Context:
In an isosceles trapezoid \(ABCD\) with \(AB = CD\), an inscribed circle is drawn. Let \(M\) be the point where the circle touches side \(CD\), \(K\) be the intersection point of the circle with segment \(AM\), and \(L\) be the intersection point of the circle with segment \(BM\). Find the value of \(\frac{AM}{AK} + \frac{BM}{BL}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
From the digits 0, 1, 2, 3, 4, 5, 6, select 2 even numbers and 1 odd number to form a three-digit number without repeating digits. The number of such three-digit numbers that are divisible by 5 is ____. (Answer with a number)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let a,b,c, and d be real numbers such that\[a^2 + 3b^2 + \frac{c^2+3d^2}{2} = a + b + c+d-1.\]Find $1000a + 100b + 10c + d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The letter T is formed by placing two $3\:\text{inch}\!\times\!5\:\text{inch}$ rectangles to form a T shape. The vertical rectangle is placed in the middle of the horizontal one, overlapping it by $1.5$ inches on both sides. What is the perimeter of the new T, in inches?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of group homomorphisms from the multiplicative group of units modulo 12, denoted as $\mathbb{Z}_{12}^{\times}$, into the multiplicative group of units modulo 8, denoted as $\mathbb{Z}_8^{\times}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a given \( n \in \mathbb{N} \), determine the values of \( p > 0 \) for which the function \( \frac{1}{|x|^p} \) is integrable over the region \( \mathbb{R}^n \setminus B_r(0) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite angles $A$, $B$, and $C$ respectively. Given the inequality $\frac{b}{{b+c}}+\frac{c}{{a+b}}≥1$, determine the range of angle $A$ in radians.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $\overrightarrow{a}$ and $\overrightarrow{b}$ are two non-collinear vectors in a plane, $\overrightarrow{AB}=m\overrightarrow{a}+2\overrightarrow{b}$, $\overrightarrow{BC}=3\overrightarrow{a}+m\overrightarrow{b}$. If points $A$, $B$, and $C$ are collinear, then $m=$____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $ a > 1$, then the sum of the real solutions of\[\sqrt{a - \sqrt{a + x}} = x\]is equal to
Please reason step by step, and put your final answer within \boxed{}.


Context:
18. Find the sum of all real roots of the equation $3 \tan ^{2} x+8 \tan x+3=0$ in the range $0<x<2 \pi$.求方程 $3 \tan ^{2} x+8 \tan x+3=0$ 在區間 $0<x<2 \pi$ 內所有實根之和。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \lim_{n \rightarrow \infty} \int_0^n \left(1 + \frac{-x}{n}\right)^n \cos\left(\frac{x}{\sqrt{n}}\right) e^{x/2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Square $ABCD$ has side length $36$.  Point $E$ is on side $AB$ a distance $12$ from $B$, point $F$ is the midpoint of side $BC$, and point $G$ is on side $CD$ a distance $12$ from $C$.  Find the area of the region that lies inside triangle $EFG$ and outside triangle $AFD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bob, having little else to do, rolls a fair $6$-sided die until the sum of his rolls is greater than or equal to $700$. What is the expected number of rolls needed? Any answer within $0.0001$ of the correct answer will be accepted.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If real numbers $a>b>1$ satisfy the inequality$$(ab+1)^2+(a+b)^2\leq 2(a+b)(a^2-ab+b^2+1)$$what is the minimum possible value of $\dfrac{\sqrt{a-b}}{b-1}$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately what fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the tetrahedron \( P-ABC \), \( PA=BC=\sqrt{6} \), \( PB=AC=\sqrt{8} \), and \( PC=AB=\sqrt{10} \). Find the radius of the circumscribed sphere of the tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a function $f(x)$ defined on $R$ such that $f(x) + f(x+4) = 23$. When $x \in (0,4]$, $f(x) = x^2 - 2^x$. Find the number of zeros of the function $f(x)$ on the interval $(-4,2023]$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $p$ for which the function \( \rho(x, y) = \int_a^b \lvert x(t) - y(t) \rvert^p \, \mathrm{d} t \) is a metric on the set of continuous functions on \([a, b]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two bullets are fired independently at the origin of an $(x,y)$ coordinate system, and the points hit, say $(X,Y)$ are random variables. The random variables $X,Y$ follow standard Normal. What is the distribution of the distance between the two points hit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
$174 S-A B C D-S^{\prime}$ is a regular octahedron with edge length 1. Sphere $O$ is outside this octahedron, tangent to face $S A B$, and tangent to the extended planes of faces $S B C$, $S A D$, and $S^{\prime} A B$. Find the radius $r$ of sphere $O$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint_{|z|=3} \frac{1}{2-z} e^{\frac{1}{z}} \,dz \) by computing the residue at the singularities inside the contour, particularly focusing on the residue at the essential singularity at \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
You are given a square  $n \times n$ . The centers of some of some  $m$  of its  $1\times 1$  cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer  $n \geq 3$ , find the largest value of  $m$  for which it is possible.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the prime ideals of the ring \( \mathbb{F}_p[x]/(x^2) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The shorter leg $AC$ of the right triangle $ABC$ has length $b$. On hypotenuse $AB$, a point $D$ is chosen such that $BD = BC$. On the leg $BC$, a point $E$ is chosen such that $DE = BE = m$. Find the perimeter of the quadrilateral $ADEC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A frog jumps on lily pads numbered 0 to 10. If it reaches pad 0, it dies, and if it reaches pad 10, it survives. When the frog is on pad $N$, where $0 < N < 10$, it jumps to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. Find the probability that the frog reaches pad 10 starting from pad 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
b takes some more days than a to finish a task. b and a start this task and a leaves the task 12 days before the task is finished. b completes 60% of the overall task. b would have taken 36 days to finish the task if he had worked independently. How many more days does b take than a to finish the task?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of the sphere that touches three faces of a regular tetrahedron with edges of unit length, as well as the three sides of the fourth face.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( S = \{1, 2, \cdots, 2025\} \). A subset of \( S \) with 51 elements is called a "good subset" if the sum of its elements is divisible by 5. Find the number of good subsets of \( S \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the symmetric group $S_5$ of permutations of five symbols. Let $S$ be the set of subgroups of $S_5$ that are isomorphic to the non-cyclic group of order 4. Two subgroups $H$ and $K$ in $S$ are considered equivalent if they are conjugate in $S_5$. Determine the number of equivalence classes in $S$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a stationary ARMA(1,1) process \(x_t = 0.7 x_{t-1} + u_t + 0.2 u_{t-1}\), where \(u_t\) is white noise with standard deviation \(\sigma(u_t) = 4\), find \( \Bbb Var(x_{t+1}|x_t, u_t)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many solutions in real numbers does the equation

$$
\sin (\sin (\sin (\sin (\sin (x)))))=x / 3 ?
$$

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


Context:
In a room lit by two bulbs, the bulbs are replaced only when both burn out. The lifetimes of the bulbs are independent and exponentially distributed with parameter \( \lambda = 1 \). What fraction of the time is the room only half-lit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the function \( f(x) = -\frac{1}{2} x^2 + \frac{13}{2} \) has a minimum value of \( 2a \) and a maximum value of \( 2b \) in the interval \( (a, b] \), find \( [a, b] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
From milk with a fat content of $5\%$, cottage cheese with a fat content of $15.5\%$ is produced, while there remains whey with a fat content of $0.5\%$. How much cottage cheese is obtained from 1 ton of milk?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the Hilbert space $l_2[0,\infty)$ with the right shift operator $S_r$ and the left shift operator $S_l$ defined as follows:

$$S_r (x_0, x_1, x_2, \dots) \mapsto (0, x_0, x_1, \dots)$$
$$S_l (x_0, x_1, x_2, \dots) \mapsto (x_1, x_2, x_3, \dots)$$

The combined operator $T = S_r + S_l$ acts as:

$$T(x_0, x_1, x_2, \dots) \mapsto (x_1, x_0 + x_2, x_1 + x_3, \dots)$$

Describe the spectrum of the operator $T$. Provide your answer as a set of complex numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values for $p$ such that the surface area of the solid generated by revolving the function $f(x) = \frac{1}{x^p}$ around the x-axis is finite.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square is drawn in the Cartesian coordinate plane with vertices at $(2, 2)$, $(-2, 2)$, $(-2, -2)$, $(2, -2)$. A particle starts at $(0,0)$. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is $\frac{1}{8}$ that the particle will move from $(x, y)$ to each of $(x, y + 1)$, $(x + 1, y + 1)$, $(x + 1, y)$, $(x + 1, y - 1)$, $(x, y - 1)$, $(x - 1, y - 1)$, $(x - 1, y)$, or $(x - 1, y + 1)$. The particle will eventually hit the square for the first time, either at one of the $4$ corners of the square or at one of the $12$ lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum possible degree of an irreducible polynomial in the ring of polynomials with real coefficients in three variables, \( \mathbb{R}[x_1, x_2, x_3] \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. 5.1. Find the largest natural number in which all digits are different and any two adjacent digits differ by 6 or 7.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Krull dimension of the ring $K[t^{-1},t]$, where $K$ is a field.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points $X$, $Y$, and $Z$ lie on a circle with center $O$ such that $XY=12$. Points $A$ and $B$ lie on segment $XY$ such that $OA=AZ=ZB=BO=5$. Compute $AB$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the line integral of the vector field \( F(x,y) = \left(\frac{x}{x^2+y^2}, \frac{-y}{x^2+y^2}\right) \) along the closed curve formed by the union of the line segments from \((1,0)\) to \((0,1)\) and from \((0,1)\) to \((-1,0)\), and the upper half-circle centered at the origin from \((-1,0)\) to \((1,0)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a sequence of independent and identically distributed (i.i.d.) random variables $X_1, X_2, \dots$, where each $X_k$ follows a distribution with $\mathbb{P}(X_k \geq x) = \frac{1}{x^a}$ for $x \in \mathbb{N} = \{1,2,3,\dots\}$ and $a > 0$. Determine the range of $a \in \mathbb{R}_+$ for which the expected value $\mu$ of $X_1$ is finite.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint_D e^{-x^2-4y^2} \, dx \, dy \), where \( D = \{(x,y) : 0 \leq x \leq 2y\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( X \) be a real-valued stochastic variable with density \( f(x) = 2xe^{-x^2} \) for \( x > 0 \). Let \( Y = X^2 \). Calculate \( E[Y^n] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum \(\sum_{r=0}^{44} \cot^2(2r+1)\) using complex numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the function \(f(x) = |x| + |x - 1| + |x - 3| + |x - 6| + \cdots + |x - (1 + 2 + \cdots + 101)|\) for all real numbers \(x\). Determine the value of \((m + n - 10)/18\), where \(m\) is the number of points at which \(f(x)\) is non-differentiable and \(n\) is the number of integer points where \(f(x)\) attains a minimum.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Approximate the expression \( \frac{n^n}{{\lfloor\alpha n \rfloor}^{\lfloor\alpha n \rfloor}{(n-\lfloor\alpha n \rfloor)}^{n-\lfloor\alpha n \rfloor}} \) using the binary entropy function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum number of monomials required to generate the ideal $I(V(x_1^2, x_2^2) \cup V(x_3^2, x_4^2))$ in $\Bbb C^4$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the curve $y = rac{4	ext{π}}{3} 	imes 	ext{sin}(x)$, let $s \in \mathbb{Q}$ with $s > 1$. The arc length $L$ from $0$ to $2\pi s$ is expressed as an elliptic integral of the second kind using the formula: $L = \int_0^{2\pi s} \sqrt{1 + \cos^2 x}\, dx$. If the period is changed to $\frac{4\pi}{3}$, determine the new amplitude of $\sin x$ so that the arc length from $0$ to $2\pi s$ remains $L$. Assume the current amplitude is $|A| = 1$. What should the new amplitude be?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that each coordinate of a k-tuple can take one of 3 values, determine the maximum number of 2-different k-tuples that can be formed.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a sufficient statistic for \( \theta \) given that \( y_1, y_2, \ldots, y_n \) are independent and identically distributed as \( U([0,\theta]) \), where \( \theta > 0 \) is unknown.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( A \) and \( B \) be two \( d \)-dimensional vectors with all elements initially equal to 1. We randomly select one element from each vector with replacement and change it to 0, repeating this process \( m \) times. If \( 2m \leq d \), what is the expected number of elements in \( A - B \) that are equal to 2?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{d \left(1 - \frac{1}{d}\right)^{2m}}


Context:
Find the coordinates $(m, n)$ of the point $P$ on the ellipse \( \frac{x^2}{8} + \frac{y^2}{18} = 1 \) such that the tangent at $P$ meets the coordinate axes at points $A$ and $B$, and the area of triangle $OAB$ is minimized, where $O$ is the origin.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \frac{1}{2\pi i}\int_{\gamma} \frac{1}{z^4(e^{2z}-1)}dz \) where \( \gamma=\{z: z=\frac{1}{2}e^{2\pi i t},0\leq t\leq1 \} \). Note that \( z=0 \) is a pole singularity of order 5.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Add $537_{8} + 246_{8}$. Express your answer in base $8$, and then convert your answer to base $16$, using alphabetic representation for numbers $10$ to $15$ (e.g., $A$ for $10$, $B$ for $11$, etc.).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two circles intersect at points $A$ and $B$ in the Cartesian coordinate system. Point $A$ is on the line $y = 3$, and point $B$ is on the line $y = 12$. The circles are tangent to the x-axis at points $P$ and $Q$, respectively. Express the distance $AB$ in terms of the distance $PQ$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cylindrical container is constructed from a material with a fixed thickness \(T\). The container has an open top and a solid circular bottom, where the bottom's radius equals the outer radius of the container. Given that the inner volume of the container is \(V\) mm\(^3\) and the volume of the material used is minimized when the inner radius is 10 mm, determine the value of \(\frac{V}{\pi T}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit superior of the sequence \( \sin(x) + \sin(\pi x) \) as \( x \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) = \left( \int_{0}^{x} e^{-t^{2}} \, dt \right)^{2} \) and \( g(x) = \int_{0}^{1} \frac{e^{-x^{2(1+t^{2})}}}{1+t^{2}} \, dt \). Find the value of \( f'(\sqrt{\pi}) + g'(\sqrt{\pi}) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sequence $(a_n)$ with $a_2 > 0$, $a_3 > 0$, and $a_{n+2}a_n = \left(1+\frac{1}{n}\right)^n a_{n+1}^2$ for $n \geq 2$, find the limit of $\sqrt[n^2]{a_n}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of continuous functions \( f(x) \) defined on the interval \([0,1]\) that satisfy the following conditions:

\[ I_1 = \int_0^1 f(x) \, dx = 1, \]

\[ I_2 = \int_0^1 x f(x) \, dx = a, \]

\[ I_3 = \int_0^1 x^2 f(x) \, dx = a^2. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the equation of the plane tangent to the surface $z = \sqrt{2x^2+y^4+1}$ that is parallel to the plane $x-y-z=0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \(\int _{ -\infty }^{ +\infty }{ { e }^{ { (ix+b) }^{ 2 } } } dx\) where \(x\) and \(b\) are real numbers and \(b\) is a constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
365. The lengths of two parallel sides of a rectangle are 1 cm. In addition, it is known that it can be divided into four rectangles by two perpendicular lines, three of which have an area of no less than $1 \mathrm{~cm}^{2}$, and the fourth no less than $2 \mathrm{~cm}^{2}$. What is the minimum length of the other two sides of the rectangle for this to be possible?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to \infty}\left[31^{1/(2n+1)}+(-93)^{1/(2n+3)}-(-75)^{1/(2n+5)}\right]^{n} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a row of $n$ opaque boxes numbered from $1$ to $n$, a robot starts in box $x$ on the first day and moves to box $(x + ky) \mod n$ on the $k$-th day, where $y$ is the robot's speed. Determine a formula or method to calculate the minimum number of days needed to guarantee finding the robot, based on $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find \( F''(\pi) \) if \( F(x) = \int_{x}^{\sin(x)} \left(\int_0^{\sin(t)}\sqrt{1+u^4}\,\text{d}u\right)\text{d}t \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Provide an example of a norm on \(\mathbb{R}^2\) that does not satisfy the parallelogram law.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For all positive integers $ n$ less than $ 2002$, let
\[ a_n = \begin{cases} 11 & \text{if }n\text{ is divisible by }13\text{ and }14 \\
13 & \text{if }n\text{ is divisible by }11\text{ and }14 \\
14 & \text{if }n\text{ is divisible by }11\text{ and }13 \\
0 & \text{otherwise} \end{cases}
\]Calculate $ \sum_{n = 1}^{2001} a_n$.
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Given a field extension $K \subseteq F$, where $K(\alpha)/K$ is an extension of degree 4 and $\alpha^2$ is not a root of the minimal polynomial $m_{(\alpha,K)}(x)$, determine the degree $[K(\alpha^2):K]$. Use the tower law for field extensions:

$$[K(\alpha^2):K] = [K(\alpha^2):K(\alpha)][K(\alpha):K]$$

It is known that $[K(\alpha):K] = 4$. Find $[K(\alpha^2):K(\alpha)]$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the cardinality of the set of all functions from \( \mathbb{N}^k \) to \( \mathbb{R} \), where \( k \geq 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Inside triangle \(ABC\), there are three circles \(k_1, k_2, k_3\), each tangent to two sides of the triangle and to its incircle \(k\). The radii of \(k_1, k_2, k_3\) are 1, 4, and 9, respectively. Determine the radius of the incircle \(k\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The diagonal of a square is 4√2 cm. What will be the diameter of another square if their area is double that of the first square?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limiting behavior of the expression \( f(n) = n^2 \left(1 - \left(\frac{c \log n}{n}\right)^2\right)^{n^2} \) as \( n \) approaches infinity, where \( c > 0 \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let set $A=\{-1, 2, 3\}$, and set $B=\{a+2, a^2+2\}$. If $A \cap B = \{3\}$, then the real number $a=$ ___.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a homomorphism \( h : \{a, b\}^* \to \{0, 1\}^* \) defined by \( h(a) = 01 \) and \( h(b) = 10 \), and a regular language \( L = L((00 + 1)^*) \), determine the preimage \( h^{-1}(L) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=\cos (2x+ \frac {\pi}{4})$, if we shrink the x-coordinates of all points on the graph of $y=f(x)$ to half of their original values while keeping the y-coordinates unchanged; and then shift the resulting graph to the right by $|\varphi|$ units, and the resulting graph is symmetric about the origin, find the value of $\varphi$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a ticket counter, females and males arrive according to independent Poisson processes with rates of 30 and 20 customers per hour, respectively. Given that exactly two customers arrived in the first five minutes, calculate the probability that both customers arrived in the first three minutes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$. What is the ratio $\frac{c_{14}}{c_1}$ for the least possible value of $r$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the positive integers \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \). Any set of four numbers selected and summed results in the set \(\{44, 45, 46, 47\}\). Determine these five numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The rightmost nonzero digit in the decimal expansion of 101 ! is the same as the rightmost nonzero digit of $n$ !, where $n$ is an integer greater than 101. Find the smallest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a Markov chain denoted by $P$, where $p_{ii} = 1 - 3a$ and $p_{ij} = a$ for all $i \neq j$, with $1 \leq i, j \leq 4$. Determine the general formula for the $(1,1)$ entry of $P^n$, denoted as $P_{1,1}^n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Color the numbers $2, 3, \cdots, 9$ with one of three colors - red, green, or blue - such that each number is colored differently from its factors. Calculate the total number of different ways this coloring can be done. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The area of the orthogonal projection of a circle with a radius of 1 on the plane $\alpha$ is 1. Find the length of the orthogonal projection of this circle on a line perpendicular to the plane $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest \( k \) such that \( 1991^{k} \) divides \( A = 1990^{1991^{1992}} + 1991^{1990^{1992}} + 1992^{1991^{1990}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
正方体的八个顶点中任取 3 个构成三角形，求该三角形是等腰三角形的概率。答案的形式为\(\frac{k}{m}\)，请计算\(k + m\)的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S$ be the graph of $y=x^3$, and $T$ be the graph of $y=\sqrt[3]{y}$. Let $S^*$ be $S$ rotated around the origin $15$ degrees clockwise, and $T^*$ be T rotated around the origin 45 degrees counterclockwise. $S^*$ and $T^*$ will intersect at a point in the first quadrant a distance $M+\sqrt{N}$ from the origin where $M$ and $N$ are positive integers. Find $M+N$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
My apologies, I misunderstood your request. I provided the answer along with the harder problem. Here is the harder problem without the solution:

Jeff has 2032 ounces of peanut butter in 16, 28, 40, and 52 ounce jars. He has an equal number of each sized jar. How many jars of peanut butter does Jeff have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A number from 1 to 144 is chosen. You are allowed to select one subset from the set of numbers from 1 to 144 and ask whether the chosen number is in that subset. For a "yes" answer, you pay 2 rubles; for a "no" answer, you pay 1 ruble. What is the minimum amount of money necessary to guarantee guessing the number correctly?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a sphere with a radius of 2, three circles with a radius of 1 are positioned, each touching the other two. Find the radius of a smaller circle, also located on the sphere, that touches each of the given circles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Through points \(A(0, 14)\) and \(B(0, 4)\), two parallel lines are drawn. The first line, passing through point \(A\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(K\) and \(L\). The second line, passing through point \(B\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(M\) and \(N\).

What is the value of \(\frac{AL - AK}{BN - BM}\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In trapezoid $ABCD$, the angle bisectors of the exterior angles at vertices $A$ and $D$ intersect at point $M$, and the angle bisectors of the exterior angles at vertices $B$ and $C$ intersect at point $K$. Given that the distance $MK$ is 15 cm, determine the perimeter of trapezoid $ABCD$. Express your answer in centimeters.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the convex pentagon \(ABCDE\), it is given that \(\angle A = \angle B = \angle D = 90^\circ\). If a circle can be inscribed in this pentagon, determine the measure of \(\angle ADB\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $a,b$ be integers greater than $1$. Find the largest $n$ which cannot be written in the form $n = 7a + 5b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( f(x)=a \sin ((x+1) \pi)+b \sqrt[3]{x-1}+2 \), where \( a \) and \( b \) are real numbers and \( f(\lg 5) = 5 \), find \( f(\lg 20) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the real numbers \( a \), \( b \), and \( c \) such that the inequality \( f(x) = a \cos x + b \cos 2x + c \cos 3x \geq -1 \) holds for any real number \( x \), determine the maximum value of \( a + b + c \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an $n<100$ such that $n\cdot 2^n-1$ is prime. Score will be $n-5$ for correct $n$, $5-n$ for incorrect $n$ ($0$ points for answer $<5$)
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many positive integers less than $2023$ cannot be expressed in the form $\lfloor {x^2} \rfloor + \lfloor {x} \rfloor$, where $x$ is a positive real number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Erin the ant starts at a given corner of a cube and crawls along exactly $7$ edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the determinant of the following 5x5 matrix $A$:
\[
A = \begin{pmatrix}
d &0 &0 &0 &0 \\
d &d &0 &x &x \\
d &d &d &0 &0 \\
d &d &d &d &0 \\
d &d &d &d &d \\
\end{pmatrix}
\] where $d$ and $x$ are constants.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that (1-2x)^2017 = a0 + a1(x-1) + a2(x-1)^2 +... + a2017(x-1)^2017 (x∈ℝ), find the value of a1 - 2a2 + 3a3 - 4a4 +... - 2016a2016 + 2017a2017. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the best integer value of $k$ to estimate the function \( e^{\sin x} = 1+x+\frac{x^2}{2!}-\frac{x^4}{4!}+\frac{x^6}{6!}-\quad... \) for \(|x|\le\frac{\pi}{2}\) as \(x \to 0\), using o-notation and retaining three terms.
Please reason step by step, and put your final answer within \boxed{}.


Context:
求最小的正整数 a，使得存在整数 b,c，满足二次三项式 ax^2+bx+c 有两个不超过 \df{1}{1000} 的不同正根.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are \( 4n \) circles of the same radius on a plane, where \( n \) is a positive integer. No two circles are tangent to each other, and each circle intersects with at least three other circles. Let \( f(n) \) represent the number of intersection points of these circles. Determine the minimum value of \( f(n) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The national security agency's wiretap recorded a conversation between two spies and found that on a 30-minute tape, starting from the 30-second mark, there was a 10-second segment of conversation containing information about the spies' criminal activities. Later, it was discovered that part of this conversation was erased by a staff member. The staff member claimed that he accidentally pressed the wrong button, causing all content from that point onwards to be erased. What is the probability that the conversation containing criminal information was partially or completely erased due to pressing the wrong button?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设非负实数 a,b,c,x,y,z 满足 a+b+c=x+y+z=1. 求 (a-x^2)(b-y^2)(c-z^2) 的最大值.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
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the inequality $x\ln x - kx > 3$, which holds true for any $x > 1$, determine the maximum value of the integer $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A fly trapped inside a cubical box with side length $1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the indefinite integral:

\[
\int \frac{3 x^4 + 3 x^3 - 5 x^2 + 2}{x(x-1)(x+2)} \, dx
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
30. [15] The function $f: \mathbb{Z}^{2} \rightarrow \mathbb{Z}$ satisfies
- $f(x, 0)=f(0, y)=0$, and
- $f(x, y)=f(x-1, y)+f(x, y-1)+x+y$
for all nonnegative integers $x$ and $y$. Find $f(6,12)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the remainder when \( \prod_{i = 1}^{1903} (2^i + 5) \) is divided by \( 1000 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \lambda \) the positive root of the equation \( {t}^{2} - {1998t} - 1 = 0 \) . It is defined the sequence \( {x}_{0},{x}_{1},{x}_{2},\ldots ,{x}_{n},\ldots \) b)
 \( {x}_{0} = 1,{x}_{n + 1} = \left\lfloor  {\lambda {x}_{n}}\right\rfloor \) for \( n = 1,2\ldots \) Find the remainder of the division of \( {x}_{1998} \) by 1998.
 Note: \( \lfloor x\rfloor \) is the greatest integer less than or equal to \( x \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the graph of the function $f(x) = |\sin(\omega x - \frac{\pi}{8})| (\omega > 0)$ is shifted to the left by $\frac{\pi}{12}$ units, and the resulting graph corresponds to an even function, then the minimum value of $\omega$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the conditions under which $n^{\aleph_0}$ equals the cardinality of the continuum, where $n$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface integral of the vector field \( \mathbf{G} = \begin{pmatrix} y \\ z \\ x \end{pmatrix} \) over a sphere \( S \) using the divergence theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a knockout tournament with $2^n$ equally skilled players, $S_1, S_2, ..., S_{2^n}$, are participating. In each round, players are divided into pairs at random, and the winner from each pair moves on to the next round. If $S_2$ reaches the semi-final, then the probability that $S_1$ wins the tournament is $\frac{1}{20}$. Find the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A train 120 meters long running at the rate of 54 kmph takes 51.99584033277338 seconds to cross a bridge of a certain length. How long is the bridge in meters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a primitive element $k$ of the field $F=\mathbb{Q}(i,\sqrt{5}+i)$. Provide your answer as a single element that generates the field.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the monic polynomial of least degree in \( \mathbb{R}[x] \) that is a square root of \(-4\) modulo the ideal generated by \((x^2 + 1)^2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of real solutions  $a$  to the equation:
\[ \left[\,\frac{1}{2}\;a\,\right]+\left[\,\frac{1}{3}\;a\,\right]+\left[\,\frac{1}{5}\;a\,\right] = a. \]
Here, if  $x$  is a real number, then  $[\,x\,]$  denotes the greatest integer that is less than or equal to  $x$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate $\displaystyle\lim_{N \rightarrow \infty}\int_{|z|=2}\frac{1}{P^3(z) - 1}dz$, where $P_N(z) = \displaystyle\sum_{n = 0}^N\frac{z^n}{n!}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a circle, $AB$ is the diameter. Tangents $AD$ and $BC$ are drawn such that lines $AC$ and $BD$ intersect at a point $E$ on the circle. Given that $|AD|=a$ and $|BC|=b$ with $a \neq b$, determine the diameter of the circle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located and crawls along a side of the square to that vertex. Given that the probability that the bug moves to its starting vertex on its eighth move is \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers, find \( p + q \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
We consider an  $n \times n$  table, with  $n\ge1$ . Aya wishes to color  $k$  cells of this table so that that there is a unique way to place  $n$  tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of  $k$  for which Aya's wish is achievable?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many lines are tangent to the circle $x^2+(y-2)^2=1$ and have equal intercepts on both coordinate axes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a point $P$ on a triangular piece of paper $ABC,\,$ consider the creases that are formed in the paper when $A, B,\,$ and $C\,$ are folded onto $P.\,$ Let us call $P$ a fold point of $\triangle ABC\,$ if these creases, which number three unless $P$ is one of the vertices, do not intersect. Suppose that $AB=36, AC=72,\,$ and $\angle B=90^\circ.\,$ Then the area of the set of all fold points of $\triangle ABC\,$ can be written in the form $q\pi-r\sqrt{s},\,$ where $q, r,\,$ and $s\,$ are positive integers and $s\,$ is not divisible by the square of any prime. What is $q+r+s\,$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all values of $x$ for which the series \( \sum^\infty_{n=1}\Big(1+\frac{1}{2}+\ldots+\frac{1}{n}\big)\frac{\sin nx}{n} \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a unit square, and let $E$ be a point on segment $AC$ such that $AE = 1$. Let $DE$ meet $AB$ at $F$ and $BE$ meet $AD$ at $G$. Find the area of $CFG$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the polynomial \( P(x) = 99(x^{101} - 1) - 101(x^{99} - 1) \), find the value of \( Q(1) \), where \( Q(x) \) is the quotient when \( P(x) \) is divided by \((x-1)^3\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square $\sqrt{5}$ away as long as it is occupied.One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability $p$. Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors.The value of $p$ that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \(4a^{2}+kab+9b^{2}\) is a perfect square, find the value of the constant \(k\). Express your answer as a numerical value, including the ± symbol if applicable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a class, there are 15 boys and 15 girls. On Women's Day, some boys called some girls to congratulate them (no boy called the same girl more than once). It turned out that the children can be uniquely divided into 15 pairs, such that each pair consists of a boy and a girl whom he called. What is the maximum number of calls that could have been made?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface integral \( \iint_S E \cdot \vec{n} \, d\sigma \) where \( S \) is the parametric surface given by \( X(s,t) = [st, s^2, t^2]^T \) for \( 0 \leq s \leq t \leq 1 \), and \( E \) is the vector field \( E(x,y,z) = [3yz, zx, 2xy]^T \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
1. Alice and Bob are playing in an eight-player single-elimination rock-paper-scissors tournament. In the first round, all players are paired up randomly to play a match. Each round after that, the winners of the previous round are paired up randomly. After three rounds, the last remaining player is considered the champion. Ties are broken with a coin flip. Given that Alice always plays rock, Bob always plays paper, and everyone else always plays scissors, what is the probability that Alice is crowned champion? Note that rock beats scissors, scissors beats paper, and paper beats rock.
Proposed by: Reagan Choi
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the function $f(x) = (m-1)x^{\alpha}$ is a power function, then the graph of the function $g(x) = \log_{a}(x-m)$ (where $a > 0$, $a \neq 1$) passes through a fixed point A, whose coordinates are ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the implicit function defined by the equation \( y - \sin y = x \), find the value of the third derivative \( y'''(\pi) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( F(x) = \int_x^{\sin x} \left(\int_0^{\sin t} \sqrt{1+u^4} \,du\right)\,dt \), find \( F''(\pi) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and \[a_{n+2} = \frac{a_n + 2006}{a_{n+1} + 1}\] for each $n \in \mathbb{N}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral: \[ \int_{0}^{2} \sqrt{1+x^3} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A tangent is drawn through a point P on the curve \(y=e^{-x}\). The tangent, the positive y-axis, and a horizontal line through P form a triangle. Determine the largest possible area of this triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a normal distribution of cheese weights X with mean 10 kg and standard deviation 2 kg, determine the weight threshold h such that the probability of a cheese weighing more than h is at least 0.991. Calculate the value of h.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=x^{2}\ln x+ax$ has a decreasing interval, the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Genesis has a wall clock in his home that rings following a specific pattern: it rings every 2 hours for the first 12 hours of the day, then every 4 hours for the remaining 12 hours. If it will first ring at 1 A.M, how many times will it ring in a day?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 $\sin x+\sin y=\frac{1}{3}, \cos x-\cos y=\frac{1}{5}$, 求 $17\sin (x-y)$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $m$ be the largest real solution to the equation $\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}=x^2-11x-4$ There are positive integers $a$, $b$, and $c$ such that $m=a+\sqrt{b+\sqrt{c}}$. Find $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a set of data: $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$ forms an arithmetic sequence with a common difference of $d$, and the variance of this set of data equals $1$, then the common difference $d$ equals ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Trent lives in a city where each block is 50 meters long. He walked 4 blocks from his house to the bus stop at a speed of 4 km/h. He then rode the bus 7 blocks to the library at a speed of 15 km/h. After spending time in the library, he picked up a bicycle from a sharing station and rode 5 blocks to the park at an average speed of 10 km/h. Later, he came home the same way, but this time he jogged 5 blocks back to the park's bike sharing station at a speed of 6 km/h, took the bus for 7 blocks and then walked the final 4 blocks to his house. In total, it took him 2 hours to complete the entire journey. What was his average speed during the entire trip?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of 5-digit numbers where the ten-thousands place is not 5, the units place is not 2, and all digits are distinct.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A vessel with a capacity of 100 liters is filled with a brine solution containing 10 kg of dissolved salt. Every minute, 3 liters of water flows into it, and the same amount of the resulting mixture is pumped into another vessel of the same capacity, initially filled with water, from which the excess liquid overflows. Determine the time at which the amount of salt in both vessels is equal.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The solution to the inequality
\[\frac{x + c}{x^2 + ax + b} \le 0\]is $x \in (-\infty,-1) \cup [1,2).$  Find $a + b + c.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tony made a sandwich with two slices of bread for lunch every day this week. On Saturday, he was extra hungry from doing yard work and made two sandwiches. Tony started with a loaf of bread with a certain number of slices and now has 6 slices left. How many slices were in the loaf Tony started with?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(x)$ be a function defined as:

$$f(x)=\int^{x}_{\gamma}\frac{\delta}{t(t^{\delta}+2+t^{-\delta})}$$

where $\gamma, \delta \in \mathbb{R^+}$. Calculate the limit of $f(x)$ as $x$ approaches infinity, expressed in terms of $\gamma$ and $\delta$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( a \) such that the equation \( a^x = |x - 4| - |2x - 8| \) has exactly one solution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
1. In a tennis tournament, 1152 schoolchildren are participating. 1 point is awarded for a win, and 0 points for a loss. Before each round, pairs are formed by drawing lots among participants with the same number of points (those who do not find a pair are awarded a point without playing). A player is eliminated after the second loss. The tournament continues as long as at least one pair of opponents can be formed. How many rounds will have to be played?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=x^3-3x-a$, find the value of $(M-N)$, where $M$ and $-N$ are the maximum and minimum values of $f(x)$ on the interval $[0, 3]$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If \( P \) is the circumcenter of \( \triangle ABC \), and
\[
\overrightarrow{P A}+\overrightarrow{P B}+\lambda \overrightarrow{P C}=\mathbf{0}, \quad \text{with} \quad \angle C=120^{\circ},
\]
determine the value of the real number \( \lambda \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$. She noticed that the pages formed only by even digits have a blue mark, and that every three pages starting from page two have a red mark. How many pages of the encyclopedia have both colors?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The main line train starts at 5:00 am and the harbor line train starts at 5:02 am. Each train has a frequency of 10 minutes. If a guy goes in the morning at a random time, what is the probability of him getting the main line train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tom and Jerry are running on a figure-eight track in the same direction at constant speeds. Initially, Jerry is directly above Tom. After 20 minutes, Tom is directly above Jerry, and neither has completed a full lap. Another 15 minutes later, Tom returns to his starting point. Determine the time from the start when Tom will catch up with Jerry.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the radius of the incircle of an equilateral triangle \(ABC\) is 2 and the center of this circle is \(I\). If point \(P\) satisfies \(P I = 1\), what is the maximum value of the ratio of the areas of \(\triangle APB\) to \(\triangle APC\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What range of values must the expression \(\lambda = \frac{\sum_{i=1}^{4} S_{i}}{S}\) satisfy, where \(S_{1}, S_{2}, S_{3}, S_{4}\) are the face areas of a tetrahedron, with the largest of these areas being \(S\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the Gromov-Witten invariant \(\langle p, p, \ell\rangle_{0, 1}\) that counts degree 1, genus 0 curves in \(\mathbb{CP}^2\), where \(p\) and \(\ell\) are the cohomology classes Poincaré dual to a point and a line, respectively?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive? Express your answer as a positive integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit:
\[ \lim_{n \to \infty} \left( \frac{(2n)!}{2^n \cdot n!} \right)^{\frac{1}{n}} \left( \tan{\left( \frac{\pi\sqrt[n+1]{(n+1)!}}{4\sqrt[n]{n!}} \right)} - 1 \right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the codimension of the commutator ideal of \( \mathfrak{gl}(V) \) in \( \mathfrak{gl}(V) \) when \( V \) is an infinite-dimensional vector space.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $M_1, M_2, . . ., M_{11}$ be $5-$element sets such that $M_i \cap M_j \neq {\O}$ for all $i, j \in \{1, . . ., 11\}$. Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( N \) be the set of natural numbers, \( R \) be the set of real numbers, and \( S \) be the set of functions \( f: N \rightarrow R \) that satisfy the following two conditions:
1. \( f(1) = 2 \);
2. \( f(n+1) \geq f(n) \geq \frac{n}{n+1} f(2n) \) for \( n = 1, 2, \ldots \)

Find the smallest natural number \( M \) such that for any \( f \in S \) and any \( n \in N \), \( f(n) < M \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the vertices of a single rectangular solid (cuboid), how many different pyramids can be formed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three numbers 1, $a$, 2 form a geometric sequence, then the real number $a$ = ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A chess tournament was held among the actors of the Karabas Barabas theater. Each participant played with each of the others exactly once. For a win, one received one soldo, for a draw - half a soldo, and for a loss - nothing. It turned out that among any three participants, there is a chess player who earned exactly 1.5 soldos in games with the other two. What is the maximum number of actors that could have participated in this tournament?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty}\frac{n}{n+1}\left(\frac{2x+1}{x}\right)^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
若复数 $z_{1}, z_{2}$ 分别满足 $\left|z_{1}-3 \mathrm{i}\right|=2,\left|z_{2}-8\right|=1$ ，则在复平面上所有由复数 $z_{1}-z_{2}$ 对应的点围成的面积是 $x \pi$。求 $x$ 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
How can I evaluate the integral $$\int_0^\frac{\pi}{2} x\ln(\sin x) \, dx$$ in terms of the Riemann zeta function $\zeta(3)$ and the natural logarithm of 2?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are \( n \) pieces of paper, each containing 3 different positive integers no greater than \( n \). Any two pieces of paper share exactly one common number. Find the sum of all the numbers written on these pieces of paper.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points $K$, $L$, $M$, and $N$ lie in the plane of the square $ABCD$ such that $AKB$, $BLC$, $CMD$, and $DNA$ are isosceles right triangles. If the area of square $ABCD$ is 25, find the area of $KLMN$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
As a seasoned executive at a mainstream headphone company, Alex is skeptical about the potential sales of a new line of niche gaming headphones. The company currently sells 100,000 units of their mainstream headphones every month. They are considering launching the niche gaming headphones and project that they could sell 5% of the mainstream units in the first month.

If the company decides to launch the gaming headphones, they plan to increase production by 2% each month, based on the initial projected sales of the gaming headphones. Calculate how many units of the gaming headphones the company expects to produce in the third month after the launch.
Please reason step by step, and put your final answer within \boxed{}.


Context:
若正整数 $n$ 使得方程 $x^{3}+y^{3}=z^{n}$ 有正整数解 $(x, y, z)$ ，称 $n$ 为 "好数"。则不超过 2019 的 "好数"的个数是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A fly and $\mathrm{k}$ spiders are moving on a $2019 \times 2019$ grid. On its turn, the fly can move by 1 square and the $k$ spiders can each move by 1 square. What is the minimal $k$ for which the spiders are guaranteed to catch the fly?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value for \(a, b \geq 1\) of the expression

$$
\frac{|7a + 8b - ab| + |2a + 8b - 6ab|}{a \sqrt{1 + b^2}}
$$
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Zang is at the point $(3,3)$ in the coordinate plane. Every second, he can move one unit up or one unit right, but he may never visit points where the $x$ and $y$ coordinates are both composite. In how many ways can he reach the point $(20, 13)$?\Based on a proposal by Ahaan Rungta
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 34 kindergartners, 48 first graders, and some second graders in Ms. Watson's class. Additionally, there are 5 third graders and 6 absent students who are a mix of first and second graders. Overall, Ms. Watson has 120 students in her class. Four new students of different grade levels are also joining the class this week. How many second graders are currently in Ms. Watson's class?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On each non-boundary unit segment of an  $8\times 8$  chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube?
Please reason step by step, and put your final answer within \boxed{}.


Context:
$\underline{\text { Senderov V.A. }}$

A thousand non-overlapping arcs are placed on a circle, and two natural numbers are written on each of them. The sum of the numbers on each arc is divisible by the product of the numbers on the next arc in the clockwise direction. What is the greatest possible value of the largest of the written numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Pablo will decorate each of $6$ identical white balls with either a striped or a dotted pattern, using either red or blue paint. He will decide on the color and pattern for each ball by flipping a fair coin for each of the $12$ decisions he must make. After the paint dries, he will place the $6$ balls in an urn. Frida will randomly select one ball from the urn and note its color and pattern. The events "the ball Frida selects is red" and "the ball Frida selects is striped" may or may not be independent, depending on the outcome of Pablo's coin flips. The probability that these two events are independent can be written as $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. What is $m?$ (Recall that two events $A$ and $B$ are independent if $P(A \text{ and }B) = P(A) \cdot P(B).$)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the sequence $\left\{\frac{1}{b_{n}}\right\}$ is a "dream sequence" defined by $\frac{1}{a_{n+1}}- \frac{2}{a_{n}}=0$, and that $b_1+b_2+b_3=2$, find the value of $b_6+b_7+b_8$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $V=[(x,y,z)|0\le x,y,z\le 2008]$  be a set of points in a 3-D space.
If the distance between two points is either  $1, \sqrt{2}, 2$ , we color the two points differently.
How many colors are needed to color all points in  $V$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of unique closed paths on the edges of a given Platonic solid, where each edge can be traversed only once, and paths that are related by a rotation of the solid are considered identical.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Seventy-five percent of claims follow a normal distribution with a mean of 3,000 and a variance of 1,000,000. The remaining 25% of claims follow a normal distribution with a mean of 4,000 and a variance of 1,000,000. Calculate the probability that a randomly selected claim exceeds 5,000.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a \neq b$ be positive real numbers and $m, n$ be positive integers. An $m+n$-gon $P$ has the property that $m$ sides have length $a$ and $n$ sides have length $b$. Further suppose that $P$ can be inscribed in a circle of radius $a+b$. Compute the number of ordered pairs $(m, n)$, with $m, n \leq 100$, for which such a polygon $P$ exists for some distinct values of $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
At the base of a quadrilateral pyramid lies a rhombus $A B C D$, in which $\angle B A D=60^{\circ}$. It is known that $S A=S C$, $S D=S B=A B$. A point $E$ is taken on the edge $D C$ such that the area of triangle $B S E$ is the smallest among the areas of all sections of the pyramid containing the segment $B S$ and intersecting the segment $D C$. Find the ratio $D E: E C$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_C (xy) \, dx + \exp(y^2) \, dy \) using Green's Theorem, where \( C \) is the triangle with vertices \((0,0)\), \((2,0)\), and \((2,1)\). Ensure your answer is positive.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Seven dwarfs stood at the corners of their garden, each at one corner, and stretched a rope around the entire garden. Snow White started from Doc and walked along the rope. First, she walked four meters to the east where she met Prof. From there, she continued two meters north before reaching Grumpy. From Grumpy, she walked west and after two meters met Bashful. Continuing three meters north, she reached Happy. She then walked west and after four meters met Sneezy, from where she had three meters south to Sleepy. Finally, she followed the rope by the shortest path back to Doc, thus walking around the entire garden.

How many square meters is the entire garden?

Hint: Draw the shape of the garden, preferably on graph paper.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the trapezoid \(ABCE\), the base \(AE\) is 16, \(CE = 8\sqrt{3}\). A circle passing through points \(A\), \(B\), and \(C\) intersects the line \(AE\) again at point \(H\), where \(\angle AHB = 60^{\circ}\). Find \(AC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( f(x) = \frac{4 e^{x-1}}{x+1} + x^2 - 3ax + a^2 - 1 \), determine the minimum possible value of the real number \( a \) such that there exists \( x_0 > 0 \) with \( f(x_0) \leq 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given vectors $\overrightarrow {a}$ = (4, -7) and $\overrightarrow {b}$ = (3, -4), find the projection of $\overrightarrow {a}$ - $2\overrightarrow {b}$ in the direction of $\overrightarrow {b}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of nontrivial cyclic subgroups in the group \( \Bbb Z_8 \oplus \Bbb Z_4 \oplus \Bbb Z_2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the rational map \( \hat{\phi} \) at the point \((1:1:0)\), where \( \phi:(t_0:t_1:t_2) \mapsto (\frac{1}{t_0}:\frac{1}{t_1}:\frac{1}{t_2}) \) and \( \hat{\phi} \) is the extension of \( \phi \) with domain \( \mathbb{P}^2-\{(1:0:0),(0:1:0),(0:0:1)\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $(3x-1)^{10} = a + a_1x + a_2x^2 + \ldots + a_{10}x^{10}$, find the value of $a_1 + a_2 + \ldots + a_{10}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the optimal fraction \( q \) of your money to bet each time in order to grow your money by \( m \)-fold in the least amount of bets, given a biased coin with probability \( p \) of landing heads. Assume \( m \) is sufficiently large to ignore finite size effects. Is there an explicit formula for the optimal \( q \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
João managed to paint the squares of an \( n \times n \) board in black and white so that the intersections of any two rows and any two columns did not consist of squares with the same color. What is the maximum value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Point \( O \) is the center of the circle circumscribed around triangle \( ABC \) with sides \( AB = 8 \) and \( AC = 5 \). Find the length of side \( BC \) if the length of the vector \(\overrightarrow{OA} + 3 \overrightarrow{OB} - 4 \overrightarrow{OC}\) is 10.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n \geq 2$ be an integer and consider the vector space $V=F^{2n}$ with the standard basis $e_1,\ldots,e_{2n}$. The second exterior power $\Lambda^2(V)$ contains the element $\omega=e_1 \wedge e_2 + e_3 \wedge e_4 + \cdots + e_{2n-1} \wedge e_{2n}$. Since $\Lambda^{2n}(V)$ is spanned by $x := e_1 \wedge \cdots \wedge e_{2n}$, the $n$-fold wedge product $\omega \wedge \cdots \wedge \omega$ is a scalar multiple of $x$. Compute the scalar $c$ such that $c x = \omega \wedge \cdots \wedge \omega$. What is the value of $c$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
B4 In a cube with edge 6, there is a sphere with diameter 6. The sphere thus touches the six faces of the cube internally. We imagine the cube divided into 216 unit cubes of $1 \times 1 \times 1$. How many of these unit cubes lie entirely within the sphere? A unit cube with a vertex on the sphere's surface and otherwise inside the sphere is counted as lying entirely within the sphere.

The work of the Stichting Nederlandse Wiskunde Olympiade is made possible by financial contributions and support from:

The Ministry of Education, Culture and Science

The Dutch Association of Mathematics Teachers

The Mathematical Society

Eindhoven University of Technology

Citogroep

Utrecht University of Applied Sciences

Educatieve Partners Nederland

Woltersgroep Groningen

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


Context:
What is the largest possible value of $8x^2 + 9xy + 18y^2 + 2x + 3y$ such that $4x^2 + 9y^2 = 8$, where $x, y$ are real numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
双曲线 $C$ 的方程为 $x^{2}-\frac{y^{2}}{3}=1$, 左、右焦点分别为 $F_{1}, F_{2}$, 过点 $F_{2}$ 作一直线与双曲线 $C$ 的右半支交于点 $P, Q$, 使得 $\angle F_{1} P Q=90^{\circ}$, 则 $\triangle F_{1} P Q$ 的内切圆半径的原始答案为$\sqrt{m}+n$的形式，请给出$m+n$的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of drawings required, with replacement, from an urn containing 18 differently-coloured balls, to achieve a 90% probability of drawing at least one ball of each colour.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the first cohomology group $H^1(\mathbb{Z}, \mathbb{Z}_2)$, where $\mathbb{Z}$ is the group of integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit as $x$ approaches 0 from the positive side: $$\lim_{x \to 0+}\frac{x^{(\sin x)^x}-(\sin x)^{x^{\sin x}}}{x^3}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the set \( A = \{1, 2, 3\} \) and a function \( f: A \rightarrow A \) which satisfies \( f(f(x)) \geq x \) for any \( x \in A \), how many such functions \( f \) exist?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of zeros, including multiplicities, of the polynomial \( f(z) = z^7 + 9z^4 - 7z + 3 \) that lie outside the disk \( \{ z : |z| \leq 2 \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximal number of crosses, each consisting of 5 squares, that can be placed on an 8x8 grid without overlapping.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If six points are chosen sequentially at random on the circumference of a circle, what is the probability that the triangle formed by the first three is disjoint from that formed by the second three?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the circles \( C_{1}: x^{2}+y^{2}-2 x+4 y-b^{2}+5=0 \) and \( C_{2}: x^{2}+y^{2}-2(a-6) x-2 a y +2 a^{2}-12 a+27=0 \), which intersect at two distinct points \( A(x_{1}, y_{1}) \) and \( B(x_{2}, y_{2}) \), find the value of \( a \) such that \( \frac{y_{1}+y_{2}}{x_{1}+x_{2}}+\frac{x_{1}-x_{2}}{y_{1}-y_{2}} = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a round rotating table with 8 white and 7 black cups seated are 15 gnomes. The gnomes are wearing 8 white and 7 black hats. Each gnome takes a cup that matches the color of their hat and places it in front of them. After this, the table rotates randomly. What is the maximum number of color matches between the cups and hats that can be guaranteed after the table rotates (the gnomes can choose how to sit, but do not know how the table will rotate)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Anthony writes the $(n+1)^2$ distinct positive integer divisors of $10^n$, each once, on a whiteboard. On a move, he may choose any two distinct numbers $a$ and $b$ on the board, erase them both, and write $\gcd(a, b)$ twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible number of moves Anthony could have made.

[i]Proposed by Andrew Wen[/i]
Please reason step by step, and put your final answer within \boxed{}.


Context:
10 camels cost as much as 24 horses, 26 horses cost as much as 4 oxen, and 6 oxen cost as much as 4 elephants. The cost of 10 elephants is Rs. 170,000. What is the cost of a camel?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the least number of colors that can be used to color all the hexagons on the given tiling such that no two adjacent hexagons share the same color. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. In a $4 \times 4$ grid, fill each cell with 0 or 1, such that the sum of the four numbers in each $2 \times 2$ subgrid is odd. There are $\qquad$ different ways to do this.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Fred had 212 sheets of paper. He received another batch of papers from Jane. Jane initially planned to give him 307 reams of paper, with 500 sheets per ream. But she ended up giving him 50% more than she originally planned. After receiving these additional papers from Jane, Fred decided to give Charles 25% of the total paper sheets he now has. How many sheets of paper does Fred have left?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the tetrahedron P-ABC, if PA=PB=BC=AC=5, PC=AB=4$\sqrt{2}$, then the surface area of its circumscribed sphere is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
\section*{Problem 6B - 131236B}

Let \(M\) be the set of all points \(P(x, y)\) in a plane rectangular Cartesian coordinate system, where \(x, y\) are rational integers, and \(0 \leq x \leq 4\) and \(0 \leq y \leq 4\).

Determine the probability that the distance between two different points chosen arbitrarily from \(M\) is a rational integer (the unit of measurement being the unit of the coordinate system).

Hint: If \(n\) is the number of different ways to choose two points and \(m\) is the number of ways in which the distance is a rational integer, then the quotient \(\frac{m}{n}\) is the probability to be determined. Two selection possibilities are considered different if and only if the sets of points (each consisting of two points) selected in them are different.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the quotient: 0.284973 / 29
Please reason step by step, and put your final answer within \boxed{}.


Context:
You have 50 dimes and 20 quarters. What percent of the value of your money is in quarters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an isosceles triangle \( ABC \) (\( AB = BC \)), a circle is inscribed. A line parallel to side \( BC \) and tangent to the circle intersects side \( AB \) at a point \( N \) such that \( AN = \frac{3}{8} AB \). Find the radius of the circle if the area of triangle \( ABC \) is 12.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of the expression \( \sqrt{\frac{16^{12} + 8^{15}}{16^5 + 8^{16}}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
An unpainted cone has radius 3 cm and slant height 5 cm. The cone is placed in a container of paint. With the cone's circular base resting flat on the bottom of the container, the depth of the paint in the container is 2 cm. When the cone is removed, its circular base and the lower portion of its lateral surface are covered in paint. The fraction of the total surface area of the cone that is covered in paint can be written as p/q where p and q are positive integers with no common divisor larger than 1. What is the value of p+q?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given in $\triangle ABC$, $\angle A \leq \angle B \leq \angle C$, and $2\angle B = 5\angle A$, the range of values for $\angle B$ is.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{0^\circ < \angle B \leq 75^\circ}


Context:
Seven people of seven diﬀerent ages are attending a meeting. The seven people leave the meeting one at a
time in random order. Given that the youngest person leaves the meeting sometime before the oldest
person leaves the meeting, the probability that the third, fourth, and ﬁfth people to leave the meeting do so in order of their ages (youngest to oldest) is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$.

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


Context:
Seven people of seven diﬀerent ages are attending a meeting. The seven people leave the meeting one at a
time in random order. Given that the youngest person leaves the meeting sometime before the oldest
person leaves the meeting, the probability that the third, fourth, and ﬁfth people to leave the meeting do so in order of their ages (youngest to oldest) is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x) = \ln(e^x + e^{-x}) + x^2$, find the range of $x$ for which $f(x) > f(2x-1)$ holds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ring homomorphisms from $\mathbb{Z}_{11}$ to $\mathbb{Z}_{13}$ that map $1$ to a non-zero element.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are rods of various lengths and quantities. The lengths are 20 cm, 21 cm, 22 cm, 23 cm, 24 cm, and 25 cm. The quantities are 23, 64, and 32 respectively. What is the average length of the rods?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Around a circular table, there are 18 girls seated, 11 dressed in blue and 7 dressed in red. Each girl is asked if the girl to her right is dressed in blue, and each one responds with either yes or no. It is known that a girl tells the truth only when both of her neighbors, the one to her right and the one to her left, are dressed in the same color. How many girls will respond yes? If there is more than one possibility, state all.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $D$, $E$, and $F$ be the feet of the altitudes from vertices $A$, $B$, and $C$ of an acute triangle $\triangle ABC$, respectively. Given that $AF = 28$, $FB = 35$, and $BD = 45$. Let $P$ be a point on segment $BE$ such that $AP = 42$. Determine the length of $CP$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the prism \(A B C A_1 B_1 C_1\), the medians of the bases \(A B C\) and \(A_1 B_1 C_1\) intersect at points \(O\) and \(O_1\) respectively. A line is drawn through the midpoint of the segment \(O O_1\) parallel to the line \(C A_1\). Find the length of the segment of this line that lies inside the prism, given that \(C A_1 = a\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Count the number of three-digit whole numbers for which the sum of the digits equals 25. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $S$  be the set of integers which are both a multiple of  $70$  and a factor of  $630{,}000$ .  A random element  $c$  of  $S$  is selected. If the probability that there exists an integer  $d$  with  $\gcd (c,d) = 70$  and  $\operatorname{lcm} (c,d) = 630{,}000$  is  $\frac mn$  for some relatively prime integers  $m$  and  $n$ , compute  $100m+n$ .

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


Context:
## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{\arccos \sqrt{2 / 3}} \frac{\tan x + 2}{\sin^2 x + 2 \cos^2 x - 3} \, dx
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S$ be the set of all rational numbers $r$ , $0<r<1$ , that have a repeating decimal expansion in the form $0.abcabcabc\ldots=0.\overline{abc}$ , where the digits $a$ , $b$ , and $c$ are not necessarily distinct. To write the elements of $S$ as fractions in lowest terms, how many different numerators are required?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Example 7 (China Mathematical Olympiad) Let $X$ be a 56-element set. Find the smallest positive integer $n$, such that for any 15 subsets of $X$, if the union of any 7 of them has at least $n$ elements, then there must exist 3 of these 15 subsets whose intersection is non-empty.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Eight teams participated in a football tournament, and each team played exactly once against each other team. If a match was drawn then both teams received 1 point; if not then the winner of the match was awarded 3 points and the loser received no points. At the end of the tournament the total number of points gained by all the teams was 61. What is the maximum number of points that the tournament's winning team could have obtained?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cube with edge length \( n \) is composed of white and black unit cubes such that each white cube shares a face with exactly three black cubes, and each black cube shares a face with exactly three white cubes. For which values of \( n \) is this possible?
Please reason step by step, and put your final answer within \boxed{}.


Context:
从一个正方体的八个顶点中取出三个.则以这三个点为顶点构成直角三角形的概率是$\underline{\quad}$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
Please reason step by step, and put your final answer within \boxed{}.


Context:
11. Let $a, b, c, d$ be four distinct real numbers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4$, and $a c=$ $b d$. Find the maximum value of $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many homomorphisms are there from the dihedral group $D_4$ to $\mathbb{Z}_4$, given that $D_4 = \{I, R, R^2, R^3, F, RF, R^2F, R^3F\}$ and the hint suggests there are between one and five homomorphisms?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the largest possible area of a rectangle whose vertices' coordinates satisfy the equation

\[ |y+1|\left(y^{2}+2y+28\right) + |x-2| = 9\left(y^{2}+2y+4\right) \]

with sides parallel to the coordinate axes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jane can arrange 25 vases of flowers in a day. If there are 378 vases of flowers to be arranged and she has 17 days to complete the task, how many vases of flowers will she arrange on the last day?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many values of $x$, where $-19 < x < 98$, satisfy the equation $\cos^2 x + 2\sin^2 x = 1$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How do I find the value of the following integral: $$\int_{0}^{\infty} \frac{\ln(x)}{x^2+x+1} dx$$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be the set $\{k^{19}-k: 1<k<20, k\in N\}$. Let $G$ be the GCD of all elements of $A$.
Then the value of $G$ is?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The trunk of the sky-high tree branches into two at a height of one foot. In the following, we consider the part between two branches as a branch, on which there is no further branching. Every branch of the sky-high tree is straight and ends one foot higher than its end closer to the ground. We consider the children of a branch to be the branches starting from the higher end of the branch, which are also considered siblings to each other. Every branch of the sky-high tree has at least two children, and if it does not have exactly two children, then it has a sibling who has exactly two children. Siblings always have a different number of children. If a branch has more than two children, then it has a sibling who has exactly one child less than it. How many branches start from the branches at a height of $n$ feet?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The base of a hill located beside a river is 300m above the seabed. This depth is a certain fraction of the vertical distance between the riverbed and the peak of the hill. The hill is 900m high. What is the ratio of the depth from the base of the hill to the seabed to the vertical distance between the riverbed and the peak of the hill?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 30 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. Calculate the area of trapezoid $EFCD$ if the height from $E$ to line $CD$ is 10 units.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,\!000$th number in the sequence?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sequence ${a_n}=C_{200}^n⋅{({\sqrt[3]{6}})^{200-n}}⋅{({\frac{1}{{\sqrt{2}}}})^n}$ for $n=1,2,\ldots,95$, calculate the number of integer terms in the sequence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A tempo is insured with fixed coverage percentages and premium rates for all three years. In the first year, it is insured to an extent of 5/7 of its original value at a premium rate of 3%. In the second year, it is insured to the same extent of its original value at a premium rate of 4%. In the third year, it is insured to an extent of 5/7 of its original value at a premium rate of 5%. If the total premium paid for all three years amounts to $950, calculate the original value of the tempo.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) If $d$ to the nearest tenth is $x$, what is the value of $10x$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sector representing a quarter of a circle has been cut out. From this sector and the remaining part of the circle, the lateral surfaces of two cones have been made. Find the ratio of the heights of these cones.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he wrote the sum of the numbers written next to its row and its column ("addition table"). What is the maximum number of sums in this table that could end up being rational numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Steve's new book sells 1,000,000 copies.  He got an advance to pay for 100,000 copies. He gets $2 for each copy of the book sold.  His agent takes 10% of that.  How much money did he keep not counting the money from the advance?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \(2n\) positive real numbers \(a_{1} \geq a_{2} \geq \cdots \geq a_{n}\), \(b_{1} \geq b_{2} \geq \cdots \geq b_{n}\) and \(0 \leq \lambda \leq 2\), define
$$
M = \sum_{i=1}^{n} \sqrt{a_{i}^2 + b_{i}^2 - \lambda a_{i} b_{i}}, \quad N = \sum_{i=1}^{n} \sqrt{a_{i}^2 + b_{i}^2 - \lambda a_{i+1} b_{i}},
$$
where \(a_{n+1} = a_{1}\). Determine the relationship between \(M\) and \(N\), specifying whether \(M \leq N\), \(M \geq N\), or if the relationship is indeterminate.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the expression \(\cos ^{4} \frac{5 \pi}{24}+\cos ^{4} \frac{11 \pi}{24}+\sin ^{4} \frac{19 \pi}{24}+\sin ^{4} \frac{13 \pi}{24}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Andrey, Maxim, Igor, and Kolya competed in a bicycle race. To the question of who took which place, they answered:

Andrey: - I was neither first nor last.

Maxim: - I was not last.

Igor: - I was first.

Kolya: - I was last.

It is known that three boys answered honestly and only one lied. Who among the boys lied?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the distance between the foci of the ellipse
\[\frac{x^2}{20} + \frac{y^2}{4} = 7.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A triangle has sides of length  $48$ ,  $55$ , and  $73$ .  A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle.  If  $c$  and  $d$  are relatively prime positive integers such that  $c/d$  is the length of a side of the square, find the value of  $c+d$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
The first digit of a string of 2002 digits is a 1. Any two-digit number formed by consecutive digits within this string is divisible by 19 or 31. What is the largest possible last digit in this string?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral $$\int_{0}^{\infty}\frac{\mathrm{d}x}{\left(x + 2\right)\left(x^{2} + 1\right)}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Select numbers $a$ and $b$ between $0$ and $1$ independently and at random, and let $c$ be their sum. Let $A, B$ and $C$ be the results when $a, b$ and $c$, respectively, are rounded to the nearest integer. Find the probability that $A + B = C$. The original answer is in \frac{k}{m} format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose a group $G$ has irreducible complex representations of dimensions $1,1,2,3$, and $d$. Find $d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $S=\{1,2, \cdots, 98\}$, 求最小正整数 $n$, 使得 $S$ 的任意一个 $n$ 元子集中都可以选出 10 个数, 无论怎样将 10 个数均分成两组, 总有一组中存在一个数与另外 4 个数互质, 而另一组中总有一个数与另外 4 个数不互质。
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bug starts at a vertex of a grid made of equilateral triangles of side length \(1\). At each step, the bug moves in one of the \(6\) possible directions along the grid lines randomly and independently with equal probability. What is the probability that after \(5\) moves the bug never will have been more than \(1\) unit away from the starting position? Provide your answer as a fraction in simplest form \(\frac{k}{m}\), and calculate the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the last three digits of the sum of all the real values of $m$ such that the ellipse $x^2 + xy + y^2 = m$ intersects the hyperbola $xy = n$ only at its two vertices, as $n$ ranges over all non-zero integers $-81 \le n \le 81$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a triangle with side lengths a, b, and c, an area equal to 1/4, and the radius of its circumcircle is 1, let s = √a + √b + √c and t = 1/a + 1/b + 1/c. Determine the relationship between s and t. Express your answer as an inequality (e.g., s < t, s > t, s = t).
Please reason step by step, and put your final answer within \boxed{}.


Context:
We call a number $10^3 < n < 10^6$ a balanced number if the sum of its last three digits is equal to the sum of its other digits. Find the sum of all balanced numbers in $\bmod{13}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The distinct positive integers $a$ and $b$ have the property that $\frac{a+b}{2}$, $\sqrt{ab}$, and $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ are all positive integers. Find the smallest possible value of $|a-b|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a certain wire service, there are reporters covering local politics in three different countries A, B, and C. The percentage of reporters covering local politics in each country are as follows: 20% cover local politics in country A, 25% cover local politics in country B, and 15% cover local politics in country C. Additionally, 20% of the reporters who cover politics for the wire service do not cover local politics in any of these three countries. What percent of the reporters for the wire service do not cover politics?
Please reason step by step, and put your final answer within \boxed{}.


Context:
All edges of a regular quadrilateral pyramid \(S A B C D\) with base \(A B C D\) are of equal length. Plane \(\alpha\) is perpendicular to line \(S A\), and plane \(\beta\) is parallel to line \(C D\). Determine the smallest possible angle between planes \(\alpha\) and \(\beta\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the vertices of a cube, the numbers \(\pm 1\) are assigned, and on its faces, the numbers are equal to the product of the numbers at the vertices of that face. Find all possible values that the sum of these 14 numbers can take. Provide their product in the answer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of homomorphisms from $S_{3} \rightarrow \mathbb{Z}_{2} \times \mathbb{Z}_{4}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
John wants to buy a $100 trouser at the store, but he thinks it's too expensive. Finally, it goes on sale for $10. What is the percent decrease in the price of the trouser?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the game of Dubblefud, yellow chips are worth a certain number of points, blue chips are worth 4 points, and green chips are worth 5 points. In a certain selection of chips, the product of the point values of the chips is 16000. The number of blue chips in this selection equals the number of green chips, and there are 4 yellow chips. How many points is a yellow chip worth?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola $\frac{x^{2}}{m} + \frac{y^{2}}{n} = 1 (m < 0 < n)$ with asymptote equations $y = \pm \sqrt{2}x$, calculate the hyperbola's eccentricity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S = \left\{A = \left(a_{1}, a_{2}, \cdots, a_{8}\right) \mid a_{i} = 0 \text{ or } 1, i = 1, 2, \cdots, 8\right\}$. For two elements $A = \left(a_{1}, a_{2}, \cdots, a_{8}\right)$ and $B = \left(b_{1}, b_{2}, \cdots, b_{8}\right)$ in $S$, denote $d(A, B) = \sum_{i=1}^{8}\left|a_{i}-b_{i}\right|$, and refer to it as the distance between $A$ and $B$. How many elements can be selected from $S$ such that the distance between any two of them is at least 5?
Please reason step by step, and put your final answer within \boxed{}.


Context:
8.7. Let's consider the equilateral triangle $A B C$. Points $D, E$, and $F$ are located on the sides $(B C),(C A)$, and, respectively, $(A B)$ such that the lines $D F$ and $A B$ are perpendicular, and the line $D E$ is the perpendicular bisector of the segment $C F$. Find the measure in degrees of the angle $D E F$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate $$\iint_{S} (x^3+y^3)\,dy\,dz + (z^3+y^3)\,dz\,dx + (x^3+z^3)\,dx\,dy$$ where $S$ is the surface of the sphere $x^2+y^2+z^2= a^2$. Spherical coordinates help me in this question, $r^2=a^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $BCB'C'$ be a rectangle, let $M$ be the midpoint of $B'C'$, and let $A$ be a point on the circumcircle of the rectangle. Let triangle $ABC$ have orthocenter $H$, and let $T$ be the foot of the perpendicular from $H$ to line $AM$. Suppose that $AM=2$, $[ABC]=2020$, and $BC=10$. Then $AT=\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$.

[i]Proposed by Ankit Bisain[/i]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2020q^2$. Find the sum of the possible values of $p$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
II. (50 points)
Let $a, b, c \in \mathbf{R}^{+}$, and $a b c + a + c = b$. Determine the maximum value of $P = \frac{2}{a^{2}+1} - \frac{2}{b^{2}+1} + \frac{3}{c^{2}+1}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111$, $R_5=11111$, etc. When $R_{24}$ is divided by $R_4$, the quotient $Q=\frac{R_{24}}{R_4}$ is an integer whose base-ten representation is a sequence containing only ones and zeros. Find the number of zeros in $Q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a sealed transparent cubic container with a volume of 6, there is a liquid inside. If the shape of the liquid surface is not a triangle regardless of how the cube is rotated, then the range of the liquid volume is __________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let set $A=\{x|x^2-2ax+a=0, x\in \mathbb{R}\}$, and set $B=\{x|x^2-4x+a+5=0, x\in \mathbb{R}\}$. If exactly one of the sets $A$ and $B$ is empty, then the range of the real number $a$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tim the Beaver can make three different types of geometrical figures: squares, regular hexagons, and regular octagons. Tim makes a random sequence \( F_{0}, F_{1}, F_{2}, F_{3}, \ldots \) of figures as follows:
- \( F_{0} \) is a square.
- For every positive integer \( i \), \( F_{i} \) is randomly chosen to be one of the 2 figures distinct from \( F_{i-1} \) (each chosen with equal probability \(\frac{1}{2}\)).
- Tim takes 4 seconds to make squares, 6 to make hexagons, and 8 to make octagons. He makes one figure after another, with no breaks in between.

Suppose that exactly 17 seconds after he starts making \( F_{0} \), Tim is making a figure with \( n \) sides. What is the expected value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer $n$ such that there exist rational numbers $a_i$ and $b_i$ ($i=1, 2, \ldots, n$) satisfying
\[
x^{2} + x + 4 = \sum_{i=1}^{n}\left(a_{i} x + b_{i}\right)^{2}.
\]
Please reason step by step, and put your final answer within \boxed{}.