Context:
Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$, with $1$ and $2$ appearing on opposite faces. Furthermore, each face is colored either red or yellow, with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.
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Context:
\( \triangle ABC \) has \( AB = 4 \) and \( AC = 6 \). Let point \( D \) be on line \( AB \) so that \( A \) is between \( B \) and \( D \). Let the angle bisector of \( \angle BAC \) intersect line \( BC \) at \( E \), and let the angle bisector of \( \angle DAC \) intersect line \( BC \) at \( F \). Given that \( AE = AF \), find the square of the circumcircle's radius' length.
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Context:
Karen's class fund contains only $10 bills and another type of bills, which amount to $120. The number of $10 bills is twice as many as the other type of bills. They have 3 of these other bills in their fund. What is the value of these other bills?
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Context:
Let  $\triangle ABC$  be a triangle with  $AB = 7$ ,  $AC = 8$ , and  $BC = 3$ . Let  $P_1$  and  $P_2$  be two distinct points on line  $AC$  ( $A, P_1, C, P_2$  appear in that order on the line) and  $Q_1$  and  $Q_2$  be two distinct points on line  $AB$  ( $A, Q_1, B, Q_2$  appear in that order on the line) such that  $BQ_1 = P_1Q_1 = P_1C$  and  $BQ_2 = P_2Q_2 = P_2C$ . Find the distance between the circumcenters of  $BP_1P_2$  and  $CQ_1Q_2$ .
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Context:
The union of sets \( A \) and \( B \), \( A \cup B = \{a_1, a_2, a_3\} \). When \( A \neq B \), pairs \((A, B)\) and \((B, A)\) are considered different. How many such pairs \((A, B)\) are there?
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Context:
Given the sets $A={x|x^2-ax+a^2-19=0}$, $B={x|x^2-5x+6=0}$, and $C={x|x^2+2x-8=0}$, if $\emptyset \subset (A\cap B)$ and $A\cap C=\emptyset$ hold simultaneously, find the value of the real number $a$.
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Context:
Among the natural numbers less than 5000, how many numbers are there that are divisible by 11 and have a digit sum of 13?
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Context:
Given the points P1(1,1), P2(1,2), P3(1/2, 1/2), and P4(2,2), determine how many of these points are intersections of an exponential function and a logarithmic function. Express your answer as a single numerical value.
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Context:
On the hypotenuse \(AB\) of the right triangle \(ABC\), a square \(ABDE\) is constructed on the side of the plane that does not include the triangle \(ABC\). Find the distance from the vertex \(C\) of the right angle of the triangle to the center of the square, given that \(BC = a\) and \(AC = b\).
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Context:
Find a basis for $\mathbb{Q}(\sqrt{3}+\sqrt{7})$ over $\mathbb{Q}(\sqrt{7})$.
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Context:
A square with a side length of 36 cm was cut into three rectangles in such a way that the areas of all three rectangles are equal and any two rectangles have a common section of the boundary. What is the total length (in cm) of the made cuts?
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Context:
Evaluate the expression with the pattern of alternating subtraction and addition of squared numbers, starting with 100 and ending with 1. What is the result of this expression?
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Context:
What is the minimum number of points that can be chosen on a circle with a circumference of 1956 so that for each of these points there is exactly one chosen point at a distance of 1 and exactly one at a distance of 2 (distances are measured along the circle)?
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Context:
A frustum of a cone with a height of 8 units contains a sphere O1 with a radius of 2 units. The center of sphere O1 lies on the axis of the frustum, and it is tangent to the upper base and the side surface of the frustum. Another sphere O2 with a radius of 3 units can be placed inside the frustum such that it is tangent to sphere O1, the lower base of the frustum, and the side surface of the frustum at one point each. Excluding sphere O2, what is the maximum number of additional spheres with a radius of 3 units that can be placed inside the frustum? Express your answer as a whole number.
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Context:
Alice draws three cards from a standard 52-card deck with replacement. Ace through 10 are worth 1 to 10 points respectively, and the face cards King, Queen, and Jack are each worth 10 points. The probability that the sum of the point values of the cards drawn is a multiple of 10 can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.
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Context:
Suppose that for a positive integer \( n \), \( 2^n + 1 \) is a prime number. What remainder can this prime have when divided by 240?
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Context:
Find the coefficient of ${x}^{2}{y}^{7}$ in the expansion of $((x+y)(x-y{)}^{8})$.
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Context:
A porter needs to transport 200 buns from the kitchen to the construction site (he is currently in the kitchen). Each time, he can carry 40 buns. However, he eats 1 bun each way, whether traveling from the kitchen to the construction site or from the construction site to the kitchen. How many buns can the porter transport to the construction site at most?
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Context:
给定正整数 $n$, 非负实数 $a_{1}$, $a_{2}, \cdots, a_{n}$ 满足 $a_{1}+a_{2}+\cdots+a_{n}=4$ 。记
$$
S=54(a_{1}+a_{1} a_{2}+\cdots+a_{1} a_{2} \cdots a_{n})-13\sqrt{13}.
$$
求 $S$ 的最大值.
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Context:
Given 15 points in space, 5 of which are collinear, what is the maximum possible number of unique planes that can be determined by these points?
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Context:
In Anchuria, there is a checkers championship that takes place over several rounds. The days and cities hosting the rounds are determined by a draw. According to the rules of the championship, no two rounds can be held in the same city, and no two rounds can be held on the same day. A lottery is conducted among the fans: the grand prize goes to the person who correctly guesses which cities and on which days all rounds will be held before the championship begins. If no one guesses correctly, the grand prize goes to the championship organizing committee. There are a total of eight cities in Anchuria, and the championship spans a total of eight days. How many rounds should be in the championship so that the organizing committee has the highest probability of winning the grand prize?
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Context:
Lindsey saved $50 in September, $37 in October, and $11 in November. Lindsey's mom said that since Lindsey had saved more than a certain amount, she would give Lindsey $25. After spending $87 on a video game, Lindsey had $36 left. What was the minimum amount Lindsey needed to save for her mom to give her $25?
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Context:
Circles $\mathcal{C}_1, \mathcal{C}_2,$ and $\mathcal{C}_3$ have their centers at $(0,0), (12,0)$, and $(24,0)$, and have radii $1, 2$, and $4$, respectively. Line $t_1$ is a common internal tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common internal tangent to $\mathcal{C}_2$ and $\mathcal{C}_3$ and has a negative slope. Given that lines $t_1$ and $t_2$ intersect at $(x,y),$ and that $x=p-q\sqrt{r},$ where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$
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Context:
Given variables x, y, and z, if x is less than y by 45% and y exceeds z by 30%, determine the percentage by which z exceeds x.
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Context:
In triangle $ABC$, angle $A$ is $40^{\circ}$. The triangle is randomly thrown onto a table.

Find the probability that vertex $A$ will be east of the other two vertices.
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Context:
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n \ge 2$. Find the smallest positive integer $k$ such that the product $a_1a_2 \cdots a_k$ is an integer.
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Context:
The set  $M= \{1;2;3;\ldots ; 29;30\}$  is  divided in  $k$ subsets such that if  $a+b=n^2, (a,b \in M, a\neq b, n$  is an
integer number  $)$ , then  $a$  and  $b$  belong different subsets.
Determine the minimum value of  $k$ .
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Context:
A frog is placed at the origin on the number line , and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of $3$, or to the closest point with a greater integer coordinate that is a multiple of $13$. A move sequence is a sequence of coordinates that correspond to valid moves, beginning with $0$ and ending with $39$. For example, $0,\ 3,\ 6,\ 13,\ 15,\ 26,\ 39$ is a move sequence. How many move sequences are possible for the frog?
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Context:
## Problem Statement

Calculate the indefinite integral:

$$
\int \frac{2 x^{3}-40 x-8}{x(x+4)(x-2)} d x
$$
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Context:
有长为 $2^{n}(n=0,1, \cdots, 1009)$ 的线段各三条, 则由这 3030 条线段能构成不全等的三角形的个数为  (用数字作答).
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Context:
In a regular pentagonal prism, a diagonal is defined as a line segment connecting two vertices that are neither on the same lateral face nor on the same base. How many diagonals does a regular pentagonal prism have? Express your answer as a whole number.
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Context:
Question 13: Given that $A$ and $B$ are two subsets of $\{1,2, \ldots, 100\}$, satisfying: $|A|=|B|, A \cap B=\emptyset$, and for any $x \in A, 2 x+2 \in B$. Try to find the maximum value of $|A \cup B|$.

---

The translation maintains the original format and line breaks as requested.
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Context:
设a、b为正整数，$a ^ { 2 } + b ^ { 2 }$除以a+b的商为q，余数为r，且$q ^ { 2 } + r = 2010$．求ab 的值.
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Context:
Let  $a,x,y$  be positive integer such that  $a>100,x>100,y>100$  and  $y^2-1=a^2(x^2-1)$   . Find the minimum value of  $\frac{a}{x}$ .
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Context:
A and B play a game with the following rules: In the odd-numbered rounds, A has a winning probability of $\frac{3}{4}$, and in the even-numbered rounds, B has a winning probability of $\frac{3}{4}$. There are no ties in any round, and the game ends when one person has won 2 more rounds than the other. What is the expected number of rounds played until the game ends?
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Context:
From point $A$ to point $B$ at 13:00, a bus and a cyclist left simultaneously. After arriving at point $B$, the bus, without stopping, returned and met the cyclist at point $C$ at 13:10. Returning to point $A$, the bus again without stopping headed towards point $B$ and caught up with the cyclist at point $D$, which is $\frac{2}{3}$ km from point $C$. Find the speed of the bus (in km/h), given that the distance between points $A$ and $B$ is 4 km and the speeds of the bus and the cyclist are constant.
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Context:
If \( a^3 + b^3 + c^3 = 3abc = 6 \) and \( a^2 + b^2 + c^2 = 8 \), find the value of \( \frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a} \).
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Context:
Let $a_n = -n^2 + 10n + 11$, then find the value of $n$ for which the sum of the sequence $\{a_n\}$ from the first term to the nth term is maximized.
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Context:
Calculate the definite integral:

$$
\int_{1}^{64} \frac{6-\sqrt{x}+\sqrt[4]{x}}{\sqrt{x^{3}}-7 x-6 \sqrt[4]{x^{3}}} d x
$$
Please reason step by step, and put your final answer within \boxed{}.

\boxed{4 \ln \left( \frac{2}{2 \sqrt{2} + 1} \right)}


Context:
The cards in a stack are numbered consecutively from 1 to $2n$ from top to bottom. The top $n$ cards are removed to form pile $A$ and the remaining cards form pile $B$. The cards are restacked by alternating cards from pile $B$ and $A$, starting with a card from $B$. Given this process, find the total number of cards ($2n$) in the stack if card number 201 retains its original position.
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Context:
On the Saturday of a weekend softball tournament, Team A plays Team D, Team B plays Team E, and Team C gets a bye (no match). The winner of Team A vs. Team D plays against Team C in the afternoon, while the winner of Team B vs. Team E has no further matches on Saturday. On Sunday, the winners of Saturday's afternoon matches play for first and second places, and the remaining teams play based on their win-loss status for third, fourth, and fifth places. There are no ties. Determine the total number of possible five-team ranking sequences at the end of the tournament.
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Context:
Problem 7.5. In each room of the hotel, no more than 3 people can be accommodated. The hotel manager knows that a group of 100 football fans, who support three different teams, will soon arrive. In one room, only men or only women can be accommodated; also, fans of different teams cannot be accommodated together. How many rooms need to be reserved to definitely accommodate all the fans?
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Context:
At a certain university, 69% of the professors are women, and 70% of the professors are tenured. If a certain percentage of the professors are women, tenured, or both, and 52% of the men are tenured, what percent of the professors are women, tenured, or both?
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Context:
A sequence of positive integers is defined by $a_{0}=1$ and $a_{n+1}=a_{n}^{2}+1$ for each $n \geq 0$. Find $\operatorname{gcd}(a_{999}, a_{2004})$.
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Context:
Determine the smallest positive integer $n$ such that there exist positive integers $x_1, x_2, \ldots, x_n$ so that each natural number from 1001 to 2021 inclusive can be expressed as the sum of one or more distinct terms $x_i$ (where $i = 1, 2, \ldots, n$).
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Context:
Example 5 Given that the function $f(n)$ is defined on the set of positive integers and takes positive integer values, is strictly monotonically increasing, and satisfies $f[f(n)]=3n$, find the value of $f(2004)$.
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Context:
Suppose two circles $\Omega_1$ and $\Omega_2$ with centers $O_1$ and $O_2$ have radii $3$ and $4$, respectively. Suppose that points $A$ and $B$ lie on circles $\Omega_1$ and $\Omega_2$, respectively, such that segments $AB$ and $O_1O_2$ intersect and that $AB$ is tangent to $\Omega_1$ and $\Omega_2$. If $O_1O_2=25$, find the area of quadrilateral $O_1AO_2B$.
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Context:
Let \( O \) be the center of the circumcircle of \( \triangle ABC \). \( P \) and \( Q \) are the midpoints of \( AO \) and \( BC \), respectively. Suppose \( \angle CBA = 4 \angle OPQ \) and \( \angle ACB = 6 \angle OPQ \). Find the size of \( \angle OPQ \) in degrees.
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Context:
15 ladies and 30 gentlemen attend a luxurious party. At the start of the party, each lady shakes hands with a random gentleman. At the end of the party, each lady shakes hands with another random gentleman. A lady may shake hands with the same gentleman twice (first at the start and then at the end of the party), and no two ladies shake hands with the same gentleman at the same time.

Let $m$ and $n$ be relatively prime positive integers such that $\frac{m}{n}$ is the probability that the collection of ladies and gentlemen that shook hands at least once can be arranged in a single circle such that each lady is directly adjacent to someone if and only if she shook hands with that person. Find the remainder when $m$ is divided by 10000.
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Context:
In a scalene triangle \( ABC \), one of the angles is equal to the difference of the other two angles, and one of the angles is twice as large as another. The angle bisectors of angles \( A \), \( B \), and \( C \) intersect the circumcircle of the triangle at points \( L \), \( O \), and \( M \) respectively. Find the area of triangle \( LOM \) if the area of triangle \( ABC \) is 2. If the answer is not an integer, round it to the nearest whole number.
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Context:
Scatterbrained Scientist walks to work every morning and back home every evening. He has two umbrellas and takes one with him under two conditions: it is raining outside or there is no umbrella at his destination. Over time, he calculates that he carries an umbrella 20% of the time. Find the probability of rain.
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Context:
A cargo ship carrying 6 kinds of items - doohickies, geegaws, widgets, yamyams, thingamajigs, and whatchamacallits - arrives at the port. Each item weighs 3, 11, 5, 7, 13, and 17 pounds, respectively, and each item is weighed as it is unloaded. Let a, b, c, d, e, and f represent the number of doohickies, geegaws, widgets, yamyams, thingamajigs, and whatchamacallits, respectively.

If the product of the individual weights of the unloaded items equals 351,125,648,000 pounds and the sum of the individual weights equals 3255 pounds, how many widgets (c) have been unloaded?

Consider the following simultaneous equations:

3a + 11b + 5c + 7d + 13e + 17f = 3255
3^a * 11^b * 5^c * 7^d * 13^e * 17^f = 351,125,648,000
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Context:
A baker bakes 5 loaves of bread an hour in one oven. He has 4 ovens. From Monday to Friday, he bakes for 5 hours, but on Saturday and Sunday, he only bakes for some hours. The baker bakes 1740 loaves of bread in 3 weeks. How many hours does the baker bake on Saturday and Sunday?
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Context:
The function $g(x)$ satisfies the equation
\[xg(y) = 2yg(x)\] for all real numbers $x$ and $y$. If $g(10) = 30$, find $g(2)$.
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Context:
Triangle $A B C$ has side lengths $A B=15, B C=18, C A=20$. Extend $C A$ and $C B$ to points $D$ and $E$ respectively such that $D A=A B=B E$. Line $A B$ intersects the circumcircle of $C D E$ at $P$ and $Q$. Find the length of $P Q$.
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Context:
If $x + \frac{1}{x} = 3$, what is $x^6 + \frac{1}{x^6}$?
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Context:
已知椭圆$\frac{{{x}^{2}}}{{{a2}}}+\frac{{{y}^{2}}}{{{b2}}}=1\left( a>b>0 \right)$的离心率为$\frac{\sqrt{2}}{2}$，右焦点为$F\left( 1,0 \right)$，三角形$ABC$的三个顶点都在椭圆上，设它的三条边$AB$、$BC$、$AC$的中点分别为$D$、$E$、$F$，且三条边所在直线的斜率分别为${{k}_{1}},{{k2}},{{k}_{3}}\left( {{k1}}{{k}_{2}}{{k3}}\ne 0 \right)$.若直线$OD$、$OE$、$OF$的斜率之和为-1（$O$为坐标原点），则$\frac{1}{{{k}_{1}}}+\frac{1}{{{k2}}}+\frac{1}{{{k}_{3}}}=$______.
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Context:
Calculate the limit of the following expression as $n$ approaches infinity, assuming $n$ is a positive integer: $$\lim_{n\to\infty}\frac{2-4+\cdots+(2n-2)-2n}{\sqrt{ (n^2+4)}+ \sqrt{ (n^2-4)}}$$
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Context:
Given \( x \in [0, 2\pi] \) and \( 2 \cos x \leqslant |\sqrt{1+\sin 2x} - \sqrt{1-\sin 2x}| \leqslant \sqrt{2} \), find the range of values for \( x \).
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\boxed{\left[ \frac{\pi}{4}, \frac{7 \pi}{4} \right]}


Context:
In a $10 \times 10$ grid square with the centers of all unit squares marked (a total of 100 points), what is the minimum number of lines, not parallel to the sides of the square, that need to be drawn to cross out all the marked points?
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Context:
Let $MATH$ be a trapezoid with $MA = AT = TH = 5$ and $MH = 11$. Point $S$ is the orthocenter of $\triangle ATH$. Compute the area of quadrilateral $MASH$.
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Context:
On one particular Wednesday, Jack worked \( t-2 \) hours and earned \( 3t-2 \) dollars per hour. His coworker Bob worked 1.5 times more hours than Jack but earned \( 2t-7 \) dollars per hour less than Jack. After paying a fixed tax of $10 each, they both netted the same amount of earnings. Determine the value of \( t \).
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Context:
4. Sofia and Viktor are playing the following game on a $2022 \times 2022$ board:

- First, Sofia covers the entire board with non-overlapping dominoes that do not extend outside the board,
- Then, Viktor writes down a natural number $n$ on a piece of paper without seeing the arrangement of the dominoes on the board, and then examines the board with the dominoes, selects $n$ dominoes, and fixes (glues) them to the board,
- After that, Sofia removes the remaining (unfixed) dominoes from the board and tries to achieve a new complete covering of the board, which will differ from the initial one.

If Sofia manages to find a new different covering, she wins; otherwise, Viktor wins.

Determine the smallest possible value of $n$ for which Viktor can always win, regardless of the initial covering.
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Context:
The king of the Lazy Kingdom issued a decree on Sunday, April 1, 2007, eliminating Fridays from all subsequent weeks. Since then, in his kingdom, Saturday always follows Thursday and the week has only six days. What day of the week will April 9, 2008, fall on in the Lazy Kingdom? (Don't forget that the year 2008 is a leap year!)
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Context:
Given a square with an area of $1$, if the square revolves around one of its sides for one full turn, the lateral surface area of the resulting geometric body is ______.
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Context:
There are five students taking a graduation photo in a row, among them, student A cannot stand next to student B, and students B and C must stand together. The number of different arrangements is \_\_\_\_\_\_ (answer in digits).
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Context:
A sequence of real numbers $x_0, x_1, x_2, \ldots$ is defined as follows: $x_0 = 1989$ and for each $n \geq 1$,
\[ x_n = -\frac{1989}{n} \sum^{n-1}_{k=0} x_k. \]
Calculate the value of $\sum^{1989}_{n=0} 2^n x_n.$
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Context:
For a positive integer \( k \), find the greatest common divisor (GCD) \( d \) of all positive even numbers \( x \) that satisfy the following conditions:
1. Both \( \frac{x+2}{k} \) and \( \frac{x}{k} \) are integers, and the difference in the number of digits of these two numbers is equal to their difference;
2. The product of the digits of \( \frac{x}{k} \) is a perfect cube.
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Context:
The rectangle $3 \times 100$ consists of 300 squares $1 \times 1$. What is the largest number of diagonals that can be drawn in the squares so that no two diagonals share endpoints? (In one square, two diagonals can be drawn without sharing endpoints. Shared internal points are allowed.)
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Context:
Given the function f(x) = 2^x - log_{1/2} x and that a is a root of f(x), if x_0 > a, determine the relationship between f(x_0) and 0. Express your answer as a comparison to 0 (e.g., > 0, < 0, = 0).
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Context:
Lucas chooses one, two or three different numbers from the list $2, 5, 7, 12, 19, 31, 50, 81$ and writes down the sum of these numbers. (If Lucas chooses only one number, this number is the sum.) How many different sums less than or equal to 100 are possible?
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Context:
Let $f(x) =\frac{x+a}{x+b}$ satisfy $f(f(f(x))) = x$ for real numbers $a, b$. If the maximum value of a is $p/q$, where $p, q$ are relatively prime integers, what is $|p| + |q|$?
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Context:
The cells of a $2 \times 2019$ table must be filled with numbers (exactly one number in each cell) according to the following rules. The top row must contain 2019 real numbers, none of which are equal to each other, and the bottom row must contain the same 2019 numbers, but in a different order. Each of the 2019 columns must contain two different numbers, and the sum of these two numbers must be a rational number. What is the maximum number of irrational numbers that could be in the first row of the table?

(S. Kudrya)
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Context:
The circle touches the sides of an angle at the points \(A\) and \(B\). A point \(M\) is chosen on the circle. The distances from \(M\) to the sides of the angle are 24 and 6. Find the distance from \(M\) to the line \(AB\).
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Context:
Five positive integers (not necessarily all different) are written on five cards. The sum of the numbers on every pair of cards is calculated, and only three different totals are obtained: 57, 70, and 83. Find the largest integer on any card. Express your answer as a single integer.
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Context:
Find the largest positive integer \( n \), such that there exists a set with \( n \) elements where exactly 1 element is divisible by \( n \), exactly 2 elements are divisible by \( n-1 \), ..., exactly \( n-1 \) elements are divisible by 2, and \( n \) elements are divisible by 1.
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Context:
For a positive integer \( n \), let \( \varphi(n) \) denote the number of positive integers less than or equal to \( n \) that are coprime to \( n \). Let \( f(n) \) denote the smallest positive integer greater than \( n \) that is not coprime to \( n \). If \( f(n) = m \) and \( \varphi(m) = n \), then the pair of positive integers \( (n, m) \) is called a friendly pair. Find all friendly pairs.
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Context:
Two ants sit at the vertex of the parabola $y = x^2$.  One starts walking northeast (i.e., upward along the line $y = x$ and the other starts walking northwest (i.e., upward along the line $y = -x$).  Each time they reach the parabola again, they swap directions and continue walking.  Both ants walk at the same speed.  When the ants meet for the eleventh time (including the time at the origin), their paths will enclose 10 squares.  What is the total area of these squares?
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Context:
In order to obtain an income of Rs. 500 from a certain percentage stock at Rs. 90, one must make an investment of Rs. 1500. What is the percentage of the stock?
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Context:
Problem 3. A clock shows exactly one o'clock. A mosquito and a fly are sitting at the same distance from the center on the hour and minute hands, respectively. When the hands coincide, the insects swap places. How many times greater is the distance that the mosquito has traveled in half a day compared to the distance the fly has traveled in the same time?
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Context:
Simplify the expression \( \left( 7 + 4 \sqrt{3}
ight)^{\frac{1}{2}} - \left( 7 - 4 \sqrt{3}
ight)^{\frac{1}{2}} \) and express your answer as a simplified expression involving square roots.
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Context:
You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2 \times 1$ and $3 \times 1$. The playing field should be completely covered with these dominoes without any overlaps, and no domino should protrude beyond the playing field. Furthermore, all dominoes must be aligned in the same direction, meaning their longer sides must be parallel to each other. How many such coverings are possible?
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Context:
In a group of 8 boys and 6 girls, a committee of 8 persons is to be formed. The committee must have an equal number of boys and girls and at least two members under the age of 18. Additionally, at least one girl must have experience in leadership and one boy must have experience in event planning. In how many different ways can the committee be formed under these conditions?
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Context:
What is the maximum number of months in a year that can have five Sundays?
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Context:
Anya, Borya, and Vasya took the same 6-question test, where each question could be answered with "yes" or "no". The answers are shown in the table:

| Question # | 1 | 2 | 3 | 4 | 5 | 6 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| Anya | no | yes | no | yes | no | yes |
| Borya | no | no | yes | yes | no | yes |
| Vasya | yes | no | no | no | yes | no |

It turned out that Anya had two incorrect answers, and Borya had only two correct answers. How many incorrect answers does Vasya have?
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Context:
设$x ^ { 2 } + y ^ { 2 } - 2 x - 2 y + 1 = 0 ( x , y \in R )$，则$F ( x , y ) = \frac { x + 1 } { y }$的最小值为
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Context:
In a convex quadrilateral $ABCD$, it is given that $\angle BAD = \angle BCD = 90^\circ$ and $BC = CD$. Let $E$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Given that $\angle AED = 123^\circ$, find the degree measure of $\angle ABD$.
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Context:
Define $a_k = 2^{2^{k-2013}} + k$ for all integers $k$. Simplify the expression:

$$(a_0 + a_1)(a_1 - a_0)(a_2 - a_1)\cdots(a_{2013} - a_{2012}).$$
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Context:
Given a sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $S_n$, $S_{n-1}$, and $S_{n+1}$ (for $n \geqslant 2$) form an arithmetic sequence, and $a_2 = -2$, then $a_4 =$ ______.
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Context:
The line $ax+my-2a=0$ ($m \neq 0$) passes through the point $(1,1)$. The angle of inclination of this line is ______.
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Context:
In each cell of a $5 \times 5$ board, there is either an X or an O, and no three Xs are consecutive horizontally, vertically, or diagonally. What is the maximum number of Xs that can be on the board?
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Context:
## Task A-4.5. (4 points)

A rectangular path with a width of $1.5 \mathrm{~m}$ and a length of $20 \mathrm{~m}$ needs to be tiled with identical tiles in the shape of an isosceles right triangle with legs of length $50 \mathrm{~cm}$, such that the legs are parallel to the sides of the rectangle. Determine the number of ways this can be done.
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Context:
Alice's white water rafting event is composed of 40 students and 10 instructors, including herself, in her class. In total, there are 4 classes participating in the event. Each student has a 40% probability of bringing their own life vest. How many more life vests will Alice need to get so that everyone attending the event has one?
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Context:
Let $a = -\sqrt{3} + \sqrt{5} + \sqrt{7}$, $b = \sqrt{3} - \sqrt{5} + \sqrt{7}$, $c = \sqrt{3} + \sqrt{5} - \sqrt{7}$. Evaluate the expression:
\[
\frac{a^4}{(a-b)(a-c)} + \frac{b^4}{(b-c)(b-a)} + \frac{c^4}{(c-a)(c-b)}.
\]
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Context:
Given complex numbers \( x \) and \( y \), find the maximum value of \(\frac{|3x+4y|}{\sqrt{|x|^{2} + |y|^{2} + \left|x^{2}+y^{2}\right|}}\).
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Context:
Find the value of the ratio $\frac{I_1}{I_2}$, where $I_1=\int_{0}^{1} \frac{\tan^{-1}x}{x} dx$ and $I_2=\int_{0}^{\pi/2} \frac{x}{\sin x}dx$.
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Context:
Eve randomly chooses two **distinct** points on the coordinate plane from the set of all $11^2$ lattice points $(x, y)$ with $0 \le x \le 10$, $0 \le y \le 10$. Then, Anne the ant walks from the point $(0,0)$ to the point $(10, 10)$ using a sequence of one-unit right steps and one-unit up steps. Let $P$ be the number of paths Anne could take that pass through both of the points that Eve chose. The expected value of $P$ is $\binom{20}{10} \cdot \frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b$.
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Context:
Circles \(\omega_{1}\) and \(\omega_{2}\) intersect at points \(A\) and \(B\), and a circle centered at point \(O\) encompasses circles \(\omega_{1}\) and \(\omega_{2}\), touching them at points \(C\) and \(D\) respectively. It turns out that points \(A, C\), and \(D\) are collinear. Find the angle \(\angle ABO\).
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Context:
An inverted frustum with a bottom diameter of 12 and height of 18, filled with water, is emptied into another cylindrical container with a bottom diameter of 24. Assuming the cylindrical container is sufficiently tall, calculate the height of the water level in the cylindrical container. Express your answer as a decimal value with one digit after the decimal point.
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Context:
在 $(x+y+z)^{8}$ 的展开式中，求所有形如 $x^{2} y^{a} z^{b}(a, b \in \mathbf{N})$ 的项的系数之和。
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Context:
James can buy a new computer for $600 and have it last a certain number of years, or buy 2 used computers for $200 each that each last 3 years. He will save $200 by picking the cheaper option. How long will the new computer last?
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Context:
Baron Munchausen told a story. "There were a whole crowd of us. We reached a crossroads. Then half of our group turned left, a third turned right, and a fifth went straight." "But wait, the Duke remarked, the sum of half, a third, and a fifth isn't equal to one, so you are lying!" The Baron replied, "I'm not lying, I'm rounding. For example, there are 17 people. I say that a third turned. Should one person split in your opinion? No, with rounding, six people turned. From whole numbers, the closest to the fraction $17 / 3$ is 6. And if I say that half of the 17 people turned, it means 8 or 9 people." It is known that Baron Munchausen never lies. What is the largest number of people that could have been in the crowd?
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Context:
If the equation $9^x + a \cdot 3^x + 4 = 0$ has a solution, then the range of the real number $a$ is \_\_\_\_\_\_.
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Context:
Let \( m, n > 2 \) be integers. One of the angles of a regular \( n \)-gon is dissected into \( m \) angles of equal size by \( (m-1) \) rays. If each of these rays intersects the polygon again at one of its vertices, we say \( n \) is \( m \)-cut. Compute the smallest positive integer \( n \) that is both 3-cut and 4-cut.
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Context:
Find the minimum value of the function
\[f(x) = \sqrt{-x^2 + 4x + 21} - \sqrt{-x^2 + 3x + 10}.\]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
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Context:
Let $x=\sqrt{37-20\sqrt{3}}$. Find the value of \(\frac{x^4-9x^3+5x^2-7x+68}{x^2-10x+19}\).
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Context:
Brian goes fishing twice as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, how many fish did they catch altogether if Chris went fishing 10 times?
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Context:
Given the function $f(x) = 1 + x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots + \frac{x^{2013}}{2013}$, and $g(x) = 1 - x + \frac{x^2}{2} - \frac{x^3}{3} + \frac{x^4}{4} + \ldots - \frac{x^{2013}}{2013}$. Let $F(x) = f(x+3)g(x-4)$ and all zeros of $F(x)$ are within the interval $[a, b]$ ($a < b$, $a, b \in \mathbb{Z}$). The minimum value of $b-a$ is \_\_\_\_\_.
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Context:
In the figure, $ABCD$ is a square, $P$, $Q$, $R$, and $S$ are midpoints of the sides $AB$, $BC$, $CD$ and $DA$ respectively. Find the ratio of the shaded area to the area of the square $ABCD$.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
2. Dan is holding one end of a 26 inch long piece of light string that has a heavy bead on it with each hand (so that the string lies along straight lines). If he starts with his hands together at the start and leaves his hands at the same height, how far does he need to pull his hands apart so that the bead moves upward by 8 inches?
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Context:
What is the smallest number of kings that must be taken so that, after being placed arbitrarily on an $8 \times 8$ chessboard, there will necessarily be two kings attacking the same square?
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Context:
If $x$, $y$, and $z$ are real numbers such that $x^2 + z^2 = 1$ and $y^2 + 2y(x + z) = 6$, find the maximum value of $y(z - x)$.
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Context:
函数 $f(x)=\frac{\sqrt{2}}{2}(\cos x-\sin x) \sin \left(x+\frac{\pi}{4}\right)-2 a \sin x+b(a>0)$ 的最大值为 1, 最小值为 -4, 求 $b$ 的值.
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Context:
An isosceles triangle with a base equal to \(a\) has an inscribed circle, and three tangents are drawn to it such that they cut off three smaller triangles from the given triangle. The sum of the perimeters of these smaller triangles is \(b\). Find the side length of the given isosceles triangle.
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Context:
Let \( A = \{1, 2, 3, \cdots, 4n+2\} \) and \( M = \{2n+1, 4n+3, 6n+5\} \). For any non-empty subset \( B \) of \( A \), \( B \) is called an \( M \)-free set if the sum of any two numbers in \( B \) does not belong to \( M \). If \( A = A_1 \cup A_2 \), \( A_1 \cap A_2 = \varnothing \), and both \( A_1 \) and \( A_2 \) are \( M \)-free sets, then the ordered pair \( (A_1, A_2) \) is called an \( M \)-partition of \( A \). Find the number of all \( M \)-partitions of \( A \).
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Context:
In triangle \(ABC\), the median \(AD\) and the angle bisector \(BE\) are perpendicular and intersect at point \(F\). It is known that \(S_{DEF} = 5\). Find \(S_{ABC}\).
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Context:
A box contains 5 white balls and 6 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate in color starting with a black ball?
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Context:
454. Several identical boxes together weigh 10 tons, with each of them weighing no more than 1 ton. What is the minimum number of three-ton trucks needed to haul away all this cargo in one trip?
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Context:
There is a rectangular prism made of 1 in cubes that has been covered in tin foil. There are exactly 128 cubes that are not touching any tin foil on any of their sides. If the width of the figure created by these 128 cubes is twice the length and twice the height, what is the measure m in inches of the width of the foil covered prism?
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Context:
The union of sets $A$ and $B$ is $A \cup B = \left\{a_{1}, a_{2}, a_{3}\right\}$. When $A \neq B$, the pairs $(A, B)$ and $(B, A)$ are considered different. How many such pairs $(A, B)$ are there?
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Context:
Fifteen students numbered $1, 2, 3, \ldots, 15$ stand in a circle facing inward. On the first turn, the student with number 1 turns around to face outward. On the second turn, the students numbered 2 and 3 turn around. On the third turn, the students numbered $4, 5, 6$ turn around, and so on. On the 15th turn, all students turn around. After the 12th turn, how many students are still facing outward?
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Context:
Determine the Löwenheim number of the extended first-order logic $L(Q^c)$, which includes Chang's quantifier $Q^c$. What is this Löwenheim number?
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Context:
Thirty people are seated at three round tables, each with ten seats, at an alumni dinner. After each time interval \( \Delta t \), everyone changes position simultaneously to sit next to different people on their left and right. What is the minimum number of seating configurations required for everyone to have sat next to every other person at least once? If it is not possible for everyone to sit next to each other just once, specify the minimum number of configurations needed for everyone to sit next to each other at least once.
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Context:
Suppose $n$ has at least two distinct odd prime factors. If $x^2 \equiv y^2 \pmod{n}$ and $\gcd(x,y) = 1$, find the probability that $\gcd(n, x-y)$ is a non-trivial factor of $n$. Express your answer as a fraction.
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Context:
An urn contains 18 black balls and 2 white balls. What is the expected number of draws (without replacement) needed to obtain two balls of different colors?
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Context:
Find the envelope of the family of straight lines given by the equation \( x \cos \alpha + y \sin \alpha = a \sec \alpha \), where \( \alpha \) is a parameter.
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Context:
Find the volume of the solid formed by the intersection of three cylinders, each with a length of 1 unit along the x, y, and z axes, starting from the origin, and each having a radius of 1 unit.
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Context:
Given a set of sentences \(\Sigma = \{\gamma_{\geq n} : n\) is even\}, where \(\gamma_{\geq n}\) is a sentence such that its models have the cardinality of at least \(n\), determine the smallest model of \(\Sigma\).
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Context:
In a game played inside the unit disc $\mathcal{D}$, player $\mathcal{A}$ chooses a point $p_0 \in \mathcal{D}$. At each step $n$, player $\mathcal{B}$ draws a line $\ell_n$ through $p_{n-1}$, and $\mathcal{A}$ chooses a point $p_n \in \mathcal{D} \cap \ell_n$. Player $\mathcal{B}$ wins if they can force the sequence $(p_n)$ to converge; otherwise, $\mathcal{A}$ wins. If both players play randomly, what is the probability that $\mathcal{B}$ wins?
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Context:
Suppose $\{A_{1},...,A_{k}\}$ is a set of linearly independent elements in an irreducible $\mathbb{R}$-algebra $D$ in $M_{n}(\mathbb{C})$. If $D$ is isomorphic to $\mathbb{H}$ and has dimension $4$ as an $\mathbb{R}$-algebra, find the value of $k$. \(\boxed{}\)
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Context:
Determine the limits of integration for the radial coordinate $r$ when using the divergence theorem to evaluate the surface integral \( I = \int \int F \cdot n \, dS \) where \( F = x^3 i + y^3 j + z^3 k \) and \( S \) is the surface of the cylinder \( x^2 + y^2 = 4 \) between \( -1 < z < 1 \).
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Context:
Calculate the volume of a regular triangular pyramid where the midpoint of its height is 2 units away from a lateral face and \(\sqrt{12}\) units away from a lateral edge. Round your answer to two decimal places if necessary.
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Context:
Given the functions \( f(x) = 3xe^{-x} \) and \( g(x) = \frac{x}{2} \), let \( v \) be the x-coordinate of their second intersection point (not 0). Find the value of \( u \), where \( 0 < u < v \), such that the area of the triangle with vertices \([0, 0], [u, f(u)], [u, g(u)]\) is maximized.
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Context:
Find a function of a natural number \( M \) that yields an odd square, such as 9 or 25, only when \( M \) is a triangular number.
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Context:
Let $K(O,R)$ be a circle with center $O$ and radious $R$ and $(e)$ to be a line thst tangent to $K$ at $A$. A line parallel to $OA$ cuts $K$ at $B, C$, and $(e)$ at $D$, ($C$ is between $B$ and $D$). Let $E$ to be the antidiameric of $C$ with respect to $K$. $EA$ cuts $BD$ at $F$.
If $R=\frac{5}{2}, AD=\frac{3}{2}$, calculate the area of $EBF$
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Context:
Evaluate the improper integral: $$\int_{0}^{\infty} \left(e^{-\frac{1}{x^2}}-e^{-\frac{4}{x^2}}\right) dx$$
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Context:
An egg plant is a hollow cylinder of negligible thickness with radius $2$ and height $h$. Inside the egg plant, there is enough space for four solid spherical eggs of radius $1$. What is the minimum possible value for $h$?
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Context:
When the celebrated German mathematician Karl Gauss (1777-1855) was nine years old, he was asked to add all the integers from 1 through 100. He quickly added 1 and 100, 2 and 99, and so on for 50 pairs of numbers each adding in 101. His answer was 50 · 101=5,050. Now find the sum of all the digits in the integers from 1 through 1,000,000 (i.e. all the digits in those numbers, not the numbers themselves).
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Context:
Find the smallest cardinality multiset \( x \) such that every element of the set \( z = \{ 1, 5, 6, 10, 11, 35, 36 \} \) can be expressed as a sum of elements from \( x \). For example, one possible multiset is \( x = \{1, 5, 5, 30\} \).
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Context:
Find the minimum value of the expression \(\frac{5x^{2}+8xy+5y^{2}-14x-10y+30}{(4-x^{2}-10xy-25y^{2})^{7/2}}\). Round your answer to the nearest hundredth if necessary.
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Context:
Determine the number of natural numbers \( n \) in the interval \([1005, 2010]\) for which the polynomial \(1 + x + x^2 + \cdots + x^{n-1}\) divides the polynomial \(1 + x^2 + x^4 + \cdots + x^{2010}\).
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Context:
Given the function $f : \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0}$ satisfying $f(1)=1$ and the functional equation \[ f\left(f(x+1) + f\left( x + f(x)\right)\right) = x+2, \] determine the value of $f(10)$.
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Context:
Find a value $p > 1$ such that the series $\sum_n f_n$ converges in $L^p(\mathbb{R})$, where the functions are defined as:
$$ f_n(x) = \frac{1}{1 + n^2\sqrt{x}}\chi_{[\exp(2n), 2\exp(2n+2)]} $$
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Context:
Determine the distribution of the sum of the squares of two independent random variables, each following a standard normal distribution \(N(0, 1)\).
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Context:
Determine the number of different solutions to the equation \(x^2_{1}+2x^2_{2}+3x^3_{3}=4x^2_{4}\) over the field with 7 elements, \(\mathbb{F}_{7}\).
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Context:
Given the unit disk $D^2$ in $\mathbb{R}^2$ and its boundary $S^1$, determine the well-known space that the quotient space $D^2/S^1$ is homeomorphic to.
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Context:
Given the differential equation \(y'(x, y_0) = y^2(x, y_0) + y(x, y_0) \sin(x)\) where \(y_0\) is the initial condition at \(x=0\), find the partial derivative \(\frac{\partial y}{\partial y_0}\) evaluated at \(y_0=0\), i.e., \(\frac{\partial y}{\partial y_0}\bigg|_{y_0=0}\).
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Context:
Suppose the random variable X has a moment generating function given by \( M(t) = \frac{e^{at}}{1-bt^2} \) for \(-1 < t < 1\). Given that the mean and variance of X are 3 and 2, respectively, find the value of \(a + b\).
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Context:
A die is rolled infinitely many times. What is the probability that the sequence (6,6) occurs before the sequence (1,2)?
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Context:
Given that \(x + x^{-1} = \frac{1}{2}(1 + \sqrt{5})\), find the value of \(x^{2000} + x^{-2000}\).
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Context:
Determine the algebraic degree of \( \tan(\pi/180) \).
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Context:
Determine the degree of \( \nu_{n,d}(C) \) for any curve \( C \subset \mathbb{P}^n \) with degree \( a \).
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Context:
Compute the exterior product \(\phi(X, Y)\) for the 2-form \(\phi = \sum_{j=1}^{m-1} y_j \wedge y_{j+1}\) on \(\mathbb{R}^m\) and the vectors \(X = (2, -1, 3, \dots, 2)\) and \(Y = (0, 2, 0, \dots, 2)\).
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Context:
Determine the generator of the quotient group \( \frac{\mathbb{Z}[a] \times \mathbb{Z}[b] \times \mathbb{Z}[c]} {\mathbb{Z}[a-c] \times \mathbb{Z}[b+c]} \) in terms of the original generators \(a\), \(b\), and \(c\).
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Context:
For a real number $\alpha>0$, consider the infinite real sequence defined by $x_1=1$ and
\( \alpha {x}_{n} = {x}_{1} + {x}_{2} + \cdots  + {x}_{n + 1}\; \) for \( n \geq  1. \)
Determine the smallest $\alpha$ for which all terms of this sequence are positive reals.
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Context:
Determine the value of \( a \in \mathbb{C} \) such that the function \( f(z) = z^2 + a\overline{z}^2 + 4z\overline{z} + 2z - 8\overline{z} + 1 + 2i \) is not holomorphic at any point in \( \mathbb{C} \).
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Context:
Determine the values of \( q \in \mathbb{R} \) for which the series \( \sum^{\infty}_{n=3}{\frac{1}{n(\log n) ^ {(\log(\log n))^q}}} \) converges.
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Context:
On a line, there are blue and red points, with at least 5 red points. It is known that on any segment with endpoints at red points that includes at least one red point inside, there are at least 3 blue points. Additionally, on any segment with endpoints at blue points that includes exactly 2 blue points inside, there are at least 2 red points. Determine the maximum number of blue points that can be on a segment with endpoints at red points, not containing other red points.
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Context:
Determine the number of conjugacy classes in a group of order 625, given that its center has an order of 25.
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Context:
The distinct symbols $\heartsuit, \diamondsuit, \clubsuit$ and $\spadesuit$ each correlate to one of $+, -, \times , \div$, not necessarily in that given order. Given that$$((((72 \,\, \,\, \diamondsuit \,\, \,\,36) \,\, \,\,\spadesuit \,\, \,\,0 ) \,\, \,\, \diamondsuit \,\, \,\, 32) \,\, \,\, \clubsuit \,\, \,\, 3)\,\, \,\,  \heartsuit \,\, \,\, 2 = \,\, \,\, 6,$$what is the value of$$(((((64 \,\, \,\, \spadesuit \,\, \,\, 8) \heartsuit \,\, \,\, 6) \,\, \,\, \spadesuit \,\, \,\, 5) \,\, \,\, \heartsuit  \,\, \,\, 1) \,\, \,\, \clubsuit \,\, \,\, 7) \,\, \,\, \diamondsuit \,\, \,\, 1?$$
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Context:
Determine the dimension of \(\mathcal{L}_F(K)\) as a vector space over \(F\), given that \(K/F\) is a finite-dimensional field extension with \([K:F]=n\).
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Context:
Compute the integral \( \oint_{\partial D(3i,5)} \frac{z}{1-e^z} \, dz \) using the Cauchy Integral Formula.
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Context:
Determine the fundamental group of \( \mathbb{R}^3 \) with the \( z \)-axis removed.
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Context:
Let $G$ be the symmetric group $S_5$ of permutations of five symbols. Consider the set $J$ of subgroups of $G$ that are isomorphic to the non-cyclic group of order $4$. Two subgroups $H$ and $K$ in $J$ are considered equivalent if they are conjugate, meaning there exists an element $g$ such that $gHg^{-1} = K$. How many equivalence classes are there in $J$?
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Context:
Given $N$ indistinguishable balls, where only one ball is heavier, and a balance scale that can determine which of two groups of balls is heavier, find the minimal number of weighings required to identify the heaviest ball. Each ball can be weighed at most $K$ times, where $K \geq 1$. For example, if $N = 19$ and $K = 2$, the answer is 3.
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Context:
Find the sum of the infinite series whose general term is given by \( \frac{n^2+2}{5^n} \).
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Context:
Determine the number of integer values of $a$ for which there exists at least one complex number $z$ that satisfies both $|z-ai|=a+4$ and $|z-2|<1$. Provide your answer as a single integer.
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Context:
Compute \( \sup_{f \in \mathcal{F}} |f(0)| \), where \( \mathcal{F} = \{ f \text{ holomorphic in } \mathbb{D} \text{ with } f(\mathbb{D}) \subset \mathbb{D}, f\left(\frac{1}{2}\right) = f'\left(\frac{1}{2}\right) = 0 \} \) and \( \mathbb{D} \) is the unit disc.
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Context:
What is the largest positive integer $r$ such that there are infinitely many elliptic curves over the rationals with rank at least $r$?
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Context:
Let \( O \) be the pole in polar coordinates. The hyperbola \(\rho=\frac{a^{2}}{5-6 \sqrt{2} \cos \theta}\) intersects the line \(\rho \cos \left(\theta-\frac{\pi}{4}\right) = a\) at points \( A \) and \( B \). Find the range of values for \( a \) such that the angle \(\angle A O B \leq \frac{2 \pi}{3}\), considering that \(\rho\) can take negative values.
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Context:
Calculate \( \lim\limits_{n \rightarrow +\infty} \int_{0}^{1} f_n(x) e^{-x^2} \, dx \), where \( f_n(x) = nxe^{-\sqrt{n}x} \) for \( x \in [0,+\infty) \). The sequence of functions \( f_n(x) \) converges pointwise to the null function on \( [0,+\infty) \) but not uniformly. However, there is uniform convergence on sub-intervals \( [a,+\infty) \) with \( a > 0 \).
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Context:
Adam has a box with $15$ pool balls in it, numbered from $1$ to $15$, and picks out $5$ of them. He then sorts them in increasing order, takes the four differences between each pair of adjacent balls, and finds exactly two of these differences are equal to $1$. How many selections of $5$ balls could he have drawn from the box?
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Context:
Given that the number of storms in a rainy season follows a Poisson distribution with a parameter \( \lambda \) that is uniformly distributed between 2 and 7, calculate the probability of having at least 5 storms in the season.
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Context:
Find the expected value of the maximum of three independent exponential random variables X, Y, and Z, with rate parameters \( \lambda = 1, 2, 3 \) respectively.
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Context:
Evaluate the limit: $$\lim_{x \to \frac{\pi}{2}} \tan \left(\frac{\pi}{4}\sin x\right)^\left({\dfrac 1{\tan(\pi \sin x)}}\right)$$
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Context:
Find the position of the string $y(x,t)$ that satisfies the standard wave equation for small amplitudes, given the initial conditions $y(x,0)=\sin x$ and $\frac{\partial y}{\partial t}(x,0)=\cos x$. Provide the solution for $y(x,t)$.
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Context:
Calculate the minimum distance of a cyclic code of length 9 over $\text{GF}(2)$ with the generator polynomial $g(x) = x^7 + x^6 + x^4 + x^3 + x + 1$. Provide your answer as a single number.
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Context:
Compute the integral \( \mathcal{J}_2 = \int_{0}^{2\pi} \cos(\cos t)\,dt \).
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Context:
Determine the smallest positive integer \( m \) for which \( \underbrace{100^{100^{\ldots^{100}}}}_m > \underbrace{3^{3^{\ldots^3}}}_{100} \).
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Context:
Determine the functional derivative \( \frac{\delta \mathfrak{F}}{\delta f} \) of the Fourier transform \( \mathfrak{F} \) given by:
\[ f \mapsto \int_{-\infty}^{\infty} f(x)e^{ix}dx. \]
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Context:
Compute the integral \( \int_{\mathbb{R}} \frac{i \omega - \omega^2}{1+\omega^6} \, d\omega \).
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Context:
Find all constants \( K > 0 \) such that for any positive measure space \((X, \Sigma, \mu)\) and any \(\mu\)-integrable function \(f: X \to \mathbb{R}\) satisfying \(\left|\int_E f\,d\mu\right| < K\) for all \(E \in \Sigma\), it follows that \(\|f\|_1 < 1\).
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Context:
Let \( X \) be an \( n \times (k+1) \) matrix such that \( X^TX \) is invertible. Determine the rank of the matrix \( I - X(X^TX)^{-1}X^T \), where \( I \) is the \( n \times n \) identity matrix.
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Context:
Compute the limit \(\lim_{x \to +\infty}\frac{\ln x}{\int_0^x \frac{|\sin t|}{t}\,dt}.\)
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Context:
Evaluate the expression \( \frac{\int_0^{\pi/2}\sin^{\sqrt{2}+1}(x) \, dx}{ \int_0^{\pi/2}\sin^{\sqrt{2}-1}(x) \, dx} \).
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Context:
Let \( f\left( x\right)  = {x}^{2} - {2x} \) . A set of real numbers \( S \) is valid if it satisfies the following:
 \( \bullet \) If \( x \in  S \) , then \( f\left( x\right)  \in  S \) .
 - If \( x \in  S \) and \( \underset{k \cdot  {f}^{\prime }s}{\underbrace{f(f(\ldots f}}\left( x\right) \ldots )) = x \) for some integer \( k \) ,then \( f\left( x\right)  = x \) .
 Compute the number of 7-element valid sets.
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Context:
B and P each deposit $100 in separate bank accounts. B's account has a nominal interest rate convertible semiannually, while P's account has a force of interest \( \delta \). After 7.25 years, each account has grown to $200. Calculate the difference \( i - \delta \), where \( i \) is the nominal interest rate for B's account.
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Context:
Let $I$ be the incenter of the triangle $ABC$, et let $A',B',C'$ be the symmetric of $I$ with respect to the lines $BC,CA,AB$ respectively. It is known that $B$ belongs to the circumcircle of $A'B'C'$.
Find $\widehat {ABC}$.
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Context:
Calculate the integral \( \int_{[0,1]} x^2 \, d\mu_F \), where \( F \) is the Cantor function. Use the following properties of the Cantor function:

- \( F(1-x) = 1 - F(x) \)
- \( F\left(\frac{x}{3}\right) = \frac{F(x)}{2} \) for all \( x \in [0,1] \)
- \( F(0) = 0 \)
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Context:
Given a positive integer $k$, there exists a positive integer $n$ such that the sum of the first $n$ positive integers can be obtained by appending $k$ digits to the right of $n$. Determine the remainder when $n$ is divided by 9.
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Context:
Find the residue of the sum \(1^{965} + 2^{965} + \ldots + 2018^{965}\) modulo 2021. What is the value of \(x\) such that:
\[
1^{965} + 2^{965} + \ldots + 2018^{965} \equiv x \pmod{2021}?
\]
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Context:
Evaluate the expression \(2\sqrt{2^3\sqrt{2^4\sqrt{2^5\sqrt{2\ldots}}}}\).
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Context:
Compute the following limit involving the hypergeometric function \(_2F_1(1,b;b+1;z)\) around the branch point at \(z=1\):
\[ \lim_{\epsilon\rightarrow 0^+} \ _2F_1(1,b;b+1;1+\epsilon)-\ _2F_1(1,b;b+1;1-\epsilon) \]
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Context:
Evaluate the integral \( \iiint_V xe^{-(x^2+y^2+z^2)^2}\,\mathrm{d} V \) using spherical coordinates, where \( V \) is the solid that lies between \( y=0 \) and the hemisphere \( x^2+y^2+z^2=4 \) in the right half-space \( (y>0) \).
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Context:
Given the process \(Y_t = Y_{t-1} + X_t - \phi^2 X_{t-2}\), where \(X_t = \phi X_{t-1} + Z_t + Z_{t-2}\) and \(Z_t\) is white noise with mean 0 and variance \(\sigma^2\), identify the parameters \((p,d,q)\) of the ARIMA process that describes \(Y_t\).
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Context:
Compute the limit: \[ \lim_{n\to\infty}\left(\prod_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n} \]
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Context:
Determine the image of the Lemniscate defined by the polar equation \(r^2 = \cos(2\theta)\) under the inversion transformation \(f(z) = \frac{1}{z}\).
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Context:
Determine the minimum number of nails required to fix any convex polygon on a table, where nails can touch the boundary but not pass through the polygon.
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Context:
Calculate the conditional probability $P(X_2 > X_3 > X_4 > X_5 | X_1 < X_2)$ for independent and identically distributed continuous random variables $X_1, X_2, X_3, X_4, X_5$ with a continuous distribution function.
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Context:
You have 100 boxes, and C of them contain a gift. You can open up to 16 boxes. What is the smallest number of C that will give you a probability greater than 0.5 of finding at least one gift? Assume that once you find a gift, you stop opening boxes.
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Context:
Determine the power to which the root \( x_0 \) of the equation \( x^{11} + x^{7} + x^{3} = 1 \) must be raised to obtain the value \( x_0^{4} + x_0^{3} - 1 \).
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Context:
Determine the maximum power of 2 that can be expressed as a sum of terms, where each term is a power of 2 multiplied by a power of 10, using only the digits 0, 1, 2, 4, and 8.
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Context:
Solve for \(x\) in the equation involving the Gaussian error function: \[ \frac{2}{\sqrt{\pi}}\int\limits_{0}^{x}e^{-t^2}dt = 0.7 \]
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Context:
Given the joint density function of two random variables \(X\) and \(Y\):
\[
f_{X,Y}(x,y)=\begin{cases}0.0009x(1+y), & \text{if } (x,y)\in \Omega \\
0, & \text{else}\end{cases}
\]
and the marginal distributions:
\[
f_X(x)=0.0036(21x-2x^2-x\sqrt{x}), \quad \text{and} \quad f_Y(y)=0.0009(1+y)\frac{y^4}{512}
\]
where
\[
\Omega=\{(x,y):4\sqrt{x}<y \quad \text{and} \quad 0<y<12\}
\]
Calculate the conditional probability \(P(X<4|Y>4)\).
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Context:
Two particles are positioned at opposite vertices of the same face of a cube. At any given time $t$, if the particles occupy the same edge, they remain there until time $t + 1$. Otherwise, one particle is randomly selected with equal probability and moves to one of its three neighboring vertices with equal probability. What is the expected value for the least amount of time it will take for the particles to be on the same edge?
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Context:
For varieties $X$ and $Y$ over an algebraically closed field and a morphism $f: X \rightarrow Y$, determine the minimum dimension of an irreducible component of the fiber $f^{-1}(y)$ for a closed point $y \in Y$. Provide your answer as a single integer.
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Context:
Given a context-free grammar with $n$ non-terminals and a maximum production rule length of $k$, determine the maximum length of the shortest word that can be generated by a grammar with $n+1$ non-terminals and a maximum production rule length of $2k$.
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Context:
Evaluate the limit: \[ L = \lim_{x\to \infty} \frac{\int_{1}^{x} t^{t-1} ( t + t\ln (t) +1 ) dt}{x^x} \]
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Context:
Calculate the remainder when \( \prod_{i=1}^{1903} (2^i + 5) \) is divided by 1000.
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Context:
Determine the radius of convergence for the power series \( f(x) = \sum_{n=1}^\infty \left(1+\frac{(-1)^n}{n}\right)^{n^2} \cdot \frac{(2x+1)^n}{n}.\)
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Context:
Charlie has a golden ticket that is a rectangle with side lengths in the ratio of the golden ratio \( \varphi = \frac{1+\sqrt{5}}{2} \). He folds the ticket so that two opposite corners coincide, creating a sharp crease. The area of the resulting shape can be expressed as \( a + b \varphi \). Find the value of \( \frac{b}{a} \).
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Context:
Let \( \phi \left( n\right) \) be the number of positive integers \( k < n \) which are relatively prime to \( n \) . For how many distinct values of \( n \) is \( \phi \left( n\right) \)
 equal to 12 ?
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Context:
In an equilateral triangle with side length 1, 8 points are placed either on the boundary or inside the triangle. Determine the maximum possible value for the shortest distance between any two of these points.
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Context:
What is the genus of the two-component unlink?
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Context:
Determine the values of \( \alpha \in \mathbb{R} \) for which the double integral \( \iint_{\mathbb{R}^2} \frac{\sin(x^2+y^2)}{(x^2+y^2+1)^\alpha} \, dx \, dy \) converges.
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Context:
Evaluate the limit: $$\lim_{n\to \infty} \frac{(n + 2){{\ln }^2}(n + 2) - (n+1){{\ln }^2}(n+1)}{\ln n}$$
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Context:
Evaluate the integral: $$\int_{-\infty}^{\infty} \left|2Af_0 \frac{\sin(2\pi f_0 t)}{2\pi f_0 t}\right|\,dt$$
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Context:
Find the function $f(x)$ such that its Maclaurin series has coefficients $f(0)=1$, $f'(0)=1$, $f''(0)=2$, $f^{(3)}(0)=3$, and so on. What is $f(x)$?
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Context:
Determine the minimum number of table arrangements required for 20 people, seated in groups of 4, so that everyone meets each other at least once.
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Context:
Given the group $G = \mathbb{Z}_6 \times \mathbb{Z}_3$, an element $a = (4, 1)$ in $G$, the subgroup $H = \langle a \rangle$, and the quotient group $G_1 = G/H$. Let $k$ denote the order of the coset of $(3, 1)$ in $G_1$, and let $\ell$ represent the order of the coset of $(2, 2)$ in $G_1$. Calculate $(k + \ell) \mod{5}$.
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Context:
Find the largest integer \( m \) such that \( a^{12} \equiv 1 \pmod{m} \) for all integers \( a \) that are coprime to \( m \).
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Context:
Let $H$ be a subgroup of a group $G$. Determine the stabilizer of the coset $aH$ for the action of $G$ on the set $G/H$ by left multiplication.
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Context:
Compute the Galois group of the splitting field of the polynomial \( t^4 - 3t^2 + 4 \) over \( \mathbb{Q} \).
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Context:
Determine the radius of convergence for the power series \( \sum_{k=0}^\infty 3^k \cdot z^{2^k} \).
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Context:
Evaluate the triple integral \( \iiint_W \sqrt{x^2 + y^2 + z^2} \, dx \, dy \, dz \) over the region \( W = \{ (x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq x \} \) using an appropriate change of variables.
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Context:
Evaluate the double integral \( \iint_Q \frac{5y-x}{y^2(y-2)^2} \, dA \), where \( Q \) is the region enclosed by the quadrilateral with vertices \((2, 4)\), \((6, 3)\), \((8, 4)\), and \((4, 8)\).
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Context:
Determine whether the series \( \sum_{n=1}^\infty \left(\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}\right) x^n \) converges using the Cauchy Root Test.
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Context:
Calculate the volume of the solid in $xyz$-space bounded by the surfaces \( z=\frac{1}{x^2+y^2+1} \) and \( z=\frac{1}{x^2+y^2+4} \).
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Context:
3. On each of the lines $x=2$ and $x=15$, there are 400 points with ordinates $1,2,3, \ldots, 400$. In how many ways can three points be chosen from the 800 marked points so that they form the vertices of a right triangle?
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Context:
Find the inverse of the polynomial \(2x^2 + x + 1\) in the quotient ring \(\frac{\mathbb{Z}_3[x]}{\langle x^3 + 2x + 2 \rangle}\).
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Context:
For positive reals \( p \) and \( q \) ,define the remainder when \( p \) and \( q \) as the smallest nonnegative real \( r \) such that \( \frac{p - r}{a} \) is an integer. For
 an ordered pair(a,b)of positive integers,let \( {r}_{1} \) and \( {r}_{2} \) be the remainder when \( a\sqrt{2} + b\sqrt{3} \) is divided by \( \sqrt{2} \) and \( \sqrt{3} \)
 respectively. Find the number of pairs(a,b) such that \( a,b \leq  {20} \) and \( {r}_{1} + {r}_{2} = \sqrt{2} \) .
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Context:
In parallelogram $ABCD$, the angle bisector of $\angle DAB$ meets segment $BC$ at $E$, and $AE$ and $BD$ intersect at $P$. Given that $AB = 9$, $AE = 16$, and $EP = EC$, find $BC$.
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Context:
Given a set of $n$ distinct positive integers, determine the number of ways to choose two disjoint subsets of size $k$ such that the absolute difference between the sums of the reciprocals of the elements in the two subsets is less than a given value $\epsilon$. Provide your answer as a single integer.
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Context:
Find the value of $a$ where the functions $\sin(x)$ and $\ln(x)$ intersect, i.e., solve $\sin(x) = \ln(x)$ for $x$. This value of $a$ is used in setting up the integral for the volume of the solid created by revolving the region bounded by $\sin(x)$, $\ln(x)$, and the $x$-axis about the line $y=5$. Provide the value of $a$.
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Context:
Evaluate the integral \( \int_{0}^{0.1} \cos \left(100 x^{2}\right) d x \) with an accuracy of \( \alpha=0.001 \).
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Context:
Find the greatest common divisor (GCD) of all numbers of the form \(p^8 - 1\), where \(p\) is a prime number greater than 5.
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Context:
Evaluate the integral \( \int _0^{2\pi}\sin^2 \left(\frac{-\pi}{6}+3e^{it} \right)dt \).
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Context:
In triangle $ABC$ , $\angle A = 45^o$ and $M$ is the midpoint of $\overline{BC}$. $\overline{AM}$ intersects the circumcircle of $ABC$ for the second time at $D$, and $AM = 2MD$. Find $cos\angle AOD$, where $O$ is the circumcenter of $ABC$.
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Context:
Given three periodic trigonometric functions $f_1(x)$, $f_2(x)$, and $f_3(x)$, where $x > 0$, the function $F(x) = f_1(x) \cdot f_2(x) \cdot f_3(x)$ is defined. It is known that $f_1(x)$ is positive for $\frac{2}{3}$ of its domain, $f_2(x)$ is positive for $\frac{1}{2}$ of its domain, and $f_3(x)$ is positive for $\frac{1}{2}$ of its domain. Determine the portion of the domain for which $F(x)$ is positive.
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Context:
Determine the Galois group of the polynomial \(X^5 - 1\) over the finite field \(\mathbb{F}_7\).
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Context:
Calculate the area enclosed by the curves $y = x$, $y = 2x$, $y = \frac{1}{x}$, and $y = \frac{3}{x}$ using the substitution $(x, y) = \left(\frac{u}{v}, uv\right)$. Provide your answer as a single numerical value.
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Context:
Determine the minimum number of lottery tickets needed to ensure that at least one cell matches the winning permutation of numbers 1 through 50.
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Context:
Find a function $f(x)$ such that \( \lim_{x\to\infty}\frac{f(x)}{x}=1 \) and \( \lim_{x\to0}\frac{f(x)}{x^2}=C \) for a constant \( C \neq 0 \).
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Context:
Find the remainder when the sum of the units digits of all the terms in the sequence \(2^{2^2}+4^{4^4}+6^{6^6}+\cdots+2016^{2016^{2016}}+2018^{2018^{2018}}\) is divided by 5.
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Context:
In quadrilateral $ABCD$, the angles are given as $\angle DAB=60^{\circ}$, $\angle ABC=90^{\circ}$, and $\angle BCD=120^{\circ}$. The diagonals $AC$ and $BD$ intersect at point $M$. If $MB=1$ and $MD=2$, determine the area of quadrilateral $ABCD$. Express your answer in simplest form.
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Context:
Calculate the gradient of the function \( L = \text{Tr}(XRX^T + YPY^T) \) with respect to the matrix \( X \), where \( R \) and \( P \) are known matrices.
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Context:
Evaluate the limit: $$ \lim_{x\to 0} \left(\frac{2+\cos x}{x^3\sin x}-\frac{3}{x^4}\right) $$
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Context:
A frog is placed at the origin on a number line, and moves according to the following rule: in a given move, the frog advanced to either the closest integer point with a greater integer coordinate that is a multiple of 3, or to the closest integer point with a greater integer coordinate that is a multiple of 13.  A [i]move sequence[\/i] is a sequence of coordinates which correspond to valid moves, beginning with 0, and ending with 39.  For example, 0, 3, 6, 13, 15, 26, 39 is a move sequence.  How many move sequences are possible for the frog?
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Context:
Find the largest non-negative value of \(a\) such that the following limit evaluates to \(\frac{1}{4}\):
\[
\lim\limits_{x \to 1} \left(\frac{-ax + \sin(x-1)+a}{x+\sin(x-1)-1}\right)^\frac{1-x}{1-\sqrt{x}}.
\]
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Context:
Let \(G = C_4 \rtimes_\phi C_3\), where \(C_4 = \langle x : x^4 = 1 \rangle\) and \(C_3 = \langle y : y^3 = 1 \rangle\), and \(\phi : C_3 \to \Aut(C_4)\) is defined by \(\phi(y^i) = f^i\) for \(i \in \mathbb{Z}\) and \(f \in \Aut(C_4)\) is defined by \(f(x) = x^3\). Find the order of the element \((x, y)\) in the group \(G\).
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Context:
Calculate the size of the stabilizer of the permutation \((1\ 2\ 3)(4\ 5\ 6)\) in the symmetric group \(S_7\) under the action defined by \(x \cdot y = xyx^{-1}\).
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Context:
Find one complex value of $x$ that satisfies the equation $\sqrt{3}x^7+x^4+2=0$.
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Context:
Let $ABCD$ be an isosceles trapezoid such that $CD >  AB = 4.$ Let $E$ be a point on line $CD$ such that $DE =2$ and $D$ lies between $E$ and $C.$ Let $M$ be the midpoint of $\overline{AE}.$ Given that points $A, B, C, D,$ and $M$ lie on a circle with radius $5,$ compute $MD.$
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Context:
Calculate the expected value $E(X)$, where $Z$ is a standard normal random variable, and $X = \min(Z+1, 0)$. Provide your answer as a single number.
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Context:
Find the number of subsets of ${1,2,3,...,10}$ that contain exactly one pair of consecutive integers. Examples of such subsets are ${1,2,5}$ and ${1,3,6,7,10}$.
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Context:
Find the smallest positive integer \( k \) such that the sum of the digits of \( k, 2k, 3k, \ldots, 2014k \) are all equal.
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Context:
Calculate the line integral \( \int_{\kappa} \arctan\left(\frac{y}{x}\right) dy - dx \), where \( \kappa \) is the part of the parabola \( y = x^2 \) from the point \((1,1)\) to \((0,0)\).
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Context:
Calculate the flux of the vector field \( F(x,y,z) = (zy^2-2x, \frac{1}{4}yz+z^2, xy+2x^2+2z) \) exiting the region \( \Omega = \{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq 16, z \leq \sqrt{x^2+y^2}\} \).
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Context:
Determine the range of values of \( \lambda \) for which the transformation \( T \) defined by \((Tf)(x) = x + \lambda \int_0^x (x-t)f(t)dt\) is a contraction with respect to the supremum norm on the space of continuous functions on the interval \([0,1]\).
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Context:
Two painters are painting a fence that surrounds 100 garden plots arranged in a single line. They paint one plot every other day in either red or green. The first painter is colorblind and cannot distinguish the color painted by the second painter, but he remembers which plots he painted. The first painter aims to maximize the number of transitions where a green plot borders a red plot. What is the maximum number of such transitions he can achieve, regardless of how the second painter acts?
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Context:
Calculate the value of the sum: \[ \sum_{k=1}^{10} k \binom{10}{k} \binom{20}{10-k} \]
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Context:
Find the radius of the inscribed sphere of a pyramid in \(\mathbb{R}^5\) with its vertex at \((1,0,0,0,0)\). The base of the pyramid is a regular 4-dimensional simplex lying in the hyperplane \(x_1=0\), which has an inscribed sphere of radius 1 centered at \((0,0,0,0,0)\).
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Context:
Let \( \{ Y_t \}_{t \ge 0} \) be defined by \( Y_t := 5+2t+2B_t \), where \( \{ B_t: t \ge 0 \} \) is a standard Brownian Motion. Let \( \{ T_t \}_{t \ge 0} \) be defined by \( T_t := e^{Y_t} \). Calculate the probability \( P(T_2 > T_4, T_2 >5) \).
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Context:
Given the three numbers $x, y=x^x, z=x^{(x^x)}$ with $.9<x<1.0$. Arranged in order of increasing magnitude, they are:
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Context:
Find the coefficient of \(x^{19}\) in the expansion of the generating function \(f(x) = \frac{(1-x^{5})^5}{(1-x)^{10}}\).
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Context:
Evaluate \( \lim_{n \to \infty} \int_0^n \frac{dx}{1 + n^2 \cos^2 x} \).
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Context:
We have an a sequence such that \( {a}_{n} = 2 \cdot  {10}^{n + 1} + {19} \) . Determine all the primes \( p \) , with \( p \leq  {19} \) , for which there exists some
 \( n \geq  1 \) such that \( p \) divides \( {a}_{n} \) .
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Context:
Evaluate the integral: \[ \int_{0}^{1}\frac{e^{2y}(4y^2-4y+2)-2}{y^2}dy \]
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Context:
Find the number of ways to order the integers $1,2,3,4,5,6,7$ from left to right so that each integer has all its divisors besides itself appearing to the left of it.
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Context:
Determine the accumulation points of the set \( A = \{ \sqrt{n} - \sqrt{m} : m, n \in \mathbb{N} \} \).
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Context:
Identify the algebraic structure defined by the following properties: Let $R$ be a ring and $G$ be a group of ring automorphisms of $R$. Consider the additive group $P=\bigoplus_{\sigma\in G}R$ and define multiplication for $x,y\in P$ as $(xy)_\omega=\sum_{\sigma\tau=\omega}x_\sigma \sigma^{-1}(y_\tau)$ for every $\omega\in G$, where the sum runs over $\sigma,\tau\in G$ such that $\sigma\tau=\omega$. What is the name of this algebraic structure?
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Context:
Find the non-zero value of $k$ such that the average area between the curves $x^2$ and $ax$ for $x \ge 0$, as $a$ varies from $0$ to $k$, equals $x^2$. What is this value of $k$?
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Context:
In a World Cup, the Brazil team has the following probabilities for each match outcome: winning with probability \( \frac{1}{2} \), drawing with probability \( \frac{1}{3} \), and losing with probability \( \frac{1}{6} \). Points are awarded as follows: 3 points for a win, 1 point for a draw, and 0 points for a loss. During the group stage, each team plays three matches, and a team needs at least 4 points to advance to the knockout stage. In the knockout stage, if a match ends in a draw, it goes to a penalty shootout, where Brazil has a \( \frac{3}{5} \) chance of winning. Calculate the probability that Brazil wins the championship with exactly one of the four knockout stage matches decided by a penalty shootout.
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Context:
Calculate the symmetric-decreasing rearrangement of the function \( f(x) = x \) on the interval \([0, 10]\), with \( f(x) = 0 \) elsewhere.
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Context:
Find the value of $n$ that maximizes the function $f(n) = \frac{\sigma_0(n)}{n}$, where $\sigma_0(n)$ is the number of divisors of $n$. Provide your answer as a single integer.
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Context:
Given that $X$ and $Y$ are normally distributed random variables with a covariance of $c$, find the covariance of $X^3$ and $Y^3$. Express your answer in terms of $c$.

\[ \text{Cov}(X^3, Y^3) = ? \]
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Context:
For which \(\alpha\) is the integral \(\int_Q \frac{1}{(x^2+2y^2)^{\alpha}}\,dx\,dy\) finite, where \(Q=[0,1]\times[0,1]\)?
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Context:
Evaluate the triple integral: $$\int_{0}^{6} \int_{0}^{12-2y} \int_{0}^{4-\frac{2y}{3}-\frac{x}{3}} x \, dz \, dx \, dy$$
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Context:
Evaluate the limit: \[ L = \lim_{x \to \infty} x^2 \times \log \left(x \cot^{-1}x\right) \]
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Context:
Evaluate the double integral: $$\int _0^{\frac{1}{4}}\int _{\sqrt{x}}^{\frac{1}{2}}\:\frac{\cos\left(\pi y\right)}{y^2}~\mathrm dy~\mathrm dx$$
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Context:
Determine the set of values for \( \alpha \) for which the improper integral \( \int_1^{+\infty} \frac{x^{\alpha} \cdot \sin x}{1+x^3} \, dx \) converges.
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Context:
Find the expected number of trials needed such that the sum of numbers picked from a uniform distribution over $[0, 1]$ is at least 1.
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Context:
Identify the group given by the presentation \( \langle a, b, c, d, e \mid cde = 1 \rangle \).
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Context:
Calculate the sum \( \sum_{k \in S(m,n)} \varphi(k) \), where \( S(a,b) = \{ k \in \mathbb{Z} \mid a \equiv a \pmod{k} + b \equiv b \pmod{k} \ge k \} \) and \( \varphi \) is Euler's totient function.
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Context:
Given real numbers \(a_{1}, a_{2}, \cdots, a_{n}\) which are all greater than 0 (where \(n\) is a natural number no less than 4) and satisfy the equation \(a_{1} + a_{2} + \cdots + a_{n} = 1\), find the maximum value of the sum \[ S = \sum_{k=1}^{n} \frac{a_{k}^{2}}{a_{k} + a_{k+1} + a_{k+2}} \] where \(a_{n+1} = a_{1}\) and \(a_{n+2} = a_{2}\).
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Context:
Consider a firm with quality \( \theta \in \{0, 1, \ldots, 100\} \), each equally likely. With probability \( \frac{1}{10} \), the firm can only send the message \( m = \varnothing \), and with probability \( \frac{9}{10} \), it can send either \( m = \varnothing \) or \( m = \theta \). A consumer observes the firm's message and forms a belief about the firm's expected quality \( b = E[\theta|m] \). The firm aims to maximize \( b \). Determine if there exists an equilibrium cutoff \( \theta^* \) such that: (i) every firm with \( \theta < \theta^* \) sends \( m = \varnothing \), and (ii) every firm with \( \theta \ge \theta^* \) sends \( m = \theta \) if possible. If such an equilibrium exists, find the value of \( \theta^* \).
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Context:
Given the vector space \( \mathbb{R}^2 \) over \( \mathbb{R} \) with the ordered basis \( \alpha = \{f_1, f_2\} \) where \( f_1 = (1,0) \) and \( f_2 = (1,1) \), consider the linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \( T((x,y)) = (-x+2y, -x+y) \). The matrix representation of \( T \) with respect to the basis \( \alpha \) is given by:
\[ [T]^{\alpha}_{\alpha} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \]
Compute \( T^{65}(f_1) \).
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Context:
Determine the number of non-conjugate $6 \times 6$ complex matrices with the characteristic polynomial $(x-5)^6=0$. Consider matrices that are not diagonalizable to the diagonal matrix $D$ with all diagonal entries equal to 5, but still have the same characteristic polynomial.
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Context:
Determine the largest number among ten consecutive numbers that can be obtained from an Antimagic arrangement of a 4x4 matrix, where each element is a unique number from 1 to 16. In an Antimagic arrangement, the sums of the rows, columns, and diagonals form an arithmetic sequence of ten consecutive numbers.
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Context:
Determine the number of solutions to Langford's problem for $n=50$.
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Context:
A pedestrian and a cyclist are moving in one direction along a highway, while a cart and a car are moving in the opposite direction. All participants move at constant speeds. The cyclist first overtakes the pedestrian at 10 o'clock, then meets the cart, and then meets the car, with equal time intervals between these events. The car first meets the cyclist, then meets the pedestrian at 11 o'clock, and then overtakes the cart, with equal time intervals between these events. At what time did the pedestrian meet the cart?
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Context:
Determine the number of different monic polynomials of degree $q$ in a finite field $F$ where the size of the field $|F|$ is greater than $q$. Provide your answer as a single number.
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Context:
Find the coefficient of \(x^{10}\) in the power series expansion of the function \(\frac{1+x^3}{1-2x+x^3-x^4}\).
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Context:
Evaluate the expression \( \left\lfloor \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{1000}} \right\rfloor \).
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Context:
We roll a fair die repeatedly. Let $X$ be the number of rolls needed to obtain a 5, and $Y$ be the number of rolls needed to obtain a 6. Calculate the expected value $E(X|Y=2)$.
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Context:
Estimate the smallest value of $r$ such that a circle of radius $r$ can contain $19$ non-overlapping circles of radius $1$. Express your answer to the nearest hundredth. For example, $11.00$, $5.60$, and $1.34$ are valid responses, but $11$ and $1.342$ are not. An invalid response will receive a score of zero.
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Context:
Calculate the expected value \( \mathbb{E}(\exp(1/X)) \) where \( X \sim N(0, \sigma_x) \).
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Context:
For every $k= 1,2, ...$ let $s_k$  be the number of pairs $(x,y)$ satisfying the equation $kx + (k+1)y = 1001 - k$ with $x$, $y$ non-negative integers. Find $s_1 + s_2 + ...+ s_{200}$.
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Context:
Evaluate the limit: $$ \lim_{x \rightarrow \infty} x\left(\frac{1}{x^2}+\frac{1}{(x+1)^2}+...+\frac{1}{(2x-1)^2}\right) $$
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Context:
4. On a plane, an infinite number of parallel lines are drawn, each separated from the next by a distance of 1. A circle with a diameter of 1 is randomly thrown onto the plane. Find the probability that a line intersecting the circle divides it into parts such that the ratio of the areas (of the smaller to the larger part) does not exceed the number \((2 \pi - 3 \sqrt{3}) : (10 \pi + 3 \sqrt{3})\).
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Context:
Let \( m = 30030 \) and let \( M \) be the set of its positive divisors which have exactly 2 prime factors. Determine the smallest positive integer \( n \) such that for any choice of \( n \) numbers from \( M \), there exist three numbers \( a \), \( b \), and \( c \) among them satisfying \( abc = m \).
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Context:
Among 10 chickens, one is false and weighs either 1kg with probability 1/2 or 3kg with probability 1/2. A machine independently picks the i-th chicken with probability 1/2 for weighing. After each weighing, all chickens are returned to the machine. After 3 weighings, the first set of chickens weighs 11kg, the second set weighs 12kg, and the third set weighs 13kg. What is the probability that the false chicken weighs 3kg?
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Context:
Evaluate the integral \( \int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}} \, dx \).
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Context:
Determine the number of homomorphisms from the symmetric group $S_3$ to the group $\mathbb{Z}_2 \times \mathbb{Z}_4$.
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Context:
Four particles are situated at the corners of a square with side length 'a' and move at a constant speed 'v'. Each particle moves towards the particle at the next corner. The time when all particles will meet is given by \( \frac{2a}{\sqrt{n} v} \). Find the value of \( n \).
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Context:
Approximate $\sqrt{f(t)}$ given that the Taylor expansion of $f(t)$ at $t=1$ is $f(t) = c^2(t-1)^2 + o(t-1)^2$, where $c$ is a constant.
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Context:
In a triangle $ABC$, it is given that $\cos 3A + \cos 3B + \cos 3C = 1$. Determine the measure of any one angle in the triangle.
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Context:
In a party attended by 2015 guests, among any 5 guests, at most 6 handshakes have been exchanged. Determine the maximal possible number of handshakes.
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Context:
Compute the correlation coefficient \( r(X_{(1)},X_{(3)}) \) for the order statistics of three independent random variables \( X_1, X_2, \) and \( X_3 \), each following an \( \text{Exp}(1) \) distribution.
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Context:
Evaluate the integral \( \int_{0}^{1}x^{m-1}\exp(-x)\,dx \).
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