binom{2014}{k}$. What is the remainder when $S$ is divided by $2017$
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Context:
Determine the cardinality of the set $S$ of all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x) \neq 0$ for only finitely many $x \in \mathbb{R}$.
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Context:
There are 100 countries participating in an olympiad. Each country is willing to communicate in exactly \( n \) languages. If each set of 20 countries can communicate in at least one common language, and no language is common to all 100 countries, what is the minimum possible value of \( n \)?
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Context:
Each time you click a toggle switch, the switch either turns from *off* to *on* or from *on* to *off*. Suppose that you start with three toggle switches with one of them *on* and two of them *off*. On each move, you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive integers such that $\frac{m}{n}$ is the probability that after four such clicks, one switch will be *on* and two of them will be *off*. Find $m+n$.
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Context:
Find the limit of the sequence \((a_n)_{n\geq1}\), where \(a_n=\frac{S_n}{n^\alpha}\) and \(S_n=\sum_{k=1}^{n^2} [\sqrt{k}]\), with \(\alpha\in\mathbb{R}\).
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Context:
On the side \(CD\) of trapezoid \(ABCD\) (where \(AD \parallel BC\)), point \(M\) is marked. A perpendicular \(AH\) is dropped from vertex \(A\) to segment \(BM\). It turns out that \(AD = HD\). Find the length of segment \(AD\) if it is known that \(BC = 16\), \(CM = 8\), and \(MD = 9\).
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Context:
1. On a $5 \times 7$ grid, 9 cells are marked. We will call a pair of cells with a common side interesting if exactly one cell in the pair is marked. What is the maximum number of interesting pairs that can be?
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Context:
Given the function f(x) = ln(x - 1/2ax^2 - x), if x = 1 is the extreme point of f(x), find the value of a. Express your answer as a single numerical value.
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Context:
On my sister's birthday, she was 126 cm in height. She had grown a certain percentage since the year before when she was 119.7 cm tall. What was the growth percentage?
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Context:
The numbers \(a, b, c, d\) belong to the interval \([-10.5, 10.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\).
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Context:
Find the largest integer $k(k \geq 2)$, for which there exists an integer $n(n \geq k)$ such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions:

1. each chosen number is not divisible by 6, by 7 and by 8;
2. the positive difference of any two different chosen numbers is not divisible by at least one of the numbers 6, 7 or 8.
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Context:
The cross-section of a regular hexagonal pyramid SABCDEF is formed by a plane passing through vertex $C$ of the base $ABCDEF$ and parallel to the median $BM$ of the lateral face $SAB$ and the apothem $SN$ of the lateral face $SAF$. The side of the base of the pyramid is 2, and the distance from vertex $S$ to the cutting plane is 1. Find the cosine of the angle between the plane of the cross-section and the plane of the base.
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Context:
Evaluate the integral \(\int_\Gamma \frac{2z+j}{z^3(z^2+1)}\,\mathrm{d}z\) where \(\Gamma\) is the positively oriented circle \(|z-1-i| = 2\). Use residue theory to find the value of the integral.
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Context:
Determine the conditions on \(\lambda\) such that the operator \(T: C([0,1]) \to C([0,1])\) defined by \((Tx)(t) = x(0) + \lambda \int_0^t x(\tau) \, d\tau\) is a contraction in the closed ball of radius \(\frac{1}{4}\) in \(C([0, 1])\).
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Context:
A function $S(m, n)$ satisfies the initial conditions $S(1, n) = n$, $S(m, 1) = 1$, and the recurrence $S(m, n) = S(m - 1, n)S(m, n - 1)$ for $m\geq 2, n\geq 2$. Find the largest integer $k$ such that $2^k$ divides $S(7, 7)$.
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Context:
Let $\|\cdot\|$ be a norm over $\mathbb{R}^2$ with $\|e_1\|=\|e_2\|=1$, where $e_1, e_2$ are the standard basis of $\mathbb{R}^2$. If $\|e_1+e_2\|=2$, find the range of possible values for $\|e_1-e_2\|$.
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Context:
Let  $G$  be the centroid of triangle  $ABC$  with  $AB=13,BC=14,CA=15$ . Calculate the sum of the distances from  $G$  to the three sides of the triangle.

Note: The *centroid* of a triangle is the point that lies on each of the three line segments between a vertex and the midpoint of its opposite side.

*2019 CCA Math Bonanza Individual Round #11*
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Context:
Determine the number of distinct ways to paint two identical regular tetrahedrons using the colors red, blue, green, and yellow, with each face of a tetrahedron painted a different color. Consider two colorings to be the same if one can be obtained from the other by rotation or reflection.
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Context:
11. (China Mathematical Olympiad) Let $S=\{1,2,3, \cdots, 98\}$, find the smallest natural number $n$, such that in any $n$-element subset of $S$, one can always select 10 numbers, and no matter how these 10 numbers are divided into two groups, there is always a number in one group that is coprime with the other four numbers, and a number in the other group that is not coprime with the other four numbers.
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Context:
Consider the integer   $$ N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}. $$   Find the sum of the digits of  $N$ .
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Context:
The Fibonacci sequence is defined as
$$
a_0 = 0, \; a_1 = a_2 = 1, \; a_{n+1} = a_n + a_{n-1} \; (n \geq 1).
$$

Find the greatest common divisor of the 1960th term and the 1988th term.
(Note: This is from the 29th International Mathematical Olympiad, 1988)
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Context:
Determine the first 1963 digits after the decimal point in the decimal expansion of \((\sqrt{26} + 5)^{1963}\).
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Context:
Grandma has 8 grandchildren and 4 different types of popsicles: 6 Vanilla, 6 Strawberry, 5 Banana, and 3 Chocolate. Each grandchild requests one popsicle of a particular flavor. What is the total number of different sets of requests that Grandma can fulfill?
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Context:
A line \( l \) passes through the point \( P(3,0) \) with slope \( k \) and intersects the right branch of the hyperbola \( C: x^{2}-\frac{y^{2}}{3}=1 \) at points \( A \) and \( B \). The point \( F \) is the right focus of the hyperbola \( C \). Given that \( |AF| + |BF| = 16 \), find the value of \( k \).
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Context:
Consider a $4 \times 4$ grid of squares, each of which are originally colored red. Every minute, Piet can jump on one of the squares, changing the color of it and any adjacent squares (two squares are adjacent if they share a side) to blue. What is the minimum number of minutes it will take Piet to change the entire grid to blue?
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Context:
Sara makes a staircase out of toothpicks as shown:
[asy] size(150); defaultpen(linewidth(0.8)); path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45); for(int i=0;i<=2;i=i+1) { for(int j=0;j<=3-i;j=j+1) { filldraw(shift((i,j))*h,black); filldraw(shift((j,i))*v,black); } } [/asy]
This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?
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Context:
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions: (a) $f(1)=1$ (b) $f(a) \leq f(b)$ whenever $a$ and $b$ are positive integers with $a \leq b$. (c) $f(2a)=f(a)+1$ for all positive integers $a$. How many possible values can the 2014-tuple $(f(1), f(2), \ldots, f(2014))$ take?
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Context:
Given an ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ with left and right foci $F_{1}$ and $F_{2}$, point $P$ is a point on the ellipse, and line segment $F_{1}P$ intersects the $y$-axis at point $Q$. If $|PQ|=2|QF_{1}|$ and triangle $\triangle PF_{1}F_{2}$ is an isosceles triangle, then the eccentricity of the ellipse is ____.
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Context:
Calculate the rate of mass flow through the sphere defined by \( \rho = 1 \) in the outward direction, given the fluid density \( f(x, y, z) = x^{2} + 2y^{2} + z^{2} \) and velocity \( v(x, y, z) = x^{2}i + xy^{2}j + zk \).
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Context:
Find the limit of the Binomial probability distribution function evaluated at \(x = k\) as \(n\) approaches infinity, given that \(p = \sqrt{\lambda/n}\). Express your answer in terms of \(\lambda\) and \(k\).
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Context:
Evaluate the double integral: $$\int_{-\pi/2}^{\pi/2} \int_{2\cos\theta}^{2}2r\sqrt{4-r^2} \,dr\,d\theta$$
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Context:
The areas of the triangles formed by the diagonals of a trapezoid and its bases are $S_{1}$ and $S_{2}$.

Find the area of the trapezoid.
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Context:
Suppose $m$ and $n$ are relatively prime positive integers. A regular $m$-gon and a regular
$n$-gon are inscribed in a circle. Let $d$ be the minimum distance in degrees (of the arc along
the circle) between a vertex of the $m$-gon and a vertex of the $n$-gon. What is the maximum
possible value of $d$?
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Context:
Determine the number of positive integers \( n < 10^4 \) such that \( 2^n - n^2 \) is divisible by 7.
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Context:
Consider two independent, identically distributed (IID) standard normal random variables $X$ and $Y$. Given that $E(X|X+Y>0) = \frac{1}{\sqrt{\pi}}$, calculate $E(XY|X+Y>0)$.
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Context:
Given two sets of points \(A = \left\{(x, y) \mid (x-3)^{2}+(y-4)^{2} \leqslant \left(\frac{5}{2}\right)^{2}\right\}\) and \(B = \left\{(x, y) \mid (x-4)^{2}+(y-5)^{2} > \left(\frac{5}{2}\right)^{2}\right\}\), the number of lattice points (i.e., points with integer coordinates) in the set \(A \cap B\) is ...
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Context:
In the game of *Winners Make Zeros*, a pair of positive integers \((m,n)\) is written on a sheet of paper. Then the game begins, as the players make the following legal moves:

- If \(m \geq n\), the player chooses a positive integer \(c\) such that \(m-cn \geq 0\), and replaces \((m,n)\) with \((m-cn,n)\).
- If \(m < n\), the player chooses a positive integer \(c\) such that \(n-cm \geq 0\), and replaces \((m,n)\) with \((m,n-cm)\).

When \(m\) or \(n\) becomes \(0\), the game ends, and the last player to have moved is declared the winner. If \(m\) and \(n\) are originally \(2007777\) and \(2007\), find the largest choice the first player can make for \(c\) (on his first move) such that the first player has a winning strategy after that first move.
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Context:
Given that $\phi=\frac{1+\sqrt{5}}{2}$. Let $$n=\frac{1}{1}+\frac{1}{1+\phi}+\frac{1}{1+\phi+\phi^2}+\frac{1}{1+\phi+\phi^2+\phi^3}+\dots$$ Find the value of $\lfloor2n\rfloor+\lceil2n\rceil$.
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Context:
If 45% of z is 120% of y, y is 75% of the difference between x and the half of w, and z is 80% of 150% of w, what percent of x is the product of w and y minus one-third of z?
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Context:
Let $X, Y, Z$ be independent and identically distributed random variables, each following a standard normal distribution, $\mathcal{N}(0,1)$. Given that $X > Z$, find the conditional probability $\mathbb{P}(Y > Z \mid X > Z)$. Express your answer as a numerical value.
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Context:
Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!"

Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I'd know for certain what it is?"

Cat says, "Nope! However, if I now told you the units digit of my favorite number, you'd know which one it is!"

Claire says, "Now I know your favorite number!" What is Cat's favorite number?
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Context:
Max and Minnie each add up sets of three-digit positive integers. Each of them adds three different three-digit integers whose nine digits are all different. Max creates the largest possible sum. Minnie creates the smallest possible sum. What is the difference between Max's sum and Minnie's sum?
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Context:
Given the parametric equations \( x = \cos^{-1} (8t^4 - 8t^2 + 1) \) and \( y = \sin^{-1} (3t - 4t^3) \), find \( \frac{dy}{dx} \) for \( 0 < t < \frac{1}{2} \).
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Context:
What is the minimum distance between  $(2019, 470)$  and  $(21a - 19b, 19b + 21a)$  for  $a, b \in Z$ ?
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Context:
Determine the largest open ball around zero where the matrix exponential on skew-symmetric $3\times3$ matrices onto $SO(3)$ is a local homeomorphism or diffeomorphism.
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Context:
Evaluate the infinite series: $$\sum_{k=1}^{\infty} \sum_{j=1, j \neq k}^{\infty} \sum_{i=1, i \neq j, i \neq k}^{\infty} \frac{1}{4^i4^j4^k}$$
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Context:
Determine the cardinality of the set of all bijective functions from the set of natural numbers to itself.
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Context:
Evaluate the limit: \[ \lim_{x \to 0} \frac{\tan^{12}x-x^{12}}{x^{14}} \]
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Context:
Find the minimum value of the function $f(x, y, z) = x^2 + y^2 + z^2$ subject to the constraints $x^2 + 2y^2 = 1$ and $xy + xz = 2$.
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Context:
Given a random variable $X$ with probability density function (pdf) \( f_X(x) = \begin{cases} \frac{3}{x^4} & \text{if } x > 1 \\ 0 & \text{if } x \leq 1 \end{cases} \), find a function of a uniform random variable \( U \sim \text{Unif}(0,1) \) that has the same distribution as \( X \).
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Context:
Find the number of nondegenerate triangles whose vertices lie in the set of points $(s, t)$ in the plane such that $0 \leq s \leq 4$, $0 \leq t \leq 4$, where $s$ and $t$ are integers.
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Context:
Determine the value of \( d \) for which a generalized Cantor set, constructed by retaining intervals of size \( d < 1 \) at both ends and removing an interval of size \( 1-2d \) at each stage, has Lebesgue measure zero. The process continues such that at the \( n^{th} \) stage, \( 2^{n-1} \) intervals of size \( 1-2d^n \) are removed. For which value of \( d \) does the following equation hold?
\[ \sum_{n=1}^{\infty} 2^{n-1} (1-2d^n) = 1. \]
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Context:
Find the vector \( \mathbf{s} = (s_1, s_2, s_3, s_4)^T \), where each \( s_i \in \{-1, 1\} \) and the number of 1's and -1's in \( \mathbf{s} \) is equal, that minimizes the maximum of the following correlation values:

\[ c_1 = \mathbf{s} \cdot (1, -1, 1, -1)^T \]
\[ c_2 = \mathbf{s} \cdot (-1, 1, -1, s_1) \]
\[ c_3 = \mathbf{s} \cdot (1, -1, s_1, s_2) \]
\[ c_4 = \mathbf{s} \cdot (-1, s_1, s_2, s_3) \]

Determine the vector \( \mathbf{s} \) that achieves this minimization.
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Context:
Find the rank of the image of the homomorphism $H_2(Q) \to H_2(\tilde{X})$ in the blow-up of $\mathbb{CP}^{3}$ at a planar nodal cubic curve, where $Q$ is the smooth quadric surface replacing the node in the blow-up.
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Context:
Provide an example of a continuous function defined on the entire complex plane, \(\mathbb{C}\), that is not holomorphic at any point.
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Context:
How many natural numbers \( x \) satisfy \( 3 \cdot 10^{13} < x < 10^{14} \) such that each digit from 1 to 9 appears exactly once in \( x \) and no zeros are adjacent to each other?
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Context:
5. A sphere with radius $\frac{4}{9}$ lies inside a regular quadrilateral pyramid $SABCD$ with a base side of 8 and height 3. This sphere touches the base plane $ABCD$ of the pyramid and the lateral faces $SBC$ and $SCD$. The plane $\gamma$ touches the sphere, passes through point $B$, the midpoint $K$ of edge $CD$, and intersects edge $SC$ at point $M$. Find the volume of the pyramid $MBCK$. (20 points)
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Context:
Minnie and Penny are cycling on the same road. Minnie's cycling speeds on flat road, downhill, and uphill are 20 km/h, 30 km/h, and 5 km/h respectively. Penny's cycling speeds on flat road, downhill, and uphill are 30 km/h, 40 km/h, and 10 km/h respectively. Minnie travels from Town A to Town B, cycling uphill for 10 km, then from Town B to Town C, cycling downhill for 15 km, and finally from Town C back to Town A on a flat road for 20 km. Penny travels the same entire distance but in a different route, starting from Town A, through Town C, then Town B, and back to Town A. Calculate the time difference in minutes between Minnie and Penny to complete the 45 km journey. Express your answer as a single numerical value in minutes.
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Context:
Bob chooses a  $4$ -digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If  $N$  is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of  $N$ . For example, if Bob’s string is  $0000$  and the stream of bits begins  $101000001 \dots$ , then  $N = 7$ .
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Context:
Given a matrix \( A \in \mathbb{R}^{5 \times 5} \) and a parameter \( \alpha \in \mathbb{R} \), consider the linear system:
\[ Ax = \begin{pmatrix} 1 \\ -1 \\ \alpha^2 + \alpha \\ 1 \\ -1 \end{pmatrix} \]
The general solution to this system is:
\[ x = \begin{pmatrix} t+u+1 \\ s+t-u \\ s-t+1 \\ -s-t+u \\ -s-u-1 \end{pmatrix} \]
Additionally, the following vectors are in the column space of \( A \):
\[ \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \\ \alpha^2-3 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \\ -1 \\ \alpha^2-2 \end{pmatrix} \]
Find the value of \( \alpha \) such that \( A^2 = 0 \).
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Context:
Points \( M \) and \( K \) are taken on the sides \( CB \) and \( CD \) of the square \( ABCD \) such that the perimeter of triangle \( CMK \) is equal to twice the side length of the square. Find the measure of angle \( MAK \).
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Context:
Determine the number of inflection points of the function \((x-2)^6(x-3)^9\).
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Context:
Evaluate the limit: \[ \lim_{n \to \infty} \int_0^1 \sqrt{\frac{1}{x} + n^2 x^{2n}} \, dx. \]
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Context:
Express the given data "$20$ nanoseconds" in scientific notation.
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Context:
Given $2020*2020$ chessboard, what is the maximum number of warriors you can put on its cells such that no two warriors attack each other.
Warrior is a special chess piece which can move either $3$ steps forward and one step sideward and $2$ step forward and $2$ step sideward in any direction.
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Context:
It takes $A$ algebra books (all the same thickness) and $H$ geometry books (all the same thickness, which is greater than that of an algebra book) to completely fill a certain shelf. Also, $S$ of the algebra books and $M$ of the geometry books would fill the same shelf. Finally, $E$ of the algebra books alone would fill this shelf. Given that $A, H, S, M, E$ are distinct positive integers, it follows that $E$ is
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Context:
Determine the number of possible orientations for a smooth manifold with $n$ maximal connected components, considering both orientable and non-orientable cases.
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Context:
$\begin{aligned}
 & \text{For certain pairs }(m,n)\text{ of positive integers with }m\geq n\text{ there are exactly }50\text{ distinct positive integers }k\mathrm{~such~that} \\
 & |\log m-\log k|<\log n.\text{ Find the sum of all possible values of the product }mn.
\end{aligned}$
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Context:
Consider a triangle $ABC$ with $AB = 7$, $BC = 8$, and $CA = 9$. Let $D$ lie on $\overline{AB}$ and $E$ lie on $\overline{AC}$ such that $BCED$ is a cyclic quadrilateral and $D, O, E$ are collinear, where $O$ is the circumcenter of $ABC$. The area of $\vartriangle ADE$ can be expressed as $\frac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. What is $m + n + p$?
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Context:
In the symmetric group $S_{13}$, determine the number of permutations $\tau$ such that $\tau\alpha\tau^{-1}=\gamma$, where $\alpha = (1,2,4,8)(3,5,7,9,11,13)(6,12)$ and $\gamma = (1,2,3,4)(5,6,7,8,9,11)(12,13)$. Provide your answer as a single integer.
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Context:
What is the maximum possible dimension of the annihilator $W°$ of the subspace $W= \{A^{m}; m \geq 0\}$ of the vector space $V= M_{n}(R)$?
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Context:
Suppose $X$ and $Y$ are independent random variables with distribution Exp(2). Define $U=e^{2X}$ and $V=X-Y$. Calculate $E(U\mid\,V=2)$. \(\boxed{\text{Your answer here}}\)
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Context:
Compute the derivative of the distribution \( f_a(x) = \frac{H(x+a) - H(x-a)}{2a} \), where \( H(x) \) is the Heaviside step function, with respect to \( a \) in the sense of distributions:

\[ \left < \frac{\partial f_a}{\partial a},\phi \right> = \lim_{h\to0} \frac{ \left <f_{a+h} ,\phi \right > - \left <f_{a} ,\phi \right > }{h} \]

Then, evaluate the limit as \( a \to 0 \):

\[ \lim_{a\to 0} \frac{\partial f_a}{\partial a} \]

for a given test function \( \phi(x) \).
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Context:
Farmer John has a (flexible) fence of length $L$ and two straight walls that intersect at a corner perpendicular to each other. He knows that if he doesn't use any walls, he call enclose an maximum possible area of $A_0$, and when he uses one of the walls or both walls, he gets a maximum of area of $A_1$ and $A_2$ respectively. If $n=\frac{A_1}{A_0}+\frac{A_2}{A_1}$, find $\lfloor 1000n\rfloor$.
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Context:
4. On each of the lines $x=0$ and $x=2$, there are 62 points with ordinates $1, 2, 3, \ldots, 62$. In how many ways can three points be chosen from the marked 124 points so that they form the vertices of a right triangle?
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Context:
The leading coefficient of the quadratic trinomial \( f(x) \) is 1. All three coefficients, in some order, form a geometric progression with a common ratio \( q \). Find all possible values of \( q \) given that it is a rational number and the difference between the roots of \( f(x) \) is \( q \).
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Context:
Calculate the expected value of \(\frac{\max\{|X|,|Y|\}}{\min\{|X|,|Y|\}}\) where \(X\) and \(Y\) are independent random variables each following a standard normal distribution.
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Context:
Let \( n > 2 \) be an integer. In a country there are \( n \) cities and every two of them are connected by a direct road. Each road is
 assigned an integer from the set \( \{ 1,2,\ldots ,m\} \) (different roads may be assigned the same number). The priority of a city is the
 sum of the numbers assigned to roads which lead to it. Find the smallest \( m \) for which it is possible that all cities have a differer
 priority.
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Context:
A circular birthday cake is divided into eight equal wedges. In how many distinct ways can red and green candles be placed on the cake, with one candle on each wedge, considering the symmetries of the circle?
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Context:
Find the sum of the two smallest possible values of \( {x}^{ \circ  } \) (in degrees) that satisfy the
 following equation if \( x \) is greater than \( {2017}^{ \circ  } \) :
 \( {\cos }^{5}{9x} + {\cos }^{5}x = {32}{\cos }^{5}{5x}{\cos }^{5}{4x} + 5{\cos }^{2}{9x}{\cos }^{2}x\left( {\cos {9x} + \cos x}\right) . \)
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Context:
Consider a group $G$ of order $105$. Determine the number of Sylow 3-subgroups of $G$.
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Context:
Find a Gröbner basis for the ideal $I \cap \mathbb{Q}[y]$ in the polynomial ring $\mathbb{Q}[x,y]$, where $I=\langle x^2-y,y^2-x,x^5-x^2\rangle$, using the lexicographic ordering $y > x$. Provide your answer as a set of polynomials.
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Context:
Determine whether the series \( \sum_{n=0}^{\infty} \frac{(-1)^n (2n)!x^n}{(n!)^2 4^n} \) converges or diverges.
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Context:
Given the differential equation \( y'(x, y_0) = y^2(x, y_0) + y(x, y_0) \sin(x) \), where \( y_0 \) is the value of \( y \) at \( x = 0 \), find \( \frac{\partial y}{\partial y_0} \bigg|_{y_0=0} \).
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Context:
Determine the maximum number of real roots of the equation $e^x + P(x) = 0$, where $P(x)$ is a polynomial of degree $n$.
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Context:
Let \( P \) be a point inside regular pentagon \( ABCDE \) such that \( \angle PAB = 48^\circ \) and \( \angle PDC = 42^\circ \). Find \( \angle BPC \), in degrees.
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Context:
Suppose $f(x)$ is a function such that $f(2x)=f(x+y)f(x-y)$ for all $x,y \in \mathbb{R}$. Given that $f(10)=4, f(0)\neq 0$, and $f'(0)=2$, find the value of $f'(10)$.
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Context:
Let $n$ be a fixed positive integer. For integer $m$ satisfying $|m| \le n$, define $S_m=\displaystyle\sum_{\substack{i-j=m \\0 \le i,j \le n}} \dfrac{1}{2^{i+j}}$. Then\[\lim_{n\to \infty} \left( S_{-n}^2 + S_{-n+1}^2 +... + S_n^2 \right)\]can be expressed in the form $\dfrac{p}{q}$ for relatively prime positive integers $p,q$. Compute $100p+q$.
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Context:
What is the coefficient of c in the equation that has the largest possible value of cc as 47 when the equation is in the form of a constant times c plus the square of the difference between a variable d and 12 equals 235?
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Context:
A $41 \times 41$ grid contains a hidden tank. A fighter plane fires at one cell at a time. If the shot hits, the tank moves to an adjacent cell (sharing a side); if the shot misses, the tank remains in place. The tank must be hit twice to be destroyed. What is the minimum number of shots needed to guarantee the destruction of the tank?
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Context:
设 $M$ 是由有限个正整数构成的集合, 且 $M=A_{1} \cup A_{2} \cup \cdots \cup A_{20}=B_{1} \cup B_{2} \cup \cdots \cup B_{20}$, 这里 $A_{i} \neq \varnothing, B_{i} \neq \varnothing, i=1,2, \cdots, 20$, 并对任意的 $1 \leq i<j \leq 20$, 都有 $A_{i} \cap A_{j}=\varnothing, B_{i} \cap B_{j}=\varnothing$.

已知对任意的 $1 \leq i \leq 20,1 \leq j \leq 20$, 若 $A_{i} \cap B_{j}=\varnothing$, 则 $\left|A_{i} \cup B_{j}\right| \geq 18$, 求集合 $M$ 的元素个数的最小值（这里， $|X|$ 表示集合 $X$ 的元素个数）.
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Context:
$\text{For how many integers}0\leq n<2013\mathrm{,is}n^4+2n^3-20n^2+2n-21\text{divisible by}2013?$
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Context:
Evaluate the triple integral: \[ \int_0^1 \int_1^2 \int_0^{x+y} 12(4x+y+3z)^2 \, dz \, dy \, dx \]
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Context:
Let $N$ be the number of non-empty subsets $T$ of $S = \{1, 2, 3, 4, . . . , 2020\}$ satisfying $\max (T) >1000$. Compute the largest integer $k$ such that $3^k$ divides $N$.
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Context:
Let $a_0, a_1, \dots, a_{19} \in \mathbb{R}$ and \( P(x) = x^{20} + \sum_{i=0}^{19}a_ix^i, x \in \mathbb{R}. \) If \( P(x)=P(-x) \) for all \( x \in \mathbb{R} \), and \( P(k)=k^2, \) for \( k=0, 1, 2, \dots, 9 \), find \( \lim_{x\rightarrow 0} \frac{P(x)}{\sin^2x}. \)
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Context:
Let  $k$  be a positive integer with the following property: For every subset  $A$  of  $\{1,2,\ldots, 25\}$  with  $|A|=k$ , we can find distinct elements  $x$  and  $y$  of  $A$  such that  $\tfrac23\leq\tfrac xy\leq\tfrac 32$ .  Find the smallest possible value of  $k$ .
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Context:
Centuries ago, the pirate Captain Blackboard buried a vast amount of treasure in a single cell of a $2 \times 4$ grid-structured island. Treasure was buried in a single cell of an $M\times N$ ($2\le M$, $N$) grid. You and your crew have reached the island and have brought special treasure detectors to find the cell with the treasure For each detector, you can set it up to scan a specific subgrid $[a,b]\times[c,d]$ with $1\le a\le b\le 2$ and $1\le c\le d\le 4$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready.
What is the minimum $Q$ required to gaurantee to determine the location of the Blackboard’s legendary treasure?
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Context:
Determine the Lebesgue measure of the Luzin set \( A \), which consists of numbers in the interval \([0,1]\) whose continued fraction expansion contains an infinite subsequence of integers such that each integer is divisible by the previous one.
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Context:
In the expansion of \((x^2+x+1)^{54}\), how many coefficients are divisible by 3?
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Context:
Elyse had 100 pieces of gum. She gave half to her brother Rick. Rick then gave a third of his to his friend Shane. Later, Shane received 10 more pieces of gum from his cousin. Shane chewed 11 pieces of gum and decided to share half of what he had left with his sister, Sarah. How many pieces of gum does Shane have left now?
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Context:
Compute the prime factorization of  $1007021035035021007001$ .  (You should write your answer in the form  $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$  where  $p_1,\ldots,p_k$  are distinct prime numbers and  $e_1,\ldots,e_k$  are positive integers.)
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Context:
Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$.
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Context:
Let \( O \) be a point in three-dimensional space, and let \( l_1, l_2, l_3 \) be mutually perpendicular straight lines passing through \( O \). Let \( S \) denote the sphere with center \( O \) and radius \( R \). For every point \( M \) on \( S \), let \( S_M \) denote the sphere with center \( M \) and radius \( R \). We denote by \( P_1, P_2, P_3 \) the intersection of \( S_M \) with the straight lines \( l_1, l_2, l_3 \), respectively. Define \( P_1 \neq O \) if \( l_1 \) meets \( S_M \) at two distinct points and \( P_1 = O \) otherwise (similarly for \( P_2 \) and \( P_3 \)). What is the set of centers of gravity of the (possibly degenerate) triangles \( P_1 P_2 P_3 \) as \( M \) runs through the points of \( S \)?
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Context:
[ Work Problems.]

Three workers are digging a pit. They work in turns, and each of them works as long as it takes the other two to dig half of the pit. Working this way, they dug the pit. How many times faster would three workers dig the same pit if they worked simultaneously?
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Context:
We have a Poisson process of intensity $\lambda = 4$. We have the following event: $A$: "Two marks appear with a separation of $\frac{1}{12}$ or less". We need to calculate the probability of exactly two "marks" appearing between $0$ and $\frac{1}{2}$, and $A$ occurring at the same time. That is, the probability of exactly two marks occurring in $(0, \frac{1}{2})$ with a separation of $\frac{1}{12}$ or less.
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Context:
Determine all non-negative integers \( n \) for which there exists a non-periodic function \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfying the functional equation \( f(x+1) + f(x-1) = \sqrt{n} f(x) \) for all \( x \in \mathbb{R} \).
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Context:
Find the smallest integer $n$ such that the inequality $0.\overline{3}_n < \log_{n}{3}$ holds for all integers greater than $n$. Provide your answer as an integer.
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Context:
Evaluate the integral \( \int_0^1 \int_0^1 \frac{xy\ln(y)}{xy-1}\ dx \ dy \).
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Context:
The sums of the three columns and the three rows of the table are the same.

| 4 | 9 | 2 |
| - | - | - |
| 8 | 1 | 6 |
| 3 | 5 | 7 |

What is the minimum number of cells in the table that must be changed so that all six new sums are different from each other?
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Context:
In a convex quadrilateral \(ABCD\), let \(M\) be the midpoint of segment \(BC\). Given that \(\angle AMD = 90^\circ\), \(\angle ADM = 15^\circ\), and \(AD = AB + CD\), find the measure of \(\angle BAD\).
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Context:
A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$ ?

[asy]
draw((0,2)--(2,2)--(2,0)--(0,0)--cycle);
draw((0,0.3)--(0.3,2)--(2,1.7)--(1.7,0)--cycle);
label("$a$",(-0.1,0.15));
label("$b$",(-0.1,1.15));
[/asy]
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Context:
A circle of radius \(r\) passes through both foci of, and exactly four points on, the ellipse with equation \(x^2 + 16y^2 = 16\). The set of all possible values of \(r\) is an interval \([a, b)\). The original answer is in the form \(\sqrt{a} + b\). Please find the value of \(a + b\).
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Context:
There are several soldiers forming a rectangular formation with exactly eight columns. If adding 120 people or removing 120 people from the formation can both form a square formation, how many soldiers are there in the original rectangular formation?
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Context:
A train 110 m long traveling at 72 kmph takes 13.998880089592832 seconds to cross a bridge of certain length. How long is the bridge?
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Context:
Seventy percent of a train's passengers are women, and fifteen percent of those women are in the luxury compartment. What is the number of women in the luxury compartment if the train is carrying 300 passengers?
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Context:
An ant starts at one vertex of a tetrahedron. Each minute it walks along a random edge to an adjacent vertex. What is the probability that after one hour the ant winds up at the same vertex it started at?
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Context:
Dan went to the grocery store to buy ingredients for several batches of cakes. He bought 9 dozen eggs, 6 pounds of flour, and 3 cases of butter with each case containing 24 sticks of butter. In addition, he purchased 12 bottles of vanilla extract with each bottle containing 1 ounce. Calculate the total number of individual items Dan bought.
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Context:
A teacher wrote a sequence of consecutive odd numbers starting from 1 on the blackboard: $1, 3, 5, 7, 9, 11, \cdots$ After writing, the teacher erased two numbers, dividing the sequence into three segments. If the sums of the first two segments are 961 and 1001 respectively, what is the sum of the two erased odd numbers?
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Context:
A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. The original answer is in the form $\frac{a\sqrt{b}-c}{d}$, please give the value of a + b + c + d.
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Context:
For an integer  $k$  let  $T_k$  denote the number of  $k$ -tuples of integers  $(x_1,x_2,...x_k)$  with  $0\le x_i < 73$  for each  $i$ , such that  $73|x_1^2+x_2^2+...+x_k^2-1$ . Compute the remainder when  $T_1+T_2+...+T_{2017}$  is divided by  $2017$ .

[i]Proposed by Vincent Huang
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Context:
For each positive integer $n$ , let $f(n) = \sum\limits_{k = 1}^{100} \lfloor \log_{10} (kn) \rfloor$ . Find the largest value of $n$ for which $f(n) \le 300$ .
Note: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$ .
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Context:
A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
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Context:
In the final round of a giraffe beauty contest, two giraffes named Tall and Spotted have made it to this stage. There are 105 voters divided into 5 districts, each district divided into 7 sections, with each section having 3 voters. Voters select the winner in their section by majority vote; in a district, the giraffe winning the majority of sections wins the district; finally, the giraffe winning the majority of districts is declared the contest winner. The giraffe named Tall won. What is the minimum number of voters who could have voted for him?
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Context:
$ABCDEFGH$ is a cube having a side length of 2. $P$ is the midpoint of $EF$, as shown. The area of $\triangle APB$ is:
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Context:
Problem 8. Given an isosceles triangle $P Q R(P Q=Q R)$ with the angle at the vertex equal to $108^{\circ}$. Point $O$ is located inside triangle $P Q R$ such that $\angle O R P=30^{\circ}$, and $\angle O P R=24^{\circ}$. Find the measure of angle $\angle Q O R$.
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Context:
Given the function f(x) = sin2x + 2√3cos²x - √3, and the function g(x) = mcos(2x - π/6) - 2m + 3 (m > 0), if there exist x₁, x₂ ∈ [0, π/4] such that f(x₁) = g(x₂) holds true, then the range of the real number m is _______.
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Context:
Calculate the areas of the figures bounded by the lines given in polar coordinates.

$$
r=\cos \phi, \quad r=\sin \phi, \quad \left(0 \leq \phi \leq \frac{\pi}{2}\right)
$$
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Context:
Initially, a natural number $n$ is written on the blackboard. Then, at each minute, [i]Neymar[/i] chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that [i]Neymar[/i] will never be able to write on the blackboard?
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Context:
For the walls of the house, John would use 12 large planks of wood and 10 small planks. If large planks together need 15 pieces of nails to be secured and small planks together need 5 nails, how many nails does John need for the house wall?
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Context:
The light glows for every some seconds. Between 1:57:58 and 3:20:47 AM, it glowed 354.92857142857144 times. What is the interval in seconds between each glow?
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Context:
How many non-isomorphic Frobenius groups of order 100 exist with a kernel of $C_{25}$?
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Context:
Given the function $f(x)=ax^{3}+2bx^{2}+3cx+4d$, where $a,b,c,d$ are real numbers, $a < 0$, and $c > 0$, is an odd function, and when $x\in[0,1]$, the range of $f(x)$ is $[0,1]$. Find the maximum value of $c$.
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Context:
Two spheres touch the plane of triangle \(ABC\) at points \(A\) and \(B\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 9, and the distance between their centers is \(\sqrt{305}\). The center of a third sphere with a radius of 7 is at point \(C\), and it externally touches each of the first two spheres. Find the radius of the circumcircle of triangle \(ABC\).
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Context:
Find the number of binary sequences of length $n$ that satisfy the pattern $x_1 \ge x_2 \le x_3 \ge x_4 \le x_5 \ge \ldots$.
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Context:
Find the closed-form expression for the series: \[ \sum_{n=1}^\infty \frac{1}{1 + 2^2 + 3^2 + \ldots + n^2} \]
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Context:
Evaluate the limit: $$\lim\limits_{x\to 0}\frac{1}{2x}\int_{0}^{x}\left[1-\cos\left(\frac{2}{u}\right)\right]du$$
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Context:
For a matrix $M \in \mathbb{R}^{n \times n}_{\geq 0}$ with nonnegative entries, define $m$ as the smallest positive integer such that all the entries of $M^m$ are strictly positive, if such an $m$ exists. Determine, as a function of $n$, the maximum possible finite value of $m$ over all possible choices of $M$. Express your answer in terms of $n$.
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Context:
A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A=(-3, 2)$ and $B=(3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths?
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Context:
Find one integer solution to the Diophantine equation \(18x + 14y + 63z = 5\).
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Context:
We define \( {\mathbb{F}}_{101}\left\lbrack  x\right\rbrack \) as the set of all polynomials in \( x \) with coefficients in \( {\mathbb{F}}_{101} \) (the integers modulo101with usual addition and
 subtraction),so that two polynomials are equal if and only if the coefficients of \( {x}^{k} \) are equal in \( {\mathbb{F}}_{101} \) for each nonnegative integer
 \( k \) . For example, \( \left( {x + 3}\right) \left( {{100x} + 5}\right)  = {100}{x}^{2} + {2x} + {15} \) in \( {\mathbb{F}}_{101}\left\lbrack  x\right\rbrack \) because the corresponding coefficients are equal modu
 101.
 We say that \( f\left( x\right)  \in  {\mathbb{F}}_{101}\left\lbrack  x\right\rbrack \) is \\emph\{lucky\} if it has degree at most 1000 and there exist \( g\left( x\right) ,h\left( x\right)  \in  {\mathbb{F}}_{101}\left\lbrack  x\right\rbrack \) such that
 \( f\left( x\right)  = g\left( x\right) \left( {{x}^{1001} - 1}\right)  + h{\left( x\right) }^{101} - h\left( x\right) \)
 in \( {\mathbb{F}}_{101}\left\lbrack  x\right\rbrack \) . Find the number of lucky polynomials.
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Context:
In triangle $ABC$, point $P$ is inside the triangle such that lines $AP$, $BP$, and $CP$ intersect sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Given that $AP = a$, $BP = b$, $CP = c$, $PD = PE = PF = 3$, and $a + b + c = 43$, find the value of $abc$.
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Context:
Determine the number of zeros of the function \( g(z) = z^4 + iz^3 + 1 \) in the region \( \{ z \in \mathbb{C} : \text{Re}(z) > 0, \text{Im}(z) > 0 \} \).
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Context:
Determine the maximal possible dimension of a subspace \( U \subseteq V \), where \( V \) is the vector space of functions from \([0,1]\) to \( \mathbb{R} \), and \( U \) consists of monotone functions.
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Context:
Determine the cardinality of a Hilbert space $H$ that has an orthonormal basis with cardinality $\mathfrak{c}$.
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Context:
Determine the number of distinct cyclic subgroups of order 10 in the group \( \mathbb{Z}_{30} \oplus \mathbb{Z}_{120} \).
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Context:
Given $n$ complete graphs, each with $m$ nodes, determine the minimum number of edges required to connect these graphs such that there exists a path of length 2 between any two nodes.
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