Context:
Let $X$ be a topological space with a subspace $Y$. Suppose there is a strong deformation retraction onto $Y$, given by a continuous map $F: X \times [0,1] \to X$ such that for all $x \in X$, $F(x,0) = x$, $F(x,1) \in Y$, and for all $y \in Y$ and $t \in [0,1]$, $F(y,t) = y$. Determine if it is true that for any $x \in X$ and $t \in [0,1]$, $F(F(x,t),1) = F(x,1)$. Answer 'yes' or 'no'.
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Context:
In the pentagon \(ABCDE\), \(AB = AE = CD = 1\), \(BC + DE = 1\), and \(\angle ABC = \angle AED = 90^\circ\). Find the area of the pentagon.
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Context:
A  $6 \times 6$  board is given such that each unit square is either red or green. It is known that there are no  $4$  adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A  $2 \times 2$  subsquare of the board is *chesslike* if it has one red and one green diagonal. Find the maximal possible number of chesslike squares on the board.

*Proposed by Nikola Velov*
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Context:
Find the sum of the infinite series

\[
\frac{1}{3^{2}-1^{2}}\left(\frac{1}{1^{2}}-\frac{1}{3^{2}}\right)+\frac{1}{5^{2}-3^{2}}\left(\frac{1}{3^{2}}-\frac{1}{5^{2}}\right)+\frac{1}{7^{2}-5^{2}}\left(\frac{1}{5^{2}}-\frac{1}{7^{2}}\right)+\ldots
\]
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Context:
For which values of \( \alpha > 0 \) is the integral \( \int_{D_{\alpha}} f \, d\lambda < + \infty \) finite, where \( D_{\alpha} = \{(x,y) \in \mathbb{R}^2 \mid 0 < y, 0 < x < y^{\alpha} < 1 \} \) and \( f(x,y) = \frac{1}{(x+y)^2} \)?
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Context:
Find the limit $$\displaystyle{\lim_{x\to4}}\frac{\left(1-\cos \left( x-4 \right)\right)^4\,\ln \left( x-3 \right)}{\left(e^{\left(x-4\right)^2}-1\right)^2\,\sin ^5\left(\pi \,x\right)}$$
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Context:
What is the value of $a+b+c+d+e+f$ for the decimal representation of $\frac{4}{37}+\frac{3}{11}+\frac{23}{9}=2.abcdef\ldots$?
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Context:
Evaluate the integral \( \frac{2}{\pi}\int_{-\pi}^{\pi}{\cos\left(4x\right) \over \sin\left(x/2\right)}\,\mathrm{d}x \) using complex analysis.
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Context:
Compute \( \mathbb{E}[ \sigma B^2_{\sigma}] \) where \( \sigma = \inf \{t \geq 0 : |B_t| = \sqrt{2} \} \) and \( B \) is Standard Brownian Motion.
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Context:
Using 5 stars (*) and 4 circles (○) to populate a $3 \times 3$ grid, how many distinct configurations are there, considering rotations and reflections that result in identical patterns as the same?
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Context:
Evaluate the limit: \[ \lim_{n\to\infty}\left(\sum_{k=1}^{n}\frac{I_{k+1}}{k}\right) \] where \( I_n=\int^1_0x^ne^x\,dx \). Is it necessary to compute the values of \( I_n \) first?
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Context:
Let $p, q, r, s$ be distinct primes such that $p q-r s$ is divisible by 30. Find the minimum possible value of $p+q+r+s$.
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Context:
What is the rank of the integer \(-4\) when represented as an equivalence class of ordered pairs of natural numbers? Is it \(\omega\) or \(\omega+1\)?
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Context:
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? Provide your answer in the form of \(\frac{m}{n}\) and find the value of m + n.
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Context:
A person invested some money. The greater part is invested at 6% and the remainder at 5%. The annual income from the greater part is $160 more than the annual income from the smaller part, which is $6,000. What is the total amount invested?
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Context:
There are  $2^{2n+1}$  towns with  $2n+1$  companies and each two towns are connected with airlines from one of the companies. What’s the greatest number  $k$  with the following property:
We can close  $k$  of the companies and their airlines in such way that we can still reach each town from any other (connected graph).
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Context:
Evaluate the complex contour integral \( \frac{1}{2 \pi i} \int_{\mid z \mid = 1} \frac{6z^{98}}{23z^{99}-2z^{81}+z^4-7}dz \) using the residue theorem.
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Context:
In how many ways can one write the numbers 1, 2, 3, 4, 5, and 6 in a row so that given any number in the row, all of its divisors (not including itself) appear to its left?
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Context:
A positive integer $N$ is given. Panda builds a tree on $N$ vertices, and writes a real number on each vertex, so that $1$ plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be $M$ and the minimal number written $m$. Mink then gives Panda $M-m$ kilograms of bamboo. What is the maximum amount of bamboo Panda can get?
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Context:
Determine whether the integral \( \int_t^\infty \exp(-x+o(x))\,dx \) approaches \( \exp(-t+o(t)) \) as \( t \to \infty \).
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Context:
Determine if 47 is the largest number that can be uniquely partitioned into five parts (15, 10, 10, 6, 6), such that no two parts are relatively prime.
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Context:
Let $G$ be a finite group with a normal subgroup $H$ such that $H \cong A_5$ and $G/H \cong A_5$. Is $G \cong A_5 \times A_5$?
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Context:
Five cards are lying on a table as shown.

$$\begin{matrix} & \qquad & \boxed{\tt{P}} & \qquad & \boxed{\tt{Q}} \\ \\ \boxed{\tt{3}} & \qquad & \boxed{\tt{4}} & \qquad & \boxed{\tt{6}} \end{matrix}$$

Each card has a letter on one side and a whole number on the other side. Jane said, "If a vowel is on one side of any card, then an even number is on the other side." Mary showed Jane was wrong by turning over one card. Which card did Mary turn over? (Each card number is the one with the number on it. For example card 4 is the one with 4 on it, not the fourth card from the left/right)

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Context:
Elodie is putting on a fashion show and has five fabulous outfits for her five fabulous fashion models. However, on the day of the show, two of the outfits were ruined in an unfortunate permanent marker incident. Regardless, the show must go on and the remaining outfits will be presented. If each outfit can only be worn by one model and there is no time for any model to wear more than one dress, how many different shows can Elodie put on? (Note: Two shows are considered the same if they contain the same models wearing the same dresses.)
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Context:
Find digits $a,b,c,x$ ($a>0$) such that $\overline{abc}+\overline{acb}=\overline{199x}$
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Context:
Brian writes down four integers \( w > x > y > z \) whose sum is 44 . The pairwise positive differences of these numbers are
 1,3,4,5,6,and 9 . What is the sum of the possible values for \( w \) ?
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Context:
Determine whether the limit \( \lim_{n \to \infty} \frac{I_n}{A_n} \) exists. If it exists, find its value.
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Context:
Exercise 4. A group of 20 dwarves coming out of the mine sit around a round table to count the gold nuggets each has mined. They make the following observations:

$\triangleright$ The difference in the number of nuggets between two neighboring dwarves is always 2 or 3.

$\triangleright$ All the dwarves have mined a different number of nuggets.

What is the greatest possible difference in nuggets between the dwarf who found the most and the one who found the least?
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Context:
Find the limit of the series \( \lim _{ n\rightarrow \infty }{ \sum _{ k=1 }^{ n }{ { k }^{ \left( \frac { 1 }{ 2k } \right) } } } \). Does the series converge, and if so, what is its limit?
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Context:
Given  $2014$  points in the plane, no three of which are collinear, what is the minimum number of line segments that can be drawn connecting pairs of points in such a way that adding a single additional line segment of the same sort will always produce a triangle of three connected points?
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Context:
Determine whether the creation operator $c^*(f)$, where $f \in H$ and $H$ is a Hilbert space representing single-particle states, is a bounded linear operator on the fermionic Fock space $\mathfrak{F}$. If it is bounded, find its norm.
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Context:
Seven students count from 1 to 1000 as follows:

•Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000.

•Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.

•Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.

•Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.

•Finally, George says the only number that no one else says.

What number does George say?

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Context:
## Zadatak B-3.4.

Riješite jednadžbu

$$
\frac{\sin \left(x-\frac{2015 \pi}{2}\right)-\cos ^{3} x}{\sin (2 x+2015 \pi)}=\frac{1}{2}
$$

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Context:
Determine the conditions on \(\lambda\) that ensure the constraint function in the given optimization problem is convex.
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Context:
$10$ cards labeled 1 through $10$ lie on a table. Kevin randomly takes $3$ cards and Patrick randomly takes 2 of the remaining $7$ cards. What is the probability that Kevin’s largest card is smaller than Patrick’s largest card, and that Kevin’s second-largest card is smaller than Patrick’s smallest card?

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Context:
Determine whether the dimension of the vector space \( V = \prod_{i \in I} \mathbb{C} \), where \( I \) is a set, is equal to the cardinality of \( I \). If not, describe the dimension of \( V \) as an infinite cardinal.
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Context:
In the plane a point  $O$  is and a sequence of points  $P_1, P_2, P_3, \ldots$  are given. The distances  $OP_1, OP_2, OP_3, \ldots$  are  $r_1, r_2, r_3, \ldots$  Let  $\alpha$  satisfies  $0 < \alpha < 1.$  Suppose that for every  $n$  the distance from the point  $P_n$  to any other point of the sequence is  $\geq r^{\alpha}_n.$  Determine the exponent  $\beta$ , as large as possible such that for some  $C$  independent of  $n$
\[r_n \geq Cn^{\beta}, n = 1,2, \ldots\]
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Context:
Given a simple graph $G$ with degree sequence $D_G$, and another sequence of numbers $D$ such that $D_G[i] \ge D[i]$ for all $i$ and $\left( \sum_{i} D_G[i] - \sum_{i} D[i] \right)$ is an even number, determine if there is a polynomial-time algorithm to realize $D$ as the degree sequence of a spanning subgraph of $G$ by removing edges from $G$. Answer 'yes' or 'no'.
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Context:
Determine whether the congruence $x^3 \equiv 5 \pmod{23}$ has a solution.
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Context:
Given a sequence {a_n} where the sum of its first n terms is S_n, and it satisfies the equation S_n = a_n^2 + bn (a and b are constants), with a_9 = π/2. Let the function f(x) = 2 + sin(2x) - 2sin^2(x/2), and denote y_n = f(a_n). Calculate the sum of the first 17 terms of the sequence {y_n}.
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Context:
Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP=5, PB=15, BQ=15,$ and $CR=10.$ What is the area of the polygon that is the intersection of plane $PQR$ and the cube?
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Context:
Determine whether the ordinary differential equation \( \frac{d^2u}{dx^2} + \frac{u}{A} = 0 \) has a solution that satisfies the following conditions: \( \left.\frac{d^2u}{dx^2}\right|_{x=0} = 0 \), \( u(x=0) = B \), and \( \left.\frac{du}{dx}\right|_{x \to \infty} = 0 \), where \( A \), \( B \), and \( C \) are constants.
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Context:
Equilateral $\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?
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Context:
\( f \) is a function that maps positive integers to positive integers, \( f(1)=5 \). It satisfies the equation \( f(m+n)=f(m)+f(n)+5mn-1 \) for all \( m, n \in \mathbb{N}_{+} \). Find \( f(n) \).
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Context:
Find the smallest positive integer \( n \) such that in any set of \( n \) irrational numbers, there always exist 3 numbers for which all pairwise sums are still irrational.
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Context:
In analytic geometry, the range of the inclination angle $\theta$ of a line is $\underline{\quad\quad}$.
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Context:
For what real \( k \) can we find a continuous function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(f(x)) = kx^9 \) for all \( x \)?
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Context:
过点$M\left( -1,0 \right)$的直线$l$与抛物线$C$：${{y}^{2}}=4x$交于$A,B$两点（$A$在$M,B$之间），$F$是$C$的焦点，点$N$满足$\overrightarrow{NF}=6\overrightarrow{AF}$，则$\vartriangle ABF$与$\vartriangle AMN$的面积之和的最小值是______.
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Context:
4. Find the number of finite increasing sequences of natural numbers $a_{1}, a_{2}, \ldots, a_{k}$, of all possible lengths $k$, for which $a_{1}=1, a_{i} \mid a_{i+1}$ for $i=1,2, \ldots, k-1$, and $a_{k}=969969$.

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Context:
Find the length of the shortest path on the surface of a unit regular tetrahedron between the midpoints of its opposite edges.
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Context:
What total cannot be achieved when throwing two dice that have been modified by replacing 1, 3, and 5 with -1, -3, and -5 respectively, and leaving the even numbers unchanged?
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Context:
A deck of cards consists of $30$ cards labeled with the integers $1$ to $30$, inclusive. The cards numbered $1$ through $15$ are purple, and the cards numbered $16$ through $30$ are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number $32$. Lilith wants to pick from the expanded deck a hand of two cards such that at least one card is green. Find the number of distinguishable hands Lilith can make with this deck.
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Context:
Suppose $Y$ is a multivariate random variable taking values in $\mathbb{R}^k$, where $k \geq 3$. If the marginal distribution function of each coordinate of $Y$ can only take values 0 and 1, does it imply that $Y$ is constant almost surely? Answer 'yes' or 'no'.
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Context:
There are $\displaystyle{2n}$ students in a school $\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}$. Each week $\displaystyle{n}$ students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{The minimum number of trips needed is 6.}


Context:
On the basis of VSEPR theory, what geometry is predicted for the central sulfur atom in SOCl₂?
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Context:
Evaluate $$\lceil\sqrt{5}\rceil + \lceil\sqrt{6}\rceil + \lceil\sqrt{7}\rceil + \cdots + \lceil\sqrt{29}\rceil$$Note: For a real number $x,$ $\lceil x \rceil$ denotes the smallest integer that is greater than or equal to $x.$
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Context:
What is the least integer whose square is 75 more than its double?
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Context:
Find all real numbers $x$ such that $x - \lfloor x \rfloor = \frac{2}{\frac{1}{x} + \frac{1}{\lfloor x \rfloor}}$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Determine if your solution set includes integers.
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Context:
Find the smallest 10-digit number such that the sum of its digits is greater than that of any smaller number.
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Context:
In a bowl, there are 19 marbles of varying sizes and weights. Out of these marbles, 5 are yellow, and the remainder are blue and red marbles in the ratio 3:4 respectively. The yellow marbles have a combined weight of 8 grams. The red marbles weigh twice as much as the yellow marbles per gram. The blue marbles are not counted for this problem. Calculate the difference in total weight between the red marbles and the yellow marbles.
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Context:
Let \(U\) be a finite set, \(D\) a set, and \(F = \{(X_\lambda, Y_\lambda)\}_{\lambda \in \Lambda}\) a finite family of pairs of subsets of \(U\). Suppose \(a \in U \setminus \cup_{\lambda \in \Lambda} (X_\lambda \cup Y_\lambda)\) and \(K \subseteq U\) satisfies that for any finite subset \(I\) of \(D^U\), \(I \vDash F\) implies \(P(I, K, U)\). Determine if \(a \in K\).
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Context:
Given that for any non-zero real number $m$, the inequality $|2m-1|+|1-m| \geqslant |m|(|x-1|-|2x+3|)$ always holds, find the range of the real number $x$.
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Context:
Let $x$ and $y$ be non-negative real numbers that sum to 1. Compute the number of ordered pairs $(a, b)$ with $a, b \in\{0,1,2,3,4\}$ such that the expression $x^{a} y^{b}+y^{a} x^{b}$ has maximum value $2^{1-a-b}$.
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Context:
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\cos t \sin( \sqrt{1+t^2})}{\sqrt{1+t^2}}dt \) using contour integration or other suitable methods.
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Context:
Given a triangle with sides 2, 3, and 4, find the radius of the smallest circle from which this triangle can be cut out.
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Context:
Find the least number \( c \) such that \( \sum_{i=1}^n x_i^2 \leq cn \) for all real numbers \( x_1, x_2, \ldots, x_n \) where each \( x_i \geq -1 \) and \( \sum_{i=1}^n x_i^3 = 0 \).
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Context:
Every cell of an $8\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. Find the maximum number of columns such that the sum of numbers inside the cells of those columns is less than $-3$.
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Context:
Determine the asymptotic equivalence of the function \( f(x) = \sum_{n=1}^\infty n^\varepsilon \frac{x^n}{n!} \) as \( x \to \infty \), where \( \varepsilon \in (0, 1) \). Find a function \( g(x) \) such that \( f(x) \sim g(x) e^x \).
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Context:
Given that the complex number $z$ and ${{(z-2)}^{2}}-8i$ are both purely imaginary, then $z=\boxed{\_\_\_\_\_}$
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Context:
Determine whether the series \( \sum_{n=0}^\infty \frac{\sin(2n+1)}{2n+1} \) converges, and if so, find its sum.
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Context:
Find the area of a wall where the base is part of the circle of radius 1 centered at the origin, lying in the first quadrant. The height of the wall is given by the function \( f(x, y) = y + 4x \). Use line integrals to determine the area.
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Context:
 Determine which inverse trigonometric function $g$ has the derivative $g'(x) = -\frac{1}{x^2+1}.$

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Context:
已知向量 \(\vec{a}=(0,1), \vec{b}=\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right), \vec{c}=\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)\)，且 \(x \vec{a}+y \vec{b}+z \vec{c}=(1,1)\)，求 \(x^{2}+y^{2}+z^{2}\) 的最小值。原始答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
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Context:
A function $f$ satisfies $f(4x) = 4f(x)$ for all positive real values of $x$, and $f(x) = 2 - |x - 3|$ for $2 \leq x \leq 4$. Find the smallest \( x \) for which \( f(x) = f(2022) \).
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Context:
NASA’s Perseverance Rover was launched on July $30,$ $2020.$ After traveling $292{,}526{,}838$ miles, it landed on Mars in Jezero Crater about $6.5$ months later. Which of the following is closest to the Rover’s average interplanetary speed in miles per hour?
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Context:
18.2. A segment 20 cm long is divided into two segments, and a square is constructed on each of them as a side. Find the lengths of the sides of these squares, if the difference in the areas of the squares is $120 \mathrm{~cm}^{2}$.

$$
\text { (7-8 grades) }
$$
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Context:
Samantha leaves her house at 7:15 a.m. to catch the school bus, starts her classes at 8:00 a.m., and has 8 classes that last 45 minutes each, a 40-minute lunch break, and spends an additional 90 minutes in extracurricular activities. If she takes the bus home and arrives back at 5:15 p.m., calculate the total time spent on the bus.
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Context:
Let $a$ and $b$ be the roots of the polynomial $x^2 - 10cx - 11d$, and let $c$ and $d$ be the roots of the polynomial $x^2 - 10ax - 11b$. Given that $a, b, c, d$ are distinct real numbers, find the value of $a + b + c + d$. \[ a + b + c + d = ? \]
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Context:
Triangle $ABC$ has $AB = AC$. Point $D$ is on side $\overline{BC}$ so that $AD = CD$ and $\angle BAD = 36^o$. Find the degree measure of $\angle BAC$.
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Context:
At $12:34$ and $12:43$, the time contains four consecutive digits. Find the next time after 12:43 that the time contains four consecutive digits on a 24-hour digital clock.

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Context:
In triangle \(ABC\), the angle bisectors at vertices \(A\) and \(C\) intersect at point \(D\). Find the radius of the circumcircle of triangle \(ABC\) if the radius of the circumcircle of triangle \(ADC\) with center at point \(O\) is \(R = 6\), and \(\angle ACO = 30^\circ\).
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Context:
Determine the largest squarefree positive integer \( n \) such that the ring \( \mathbb{Q}(\sqrt{-n}) \cap \overline{\mathbb{Z}} \) is a unique factorization domain (UFD).
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Context:
Determine the parameters \((p, d, q)\) for the \(ARIMA(p, d, q)\) process of the time series \(Y_t\), given the stationary process \(X_t\) defined by the equation \(X_t = \phi X_{t-1} + Z_t + Z_{t-2}\), where \(Z_t\) is white noise with mean 0 and variance \(\sigma^2\), and the process \(Y_t\) defined as \(Y_t = Y_{t-1} + X_t - \phi^2 X_{t-2}\).
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Context:
Find the orthogonal trajectories of the family of curves given by the equation \(3x^2 - y^2 = c\).
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Context:
Mia buys 10 pencils and 5 erasers for a total of $2.00. Both a pencil and an eraser cost at least 3 cents each, and a pencil costs more than an eraser. Determine the total cost, in cents, of one pencil and one eraser.
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Context:
Evaluate the double integral \( \int_0^1 \int_0^x \left( 1 - \frac{\sqrt{3}}{2 + \cos(2\pi t)}\right) dt \, dx \).
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Context:
Let \( A B C D \) be a quadrilateral and \( P \) the intersection of \( (A C) \) and \( (B D) \). Assume that \( \widehat{C A D} = 50^\circ \), \( \widehat{B A C} = 70^\circ \), \( \widehat{D C A} = 40^\circ \), and \( \widehat{A C B} = 20^\circ \). Calculate the angle \( \widehat{C P D} \).
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Context:
Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of side lengths $9$ and $10$ or of side lengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of side lengths $1$ and $12$ or $9$ and $10$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?
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Context:
Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits?

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Context:
Real numbers $x, y, z$ satisfy $$x+x y+x y z=1, \quad y+y z+x y z=2, \quad z+x z+x y z=4$$ The largest possible value of $x y z$ is $\frac{a+b \sqrt{c}}{d}$, where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\operatorname{gcd}(a, b, d)=1$. Find $1000 a+100 b+10 c+d$.
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Context:
Determine the cardinality of the set of functions from the set of natural numbers \( \mathbb{N} \) to the set of all subsets of the natural numbers \( \mathcal{P}(\mathbb{N}) \).
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Context:
Determine the number of edges in the graph $G_n$, where the vertex set $V(G_n)$ consists of all binary strings of length $n$ with at most one continuous block of $1$'s, and two vertices are adjacent if they differ in exactly one position.
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Context:
Determine the value of \( n = m \) such that the width of the 95% confidence interval for \( p_1 - p_2 \) is at most 0.1, regardless of the sampling results. The confidence interval is given by:
\[ \hat{p}_1 - \hat{p}_2 \pm z_{0.025} \sqrt{\frac{\hat{p}_1\hat{q}_1}{m} + \frac{\hat{p}_2\hat{q}_2}{n} } \]
where \( \hat{p}_1 = \frac{X}{m} \) and \( \hat{p}_2 = \frac{Y}{n} \), with \( X \sim \text{Binomial}(m, p_1) \) and \( Y \sim \text{Binomial}(n, p_2) \).
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Context:
Compute \( \mathbb{E}\left[\left(X_1 + 2X_2 + 3X_3\right)^2\right] \), where \( X_t \) is a Wiener process.
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Context:
Let $G$ be a simple connected graph with 2016 vertices and $k$ edges. Determine the maximal value of $k$ such that there exists a set of vertices whose removal disconnects the graph, regardless of the arrangement of the $k$ edges.
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Context:
Sam and Ben are each flipping fair coins. If Sam flips a single coin until he gets a tails, and Ben flips $10$ coins in total, what is the probability Sam and Ben get the same number of heads?
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Context:
Evaluate the expression: \[ \frac{2012!}{2^{2010}} - \sum_{k=1}^{2010} \frac{k^2 k!}{2^k} - \sum_{k=1}^{2010} \frac{k \cdot k!}{2^k} \]
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Context:
Determine the value of $n$ such that the limit \( \lim\limits_{x \to 0} \frac{x^n}{\sin(\cos x) -\sin x} \) is a non-zero finite real number.
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Context:
Identify the elements of the unit group of the polynomial ring $F[x]$, where $F$ is a field.
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Context:
Determine the number of points \((x, y)\) over the finite field \(\mathbb{F}_p\), where \(p\) is a prime number such that \(p \equiv 2 \pmod{3}\), that satisfy the equation \(y^2 = x^3 + 1\).
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Context:
Given $n$ balls, where $\frac{n}{3}$ are blue and $\frac{2n}{3}$ are red, each ball is thrown randomly into one of $k$ bins. Define $M_k$ as 1 if a bin has more red balls than blue balls, and 0 otherwise. Let $A = \sum_{i=0}^k M_i n_i$, where $n_i$ is the number of balls in bin $i$. What is the upper bound of $A$?
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Context:
Simplify the expression: \[ \frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}} \]
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Context:
Let $ABC$ be a triangle with $AB = 20$ and $AC = 18$. $E$ is on segment $AC$ and $F$ is on segment $AB$ such that $AE = AF = 8$. Let $BE$ and $CF$ intersect at $G$. Given that $AEGF$ is cyclic, then $BC = m\sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m + n$.
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Context:
Let \( P(z) \) be a monic polynomial with complex coefficients of degree \( n \) such that \( |P(z)| \le 1 \) for all \( |z| \le 1 \). Is it true that \( P(z) = z^n \)?
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Context:
Given that the function $f(x) = -x^3 + 1 + a$ $(\frac{1}{e} \leq x \leq e, e$ is the base of natural logarithms$)$ and the graph of $g(x) = 3\ln(x)$ have points that are symmetric about the $x$-axis, find the range of the real number $a$.
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Context:
Apply the operator \( \hat{u} \) to the derivative \( \dfrac{\partial f(x)}{\partial x} \), given that \( \hat{u}(f(x)) = f(u(x)) \) for any function \( f(x) \) over the real numbers \( \mathbb{R} \).
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Context:
The distance (as the crow flies, as are all distances in this problem) from Chalons to Vitry is 30 km, from Vitry to Chaumont is 80 km, from Chaumont to Saint-Quentin is 236 km, from Saint-Quentin to Reims is 86 km, from Reims to Chalons is 40 km. Calculate the distance from Reims to Chaumont in this closed polygon.
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Context:
A girl and a guy are going to arrive at a train station. If they arrive within 10 minutes of each other, they will instantly fall in love and live happily ever after. But after 10 minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between 8 AM and 9 AM with equal probability. The guy will arrive between 7 AM and 8:30 AM, also with equal probability. Let $\frac{p}{q}$ for $p$, $q$ coprime be the probability that they fall in love. Find $p + q$.
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Context:
Find the value of $\tan{t}$ if the series $\sum_{i=1}^{\infty} \tan^{-1}\left(\frac{1}{2i^{2}}\right)$ is equal to $t$.
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Context:
For the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$, if it is an increasing function on the interval $(0, +\infty)$, then $m = \_\_\_\_\_\_$.
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Context:
试求正整数k，使$f ( x ) = \sin k x \cdot \sin ^ { k } x + \cos k x \cdot \cos ^ { k } x - \cos ^ { k } 2 x$的值不依赖于x.
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Context:
Let $V$ be an $n$-dimensional real vector space with two different bases $B=\{v_1,\ldots,v_n\}$ and $B'=\{w_1,\ldots,w_n\}$. Consider an endomorphism $f:V\to V$ such that $f(v_i)=w_i$ for every $i=1,\ldots,n$. Is the matrix $M_{B',B}(f)$, associated with $f$ from the basis $B$ to the basis $B'$, equal to the identity matrix $I_n$?
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Context:
Two decimals are multiplied, and the resulting product is rounded to 27.6. It is known that both decimals have one decimal place and their units digits are both 5. What is the exact product of these two decimals?
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Context:
Let \( S_{n} \) denote certain subsets of the set of positive integers \( \{1, 2, 3, \cdots, 100\} \) that satisfy the condition: no number in the subset is twice any other number. What is the maximum number of elements that such a subset can contain?
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Context:
A noodle chef first rolls a piece of dough into a 1.6-meter-long cylindrical noodle stick, folds it in half, and then stretches it back to 1.6 meters; folds it in half again, and stretches it back to 1.6 meters;... doing this 10 times, and stretching it back to 1.6 meters each time, he makes noodles. At this point, if the noodles in his hands are stretched out and considered as a cylinder of uniform thickness, find the ratio of the final thickness (diameter) to the original thickness (diameter) of the noodle stick. Express your answer as a fraction.
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Context:
Let \( \{x_n\} \) be a non-decreasing sequence of integers such that \( \lim_{n\to\infty} \left( \frac{\text{number of elements in } \{x_i\}_{i\leq n} \text{ that are } \leq n }{n} \right) = \frac{1}{2} \). Determine if \( \lim_{n\to\infty} \left( \frac{\sum_{i=1}^n x_i}{n^2} \right) = 1 \).
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Context:
Determine the number of non-constant polynomial factors with leading coefficient 1 of the polynomial \(x^{2015} + 18\).
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Context:
The full time score of a football match was $3$-$2$. How many possible half time results can we have for this match?

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Context:
A bug moves along a circle of radius 200 cm centered at the origin in the $xy$-plane with a constant speed of 3 cm/s. The temperature at any point $(x, y)$ is given by the function $T(x, y)$. Calculate the rate of change of temperature experienced by the bug at time $t = \frac{\pi}{3}$, given that the circle is parameterized by $\gamma(t) = \langle 200 \cos t, 200 \sin t \rangle$. Provide your answer in cm/s.
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Context:
For every real number $x$, let $[x]$ be the greatest integer which is less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always

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Context:
Combine the following radicals: $\sqrt{a^{4} b^{3} c^{4}} \sqrt[3]{a b^{2} c}$
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Context:
Elizabetta wants to write the integers 1 to 9 in the regions of the shape shown, with one integer in each region. She wants the product of the integers in any two regions that have a common edge to be not more than 15. In how many ways can she do this?
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Context:
Workers were laying down a floor of size \( n \times n \) (\(10 < n < 20\)) with two types of tiles: \( 2 \times 2 \) and \( 5 \times 1 \). It turned out that they managed to completely cover the floor using the same number of tiles of each type. For which \( n \) could this happen? (Cutting the tiles and overlapping them is not allowed.)
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Context:
In order to pass $B$ going $40$ mph on a two-lane highway, $A$, going $50$ mph, must gain $30$ feet. Meanwhile, $C$, who is $210$ feet from $A$, is headed toward him at $50$ mph. If $B$ and $C$ maintain their speeds, then, in order to pass safely, $A$ must increase his speed by how many mph?
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Context:
Let $X$ be a $p \times q$ matrix with characteristic polynomial $(x-c)^r(x-d)^s$, where $c$ and $d$ are distinct complex numbers. Determine the dimension of the complex vector space $W$ of all $q \times p$ matrices $Y$ such that $XY=YX$. Provide your answer as a single integer.
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Context:
Consider a rectangular grid of  $ 10 \times 10$  unit squares. We call a *ship* a figure made up of unit squares connected by common edges. We call a *fleet* a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the least number of squares in a fleet to which no new ship can be added.
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Context:
Samuel is driving to San Francisco's Comic-Con in his car and he needs to travel 1,200 miles to the hotel where he made a reservation. At first, he drives at 60 miles/hour for 2 hours straight. Heavy rain starts, forcing Samuel to slow down to 40 miles/hour for the next hour. After the rain stops, he takes a 30-minute break.

He then speeds up to 70 miles/hour for 3 hours, but due to traffic, he only maintains this speed for 2 hours and 30 minutes.  He takes a 1 hour break before switching to a different car. In the new vehicle, he drives at 50 miles/hour for 4 hours before stopping at a friend's place for a 2-hour break.

Upon leaving his friend's place, Samuel switches to a motorcycle and maintains a speed of 80 miles/hour for the first hour. However, the motorcycle begins to malfunction, and he is forced to reduce his speed to 60 miles/hour for the next 3 hours. How far is he from the hotel?
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Context:
In the figure below, there are 4 identical circles placed inside an equilateral triangle. If the radii of the circles are 1 unit, what is the value of the area of the triangle?
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Context:
Determine the center, $Z(G)$, of the group $G$ generated by two elements, $\sigma$ and $\tau$, with orders 5 and 4, respectively, given that $\tau\sigma\tau^{-1}=\sigma^2$. The group $G$ has order 20, and every element $g \in G$ can be uniquely expressed as $g=\sigma^k\tau^m$ for $0 \le k < 5$ and $0 \le m < 4$. The commutator subgroup of $G$, $[G:G]$, is equal to $\langle\sigma\rangle$. What is $Z(G)$?
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Context:
Determine the number of roots of the equation $\displaystyle \frac{\mathrm d^{40}}{\mathrm dx^{40}}\left(x^3-8\right)^{20}=0$ within the interval $[-5,5]$.
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Context:
Kiera is making trail mix out of 16 bags of nuts and some bags of dried fruit. She wants each new portion of trail mix to be identical, containing the same combination of bags of nuts and bags of dried fruit, with no bags left over. The greatest number of portions of trail mix Kiera can make is 2. How many bags of dried fruit does Kiera have?
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Context:
On a board, the product \(a_{1} a_{2} \ldots a_{100}\) is written, where \(a_{1}, \ldots, a_{100}\) are natural numbers. Consider 99 expressions, each obtained by replacing one of the multiplication signs with an addition sign. It is known that exactly 32 of these expressions are even. What is the maximum number of even numbers among \(a_{1}, a_{2}, \ldots, a_{100}\)?
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Context:
Given 3 distinct natural numbers whose sum is 55, and the sum of any two of these numbers is a perfect square, what are the three natural numbers?
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Context:
Calculate the area of the region bounded by the curves $xy = 4$, $xy = 8$, $xy^3 = 5$, and $xy^3 = 15$. Express your answer in terms of natural logarithms.
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Context:
As shown in the diagram, \( D \), \( E \), and \( F \) are points on the sides \( BC \), \( CA \), and \( AB \) of \(\triangle ABC\), respectively, and \( AD \), \( BE \), \( CF \) intersect at point \( G \). Given that the areas of \(\triangle BDG\), \(\triangle CDG\), and \(\triangle AEG\) are 8, 6, and 14 respectively, find the area of \(\triangle ABC\).
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Context:
A standard deck of 52 cards is shuffled and dealt. Let \(X_{1}\) be the number of cards appearing before the first ace and \(X_{2}\) be the number of cards between the first and second ace (not counting either ace). Calculate the probability \(P(X_{2} = 10 | X_{1} = 15)\).
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Context:
In an acute-angled triangle \(ABC\), heights \(AP\) and \(CQ\) are drawn from vertices \(A\) and \(C\) to sides \(BC\) and \(AB\), respectively. Given that the area of triangle \(ABC\) is 18, the area of triangle \(BPQ\) is 2, and \(PQ = 2\sqrt{2}\), find the radius of the circumcircle of triangle \(ABC\).
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Context:
A 600 meter long train crosses a signal post in 40 seconds. How long will it take to cross a bridge of a certain length at the same speed if it takes 6 minutes to cross the bridge?
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Context:
Let $X_k$, $k \geq 0$, be iid random variables with pmf $p_j = P(X_1 = j)$ for $1 \leq j \leq m$, where $\sum_{j = 1}^{m} p_j = 1$. Let $T = \min\{n > 0 : X_n = X_0\}$ denote the first time we observe the initial value again. Compute $E[T]$. \(\text{Express your answer in terms of } p_j.\)
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Context:
Evaluate the integral \( \int_{0}^{\infty} x \exp\left(-\frac{x^2+a^2}{2}\right) I_0(ax) \, dx \), where \( I_0(x) \) is the modified Bessel function of the first kind of order zero.
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Context:
Let \( M = \{1, 2, 3, \cdots, 1995\} \) and \( A \subseteq M \), with the constraint that if \( x \in A \), then \( 19x \notin A \). Find the maximum value of \( |A| \).
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Context:
The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum?
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Context:
Find an integer solution to the quadratic congruence equation \(x^2 \equiv 7 \pmod{787}\), where \(1 \leq x \leq 786\).
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Context:
Nyusha has 2022 coins, and Barash has 2023. Nyusha and Barash toss all their coins simultaneously and count how many heads each gets. The one who gets more heads wins, and in case of a tie, Nyusha wins. What is the probability that Nyusha wins?
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Context:
Two concentric circles have radii 2006 and 2007. $ABC$ is an equilateral triangle inscribed in the smaller circle and $P$ is a point on the circumference of the larger circle. Given that a triangle with side lengths equal to $PA, PB$, and $PC$ has area $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime, find $a+b+c$.
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Context:
Let $m$ be the largest real solution to the equation
$\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}=x^2-11x-4$
 There are positive integers $a$, $b$, and $c$ such that $m=a+\sqrt{b+\sqrt{c}}$. Find $a+b+c$.
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Context:
Rthea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea?
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Context:
The red segments in the diagram have equal lengths. They overlap by identical small segments of length \( x \) cm. What is \( x \) in centimeters?
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Context:
In the acute-angled triangle $ABC$ the angle$ \angle B = 30^o$, point $H$ is the intersection point of its altitudes. Denote by $O_1, O_2$ the centers of circles inscribed in triangles  $ABH ,CBH$  respectively. Find the degree of the angle between the lines $AO_2$ and $CO_1$.
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Context:
Calculate the expected number of rolls required when rolling a fair 6-sided die repeatedly until the product of the last two rolls equals 6, using an absorbing Markov chain. Note that the process restarts if the previous roll was a 4 or 5.
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Context:
Find the minimum value of the expression \( \frac{x^4 + y^4 + z^2}{xyz} \) for real numbers \(x\), \(y\), and \(z\).
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Context:
Determine the number of ring homomorphisms from \( \mathbb{Z}[X]/(X^2-X) \) to \( \mathbb{Z}/6\mathbb{Z} \), excluding the zero homomorphism. Assume that any non-zero homomorphism satisfies \( f(1) = 1 \mod 6 \).
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Context:
On the map shown, there are a number of routes from Mathville to Algebratown. Each route must travel along the roads in the direction marked by the arrows. What is the total number of routes from Mathville to Algebratown? Express your answer as a single integer.
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Context:
Determine the cardinality of the set \( S = \{ f: \mathbb{R} \to \mathbb{R} \mid f(x) \neq 0, \text{ only for finitely many } x \in \mathbb{R} \} \).
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Context:
Identify the set of non-invertible elements in the formal power series ring $K[[x]]$ where $K$ is a field.
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Context:
A social club has $2k+1$ members, each of whom is fluent in the same $k$ languages.  Any pair of members always talk to each other in only one language.  Suppose that there were no three members such that they use only one language among them.  Let $A$ be the number of three-member subsets such that the three distinct pairs among them use different languages.  Find the maximum possible value of $A$.
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Context:
The equation \( ax^{2} - bx + c = 0 \) has two distinct roots \( x_{1}, x_{2} \). What is the maximum number of elements in the set \( \{a, b, c, x_{1}, x_{2}\} \) that can be prime (and hence natural) numbers? If a prime number appears twice in the set, it must be counted twice. Note that the number 1 is not considered a prime number.
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Context:
Let $A$ be the set $\{1,2,\ldots,n\}$, $n\geq 2$. Find the least number $n$ for which there exist permutations $\alpha$, $\beta$, $\gamma$, $\delta$ of the set $A$ with the property:
设 $A$ 为集合 $\{1,2,\ldots,n\}$ ， $n\geq 2$ 。 查找具有以下属性的集合 $A$ 中存在排列 $\alpha$ ， $\beta$ ， $\gamma$ ， $\delta$ 的最小数 $n$ ：\[ \sum_{i=1}^n \alpha(i) \beta (i) = \dfrac {19}{10} \sum^n_{i=1} \gamma(i)\delta(i) . \]
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Context:
Suppose that  $ABCD$  is a rectangle with sides of length  $12$  and  $18$ . Let  $S$  be the region of points contained in  $ABCD$  which are closer to the center of the rectangle than to any of its vertices. Find the area of  $S$ .
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Context:
In the triangle \(ABC\), \(AD = 2BD\), \(AD = EC\), and \(BC = 18\). The area of triangle \(AFC\) is equal to the area of quadrilateral \(DBEF\). What is the length of \(AB\)?
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Context:
Find the increments of the argument and the function for \( y = 2x^2 + 1 \) when the argument \( x \) changes from 1 to 1.02.
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Context:
Determine the Galois group of the splitting field of the polynomial \( f(x) = x^8 - 1 \) over the finite field \( \mathbb{F}_3 \).
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Context:
A $3$ by $3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant.
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Context:
Consider all functions  $f: \mathbb{Z} \rightarrow \mathbb{Z}$  satisfying \[f(f(x) + 2x + 20) = 15. \]

Call an integer  $n$   $\textit{good}$  if  $f(n)$  can take any integer value. In other words, if we fix  $n$ , for any integer  $m$ , there exists a function  $f$  such that  $f(n) = m.$  Find the sum of all good integers  $x.$
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Context:
In triangle $\triangle ABC$, $AB=4cm$, $AC=6cm$. If the length of $BC$ is an integer, apply the triangle inequality theorem to find the possible length(s) of $BC$. Express your answer as a list or range of integers in centimeters (cm).
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Context:
Alice can bake a pie in 5 minutes. Bob can bake a pie in 6 minutes. Charlie can bake a pie in 7 minutes. Compute the difference between how many pies Alice can bake compared to Bob and Charlie in 90 minutes.
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Context:
A digital display shows the current date as an 8-digit integer, consisting of a 4-digit year, followed by a 2-digit month, followed by a 2-digit date within the month. For how many dates in 2023 will each digit appear an even number of times in the digital display for that date?
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Context:
What is the volume of the pyramid whose net is shown, if the base is a square with a side length of $1$?
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Context:
In the geometric sequence {a_n}, the product of the first three terms is 27, and the sum of the second and fourth terms is 30. Find the common ratio q. Express your answer as a numerical value, including the ± symbol if necessary.
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Context:
In the pyramid \(S A_{1} A_{2} \ldots A_{n}\), point \(O\) is located at the base. It is given that \(S A_{1}=S A_{2}=\ldots=S A_{n}\) and \(\angle S A_{1} O = \angle S A_{2} O = \ldots = \angle S A_{n} O\). Determine the smallest value of \(n\) such that \(SO\) is the height of the pyramid.
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Context:
Hermione wants to maximize her earnings from selling badges to support the Society for the Promotion of Elfish Welfare (S.P.E.W.). The production cost for the badges is given by the function \( C(Q) = 5Q^2 \), where \( Q \) is the number of badges produced. The demand for badges from all houses except Slytherin is \( q_{ws} = 26 - 2P \), and the demand from Slytherin is \( q_s = 10 - P \). If the price of the badges must be the same for all houses, what is the maximum amount Hermione can earn from selling the badges?
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Context:
Consider the sequences \((a_{n})\) and \((b_{n})\) defined by the initial conditions \(a_{1} = \alpha\) and \(b_{1} = \beta\), and the recurrence relations \(a_{n+1} = \alpha a_{n} - \beta b_{n}\) and \(b_{n+1} = \beta a_{n} + \alpha b_{n}\) for all \(n > 0\). Determine the number of pairs \((\alpha, \beta)\) of real numbers such that \(a_{1997} = b_{1}\) and \(b_{1997} = a_{1}\).
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Context:
Ludwig works 7 days a week with varying daily salaries depending on the day of the week. He earns $12 on weekdays (Monday to Thursday) and $15 on weekends (Friday to Sunday). He usually works half of the day on weekends but if he works more than 48 hours a week, he is also entitled to an overtime pay of 1.5 times the normal daily rate for any additional hours worked beyond that. In a given week, Ludwig worked 52 hours. Calculate his total earnings for that week considering his daily rate, weekend rate, and overtime pay.
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Context:
Find the set of all points where the Taylor series of the function \( f(x)=\sum_{n=0}^{\infty} \frac {x^2}{(1+x^2)^n} \) around the point \( x=e \) converges to \( f(x) \).
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Context:
Three identical circles are packed into a unit square. Each of the three circles is tangent to each other and tangent to at least one side of the square. If $r$ is the maximum possible radius of the circle, what is $(2-\frac{1}{r})^2$?
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Context:
A pebble is shaped as the intersection of a cube of side length 1 with the solid sphere tangent to all of the cube's edges. What is the surface area of this pebble?
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Context:
Calculate the volume of the set \( \{ \begin{pmatrix} 2 & 33 & 444 \\ 0 & 3 & 4 \\ 0 & 0 & 4 \end{pmatrix}\begin{pmatrix}x \\ y \\ z \end{pmatrix} : x^2 + y^2 + z^2 \leq 1 \} \) in \( \mathbb{R}^3 \).
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Context:
Evaluate the definite integral \( \int_{-\pi/2}^{\pi/2} \sqrt{\sin^3x - \sin^5x} \, dx \).
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Context:
Evaluate the limit of the integral: \[ \lim_{n\to\infty}\int_0^\infty\frac{n\cos^2(x/n)}{n+x^4}dx. \] Determine whether the integral converges to a finite value or diverges to infinity.
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Context:
Find all real numbers $x$ between 0 and 360 such that $\sqrt{3} \cos 10^\circ = \cos 40^\circ + \sin x^\circ.$  Enter all the solutions, separated by commas.
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Context:
Company P had 15 percent more employees in December than it had in January, Company Q had 10 percent more employees in December than in January, and Company R had 20 percent more employees in December than in January. If Company P had 500 employees in December, Company Q had 550 employees in December, and Company R had 600 employees in December, how many employees did these three companies have combined in January?
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Context:
Evaluate the infinite-nested radical expression \(\sqrt{9-5\sqrt{3-\sqrt{9-5\sqrt{3-\sqrt{9-5\sqrt{3-\cdots}}}}}}\) and determine if it converges to a specific value or oscillates between two values.
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Context:
Find the dimension of an irreducible finite-dimensional representation of the algebra \( A=\frac{k[x,y]}{\langle yx-xy-1\rangle} \), where \( k \) is an algebraically closed field of characteristic \( p \), and the action of \( x \) and \( y \) is given by the Jordan normal form of \( y \).
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Context:
In a 4-dimensional space divided into 16 equal regions about the origin, determine the maximum number of regions that a plane can intersect.
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Context:
Determine a formula for the total number of bit changes when counting from 1 to a power of 2, n, in binary. The time to add 1 to the current number i is proportional to the number of bits that change from i to i + 1 in their binary representation. For example, transitioning from 0000 to 0001 involves changing one bit, and from 0001 to 0010 involves changing two bits. What is the formula for the total number of bit changes as we increment from 1 to n?
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Context:
Determine the cardinality of the set of all different partitions of the set of natural numbers \( \mathbb{N} \).
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Context:
Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$. He knows the die is weighted so that one face comes up with probability $1/2$ and the other five faces have equal probability of coming up. He unfortunately does not know which side is weighted, but he knows each face is equally likely to be the weighted one. He rolls the die 5 times and gets a $1$, $2$, $3$, $4$ and $5$ in some unspecified order. Compute the probability that his next roll is a $6$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \(\oint_{C}\frac{e^z+\sin z}{z^5}dz,\) where \(C\) is the path defined by \(r(t) = 3 + e^{2\pi i t}\) for \(t \in [0,1]\), using Cauchy's integration formula.
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Context:
A policeman is trying to catch a thief who is initially $a$ meters away. The thief moves with a constant velocity $v$ in a straight line to the right, starting at $(0, a)$. The policeman, starting at $(0, 0)$, moves with a constant velocity $k \cdot v$, where $k > 1$, and always runs directly towards the thief. The thief is caught at the point $(a, a)$. Determine the value of $k$.
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Context:
Consider a $(0,1)$-matrix $A$ with $m$ rows and $m^2$ columns, where each column contains exactly three $1$'s. What is the minimum number of rows that need to be removed to ensure that no column has more than two $1$'s?
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Context:
In the diagram, \( AB \parallel EF \parallel DC \). Given that \( AC + BD = 250 \), \( BC = 100 \), and \( EC + ED = 150 \), find \( CF \).
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Context:
We call a number beautiful if it reads the same from left to right as from right to left. For example, the number 12321 is beautiful. Find all beautiful five-digit numbers that use only the digits 1 and 0.
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Context:
In a class of 20 students, all but 4 of the students put their names on a typed assignment. If the teacher randomly guesses, what is the probability that she correctly guesses which paper belongs to each of the four remaining students? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Let point P be a common focus $F_1(-2, 0)$, $F_2(2, 0)$ of an ellipse and a hyperbola, and $\cos\angle F_1PF_2 = \frac{3}{5}$. It is known that the length of the major axis of the ellipse is four times the length of the real axis of the hyperbola. Calculate the eccentricity of the hyperbola.
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Context:
Let $P A B C$ be a tetrahedron such that $\angle A P B=\angle A P C=\angle B P C=90^{\circ}, \angle A B C=30^{\circ}$, and $A P^{2}$ equals the area of triangle $A B C$. Compute $\tan \angle A C B$.
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Context:
A group \( G \) of order 15 acts on a set of order 22 with no fixed points. Determine the number of orbits in this group action.
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Context:
Determine the Galois group of the polynomial \( f(x) = x^3 + 2x + 2 \) over the field \( \mathbb{Q}(i\sqrt{19}) \).
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Context:
Consider a Markov chain \( \{X_n\} \) on \( \mathbb{N}_0 \) with transition probabilities \( p(x,x+1) = p(x,0) = \frac{1}{2} \) for all \( x \in \mathbb{N}_0 \). Find the expected time \( E(T_0 \mid X_0 = 1) \), where \( T_0 = \inf\{k \geq 0 : X_k = 0\} \).
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Context:
若 $A B$ 为过抛物线 $y^{2}=4 x$ 的焦点 $F$ 的弦, $O$ 为坐标原点, 且 $\angle O F A=135^{\circ}$, $C$ 为抛物线的准线与 $x$ 轴的交点, 求 $\angle A C B$ 的正切值。原始答案的形式为 $k \sqrt{m}$，请给出 k + m 的值。
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Context:
Evaluate the series \( \sum_{n=1}^{\infty}\frac{f(n)}{n(n+1)} \), where \( f(n) \) is the number of ones in the binary representation of \( n \).
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Context:
Find the analytic extension of the function $f(x) = |x|$ from the real numbers to the complex plane, excluding zero.
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Context:
In quadrilateral $ABCD$, $\angle DAB=\angle ABC=110^{\circ}$, $\angle BCD=35^{\circ}$, $\angle CDA=105^{\circ}$, and $AC$ bisects $\angle DAB$. Find $\angle ABD$.
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Context:
Let \( \varphi: \mathbb{R}^n \rightarrow \mathbb{R}^{n+k} \) be a global parametrization for the differentiable manifold \( M = \varphi(\mathbb{R}^n) \subseteq \mathbb{R}^{n+k} \). The inner product \( \langle , \rangle_{\varphi(0)} \) in \( T_{\varphi(0)}M \) is such that \( \{\frac{\partial}{\partial x_1}\vert_{\varphi(0)}, \ldots, \frac{\partial}{\partial x_n}\vert_{\varphi(0)}\} \) is an orthonormal basis. For \( a \in \mathbb{R}^n \), consider the map \( L_a: M \rightarrow M \) defined by \( L_a(\varphi(x)) = \varphi(x+a) \). Choose \( \langle , \rangle_{\varphi(a)} \) as an inner product in \( T_{\varphi(a)}M \) such that \( DL_a(\varphi(0)): T_{\varphi(0)}M \rightarrow T_{\varphi(a)}M \) is an Euclidean isometry. Find the components \( g_{ij} \) of the metric tensor \( (M, \langle , \rangle) \).
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Context:
Determine the value(s) of \(\alpha\) in the matrix \(A\) such that the geometric multiplicity of the eigenvalue \(\lambda = 4\) is 2, where \(A\) is a \(4 \times 4\) matrix.
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Context:
A collection of 8 cubes consists of one cube with edge-length $ k $ for each integer $ k, 1 \le k \le 8. $ A tower is to be built using all 8 cubes according to the rules:

* Any cube may be the bottom cube in the tower.
* The cube immediately on top of a cube with edge-length $ k $ must have edge-length at most $ k+2. $

Let $ T $ be the number of different towers than can be constructed. What is the remainder when $ T $ is divided by 1000?
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Context:
Find the smallest possible value of \( n \) such that there are exactly 77,000 ordered quadruples \((a, b, c, d)\) with \( \gcd(a, b, c, d) = 77 \) and \( \mathrm{lcm}(a, b, c, d) = n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^\infty \Gamma(1+ix)\Gamma(1-ix)\,dx \).
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Context:
Determine the convergence rate of Newton's method when the slope is not updated in each iteration.
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Context:
Determine the cardinality of the set $X$, where $X$ consists of all non-decreasing functions from $\mathbb{R}$ to $\{0,1\}$.
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Context:
In triangle $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}$ and $\overline{AB}$, respectively. Let $P$ be the intersection of $\overline{BE}$ and $\overline{CF}$. Given that $PB = 14$, $PC = 4$, $PE = 7$, and $PF = 2$, compute the maximum possible area of $\triangle ABC$. \(\text{Proposed by Eugene Chen}\)
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Context:
Let $A$ be an $R$-right module, $N$ be a submodule of the $R$-left module $M$, and $\pi: M \rightarrow M/N$ be the natural epimorphism. Determine $\ker(\pi \otimes 1_A)$, where $1_A$ is the identity map on $A$. Provide your answer in terms of submodules of $M \otimes_R A$.
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Context:
9. [55] Let $N$ be the smallest positive integer for which
$$
x^{2}+x+1 \quad \text { divides } \quad 166-\sum_{d \mid N, d>0} x^{d} \text {. }
$$

Find the remainder when $N$ is divided by 1000 .
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Context:
Given the function $f(x)=\frac{1}{1+{2}^{x}}$, if the inequality $f(ae^{x})\leqslant 1-f\left(\ln a-\ln x\right)$ always holds, then the minimum value of $a$ is ______.
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Context:
Select 3 teachers from 6 to teach in 3 different poverty-stricken areas, one in each area. Teachers A and B cannot go to the same area, and teachers A and C must either both go or both not go. How many different selection plans are there?
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Context:
Grisha wrote 100 numbers on the board. Then he increased each number by 1 and noticed that the product of all 100 numbers did not change. He increased each number by 1 again, and again the product of all the numbers did not change, and so on. Grisha repeated this procedure $k$ times, and each of the $k$ times the product of the numbers did not change. Find the largest possible value of $k$.
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Context:
已知椭圆的中心在原点 $O$, 焦点在 $x$ 轴上, 离心率为 $\frac{\sqrt{3}}{2}$, 且过点 $\left(\sqrt{2}, \frac{\sqrt{2}}{2}\right)$. 设不过原点 $O$ 的直线 $l$ 与该椭圆交于点 $P$ 和 $Q$, 且直线 $O P, P Q, O Q$ 的斜率构成等比数列, 求 $\triangle O P Q$ 面积的取值范围.
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Context:
8. (5 points) The founder of a noble family received a plot of land. Each man in the family, upon dying, divided the land he inherited equally among his sons. If he had no sons, the land went to the state. No other members of the family gained or lost any land in any other way. In total, there were 200 people in the family. What is the smallest fraction of the original plot that any member of the family could have received?
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Context:
Suppose that the edges of a graph with 10 vertices can be colored with two colors such that the graph does not contain a monochromatic triangle. What is the maximum number of edges the graph can have?
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Context:
In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam.
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Context:
Suppose you forgot one digit of the phone number you need and you are dialing it at random. What is the probability that you will need to make no more than two calls?
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Context:
Determine the number of values of $x$ for which the function $f(x) = \cos x + \cos(\sqrt{2}x)$ attains its absolute minimum.
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Context:
Find the number of ring homomorphisms from \( \mathbb{Z}_{12} \) to \( \mathbb{Z}_{18} \), excluding the zero homomorphism.
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Context:
Evaluate the definite integral \( \int_{-2}^{2}\int_{0}^{\sqrt{1-x^2}}\sqrt[3]{x^2+y^2} \, dx \, dy \).
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Context:
已知函数 $f:\{1,2, \cdots, 10\} \rightarrow\{1,2, \cdots, 5\}$ ，且对一切 $k=1,2, \cdots, 9$ ，有
$|f(k+1)-f(k)| \geqslant 3$.
则符合条件的函数 $f$ 的个数是 $\qquad$.
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Context:
For \( n = 2^k \) where \( k \ge 1 \), find a formula for the number of silver matrices of size \( n \). A silver matrix of size \( n \) is an \( n \times n \) matrix where the \( i \)-th row and \( i \)-th column contain all the numbers in \( \{1, 2, \ldots, 2n-1\} \).
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Context:
There is a board of size 7×12 cells and a cube, the side of which is equal to a cell. One face of the cube is painted with non-drying paint. The cube can be placed in a certain cell of the board and rolled over an edge to an adjacent face. The cube cannot be placed twice in the same cell. What is the maximum number of cells that the cube can visit without soiling the board with paint?
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Context:
Let  $A$  be a subset of  $\{1, 2, \dots , 1000000\}$  such that for any  $x, y \in A$  with  $x\neq y$ , we have  $xy\notin A$ . Determine the maximum possible size of  $A$ .
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Context:
Given that point $P$ moves on the parabola $y^2 = 8x$, $F$ is the focus of the parabola, and point $A$ has coordinates $(5,2)$. Find the minimum value of $PA + PF$.
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Context:
Two \(4 \times 4\) squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. Find the area of the shaded region created by removing the circle from the squares. The answer is in the form \(k - m\pi\), please provide the value of \(k + m\).
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Context:
Given that Figure 1 is the graph of an even function \( f(x) \) and Figure 2 is the graph of an odd function \( g(x) \).

Let \(a\), \(b\), \(c\), and \(d\) be the number of real roots of the equations \( f(f(x)) = 0 \), \( f(g(x)) = 0 \), \( g(g(x)) = 0 \), and \( g(f(x)) = 0 \) respectively.

Then, find the value of \( a + b + c + d \).
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Context:
There are eight points on a circle that divide the circumference equally. Count the number of acute-angled triangles or obtuse-angled triangles that can be formed with these division points as vertices.
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Context:
In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice that of angle $B$, and the measures of segments $AD$ and $CD$ are $a$ and $b$ respectively. Find the measure of $AB$ in terms of $a$ and $b$.
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Context:
Calculate \(\int \int \vec{F} \cdot d\vec{S}\) over the surface defined by \(x = u^2\), \(y = uv\), \(z = \frac{1}{2} v^2\) bounded by the curves \(u = 0\), \(u = 1\), \(v = 0\), and \(v = 3\), for the vector field \(\vec{F} = y \vec{i} - x \vec{j} + xy \vec{k}\).
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Context:
Consider a rectangular array of single digits \(d_{i,j}\) with 10 rows and 7 columns, such that \(d_{i+1, j} - d_{i, j}\) is always 1 or -9 for all \(1 \leq i \leq 9\) and all \(1 \leq j \leq 7\). For \(1 \leq i \leq 10\), let \(m_{i}\) be the median of \(d_{i,1}, \ldots, d_{i, 7}\). Determine the least and greatest possible values of the mean of \(m_{1}, m_{2}, \ldots, m_{10}\).
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Context:
A person can be in one of three states: employed, unemployed, or early retirement. The transition rates per year are as follows: an employed person becomes unemployed with a rate of 0.2, an unemployed person becomes employed with a rate of 4, and an unemployed person retires early with a rate of 0.5. A working person never retires directly, and a retired person stays retired forever. A person starts employed, earning $280,000 per year while employed and $224,000 per year while unemployed, with no earnings in retirement. Calculate the expected total earnings before the person retires early.
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Context:
Find the positive integer $k$ for which the coefficient of $x^k$ in the expansion of $(x + 3)^{50}$ is maximized.
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Context:
Alice and Bob are playing a coin flipping game where they flip coins until one of them wins. Alice wins when the sequence "h, h, h" appears, and Bob wins if the sequence "t, h, t" appears. Find the expected number of coin flips, $E[X]$, until one of them wins.
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Context:
Given a fixed number $c$ and random variables $X_i$ and $Y_i$ for $i=1,2,...,n$, where $|P(X_i<c)-P(Y_i<c)| \leq \alpha$ for any $i$ and $\alpha$ is a constant, find an upper bound for $|P(\max_i X_i<c)-P(\max_i Y_i<c)|$. Assume that $X_i$ and $Y_i$ are not independent and identically distributed.
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Context:
Evaluate the integral \( \iint_R y\ dx\ dy \) where \( R \) is the region bounded by the \( x \)-axis and the parabolas \( y^2 = 4-4x \) and \( y^2=4+4x \).
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Context:
Let $n$ be the maximum number of bishops that can be placed on the squares of a $6 \times 6$ chessboard such that no two bishops are attacking each other. Let $k$ be the number of ways to put $n$ bishops on an $6 \times 6$ chessboard such that no two bishops are attacking each other. Find $n+k$. (Two bishops are considered to be attacking each other if they lie on the same diagonal. Equivalently, if we label the squares with coordinates $(x, y)$, with $1 \leq x, y \leq 6$, then the bishops on $(a, b)$ and $(c, d)$ are attacking each other if and only if $|a-c|=|b-d|$.)
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Context:
A point is “bouncing” inside a unit equilateral triangle with vertices(0,0),(1,0), and \( \left( {1/2,\sqrt{3/2}}\right) \) . The point moves in straight
 lines inside the triangle and bounces elastically off an edge at an angle equal to the angle of incidence. Suppose that the point
 starts at the origin and begins motion in the direction of(1,1). After the ball has traveled a cumulative distance of \( {30}\sqrt{2} \) ,compute its distance from the origin.
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Context:
Calculate the expected number of solutions found in an infinitely large, balanced search tree using a depth-first search algorithm, given that the probability of finding a solution at level $l$ is $p^l$.
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Context:
Let $S$ be the set of all positive integers between 1 and 2017, inclusive. Suppose that the least common multiple of all elements in $S$ is $L$. Find the number of elements in $S$ that do not divide $\frac{L}{2016}$.
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Context:
Using the systematic sampling method to select 32 people for a questionnaire survey from 960 people, determine the number of people among the 32 whose numbers fall within the interval [200, 480].
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Context:
The real number $x$ satisfies
\[3x + \frac{1}{2x} = 3.\]Find
\[64x^6 + \frac{1}{729x^6}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Let $A, B, C$ be points in that order along a line, such that $A B=20$ and $B C=18$. Let $\omega$ be a circle of nonzero radius centered at $B$, and let $\ell_{1}$ and $\ell_{2}$ be tangents to $\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\ell_{1}$ and $\ell_{2}$. Let $X$ lie on segment $\overline{K A}$ and $Y$ lie on segment $\overline{K C}$ such that $X Y \| B C$ and $X Y$ is tangent to $\omega$. What is the largest possible integer length for $X Y$?
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Context:
A cube has six faces painted with six different colors: red, blue, green, yellow, purple, and orange. The cube is then cut into 125 small equal cubes. Calculate the number of small cubes with only one face colored for each of the six different colors.
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Context:
In how many ways can 10 men and 4 women be seated at a round table such that no three women sit consecutively?
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Context:
有 2022 个格排成一行，A,B 两人在玩游戏. 初始时，在每个左起奇数号格中填入 A、偶数号格中填入 B. 接下来从 A 开始，两人轮流进行以下操作：选择两个写有自己名字的格，这两个格不相邻，且它们之间的所有格都写有另一人的名字，然后将这些格都改为自己的名字. 当某人不能操作时游戏结束. 求最大的正整数 m，使得无论 B 如何操作，A 能保证游戏结束时总有至少 m 个格写有 A 的名字.
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Context:
Let \(i_{1}, i_{2}, \cdots, i_{10}\) be a permutation of \(1, 2, \cdots, 10\). Define
\[
S = \left|i_{1} - i_{2}\right| + \left|i_{3} - i_{4}\right| + \cdots + \left|i_{9} - i_{10}\right|,
\]
find all possible values of \(S\).
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Context:
Determine the number of integer solutions \((x_1, x_2, x_3, x_4, x_5)\) to the system of equations \(\sum_{i=1}^5 x_i = 10\) and \(\sum_{i=1}^5 x_i^2 = 40\), where each \(x_i\) is an integer.
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Context:
Determine the smallest number \( k \) of elements \( s_1, \ldots, s_k \in M \), where each \( s_i = A_i\text{SO}(3) \) for some \( A_i \in \text{SL}(3,\mathbb{R}) \), such that the simultaneous stabilizer of all \( s_i \) in \( \text{SL}(3,\mathbb{R}) \) is the identity matrix.
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Context:
Calculate the number of ways to arrange 4 Americans, 3 Russians, and 5 Chinese into a queue such that no nationality forms a single consecutive block. Use the inclusion-exclusion principle to find the solution.
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Context:
Determine the range of values of $m$ for which there exists a $p > 0$ such that \( f(m,p) = \frac{2}{m^2} - \sum_{k=m}^{m+p} \frac{1}{(2k-1)^2} < 0 \).
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Context:
Determine the isomorphism class of the quotient ring \( \mathcal{O}_K / \langle \sqrt{6} - 1 \rangle \) for the number field \( K = \mathbb{Q}(\sqrt{2}, \sqrt{3}) \).
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Context:
Suppose  $a,b,c,x,y,z$  are pairwisely different real numbers. How many terms in the following can be  $1$  at most:   $$ \begin{aligned}
&ax+by+cz,&&&&ax+bz+cy,&&&&ay+bx+cz,
&ay+bz+cx,&&&&az+bx+cy,&&&&az+by+cx?
\end{aligned} $$
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Context:
Given an $\mathfrak{sl}(2,\mathbb{C})$-module $V$ with eigenvalues of the action $h$ being 6, 4, 2, 0, -2, -4, and -6, determine the number of irreducible submodules isomorphic to $V_4$.
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Context:
Students in a class participate in a tree-planting activity. There are three different types of saplings, labeled A, B, and C, in the nursery. They are to be planted in a row of 5 pits, with the condition that the same type of sapling cannot be adjacent, and only type A saplings can be planted in the first and the fifth pits. Calculate the total number of planting methods. Express your answer as a single integer.
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Context:
4.9. Each diagonal of the convex pentagon $A B C D E$ cuts off a triangle of unit area from it. Calculate the area of the pentagon $A B C D E$.
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Context:
Let \(f(x)\) be a twice differentiable function on \([1,3]\) such that \(f(1) = f(3)\) and \(|f''(x)| < 2\) for all \(x\) in \([1,3]\). Determine the range of \(|f'(x)|\) on the interval \([1,3]\).
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Context:
A square with a side of 36 cm was cut into three rectangles such that the areas of all three rectangles are equal and any two rectangles have a common boundary section. What is the total length (in cm) of the cuts made?
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Context:
Determine the number of quadratic subfields in the splitting field of the polynomial $x^4-20$.
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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:
Determine the constant $d$ such that
$$\left(3x^3 - 2x^2 + x - \frac{5}{4}\right)(ex^3 + dx^2 + cx + f) = 9x^6 - 5x^5 - x^4 + 20x^3 - \frac{25}{4}x^2 + \frac{15}{4}x - \frac{5}{2}$$
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Context:
Three flower beds overlap as shown. Bed $A$ has $500$ plants, bed $B$ has $450$ plants, and bed $C$ has $350$ plants. Beds $A$ and $B$ share $50$ plants, while beds $A$ and $C$ share $100$. What is the total number of plants?
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Context:
Count the number of different shapes of isosceles triangles with all sides being positive numbers and their perimeter not exceeding 10. Express your answer as a whole number.
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Context:
A student correctly added the two two-digit numbers on the left of the board and got the answer 137. What answer will she obtain if she adds the two four-digit numbers on the right of the board?
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Context:
Given a convex hexagon $A B C D E F$ with all six side lengths equal, and internal angles $\angle A$, $\angle B$, and $\angle C$ are $134^{\circ}$, $106^{\circ}$, and $134^{\circ}$ respectively. Find the measure of the internal angle $\angle E$.
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Context:
Determine the value of $x$ for which all eigenvalues of the matrix \( A = \begin{pmatrix} 10 & 5+i & 4 \\ x & 20 & 2 \\ 4 & 2 & -10 \end{pmatrix} \) are real.
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Context:
Find all pairs of natural numbers  $ (a, b)$  such that  $ 7^a \minus{} 3^b$  divides  $ a^4 \plus{} b^2$ .

*Author: Stephan Wagner, Austria*
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Context:
满足方程 $\frac{1}{k_{1}}+\frac{1}{k_{2}}+\cdots+\frac{1}{k_{n}}=1, k_{1} k_{2} \cdots k_{n} \leq 200$ 的所有有序正整数组 $\left(n, k_{1}, \cdots, k_{n}\right)$的组数为 $\qquad$.
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Context:
Find the least number that must be subtracted from 899830 so that the remaining number is divisible by a certain number. The least number is 6. What is the divisor?
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Context:
Determine the number of Hahn-Banach extensions of a non-zero continuous linear functional \( f:(c_{00},\|\cdot\|_1)\to \mathbb{C} \) to the entire space \( (\ell^1,\|\cdot\|_1) \). Here, \( c_{00} \) is the set of all sequences with finitely many non-zero terms, and \( \|x\|_1 = \sum_{n=1}^\infty |x_n| \) for \( x = (x_1, x_2, \ldots) \in c_{00} \) or \( \ell^1 \), where \( \|x\|_1 < \infty \) for \( x \in \ell^1 \). Use the Hahn-Banach theorem on normed linear spaces, which states that a linear functional defined on a subspace can be extended to the whole vector space while preserving the norm.
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Context:
某足球赛有16个城市参加，每市派出2个队，根据比赛规则，每两队之间至多赛一场，同城两队之间不进行比赛.赛过一段时间后，发现除A城甲队外，其他各队已赛过的场数各不相同.问A城乙队已赛过几场？
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Context:
Transform the equation \(x_{uu} + 2x_{uv} + x_{vv} = 0\) using the substitutions \(p = u + v\) and \(q = u - v\) to express it in terms of \(p\) and \(q\).
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Context:
A regular dodecahedron is projected orthogonally onto a plane, and its image is an $n$-sided polygon. What is the smallest possible value of $n$?
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Context:
In a government institute, 20 staff members are employed. In a festival season, their boss gives a festival allowance to all. He instructs his accountant to calculate for 30 days at a certain rate. He also says the balance payment will be given to the driver and cleaning staff and to be adjusted from petty cash. He had given Rs. 65000 to the accountant. The amount taken from petty cash was Rs. 1000. What was the daily rate for the festival allowance?
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Context:
Find the smallest integer $k$ for which the conditions
 (1) $a_1,a_2,a_3\cdots$ is a nondecreasing sequence of positive integers
 (2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$
 (3) $a_9=k$
 are satisfied by more than one sequence.
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Context:
If $$A=\int_0^1(1-x^{2023})^{\frac{1}{2022}}dx$$ and $$B=\int_0^1(1-x^{2022})^{\frac{1}{2023}}dx$$ then find the value of $S$ if $$S=\frac{A^3-B^3}{B^3}+\frac{A^4+B^4}{B^4}$$
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Context:
Given a set $A$ which contains $n$ elements. For any two distinct subsets $A_{1}$, $A_{2}$ of the given set $A$, we fix the number of elements of $A_1 \cap A_2$. Find the sum of all the numbers obtained in the described way.
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Context:
Given a point  $P$  on a triangular piece of paper  $ABC,$  consider the creases that are formed in the paper when  $A, B,$  and  $C$  are folded onto  $P.$   Let us call  $P$  a fold point of  $\triangle ABC$  if these creases, which number three unless  $P$  is one of the vertices, do not intersect.  Suppose that  $AB=36, AC=72,$  and  $\angle B=90^\circ.$   Then the area of the set of all fold points of  $\triangle ABC$  can be written in the form  $q\pi-r\sqrt{s},$  where  $q, r,$  and  $s$  are positive integers and  $s$  is not divisible by the square of any prime.  What is  $q+r+s$ ?
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Context:
Ann made a 3-step staircase using 18 toothpicks. To complete a 5-step staircase, how many additional toothpicks does she need? Express your answer as a single integer.
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Context:
Let $A_{1} A_{2} \ldots A_{6}$ be a regular hexagon with side length $11 \sqrt{3}$, and let $B_{1} B_{2} \ldots B_{6}$ be another regular hexagon completely inside $A_{1} A_{2} \ldots A_{6}$ such that for all $i \in\{1,2, \ldots, 5\}, A_{i} A_{i+1}$ is parallel to $B_{i} B_{i+1}$. Suppose that the distance between lines $A_{1} A_{2}$ and $B_{1} B_{2}$ is 7 , the distance between lines $A_{2} A_{3}$ and $B_{2} B_{3}$ is 3 , and the distance between lines $A_{3} A_{4}$ and $B_{3} B_{4}$ is 8 . Compute the side length of $B_{1} B_{2} \ldots B_{6}$.
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Context:
In triangle \(ABC\), the angle bisectors \(AK\) and \(CL\) and the median \(BM\) are drawn. It turns out that \(ML\) is the angle bisector of \(\angle AMB\) and \(MK\) is the angle bisector of \(\angle CMB\). Find the angles of triangle \(ABC\).
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Context:
A random variable $X_1$ has a uniform distribution on the interval $(0, 1)$. Another random variable $X_2$ has a uniform distribution on the interval $(0, X_1)$. A third random variable $X_3$ has a uniform distribution on the interval $(0, X_2)$. Calculate the variance of $12(X_1 + X_2 + X_3)$. \( \text{Provide your answer as a single numerical value.} \)
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Context:
On the board, the numbers 25 and 36 are written. Two students play alternately: on each turn, they choose any two numbers already written on the board and add a new number equal to the absolute value of their difference (it is not allowed to rewrite a number that is already on the board). The player who cannot make a move loses. Who has a winning strategy: the first or the second player?
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Context:
Determine the number of elements of order 15 in the symmetric group $S_{13}$. Provide your answer as a single integer.
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Context:
Determine the number of ways to arrange the five vowels—A, E, I, O, U—along with 20 X's in a row such that no X is at an extreme position and there are at least 4 X's between any two vowels.
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Context:
How many of the first 1000 positive integers can be written as the sum of finitely many distinct numbers from the sequence $3^{0}, 3^{1}, 3^{2}, \ldots$?
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Context:
A deer passed a certain tree at a constant speed of 50 miles per hour while being chased by a cheetah. Then, 2 minutes later, the cheetah passed the same tree at a certain speed. If both animals maintained their speeds, the cheetah caught up with the deer 1 minute after the cheetah passed the tree. What was the speed of the cheetah?
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Context:
Square $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\sqrt {50}$ and $BE = 1$. What is the area of the inner square $EFGH$?
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Context:
There are 96 matches divided into three piles. First, take from the first pile the same number of matches as in the second pile and add them to the second pile. Then, take from the second pile the same number of matches as in the third pile and add them to the third pile. Finally, take from the third pile the same number of matches as in the first pile and add them to the first pile. At this point, all three piles have the same number of matches. Determine how many matches were originally in the first pile, second pile, and third pile.
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Context:
若$2$是${{4}^{a}}$与${{2b}}$的等比中项，则$a\left( a+b \right)$的最大值为______.
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Context:
Let $N=30^{2015}$.  Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$.
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Context:
A pyramid $SABC$ has a triangular base $ABC$ with a height of $9\sqrt{3}/2$. Triangle $SAB$ is equilateral with side length $6\sqrt{3}$, and the other sides of the pyramid are equal ($SC=AC=BC$). Calculate the volume of the pyramid.
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Context:
已知 \( \frac{1-4 \sin \alpha}{\tan \alpha}=\sqrt{3}(\alpha \in\left(0, \frac{\pi}{2}\right)) \). 原始答案为 \( \frac{m\pi}{n} \) 的形式，请给出 m + n 的值.
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Context:
Let \( K = \mathbb{R}(X) \) be the rational function field, and \( F = \mathbb{R}(X^4 - \frac{1}{X^4}) \). Let \( L \) be the Galois closure of the extension \( K/F \). Determine the extension degree \([L:F]\).
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Context:
The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle, with the vertices around the circumference in the given order.  Given that the polygon $P_1P_3P_5P_7$ is a square of area 5, and the polygon $P_2P_4P_6P_8$ is a rectangle of area 4, find the maximum possible area of the octagon.
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Context:
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form $y=ax+b$ with a and b integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
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Context:
Determine the number of countable non-isomorphic models of the theory $\text{ACF}_p$. Note that $\text{ACF}_p$ is $\kappa$-categorical for all uncountable $\kappa$. Provide your answer as a single number.
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Context:
On the side \( BC \) of triangle \( ABC \), point \( K \) is marked.

It is known that \( \angle B + \angle C = \angle AKB \), \( AK = 4 \), \( BK = 9 \), and \( KC = 3 \).

Find the area of the circle inscribed in triangle \( ABC \).
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Context:
A permutation $a_1, a_2, ..., a_{13}$ of the numbers from $1$ to $13$ is given such that $a_i > 5$ for $i=1,2,3,4,5$. Determine the maximum possible value of\[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \]Evan Chen
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Context:
Toss a die 6 times, and let the number obtained on the $i$-th toss be $a_{i}$. If there exists a positive integer $k$ such that $\sum_{i=1}^{k} a_{i}=6$, what is the probability $p=\frac{n}{m}$, where $m$ and $n$ are coprime positive integers. Find the value of $\log _{6} m - \log _{7} n$.
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Context:
Consider an infinite sequence of tosses of a fair coin. Using a Markov Chain approach with a modified state-change matrix, calculate the expected number of tosses until the pattern HHTT appears.
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