Context:
Consider an angular sector on a plane with its vertex at the origin and an angle of 30 degrees. Place a ball (a material point of zero diameter) on one of the sides of the angle (ray) and launch it inside the sector with a constant speed at an angle of 110 degrees to the considered ray. The ball will fly within the sector, elastically bouncing off its sides, preserving the magnitude of its speed: the angle of incidence is equal to the angle of reflection. How many times will it hit the sides of the sector before flying off to infinity?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the parabola $x^2 = 4y$ with focus $F$ and point $A(-1, 8)$, let $P$ be a point on the parabola. Find the minimum value of $|PA| + |PF|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each of the three aluminum cans holds an integer number of liters of water. The second can holds 1.5 times more than the first, and the volume of the third can is equal to \(\frac{4^3}{3}\) times the volume of the first can. What is the total number of liters of water that the three cans together can hold, given that this number is less than 30?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0, b>0)$ with the left vertex $A$, $P$ is a point on one of the asymptotes of $C$, and $Q$ is the intersection point of $AP$ and the other asymptote of $C$. If the slope of the line $AP$ is $1$ and $A$ is the trisection point of $PQ$, then the eccentricity of $C$ is ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of non-abelian groups of order $p^n$ (where $n > 2$) that cannot be expressed as a direct product of any of their two subgroups.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Complex numbers \( a \), \( b \), and \( c \) form an equilateral triangle with side length 24 in the complex plane. If \( |a + b + c| = 48 \), find \( |ab + ac + bc| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( \lim\limits_{n\to\infty} \sup \exp\left( \frac{k}{n} \ln(n^2) \right) \) for \( k \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A train 1500 m long can cross a pole in 8 seconds. Find the speed of the train.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the VC dimension of a Support Vector Machine (SVM) using the polynomial kernel \(k(x,x')=(1+\langle x,x'\rangle_{\mathbb{R}^2})^2\) for binary classification in \(\mathbb{R}^2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For what values of the real parameter \( t \) does the system of inequalities
\[ x \geq y^2 + t y \geq x^2 + t \]
have exactly one solution in the set of real number pairs?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose we have a sequence defined by $a_1 > 0$ and the recursive relation $a_{n+1} = \ln\left[\frac{\exp(a_n)-1}{a_n}\right]$. Determine the value of the infinite sum $a_1 + a_1 a_2 + a_1 a_2 a_3 + a_1 a_2 a_3 a_4 + \cdots$, given that the sequence is decreasing and converges to zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_\gamma \left(\frac{1}{x+1}-\frac{y}{(x+y)^2}\right)dx+\left(\frac{1}{y+1}-\frac{x}{(x+y)^2}\right)dy \) where \( \gamma \) is the path along the circle from \((1,0)\) to \((0,1)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the length of a normal chord in the parabola \(y^2 = 4x\) that forms a right angle with the vertex \(O(0,0)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three cones with vertex $A$ and a slant height of 6 touch each other externally. For two of the cones, the angle between the slant height and the axis of symmetry is $\frac{\pi}{8}$, and for the third cone, this angle is $\frac{\pi}{4}$. Find the volume of the pyramid $O_{1} O_{2} O_{3} A$, where $O_{1}, O_{2}, O_{3}$ are the centers of the bases of the cones.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An integer  $n\ge1$  is *good* if the following property is satisfied:
If a positive integer is divisible by each of the nine numbers  $n + 1, n + 2, ..., n + 9$ , this is also divisible by  $n + 10$ .
How many good integers are  $n\ge 1$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to\infty}\int_0^\infty nx^2\sin\left(\frac{1}{nx}\right)dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to\infty} \left(\sqrt[3]{n^{48}+n} - \sqrt[3]{n^{48}+n^2}\right) \left((n^3 +3)^{12} - (n^4+4n)^9\right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the tetrahedron \( D-ABC \), \( DA \) is perpendicular to the plane \( ABC \), and \( \triangle ABC \) is an equilateral triangle with side length 2. Given that the tangent of the dihedral angle \( A-BD-C \) is 2, find the volume \( V \) of the tetrahedron \( D-ABC \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \(x = 3^{a_1}5^{a_2}7^{a_3}\) and \(y = 3^{b_1}5^{b_2}7^{b_3}\), where \(a_i, b_i\) are nonnegative integers with \(a_k + b_k \leq 5\) for all \(k\), determine the sum of the greatest common divisors (gcd) of all possible pairs \((x, y)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $sn(\alpha+\beta) = \frac{1}{2}$, $si(\alpha-\beta) = \frac{1}{3}$, find the value of $\lg 5 \frac{ta\alpha}{ta\beta}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a triangle $ABC$, points $M$ and $N$ are on sides $AB$ and $AC$, respectively, such that $MB = BC = CN$. Let $R$ and $r$ denote the circumradius and the inradius of the triangle $ABC$, respectively. Express the ratio $MN/BC$ in terms of $R$ and $r$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \iint_\Sigma \mathbf{f} \cdot d \mathbf{a} \) where \( \mathbf{f}(x,y,z)=(x^2,xy,z) \) and \( \Sigma \) is the part of the plane \( 6x+3y+2z=6 \) with \( x,y,z\geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the derivative with respect to $x$ of the expression:

$$\frac{1}{\left( \int_{a}^{x+h} f(x') \, dx' \right)^{-1}}$$

where $a$ is a constant and $h$ approaches zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the vector $w$ such that the reflection $r_w$ about the orthogonal space $w^\perp$ maps a vector $u$ of length $l$ to a vector $v$ of the same length, given that $u \ne v$. What is the vector $w$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S=2^3+3^4+5^4+7^4+\cdots+17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find the remainder when $S$ is divided by $240$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an isosceles triangle $ABC$ with $AB = AC$, the angle bisector from $B$ intersects $AC$ at $D$ such that $AD + BD = BC$. Find the measure of angle $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a linear operator \( \psi : M \to M \) with minimal polynomial \( m_\psi (x) = (x-2)^2x^5 \), find the minimal polynomial of \( \psi^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The cubic polynomials $p(x)$ and $q(x)$ satisfy the following conditions:
1. $p(1) = q(2)$
2. $p(3) = q(4)$
3. $p(5) = q(6)$
4. $p(7) = q(8) + 13$

Find $p(9) - q(10)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Climbing a staircase with four steps, how many distinct arrangements of steps are possible, considering different step sequences as distinct? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A polygon has a ratio of 7:4 for the number of diagonals to the number of sides. If the sum of its diagonals and sides is an even number and greater than 50, find the smallest number of sides that the polygon can have.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A point $P$ is chosen uniformly at random inside a square of side length 2. If $P_{1}, P_{2}, P_{3}$, and $P_{4}$ are the reflections of $P$ over each of the four sides of the square, find the expected value of the area of quadrilateral $P_{1} P_{2} P_{3} P_{4}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit as \( n \to \infty \): \[ \lim_{n\to\infty}\frac {3^{3n} \binom{2n}{n}^2} {\binom{3n}{n} \binom{6n}{3n}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the function \( \phi(d, n) = k \) that gives the minimum number of real numbers \( k \) required to define a flat subspace of dimension \( d \) in the \( n \)-dimensional Euclidean space \( \mathbb{R}^n \) in a canonical way. Here, "canonical" means using a continuous and surjective function \( f: \mathbb{R}^k \rightarrow \) set of flat subspaces of dimension \( d \) in \( \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the chromatic number of the complement of the Petersen graph.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic behavior of the function $f(n)$, where $f(n)$ is the minimal length such that there exist $n$ binary sequences of length $f(n)$, and any pair of these sequences has a unique Hamming distance.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( ABCDEF \) be a regular hexagon with side length 2. A circle with radius 3 and center at \( A \) is drawn. Find the area inside quadrilateral \( BCDE \) but outside the circle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flow of fluid through the closed surface S, which is the boundary of the region bounded by the half-spheres $z = -\sqrt{4-x^{2}-y^{2}}$ and $z = \sqrt{9-x^{2}-y^{2}}$, and the plane $z = 0$. The velocity field of the fluid is given by $\vec{v}(x,y,z)= (y^{3}e^{-z^{2}} + x)\vec{i} + (ze^{x} + y^{2})\vec{j} + (\cos(x^{2}+ y^{2}) +2z)\vec{k}$. Use the divergence theorem to find the flow of fluid through the surface.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A street has 20 houses on each side, for a total of 40 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 4, 10, 16, etc., and on the north side they are 3, 9, 15, etc. A sign painter paints house numbers on a house for $\$1$ per digit. If he paints the appropriate house number once on each of these 40 houses, how many dollars does he collect?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two positive integers \( x \) and \( y \) are such that:

\[ \frac{2010}{2011} < \frac{x}{y} < \frac{2011}{2012} \]

Find the smallest possible value for the sum \( x + y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a round-robin tournament among $8$ chess players (each pair plays one match), the scoring rules are: the winner of a match earns $2$ points, a draw results in $1$ point for each player, and the loser scores $0$ points. The final scores of the players are all different, and the score of the player in second place equals the sum of the scores of the last four players. What is the score of the second-place player?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the absolute value of \(p^3 + q^3 + r^3\) given that \(p, q, r\) are distinct integers such that \(p + q + r = 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( x, y, z \) be the lengths of the midline segments connecting the midpoints of opposite edges of a tetrahedron \( P-ABC \). If the sum of the squares of the six edge lengths of the tetrahedron is 300, find \( x^{2} + y^{2} + z^{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Last year, Isabella took 8 math tests and received 8 different scores, each an integer between 91 and 100, inclusive. After each test, she noted that the average of her test scores was an integer. Her score on the seventh test was 97. What was her score on the eighth test?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given 6 points in space such that no 4 points are coplanar, find the smallest positive integer $n$ such that whenever 10 line segments are drawn connecting these points, it is always possible to find $n$ distinct pairs of triangles, each pair sharing a common edge.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\pi/2}^{\pi/2} [\cot x] \,\text{d}x \) where \([\quad]\) represents the floor function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the dimension of the centralizer of the matrix \( A \), where \( A = \begin{bmatrix}8 & -9 & -2 \\ 6 & -7 & -2 \\ -6 & 9 & 4 \end{bmatrix} \). The centralizer \( V(A) \) is defined as \( \{X \in M_{3}(\mathbb{R}) : AX = XA \} \), where \( M_{3}(\mathbb{R}) \) is the vector space of \( 3 \times 3 \) real matrices. Determine the dimension of \( V(A) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the number of nonnegative integer triples \((x, y, z)\) which satisfy the inequality:
\[ 4x + 2y + z \leq 36 \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the infinite series: $$\sum_{n = 1}^{\infty} \frac{\ln(n^3+1)}{n^3+1} \frac{1}{4^{n-1}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum possible value of \( k \) for which there exist distinct reals \( {x}_{1},{x}_{2},\ldots ,{x}_{k} \) greater than 1 such that for all
 \( 1 \leq  i,j \leq  k \) ,
 \( {x}_{i}^{\left\lfloor  {x}_{j}\right\rfloor  } = {x}_{j}^{\left\lfloor  {x}_{i}\right\rfloor  }. \)
Please reason step by step, and put your final answer within \boxed{}.

\boxed{
The maximum possible value of \( k \) is 4.}


Context:
Given a Markov chain with the following transition matrix and times for each state:

$$\begin{bmatrix}3/4&1/4&0&0\\4/5&0&1/5&0\\6/7&0&0&1/7\\0&0&0&1\end{bmatrix}$$

State times: \(t_0 = 5\), \(t_1 = 10\), \(t_2 = 15\), \(t_3 = 20\).

Starting at state \(t_0\), calculate the expected time to reach the absorbing state \(t_3\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the variance of the hitting time \( \tau = \inf \{t>0: |B_t|=1\} \) for a reflected Brownian motion \( Z_t = |B_t| \), where \( B_t \) is a standard Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the signature of the quadratic form \( f(x, y, z) = 2xy + 2yz + 2xz + x^2 - z^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the fixed subfield of the field of rational functions \( \mathbb{K}(x) \) over a field \( \mathbb{K} \) of characteristic zero, under the action of the subgroup \( G \) generated by the automorphism \( \sigma: x \mapsto x+1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(f(x)) = 3 + 2x$ and the condition $f(0) = 3$, find the value of $f(3)$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the conditional expectation \( E(N_2 \mid N_1) \) where \( N_t \) is a Poisson process with rate \( \lambda = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the expected value \( E[X_{n+1} | X_0, \dots, X_n] \) given that \( X_n = X_0 e^{\mu S_n} \), where \( X_0 > 0 \), \( S_n \) is a symmetric random walk, and \( \mu > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three different real numbers $a$, $b$, $c$ form an arithmetic sequence, and $a$, $c$, $b$ form a geometric sequence. Find the ratio of $a:b:c$. Express your answer in the form $a:b:c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many solutions does the equation \((a-1)(\sin 2 x + \cos x) + (a+1)(\sin x - \cos 2 x) = 0\) (where the parameter \(a < 0\)) have in the interval \((-π, π)\) ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit \( \lim\limits_{n\to \infty }\frac {(2n)!}{(2^n(n)!)^2} \) using Stirling's formula or other methods.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A school provides apples, bananas, and pears as snacks for students. Each student must choose at least one fruit, but can select multiple. The statistics show that 70% of the students choose apples, 40% choose bananas, and 30% choose pears. What is the maximum percentage of students who chose all three fruits?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A quadrilateral is divided into 1000 triangles. What is the maximum number of distinct points that can be the vertices of these triangles?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sum of one hundred numbers is 1000. The largest of them was doubled, and another number was decreased by 10. It turned out that the sum did not change. Find the smallest of the original numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_C F \cdot dr \) where \( F = (5xy^3)i + (3x^2y^2)j \) along the curve \( C \) consisting of the x-axis from \( x=0 \) to \( x=4 \), the parabola \( y=16-x^2 \) up to the y-axis, and the y-axis down to the origin.
Please reason step by step, and put your final answer within \boxed{}.


Context:
One angle of an isosceles triangle is 30 degrees. Find the possible difference between the other two angles. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the flux of the vector field \( \mathbf{a} = \mathbf{i} - \mathbf{j} + xyz \mathbf{k} \) through the circle \( S \), obtained by the intersection of the sphere \( x^2 + y^2 + z^2 \leq R^2 \) with the plane \( y = x \). Take the side of the circle facing the positive part of the \( O_x \)-axis.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The random variable $X$ follows a normal distribution $N(1, 4)$, where the mean $\mu = 1$ and the variance $\sigma^2 = 4$. Given that $P(X \geq 2) = 0.2$, calculate the probability $P(0 \leq X \leq 1)$. Express your answer as a decimal value rounded to 1 decimal place.
Please reason step by step, and put your final answer within \boxed{}.


Context:
When studying the operation of a new type of cyclic thermal engine, it was found that during part of the period it receives heat, and the absolute power of heat supply is expressed by the law:
\[ P_{1}(t)=P_{0} \frac{\sin (\omega t)}{100+\sin (t^{2})}, \quad 0<t<\frac{\pi}{\omega}. \]
The gas performs work, developing mechanical power
\[ P_{2}(t)=3 P_{0} \frac{\sin (2 \omega t)}{100+\sin (2 t)^{2}}, \quad 0<t<\frac{\pi}{2 \omega}. \]
The work on the gas performed by external bodies is \( \frac{2}{3} \) of the work performed by the gas. Determine the efficiency of the engine.
Please reason step by step, and put your final answer within \boxed{}.


Context:
. Let $n$ be an integer with $n \geq 3$. Consider all dissections of a convex $n$-gon into triangles by $n-3$ non-intersecting diagonals, and all colourings of the triangles with black and white so that triangles with a common side are always of a different colour. Find the least possible number of black triangles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Laplacian of $\frac{z}{s}$, where $s = \sqrt{x^2 + y^2 + z^2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $q(z)=z^5+z(z-2)^2+1$. Find the number of zeros of $q(z)$ in the disk $|z-2|<1$, considering multiplicities.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using systematic sampling, extract a sample of size 12 from a population of 123 individuals. The sampling interval is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The Absent-Minded Scientist commutes to work on weekdays using the Ring Line of the Moscow Metro from "Taganskaya" station to "Kievskaya" station, and back in the evening. When he enters the station, he boards the first arriving train. It is known that in both directions, trains run at approximately equal intervals. The train traveling via the northern route (through "Belorusskaya") takes 17 minutes to go from "Kievskaya" to "Taganskaya" or vice versa, while the train traveling via the southern route (through "Paveletskaya") takes 11 minutes.

According to his long-term observations, the train traveling counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after the clockwise train arrives. The same is true for "Taganskaya." Moreover, the scientist notes that his average travel time from home to work is 1 minute less than his average travel time from work to home. Find the expected interval between trains traveling in one direction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
7. Problem: Let $a, b, c, d$ be integers such that $a>b>c>d \geq-2021$ and
$$
\frac{a+b}{b+c}=\frac{c+d}{d+a}
$$
(and $b+c \neq 0 \neq d+a)$. What is the maximum possible value of $a c$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the  $33$ rd number after the decimal point of $(\sqrt{10} + 3)^{2001}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 f(x) 是定义域为 \bR、最小正周期为 5 的函数. 若函数 g(x)=f(2^x) 在区间 [0,5) 上的零点个数为 25，则 g(x) 在区间 [1,4) 上的零点个数为__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Gunner has a box of mittens and a box of plugs. The box of mittens has 150 pairs of mittens, and he decides to put 30 more pairs of plugs into the box of plugs. Now, there are 400 plugs. How many fewer pairs of mittens are there compared to the original number of pairs of plugs?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ring isomorphisms from $\mathbb{Z}^n$ to $\mathbb{Z}^n$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which values of the parameter \( a \), the equation

\[
\log _{2}\left(2 x^{2}-x-2 a-4 a^{2}\right)+3 \log _{1 / 8}\left(x^{2}-a x-2 a^{2}\right)=0
\]

has two distinct roots, the sum of whose squares belongs to the interval \( (4, 8) \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral \( \int_1^\infty \frac{1}{y^2}\left(\left\lfloor \beta y\right\rfloor-\beta\left\lfloor y\right\rfloor\right) \mathrm{d}y \) for \( \beta \in (0, 1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $165$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sequence is recursively defined as: \( t_{1} = 1 \). For \( n > 1 \), if \( n \) is even, \( t_{n} = 1 + t_{\frac{n}{2}} \); if \( n \) is odd, \( t_{n} = \frac{1}{t_{n-1}} \). Given that \( t_{n} = \frac{19}{87} \), find the sum of the digits of \( n \). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An infinite sequence of integers, $a_0, a_1, a_2, \dots$, with $a_0 > 0$, has the property that for $n \ge 0$, $a_{n+1} = a_n - b_n$, where $b_n$ is the number having the same sign as $a_n$, but having the digits written in reverse order. For example, if $a_0 = 1210$, $a_1 = 1089$, and $a_2 = -8712$, etc. Find the smallest value of $a_0$ such that $a_n \neq 0$ for all $n \ge 1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a game involving the polynomial equation $x^3 + Ax^2 + Bx + C = 0$, two players, Boy and Girl, take turns as follows: Boy chooses a real number, and Girl assigns it to one of the coefficients $A$, $B$, or $C$. This process is repeated until all coefficients are determined. If the resulting polynomial has three distinct integer roots, Boy wins; otherwise, Girl wins. Determine if either player has a winning strategy, and if so, identify the player.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \cdots, T_n$ returns the point $(1,0)$ back to itself?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An urn contains $w$ white and $b$ black balls. $n$ extractions without replacement are made. What is the conditional probability of drawing black on the $3$rd draw and white on the $6$th draw given that a black ball was drawn on the $2$nd draw and a white ball on the $4$th draw?
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Context:
Let \( a, b, c \in \mathbb{R}^{+} \) and \( a b c + a + c = b \). Determine the maximum value of \( P = \frac{2}{a^{2} + 1} - \frac{2}{b^{2} + 1} + \frac{3}{c^{2} + 1} \).
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Context:
Example 10 Let real numbers $x, y, z, w$ satisfy
$$
\left\{\begin{array}{l}
\frac{x^{2}}{2^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}+\frac{z^{2}}{2^{2}-5^{2}}+\frac{w^{2}}{2^{2}-7^{2}}=1, \\
\frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}+\frac{z^{2}}{4^{2}-5^{2}}+\frac{w^{2}}{4^{2}-7^{2}}=1, \\
\frac{x^{2}}{6^{2}-1^{2}}+\frac{y^{2}}{6^{2}-3^{2}}+\frac{z^{2}}{6^{2}-5^{2}}+\frac{w^{2}}{6^{2}-7^{2}}=1, \\
\frac{x^{2}}{8^{2}-1^{2}}+\frac{y^{2}}{8^{2}-3^{2}}+\frac{z^{2}}{8^{2}-5^{2}}+\frac{w^{2}}{8^{2}-7^{2}}=1 .
\end{array}\right.
$$

Try to find the value of $x^{2}+y^{2}+z^{2}+w^{2}$.
(From the 2nd American Invitational Mathematics Examination)
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Context:
Initially the equation
$$\star \frac{1}{x-1} \star \frac{1}{x-2} \star \frac{1}{x-4} ... \star \frac{1}{x-2^{2023}}=0$$
is written on the board. In each turn Aslı and Zehra deletes one of the stars in the equation and writes $+$ or $-$ instead. The first move is performed by Aslı and continues in order. What is the maximum number of real solutions Aslı can guarantee after all the stars have been replaced by signs?
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Context:
Square  $CASH$  and regular pentagon  $MONEY$  are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have?
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Context:
Triangle $ABC$ with $\angle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\frac{BE}{EF} = \frac{2}{3}$, then $\frac{CF}{FE} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Context:
Given a six-digit decimal number formed by six positive integers, where the unit digit is a multiple of 4, the tens and hundreds digits are multiples of 3, and the sum of the digits of the six-digit number is 21. How many six-digit numbers satisfy the above conditions?
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Context:
We draw a circle with radius $5$ on a gridded paper where the grid consists of squares with sides of length $1$. The center of the circle is placed in the middle of one of the squares. Find the number of squares through which the circle passes. Please provide your answer as an integer.
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Context:
Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is monotonically bounded if $1\leq a_i\leq i$ for all $1\leq i\leq 2^{16}$. We say that a term $a_k$ in the sequence with $2\leq k\leq 2^{16}-1$ is a mountain if $a_k$ is greater than both $a_{k-1}$ and $a_{k+1}$. Evan writes out all possible monotonically bounded sequences. Let $N$ be the total number of mountain terms over all such sequences he writes. Find the remainder when $N$ is divided by $p$.
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Context:
In a kindergarten, each child was given three cards, each of which has either "MA" or "NY" written on it. It turned out that 20 children can arrange their cards to spell the word "MAMA", 30 children can arrange their cards to spell the word "NYANYA", and 40 children can arrange their cards to spell the word "MANYA". How many children have all three cards that are the same?
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Context:
Suppose that $x$ is a real number such that \( \frac{27 \times 9^x}{4^x} = \frac{3^x}{8^x} \). Find the value of \( 2^{-(1+\log_2 3)x} \).
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Context:
There are 9 representatives from different countries, with 3 people from each country. They sit randomly around a round table with 9 chairs. What is the probability that each representative has at least one representative from another country sitting next to them?
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Context:
If $a=( \frac {1}{2})^{ \frac {2}{3}}$, $b=( \frac {1}{5})^{ \frac {2}{3}}$, $c=( \frac {1}{2})^{ \frac {1}{3}}$, determine the relationship in size between $a$, $b$, and $c$ (connect the variables using the "<" symbol).
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Context:
How many sets containing at least two elements of consecutive positive integers exist, where the sum of the elements is 100?
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Context:
Given a sequence ${a_n}$ where all terms are positive numbers, $a_1=2$, and $a_{n+1} - a_n = \frac{4}{a_{n+1}+a_n}$, if the sum of the first $n$ terms of the sequence ${\frac{1}{a_{n+1} + a_n}}$ is 5, calculate the value of $n$. Express your answer as a whole number.
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Context:
A dog is being treated for fleas. The effectiveness of each flea treatment varies as follows: the first treatment eliminates 60% of fleas, the second treatment eliminates 45% of the remaining fleas, the third treatment eliminates 30% of the remaining fleas, the fourth treatment eliminates 20% of the remaining fleas, and the fifth treatment eliminates 10% of the remaining fleas. Additionally, the dog spends time with three other dogs, each of which has a different amount of fleas which could transfer to the first dog. The dog also spends time in an environment that could cause some fleas removed by treatment to return. After all five treatments, the dog has 25 fleas left. Taking into account these factors, how many more fleas did the dog have before the flea treatments than after?
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Context:
Evaluate the integral \( \oint_{|w|=2} \frac{1}{w(5w^8-10w^4+4w^2+2iw+10)} \, dw \) given that all the zeros of the denominator lie within the circle \(|w|<1\).
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Context:
if 4 consecutive letters are selected at random from the english alphabet , then the probability that one of the letters is a vowel is ?
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Context:
In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. It is known that $\sin ^{2}B+\cos ^{2}A-\cos ^{2}C= \sqrt {3}\sin B\sin C$, and the area of the circumcircle of triangle $ABC$ is $4\pi$. Find the value of $a$.
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Context:
The function $y=\log_{\frac{1}{2}}(3x^2-ax+5)$ is decreasing on the interval $[-1,+\infty)$. The range of the real number $a$ is ______.
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Context:
How many 4x3 matrices of rank 3 can be formed where the entries are from a field with 3 elements?
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Context:
Ayu is about to open the suitcase but she forgets the key. The suitcase code consists of nine digits, namely four $0$s (zero) and five $1$s. Ayu remembers that no four consecutive numbers are the same. How many codes might have to try to make sure the suitcase is open?
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Context:
Every pair of communities in a county are linked directly by one mode of transportation; bus, train, or airplane. All three methods of transportation are used in the county with no community being serviced by all three modes and no three communities being linked pairwise by the same mode. Determine the largest number of communities in this county.
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Context:
Let \(a, b, c, d\) be integers such that \(\frac{a}{c} \in \mathbb{Q}^+ \cap \mathbb{Z}^+\) and \(\frac{b}{d} \in \mathbb{Q}^- \cap \mathbb{Z}^-\). Determine the number of solutions to the equation \(|ad - bc| = 1\).
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Context:
What is the name of the algebraic structure where the additive group of a ring is non-abelian, and only right distributivity is required?
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Context:
Six chairs sit in a row.  Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left.  There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right.  The probability that this will **not** happen is given by  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .
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Context:
Let $z_1,$ $z_2,$ $\dots,$ $z_{20}$ be the twenty (complex) roots of the equation
\[z^{20} - 4z^{19} + 9z^{18} - 16z^{17} + \dots + 441 = 0.\]Calculate $\cot \left( \sum_{k = 1}^{20} \operatorname{arccot} z_k \right).$  Note that the addition formula for cotangent is still valid when working with complex numbers.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:
Find the smallest positive integer \( n \) such that in any 2-coloring of \( K_n \), there exist two monochromatic triangles of the same color that do not share any edge.
(1991 China National Training Team Problem)
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Context:
The side \( AD \) of the rectangle \( ABCD \) is equal to 2. On the extension of the side \( AD \) beyond point \( A \), a point \( E \) is taken such that \( EA = 1 \), and \( \angle BEC = 30^\circ \). Find \( BE \).
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Context:
Given a regular pentagon \( A B C D E \). On side \( A E \), point \( K \) is marked and on side \( C D \), point \( L \) is marked. It is known that \( \angle L A E + \angle K C D = 108^\circ \) and \( \frac{A K}{K E} = \frac{3}{7} \). Find the ratio \( \frac{C L}{A B} \).

Note: A regular pentagon is a pentagon where all sides are equal and all angles are equal.
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Context:
In a pentagon \(ABCDE\) inscribed in a circle with radius \(R\), it is given that \(\angle B = 110^\circ\) and \(\angle E = 100^\circ\). Determine the length of side \(CD\).
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Context:
Consider the numbers 1 through 1000 arranged in a line. Each number, except the rightmost, must differ by 1 from at least one number to its right. Determine the total number of such possible arrangements.
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Context:
You are given  $n$  not necessarily distinct real numbers  $a_1, a_2, \ldots, a_n$ . Let's consider all  $2^n-1$  ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to  $1$ ?

For example, if  $a = [-1, 2, 2]$ , then we got  $3$  once,  $4$  once,  $2$  twice,  $-1$  once,  $1$  twice, so the total number of ones here is  $2$ .
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Context:
Evaluate the expression:
\[2 \times (2 \times (2 \times (2 \times (2 \times (2 \times 2 - 2) - 2) - 2) - 2) - 2) - 2.\]
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Context:
Find the sum of the non-repeated roots of the polynomial $P(x) = x^6 - 5x^5 - 4x^4 - 5x^3 + 8x^2 + 7x + 7$.
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Context:
Determine the value of $P$ at which the function \( \frac{dP}{dt}=c P\left(\frac{\ln(k)}{P}\right) \) grows fastest, where \( c \) is a constant and \( k \) is a carrying capacity.
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Context:
Find the smallest positive integer \( k \) such that \( (k-10)^{5026} \geq 2013^{2013} \).
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Context:
In one year, what is the maximum number of months that can have five Sundays? Express your answer as a whole number.
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Context:
The smallest number which, when diminished by 6, is divisible by some numbers including 12, 18, 21, and 28 is 1014. What is the second number that the diminished value is divisible by?
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Context:
4. All possible 6-digit numbers, in each of which the digits occur in non-increasing order (from left to right, e.g., 877550) are written as a sequence in increasing order. Find the 2005 -th number in this sequence.

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Context:
Determine the lower bound of the radius of convergence for the power series \( \sum_{n \ge 1} a_n z^n \), where \( a_n \) is the number of divisors of \( n^{50} \).
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Context:
Let the circles $S_1$ and $S_2$ meet at the points $A$ and $B$. A line through $B$ meets $S_1$ at a point $D$ other than $B$ and meets $S_2$ at a point $C$ other than $B$. The tangent to $S_1$ through $D$ and the tangent to $S_2$ through $C$ meet at $E$. If $|AD|=15$, $|AC|=16$, $|AB|=10$, what is $|AE|$?
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Context:
Calculate the limit of the function:

\[
\lim_{{x \to 2}} \left(2 e^{x-2} - 1\right)^{\frac{3x+2}{x-2}}
\]
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Context:
Let \( OP \) be the diameter of the circle \( \Omega \), and \( \omega \) be a circle centered at point \( P \) with a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of the circle \( \Omega \) intersects the second circle at point \( A \). Find the length of segment \( AB \), if \( BD \cdot BC = 5 \).
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Context:
Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The minimum possible area of the resulting shape is $A$. Find the integer closest to $100A$.

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Context:
Suppose a tangent to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) intersects the ellipse \(\frac{x^2}{c^2} + \frac{y^2}{d^2} = 1\) at points \(P\) and \(Q\). If the tangents drawn to the second ellipse at these points are perpendicular, find the value of \(\frac{a^2}{c^2} + \frac{b^2}{d^2}\).
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Context:
For certain ordered pairs $(a,b)\,$ of real numbers, the system of equations
\[\begin{aligned} ax+by&=1 \\ x^2 + y^2 &= 50  \end{aligned}\]has at least one solution, and each solution is an ordered pair $(x,y)\,$ of integers. How many such ordered pairs $(a,b)\,$ are there?
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Context:
Evaluate the real integral \( \int_{-\infty}^{\infty}{\frac{\sin x}{x^4-6x^2+10}\,\mathrm dx} \) using the Residue Theorem.
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Context:
In acute \( \bigtriangleup  {ABC} \) ,let \( I \) denote the incenter and suppose that line \( {AI} \) intersects segment \( {BC} \) at a point \( D \) . Given that
 \( {AI} = 3,{ID} = 2 \) ,and \( B{I}^{2} + C{I}^{2} = {64} \) ,compute \( B{C}^{2} \) .
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Context:
Two circles $F$ and $G$ with radius $10$ and $4$ respectively are externally tangent. A square $ABMC$ is inscribed in circle $F$ and equilateral triangle $MOP$ is inscribed in circle $G$ (they share vertex $M$). If the area of pentagon $ABOPC$ is equal to $a + b\sqrt{c}$, where $a$, $b$, $c$ are integers $c$ is square free, then find $a + b + c$.

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Context:
A batch of one hundred parts is subjected to a sampling inspection. The entire batch is considered unacceptable if at least one defective part is found among the four parts that are tested. What is the probability that this batch will not be accepted if it contains $3\%$ defective parts?
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Context:
A [i]lame king[/i] is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell.

A lame king is placed in the top-left cell of a $7\times 7$ grid. Compute the maximum number of cells it can visit without visiting the same cell twice (including its starting cell).
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Context:
Consider the endomorphism \( \phi: C(\mathbb{R}) \to C(\mathbb{R}) \) defined by \( \phi(f)(x) = x^2 f(x) \). Determine whether this endomorphism has an infinite norm, where the norm of a function \( f \) is defined as \( \|f\| = \sup_{x \in \mathbb{R}} |f(x)| \).
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Context:
Evaluate the limit: $$\lim _{x\to \infty }\frac{\left((x+2)!\right)^2\left(2\left(x-1\right)\right)!}{\left(\left(x+1\right)!\right)^2\left(2x\right)!}$$.
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Context:
Given the equation \( \bar{L}\bar{C}=T^{-1}LCT \), where \( T \) is a coordinate change matrix, \( L \) is a column vector, and \( C \) is a row vector of compatible dimension, solve for the vector \( L \).
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Context:
Determine the fundamental group of a 2-sphere with the identification $x = -x$. What is the fundamental group of this space?
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Context:
Let $T$ denote the sum of all three-digit positive integers where each digit is different and none of the digits are 5. Calculate the remainder when $T$ is divided by $1000$.
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Context:
If the exponential function $y=(a^{2}-1)^{x}$ is a decreasing function on $(-\infty,+\infty)$, then the range of the real number $a$ is __________.
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Context:
Determine a suitable substitution to simplify the integral \( \iint_R (x+y+1) \, dy \, dx \), where the region \( R \) is bounded by the curves \( y=1-x \), \( y=2-x \), \( y=1+x \), and \( y=-1+x \).
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Context:
Let $V,W \subset \mathbb{C}$ be open sets. Let $f\colon V \to W$ be a bijective holomorphic map. Determine if the set $Z:= \left\{v\in V \colon f'(v)=0\right\}$ is empty.
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Context:
Consider the following 100 sets of 10 elements each:
 \( \{ 1,2,3,\cdots ,{10}\} \) ,
 \( \{ {11},{12},{13},\cdots ,{20}\} \) ,
 \( \{ {21},{22},{23},\cdots ,{30}\} \) ,
 \( \vdots \)
 \( \{ {991},{992},{993},\cdots ,{1000}\} \) .
 How many of these sets contain exactly two multiples of 7 ?
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Context:
In triangle $\triangle ABC$, given that $\cos A \cos B \cos C = \frac{1}{3}$, determine if the value of $\tan A \tan B + \tan B \tan C + \tan C \tan A$ is constant.
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Context:
Determine whether the sequence \((a_n)_{n \in \mathbb{N}}\) defined by \(a_n = \sqrt[n]{|\sin(n)|}\) converges to 1.
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Context:
Several irrational numbers are written on a blackboard. It is known that for every two numbers  $ a$  and  $ b$  on the blackboard, at least one of the numbers  $ a\over b\plus{}1$  and  $ b\over a\plus{}1$  is rational. What maximum number of irrational numbers can be on the blackboard?

*Author: Alexander Golovanov*
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Context:
计算下列实变函数定积分．

$\int _ { 0 } ^ { 2 \pi } \frac { d x } { ( 1 + \varepsilon \cos x ) ^ { 2 } } ( 0 < \varepsilon < 1 ) ,$
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Context:
In the Cartesian coordinate plane \( xOy \), there are infinitely many circles inside the parabola \( y = ax^2 \) (where \( a \) is a constant). The centers of the circles \( O_1, O_2, \cdots \) are all on the y-axis. For each integer \( n > 1 \), the circle \( \odot O_n \) is tangent to the parabola and also externally tangent to the circle \( \odot O_{n-1} \). If the radius of \( \odot O_1 \) is \( \frac{1}{2a} \) and it passes through the vertex of the parabola, find the radius of \( \odot O_{2016} \).
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Context:
A school decides to check the homework of students whose student ID numbers end in 5 from each class. What type of sampling method is being applied? Provide your answer as the name of the sampling method.
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Context:
At a certain high school, the senior class is twice the size of the junior class. If 3/8 of the seniors study Japanese and 1/4 of the juniors study Japanese, while an additional 1/6 of the seniors and 1/12 of the juniors study both Japanese and French, what fraction of the students in both classes study Japanese, accounting for those who study both languages?
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Context:
In the right triangle \( \triangle ABC \), \( \angle B = 90^\circ \). Point \( P \) is on the angle bisector of \( \angle A \) within \( \triangle ABC \). Point \( M \) (distinct from \( A \) and \( B \)) is a point on side \( AB \). The lines \( AP \), \( CP \), and \( MP \) intersect sides \( BC \), \( AB \), and \( AC \) at points \( D \), \( E \), and \( N \) respectively. Given that \( \angle MPB = \angle PCN \) and \( \angle NPC = \angle MBP \), find the ratio \( \frac{S_{\triangle APC}}{S_{ACDE}} \).
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Context:
Someone draws a 2400 franc annuity 8 times at the end of every third year, what is the present value of the annuity at $4 \%$?
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Context:
Determine if there exists a largest nested prime number, and if so, find it.
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Context:
If $11 = x^6 + \frac{1}{x^6}$, find the value of $x^3 + \frac{1}{x^3}$.
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Context:
G3.3 $A B C D$ is a cyclic quadrilateral. $A C$ and $B D$ intersect at $G$. Suppose $A C=16 \mathrm{~cm}, B C=C D=8 \mathrm{~cm}$, $B G=x \mathrm{~cm}$ and $G D=y \mathrm{~cm}$. If $x$ and $y$ are integers and $x+y=c$, find the value of $c$.
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Context:
Let \( l = \lim_{x \rightarrow a}\frac{x^x-a^x}{x-a} \) and \( m = \lim_{x \rightarrow a}\frac{a^x-x^a}{x-a} \) where \( a > 0 \). If \( l = m \), find the value of \( a \).
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Context:
The parabola $y=-\frac{1}{3}x^{2}+2$, when $-1\leqslant x\leqslant 5$, the range of $y$ is ______.
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Context:
4. The base of the pyramid is a parallelogram. A plane is drawn through one side of the base and the midline of the opposite lateral face. Into what ratio does this plane divide the volume of the pyramid?
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Context:
Determine the cardinality of the set \( A = \{ [a, b] \mid a, b \in \mathbb{R}, \frac{a+b}{2} \in \mathbb{Q} \} \).
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Context:
Determine the most general form of the entire function $f(z)$ that satisfies the limit condition: \[ \lim_{z \to \infty} \frac{f(z)^2}{z^3} = 0. \]
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Context:
Determine the number of homomorphisms from the group $\Bbb Z/3\Bbb Z \times \Bbb Z/3\Bbb Z$ to the symmetric group $S_4$. Provide your answer as a single integer.
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Context:
There are two piles of stones, one containing 15 stones and the other containing 20 stones. Two players take turns in the following game: on each turn, a player can take any number of stones, but only from one pile. The player who cannot make a move loses. Who wins with optimal play?
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Context:
How many integer triples $(x,y,z)$ are there satisfying $x^3+y^3=x^2yz+xy^2z+2$ ?

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Context:
Call a polynomial $f$ with positive integer coefficients "triangle-compatible" if any three coefficients of $f$ satisfy the triangle inequality. For instance, $3x^3 + 4x^2 + 6x + 5$ is triangle-compatible, but $3x^3 + 3x^2 + 6x + 5$ is not. Given that $f$ is a degree $20$ triangle-compatible polynomial with $-20$ as a root, what is the least possible value of $f(1)$?
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Context:
Evaluate the limit: $$\lim_{n \rightarrow \infty} \sqrt{n}\int_{-1/\sqrt{n}}^0  e^{x^2/2} \text{erfi}\left(\frac{1}{\sqrt{2n}}\right) dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many elements in the set \( S = \{0, 1, \ldots, 5399\} \) are quadratic residues modulo 5400?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( a = 19911991 \cdots \cdots 1991 \) (a repeated sequence of 1991), what is the remainder when \( a \) is divided by 13?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Last week, a farmer shipped 15 boxes of pomelos which had 360 pomelos in total. The boxes were of three types: small boxes containing 10 pomelos each, medium boxes containing 20 pomelos each, and large boxes containing 30 pomelos each. This week, the farmer shipped 25 boxes: 10 of them were small boxes, 8 were medium boxes, and 7 were large boxes. During transportation, 3 small boxes, 2 medium boxes, and 2 large boxes were damaged, causing a loss of 10% of pomelos for small boxes, 15% for medium boxes, and 20% for large boxes. How many dozens of pomelos did the farmer successfully ship in total for both weeks?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A position of the hands of a (12-hour, analog) clock is called valid if it occurs in the course of a day. For example, the position with both hands on the 12 is valid; the position with both hands on the 6 is not. A position of the hands is called bivalid if it is valid and, in addition, the position formed by interchanging the hour and minute hands is valid. Find the number of bivalid positions.
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Context:
Wendy is playing darts with a circular dartboard of radius 20. Whenever she throws a dart, it lands uniformly at random on the dartboard. At the start of her game, there are 2020 darts placed randomly on the board. Every turn, she takes the dart farthest from the center, and throws it at the board again. What is the expected number of darts she has to throw before all the darts are within 10 units of the center?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two concentric circles have radii of 1 and 2 units, respectively. Determine the minimum possible area of a cyclic quadrilateral that can be inscribed in the larger circle while also containing the smaller circle.
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Context:
Find \( F''(\pi) \) if \[ F(x) = \int_x^{\sin x}  \left(\int_0^{\sin t} (1+u^4)^{0.5} \,du\right)\,dt \]
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Context:
We have a half-circle with endpoints $A$ and $B$ and center $S$. The points $C$ and $D$ lie on the half-circle such that $\angle BAC = 20^\circ$ and the lines $AC$ and $SD$ are perpendicular to each other. If the angle between the lines $AC$ and $BD$ is $x^\circ$, what is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three cones with bases touching each other are placed on a table. The heights of the cones are equal, and the radii of their bases are 1, 2, and 3. A sphere is placed on the table such that it touches all three cones. It turns out that the center of the sphere is equidistant from all points of contact with the cones. Find the radius of the sphere.
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Context:
A right pyramid has a rectangular base with dimensions 8 cm by 15 cm. Its peak is 10 cm vertically above the center of its base. Calculate the sum of the lengths of the pyramid's eight edges. Round your answer to the nearest whole number.
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Context:
Find the sum of the series $$\sum_{n=1}^{\infty}\frac{(-n)^{n-1}e^{-n}}{n!}$$ using the Lambert W function.
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Context:
The Planar National Park is an undirected 3-regular planar graph (i.e., all vertices have degree 3). A visitor walks through the park as follows: she begins at a vertex and starts walking along an edge. When she reaches the other endpoint, she turns left. On the next vertex, she turns right, and so on, alternating left and right turns at each vertex. She does this until she gets back to the vertex where she started. What is the largest possible number of times she could have entered any vertex during her walk, over all possible layouts of the park?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the side length of the square base of a square pyramid with equilateral triangular faces that can completely contain a cuboid with dimensions 25 (height) x 50 (width) x 50 (length), given that the bottom of the cuboid is 26 units above the base of the pyramid.
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Context:
Let \( F_1 \) and \( F_2 \) be the left and right foci of the ellipse \( C \):

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b > 0) \]

If there exists a point \( P \) on the ellipse \( C \) such that \( P F_1 \) is perpendicular to \( P F_2 \), then determine the range of the eccentricity \( e \) of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left[ \frac{\sqrt{2}}{2}, 1 \right)}


Context:
Given that $U$ and $V$ are independent standard normal variables, calculate $\text{Cov}(U^2 - V^2 \mid U > 0)$. Provide your answer as a single value.
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Context:
Transform a rectangle with a length of 10 cm and a width of 8 cm into a parallelogram with a height of 9 cm. The area of this parallelogram is \_\_\_\_\_\_ square centimeters.
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Context:
In heptagon $GEOMETRY$, $\angle G \cong \angle E \cong \angle T \cong \angle R$ and $\angle M$ is supplementary to $\angle Y$, $\angle J$ is supplementary to $\angle O$. Find the measure of $\angle T$.
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Context:
Find the maximum number of colors that can be used to paint the edges of a cube (each edge in one color) such that for every pair of colors, there exist two adjacent edges painted in these colors. Edges are considered adjacent if they share a common vertex.
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Context:
Given a language with $n+1$ atomic propositions, determine the number of non-equivalent propositions that can be expressed using a combinatorial approach.
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Context:
Find the minimum value of the function \( f(x) = \sum_{i=1}^{2017} i |x - i| \) for \( x \in [1, 2017] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sum \( \sum_{j=1}^M c_M(j) \), where \( \lim_{M\to \infty}c_M(j)=0 \) and there exists a sequence \( d_j \ge 0 \) with \( \sum_{j=1}^\infty d_j < \infty \) such that \( |c_M(j)| \le d_j \) for all \( j \) and \( M \), determine if the sum converges to zero as \( M\to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \int_0^\pi \int_0^\pi |\cos(x+y)| \,dx\,dy \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \iiint_R z(x^2 + y^2 + z^2)^{-3/2} \, dV \) over the region \( R \) within the sphere \( x^2 + y^2 + z^2 \leq 81 \) and where \( z \geq 4.5 \), using spherical coordinates.
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Context:
Calculate the volume of the body bounded by the surface defined by the inequality:
\[ 0 \leq z \leq c \cdot \sin\left(\pi \sqrt{\frac{x^2}{a^2} + \frac{y^2}{b^2}}\right) \]
where \(a\), \(b\), and \(c\) are positive parameters.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^\infty \frac{\sin^2(4\pi x)}{x^{3/2}}dx. \)
Please reason step by step, and put your final answer within \boxed{}.


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 example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flow of the vector field \( F(x,y,z) = (xz, yz, -z^2) \) out of the surface \( S = \{(x,y,z) \in \mathbb{R}^3 : z^2 = 1 + x^2 + y^2, 2 \leq z \leq 3\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the expected size of the largest connected component of the graph $H(s_1, s_2)$ as a function of $n$, where $s_1$ and $s_2$ are two binary strings of length $n$. The graph $H(s_1, s_2)$ is constructed based on the principles of Hitomezashi Sashiko, a form of decorative mending.
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Context:
Solve for \( x \) in the equation: \[ \frac{(1+x)^{36} -1}{x} = \frac{20142.9}{420} \]
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Context:
Let \( f: \mathbb{C}[x,y] \rightarrow \mathbb{C}[t] \) be a non-zero homomorphism such that \( \ker f \) strictly contains a prime ideal \( P \neq 0 \). Determine whether \( \text{im} \ f = \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the variance of the random variable $X$ given that $Y=25$ for the joint probability density function $f(x,y) = \frac{1}{1250}$, where $0 < x < y < 50$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Lebesgue measure of the set \( \left\{ \frac{x}{y} : x, y \in I_\alpha^* \right\} \), where \( I_\alpha \subset [0,1] \) is an \( \alpha \)-Cantor set of Lebesgue measure \( \alpha \) and \( I_\alpha^* = I_\alpha \setminus \{0\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the context of Fourier series, can the coefficients of the full Fourier series always be obtained from the complex form coefficients \( C_n \)? Answer "yes" or "no."
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Context:
How many sequences of words (not necessarily grammatically correct) have the property that the first word has one letter, each word can be obtained by inserting a letter somewhere in the previous word, and the final word is CCAMT? Here are examples of possible sequences:
\[C,CA,CAM,CCAM,CCAMT.\]
\[A,AT,CAT,CAMT,CCAMT.\]

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Context:
Consider the function \( f(y) = \int_0^\epsilon \frac{\left(\frac{p-t}{p(1-t)}\right)^y-(1-t)^{y(1-p)/p}}{t\log(1-t)}dt \) for \( p \in (0,1) \) and \( \epsilon > 0 \) a small constant. Is it true that \( f(y) = O(1) \) for large \( y \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Example 5 If for any positive real numbers, $\frac{a^{2}}{\sqrt{a^{4}+3 b^{4}+3 c^{4}}}+\frac{k}{a^{3}} \cdot\left(\frac{c^{4}}{b}+\frac{b^{4}}{c}\right) \geqslant \frac{2 \sqrt{2}}{3}$. Always holds, find the minimum value of the real number $k$.

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Context:
Suppose we have a probability space \((\Omega, \mathcal{F}, \mu)\), where \(\Omega\) is a set, \(\mathcal{F}\) is a \(\sigma\)-algebra, and \(\mu\) is a probability measure. Let \(f: \Omega \rightarrow \mathbb{R}\) be a non-negative bounded function. Consider a sequence of sets \(A_i\), \(i=1, 2, \dots\), in \(\mathcal{F}\) such that \(\int_{A_i} f\ d\mu \leq a \mu(A_i)\) for each \(i\) and \(\mu\left(\bigcup_{i=1}^{\infty} A_i\right) = 1\). Is \(\int_{\Omega} f\ d\mu\) upper bounded by a scalar multiple of \(a\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Cartesian plane, let $\omega$ be the circle centered at $(24, 7)$ with radius $6$. Points $P, Q$, and $R$ are chosen in the plane such that $P$ lies on $\omega$, $Q$ lies on the line $y = x$, and $R$ lies on the $x$-axis. The minimum possible value of $PQ+QR+RP$ can be expressed in the form $\sqrt{m}$ for some integer $m$. Find m.

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Context:
There are 103 numbers written in a circle. It is known that among any five consecutive numbers, there will be at least two positive numbers. What is the minimum number of positive numbers that can be among these 103 written numbers?
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Context:
Find the point of intersection of the lines defined by the equations \( \vec{r}\times\vec{a}=\vec{b}\times\vec{a} \) and \( \vec{r}\times\vec{b}=\vec{a}\times\vec{b} \), where \( \vec{a}=\langle 2,-1,3 \rangle \) and \( \vec{b}=\langle 1,0,1 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
4 / 6 of the population of the country of Venezia lives in Montague province, 1 / 6 lives in Capulet province, and the remaining 1 / 6 lives in Verona province. In the upcoming election, 80% of Montague residents support Romeo, while 70% of Capulet residents and 60% of Verona residents support Juliet. Each resident of Venezia supports exactly one of these two candidates. Rounded if necessary to the nearest percent, the probability that a Juliet supporter chosen at random resides in Capulet is
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the probability that a Markov chain, starting in state (Aa, Aa), will eventually reach state (AA, AA).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate whether the integral \( \int_{\mathbb{R}^2} \frac{1}{\sqrt{x^2 + y^4}} \, dx \, dy \) is finite.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the sequence: \[ \lim_{n\to \infty} \frac{\prod_{i = 0}^{n}\left(i + \frac{2}{3}\right)}{\prod_{j = 0}^{n}\left(j + \frac{1}{3}\right)} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 \(A\), \(B\), \(C\) 是平面上三个不同点，且 \(BC = a\), \(CA = b\), \(AB = c\)，求 \(\frac{c}{a+b}+\frac{b}{c}\) 的最小值。答案应为\(\sqrt{k} - \frac{m}{n}\)的形式，请给出k+m+n的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ordered pairs $(A, B)$ such that the following conditions hold: $A$ and $B$ are disjoint subsets of $\{1,2, \ldots, 50\}$, $|A|=|B|=25$, and the median of $B$ is 1 more than the median of $A$.
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Context:
设正三棱雉 $P-A B C$ 的高为 $P O, M$ 为 $P O$ 的中点, 过 $A M$ 作与棱 $B C$ 平行的平面, 将三棱雉截为上、下两部分，试求此两部分的体积比。请给出比值中两个数之和的值。
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Context:
In the triangular prism \(P-ABC\), \(\triangle ABC\) is an equilateral triangle with side length \(2\sqrt{3}\), \(PB = PC = \sqrt{5}\), and the dihedral angle \(P-BC-A\) is \(45^\circ\). Find the surface area of the circumscribed sphere around the triangular prism \(P-ABC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{C(1)} (z-a)^{-n} (z-b)^{-n} \, dz \) where \(|a|<|b|<1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute $\left|\sum_{i=1}^{2022} \sum_{j=1}^{2022} \cos\left(\frac{ij\pi}{2023}\right)\right|.$
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Context:
Given a differentiable function \( f: \mathbb{R}^2 \to \mathbb{R} \) satisfying \( f(t,t) = t^3 + t \) and \( f(t,-2t) = 2t \) at the origin, calculate the directional derivative \( D_v f(0,0) \) for the vector \( v = (1,3) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an increasing sequence $(a_n)$ that diverges to infinity. If for any $\delta > 0$, $\lim_{n \to \infty} \frac{a_n}{e^{\delta n}} = 0$, does it follow that $\lim_{n \to \infty} \frac{a_n}{\sum_{k=1}^n a_k} = 0$?
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Context:
Sitting at a desk, Alice writes a nonnegative integer $N$ on a piece of paper, with $N \le 10^{10}$. Interestingly, Celia, sitting opposite Alice at the desk, is able to properly read the number upside-down and gets the same number $N$, without any leading zeros. (Note that the digits 2, 3, 4, 5, and 7 will not be read properly when turned upside-down.) Find the number of possible values of $N$.
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Context:
Marnie the Magical Cello is jumping on nonnegative integers on number line. She starts at $0$ and jumps following two specific rules. For each jump she can either jump forward by $1$ or jump to the next multiple of $4$ (the next multiple must be strictly greater than the number she is currently on). How many ways are there for her to jump to $2022$? (Two ways are considered distinct only if the sequence of numbers she lands on is different.)
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Example 5 Let $a, b, c$ be positive real numbers, find the minimum value of
$$\frac{a+3 c}{a+2 b+c}+\frac{4 b}{a+b+2 c}-\frac{8 c}{a+b+3 c}$$
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Context:
What positive real number can be expressed as both $\frac{b}{a}$ and $a:b$ in base $10$ for nonzero digits $a$ and $b$? Express your answer as a decimal.

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Context:
在计算器上分别按3、8、+、3、2、*、1、5、=，最后在屏幕上显示的是多少
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Context:
Given the sequence $\{a_n\}$ that satisfies $a_1=1$, $a_2=3$, and $|a_{n+1}-a_n|=2^{n}$ $(n\in\mathbb{N}^*)$, and $\{a_{2n-1}\}$ is an increasing sequence while $\{a_{2n}\}$ is a decreasing sequence, find the limit $\lim\limits_{n\to+\infty} \dfrac{a_{2n-1}}{a_{2n}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points  $P_1, P_2, P_3,$  and  $P_4$  are  $(0,0), (10, 20), (5, 15),$  and  $(12, -6)$ , respectively. For what point  $P \in \mathbb{R}^2$  is the sum of the distances from  $P$  to the other  $4$  points minimal?
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Context:
If k ^ 3 is divisible by a certain number, and the least possible value of integer k is 84, what is the number by which k ^ 3 is divisible?
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Context:
Let $P_i(x) = x^2 + b_i x + c_i$, where $i = 1, 2, \ldots, n$, be pairwise distinct polynomials of degree $2$ with real coefficients. It is given that for any $0 \le i < j \le n$, $i, j \in \mathbb{N}$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root. Find the greatest possible value of $n$.
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Context:
Suppose $a : \mathbb{R}_+ \to \{-1,1\}$ is a measurable function. Let $X_0 = \frac{1}{2}$. A particle moves on the $X$-axis according to the equation:
\[ X_t = X_0 + \int_0^t a_s \, ds \]
where the integral is a Lebesgue integral. Fix $T = \frac{1}{2}$, so $X_t \in [0,1]$ for all $t \le T$. Let $S \subset [0,1]$ be a set such that $\ell(S) = 1$, where $\ell(\cdot)$ denotes the Lebesgue measure. Define:
\[ G := \{t \le T : X_t \in S\} \]
Is it true that $\ell(G) = \ell([0,T]) = \frac{1}{2}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
求最大的实数 c，使得对平面上任意两个单位向量 \overrightarrow{v_1}, \overrightarrow{v_2}，均存在单位向量 \overrightarrow{v}，满足|\overrightarrow{v}\cdot\overrightarrow{v_1}|\cdot|\overrightarrow{v}\cdot\overrightarrow{v_2}|\geq c.（胡珏伟供题）原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
We call an even positive integer $n$ nice if the set $\{1, 2, \dots, n\}$ can be partitioned into $\frac{n}{2}$ two-element subsets, such that the sum of the elements in each subset is a power of $3$. For example, $6$ is nice, because the set $\{1, 2, 3, 4, 5, 6\}$ can be partitioned into subsets $\{1, 2\}$, $\{3, 6\}$, $\{4, 5\}$. Find the number of nice positive integers which are smaller than $3^{2022}$.
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Context:
Find all real $x$ such that $(2^x-4)^3+(4^x-2)^3=(4^x+2^x-6)^3.$

Enter all the solutions, separated by commas.
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Context:
2. (10 points) YY's family of 3, plus YY's cousin, have a total age of 95 years this year. Dad is 4 years older than Mom, and YY is 3 years older than the cousin. 8 years ago, the sum of their ages was 65 years. Then Dad is $\qquad$ years old this year.
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Context:
$\overline{AB}$ is a diameter of a circle. Tangents $\overline{AD}$ and $\overline{BC}$ are drawn so that $\overline{AC}$ and $\overline{BD}$ intersect in a point on the circle. If $\overline{AD}=a$ and $\overline{BC}=b$, $a \not= b$, the diameter of the circle is:
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Context:
Evaluate the limit: \[ L = \lim_{n\to\infty} e^{\frac{n}{4}}n^{-\frac{n+1}{2}}\left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Carlos is doing his laundry along with that of his two other household members.

For Carlos:
He needs to wash five loads, which take 30 minutes for the first load, 45 minutes for the second load, 40 minutes for the third load, 50 minutes for the fourth load, and 35 minutes for the fifth load. He can combine all his loads and dry them in two separate drying cycles, with the first cycle taking 85 minutes and the second cycle taking 95 minutes.

For his sister, Maria:
Maria needs to wash three loads of laundry, which take 25 minutes, 55 minutes, and 40 minutes for the first, second, and third load, respectively. She can combine her loads and dry them in one drying cycle for 80 minutes.

For their father, José:
José needs to wash four loads of laundry, which take 20 minutes, 45 minutes, 35 minutes, and 60 minutes for first, second, third, and fourth load, respectively. He can combine his loads and dry them in one drying cycle for 90 minutes.

Carlos is responsible for coordinating the laundry process for himself and his family members. How much time will it take to complete all the laundry, keeping in mind that only one washing machine and one dryer are available and they can't work simultaneously?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are $2 n$ students in a school $(n \in \mathbb{N}, n \geq 2)$. Each week $n$ students go on a trip. After several trips the following condition was fulfilled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?
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Context:
Determine the number of ten-digit positive integers with the following properties:
$\bullet$ Each of the digits $0, 1, 2, . . . , 8$ and $9$ is contained exactly once.
$\bullet$ Each digit, except $9$, has a neighbouring digit that is larger than it.
(Note. For example, in the number $1230$, the digits $1$ and $3$ are the neighbouring digits of $2$  while $2$ and $0$ are the neighbouring digits of $3$. The digits $1$ and $0$ have only one neighbouring digit.)

[i](Karl Czakler)[/i]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a game, player A selects a number n from the set S = \{1, 2, \ldots, 1001\}. Player B must guess the value of n by choosing subsets of S. After each guess, player A reveals the number of subsets that contain n. Player B makes three guesses, with the number of subsets in each guess denoted by k_1, k_2, and k_3, respectively. Determine the minimum possible value of k_1 + k_2 + k_3 such that player B can always correctly identify n.
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Context:
The word "папаха" is composed of letters from a cut-out alphabet. The cards with the letters are thoroughly shuffled. Four cards are drawn in succession and laid out in a row. What is the probability of forming the word "nana" this way?
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Context:
Find the value of \( a \) such that \( a > 0 \) and \( \lim_{x \to a}\frac{x^x-a^x}{x-a} = \lim_{x \to a}\frac{a^x-x^a}{x-a} \).
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Context:
Flights are arranged between 13 countries. For $ k\ge 2$, the sequence $ A_{1} ,A_{2} ,\ldots A_{k}$ is said to a cycle if there exist a flight from $ A_{1}$ to $ A_{2}$, from $ A_{2}$ to $ A_{3}$, $ \ldots$, from $ A_{k - 1}$ to $ A_{k}$, and from $ A_{k}$ to $ A_{1}$. What is the smallest possible number of flights such that how the flights are arranged, there exist a cycle?
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Context:
Given that the even function $f(x)$ is monotonically increasing on $(0, +\infty)$, determine the relationship among $f(-3)$, $f(-1)$, and $f(2)$.
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Context:
Four points are independently chosen uniformly at random from the interior of a regular dodecahedron. What is the probability that they form a tetrahedron whose interior contains the dodecahedron's center?
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Context:
Find the derivative.

\[ y=\arcsin \frac{1}{2 x+3}+2 \sqrt{x^{2}+3 x+2}, \quad 2 x+3>0 \]
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Context:
Find the value of $x$ at the inflection point of the function $f(x) = 10 \frac{\ln(\ln(x))}{\ln(x)}$. \(x > 1\)
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Context:
Determine the number of zeroes of the polynomial \( f(z) = z^{10} + 10z + 9 \) within the annulus \( \{ z : 1 < |z| < 2 \} \).
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Context:
A bookstore has 11 types of magazines, 8 types priced at 2 yuan each, and 3 types priced at 1 yuan each. Xiao Zhang has 10 yuan to spend on magazines (at most one of each type, exactly using up the 10 yuan). How many different purchasing methods are there? (Answer with a number).
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Context:
Let $P=\left\{x\mid x+2\geqslant x^{2}\right\}$, $Q=\left\{x\in \mathbb{N}\mid |x|\leqslant 3\right\}$. Find the intersection of sets P and Q, and express your answer as a set of numbers.
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Context:
A block wall $100$ feet long and $7$ feet high will be constructed using blocks that are $1$ foot high and either $2$ feet long or $1$ foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall?
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Context:
In a test, $3n$ students participate, who are located in three rows of $n$ students in each. The students leave the test room one by one. If $N_1(t), N_2(t), N_3(t)$ denote the numbers of students in the first, second, and third row respectively at time $t$, find the probability that for each t during the test\[|N_i(t) - N_j(t)|＜ 2, i \neq  j, i, j = 1, 2, \dots .\]
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Context:
Write  $2017$  following numbers on the blackboard:  $-\frac{1008}{1008}, -\frac{1007}{1008}, ..., -\frac{1}{1008}, 0,\frac{1}{1008},\frac{2}{1008}, ... ,\frac{1007}{1008},\frac{1008}{1008}$  .
One processes some steps as: erase two arbitrary numbers  $x, y$  on the blackboard and then write on it the number  $x + 7xy + y$ . After  $2016$  steps, there is only one number. The last one on the blackboard is
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Context:
Carolyn buys a silverware set with 6 knives, 12 forks, and three times as many spoons as knives. Then her friend trades her 10 knives for 6 spoons. What percentage of Carolyn's silverware is knives?
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Context:
Joe likes to play video games and he normally spends $50 a month on video games. Joe also sells his games for $30 each once he is done playing them. If he can buy and sell games for 12 months before he is out of money, how much money does he start with?
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Context:
Consider the permutation $a \in S_9$ given by $a=(1\ 6)(2\ 5\ 7)(3\ 8\ 4\ 9)$. Determine the number of permutations $b \in S_9$ such that $a \circ b = b \circ a$. Provide your answer as a single integer.
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Context:
Evaluate the integral \( \int _{ c } \frac { \sin(iz) }{ { z }^{ 3 }({ z }^{ 2 }+2i) } dz \) using residue calculus.
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Context:
2. Determine the number of all quadratic polynomials $P(x)$ with integer coefficients such that for every real number $x$ the following holds:

$$
x^{2}+2 x-2023<P(x)<2 x^{2} \text {. }
$$

(Ján Mazák, Michal Rolínek)
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Context:
Given that the function \[ f(x) = \lim_{n\to\infty} \frac{(1-\cos(1-\tan(\frac{\pi}{4}-x)))(1+x)^n+\lambda\sin((n-\sqrt{n^2-8n})x)}{x^2(1+x)^n+x} \] is continuous at $x=0$, find the value of $f(0) + 2\lambda$.
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Context:
A solid right prism $ABCDEF$ has a height of 20, as illustrated. The bases are equilateral triangles with a side length of 14. Points $X$, $Y$, and $Z$ are the midpoints of edges $AC$, $BC$, and $DC$, respectively. Determine the perimeter of triangle $XYZ$.
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Context:
Let \( P(x) = x^3 + ax^2 + bx + c \) be a polynomial where \( a, b, c \) are integers and \( c \) is odd. Let \( p_i \) be the value of \( P(x) \) at \( x = i \). Given that \( p_1^3 + p_2^3 + p_3^3 = 3p_1p_2p_3 \), find the value of \( p_2 + 2p_1 - 3p_0 \).
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Context:
Determine the radius of convergence of the power series \(\sum_{n=0}^{\infty} (n^2+1)a_n z^n\), given that \(\sum_{n=0}^{\infty} a_n z^n\) is a convergent power series with \(\lim_{n \to \infty}\frac{a_{n+1}}{a_n}=R>0\).
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Context:
1. How many distinct permutations of the letters of the word REDDER are there that do not contain a palindromic substring of length at least two? (A substring is a contiguous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.)
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Context:
How to calculate $$\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{t^2}{2x^2}-\frac{x^2}{2}\right)\,dx$$
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Context:
On the Island of Misfortune with a population of 96 people, the government decided to carry out five reforms. Each reform is opposed by exactly half of all citizens. A citizen will go to a rally if they are dissatisfied with more than half of all the reforms. What is the maximum number of people the government can expect at the rally?
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Context:
A gardener is preparing to plant a row of 20 trees, with the choice between two types of trees: maple trees and sycamore trees. The number of trees between any two maple trees (excluding these two maple trees) cannot be equal to 3. What is the maximum number of maple trees that can be planted in these 20 trees?
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Context:
Determine the dimension of the subspace of quadric hypersurfaces in $\mathbb{P}^4$ that vanish at eight points in general position on a rational normal curve of degree 4.
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Context:
A boy squirrel gathers 20 walnuts and carries them to his burrow in the tree, adding to the 30 already there, but he drops 4 on the way and hides 8 in another location. Unfortunately, he forgets where he hid 3 of them. The girl squirrel brings 15 more walnuts to their burrow, eats 5, gives 4 to another squirrel, and loses 3 while playing. Later, she accidentally knocks 2 out of the burrow. A third squirrel gathers 10 walnuts and drops 1 on the way to the burrow. This squirrel also hides 3 walnuts before returning 6 to the main burrow and giving 1 away to a friend. How many walnuts are left in the main burrow of the boy and girl squirrel?
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Context:
Given real numbers \( x_{1}, x_{2}, \cdots, x_{1991} \) that satisfy the condition
\[
\sum_{i=1}^{1990} \left| x_{i} - x_{i+1} \right| = 1991,
\]
and \( y_{k} = \frac{1}{k} \sum_{i=1}^{k} x_{i} \) for \( k = 1, 2, \cdots, 1991 \), determine the maximum value of \( \sum_{i=1}^{1990} \left| y_{i} - y_{i+1} \right| \).
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Context:
The sequence $(x_n)$ is defined by $x_1 = 150$ and $x_k = x_{k - 1}^2 - x_{k - 1}$ for all $k \ge 2.$ Compute
\[\frac{1}{x_1 + 1} + \frac{1}{x_2 + 1} + \frac{1}{x_3 + 1} + \dots.\]
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Context:
How many 7-digit numbers divisible by 9 are there, where the second to last digit is 5?
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Context:
2019 points are chosen independently and uniformly at random on the interval $[0,1]$. Tairitsu picks 1000 of them randomly and colors them black, leaving the remaining ones white. Hikari then computes the sum of the positions of the leftmost white point and the rightmost black point. What is the probability that this sum is at most 1 ?
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Context:
已知复数 $z_{1} 、 z_{2} 、 z_{3}$ 的辐角分别为 $\alpha 、 \beta 、 \gamma$, 且满足:  $$  \left|z_{1}\right|=1,\left|z_{2}\right|+\left|z_{3}\right|=2, z_{1}+z_{2}+z_{3}=0  $$  求 $\cos (\alpha-\beta)+2 \cos (\beta-\gamma)+3 \cos (\gamma-\alpha)$的取值范围。请给出该范围的起点和终点的和。
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Context:
In a class of 31 students, three of them have exactly three friends each, the next three have six friends each, the next three have nine friends each, and so on, until the last three have thirty friends each. How many friends does the 31st student have? (Friendship between people is mutual.)
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Context:
Example 6 Given 2014 real numbers $x_{1}, x_{2}, \cdots, x_{2014}$ satisfy the system of equations
$$
\sum_{k=1}^{2014} \frac{x_{k}}{n+k}=\frac{1}{2 n+1}(n=1,2, \cdots, 2014) .
$$

Try to calculate the value of $\sum_{k=1}^{2014} \frac{x_{k}}{2 k+1}$.
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Context:
In a basketball tournament, every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$, and there are no draws. A single team wins the tournament with $26$ points, and exactly two teams share the last position with $20$ points. How many teams participated in the tournament?
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Context:
The radius of the circle inscribed in an isosceles right triangle is $r$, and half of the perimeter is $p$. Find the hypotenuse.
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Context:
5. A sphere $Z$ with radius $R$ has a spherical segment of height $v (v < R)$ removed. A sphere $K$ with radius $\frac{1}{2} v$ is inscribed in this segment. Additionally, eight identical smaller spheres are inscribed in the segment, each touching the sphere $K$. No two of these smaller spheres share an internal point, and each touches exactly two others. Determine the ratio $v: R$.
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Context:
Two thirds of a pitcher is filled with orange juice and the remaining part is filled with apple juice. The pitcher is emptied by pouring an equal amount of the mixture into each of 6 cups. Calculate the percentage of the total capacity of the pitcher that each cup receives.
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Context:
In a sample of 1,200 high school students, all students are either freshmen, sophomores, juniors, or seniors. In this sample, 23 percent are juniors, 70 percent are not sophomores, and there are 160 seniors. Furthermore, the gender distribution within each grade level is as follows: 55 percent males among freshmen, 60 percent males among sophomores, 48 percent males among juniors, and 52 percent males among seniors. Moreover, students can be members of up to three different clubs - the Science Club, the Drama Club, or the Debate Club. The percentage of students joining each club is: 30 percent of freshmen are in the Science Club, 20 percent of sophomores are in the Drama Club, and 25 percent of juniors are in the Debate Club. Considering the given information, how many more male freshmen who are also in the Science Club than male sophomores who are in the Drama Club are there among the sample of students?
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Context:
If an exterior angle of a triangle is equal to its adjacent interior angle, then this triangle is a \_\_\_\_\_\_ triangle.
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Context:
Mr. Willson worked on making his furniture for some time on Monday. On Tuesday, he worked for half an hour. Then he worked for 2/3 an hour on Wednesday and 5/6 of an hour on Thursday. He worked for 75 minutes on Friday. In all, he worked for 4 hours from Monday to Friday. How long did he work on Monday?
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Context:
What is the probability that two randomly selected two-digit positive integers share at least one digit?
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Context:
设非负实数 x_1,x_2,\cdots,x_{2018} 满足 \ds{1\leq i<j\leq 2018}{}x_ix_j=1. 求 \ds{i=1}{2018}\df{1}{s-x_i} 的最小值，其中 s=\ds{i=1}{2018}x_i.（赵斌、王坤供题）原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
设$4 ^ { 2 7 } + 4 ^ { 5 0 0 } + 4 ^ { n }$是平方数（整数的平方），求整数n的最大值.
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Context:
At a restaurant, there are 3 types of adult meals, which cost $5, $7, and $9, respectively. Kids eat free, but beverages cost $2 for adults and $1 for kids. If a group of 14 people came in, including 7 kids, and decided to purchase the following: 4 adult meals of the $5 meal, 2 adult meals of the $7 meal, and 1 adult meal of the $9 meal, along with 9 beverages, calculate the total cost for the group to eat.
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Context:
The result of the calculation \(1! \times 3 - 2! \times 4 + 3! \times 5 - 4! \times 6 + \cdots + 2009! \times 2011 - 2010! \times 2012 + 2011!\) is \(\qquad\).
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Context:
in a soap company a soap is manufactured with 11 parts . for making one soap you will get 1 part as scrap . at the end of the day you have 251 such scraps . from that how many soaps can be manufactured ?
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Context:
Ashwin the frog is traveling on the $xy$-plane in a series of $2^{2017}-1$ steps, starting at the origin. At the $n^{\text{th}}$ step, if $n$ is odd, Ashwin jumps one unit to the right. If $n$ is even, Ashwin jumps $m$ units up, where $m$ is the greatest integer such that $2^{m}$ divides $n$. If Ashwin begins at the origin, what is the area of the polygon bounded by Ashwin's path, the line $x=2^{2016}$, and the $x$-axis?
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Context:
In a certain country, 1/5 of 8 equals 4. Assuming the same proportion, what would be the value of 1/4 of a certain number if the result is 10?
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Context:
In triangle  $ABC$ ,  $AB = 28$ ,  $AC = 36$ , and  $BC = 32$ . Let  $D$  be the point on segment  $BC$  satisfying  $\angle BAD = \angle DAC$ , and let  $E$  be the unique point such that  $DE \parallel AB$  and line  $AE$  is tangent to the circumcircle of  $ABC$ . Find the length of segment  $AE$ .

*Ray Li*
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Context:
Given a right triangular prism $ABC-A_{1}B_{1}C_{1}$ whose side edge length is equal to the base edge length, find the sine value of the angle formed by $AB_{1}$ and the side face $ACC_{1}A_{1}$.
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Context:
A man lost some percentage by selling oranges at the rate of 16 a rupee. To gain 28%, he must sell them at 12 a rupee. What was the percentage of his loss?
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Context:
Determine the number of revolutions a wheel, with a fixed center and with an outside diameter of 8 feet, would require to cause a point on the rim to travel one mile.
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Context:
Let $ABCD$ be a rectangle inscribed in circle $\Gamma$, and let $P$ be a point on the minor arc $AB$ of $\Gamma$. Suppose that $PA \cdot PB = 2$, $PC \cdot PD = 18$, and $PB \cdot PC = 9$. The area of rectangle $ABCD$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers and $b$ is a squarefree positive integer. Compute $100a + 10b + c$.
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Context:
[ Combinatorics (miscellaneous).]
$[$ Estimation + example $]$

In a pond, 30 pikes were released, which gradually eat each other. A pike is considered full if it has eaten at least three pikes (full or hungry). What is the maximum number of pikes that can become full?
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Context:
Each of the 33 warriors either always lies or always tells the truth. It is known that each warrior has exactly one favorite weapon: a sword, a spear, an axe, or a bow.

One day, Uncle Chernomor asked each warrior four questions:
- Is your favorite weapon a sword?
- Is your favorite weapon a spear?
- Is your favorite weapon an axe?
- Is your favorite weapon a bow?

13 warriors answered "yes" to the first question, 15 warriors answered "yes" to the second question, 20 warriors answered "yes" to the third question, and 27 warriors answered "yes" to the fourth question.

How many of the warriors always tell the truth?
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Context:
Let $\vartriangle ABC$ be a triangle with $AB = 5$, $BC = 8$, and, $CA = 7$. Let the center of the $A$-excircle be $O$, and let the $A$-excircle touch lines $BC$, $CA$, and,$ AB$ at points $X, Y$ , and, $Z$, respectively. Let $h_1$, $h_2$, and, $h_3$ denote the distances from $O$ to lines $XY$ , $Y Z$, and, ZX, respectively. If $h^2_1+ h^2_2+ h^2_3$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.
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Context:
A point $P$ lies at the center of square $A B C D$. A sequence of points $\left\{P_{n}\right\}$ is determined by $P_{0}=P$, and given point $P_{i}$, point $P_{i+1}$ is obtained by reflecting $P_{i}$ over one of the four lines $A B, B C, C D, D A$, chosen uniformly at random and independently for each $i$. What is the probability that $P_{8}=P$ ?
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Context:
Given that the negative of one of the solutions of x^2 - 3x + c = 0 is a solution of x^2 + 3x - c = 0, find the solutions of x^2 - 3x + c = 0. Express your answer as two numerical values.
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Context:
Given real numbers $x$ and $y$ satisfying $x^2+y^2=4$, find the range of values for $4(x- \frac {1}{2})^2+(y-1)^2+4xy$.
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Context:
If two distinct members of the set $\{ 3, 7, 21, 27, 35, 42, 51 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 63? 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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