Context:
There are 4 spheres in space with radii 2, 2, 3, and 3, respectively. Each sphere is externally tangent to the other 3 spheres. Additionally, there is a small sphere that is externally tangent to all 4 of these spheres. Find the radius of the small sphere.
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Context:
Given that point P is on the curve $y=\frac{1}{2}{{e}^{x}}$ and point Q is on the curve $y=x$, determine the minimum value of the distance $|PQ|$.
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Context:
Paint three faces of an $8 \times 8$ cube red and three faces blue (requiring that any three faces sharing a common vertex cannot all be the same color), then cut it into 512 $1 \times 1 \times 1$ small cubes. Among these 512 small cubes, there are $\qquad$ small cubes that have both red and blue faces.
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Context:
Let \( f(x) = \sqrt{x + \sqrt{0 + \sqrt{x + \sqrt{0 + \sqrt{x + \dots}}}}} \,.\) If \( f(a) = 4 \), find \( f'(a) \).
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Context:
For real numbers $$ a $$ and $$ b $$, define $$ a \$b = (a - b)^2 $$. What is $$ (x - y)^2 \$(y - x)^2 $$?

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Context:
已知$\vartriangle ABC$的内角$A$,$B$,$C$的对边分别为$a$,$b$,$c$,且${{\left( a+b+c \right)}^{2}}+{{\left( a+b-c \right)2}}=4{{c}^{2}}+2ab$,$B=30{}^\circ $,$a=2$,则$\vartriangle ABC$的面积为______.
______.
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Context:
Calculate the Euler characteristic of the surface given by the presentation \( \langle a,b,c \mid abcb^{-1}a^{-1}b^{-1}\rangle \) and determine the type of surface it represents.
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Context:
Let  $\triangle ABC$  be a right-angled triangle and  $BC > AC$ .  $M$  is a point on  $BC$  such that  $BM = AC$  and  $N$  is a point on  $AC$  such that  $AN = CM$ . Find the angle between  $BN$  and  $AM$ .

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Context:
Calculate the last two digits of the sum $f(17) + f(18) + f(19) + f(20)$, where $f(x) = x^{x^{x^x}}$. Given that $f(20)$ ends with $00$, determine the last two digits of $f(17), f(18),$ and $f(19)$ using modular arithmetic, and find the last two digits of the entire sum.
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Context:
Given two independent random variables $X$ and $Y$ with moment-generating functions (mgfs) $M_{X}(t) = (1-t)^{-1}$ for $t<1$ and $M_{Y}(t) = (1-2t)^{-1}$ for $t<0.5$, respectively. Let $U = \frac{1}{2}(X+Y)$ and $V = \frac{1}{2}(X-Y)$. Determine the covariance, $\text{Cov}(U,V)$. Use the relationships $U+V = X$ and $U-V = Y$ if they simplify the calculation.
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Context:
Alissa used discarded A4-size paper to make notepads for her friends. She would fold 8 pieces of A4-size paper 4 times then cut along the edges to form smaller note papers. She would then stack these smaller note papers and staple them together. How long would 1 notepad last if someone wrote 15 notes per day on both sides of the note papers?
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Context:
There is a billiard table in shape of rectangle $2 \times 1$, with pockets at its corners and at midpoints of its two largest sizes. Find the minimal number of balls one has to place on the table interior so that any pocket is on a straight line with some two balls. (Assume that pockets and balls are points).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{
The minimal number of balls required is 4.

}


Context:
[ Ordinary fractions ]  [ Inclusion-exclusion principle ]

How many irreducible fractions with numerator 2015 are there, which are less than $1 / 2015$ and greater than $1 / 2016$?
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Context:
Given the function $f(x)=\cos ^{2}\omega x+2\sin \omega x\cos \omega x-\sin ^{2}\omega x$ ($\omega\ \gt 0$), which has a maximum value and no minimum value in the interval $\left(\frac{π}{{12}},\frac{π}{3}\right)$, find the range of values for $\omega$.
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Context:
A palindrome number is a positive integer that reads the same forward and backward. For example, $1221$ and $8$ are palindrome numbers whereas $69$ and $157$ are not.  $A$ and $B$ are $4$-digit palindrome numbers.  $C$ is a $3$-digit palindrome number.  Given that $A-B=C$, what is the value of $C$?
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Context:
4. At a rectangular table, eight chairs are placed, four on one side and four opposite them on the other side. In how many ways can eight friends be seated at this table so that Ana and Bane sit opposite each other, and Vesna and Goran sit next to each other? (It is known that all friends have distinct names.)

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Context:
G7 Let $A B C$ be an isosceles triangle with $A C=B C$. The point $D$ lies on the side $A B$ such that the semicircle with diameter $[B D]$ and center $O$ is tangent to the side $A C$ in the point $P$ and intersects the side $B C$ at the point $Q$. The radius $O P$ intersects the chord $D Q$ at the point $E$ such that $5 \cdot P E=3 \cdot D E$. Find the ratio $\frac{A B}{B C}$.

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Context:
Consider a sequence of subdivisions \(\{x_i^n\}_{i=0}^{m_n-1}\) of the interval \([0,1]\) such that \(\max_{i=1,...,m_n-1}\Delta x_i^n\to 0\). Determine whether the following limit always holds: \[ \lim_{n\to \infty }\sum_{i=0}^{m_n-1}o(\Delta x_i^n)=0. \]
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Context:
Find the limit of the sequence: \[ \lim_{{n}\to{\infty}} (n+1)!(e-1-{1\over2}-\cdots-{1\over n!}) \]
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Context:
Find the digits $x, y, z$ that satisfy the equation $\overline{x 5} \cdot \overline{3 y z}=7850$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum integer $k \ge 7$ such that there exists an integer $n$ for which the digit sums of the squares of $n+1, n+2, \ldots, n+k$ are all perfect squares. What is the largest such $k$?
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Context:
Let $a, b, c$ be positive integers and $p$ be a prime number. Assume that
$$a^n(b + c) + b^n(a + c) + c^n(a + b) \equiv 8 \pmod{p}$$
for each nonnegative integer $n$. Let $m$ be the remainder when $a^p + b^p + c^p$ is divided by $p$, and $k$ the remainder when $m^p$ is divided by $p^4$. Find the maximum possible value of $k$.

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


Context:
Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n \ge 3$ be a fixed integer. The numbers $1,2,3, \cdots , n$ are written on a board. In every move one chooses two numbers and replaces them by their arithmetic mean. This is done until only a single number remains on the board.

Determine the least integer that can be reached at the end by an appropriate sequence of moves.
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Context:
In Smithtown, the ratio of right-handed people to left-handed people is 3 to 1 and the ratio of men to women is some ratio. If the number of right-handed men is maximized, then 25.00000000000001 percent of all the people in Smithtown are left-handed women. What is the ratio of men to women?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of integer solutions to the equation \(\frac{xy}{z}+\frac{xz}{y}+\frac{yz}{x} = 6\), where \(x, y, z\) are non-zero integers.
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Context:
Find the smallest integer value of \(a\) such that the function \(f(x) = \int_{0}^{x} |\log_{2}(\log_{3}(\log_{4}(\cos t + a)))| \, dt\) is increasing for all \(x \in \mathbb{R}\).
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Context:
It is known that the optimal amount of a certain material to be added is between 100g and 1100g. If the 0.618 method is used to arrange the experiment and the first and second trials are at points $x_1$ and $x_2$ ($x_1 > x_2$), then when $x_2$ is considered the better point, the third trial point $x_3$ should be __g (answer with a number).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the NEMO, Kevin needs to compute the product:
\[
9 \times 99 \times 999 \times \cdots \times 999999999.
\]
Kevin takes exactly $ab$ seconds to multiply an $a$-digit integer by a $b$-digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the proposition "For all $x$ in the interval $[-1, 1]$, the inequality $1 + 2^x + a \cdot 4^x < 0$ is false," then the minimum value of the real number $a$ is ______.
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Context:
Evaluate the integral \( \int_C \frac{\cos(z)}{z^2 - 4z + 5} \, dz \), where \( C \) is the contour consisting of the real line segment from \(-r\) to \(r\) and the upper semi-circle of radius \(r\).
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Context:
Two trains of different lengths are running in opposite directions, one at the rate of 40 km and the other at the rate of 50 kmph. The first train is 111 meters long. They will be completely clear of each other from the moment they meet in 11.039116870650348 seconds. What is the length of the second train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum number of distinct numbers from 1 to 1000 that can be selected so that the difference between any two selected numbers is not equal to 4, 5, or 6?
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Context:
Given a positive integer $\mathrm{n}$, write the $\mathrm{n}$ positive integers $1, 2, \ldots, \mathrm{n}$ on paper, where some numbers are written in red and some in blue, with both colors used, and each number is only written in one color. For $1 \leq i \leq j \leq n$, if there are an odd number of blue numbers among $i, i+1, \ldots, j$, then the pair of numbers $(i, j)$ is called "good." Find the maximum possible number of good pairs.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the recurrence relation \(a_{n+1}a_{n} = a_{n}^{2} + 1\) with \(a_{0} = 5\), determine \(\left \lfloor{a_{1000}}\right \rfloor\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
[ Area and orthogonal projection ]

Find the area of the section of a regular triangular prism ABCA1B1C1 by a plane passing through vertex C and the midpoint of side B1C1 of the base A1B1C1, and parallel to the diagonal AC1 of the lateral face AA1C1C, if the distance between the line AC1 and the cutting plane is 1, and the side of the base of the prism is $\sqrt{14}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of \( \overline{AP} + \overline{AQ} \), where \( A(-3,2) \) is a point on the coordinate plane. A line \( L \) is perpendicular to the line \( x = y \). Line \( L \) intersects the curve \( y = 2^x \) at point \( P \) and the curve \( y = \log_2 x \) at point \( Q \). Determine the minimum value of the sum of the distances from \( A \) to \( P \) and \( Q \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Isosceles trapezoid \(ABCD\) has parallel sides \(\overline{AD}\) and \(\overline{BC},\) with \(BC < AD\) and \(AB = CD.\) There is a point \(P\) in the plane such that \(PA=1, PB=2, PC=3,\) and \(PD=4.\) Find the value of \(\frac{BC}{AD}.\) The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is in simplest form. Please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
From how many of the vertices of a convex $n$-gon can you choose four such that the quadrilateral they form has sides that are diagonals of the $n$-gon?
Please reason step by step, and put your final answer within \boxed{}.


Context:
$A,B,C$ are playing backgammon tournament. At first, $A$ plays with $B$. Then the winner plays with $C$. As the tournament goes on, the last winner plays with the player who did not play in the previous game. When a player wins two successive games, he will win the tournament. If each player has equal chance to win a game, what is the probability that $C$ wins the tournament?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of L-structures of size \( n \) for a language \( L \) that contains a single binary function symbol \( + \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a Markov jump process for a gas station with a single gas pump, cars arrive at a rate of 20 vehicles per hour. Each car takes 6 minutes (1/10 hour) to fill up. The system states are 0, 1, or 2 cars at the pump. The transition rate matrix is:
$$
Q = \begin{bmatrix}
-20 & 20 & 0 \\
10 & -30 & 20 \\
0 & 10 & -10
\end{bmatrix}
$$
The stationary distribution is $(u_1, u_2, u_3) = \left(\frac{1}{7}, \frac{2}{7}, \frac{4}{7}\right)$. Calculate the average number of cars served by the gas station per hour.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three numbers form a geometric progression; the sum of the numbers is 65, and their product is 3375. What are these numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $c$ such that $6x^2 + cx + 16$ equals the square of a binomial.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Any right triangle has the property that the sum of the areas of the squares on its two legs is equal to the area of the square on its hypotenuse. This is the famous Pythagorean theorem, also known in the West as the Pythagorean theorem. The Pythagorean theorem has a history of 4000 years, with hundreds of different proofs. The ancient Chinese mathematician Liu Hui from the Wei and Jin periods provided a concise and beautiful proof method shown in the left diagram below. The right diagram is the Liu Hui block puzzle based on this method:

The five blocks of the Liu Hui block puzzle can be assembled into the parallelogram shown on the right. If the right triangle's legs are 3 cm and 4 cm, what is the perimeter of the parallelogram in $\qquad$ centimeters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two circles with equal radii intersect as shown. The area of the shaded region equals the sum of the areas of the two unshaded regions. If the area of the shaded region is $216\pi$, what is the circumference of each circle?
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Context:
The worm consists of a white head and several segments, as shown in the figure.

When the worm is born, it has a head and one white segment. Each day, a new segment is added to the worm in one of the following ways:

- Either a white segment splits into a white and a gray segment.
- Or a gray segment splits into a gray and a white segment.

(On the fourth day, the worm reaches maturity and stops growing - its body consists of a head and four segments.)

How many different color variants of mature worms of this species can exist?

Hint: What could a two-day-old worm look like?
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Context:
What is the weight of 3.75 moles of Na2Ca(CO3)2?
Please reason step by step, and put your final answer within \boxed{}.


Context:
You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps. What is the remainder when $m$ is divided by $19$?
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Context:
A hollow cylinder with a very thin wall and a block are placed at rest at the top of a plane with inclination  $\theta$  above the horizontal. The cylinder rolls down the plane without slipping, and the block slides down the plane; it is found that both objects reach the bottom of the plane simultaneously. Find the coefficient of kinetic friction between the block and the plane in terms of  $\theta$.
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Context:
Calculate \( \lim _{ n\to \infty } \int _{ |x|<n }{ \int _{ |y|<n }{ \sin(x^2+y^2)\,dx\,dy } } \).
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Context:
Determine the number of automorphisms of the group \( \Bbb Z \oplus \Bbb Z_3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f$ and $g$ be meromorphic functions. Suppose $f$ has a pole of order $k$ at $z_0$ and $g$ has a zero of order $m$ at $z_0$. Determine the condition on $k$ and $m$ for the function $f(z)g(z)$ to have a removable singularity at $z_0$.
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Context:
Determine the range of values for \( \beta > 0 \) such that the random walk defined by \( S_0 = 0 \) and \( S_n = \varepsilon_1 + \cdots + \varepsilon_n, n \geq 1 \), with i.i.d increments \( \lbrace \varepsilon_n \rbrace \) having the distribution \( P(\varepsilon_1 = -1) = 1 - C_{\beta} \) and \( P (\varepsilon_1 > t) = C_{\beta}e^{-t^{\beta}}, t \geq 0 \), where \( C_{\beta} \in (0,1) \) and \( \mathbb{E}\varepsilon_1 = -1/2 \), is heavy-tailed.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For composite number $n$, let $f(n)$ denote the sum of the least three divisors of $n$, and $g(n)$ the sum of the greatest two divisors of $n$. Find all composite numbers $n$, such that $g(n)=(f(n))^m$ ($m\in N^*$).
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Context:
Let $M$ be the intersection of the diagonals $AC$ and $BD$ of cyclic quadrilateral $ABCD$. If $|AB|=5$, $|CD|=3$, and $m(\widehat{AMB}) = 60^\circ$, what is the circumradius of the quadrilateral?
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Context:
Find all positive integers \( n \) such that \( f(n^2+2) + f(n^2+5) = 2n-4 \), where \( f(m) \) is the \( \left\lfloor \frac{p(m)+1}{2} \right\rfloor \)th smallest prime divisor of \( m \) and \( p(m) \) is the number of distinct prime divisors of \( m \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a \) be an element of a group with infinite order. Determine the number of generators of the cyclic group \( \langle a \rangle \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi{\det}^2\begin{Vmatrix}\sin \alpha x&\sin \alpha y&\sin \alpha z\\\sin \beta x&\sin \beta y&\sin \beta z\\\sin \gamma x&\sin \gamma y&\sin \gamma z\end{Vmatrix} \text{d}x\,\text{d}y\,\text{d}z \) where \(\alpha,\beta,\gamma\) are integers.
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Context:
Let $ABC$ be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let $N = 2017$, and define the $A$-ntipodes to be the points $A_1,\dots, A_N$ to be the points on segment $BC$ such that $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$, and similarly define the $B$, $C$-ntipodes. A line $\ell_A$ through $A$ is called a qevian if it passes through an $A$-ntipode, and similarly we define qevians through $B$ and $C$. Compute the number of ordered triples $(\ell_A, \ell_B, \ell_C)$ of concurrent qevians through $A$, $B$, $C$, respectively.
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Context:
Determine the cardinality of the set of chains of \((\mathcal{P}(\mathbb{N}), \subseteq)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $x$ for which the series \( \sum\limits_{n=1}^\infty \frac{\sin(2nx+x)-\sin(2nx-x)}{n} \) converges.
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Context:
There are three alloys made of nickel, copper, and manganese. The first alloy contains 30% nickel and 70% copper, the second one has 10% copper and 90% manganese, and the third one consists of 15% nickel, 25% copper, and 60% manganese. A new alloy of these three metals, containing 40% manganese, needs to be created. What values can the percentage content of copper take in the new alloy?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct topologies on a set with 4 elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two trains of different lengths run at the speed of 68 kmph and 40 kmph in opposite directions in parallel tracks. The time which they take to cross each other is 11.999040076793857 seconds. If the longer train is 200 m long, how long is the shorter train?
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Context:
Determine the smallest number of same-suited cards, n, out of 9 cards (2 cards for each of 2 players and 5 community cards) such that there is a 50% probability that at least one player has a flush.
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Context:
Given that $\tan x + 2\tan y + 3\tan z = 40$ for $x, y, z \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$, find the minimum value of $\tan^2 x + \frac{\tan^2 y}{4} + \frac{\tan^2 z}{9}$.
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Context:
Given the block matrix \( \pmatrix{A&B\\ C&D} \), determine the expression for its determinant if both pairs \((A,C)\) and \((A,B)\) commute.
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Context:
In the 10 by 10 grid of squares shown, point \( P \) can be at any of the 41 points of intersection of pairs of gridlines inside (and not on) \(\triangle F G H\). For each possible location of \( P \), exactly three triangles are formed: \(\triangle F P G\), \(\triangle G P H\), \(\triangle H P F\). How many of these triangles have an area that is exactly half of the area of \(\triangle F G H\)? Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The odd positive integers $1,3,5,7,\cdots,$ are arranged into in five columns continuing with the pattern shown on the right. Determine the column number in which 1985 appears.

[asy]
int i,j;
for(i=0; i<4; i=i+1) {
label(string(16*i+1), (2*1,-2*i));
label(string(16*i+3), (2*2,-2*i));
label(string(16*i+5), (2*3,-2*i));
label(string(16*i+7), (2*4,-2*i));
}
for(i=0; i<3; i=i+1) {
for(j=0; j<4; j=j+1) {
label(string(16*i+15-2*j), (2*j,-2*i-1));
}}
dot((0,-7)^^(0,-9)^^(2*4,-8)^^(2*4,-10));
for(i=-10; i<-6; i=i+1) {
for(j=1; j<4; j=j+1) {
dot((2*j,i));
}}
[/asy]

Express your answer as a column number (e.g., first, second, third, etc.).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the variance of the integral \( \int_0^1 X(t) \, dt \), where \( X(t) \) is a stationary random process with expected value \( E[X(t)] = m \) and covariance function \( r_X(\tau) = 2e^{-\lvert \tau \rvert} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(\zeta = e^{2\pi i / 7}\). Determine the degree of the field extension \([ \mathbb{Q}(\zeta) : \mathbb{Q}(\zeta) \cap \mathbb{R}]\].
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Context:
Simplify $(\sqrt{32})(\sqrt[5]{64})$ to the simplest radical form. Express any number inside a radical by its prime factorization.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given Abby finished the softball season with a total of 45 hits, among which were 2 home runs, 3 triples, and 7 doubles, calculate the percentage of her hits that were singles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ah Hua and Ah Hua did the same number of questions. Each correct answer adds 10 points, and each wrong answer deducts 5 points. In the end, Ah Hua's score is 30 points higher than Ah Hua's score. Given that Ah Hua got 5 questions correct, how many questions did Ah Hua get correct?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $X$ and $Y$ are jointly distributed continuous random variables with $Y$ being positive and $E[Y^3]=27$, and the conditional distribution of $X$ given $Y=y$ is uniform on $(1-y, 1+y)$, find the expected value $E[X^3]$. Express your answer as a single number.
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Context:
Given 8 planes in space, for each pair of planes, the line of their intersection is noted. For each pair of these noted lines, the point of their intersection is noted (if the lines intersect). What is the maximum number of noted points that could be obtained?
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Context:
Find a generator of the ideal generated by $3$ and $2 - 2\sqrt{-2}$ in the ring $\mathbb{Z}[\sqrt{-2}]$. The generator should be an element of the ideal with minimal norm. The norms of $3$ and $2 - 2\sqrt{-2}$ are $9$ and $12$, respectively. Determine this generator.
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Context:
A clock shows the time as some hour in the morning. If the minute hand gains 7 minutes every hour, the clock will gain 63 minutes by 6 p.m. What time does the clock show initially?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ u = \int_0^\infty \frac{dx}{x^4 + 7x^2 + 1} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Michelle is at the bottom-left corner of a $6 \times 6$ lattice grid, at $(0,0)$. The grid also contains a pair of one-time-use teleportation devices at $(2,2)$ and $(3,3)$. The first time Michelle moves to one of these points, she is instantly teleported to the other point and the devices disappear. If she can only move up or to the right in unit increments, in how many ways can she reach the point $(5,5)$?
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Context:
Ten people form a circle.  Each picks a number and tells it to the two neighbors adjacent to them in the circle.  Then each person computes and announces the average of the numbers of their two neighbors. The figure shows the average announced by each person (not the original number the person picked.)

The number picked by the person who announced the average $6$ was
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Context:
Find the expected number of additional coin flips needed to achieve a run of \( n+1 \) successive heads, given that it took \( k \) flips to achieve a run of \( n \) successive heads. Express this expectation as \( E(X_{n+1} | X_n = k) \).
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Context:
$ABCD$ is a rectangle with an area of 24. Find the area of the shaded part if $DQ = QC$ and $AP:PD = 1:2$.
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Context:
Determine the convergence domain of the series \(\sum_{n=1}^\infty \frac{z^n - (\overline{z})^n}{n}.\)
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\boxed{\{ z \in \mathbb{C} \mid |z| \leq 1 \}}


Context:
Fill the numbers 1 through 6 into the three equations in the diagram, such that each number is used exactly once, $\mathrm{A}+\mathrm{B}$ is a multiple of 2, $\mathrm{C}+\mathrm{D}$ is a multiple of 3, $\mathrm{E}+\mathrm{F}$ is a multiple of 5. Find the smaller number between $\mathrm{C}$ and $\mathrm{D}$. (Enter specific value)
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Context:
In the diagram, use the given angle measurements and geometric properties to calculate the value of \(x\). Express your answer in degrees as a whole number.
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Context:
A random walker on the real line starts at position 25 and takes independent normal steps with mean 0 and variance 4. What is the probability that the walker will be at a position greater than 32 at 12 seconds? Express this probability as \( P(X_{12} > 32) \), where each step \( \Delta X_i \) is an independent normally distributed random variable with mean 0 and variance 4.
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Context:
How many times does the digit 9 appear in the list of all integers from 1 to 1000?
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Context:
A school has carried out a variety of colorful club activities. Some students in a class signed up for two clubs, calligraphy and chess. The class monitor went to the mall to buy brushes and chess sets (both items) for the students, spending a total of $360. Each brush costs $15, and each chess set costs $20. Calculate the number of different purchasing plans that satisfy the given conditions.
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Context:
Let the sequence of real numbers \(a_{0}, a_{1}, \cdots\) satisfy \(a_{0} = 0\) and \(a_{1} = 1\). For every positive integer \(n \geq 2\), there exists a positive integer \(k\) (where \(1 \leq k \leq n\)) such that
\[
a_{n} = \frac{a_{n-1} + a_{n-2} + \cdots + a_{n-k}}{k}.
\]

Find the maximum value of \(a_{2018} - a_{2017}\).
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Context:
Consider a process where you roll an n-sided die, and the result is m. You then roll an m-sided die, and continue this process until you roll a 1. What is the expected number of rolls needed to reach a result of 1, starting with an n-sided die?
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Context:
10,11 |

A sphere is constructed with the height of a cone as its diameter. The surface area of the part of the sphere lying inside the cone is equal to the surface area of the part of the cone lying inside the sphere. Find the angle in the axial section of the cone.
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Context:
Find all the right triangles that have an integer leg and a integer hypotenuse knowing that the other leg has length $\sqrt{1989}$.
Note: $1989 = 9 \times 13 \times 17$
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Context:
Let \(\triangle XYZ\) be a right triangle with \(\angle XYZ = 90^\circ\). Suppose there exists an infinite sequence of equilateral triangles \(X_0Y_0T_0\), \(X_1Y_1T_1\), \ldots such that \(X_0 = X\), \(Y_0 = Y\), \(X_i\) lies on the segment \(XZ\) for all \(i \geq 0\), \(Y_i\) lies on the segment \(YZ\) for all \(i \geq 0\), \(X_iY_i\) is perpendicular to \(YZ\) for all \(i \geq 0\), \(T_i\) and \(Y\) are separated by line \(XZ\) for all \(i \geq 0\), and \(X_i\) lies on segment \(Y_{i-1}T_{i-1}\) for \(i \geq 1\).

Let \(\mathcal{P}\) denote the union of the equilateral triangles. If the area of \(\mathcal{P}\) is equal to the area of \(\triangle XYZ\), find \(\frac{XY}{YZ}\).
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Context:
Given the set $$P = \{-\frac{1}{2}, \frac{1}{3}, \frac{1}{2}, 1, 2\}$$, let the non-empty subsets of $P$ be sequentially denoted as $M_1, M_2, \ldots, M_{31}$. Let $m_1, m_2, \ldots, m_{31}$ respectively be the product of the elements within each subset. If in $P$, there is only one element whose product equals to itself (that is, among the subsets of $P$, there is only one subset whose product of elements equals that element itself), then find the value of $1+m_2+\ldots+m_{31}$.
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Context:
Two distinct squares of the $8\times8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common.
Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$ there exist twoadjacent squares whose numbers differ by at least $g$. Determine the largest $C$-gap $g$.
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Context:
Five identical circles are placed in a line inside a larger one as shown. If the shown chord has length  $16,$  find the radius of the large circle.
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Context:
In XYZ business school, some percentage of students are taking their specialization as finance at the same time 76% taking their specialization as marketing. If 90% are taking their specialization as finance, then the percentage of marketing specialization students is 43.333333333333336%. What is the original percentage of finance specialization students?
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Context:
Find the number of $10$-tuples $(a_1,a_2,\dots,a_{10})$ of integers such that $|a_1| \leq 1$ and
\[a_1^2 + a_2^2 + a_3^2 + \cdots + a_{10}^2 - a_1a_2 - a_2a_3 - a_3a_4 - \cdots - a_9a_{10} - a_{10}a_1 = 2.\]
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Context:
In the Bank of Shower, a bored customer lays $n$ coins in a row. Each second, the customer performs "The Process." In The Process, all coins with exactly one neighboring coin heads-up before The Process are placed heads-up (in its initial location), and all other coins are placed tails-up. The customer stops once all coins are tails-up.

Define the function $f$ as follows: If there exists some initial arrangement of the coins so that the customer never stops, then $f(n) = 0$. Otherwise, $f(n)$ is the average number of seconds until the customer stops over all initial configurations. It is given that whenever $n = 2^k-1$ for some positive integer $k$, $f(n) > 0$.

Let $N$ be the smallest positive integer so that
\[
M = 2^N \cdot \left(f(2^2-1) + f(2^3-1) + f(2^4-1) + \cdots + f(2^{10}-1)\right)
\]
is a positive integer. If $M = \overline{b_kb_{k-1}\cdots b_0}$ in base two, compute $N + b_0 + b_1 + \cdots + b_k$.
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Context:
The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $150$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?
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Context:
Quarter circles of radius 1' form a pattern as shown below. What is the area, in square feet, of the shaded region in a 3-foot length of this pattern? The quarter circles' flat sides face outward, alternating top and bottom along the length.
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Context:
In the example 4-10, as shown in Figure 4-12(a), $v_{1} v_{2} v_{3}$ are three equidistant points on a circle. These points are to be decorated with beads in three different colors: red, blue, and green. How many distinct ways can this be done?
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Context:
Circles  $\omega_1$ ,  $\omega_2$ , and  $\omega_3$  are centered at  $M$ ,  $N$ , and  $O$ , respectively.  The points of tangency between  $\omega_2$  and  $\omega_3$ ,  $\omega_3$  and  $\omega_1$ , and  $\omega_1$  and  $\omega_2$  are tangent at  $A$ ,  $B$ , and  $C$ , respectively.  Line  $MO$  intersects  $\omega_3$  and  $\omega_1$  again at  $P$  and  $Q$  respectively, and line  $AP$  intersects  $\omega_2$  again at  $R$ .  Given that  $ABC$  is an equilateral triangle of side length  $1$ , compute the area of  $PQR$ .
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Context:
求所有的整数 $k$, 使得存在正整数 $a$和 $b$, 满足  $$  \frac{b+1}a+\frac{a+1}b=k.$$ 请给出所有满足条件的整数 $k$ 的和。
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Context:
Evaluate the sum of the series: \[ \sum_{n=0}^\infty \frac{(-1)^{n+1}n}{(2n+1)!} \cdot \pi^{2n+1} \] given that the ratio test confirms convergence.
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Context:
Given the system of equations:
\[\begin{cases}
ab+c+d = 13\\
bc+d+a = 27\\
cd+a+b = 30\\
da+b+c = 17
\end{cases}\]
Compute the value of \(a+b+c+d\).
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Context:
Person A has six coupons that can be used to get a free ice cream cone at a dessert shop. Person A decides to use one coupon every 10 days until all six coupons are used. It is known that the dessert shop is closed on Sundays. Person A marks the dates on the calendar when they will use the coupons and finds that none of these dates fall on a Sunday. Determine the day of the week when Person A first uses a coupon. Express your answer as a full day name (e.g., Monday, Tuesday, etc.).
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Context:
Simplify the algebraic fraction: $$\frac{(2^4+2^2+1)(4^4+4^2+1)(6^4+6^2+1)(8^4+8^2+1)(10^4+10^2+1)}{(3^4+3^2+1)(5^4+5^2+1)(7^4+7^2+1)(9^4+9^2+1)(11^4+11^2+1)}$$
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Context:
Evaluate the integral: $$\int_{-\infty}^\infty \frac{e^{-x} \, dx}{1-e^{-2x}}$$
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Context:
Find all positive integers $x,y$ satisfying the equation \[9(x^2+y^2+1) + 2(3xy+2) = 2005 .\]
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Context:
Given that the number of parcels received by a person in the months from January to May are $1$, $3$, $2$, $2$, $2$ respectively, find the variance ($s^{2}=$ ___) of these $5$ numbers.
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Context:
设 b,c 为实数，满足关于 x 的方程 f(x)^2+bf(x)+c=0 有 6 个互不相等的实数解，其中 f(x)=\left|x-\df{1}{x}\right|-\left|x+\df{1}{x}\right|+2，则 f(2025b)+f(c+2024) 的最小值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Solve in natural numbers the equation \(x^y = y^x\) for \(x \neq y\).
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Context:
Micheal and Adam can do together a piece of work in 20 days. After they have worked together for 15 days, Micheal stops and Adam completes the remaining work in some days. In how many days can Micheal complete the work separately if it takes him 40 days?
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Context:
In the given figure, by connecting every other vertex of a regular octagon with each side measuring 12 cm, two squares can be formed. The area of the shaded region is $\quad$ square centimeters.
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Context:
Let $A B C$ be a triangle whose incircle has center $I$ and is tangent to $\overline{B C}, \overline{C A}, \overline{A B}$, at $D, E, F$. Denote by $X$ the midpoint of major arc $\widehat{B A C}$ of the circumcircle of $A B C$. Suppose $P$ is a point on line $X I$ such that $\overline{D P} \perp \overline{E F}$. Given that $A B=14, A C=15$, and $B C=13$, compute $D P$.
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Context:
Evaluate the limit: \[ \lim_{x\to\infty}4 x \left(1 + \frac{\log\frac{1 + e^3}{1 + e^x}}{x}\right) \]
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Context:
给定正整数 $a, b(a \leqslant b)$. 数列 $\left\{f_{n}\right\}$ 满足:
$$
f_{1}=a, f_{2}=b, f_{n+2}=f_{n+1}+f_{n}(n=1,2, \cdots)
$$
若对任意的正整数 $n$, 都有 $\left(\sum_{k=1}^{n} f_{k}\right)^{2} \leqslant \lambda \cdot f_{n} f_{n+1}$, 求实数 $\lambda$ 的最小值。原始答案是 $2 + \sqrt{5}$，请给出数字部分的和。
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Context:
Find the spectrum of an element $y$ in a unital $C^*$-algebra $B$ given that $y^* = y$ and $y^2 = y^4$.
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Context:
IMO 1995 Problem B3 Let p be an odd prime number. How many p-element subsets A of {1, 2, ... , 2p} are there, the sum of whose elements is divisible by p?
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Context:
Given the Monge circle of the ellipse $\Gamma$ as $C: x^{2}+y^{2}=3b^{2}$, calculate the eccentricity of the ellipse $\Gamma$. Express your answer as a fraction with a square root, if necessary.
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Context:
In the figure, \( AP \) bisects \( \angle BAC \). Given that \( AB = c \), \( BP = d \), \( PC = 75 \), and \( AC = 150 \), find \( d \).
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Context:
设集合 $S$ 是 $\{0,1, \cdots, 98\}$ 的 $m(m \geqslant 3)$元子集，满足对任意的 $x, y \in S$ ，均存在 $z \in$ $S$, 使得 $x+y \equiv 2 z(\bmod 99)$. 求 $m$ 的所有可能值。请给出所有可能值之和。
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Context:
The following diagram shows an equilateral triangle $\triangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\triangle ABC$ a distance $\frac{1}{8} AB$ along a side of $\triangle ABC$ in the directions from $A$ to $B$, from $B$ to $C$, and from $C$ to $A$. The shaded region inside all four of the triangles has an area of $300$. Find the area of $\triangle ABC$.
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Context:
Found the smaller multiple of  $2019$  of the form  $abcabc\dots abc$ , where  $a,b$  and  $c$  are digits.
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Context:
Given a linear transformation $L: \mathbb{R}^4 \rightarrow \mathbb{R}^4$ with characteristic polynomial $t^4+1$, determine the total number of invariant subspaces of $L$. Consider the factorization of the polynomial and the structure of the module over PIDs to identify all possible invariant subspaces.
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Context:
Determine all values of \( M \) such that the expression \( \langle u, v \rangle = ac - 3ad - 3bc + Mbd \) defines an inner product on \( \mathbb{R}^2 \), where \( u = \langle a, b \rangle \) and \( v = \langle c, d \rangle \).
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Context:
Evaluate the integral \( \int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos\theta}\,\mathrm{d}\theta \).
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Context:
A block of mass $m = 4.2 \text{ kg}$ slides on a frictionless table with speed $v$ and collides with an identical block of mass $m$, initially at rest, hanging on a pendulum. The collision is perfectly elastic, and the pendulum block swings up to an angle $\theta = 12^\circ$. It takes $t = 1.0 \text{ s}$ for the block to reach this peak angle. Find $10v$ in $\text{m/s}$, rounded to the nearest integer. Use the small-angle approximation for the pendulum's period, but do not approximate $\theta \approx 0$.
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Context:
Determine the radius of convergence of the Laurent series for the function \( f(z) = \frac{\sin z}{(z-\pi)^2} \) centered at \( z = \pi \).
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Context:
Let \( P\left( A\right) \) be the arithmetic-means of all elements of set \( A = \left\{  {{a}_{1},{a}_{2},\ldots ,{a}_{n}}\right\} \) ,namely
 \( P\left( A\right)  = \frac{1}{n}\mathop{\sum }\limits_{{i = 1}}^{n}{a}_{i} \) . We denote \( B \) "balanced subset" of \( A \) ,if \( B \) is a non-empty subset of
 \( A \) and \( P\left( B\right)  = P\left( A\right) \) . Let set \( M = \{ 1,2,3,4,5,6,7,8,9\} \) .
 Find the number of all "balanced subset" of \( M \) .
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Context:
Calculate the expected number of trials needed to obtain a single success, given that the probability of success on the first trial is 5%. If the first trial is a failure, the probability of success on the next trial increases by 5%, and this pattern continues for subsequent trials. If a trial is successful, the probability resets to 5% for the next trial. Assume this process continues for an arbitrary number of trials.
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Context:
Triangle $ABC$ with $AB < AC$ is inscribed in a circle. Point $D$ lies on the circle and point $E$ lies on side $AC$ such that $ABDE$ is a rhombus. Given that $CD = 4$ and $CE = 3$, compute $AD^2$.
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Context:
Find an approximation for the number of prime factors in the factorization of $N!$ that is less than $4N$.
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Context:
Evaluate the real convergent improper integral using residue theory: \[ \int_0^\infty \frac{\sin(\pi x)}{x(1-x^2)} \, \textrm{d}x \]
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Context:
12.235. The angle between a lateral edge of a regular quadrilateral pyramid and the plane of the base is equal to the magnitude of the plane angle at the vertex of the pyramid. Find the angle between a lateral face and the plane of the base.
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Context:
Evaluate the expression \( y = \sin^{-1}(\sin 8) - \tan^{-1}(\tan 10) + \cos^{-1}(\cos 12) - \sec^{-1}(\sec 9) + \cot^{-1}(\cot 6) - \csc^{-1}(\csc 7) \). If \( y \) simplifies to \( y = a\pi + b \), find the value of \( a - b \).
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Context:
Find a 3-regular graph with the smallest number of vertices that contains a path of 6 vertices (P6) as an induced subgraph.
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Context:
Determine the smallest number of doors, $N$, that must be closed to ensure at least one compartment of a spaceship remains habitable after a toxic leak.
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Context:
Let $k$ be a field and consider the ring $R = k[x,y,z,t]/(xy-z^3,z^5,x^2t^3+y^2)$. Determine the Krull dimension of $R$. Provide your answer as a single integer.
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Context:
Evaluate the derivative of the following double integral with respect to $x$, given that $f(x,y)$ is a joint density function such that $f(\pm \infty, y) = 0$ for all $y$:
\[ \frac{\partial}{\partial x} \left( \int_{-\infty}^\infty \int_{-\infty}^{g(x)} f(x,y) \, dy \, dx \right) \]
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Context:
Given the polynomial equation \(x^5 + 5x + 1 = 0\), find the sum of the 11th powers of all its roots. The roots are denoted by the set \(R = \{\alpha, \beta, \gamma, \delta, \mu\}\). It is given that for any root \(x \in R\), the 11th power can be expressed as:

\[x^{11} = 25x^3 + 10x^2 + x\]

Calculate the sum \(\alpha^{11} + \beta^{11} + \gamma^{11} + \delta^{11} + \mu^{11}\).
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Context:
In a cylinder with a height of 10 and a base radius of 1, two spheres are attached to the upper and lower bases. The sphere on the lower base is tangent to the bottom and the side of the cylinder, and the sphere on the upper base is tangent to the top and the side of the cylinder. A plane is tangent to both spheres. Determine the area of the ellipse formed by the intersection of this plane with the edge of the cylinder.
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Context:
A cauldron has the shape of a paraboloid of revolution with a base radius of 3 meters and a depth of 5 meters. The cauldron is filled with a liquid that has a specific weight of 0.8 grams per cubic centimeter. Calculate the work required to pump all the liquid out of the cauldron.
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Context:
Determine the values of \( \beta \) for which the function \( g(x) = \sum_{n=1}^\infty n^{-\beta} f(nx) \) is Lebesgue-integrable for almost all \( x \in \mathbb{R} \), given that \( f: \mathbb{R} \to [0,\infty] \) is a Lebesgue-integrable function.
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Context:
Compute the product \((a^4 + a^2)(a^3 + a + 1)\) in the finite field \(\Bbb F_{32} = \Bbb F_2[X]/(X^5 + X^2 + 1)\), where \(a = [X]\) is the congruence class of \([X]\). Reduce the result to a polynomial of degree less than 5.
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Context:
Find the derivative of the trace function \( \text{Tr} (X^TP^TPX) \) with respect to the matrix \( P \), where \( X \) and \( P \) are real matrices.
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Context:
Find the length of the latera recta of the ellipse given by the equation \(3x^2+3y^2+2xy-12x+12y+4=0\).
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Context:
Evaluate the limit: $$\lim_{x \to 0}\left(\frac{1}{\log(\cos(x))}+\frac{2}{\sin^2(x)}\right)$$
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Context:
Evaluate the triple integral \( \int_{0}^{2} \int_{0}^{\sqrt{2x - x^2}} \int_{0}^{1} z \sqrt{x^2 +y^2} \, dz\,dy\,dx \) by converting it to cylindrical coordinates.
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Context:
Use a line integral to find the area of the surface that extends upward from the semicircle \( y=\sqrt{4-x^2} \) in the \( xy \)-plane to the surface \( z=3x^4y \).
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Context:
Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$, and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\theta$. The maximum possible value for $\tan \theta$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
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Context:
Find a function \( f(x) \) such that \( \int_{1}^{\infty} f(x) \, dx \) converges, but \( \lim_{x \to \infty} f(x) \) does not exist.
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Context:
Let \( n \geq 3 \) be a positive integer, and consider \( n \) numbers \( a_{1}, a_{2}, \cdots, a_{n} \). Denote the sums of every pair of these numbers by \( b_{ij} = a_{i} + a_{j} \) for \( i > j \). If selecting any \( k \) numbers from these pairwise sums can uniquely determine the \( n \) numbers \( a_{1}, a_{2}, \cdots, a_{n} \), find the smallest value of \( k \).
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Context:
Determine an asymptotic upper bound for the recurrence relation \( T(n) = (T(\sqrt{n}))^2 \), with the base case \( T(n) = n \) for \( n \leq 2 \).
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Context:
Evaluate the contour integral \( \oint_C \frac{\sinh \frac{1}{z}}{z^2+1} \, dz \) where \( C \) is the circle \( |z|=2 \).
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Context:
In the following right-angled triangle \( ABC, \triangle C = BC = 1 \) and \( DEF \) is an arc of a circle with center \( A \). Suppose the shaded areas \( BDE \) and \( CEF \) are equal and \( AD = \frac{x}{\sqrt{\pi}} \). Find the value of \( x \). Express your answer as a single integer.
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Context:
Let $f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4$ and let $\zeta = e^{2\pi i/5} = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}$. Find the value of the following expression: $$f(\zeta)f(\zeta^2)f(\zeta^3)f(\zeta^4).$$
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Context:
将一枚骰子连续投掷五次，则事件"五次出现的点数既不全相同，也不两两互异，且从第二次起每一次的点数都不小于前一次的点数"的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Given that the domain of the function $f(x)$ is $D$, and $f(X)=\{y|y=f(x),x∈X⊆D\}$, $f^{-1}(Y)=\{x|f(x)∈Y,x∈D\}$. If $f(x)=2\sin (ωx+ \frac {5π}{6})(ω > 0)$, $D=\[0,π\]$, and $f(f^{-1}(\[0,2\])=\[0, 2\]\)$, find the range of $ω$.
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Context:
Calculate the limit $$ \lim_{x\rightarrow 0}\left( \frac{1}{x\sin(x)\tan(x\sin(x))}-\frac{1}{x^2\sin^2(x)} \right) $$
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Context:
What is the % change in the area of a rectangle when its length increases by 30 % and its width decreases by 30 %?
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Context:
While playing a video game, Ned lost thirteen lives but collected a certain number of power-ups, each giving him five extra lives. After these events, he ended up with seventy lives. If we let x be the initial number of lives he had and y be the number of power-ups collected, how many lives did Ned have at the start?
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Context:
A plane-parallel glass plate is normally illuminated by a thin beam of light. Behind the plate, at some distance from it, there is an ideal mirror (with a reflection coefficient of one). The plane of the mirror is parallel to the plate. It is known that the intensity of the beam passing through this system is 256 times less than the intensity of the incident beam. Assume that the reflection coefficient at the glass-air interface is constant regardless of the direction of the beam. Ignore the absorption and scattering of light in the air and glass. Find the reflection coefficient at the glass-air interface under these conditions. (10 points)
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Context:
Evaluate the double integral $\int\int 2xy \, dA$ over the region bounded by $y = 1 - x^2$, $y = 1 + x$, and $y = 0$ in the $xy$ plane.
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Context:
The dollar is now worth  $\frac{1}{980}$  ounce of gold. After the  $n^{th}$  7001 billion dollars bailout package passed by congress, the dollar gains  $\frac{1}{2{}^2{}^{n-1}}$  of its  $(n-1)^{th}$  value in gold. After four bank bailouts, the dollar is worth  $\frac{1}{b}(1-\frac{1}{2^c})$  in gold, where  $b, c$  are positive integers. Find  $b + c$ .
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Context:
Let  $T$  denote the  $15$ -element set  $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$ . Let  $S$  be a subset of  $T$  in which all six digits  $1,2,\ldots ,6$  appear and in which no three elements together use all these six digits. Determine the largest possible size of  $S$ .
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Context:
The natural numbers from 1951 to 1982 are arranged in a certain order one after another. A computer reads two consecutive numbers from left to right (i.e., the 1st and 2nd, the 2nd and 3rd, etc.) until the last two numbers. If the larger number is on the left, the computer swaps their positions. Then the computer reads in the same manner from right to left, following the same rules to change the positions of the two numbers. After reading, it is found that the number at the 100th position has not changed its position in either direction. Find this number.
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Context:
Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, then turns 3 random lamps on. What is the probability that the leftmost lamp on the shelf is red, and the leftmost lamp which is turned on is also red?
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Context:
In a grid of numbers and letters, starting from the letter A in the middle of the grid, how many different paths can spell out "AMC8"? Each path must move to an adjacent (up, down, left, or right) letter or number only once (no diagonal moves allowed). Provide your answer as a single integer (number of paths).
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Context:
Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10$ ?
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Context:
Example 9 Given a positive integer $k$, when $x^{k}+y^{k}+z^{k}=1$, find the minimum value of $x^{k+1}+y^{k+1}+z^{k+1}$.
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Context:
In an electric circuit, two resistors are connected in parallel. One resistor has a resistance of 9 ohms, and the other has a resistance that results in a combined resistance of 4.235294117647059 ohms. What is the resistance of the first resistor?
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Context:
Evaluate the limit: $$ \lim_{x\to0} \sum_{n=1}^\infty \frac{\sin x}{4+n^4x^4} $$
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Context:
Determine the number of solutions to the equation $X^{10} - I = 0$ in $M_2(\mathbb{R})$, the set of $2 \times 2$ real matrices.
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Context:
Solve the recurrence relation \( T(n) = n \cdot T(n/2)^2 \). Determine the asymptotic behavior of \( T(n) \).
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Context:
Let $X \subset \mathbb{P}^3_\mathbb{C}$ be a smooth algebraic surface of degree $d$, and let $L \subset X$ be a line on this surface. Compute the self-intersection number $L^2$ of the line $L$ on the surface $X$.
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Context:
Find the result of dividing \(2^{2^k} - b^{2^k}\) by the product \((a+b)(a^2+b^2)(a^4+b^4) \cdots (a^{2^{k-1}}+b^{2^{k-1}})\).
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Context:
Let \( f(n) = \frac{n}{3} \) if \( n \) is divisible by 3, and \( f(n) = 4n - 10 \) otherwise. Find the sum of all positive integers \( c \) such that \( f^5(c) = 2 \). (Here, \( f^5(x) \) means \( f(f(f(f(f(x))))) \).)
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Context:
Determine the conditional expectation \( \mathbb{E}(X|\mathcal{G}) \) where \( X(\omega) = \cos(\pi \omega) \) and \( \mathcal{G} \) is the \( \sigma \)-algebra of countable and co-countable subsets of \((0,1)\). Assume \( \lambda \) is the Lebesgue measure restricted to \((0,1)\).
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Context:
In a party attended by $2015$ guests among any $5$ guests at most $6$ handshakes had been exchanged. Determine the maximal possible total number of handshakes.
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Context:
Determine the cardinality of the set of all root paths in an infinite complete infinitary tree where the branching factor k is infinite.
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Context:
Determine \( \phi'(0) \) for the function \( \phi(x) = \int_{0}^{x}\sin\left(\frac{1}{t}\right)dt \), where \( \phi: [0, \infty) \to \mathbb{R} \). Note that the function is undefined at \( t=0 \) and the limit does not exist at that point.
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Context:
Evaluate the integral \( \int_{25\pi/4}^{53\pi/4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx \).
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Context:
Evaluate the surface integral \( \int_S { \vec{D} \cdot} d\vec{s} \) over the surface of the cube defined by the vector field \( \vec{D} = 2y\vec{a_x} +  xz\vec{a_y} +z\vec{a_z} \), where the cube's coordinates are from \(x, y, z \in (0, 2)\).
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Context:
Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k = 1}^{n^2+1}\frac{1}{\sqrt{n^{2} + k}}. \]
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Context:
Find the limit of the function \( g_n(x) = \frac{1}{\sqrt{2\pi n^{-5}}}\exp\left(-\frac{(x+\frac{1}{n})^2}{2n^{-5}}\right) \) as \( n \to \infty \).
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Context:
Let $T = \left\{(\lambda_n)_{n} \in l^2 : \sum_{n=1}^{\infty} \frac{\lambda_n}{\sqrt{n}} = 0\right\}$. Determine $T^{\perp}$, the orthogonal complement of $T$ in $l^2$. Assume $\mathbb{K} = \mathbb{R}$. Provide your answer in terms of a description of the subspace $T^{\perp}$.
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Context:
Determine the cardinality of the set of equivalence classes \( P(\mathbb{N} \times \mathbb{N})/_R \), where \( R \subseteq P(\mathbb{N} \times \mathbb{N})^2 \) is defined by the relation \( \langle r,s\rangle \in R \Leftrightarrow (\exists \pi : \mathbb{N} \to \mathbb{N} \forall x,y \in \mathbb{N} (\langle x,y\rangle \in r \leftrightarrow \langle\pi(x),\pi(y)\rangle \in s)) \). Here, \( \pi \) is a bijection, and \( A/_R \) denotes the set of equivalence classes of \( A \) under \( R \).
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Context:
How many ways, without taking order into consideration, can 2002 be expressed as the sum of 3 positive integers (for instance, $1000+1000+2$ and $1000+2+1000$ are considered to be the same way)?
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Context:
Source: 1976 Euclid Part B Problem 1
-----
Triangle  $ABC$  has  $\angle{B}=30^{\circ}$ ,  $AB=150$ , and  $AC=50\sqrt{3}$ . Determine the length of  $BC$ .
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Context:
Let the number  $x$ . Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent  of the number  $x$  (for example,  $x\cdot x=x^{2}$ ,  $x^{2}\cdot x^{2}=x^{4}$ ,  $x^{4}: x=x^{3}$ , etc). Determine the minimal number of operations needed for calculating  $x^{2006}$ .
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Context:
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
3. The continuation of the height $B H$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$ (points $B$ and $D$ lie on opposite sides of line $A C$). The degree measures of arcs $A D$ and $C D$, not containing point $B$, are $60^{\circ}$ and $90^{\circ}$, respectively. Determine in what ratio the segment $B D$ is divided by side $A C$.
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Context:
An ant is walking on the edges of an icosahedron of side length  $1$ . Compute the length of the longest path that the ant can take if it never travels over the same edge twice, but is allowed to revisit vertices.

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Context:
Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is
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Context:
Find the radius of convergence and the convergence at the end points of the series: $$\sum_{n=1}^\infty(2+(-1)^n)^nx^n$$
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Context:
Find all prime numbers $ p,q$ less than 2005 and such that $ q|p^2 \plus{} 4$, $ p|q^2 \plus{} 4$.
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Context:
A number $n$ is called multiplicatively perfect if the product of all the positive divisors of $n$ is $n^2$. Determine the number of positive multiplicatively perfect numbers less than $100$.
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Context:
Determine the power of the group of languages A, where A consists of languages L such that L = L* and L is defined over the alphabet Σ = {1}.
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Context:
Determine the largest natural number \( n \) such that

\[ 4^{995} + 4^{1500} + 4^{n} \]

is a square number.
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Context:
Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\frac{3}{5}$. If he flips heads, he adds $1$ to his score. A player wins the game if their score is higher than the other player's score.

The probability that Dukejukem defeats Trollner to win WASS equals $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers. Compute $m+n$.
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Context:
Calculate the expected value \( \mathbb{E}(X^3) \) for the stick breaking problem.
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Context:
Determine the number of non-isomorphic quotient groups of the symmetric group $S_4$.
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Context:
radha bought a watch for rs . 144 and got a percentage of profit equal to the cost price of the watch . what is the cost price of the watch ?
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Context:
In the given figure, hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?
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Context:
Triangle  $ABC$  has side lengths  $AB=12$ ,  $BC=25$ , and  $CA=17$ . Rectangle  $PQRS$  has vertex  $P$  on  $\overline{AB}$ , vertex  $Q$  on  $\overline{AC}$ , and vertices  $R$  and  $S$  on  $\overline{BC}$ . In terms of the side length  $PQ=w$ , the area of  $PQRS$  can be expressed as the quadratic polynomial
\[\text{Area}(PQRS)=\alpha w-\beta\cdot w^2\]
Then the coefficient  $\beta=\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .
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Context:
How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n$?
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Context:
求函数$f ( x ) = \sqrt { \frac { 1 } { 4 } x ^ { 4 } - 2 x ^ { 2 } - 4 x + 1 3 } + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 }$的最小值.
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Context:
Rectangle \(ABCD\) has area 2016. Point \(Z\) is inside the rectangle and point \(H\) is on \(AB\) so that \(ZH\) is perpendicular to \(AB\). If \(ZH : CB = 4 : 7\), what is the area of pentagon \(ADCZB\)?
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Context:
Robert read a book for 10 days. He read an average of 25 pages per day for the first 5 days and an average of 40 pages per day for the next 4 days, and read 30 more pages on the last day. Calculate the total number of pages in the book.
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Context:
Petrov writes down the odd numbers: $1, 3, 5, \ldots, 2013$, and Vasechkin writes down the even numbers: $2, 4, \ldots, 2012$. Each of them calculates the sum of all digits of all their numbers and tells the result to their classmate Masha. Masha subtracts Vasechkin's result from Petrov's result. What is the final result?
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Context:
In the given figure, \(ABCD\) is a rectangle and triangles \(\triangle ABE\) and \(\triangle CDF\) are right triangles. The area of triangle \(\triangle ABE\) is \(150 \mathrm{~cm}^{2}\) and the lengths of segments \(AE\) and \(DF\) are 15 cm and 24 cm, respectively. What is the length of segment \(CF\)?
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Context:
设 $a_{n}$ 是与 $\sqrt{\frac{n}{2}}$ 的差最小的整数， $b_{n}$ 是与 $\sqrt{2 n}$ 的差最小的整数，记 $\left\{\frac{1}{a_{n}}\right\}$ 的前 $n$ 项和为 $S_{n},\left\{\frac{1}{b_{n}}\right\}$的前 $n$ 项和为 $T_{n}$ ，则 $2 T_{100}-S_{100}$ 。
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Context:
Find the sides of a triangle with an area of 84, given that the sides are three consecutive integers.
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Context:
Three positive reals  $x , y , z $  satisfy  $x^2 + y^2 = 3^2
y^2 + yz + z^2 = 4^2
x^2 + \sqrt{3}xz + z^2 = 5^2 .$
Find the value of  $2xy + xz + \sqrt{3}yz$
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Context:
A teacher must divide 221 apples evenly among 403 students. What is the minimal number of pieces into which she must cut the apples? (A whole uncut apple counts as one piece.)
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Context:
By dividing  $2023$  by a natural number  $m$ , the remainder is  $23$ . How many numbers  $m$  are there with this property?
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Context:
Let $n$ be a positive integer, $n \ge 2$, and put $\theta = 2 \pi / n$. Define points $P_k = (k,0)$ in the $xy$-plane, for $k = 1, 2 , \dots, n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$. Let $R$ denote the map obtained by applying, in order, $R_1$, then $R_2, \dots$, then $R_n$. For an arbitrary point $(x,y)$, find, and simplify, the coordinates of $R(x,y)$.
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\boxed{The coordinates of $R(x,y)$ are $(x+n,y)$.}


Context:
The cells of a  $2021\times 2021$  table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate  $(1, 1)$ , contains the number  $0$ . For every other cell  $C$ , we consider a route from  $(1, 1)$  to  $C$ , where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell  $C$ . For example, the cell with coordinate  $(2, 1)$  contains  $1 = 1 + 0$ , the cell with coordinate  $(3, 1)$  contains  $3 = 2 + 0 + 1$ , and the cell with coordinate  $(3, 2)$  contains  $7 = 3 + 0 + 1 + 3$ . What is the last digit of the number in the cell  $(2021, 2021)$ ?
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Context:
Find the derivative of the specified order.

\[ y = \frac{\log_{3} x}{x^{2}}, \quad y^{(IV)} = ? \]
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Context:
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$?
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Context:
一个四位数，它的个位数字与百位数字相同，如果把这个四位数的数字顺序颠倒过来（即千位数字与个位数字互换，百位数字与十位数字互换），所得的新数减去原数得7812，求原来的四位数．
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Context:
If the square root of the length of the hypotenuse of a right triangle is 2 units, what is the sum of the squares of the length of the two other sides?
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Context:
Calculate the value of \((2\vec{a}+\vec{b})\cdot[(\vec{a}\times\vec{b})\times(\vec{a}-2\vec{b})]\) given \(\vec{a} = \frac{\hat{i}-2\hat{j}}{\sqrt{5}}\) and \(\vec{b} = \frac{2\hat{i}+\hat{j}+3\hat{k}}{\sqrt{14}}\).
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Context:
The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$.

$ \begin{array}{ccccccc}     & 7 & 4 & 2 & 5 & 8 & 6 \\+    & 8 & 2 & 9 & 4 & 3 & 0 \\\hline1    & 2 & 1 & 2 & 0 & 1 & 6 \\\end{array} $
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Context:
Find the derivative of the function \( f(M) = \text{Tr}(M\log M - M) \), where \( M \) is a positive definite matrix in \( \mathbb{S}_+^n \) and \( \log(M) = \sum_{i=1}^{n}\log(\lambda_i)v_iv_i^T \) with \((\lambda_i, v_i)\) as eigenpairs of \( M \).
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Context:
On Monday, Millie puts a quart of seeds, $25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
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Context:
Find the largest real  $ T$  such that for each non-negative real numbers  $ a,b,c,d,e$  such that  $ a\plus{}b\equal{}c\plus{}d\plus{}e$ : \[ \sqrt{a^{2}\plus{}b^{2}\plus{}c^{2}\plus{}d^{2}\plus{}e^{2}}\geq T(\sqrt a\plus{}\sqrt b\plus{}\sqrt c\plus{}\sqrt d\plus{}\sqrt e)^{2}\]
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Context:
Arrange 3 volunteer teachers to teach in 6 schools, with at most 2 people per school. How many different distribution plans are there? (Answer with a number)
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Context:
Inside a circle with radius $6$ lie four smaller circles with centers $A$, $B$, $C$, and $D$. These circles touch each other as shown. The point where the circles with centers $A$ and $C$ touch each other is the center of the large circle. Calculate the area of quadrilateral $ABCD$.
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Context:
$f(x)=\sum_{n=0}^{\infty} 2^{-n} \cos (n x)$, 记 $A=\int_{0}^{2 \pi} f^{2}(x) \mathrm{d} x$, 则 [10A] $=$
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Context:
A problem-solving robot with an initial IQ balance of 25 solves a set of questions with point values ranging from $1$ to $10$. Each time it correctly solves a problem, the robot’s IQ balance decreases by the point value of the problem and then increases by 1. If the robot's IQ balance becomes less than the point value of a problem, it fails to solve that problem. What is the maximum score the robot can achieve?
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Context:
Determine the number of homomorphisms from $(\mathbb{Z} / p) [x] / (x^2 - x)$ to $\mathbb{Z} / q$, where $p$ and $q$ are primes.
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Context:
A train is travelling at a certain speed and crosses a pole in 6 seconds. The length of the train is 200.016 meters. What is the speed of the train in kmph?
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Context:
Let \( {a}_{0},{a}_{1},\ldots ,{a}_{19} \in  \mathbb{R} \) and
 \( P\left( x\right)  = {x}^{20} + \mathop{\sum }\limits_{{i = 0}}^{{19}}{a}_{i}{x}^{i},x \in  \mathbb{R}. \)
 If \( P\left( x\right)  = P\left( {-x}\right) \) for all \( x \in  \mathbb{R} \) ,and
 \( P\left( k\right)  = {k}^{2}, \)
 for \( k = 0,1,2,\ldots ,9 \) then find
 \( \mathop{\lim }\limits_{{x \rightarrow  0}}\frac{P\left( x\right) }{{\sin }^{2}x} \)
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Context:
Find all $t$ such that $x-t$ is a factor of $6x^2+13x-5.$

Enter your answer as a list separated by commas.
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Context:
Roll a die twice. Let $X$ be the maximum of the two numbers rolled. Which of the following numbers is closest to the expected value $E(X)$?
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Context:
Find all natural numbers \( n \) such that the polynomial \((X^2+1)^{6n+2}+X^4+1\) divides the polynomial \(X^2+X+1\).
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Context:
Find the value of x. (47% of some number - 36% of 1412) + 63 = x. If x is 3, what is the number?
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Context:
Points $A, B, Q, D,$ and $C$ lie on the circle shown. The measures of arcs $\widehat{BQ}$ and $\widehat{QD}$ are $42^\circ$ and $38^\circ$, respectively. Find the sum of the measures of angles $P$ and $Q$ in degrees.
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Context:
If the product of a and b is 240, the average (arithmetic mean) of b and c is 60, and the average of c and d is 90, what is the value of d - a, given that a, b, c, and d are all positive integers?
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Context:
A function \( g \) is ever more than a function \( h \) if, for all real numbers \( x \), we have \( g(x) \geq h(x) \). Consider all quadratic functions \( f(x) \) such that \( f(1)=16 \) and \( f(x) \) is ever more than both \( (x+3)^{2} \) and \( x^{2}+9 \). Across all such quadratic functions \( f \), compute the minimum value of \( f(0) \).
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Context:
Find the asymptotic bounds for the recurrence relation \(T(n)=T(n/4+n^{1/3})+n^{1/2}\) given that \(T(n)=1\) for \(n\le 5\).
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Context:
The ones digit of the product of four consecutive positive integers is 4. If this product is greater than 1000, what is the sum of the four smallest such integers?
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Context:
In the figure below, $ADE$ is a triangle with $\angle AED = 120^\circ$, and $B$ and $C$ are points on side $AD$ such that $BCE$ is an equilateral triangle. If $AB = 4$ cm, $CD = 16$ cm, and $BC = x$ cm, find the value of $x$.
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Context:
In the diagram, points P, Q, and R lie on a circle with center O and radius 12, and point S lies on OR. If angle POR = 135 degrees, calculate the exact area of trapezoid OPQS.
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Context:
In the figure, letters represent different digits within the range 1-9. Adjacent cells can form 24 two-digit numbers (e.g., $\overline{\mathrm{AB}}, \overline{\mathrm{BA}}, \overline{\mathrm{GD}}$), and three digits in the same row or column can form 12 three-digit numbers in sequence (e.g., $\overline{\mathrm{ABC}}, \overline{\mathrm{BEH}}, \overline{\mathrm{HEB}}$). Among these 36 numbers, the maximum number of composite numbers is $\qquad$.
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Context:
Find all positive integers  $k$ , for which the product of some consecutive  $k$  positive integers ends with  $k$ .
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Context:
There are  $1,000,000$  piles of  $1996$  coins in each of them, and in one pile there are only fake coins, and in all the others - only real ones. What is the smallest weighing number that can be used to determine a heap containing counterfeit coins if the scales used have one bowl and allow weighing as much weight as desired with an accuracy of one gram, and it is also known that each counterfeit coin weighs  $9$  grams, and each real coin weighs  $10$  grams?
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Context:
A starship enters an extraordinary meteor shower. Some of the meteors travel along a straight line at the same speed, equally spaced. Another group of meteors travels similarly along another straight line, parallel to the first, with the same speed but in the opposite direction, also equally spaced. The ship travels parallel to these lines. Astronaut Gavrila recorded that every 7 seconds the ship encounters meteors coming towards it, and every 13 seconds it overtakes meteors traveling in the same direction. He wondered how often the meteors would pass by if the ship were stationary. He thought that he should take the arithmetic mean of the two given times. Is Gavrila correct? If so, write down this arithmetic mean. If not, indicate the correct time in seconds, rounded to the nearest tenth.
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Context:
Let \( n \) be a positive integer and \( \mathbf{A} \) an \( n \)-by-\( n \) matrix with integer entries such that \( k = \big|\det(\mathbf{A})\big| \neq 0 \). Determine the number of \( n \)-by-\( 1 \) column vectors \( \mathbf{x} \in \mathbb{Q}^n \) with entries in the interval \([0,1)\) that satisfy \( \mathbf{A} \mathbf{x} = \mathbf{b} \) for some \( n \)-by-\( 1 \) column vector \( \mathbf{b} \in \mathbb{Z}^n \).
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Context:
Given a positive integer $n>1$. In the cells of an $n\times n$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least two cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.
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Context:
Compute the sum of all 2-digit prime numbers  $p$  such that there exists a prime number  $q$  for which  $100q + p$  is a perfect square.
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Context:
Find the least natural number $n$, such that the following inequality holds:$\sqrt{\dfrac{n-2011}{2012}}-\sqrt{\dfrac{n-2012}{2011}}<\sqrt[3]{\dfrac{n-2013}{2011}}-\sqrt[3]{\dfrac{n-2011}{2013}}$
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Context:
What is the range of the function \( y = \arcsin[\sin x] + \arccos[\cos x] \) (for \( x \in [0, 2\pi) \) and where \([x]\) denotes the greatest integer less than or equal to \( x \))?
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Context:
Among 100 points in the plane, no three are collinear, and exactly 4026 pairs are connected by line segments. Each point is randomly assigned a distinct integer from 1 to 100. Find the expected value of the number of segments that join two points whose labels differ by at least 50.
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Context:
The number $12ab9876543$ is divisible by $101$, where $a, b$ represent digits between $0$ and $9$. What is $10a + b$?
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Context:
Determine the number of non-invertible elements in the multiplicative group $(\mathbb{Z}/p^n\Bbb{Z})^\times$, where $p$ is a prime number.
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Context:
Calculate the moment of inertia about the x-axis for the region bounded by the curves \( x = 2y - y^2 \) and \( x = y^2 \), given that the density at any point \((x, y)\) is \( y + 1 \). Use double integration to find your answer.
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Context:
Find the volume bounded by the surfaces \(x^2+z=1\) and \(y^2+z=1\) in the first octant.
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Context:
Compute the Cauchy principal value of the integral \( \int_{-\infty}^{\infty} \frac{x}{(x^{2}+4)(x-1)} \, dx \).
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Context:
How can we calculate $|M|^2$ for the complex expression $M=\frac{e^{2(1+{\rm i})l}-e^{-2(1+{\rm i})l}}{e^{2(1+{\rm i})x}-e^{-2(1+{\rm i})x}}$, where $l$ and $x$ are real numbers?
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Context:
Rectangle $ABCD$, pictured below, shares $50\%$ of its area with square $EFGH$. Square $EFGH$ shares $20\%$ of its area with rectangle $ABCD$. What is $\frac{AB}{AD}$?
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Context:
Find the maximum number $E$ such that the following holds: there is an edge-colored graph with 60 vertices and $E$ edges, with each edge colored either red or blue, such that in that coloring, there is no monochromatic cycles of length 3 and no monochromatic cycles of length 5 .
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Context:
Deanna has $44$ pieces of candy. What is the largest number of kids whom she can give an equal number of candies such that every kid gets at least $3$ pieces?
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Context:
What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base-$5$ representation?
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Context:
Determine the value of $-1+2+3+4-5-6-7-8-9+...+10000$, where the signs change after each perfect square.
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Context:
Determine the radius of convergence of the power series \( \sum_{n=0}^{\infty} \left(\sqrt{4^n + 3^n} + (-1)^n \sqrt{4^n - 3^n}\right)x^n \).
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Context:
Find the value of $\sin(x)$ that maximizes the expression $(5+3\sin(x))^2 (7-3\sin(x))^3$. What is this value of $\sin(x)$?
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Context:
Determine the number of surjective ring homomorphisms from \(\mathbb{Z}[i]\) to \(\mathbb{F}_{11^2}\).
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Context:
Compute the Lie bracket of the vector fields \(\xi = \left(\frac{x}{\sqrt{x^2 + y^2}}, \frac{y}{\sqrt{x^2+y^2}} \right)\) and \(\eta = (-y, x)\) in \(\mathbb{R}^2\).
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Context:
Consider $h+1$ chess boards. Number the squares of each board from 1 to 64 such that when the perimeters of any two boards are aligned in any possible manner, no two squares in the same position have the same number. What is the maximum value of $h$?
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Context:
Two robots, A and B, start from the same point on a circular track with a circumference of 8 meters and move in opposite directions. Robot A walks 5 meters per minute, while Robot B walks 3 meters per minute. When they meet head-on, Robot B turns around. If Robot A catches up with Robot B from behind, Robot A turns around. Determine the distance from the starting point to the location where they meet for the 10th time. (Catching up from behind is also considered a meeting.)
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Context:
Given the hyperbola $C$: $\frac{x^{2}}{4} - \frac{y^{2}}{b^{2}}=1 (b > 0)$, the distance from its focus to the asymptote is $3$. Find the length of the conjugate axis of the hyperbola $C$. Express your answer as a numerical value.
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Context:
From a \(6 \times 6\) checkered square, gray triangles were cut out. What is the area of the remaining figure? The side length of each cell is 1 cm. Give the answer in square centimeters.
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Context:
Two sides of a regular \( n \)-gon are extended to meet at a \( 28^\circ \) angle. What is the smallest possible value for \( n \)?
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Context:
Determine the order of the normalizer of the subgroup \( H = \langle (1,2)(3,4),(1,3)(2,4) \rangle \) in the alternating group \( A_5 \).
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Context:
In $\Delta XYZ$, $\overline{MN} \parallel \overline{XY}$, $XM = 5$ cm, $MY = 8$ cm, and $NZ = 9$ cm. What is the length of $\overline{YZ}$?
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Context:
Evaluate the integral: $$\int_{0}^{1} \left( 1 + \sin^2 \left( \frac{1}{x} \right) + \frac{1}{x^2} \cos^2 \left( \frac{1}{x} \right) - \frac{1}{x} \sin \left( \frac{2}{x} \right) \right) dx$$
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Context:
Van Helsing gets paid by the town to remove all the vampires and werewolves. He gets $5 per vampire and $10 per werewolf. He removes half the vampires and removes 8 werewolves. There were 4 times as many werewolves as vampires. How much did he earn in total?
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Context:
The numbers from 1 to 8 are placed at the vertices of a cube such that the sum of the numbers at any three vertices lying on the same face is at least 10. What is the minimum possible sum of the numbers at the vertices of one face?
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Context:
Given the equation \( z - c = \rho\frac{1 + it}{1 - it} \), where \( c \) is a complex number, \( \rho \) is a real number, and \( t \) is a real parameter, find the relationship between \( z \), \( c \), \( \rho \), and \( t \). Assume \( z = x + iy \) and \( c = a + ib \), where \( x \), \( y \), \( a \), and \( b \) are real numbers.
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Context:
Evaluate the limit: \[ \lim_{n\to\infty}\int_0^1 n\ln\left(1+\frac{1}{n\sqrt{x}}\right)\, dx \]
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Context:
Calculate the area of the region enclosed by the curves $xy=10$, $xy=16$, $xy^4=3$, and $xy^4=9$. Provide your answer as a single numerical value.
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Context:
The cook, engineer, and sailor have a conversation about their ages:

- Cook: "When I was as old as the sailor is now, I was twice as old as him."
- Engineer: "I am only 4 years older than the sailor."
- Sailor: "Only the cook's age is an odd number, and the least common multiple of our three ages is the captain's year of birth."

Determine the captain's year of birth.
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Context:
A father tells his son: "Today is both of our birthdays, and you are exactly 2 times younger than me. This is the eighth time in your life that you have been younger than me by an integer multiple." If the father is no older than 75 years, how old is the son?
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Context:
Albert has a very large bag of candies and he wants to share all of it with his friends. At first, he splits the candies evenly amongst his $20$ friends and himself and he finds that there are five left over. Ante arrives, and they redistribute the candies evenly again. This time, there are three left over. If the bag contains over $500$ candies, what is the fewest number of candies the bag can contain?
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Context:
Quadrilateral \(ABCD\) is inscribed in a circle. At point \(C\), a tangent \(\ell\) to this circle is drawn. Circle \(\omega\) passes through points \(A\) and \(B\) and touches the line \(\ell\) at point \(P\). Line \(PB\) intersects segment \(CD\) at point \(Q\). Find the ratio \(\frac{BC}{CQ}\), given that \(B\) is tangent to circle \(\omega\).
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Context:
A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn.

How many ways are there to put the cards in the three boxes so that the trick works?
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Context:
The ten smallest positive odd numbers \( 1, 3, \cdots, 19 \) are arranged in a circle. Let \( m \) be the maximum value of the sum of any one of the numbers and its two adjacent numbers. Calculate the minimum value of \( m \). Express your answer as a single integer.
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Context:
From the numbers 1, 2, …, 9, select $n$ numbers, among which there must be several numbers (at least one, or all of them) whose sum is divisible by 10. Find the minimum value of $n$.
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Context:
100 thimbles are arranged in a circle, and a coin is hidden under one of them. In one move, you can flip four thimbles to check if the coin is under one of them, then return them to their original position. After each move, the coin moves to a neighboring thimble. What is the minimum number of moves required to guarantee finding the coin?
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Context:
Determine the values of \( n \) for which a cube with edge length \( n \) can be composed of white and black unit cubes such that each white cube shares a face with exactly three black cubes, and each black cube shares a face with exactly three white cubes.
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Context:
Given a monotonically increasing sequence, $\{a_n\}$, satisfying $a_n + a_{n+1} + a_{n+2} = 3n - 6$, and $a_2 = \frac {1}{2}a_1$, determine the range of possible values for $a_1$.
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Context:
Jessica had 7 roses in the vase. She went to her flower garden, which had a total of 84 roses, and cut some roses. However, her friend Sarah, who initially gave her 15 roses, wanted 10 of the roses from the garden back. After giving Sarah her roses, Jessica put the remaining roses into the vase. There are now 37 roses in the vase. How many roses did Jessica cut from her garden?
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Context:
In triangle \( ABC \), \( AB = 33 \), \( AC = 21 \), and \( BC = m \), where \( m \) is a positive integer. If point \( D \) can be found on \( AB \) and point \( E \) can be found on \( AC \) such that \( AD = DE = EC = n \), where \( n \) is a positive integer, what must the value of \( m \) be?
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Context:
In a convex 13-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides among them. What is the greatest number of sides that it can have?
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Context:
Four points \( B, A, E, L \) are on a straight line. \( G \) is a point off the line such that \(\angle B A G = 120^\circ\) and \(\angle G E L = 80^\circ\). If the reflex angle at \( G \) is \( x^\circ \), then \( x \) equals:
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Context:
A fair coin is tossed. If the first toss shows heads, it is tossed again: if the second toss shows heads, record 0; if the second toss shows tails, record 1. If the first toss shows tails, record any random real number in the closed interval [0, 1]. This process is repeated two independent times, obtaining two real numbers x and y. What is the probability that |x-y| > 1/2?
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Context:
Calculate the Gaussian curvature $K$ for the set $S = \{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1\}$, given that the first fundamental form is $ds^2 = \frac{4}{(1-x^2-y^2)^2}(dx^2 + dy^2)$. The coefficients of the first fundamental form are $E = G = \frac{4}{(1-x^2-y^2)^2}$ and $F = 0$. Express your answer in terms of $x$ and $y$.
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Context:
Find the limit of $P_{n-t}(n)$ as $n$ approaches infinity, where $P_{k}(n)$ is the number of partitions of $n$ into $k$ parts and $t$ is a fixed non-negative integer.
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Context:
Given a pentagon $ABCDE$ with the following properties:

1. Angles $\angle A$ and $\angle C$ are right angles.
2. The side lengths are $|AB| = 4$, $|BC| = 5$, $|CD| = 10$, and $|DE| = 6$.
3. Point $C'$ is the reflection of point $C$ across line $BD$ and lies on line segment $AE$.

Find the measure of angle $\angle E$.
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Context:
The squares in a  $7\times7$  grid are colored one of two colors: green and purple. The coloring has the property that no green square is directly above or to the right of a purple square. Find the total number of ways this can be done.
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