Context:
Calculate the normal stress at a point on an interface with a normal unit vector \( n = \begin{bmatrix} 0.8 & -0.6 & 0 \end{bmatrix} \) and a stress tensor \( \tau_{ij} = \begin{bmatrix} 1500 & 200 & -500 \\ 200 & 2500 & 1500 \\ -500 & 1500 & -3000 \end{bmatrix} \) by rotating the stress tensor to align the rotated \( x \) axis (\( x' \)) with the normal vector \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four entities are traveling around a 1280-meter lap. The first entity travels at 1 m/s starting at 1 meter, the second at 2 m/s starting at 2 meters, the third at 3 m/s starting at 8 meters, and the fourth at 4 m/s starting at 16 meters. Determine the number of seconds it will take for all four entities to be at the same point on the track simultaneously.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the inverse Fourier transform given by the integral:
\[ f(x) = \int_{-\infty}^\infty \frac{dk}{2\pi} e^{-ikx} \frac{\sin^2(k/2)}{(k/2)^2} \]
Determine the function \( f(x) \) for \( x \in [-1, 1] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(N\) be the smallest positive integer such that \(f(f(f(N))) = 97\), where \(f(n) = n - g(n)\) and \(g(n)\) is the largest positive proper divisor of \(n\). Find the largest integer not exceeding \(\sqrt{N}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$, $b$, and $c$ be complex numbers. Let $a_1$ be the image of $a$ when it is rotated $90^\circ$ clockwise around $b$. Let $a_2$ be the image of $a_1$ when it is rotated $90^\circ$ counter-clockwise around $c$.

Then $a_2$ can be written in the form
$$a_2 = pa + qb + rc$$
for some complex numbers $p$, $q$, and $r$. Find $p^2 + q^2 + r^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider non-negative real numbers \(a, b,\) and \(c\) such that \(a \leq b + c\). For which positive real values of \(n\) does the inequality \(a^n \leq b^n + c^n\) hold?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An equilateral $12$-gon has side length $10$ and interior angle measures that alternate between $90^\circ$, $90^\circ$, and $270^\circ$. Compute the area of this $12$-gon.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \Omega \) be a circle with radius18and let \( \mathcal{S} \) be the region inside \( \Omega \) that the centroid of \( \bigtriangleup {XYZ} \) sweeps through as \( X \) varies along
 all possible points lying outside of \( \Omega \) , \( Y \) varies along all possible points lying on \( \Omega \) and \( {XZ} \) is tangent to the circle. Compute the
 greatest integer less than or equal to the area of \( \mathcal{S} \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider arrangements of the9numbers \( 1,2,3,\ldots ,9 \) in a \( 3 \times  3 \) array. For each such arrangement,let \( {a}_{1} \) , \( {a}_{2} \) ,and \( {a}_{3} \) be the medians of the numbers in rows 1,2,and 3 respectively,and let \( m \) be the median of \( \left\{  {{a}_{1},{a}_{2},{a}_{3}}\right\} \) . Let \( Q \) be the number of arrangements for which \( m = 5 \) . Find the remainder when \( Q \) is divided by1000.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the derivative of the function \( f(W) = x^TWW^Tx \) with respect to the matrix \( W \), where \( x \) is an \( n \times 1 \) vector and \( W \) is an \( n \times m \) matrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of the spherical triangle \( T = \{ x \in S^2 : d_S(x,u) \le \frac{\pi}{2}, d_S(x,v) \le \frac{\pi}{2}, d_S(x,w) \le \frac{\pi}{2} \} \), where \( u = \frac{1}{\sqrt{2}}(1,-1,0) \), \( v = \frac{1}{\sqrt{2}}(1,1,0) \), and \( w = \frac{1}{\sqrt{2}}(0,1,1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$ \frac{2}{\pi}\int_{-\pi}^\pi\frac{\sin\frac{9x}{2}}{\sin\frac{x}{2}}dx $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given positive numbers \( h, s_{1}, s_{2} \) and a triangle \( ABC \) in space, determine the number of ways to choose a point \( D \) such that in the tetrahedron \( ABCD \), the height from vertex \( D \) is \( h \), and the areas of faces \( ACD \) and \( BCD \) are \( s_{1} \) and \( s_{2} \) respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of $0^0$ such that the function $f(x) = \frac{x}{x^x}$ is continuous at $x = 0$. What is this value?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set of all complex numbers $w$ that satisfy the inequality $|z-w|^2 \geq |1-\overline{w}z|^2$ for all complex numbers $z$ with $|z|=1$. Provide your answer as a set of complex numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of zeros of the function \( q(z) = z^3 + 2z^2 - 5 \) within the annulus \( A(0, 2, 3) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the probability that a randomly chosen word of this sentence has exactly four letters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim _{n \to \infty} \int_{0}^1 \arcsin(\sin(nx)) \, dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the index of the subgroup \( H \) in \( \mathrm{GL}_2(\mathbb{Z}) \), where \( H \) is defined as:
\[
H = \left\{\begin{bmatrix} a & b \\\ c & d \end{bmatrix} \mid a\equiv 1 \pmod{3},\ b\equiv 0 \pmod{3},\ c\equiv 0 \pmod{3},\ d\equiv 1 \pmod{3} \right\}.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $T \sim t_7$, find the value of $t$ such that $$P(-t < T) + P(0 < T) + P(t < T) + P(2t < T) = 1.75.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a tight upper bound for the function \( f(m,n) \) given the following conditions: \( f(m,n) \leq \frac{f(m-1,n-1) + f(m,n-2)}{2} \), \( f(m,m-r) = 0 \) for integer \( r \geq 1 \), and \( f(0,n) = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the first three digits of \( \frac{1}{x} \) in \( \mathbb{Z}/7^3\mathbb{Z} \), where \( x = 1+2\cdot7+4\cdot7^2 +1\cdot7^3 +\ldots \in \mathbb{Z}_7 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(x_{n})_{n \ge 1}$ be a sequence of real numbers such that \( \lim_{n\to\infty} x_n \sum_{k=1}^{n}x^2_{k}=1 \). Compute \( \lim_{n\to\infty} (3n)^{1/3} x_n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct values that \( x^9 \) can take modulo 999.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bus tickets from a transportation company are numbered with six digits, ranging from 000000 to 999999. A ticket is considered "lucky" if the sum of the first three digits equals the sum of the last three digits. For example, ticket 721055 is lucky, whereas 003101 is not.

Determine how many consecutive tickets a person must buy to guarantee obtaining at least one lucky ticket, regardless of the starting ticket number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of distinct sets that can be generated from a single set \( B \subseteq \mathbb{R} \) by taking a finite number of closures and interiors.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The area of triangle $\triangle OFA$, where line $l$ has an inclination angle of $60^\circ$ and passes through the focus $F$ of the parabola $y^2=4x$, and intersects with the part of the parabola that lies on the x-axis at point $A$, is equal to $\frac{1}{2}\cdot OA \cdot\frac{1}{2} \cdot OF \cdot \sin \theta$. Determine the value of this expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $b$ for which the system of differential equations \( \begin{cases} x'(t)=by(t) \\ y'(t)=x(t)+y(t) \end{cases} \) has a unique non-zero solution \((x(t),y(t))\) that approaches \((0,0)\) as \(t\) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three shepherds met on a large road, each driving their respective herds. Jack says to Jim:

- If I give you 6 pigs for one horse, your herd will have twice as many heads as mine.

And Dan remarks to Jack:

- If I give you 14 sheep for one horse, your herd will have three times as many heads as mine.

Jim, in turn, says to Dan:

- And if I give you 4 cows for one horse, your herd will become 6 times larger than mine.

The deals did not take place. Determine the total number of heads of livestock in the three herds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \(\lim_{x\to\infty}(x!)^{1/x}-\frac{x}{e}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a multiset of integers, $M1 = \{6,6,4,4,4,2,2\}$, and another multiset, $M2 = \{16,8,4\}$, determine the number of distinct ways to form $M2$ by picking integers without replacement from $M1$. Each element of $M2$ must be a sum of elements from $M1$. What is the count of such distinct combinations?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the set operation that generates the group \(\mathbb{Z}_4\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values for \( a + b \) such that the inequality \( a \cos x + b \cos 2x \geq -1 \) holds for all \( x \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the gradient of the function \( f(A) = ||Ax - y||^2 \) with respect to the matrix \( A \), where \( A \in \mathbb{R}^{n\times n} \), \( x,y \in \mathbb{R}^n \), and \( ||\cdot|| \) denotes the Euclidean norm.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the relationship between the determinants of the matrices \( M_1 = \begin{vmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{vmatrix} \) and \( M_2 = \begin{vmatrix} x_1+ay_1 & y_1+bz_1 & z_1+cx_1 \\ x_2+ay_2 & y_2+bz_2 & z_2+cx_2 \\ x_3+ay_3 & y_3+bz_3 & z_3+cx_3 \end{vmatrix} \), where \( a \), \( b \), and \( c \) are constants.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a biased coin with a probability of \( p = \frac{1}{3} \) of landing heads and \( \frac{2}{3} \) of landing tails. The coin is flipped \( n \) times. Let \( X_i \) be a random variable denoting the \( i \)-th flip, where \( X_i = 1 \) means heads, and \( X_i = 0 \) means tails. Determine the smallest value of \( n \) such that the probability that more than half of the coin flips result in heads is less than 0.001, using the Chernoff bound.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximal length of a sequence of numbers consisting of only $+1$ and $-1$ such that the sum of every 10 consecutive numbers is 0, but the sum of every 12 consecutive numbers is not 0.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of consecutive composite integers that follow \( k! + 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of all ordered triplets of positive integers $(a, b, c)$, which satisfy the equalities:
$$[a, b] =1000, [b, c] = 2000, [c, a] =2000.$$
([x, y]represents the least common multiple of positive integers x,y)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \iint \frac{x^2}{y^3} \, dA \) over the region defined by the curves \( y=2x, \; y=x, \; y=x^2, \; y=2x^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can the sides of an icosahedron be numbered from 1 to 20, considering the sides indistinguishable?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $F=\langle (1,0,0,2),(0,1,0,0) \rangle$ and $G=\langle (-1,1,0,0),(1,0,1,0),(0,0,0,1) \rangle$. Given that the basis of $F \cap G$ is $\{(-1,1,0,-2)\}$, find the basis of the quotient space $F/(F \cap G)$. Provide your answer as a set of vectors.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the definite integral \( \int_{1}^{2} \frac{e^x}{x} \, dx \) to five decimal places using the Taylor series expansion for \( \frac{e^x}{x} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the difference between the sum of real roots of the function \( f_{10}(x) = 1 - \sum_{k=0}^{10} \frac{x(1-x)\cdots(k-x)}{(k+1)!} \) and the sum of real roots of its derivative \( f'_{10}(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the range of values of $q$ for which the series \( \sum_{n=1}^{\infty} \frac{n^2}{(n+1)^q} \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The persons $A$, $B$, and $C$ are playing a game where the person with the highest points wins. The points are distributed as follows:

- $A$'s points are a Poisson distributed random variable with parameter 2.
- $B$'s points are a Poisson distributed random variable with parameter 3.
- $C$'s points are a Poisson distributed random variable with parameter 4.

The points of each player are independent of each other. The outcome of the points was 1, 2, and 3, but it is unknown who received which score. What is the conditional probability that person $C$ won the game?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the interval of convergence for the series \( \sum\limits_{n=1}^{\infty} \frac{n+3}{2n^2+2} x^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint_{|z|=2} \frac{dz}{z\sqrt{1-\frac{1}{z^2}}} \) using the principal branch of the square root, where \(|z|=2\) is a circle centered at the origin with radius 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the board, there is a function $\sin x + \cos x$. You are allowed to write the derivative of any previously written function, as well as the sum and product of any two previously written functions. This process can be repeated many times. Determine the constant $c$ such that a function equal to $c$ for all real $x$ can appear on the board.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the general form of the function $f(r)$ that satisfies the differential equation $$\nabla^2 f(r) = \frac{d^2f}{dr^2} + \frac{2}{r} \frac{df}{dr} = 0,$$ where $a$ and $b$ are constants.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a positive number \( \delta \) such that if \( 0 < |x-1| < \delta \), then \( |f(x)g(x) - h(x) - 1| < \frac{1}{1000} \), given that \( |f(x) - 2| < |x-1|^2 \), \( |g(x)-3| < 2|x-1| \), and \( |h(x)-5| < |x-1| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the Krull dimension of the ring \( k[x,y,z]/(xz, yz) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(f(x)\) and \(g(x)\) be continuous, positive functions such that \(f(-x) = g(x) - 1\), \(f(x)=\frac{g(x)}{g(-x)}\) and \(\int_{ - 20}^{20} {f\left( x \right)dx} = 2020\). Find the value of \(\int_{ - 20}^{20} {\frac{{f\left( x \right)}}{{g\left( x \right)}}dx} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x > 0$. Solve the inequality: $$ \Gamma(x) \le \frac{1}{x} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many trailing zeros does the value
\[300\cdot305\cdot310\dots1090\cdot1095\cdot1100\]end with?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set \( B = \{ f \in \mathbb{R}^\mathbb{N} \mid f(\mathbb{N}) \text{ is a well-ordered subset of } \mathbb{R} \text{ with respect to the standard ordering} \}. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^e{W(x)}\,\mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the contour integral \( \int_{|z|=3} \frac{z^{17}}{(z^2+2)^3(z^3+3)^4}dz \). Assume the singular points are within the circle \(|z|=3\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the distribution of the statistic \( T = \frac{S_{n-1}^2/\sigma_{1}^2}{S_{m-1}^2/\sigma_{2}^2} \), where \( X \sim N(\mu_1, \sigma_1) \) and \( Y \sim N(\mu_2, \sigma_2) \). Given two simple random samples of sizes \( n \) and \( m \) from \( X \) and \( Y \) respectively, determine the distribution of \( T \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of graphs with 5 vertices labeled a, b, c, d, and e, such that exactly one vertex has degree 4.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{\pi} \sin(A\sin(x))\sin(x) \,dx \) where \( A \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X=(X_1,X_2,X_3)$ be a continuous random vector with the joint probability density function given by \( f(x_1,x_2,x_3)= 12x_2 \) for \( 0<x_3<x_2<x_1<1 \) and \( 0 \) elsewhere. Calculate the probability of the event \( B = \{ x_3 \leq \frac{1}{3} \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of \( \frac{dy}{dx} \) at \( x = -1 \) for the equation:
\[
(\sin y)^{\sin(\frac{\pi x}{2})}+ \frac{\sqrt{3}}{2}\sec^{-1}(2x)+ 2^x\tan (\log_e(x+2))=0
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the following expression as \( n \to \infty \):
\[ M_{Y_n}(t) = e^{-t\sqrt{n}} \left(\frac{1}{1 - t/\sqrt{n}}\right)^n. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(ABC\) be a triangle with \(AB=13\), \(BC=14\), and \(CA=15\). The altitude from \(A\) intersects \(BC\) at \(D\). Let \(\omega_1\) and \(\omega_2\) be the incircles of \(ABD\) and \(ACD\), and let the common external tangent of \(\omega_1\) and \(\omega_2\) (other than \(BC\)) intersect \(AD\) at \(E\). Compute the length of \(AE\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle is inscribed in an equilateral triangle with side length $s$. Points $A$,$B$,$C$,$D$,$E$,$F$ lie on the triangle such that line segments $AB$, $CD$, and $EF$ are parallel to a side of the triangle, and tangent to the circle. If the area of hexagon $ABCDEF = \frac{9\sqrt3}{2}$ , find $s$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\{X(t), t \geq 0\}$ be a Brownian motion with drift parameter $\mu = -2$ and variance parameter $\sigma^2 = 16$. If $X(0) = 5$, calculate $P(X(1) < 0)$. \(\text{Express your answer as a probability.}\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of \(x^{10}\) in the expansion of \((1+2x+3x^2+4x^3+5x^4+6x^5)^4\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), the equation of side \(BC\) is given by \(2x - y = 3\). The circumcenter of the triangle is at \((2, 4)\) and the orthocenter is at \((1, 2)\). Determine the value of \(\tan B \tan C\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( m \) such that the polynomial \( p(x) = (x + 1)^m - x^m - 1 \) is divisible by \( q(x) = (x^2 + x + 1)^2 \) without a remainder.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to arrange the letters in the word 'INTELLIGENT' such that there are at least two consecutive pairs of identical letters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to \infty}{\sqrt{(1-\cos(1/n))\sqrt{(1-\cos(1/n))\dots}}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \left\lVert A-B \right\rVert = (x_A-x_B)^2+(y_A-y_B)^2 \) for every two points \( A = (x_A, y_A) \) and \( B = (x_B, y_B) \) in the plane. Let \( S \) be the set of points \((x,y)\) in the plane for which \( x,y \in \{ 0,1,\dots,100 \} \). Find the number of functions \( f : S \to S \) such that \( \left\lVert A-B \right\rVert \equiv \left\lVert f(A)-f(B) \right\rVert \pmod{101} \) for any \( A, B \in S \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the product as \( n \to \infty \):
\[
P = \lim_{n\to \infty} \prod_{i=1}^n \frac{1 - (2i + 1)a/(2n)}{1 - ia/n}
\]
where \( 0 < a < 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the series as \( n \to \infty \):
\[ x(n) = \sum\limits_{k=1}^n \frac{k^2}{n^3+k^2} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(n\) and \(k\) be positive integers such that \(n \geq 2k > 3\) and \(A = \{1, 2, \ldots, n\}\). Determine all pairs \((n, k)\) such that the number of \(k\)-element subsets of \(A\) is \(2n - k\) times the number of 2-element subsets of \(A\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $2017$ is a prime number, find the remainder when the sum $S = \sum_{k=0}^{62} \binom{2014}{k}$ is divided by $2017$. Simplify the expression for $S$ using combinatorial identities and modulo arithmetic.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the last digit of \(AD^{AD}\) in hexadecimal when expressed in octal.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \(a, b, c > 0\) and \(a + b + c = abc\), find the maximum value of \((a-1)(b-1)(c-1)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation $8GX = XX^T$, where $G = \begin{pmatrix} 5 & 4 \\ 3 & 2 \end{pmatrix}$, find the determinant of the matrix $X$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the units in the polynomial ring $R[x]$, where $R$ is a domain.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the gradient of the function \( f(x) = \|x\|^p \), where \( f(x) = (x_1^2 + x_2^2 + \ldots + x_N^2)^{p/2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum perimeter of a triangle that can be inscribed in a regular septagon with each side measuring 1 unit. Provide your answer rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint_G (x^{4}-y^{4})xy\,dxdy \), where \( G \) is the region defined by \( 0 \leq x \), \( 1 \leq x^{2}-y^{2} \leq 2 \), and \( 1 \leq xy \leq 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum value of the expression \( \left|7x-1\right|+\left|7y-5\right|+\left|7z-1\right| \) for non-negative real numbers \(x\), \(y\), and \(z\) such that \(x + y + z = 1\) and \(y^2 \leq 3xz\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$ \lim_{n \to \infty} \frac{\int_{0}^{1} \frac{e^{-x^2} dx}{(1+x^{2})^{n}}}{\frac{1}{n^2}} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a laboratory, a chemist adds sulfuric acid to flasks held by four groups, each consisting of 3 people. For the first group, he adds $a_1$, $a_2$, and $a_3$ drops to the flasks such that $a_1 + a_2 + a_3 = 4$. For the second group, he adds $b_1$, $b_2$, and $b_3$ drops such that $b_1 + b_2 + b_3 = 3$. For the third group, he adds $c_1$, $c_2$, and $c_3$ drops such that $c_1 + c_2 + c_3 = 2$. For the fourth group, he adds $d_1$, $d_2$, and $d_3$ drops such that $d_1 + d_2 + d_3 = 1$. Assuming the drops are of equal size and randomly distributed, what is the probability that the flask with the most acid belongs to the first group? Express your answer as a percentage.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of surjective ring homomorphisms from the ring of Gaussian integers $\mathbb{Z}[i]$ onto the finite field $\mathbb{F}_{11^2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a least-squares problem \(||Ax-b||=0\) solved using a backward stable algorithm (QR decomposition with Householder projectors), and a condition number \(\kappa(A)=10^5\), determine how many digits of accuracy the solution \(x\) should have when using double-precision floating-point arithmetic (\(10^{-16}\)).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all positive real numbers \( p \) such that the inequality \( \sqrt{a^2 + pb^2} + \sqrt{b^2 + pa^2} \ge a + b + (p - 1) \sqrt{ab} \) holds for any positive real numbers \( a \) and \( b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
We say that an integer $x \in \{1, \dots, 102\}$ is \textit{square-ish} if there exists some integer $n$ such that $x \equiv n^2 + n \pmod{103}$. Compute the product of all \textit{square-ish} integers modulo $103$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of homomorphisms from the group \( \Bbb{Z}_{10} \times \Bbb{Z}_{25} \) to the symmetric group \( S_4 \). Here, \( \Bbb{Z}_n \) denotes the integers from 0 to \( n-1 \) with addition modulo \( n \), and \( S_4 \) is the group of permutations of the set \( \{1, 2, 3, 4\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Krull dimension of the ring \( R = \mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two circles \(\gamma_{1}\) and \(\gamma_{2}\) intersect at points \(P\) and \(Q\). The tangent to \(\gamma_{2}\) at \(P\) intersects \(\gamma_{1}\) at point \(A\), and the tangent to \(\gamma_{1}\) at \(P\) intersects \(\gamma_{2}\) at point \(B\), with \(A\) and \(B\) distinct from \(P\). Given that \(PQ \cdot O_{1}O_{2} = PO_{1} \cdot PO_{2}\) and \(\angle APB\) is acute, find the measure of \(\angle APB\) in degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a boundary value problem with the operator $\mathscr{L}:=-\Delta$ and the boundary condition $u=0$. If $u$ satisfies the inequality $u^5-u \le 0$ in the domain $\Omega$, determine an upper bound for $\sup_{\Omega} |u|$. Provide your answer as a specific numerical value or expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(b_i)_{i\in \Bbb{N}}$ be a sequence of positive integers such that $b_3 = 7$, $b_{2020} = 2021$, and $b_n=b_{b_{n-1}}$ for all other positive $n$. Determine the possible values of $b_{2021}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), the angle bisector \(AL\) (where \(L \in BC\)) is drawn. Points \(M\) and \(N\) lie on the other two angle bisectors (or their extensions) such that \(MA = ML\) and \(NA = NL\). Given that \(\angle BAC = 50^\circ\), find the measure of \(\angle MAN\) in degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum distance $d(C)$ of the ternary linear code $C$ with the following parity check matrix:
$$H=\begin{pmatrix} 2 & 0 & 1 & 1 & 1 \\ 0 & 2 & 2 & 0 & 0 \\ 0 & 0 & 2 & 2 & 1 \\ 1 & 1 & 2 & 2 & 0 \end{pmatrix}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $n \equiv 0 \,(\mathrm{mod}\,4)$, determine the minimum value of $m$ such that there exists an $m \times n$ partial Hadamard matrix that cannot be extended to an $(m+1) \times n$ partial Hadamard matrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a regular triangle \( ABC \), three nails are driven into the floor, one near each side, preventing the triangle from rotating without lifting. The first nail divides side \( AB \) in the ratio \( 1:3 \) from vertex \( A \), and the second nail divides side \( BC \) in the ratio \( 2:1 \) from vertex \( B \). Determine the ratio in which the third nail divides side \( AC \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \iint_S \vec{A} \cdot \hat{n} \, dS \), where \( \vec{A} = 18z\hat{i} - 12\hat{j} + 3y\hat{k} \) and \( S \) is the part of the plane \( 2x + 3y + 6z = 12 \) located in the first octant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given 365 unique days in a year, how many friends would you need on average to ensure that at least one friend's birthday falls on each day? Assume that birthdays are equally distributed across all days of the year.
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Context:
Compute the integral \( \int_{C} \log(z-1) \, dz \) over a circle centered at the origin with a radius of 3, oriented counterclockwise.
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Context:
In an $n \times n$ grid, each cell contains a student facing one of the four cardinal directions: up, down, left, or right. The students on the grid's borders do not face each other directly, meaning no student looks directly at another student's face; they can only see another student's ear or back. Determine the minimum number of students who can see the ear of another student. For example, in a $5 \times 5$ grid, the minimum number of students with this view is $n+4$. What is the minimum number of students who see the ear of another student for a general $n \times n$ grid?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of equivalence relations \(S\) over the power set of natural numbers \(\mathcal P(\mathbb N)\) such that the equivalence class of \(\{8\}\) under \(S\) is \(\{A \in \mathcal P(\mathbb N) \mid A \neq \{1\} \wedge A \neq \{2\}\}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of the splitting field of the polynomial \(x^8 - 1\) over the field with 4 elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a 2015x2015 square table filled with positive numbers, the product of the numbers in each row and each column is 2, and the product of the numbers in any 3x3 sub-square is 1. Determine the number located at the center of the table.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate $ \int_0^1 \frac{(1-2x)e^{x}+(1+2x)e^{-x}}{(e^x+e^{-x})^3}\ dx.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x\to 0}\left\{\frac{e^{-x^2}\sin|x|-|x|}{|x^3|}\right\}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive number \( \lambda \) such that for any positive numbers \( a_{1}, a_{2}, \ldots, a_{n} \) and any \( n \) numbers \( b_{1}, b_{2}, \ldots, b_{n} \) from the interval \( \left[0, \frac{1}{2}\right] \), if \( a_{1}+a_{2}+\cdots+a_{n}=b_{1}+b_{2}+\cdots+b_{n}=1 \), then \( a_{1} a_{2} \cdots a_{n} \leq \lambda\left(a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct values that the integral \( \oint_\gamma \frac{dz}{(z-a_1)(z-a_2)...(z-a_n)} \) can take, where \( \gamma \) is a simple closed curve not passing through any of the points \( a_1, a_2, \ldots, a_n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
We write the digits of $1995$ in the following way:
$199511999955111999999555......$
Which digit is in position number $1995$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest positive integer $r$ such that in any 500-element subset of the set $\{1, 2, \ldots, 1000\}$, there exist two subsets whose intersection contains at least $r$ elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the integral \( \int_{0}^{1}|t^3-a-bt-ct^2|^2dt \) by using the projection of \( t^3 \) onto the subspace of polynomials of degree at most 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of conjugacy classes of the group \( G = \langle s,t \mid s^8=t^2=1, st=ts^3 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the common ring to which the quotient ring \( \mathbb{Z}_6[x]/\langle 2x-3 \rangle \) is isomorphic.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $A$ and $B$ are matrices with dimensions $n \times K$ and $n \times M$ respectively, where $n > K$ and $\text{rank}(A) = K$. If $\text{rank}(A + B) = K - s$ for some $s > 0$, determine the minimum possible dimension of the column space of $B$. \(\text{Express your answer as an integer.}\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a particular solution to the differential equation \( y(x) = xy' + \ln y' \) given the initial condition \( y(-1) = -1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the group of grouplike elements in the Hopf algebra $U(L)$, where $U(L)$ is the universal enveloping algebra of a Lie algebra $L$. The algebra $U(L)$ is constructed as $U(L) = T(L) / I(L)$, with $I(L)$ being the two-sided ideal generated by elements of the form $xy - yx - [x,y]$, and $T(L)$ is the tensor algebra of $L$. A grouplike element in a coalgebra is an element $x \neq 0$ with $\Delta(x) = x \otimes x$. The set of grouplike elements forms a group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple summation \( S = \sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\frac{1}{3^{i+j+k}} \) where \( i \neq j \neq k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
As $n_1 \to \infty$ and $n_2 \to \infty$, does the statistic \( T = \dfrac{\bar{X}_{1, n_1} - \bar{X}_{2, n_2}}{\sqrt{\dfrac{S_1^2}{n_1} + \dfrac{S_2^2}{n_2}}} \) converge in distribution to a random variable? If so, identify the distribution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Laplacian of the function \( u(x,y,z) = \frac{1}{\sqrt{x^2 + y^2 + z^2}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the largest possible number of inversions in a sequence of positive integers whose sum is 2014?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum percentage increase in the number of basic solutions for a bounded linear programming problem when an additional constraint is added.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x_1, x_2, \ldots, x_{2560}$ be the roots of the polynomial $x^{2560} + 2x^{2559} + 3x^{2558} + \ldots + 2561$. Determine the value of the expression:
$$(x_1^2 + 1)(x_2^2 + 1)(x_3^2 + 1)\ldots(x_{2560}^2 + 1)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Provide an example of a bipartite connected graph with an even number of vertices that has an Eulerian circuit but does not have a perfect matching.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit \( \lim_{n\to \infty}\left(\dfrac{f\left(a+\frac{1}{n}\right)}{f(a)}\right)^{1/n} \) given that \( f:\mathbb{R}\to \mathbb{R} \) is differentiable at \( a \) and \( f(a) > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit superior of the sequence \( a_j = |\sin j|^{\sin j} \) for all \( j \in \mathbb{N} \). Determine \( \limsup_{j \to \infty} \{ a_j \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of permutations \( \pi \) of \([n]\) such that the numbers 1, 2, and 3 appear in the same cycle of \( \pi \), while the number 4 appears in a different cycle. Provide a general formula for any \( n \geq 4 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected number of collisions when hashing 80,000 unique entries using a 36-bit hash function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct sets of eight numbers such that each number in the set is the product of two other numbers from the same set. Sets that differ only in the order of the numbers are considered identical.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the points of continuity for the function \( f:(0,1)\rightarrow \mathbb{R} \) defined by \( f(x)= x^{2} \) if \( x \) is rational and \( f(x) = 2-x^{2} \) if \( x \) is irrational.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the triangle $ABC$ we have $\angle ABC=100^\circ$, $\angle ACB=65^\circ$, $M\in AB$, $N\in AC$, and $\angle MCB=55^\circ$, $\angle NBC=80^\circ$. Find $\angle NMC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a grand hall, there are 1000 people having dinner. One person is initially sick, while the other 999 are healthy. Each minute, every person talks to one other person in the room at random, ensuring they do not talk to someone they have previously talked to. If a sick person talks to a healthy person, the healthy person becomes sick. Once a person becomes sick, they remain sick. Determine the maximum number of minutes until all 1000 people in the hall are sick.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int^{+\infty}_0 \frac{e^{-x^2}}{(x^2+\frac{1}{2})^2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct ways to select four 3-combinations from a set of six elements, such that each pair of combinations shares exactly one common element. Consider two sets of combinations equivalent if one can be transformed into the other by an element of the dihedral group D6.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each cell of a 2 × 5 grid of unit squares is to be colored white or black. Compute the number of such colorings for which no 2 × 2 square is a single color.

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


Context:
Evaluate the limit: \( \lim_{n\to\infty}\sqrt[n]{c_n^n+d_n^n+2023} \), given that \( c_n \rightarrow c > 1 \) and \( d_n \rightarrow d > 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( \Phi \), the infinite product of successive approximations to the golden ratio \( \phi \), defined as \[ \Phi = \frac{\sqrt{1}}{\phi} \times \frac{\sqrt{1+\sqrt{1}}}{\phi} \times \frac{\sqrt{1+\sqrt{1+\sqrt{1}}}}{\phi} \times \dots \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest natural number $b$ such that the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots is 20.
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Context:
Find an expression for the function \( h(z) = \sum_{n=1}^{\infty}\frac{(z-2)^n}{n} \) as an elementary function. Consider using techniques similar to those used for the related function \( f(z) = \sum_{n=1}^{\infty} n(z-2)^n = \frac{z+3}{(z+2)^2} \). Knowledge of logarithms in the complex plane may be required.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{|z|=3\pi/2} \frac{2\cosh(z)}{1-\sinh(z)} \, dz \) by considering the singularities of the integrand within the contour.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the minimum value of the integral \(\displaystyle\int_{0}^{\infty} \left | x^3-a-bx-cx^2 \right |^2e^{-x}\, dx\) subject to the constraints \(a,b,c \in \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a sequence of independent and identically distributed (i.i.d.) Gaussian matrices $A_i \in \mathbb{R}^{p \times n}$ with $p < n$. Each matrix $A_i$ can be decomposed into its reduced singular value decomposition (SVD) as $A_i = V_i \Sigma_i U_i^T$. When we concatenate the column spaces of these matrices by forming $\sum_{i=1}^m U_i U_i^T$, what is the minimum value of $m$ for this sum to be a full-rank matrix, assuming that the matrices $U_i$ follow the Haar measure?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest constant \( k > 0 \) such that \[ \max\left(\frac{5}{5-3c},\frac{5b}{5-3d}\right) \geq k\cdot\frac{2+3b}{5-c-2d} \] for all \( 0 \leq b \leq 1 \) and \( 0 \leq c \leq d \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the term with the highest power of \(x\) in the expression \((x^3-2)^{16}-(x^4 + 3)^{12}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all values of the parameter $m$ such that the system of equations \( x^2 = 2^{|x|} + |x| - y - m \) and \( 1 - y^2 = 0 \) has exactly one solution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the condition under which equality holds for the inequality \(|x+y| \leq |x| + |y|\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
64 people are in a single elimination rock-paper-scissors tournament, which consists of a 6-round knockout bracket. Each person has a different rock-paper-scissors skill level, and in any game, the person with the higher skill level will always win. For how many players \( P \) is it possible that \( P \) wins the first four rounds that he plays?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( x \) for which all eigenvalues of the matrix \( A \) are real, where:
\[ A=\begin{pmatrix} 10 & 5+i & 4 \\ x & 20 & 2 \\ 4 & 2 & -10 \end{pmatrix} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find \( \lim_{n \to \infty} \int_{1}^{n}\frac{nx^{1/2}}{1+nx^2}dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$ with sides $AB = 30$, $BC = 51$, and $CA = 63$, points $P$, $Q$, $R$, and $S$ are on sides $BC$, $CA$, and $AB$, respectively, such that $PQRS$ forms a parallelogram with its center at the centroid of triangle $ABC$. Determine the area of parallelogram $PQRS$ using an analytic approach.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $X_1, X_2, \ldots, X_n$ are independent random variables, each uniformly distributed over the interval $[0, 1]$. Let $X_{(n)}$ be the largest and $X_{(n-1)}$ be the second largest of these variables. Calculate the expected value of the ratio $\frac{X_{(n-1)}}{X_{(n)}}$. In other words, find $\mathbb{E}\left[\frac{X_{(n-1)}}{X_{(n)}}\right]$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( r \), where \( r > \frac{1}{2} \), such that two non-vertical normal lines from the point \((r, 0)\) to the parabola \(x = y^2\) intersect perpendicularly. One of these normal lines is the x-axis. The derivative of the parabola is \(\frac{dy}{dx} = \frac{1}{2\sqrt{x}}\), so the gradient of the normal is \(-2\sqrt{x}\). The equation of the normal can be expressed as \(y = -2\sqrt{x}(x - r)\). Determine the correct value of \( r \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Assume a problem $R \in \mathit{NP}$ can be solved with a verifier $M(x,y)$ that runs in $O(n^3)$ time, where $y$ is additional information no longer than $3\log n$ bits. Determine the function $f(n)$ such that $R \in \mathit{TIME}(f(n))$. Provide your answer as a function of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_1, \ldots, X_{735}$ and $Y_1, \ldots, Y_{880}$ be independent random variables such that $P(X_i=0)=\frac{3}{7}$, $P(X_i=1)=\frac{4}{7}$, and $P(Y_i=0)=P(Y_i=1)=\frac{1}{2}$. Using the Central Limit Theorem, find $P\left(\sum_{i=1}^{735} X_i < \sum_{i=1}^{880} Y_i\right)$. Express your answer as a probability.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$ where $n$ and $d$ are integers with $1\leq d\leq 5$. What is the probability that\[(\cos(a\pi)+i\sin(b\pi))^4\]is a real number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ I = \int_{0}^{\frac{\pi}{2}} \ln(2468^{2} \cos^2x + 990^2 \sin^2x) \, dx \]
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Context:
Calculate the expectation value for the function: $$ f(x)=\begin{cases} \frac{x^2}{2\beta^{3}}e^{-\frac{x}{\beta}} & x>0\\ 0 & \text{otherwise} \end{cases}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of non-isomorphic groups of order 30, given that every Sylow 3 and Sylow 5 subgroup is normal in the group, and the group has a normal subgroup of order 15.
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Context:
Find the constant $p$ such that the product of any positive number $N_0$ multiplied by successive random numbers between $0$ and $p$ will, on average, neither diverge to infinity nor converge to zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the smallest number of vertices, $n$, in a polygon such that there exists an interior point $p$ where the distance from any vertex $b$ to $p$ is greater than the distance from $b$ to either of its adjacent vertices?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimum number of combination attempts required to guarantee opening a safe with 3 wheels, each having 8 positions, if the safe opens when any two of the three wheels are set correctly? Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} = \frac 1m + \frac 1n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the degree of the differential equation \((y''')^\frac{4}{3} + (y')^\frac{1}{5} + 4=0\) after rationalizing the indices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{h \to 0} \frac{2\sin\left(\frac{1}{2}\ln\left(1+\frac{h}{x}\right)\right)}{h} \) without using L'Hopital's rule, given that \( \lim_{x \to 0} \frac{\ln(1+x)}{x} = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a_{1}, a_{2}, \cdots, a_{2014} \) be a permutation of the positive integers \( 1, 2, \cdots, 2014 \). Define
\[ S_{k} = a_{1} + a_{2} + \cdots + a_{k} \quad (k=1, 2, \cdots, 2014). \]

Determine the maximum number of odd numbers among \( S_{1}, S_{2}, \cdots, S_{2014} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of the group $G$ given by the presentation $G = \langle a, b \mid a^3 = b^2 = (ab)^3 = e \rangle$, where $e$ is the identity element.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence of the power series \( \sum_{k=0}^{\infty} a_k x^{k^2} \), where \( a_k = (2k)! \) if \( k \) is a perfect square, and \( a_k = 0 \) otherwise.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the function \( f(x,y,z) = \left|7x-1\right|+\left|7y-5\right|+\left|7z-1\right| \) given that \( x, y, z \) are non-negative real numbers satisfying \( x+y+z=1 \) and \( y^2 \le 3xz \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A battery has a lifetime of 24 hours and is used for a maximum of three days. Each day, the battery is used for a random number of hours, denoted by \( K \), where \( K \) is uniformly distributed over \([0, 24]\) and independent of usage on other days. Calculate the expected remaining lifetime of the battery at the end of the third day.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum value of \((f^{-1})'f(x)\) for an invertible increasing function \(f(x)\) that satisfies \(f''(x)+2f'(x)\geq 2\) and \(f'(0)=2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( I = \frac{1}{2\pi i}\oint_{|z|=1}\frac{2(1-\cos(z))e^z}{z^4}dz \) using the residue theorem by finding the coefficient \( a_{-3} \) in the Laurent series expansion around \( z_0 = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all values of the parameter \( a \) for which there exists a number \( b \) such that the system of equations \( x^{2}+y^{2}+2a(a-x-y)=64 \) and \( y=\frac{7}{(x+b)^{2}+1} \) has at least one solution \( (x, y) \).
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Context:
Given a triple of digits \((a, b, c)\), where \(a, b, c \in \{0, 1, \dots, 9\}\), you can guess a triple of digits \((m, n, p)\) each turn. You will be informed whether the statement \(a = m\) or \(b = n\) or \(c = p\) is true, but not which specific equality holds. What is the minimum number of turns required to guarantee determining the triple \((a, b, c)\)?
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Context:
Use Stokes' theorem to evaluate the line integral \( \int_C{ \vec{F} \cdot \vec{dr}} \), where \( \vec{F}(x, y, z) = \langle 2 x^2 y , \frac{2 x^3}{3}, 2xy \rangle \) and \( C \) is the curve of intersection of the hyperbolic paraboloid \( z = y^2 - x^2 \) and the cylinder \( x^2 + y^2 = 1 \), oriented counterclockwise as viewed from above.
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Context:
Let $p$ be a prime number. Given a positive integer $n$, an $n \times n$ matrix $A$ is formed with each element $(a_{ij})$ randomly selected, with equal probability, from $[0, 1, \ldots, p - 1]$. Let $q_n$ be the probability that $\text{det}A \equiv 1 \pmod{p}$. Find $\lim_{n \to \infty} q_n$. \( \text{Express your answer as a single value.} \)
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Context:
During a vacation, I dined with my 6 friends under the following conditions: I dined with all 6 friends on 1 day, with every 5 of them on 2 days, with every 4 of them on 3 days, with every 3 of them on 4 days, and with every 2 of them on 5 days. Additionally, each friend was present at 7 dinners and absent at 7 dinners. How many dinners did I have alone?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimal number of vertices required to triangulate a Klein bottle.
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Context:
Calculate the expectation of the random variable $f(w) = 1 - w$ on the probability space $(\Omega, B, \mu)$, where $\Omega = [0,1]$, $\mu$ is the Lebesgue measure, and $B$ is the Borel $\sigma$-algebra. Use the measure $\mu_f$ defined by:

$$\mu_f = \begin{cases} 0 & x < 0 \\ x & 0 \leq x \leq 1 \\ 1 & x > 1 \end{cases}$$

Express the expectation as:

$$E(f) = \int_{\mathbb R}x\mu_f(dx) = \int_{[0,\infty)}x\mu_f(dx) - \int_{(-\infty,0)}x\mu_f(dx)$$

What is the value of $E(f)$?
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Context:
Evaluate the integral: $$\int^1_0 \left (\sum^\infty_{n=0}\frac{\left\lfloor 2^nx\right\rfloor}{3^n} \right )^2{\rm d}x$$
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Context:
Evaluate the integral: \[ \int_{0}^{2\pi} e^{\cos\theta} \cos(n\theta - \sin\theta) \, d\theta \]
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Context:
Evaluate the limit: $$\lim_{x \rightarrow 0^+} \frac{\arctan(e^x+\arctan x)-\arctan(e^{\sin x}+\arctan(\sin x))}{x^3}$$
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Context:
Through vertex $A$ of parallelogram $ABCD$, a line is drawn that intersects diagonal $BD$, side $CD$, and line $BC$ at points $E$, $F$, and $G$, respectively. Find the ratio $BE:ED$ if $FG:FE=4$. Round your answer to the nearest hundredth if needed.
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Context:
Determine the maximum value of \( k \) such that any good number can be transformed into any other good number by appending a special number to either end, inserting it between any two digits, or erasing a special number from its representation.
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Context:
Evaluate the integral: $$\int_0^{\pi/4}\frac{\ln(\tan x)}{\sin 2x}dx$$
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Context:
Provide an example of a metric space that is complete but not pre-compact.
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Context:
A fair standard six-sided die is tossed three times. Given that the sum of the first two tosses equals the third, what is the probability that at least one "2" is tossed?
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Context:
Calculate the probability that a sequence \((x_n)_{n \ge 1}\), defined by \(x_1 = x_2 = 1\) and \(x_{n+2} = ax_{n+1} - bx_n\), where \(a\) and \(b\) are distinct numbers chosen randomly from the set \(\{1, 2, \ldots, 10\}\), converges.
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Context:
Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and point $P$ is a point on the right branch of the hyperbola. $M$ is the incenter of $\triangle PF\_1F\_2$, satisfying $S\_{\triangle MPF\_1} = S\_{\triangle MPF\_2} + \lambda S\_{\triangle MF\_1F\_2}$. If the eccentricity of this hyperbola is $3$, then $\lambda = \_\_\_\_\_\_$.
(Note: $S\_{\triangle MPF\_1}$, $S\_{\triangle MPF\_2}$, $S\_{\triangle MF\_1F\_2}$ represent the area of $\triangle MPF\_1$, $\triangle MPF\_2$, $\triangle MF\_1F\_2$ respectively.)
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Context:
Determine the values of \( x \) for which the series \( \sum_{n=0}^{\infty} \left(\frac{n}{2n+1}\right)^{2n-1}x^n \) converges.
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Context:
Find the linear function $f(x)$ such that $f(f(f(\cdots f(x)))) \geqslant 1024x + 1023$, where $f$ is applied multiple times.
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Context:
A circle passes through the vertices \( A \) and \( B \) of triangle \( ABC \) and intersects its sides \( AC \) and \( BC \) at points \( Q \) and \( N \) respectively. Given \( AQ : QC = 5 : 2 \) and \( CN : NB = 5 : 2 \), find the length of \( AB \) if \( QN = 5\sqrt{2} \).
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Context:
A sequence of numbers is defined by $D_0=0$, $D_1=0$, $D_2=1$ and $D_n=D_{n-1}+D_{n-3}$ for $n\geq3$. What are the parities (evenness or oddness) of the triple of numbers $(D_{2021}, D_{2022}, D_{2023})$, where E denotes even and O denotes odd?
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Context:
Given a set \( \Omega_3 = \{1, 2, 3\} \), a collection of subsets \( A_1, A_2, \ldots, A_r \) is called a covering of \( \Omega_3 \) if \( \bigcup_{i=1}^r A_i = \Omega_3 \), \( \emptyset \neq A_i \subset \Omega_3 \), and \( A_i \neq A_j \) for all \( i \neq j \). Denote by \( C_3 \) the count of different coverings of \( \Omega_3 \). Given that \( C_1 = 1 \) and \( C_2 = 5 \), determine the value of \( C_3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum number of moves required to transform one position of a Rubik's cube into another?
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Context:
Find the coefficient of \(x^{203}\) in the expansion of \((x-1)(x^2 - 2)(x^3-3)\dots(x^{20} - 20)\).
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Context:
Determine the number of countable models of ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice).
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Context:
Find the range of values for $p$ such that the inequality \(2r^2+(2p+2q-pq)r+2(p^2+q^2+pq)\ge0\) holds for all real numbers $q$ and $r$.
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Context:
What is the size of the largest subset $S'$ of $S = \{2^x 3^y 5^z : 0 \le x,y,z \le 4\}$ such that there are no distinct elements $p,q \in S'$ with $p \mid q$?
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Context:
设实数 $x_{1}, x_{2}, \cdots, x_{21}$ 满足 $0 \leqslant x_{i} \leqslant 1(1,2, \cdots, 21)$, 求 $\sum_{i=1}^{21} \sum_{k=1}^{21}\left|x_{i}-x_{k}\right|$ 的最大值。
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Context:
Triangle  $ABC$  has  $AB=25$ ,  $AC=29$ , and  $BC=36$ .  Additionally,  $\Omega$  and  $\omega$  are the circumcircle and incircle of  $\triangle ABC$ .  Point  $D$  is situated on  $\Omega$  such that  $AD$  is a diameter of  $\Omega$ , and line  $AD$  intersects  $\omega$  in two distinct points  $X$  and  $Y$ .  Compute  $XY^2$ .

*Proposed by David Altizio*
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Context:
You are given two line segments of length \(2^{n}\) for each integer \(0 \leq n \leq 10\). How many distinct nondegenerate triangles can you form with three of the segments? Two triangles are considered distinct if they are not congruent.
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Context:
Find the number of distinct numbers in the list
\[\left\lfloor \frac{1^2}{1000} \right\rfloor, \ \left\lfloor \frac{2^2}{1000} \right\rfloor, \ \left\lfloor \frac{3^2}{1000} \right\rfloor, \ \dots, \ \left\lfloor \frac{1000^2}{1000} \right\rfloor.\]
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Context:
Brad wanted to set up a lemonade stand to earn some money. He decided to sell lemonade in three different sizes: small, medium, and large. The cost of ingredients and the prices for each size varied. He also considered stand setup costs and the cost of advertising.

Every gallon of lemonade would yield 16 small glasses, 10 medium glasses, or 6 large glasses. Brad estimated the following costs for each size:
- Small: $2.00 per gallon
- Medium: $3.50 per gallon
- Large: $5.00 per gallon

He also decided to charge the following prices for each size:
- Small: $1.00 per glass
- Medium: $1.75 per glass
- Large: $2.50 per glass

Brad made 2 gallons of lemonade for each size, which resulted in:
- 32 small glasses
- 20 medium glasses
- 12 large glasses

During the sales process, several size-specific challenges arose:
- Brad drank 4 small glasses himself at the stand
- His best friend bought 3 medium glasses, but one glass accidentally spilled
- Brad sold all the large glasses except for 2, due to their demand

Furthermore, the expenses involved were:
- Stand setup cost: $15.00
- Advertisement cost: $10.00

Taking into account the various expenses, glasses sold, and unsold glasses of lemonade, calculate the net profit Brad earned from his lemonade stand.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest natural number written in the decimal system with the product of the digits equal to  $10! = 1 \cdot 2  \cdot 3\cdot  ... \cdot9\cdot10$ .
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Context:
7. Suppose that $x, y$, and $z$ are complex numbers of equal magnitude that satisfy
$$
x+y+z=-\frac{\sqrt{3}}{2}-i \sqrt{5}
$$
and
$$
x y z=\sqrt{3}+i \sqrt{5} \text {. }
$$

If $x=x_{1}+i x_{2}, y=y_{1}+i y_{2}$, and $z=z_{1}+i z_{2}$ for real $x_{1}, x_{2}, y_{1}, y_{2}, z_{1}$, and $z_{2}$, then
$$
\left(x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2}\right)^{2}
$$
can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.
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Context:
Given the function $f(x)=\log_{2}(x^{2}-ax-3a)$, if it is a decreasing function on the interval $(-\infty, -2]$, then the range of values for $a$ is.
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Context:
Tim and Allen are playing a match of tenus. In a match of tenus, the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability $3 / 4$, and in the even-numbered games, Allen wins with probability $3 / 4$. What is the expected number of games in a match?
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Context:
$甲、乙、丙三位同学进行羽毛球比赛，约定赛制如下：累计负两场者被淘汰；比赛前抽签决定首先比赛的两人，另一人轮空；每场比赛的胜者与轮空者进行下一场比赛，负者下一场轮空，直至有一人被淘汰；当一人被淘汰后，剩余的两人继续比赛，直至其中一人被淘汰，另一人最终获胜，比赛结束.经抽签，甲、乙首先比赛，丙轮空.设每场比赛双方获胜的概率都为  \frac{1}{2}  .$
求丙最终获胜的概率.
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Context:
$5$ people stand in a line facing one direction. In every round, the person at the front moves randomly to any position in the line, including the front or the end. Suppose that $\frac{m}{n}$ is the expected number of rounds needed for the last person of the initial line to appear at the front of the line, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
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Context:
Compute the dimension of the spline space \(\mathcal{S}_\Delta^{3,2}\) on the interval \([a,b]\) with \(n\) subintervals, where \(\Delta\) denotes the grid on \([a,b]\).
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Context:
Let $a$ and $b$ be real numbers such that $a + b \ne 0$. Solve the equation
$$\frac{1}{(x + a)^2 - b^2} +\frac{1}{(x +b)^2 - a^2}=\frac{1}{x^2 -(a + b)^2}+\frac{1}{x^2-(a -b)^2}$$
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Context:
A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_n$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10} = .4$ and $a_n\le.4$ for all $n$ such that $1\le n\le9$ is given to be $p^aq^br/\left(s^c\right)$ where $p$, $q$, $r$, and $s$ are primes, and $a$, $b$, and $c$ are positive integers. Find $\left(p+q+r+s\right)\left(a+b+c\right)$.
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Context:
A function \(g\) is defined by \(g(w)=(3-2i)w^2+c w+d\) for all complex numbers \(w\), where \(c\) and \(d\) are complex numbers. If \(g(2)\) and \(g(-i)\) are both purely imaginary, find the minimum value of \(|c|+|d|\).
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Context:
Determine the order of the function \( f(n,k) = \sum_{i=1}^{n-1}(n-i)(k-i)\log(k-i) \) in terms of Big O notation, given that \( n < k \).
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Context:
The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5{.}$ The numbers in positions $(5, 5), \,(2,4),\,(4,3),$ and $(3, 1)$ are $0, 48, 16,$ and $12{,}$ respectively. What number is in position $(1, 2)?$
\[ \begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}\]
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Context:
Determine the Krull dimension of $\mathbb{C}[x,y,z]/I$ where $I=(x^2-yz,xz-x)$ using the fact that $p_1=(x,y)$, $p_2=(x,z)$, and $p_3=(x^2-y,z-1)$ are prime minimal over $I$.
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Context:
In a biregular graph with 40 red vertices each of degree 7 and 30 blue vertices each of degree 9, determine the minimum number of additional blue vertices required so that the minimum distance between any two distinct red vertices is 2.
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Context:
Evaluate the integral \( \int_0^\infty \frac{\sqrt{x}}{x^2+4} \, dx \).
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Context:
Let \( \mathcal{A} = A_0A_1A_2A_3 \cdots A_{2013}A_{2014} \) be a regular 2014-simplex, meaning the 2015 vertices of \( \mathcal{A} \) lie in 2014-dimensional Euclidean space and there exists a constant \( c > 0 \) such that \( A_iA_j = c \) for any \( 0 \le i < j \le 2014 \). Let \( O = (0,0,0,\dots,0) \), \( A_0 = (1,0,0,\dots,0) \), and suppose \( A_iO \) has length 1 for \( i=0,1,\dots,2014 \). Set \( P=(20,14,20,14,\dots,20,14) \). Find the remainder when \[PA_0^2 + PA_1^2 + \dots + PA_{2014}^2 \] is divided by \( 10^6 \).
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Context:
What is the maximum number of sides that the projection of a convex polyhedron with $n$ faces can have?
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Context:
Determine the range of real values for \( n \) such that the integral \( I(n) = \int^{\infty}_{0} \frac{\ln(x)}{x^n(1+x)}\, dx \) converges.
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Context:
In circle $k$ with center $O$, points $A, B, C, D$ are consecutive points on the circumference such that $\angle AOB = \angle BOC = \angle COD = \alpha < 60^\circ$. Let $E$ be the projection of $D$ onto the diameter $AO$, and $F$ be the trisection point of segment $DE$ closer to $E$. If $G$ is the intersection point of $AO$ and $BF$, find the measure of angle $OGD$ as $\alpha$ approaches $60^\circ$. \( \alpha \rightarrow 60^\circ \).
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Context:
Determine the structure of the factor ring \( \mathbb{Z}_3[x]/(x^3+2x+1) \).
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Context:
In triangle $ABC$, the angle at $C$ is $30^o$, side $BC$ has length $4$, and side $AC$ has length $5$. Let $ P$ be the point such that triangle $ABP$ is equilateral and non-overlapping with triangle $ABC$. Find the distance from $C$ to $ P$.
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Context:
Find the number of solutions in natural numbers for the equation \(\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1\).
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Context:
9. (3 points) Using the digits $0, 1, 2, 3, 4$, the number of even numbers that can be formed without repeating any digit is $\qquad$.
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Context:
Let  $a=256$ . Find the unique real number  $x>a^2$  such that
\[\log_a \log_a \log_a x = \log_{a^2} \log_{a^2} \log_{a^2} x.\]
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Context:
已知过点 $(0,1)$ 的直线 $l$ 与曲线 $C: y=x+\frac{1}{x}(x>0)$ 交于两个不同点 $M$ 与 $N$, 求曲线 $C$ 在点 $M 、 N$ 处切线的交点轨迹。请给出该轨迹线段两个端点坐标的和，该和为\frac{m}{n}形式，请给出m+n的值。
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Context:
A pentagon $ABCDE$ is circumscribed about a circle. The angles at the vertices $A{}$, $C{}$ and $E{}$ of the pentagon are equal to $100^\circ$. Find the measure of the angle $\angle ACE$.
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Context:
Evaluate the integral: $$ \frac{\exp\left(-\frac{N}{2}\right)}{2\pi N} \iint \exp\left(-\left( \frac{x^2-2xy-2Ny+2y^2}{2N}\right)\right) \,dx\,dy $$ where $N > 0$. Provide your answer in terms of $N$.
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Context:
Determine the shortest possible length of an algorithm that generates a harlequin pattern on a Rubik's cube. The pattern should use at most 6 colors, with no more than 2 facelets of the same color, and no adjacent facelets of the same color.
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Context:
Find the rightmost nonzero digit of $ \frac{100!}{5^{20}}$ (here $ n!=1\cdot2\cdot3\cdot\ldots\cdot
n$).
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Context:
Find the values of \( x \) for which the distance between the vector \( A = (1, 1, 1) \) and the subspace \( L \) spanned by the vectors \( (1, 0, -1) \) and \( (3, 5, x) \) in \( \mathbb{R}^3 \) is maximal.
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Context:
Evaluate the infinite sum: $$\sum_{n=0}^\infty \binom{2n+2}{n}\left(\frac{2}{3}\right)^n\left(\frac{1}{3}\right)^{n+2}$$
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Context:
How many of the first $500$ positive integers can be expressed in the form
\[\lfloor 3x \rfloor + \lfloor 6x \rfloor + \lfloor 9x \rfloor + \lfloor 12x \rfloor\]
where \( x \) is a real number?
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Context:
An $n$-digit positive integer is cute if its $n$ digits are an arrangement of the set $\{1,2,\ldots,n\}$ and its first $k$ digits form an integer that is divisible by $k$, for $k = 1,2,\ldots,n$. For example 321 is a cute 3-digit integer because 1 divides 3, 2 divides 32, and 3 divides 321. How many cute 6-digit integers are there?
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Context:
The lateral face of a regular quadrangular pyramid forms a $45^{\circ}$ angle with the base plane. Find the angle between the opposite lateral faces.
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Context:
Let $\triangle A_1B_1C_1$ be an equilateral triangle with an area of $60$. Chloe constructs a new triangle $\triangle A_2B_2C_2$ as follows:

- She flips a coin.
  - If it lands on heads, she constructs point $A_2$ such that $B_1$ is the midpoint of $\overline{A_2C_1}$.
  - If it lands on tails, she constructs $A_2$ such that $C_1$ is the midpoint of $\overline{A_2B_1}$.

- She performs similar operations for $B_2$ and $C_2$.

What is the expected value of the area of $\triangle A_2B_2C_2$?
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Context:
Assume that $X$ has five path components. If $CX$ is the cone on $X$, determine the relative homology group $H_1(CX,X)$. Provide your answer in terms of an isomorphism to a known group.
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Context:
Let $A B C$ be a triangle with $A B=A C=\frac{25}{14} B C$. Let $M$ denote the midpoint of $\overline{B C}$ and let $X$ and $Y$ denote the projections of $M$ onto $\overline{A B}$ and $\overline{A C}$, respectively. If the areas of triangle $A B C$ and quadrilateral $A X M Y$ are both positive integers, find the minimum possible sum of these areas.
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Context:
Find the original function \( f(x, y) \) given its gradient components: \( \frac{\partial f}{\partial x} = 2xy - y \) and \( \frac{\partial f}{\partial y} = x^2 - x \).
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Context:
Three metal cubes with edges of 3 dm, 4 dm, and 5 dm, along with a metal cylinder of height 7 dm and a base diameter of 4 dm, and a metal rectangular prism of dimensions 2 dm x 3 dm x 6 dm, are melted and formed into a single cube. Find the edge of the new cube.
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Context:
Determine the values of $x$ for which the expression $(A^TA)^x$ is well-defined, where $A$ is a real matrix.
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Context:
Let $x_1, x_2, x_3$ be the solutions to $(x - 13)(x - 33)(x - 37) = 1337$. Find the value of
$$\sum_{i=1}^3 \left[(x_i - 13)^3 + (x_i - 33)^3 + (x_i - 37)^3\right].$$
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Context:
Compute the limit \( \lim_{n \rightarrow \infty}{ \int_0^{\infty}\left(\frac{nx^{1/n}}{ne^x+\sin(nx)}\right)dx} \) using the relation between Riemann and Lebesgue integrals and Lebesgue’s Dominated Convergence Theorem.
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Context:
Evaluate the limit: \[ \lim_{n \to \infty} 3^{3n-1}\left(\sqrt[4]{27^n+5}-\sqrt[4]{27^n-5}\right) \]
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Context:
Determine the number of distinct values that can be obtained by inserting the operators $+, -, \times, \div$ and parentheses into the sequence $2, 2, 2, \ldots, 2$ (with $n$ occurrences of the number 2) and evaluating the resulting expression.
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Context:
Consider a hash function \( H(x) \rightarrow \{ 0,1 \}^{160} \) that maps inputs to 160-bit strings. What is the minimum number of attempts required to find a collision, i.e., two different inputs \( x_1 \) and \( x_2 \) such that \( H(x_1) = H(x_2) \)?
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Context:
Determine the complexity of an algorithm that tests whether a given polynomial is a square in a finite field with small characteristic.
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Context:
Triangle $ABC$ has $\overline{AB} = \overline{AC} = 20$ and $\overline{BC} = 15$. Let $D$ be the point in $\triangle ABC$ such that $\triangle ADB \sim \triangle BDC$. Let $l$ be a line through $A$ and let $BD$ and $CD$ intersect $l$ at $P$ and $Q$, respectively. Let the circumcircles of $\triangle BDQ$ and $\triangle CDP$ intersect at $X$. The area of the locus of $X$ as $l$ varies can be expressed in the form $\tfrac{p}{q}\pi$ for positive coprime integers $p$ and $q$. What is $p + q$?
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Context:
Given that the weight of each gum ball (in centigrams) follows a gamma distribution with parameters \( \alpha = 25 \) and \( \beta = 2 \), calculate the probability that the total weight of 100 gum balls exceeds 52 grams.
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Context:
给定 2003 个集合,每个集合都恰含 44 个元素, 并且每两个集合恰有一个公共元. 试求这 2003 个集合的并集中元素的个数.
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Context:
Evaluate the limit: $$\lim\limits_{n \to \infty} \frac{n^{1/n^2+x}−1}{\ln(n)\cdot \frac{1}{n^2}}$$
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Context:
Calculate the gradient with respect to the matrix $X$ of the function: $$ \log \mathrm{det}\, X^{-1} $$ where $X$ is a positive definite matrix.
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Context:
Find the smallest integer value of \( n \) such that the remainder \( R_{n}f(x) \) of the Taylor polynomial of degree \( n \) for \( \sin(x) \) around \( a=0 \) satisfies \( R_{n}f(x) \leq 0.01 \) for all \( x \in [0, 1/3] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the positive divisors of a number $n$ are $d_1, d_2, \ldots, d_r$ such that $1 = d_1 < d_2 < \ldots < d_r = n$, and that $(d_7)^2 + (d_{15})^2 = (d_{16})^2$, find all possible values of $d_{17}$.
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Context:
Suppose we have a Bernoulli random variable $\Theta$ representing two hypotheses, where the probability of $\Theta = 1$ is $p$. Under hypothesis $\Theta = 0$, the random variable $X$ follows a uniform distribution on $[0,1]$. For the alternative hypothesis $\Theta = 1$, the probability density function (PDF) of $X$ is $f_{X|\Theta}(x|1) = 2x$ if $0 \leq x \leq 1$, and $0$ otherwise. When deciding between the two hypotheses based on the maximum a posteriori (MAP) rule with observed data $X = x$, find the probability of error as a function of $p$. Specifically, evaluate $P(\text{error})$ when $p \geq \frac{1}{3}$.
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Context:
Calculate the probability $P(\max(X_1, X_2, \ldots, X_n) > X_{n+1})$ where $X_1, X_2, \ldots, X_n, X_{n+1}$ are independently chosen from the same distribution.
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Context:
$\mathrm{Let~}S=\{1,2,...,8\}.\text{ How many ways are there to select two disjoint subsets of }S?$
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Context:
Find a non-trivial lower bound for the conditional expectation \(E(X \mid X \neq 0)\), where \(X\) is the number of people who get their own hat back at a party with \(N\) people, and each person grabs a hat at random when leaving.
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Context:
Determine the number of homomorphisms from the group \( \mathbb{Z}/3 \times \mathbb{Z}/3 \) to the group \( A_4 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $U$ and $W$ be subspaces of polynomials over $\mathbb{R}$ defined as follows:

\[ U = \text{Span}(t^3 + 4t^2 - t + 3, t^3 + 5t^2 + 5, 3t^3 + 10t^2 -5t + 5) \]
\[ W = \text{Span}(t^3 + 4t^2 + 6, t^3 + 2t^2 - t + 5, 2t^3 + 2t^2 -3t + 9) \]

Determine the dimension of the intersection $\dim(U \cap W)$. Provide your answer as a single integer.
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Context:
Given 155 birds perched on a circle \( C \), where birds \( P_{i} \) and \( P_{j} \) are considered mutually visible if \(\overparen{P_{i} P_{j}} \leqslant 10^{\circ}\), find the minimum number of pairs of mutually visible birds.
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Context:
Evaluate the line integral \( \int_C F \, dr \) using Green's Theorem, where the vector field is given by \( F=\langle e^{\sin{x}}+y^2, x^2+y^2 \rangle \) and the curve \( C \) is oriented as follows: it starts at point \((1,2)\), moves along a line segment to \((1,4)\), then to \((2,4)\), and finally back to \((1,2)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a sequence of positive real numbers \( \{y_n\} \) such that \( \frac{y_n}{\sqrt{n}} \to \infty \) as \( n \to \infty \), but \( \frac{y_n}{\sqrt{n^\beta}} \to 0 \) for any \( \beta > 1 \).
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Context:
Find all positive integer solutions to the equation \((n+1)^n = 2n^k + 3n + 1\).
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Context:
Evaluate\[  \frac{1}{729} \sum_{a=1}^{9} \sum_{b=1}^9 \sum_{c=1}^9 \left( abc+ab+bc+ca+a+b+c \right). \]
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Context:
Find the coefficient of $x^2$ in the expansion of $(1+2x-3x^2+4x^3)^{-1/2}$.
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Context:
Find the probability distribution function of the random variable \( z = |x - y| \), where \( x \) and \( y \) are independent random variables uniformly distributed between 0 and 1.
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Context:
What is the expected number of non-empty columns (or sampled dishes) $\Lambda_t$ in an Indian Buffet Process with $T$ customers and concentration parameter $\alpha$?
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Context:
Let $A$ be an integer and $A = 2 + 20 + 201 + 2016 + 20162 + ... + \underbrace{20162016...2016}_{40\,\, digits}$
Find the last seven digits of $A$, in order from millions to units.
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Context:
Determine the largest possible determinant of a matrix whose elements are either $\pm1$ or $0$. Provide your answer as a single number or expression.
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Context:
Calculate the product modulo 1234 of all numbers less than 1234 that are relatively prime to 1234, excluding those numbers that are their own multiplicative inverses.
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Context:
In a competition with 37 contestants, each contestant has a non-negative integer score. The sum of the scores of any 17 contestants is at least 45, and the total score of all 37 contestants is 150. What is the maximum score that a single contestant can achieve?
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Context:
Evaluate the integral \( \int_{-\infty}^{\infty} e^{-\pi(x - i\alpha)^2}dx \) where \( \alpha \) is a real constant.
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Context:
How many permutations of the string "ABCDEF" exist such that no two adjacent characters in the original string are adjacent in the permutation?
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Context:
In a convex pentagon $ABCDE$ the areas of the triangles $ABC, ABD, ACD$ and $ADE$ are all equal to the same value x. What is the area of the triangle $BCE$?
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Context:
In a graph where each vertex has a degree $d$, determine the minimum length of a path that can be constructed using a greedy algorithm.
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Context:
Determine the number of distinct colorings of the vertices of a cycle graph $C_{8}$ using 10 colors, where each color can be used multiple times. Two colorings are considered the same if one can be transformed into the other by an isometry of the octagon. Use Burnside's Lemma to find the number of distinct colorings.
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Context:
Find a primitive root of 338.
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Context:
Determine the values of $n \geq 2$ for which there exists a permutation $\sigma$ in the alternating group $A_n$ such that the order of $\sigma$ is greater than $n$.
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Context:
Consider two Gaussian distributions, where one has a variance of zero, effectively making it a unit impulse function. Determine the Overlapping Coefficient between these two distributions.
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Context:
Let $Y$ be a connected covering space of a bouquet of 5 circles with a degree of 5. Determine the rank of the free group $\pi_{1}(Y)$.
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Context:
Using the Cauchy-Schwarz inequality, determine the minimum value of \( \int_0^1 f^2(x) \, \mathrm{d}x \) for a non-negative Riemann integrable function \( f(x) \) satisfying \( \int_0^1 f(x) \, \mathrm{d}x = 1 \) and \( \int_0^1 x f(x) \, \mathrm{d}x = \frac{1}{6} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( f(x) = x^3 \), find the value of \( \frac{\partial^2 f}{\partial x \partial f} \).
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Context:
On a street with 14 houses, the odd-numbered houses are on one side and the even-numbered houses are on the opposite side. Families on one side have surnames that are colors, and on the other side, surnames indicate professions. Szabó and Fazekas live opposite Zöld and Fehér, respectively, who are neighbors of Fekete. Kovács is the father-in-law of Lakatos. Lakatos lives in a higher-numbered house than Barna, and the sum of their house numbers equals the sum of Fehér and Fazekas's house numbers. Kádárné's house number is twice that of her sister, Kalaposné. Sárga lives opposite Pék. Bordóné's house number is two-digit, and she lives opposite her sister, Kádárné. What is the house number of Mr. Szürke?
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Context:
Find the maximum number of different permutations of the projections of 8 points onto all possible directed lines on a plane. Let this maximum number be denoted as \( N_8 \). What is the value of \( N_8 \)?
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Context:
Find the limit of the function \( \frac{\ln(\frac{1+x^2}{1-x^2}) - \arctan(2x^3) + 2x\sin^2(x)}{x\exp(-\frac{1}{x}) - 2x^2 + \ln(\frac{1+x^2}{1-x^2})} \) as \( x \) approaches 0 from the positive side.
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Context:
Evaluate the integral \( \int\int\int_E z \, dV \), where \( E \) is the region above the paraboloid \( z = x^2 + y^2 \) and below the plane \( z = 6y \). Round your answer to the nearest hundredth.
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Context:
Evaluate the integral \( \int_0^1 \ln(\sin(2\pi z)) \, \mathrm{d}z \) using the properties of the Gamma function and the identity \( \sin(\pi z) = \pi z \prod_{k=1}^\infty \left(1 - \frac{z^2}{k^2}\right) \).
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Context:
Let $X$ be a random variable with probability density function \( f(x)=\frac{1}{2} e^{-|x|},\; -\infty<x<\infty. \) Calculate the expected value of \( \lfloor X \rfloor \), where \( \lfloor X \rfloor \) denotes the greatest integer less than or equal to \( X \).
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Context:
Find the spectrum of the operator \( A: C[0,1] \rightarrow C[0,1] \) defined by \( (Af)(x) = f(x) + \int_0^x f(t)dt \).
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Context:
Consider the set \( S = \{1, 2, \ldots, 100\} \). A partition of \( S \) is defined as \( S_1, S_2, \ldots, S_n \) for some \( n \), where each \( S_i \) is nonempty, pairwise disjoint, and \( S = \bigcup_{i=1}^n S_i \). Let \( a_i \) be the average of the elements in the set \( S_i \). The score of this partition is given by \( \frac{a_1 + a_2 + \ldots + a_n}{n} \). Determine the minimum possible score among all possible partitions of \( S \).
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Context:
A short-sighted rook is a rook that beats all squares in the same column and in the same row for which he can not go more than
 60-steps.
 What is the maximal amount of short-sighted rooks that don’t beat each other that can be put on a \( {100} \times  {100} \) chessboard.
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Context:
Determine the limit of the sequence \( f_n(a) = a^{f_{n-1}(a)} \) as \( n \) approaches infinity, where \( f_0(a) = a \) and \( a = 1.1 \).
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Context:
Find an upper bound for $l_n$, the length of the curve $y=x^n$ in the interval $[0,1] \times [0,1]$, using the function $f_n(x) = \sqrt{1+n^2(1-x)^{2n-2}}$ and the approximation provided in the original answer.
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Context:
Determine the greatest positive real number \( M \) such that for every positive real sequence \((a_n)\) and for every real number \( m < M \), there exists an index \( n \geq 1 \) satisfying the inequality \( a_1 + a_2 + \cdots + a_n + a_{n+1} > m a_n \).
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Context:
Observing the following expressions: $$1+ \frac {1}{2^{2}} < \frac {3}{2}, 1+ \frac {1}{2^{2}}+ \frac {1}{3^{2}} < \frac {5}{3}, 1+ \frac {1}{2^{2}}+ \frac {1}{3^{2}}+ \frac {1}{4^{2}} < \frac {7}{4},…$$, what could be the conjectured conclusion when $$n \in \mathbb{N}$$ and $$n \geqslant 2$$?
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Context:
If $f(x)$ is a differentiable function such that \( f(x) = \left(1 + \frac{x^3}{3}\right) + \int_0^x e^{-t} f(x-t) \, dt \), find the value of \( \int_0^1 f(x) \, dx \).
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Context:
数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=1, \frac{a_{n+1}-a_{n}}{a_{n}}=\frac{a_{n+2}-a_{n+1}}{a_{n+2}}\left(n \in N^{*}\right)$, 若 $a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{6} a_{7}=3$, 则 $a_{2024}=$ $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
In a Poisson process with parameter \( \lambda \), let \( S_1, S_2, \ldots \) be the arrival times. Given the time of the \( n \)-th arrival, find the expected value \( E(S_1|S_n) \).
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Context:
Evaluate the contour integral \( \oint_{\gamma} \frac{f'(z)}{f(z)} dz \), where \( f(z) = f(0) \prod_{\rho} \left(1 - \frac{z}{\rho}\right) \) and \( \rho \) are the roots of \( f(z) \). The contour \( \gamma \) is a rectangle with vertices at \( a \), \( a + b \), \( a + b + it \), and \( a + it \), traversed counterclockwise. Assume all roots of \( f(z) \) lie within the rectangle. Determine the value of the integral.
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Context:
In a convex equilateral hexagon \(A B C D E F\), lines \(B C\), \(A D\), and \(E F\) are parallel. Let \(H\) be the orthocenter of triangle \(A B D\). Given that the smallest interior angle of the hexagon is 4 degrees, find the smallest angle of triangle \(H A D\) in degrees.
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Context:
Let $k:\Bbb R\to\Bbb R$ be a twice differentiable function such that $k(a)=k(e)=0$, $k(b)=k(d)=2$, and $k(c)=1$, where $a<b<c<d<e$. Determine the minimum number of solutions to the equation $(k'(x))^2+k(x)k''(x)=0$ in the interval $[a,e]$. \( \text{Provide your answer as a single integer.} \)
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Context:
Determine the average length of a cycle in a permutation of the set \( \{1, 2, 3, \dots, n\} \).
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Context:
In a quadrilateral pyramid with a rhombus base $ABCD$, where $\angle BAD = 60^\circ$, it is given that $SA = SC$ and $SD = SB = AB$. A point $E$ is located on edge $DC$ such that the area of triangle $BSE$ is minimized among all sections of the pyramid containing segment $BS$ and intersecting segment $DC$. Determine the ratio $DE:EC$. \( \text{Express your answer as a simplified fraction.} \)
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Context:
Determine the condition on $a(t)$ such that the stochastic process $X(t)$, defined by the stochastic differential equation $$ dX = a(t)Xdt + b(t)XdW(t), $$ is a $P$-martingale, where $b(t)$ is non-zero almost surely and $W(t)$ is a Wiener process.
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Context:
Let $X$ and $Y$ be independently distributed central chi-squared random variables with degrees of freedom $m(\geq 3)$ and $n(\geq 3)$, respectively. Given that $E \left(\frac{X}{Y}\right)=3$ and $m+n=14$, find $E \left(\frac{Y}{X}\right)$. Provide your answer as a single number.
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Context:
Given an $m \times n$ board where $m,n > 2$ are even integers, each cell is colored either black or white. The Guesser can ask the Oracle whether two adjacent cells (sharing an edge) have the same color. What is the minimum number of inquiries the Guesser needs to make to determine the parity of the number of adjacent cell-pairs with different colors?
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Context:
 Let $f_n(x) = ln(1 + x^{2^n}+ x^{2^{n+1}}+ x^{3\cdot 2^n})$. Compute $\sum^{\infty}_{k=0} f_{2k} \left( \frac12 \right)$.
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Context:
A driver travels from point A to point B, which are 100 km apart. Initially, the driver travels at a constant speed. When the navigator indicates 30 minutes of travel remain, the driver reduces speed by 10 km/h. When the navigator indicates 20 minutes of travel remain, the driver reduces speed by another 10 km/h. The car travels 5 minutes longer at the reduced speed than it would have at the twice-reduced speed. Determine the initial speed of the car.
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Context:
Determine the range of \( t \) such that for any \( a \in (0, 4) \), there exists an \( x_0 \in [0, 2] \) satisfying \( t \leq |f(x_0)| \), where \( f(x) = x^2 - ax + a - 1 \).
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Context:
Determine the smallest integer $n$ such that the approximation of $e^x$ by the series $s_n(x) = \sum_{k=0}^{n} \frac{x^k}{k!}$ satisfies the inequality \(|e^x - s_n(x)| \leq \frac{|e^x|}{10^{16}}\) for all $x \in \left(-\frac{1}{2}, \frac{1}{2}\right)$. Provide your answer as the smallest $n$ that meets this condition.
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Context:
Determine the number of different ring homomorphisms from \(\mathbb{Z} \times \mathbb{Z}\) to \(\mathbb{Z} \times \mathbb{Z}\), given that a ring homomorphism must map identities to identities.
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Context:
Given a simple polygon with \( n \) vertices, where each vertex is exactly visible to four other vertices, determine all possible values of the positive integer \( n \).
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Context:
Evaluate the integral: $$\int^{1/2}_0 \int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx $$
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Context:
Let $a$ be a positive real number, $n$ a positive integer, and define the *power tower* $a\uparrow n$ recursively with $a\uparrow 1=a$, and $a\uparrow(i+1)=a^{a\uparrow i}$ for $i=1,2,3,\ldots$. For each positive integer $k$, let $x_k$ denote the unique positive real number solution of the equation $x\uparrow k=10\uparrow (k+1)$. Determine which is larger: $x_{42}$ or $x_{43}$?
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Context:
Evaluate the integral $$\int_0^{\infty} \frac{\cos(2\alpha q)}{\sqrt{1+q^2}} \, dq.$$
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Context:
Seven fishermen stand in a circle, each with a unique integer exaggeration factor. When asked, "How many fish did your left neighbor catch?", the responses were $12, 12, 20, 24, 32, 42,$ and $56$. When asked, "How many fish did your right neighbor catch?", six of the fishermen responded $12, 14, 18, 32, 48,$ and $70$. What did the seventh fisherman respond?
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Context:
Let $G$ be a finite group and let $x_1, \ldots, x_n$ be an enumeration of its elements. Consider the matrix $(a_{ij})_{1 \le i,j \le n}$, where $a_{ij} = 0$ if $x_i x_j^{-1} = x_j x_i^{-1}$, and $a_{ij} = 1$ otherwise. Determine the parity of the integer $\det(a_{ij})$. \(\text{(Even or Odd)}\)
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Context:
Find the probability density function (pdf) of the random variable \( Y = \frac{T_1}{T_3} \), where \( \{N(t): t \geq 0\} \) is a Poisson process with rate 1, and \( T_1, T_2, \ldots \) denote the event times.
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Context:
Find the constant \( c \) such that the random variable \( Y = \frac{c(X_1 + X_2)}{\sqrt{(X_3)^2 + (X_4)^2}} \) has a t-distribution.
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Context:
Determine the probability that the interval \( \left[\frac{Y}{124.34},\frac{Y}{77.93}\right] \) contains \( \sigma^2 \), where \( Y = \sum_{i=1}^{100}(X_i-\mu)^2 \) and \( \mu \) is known for a normally distributed population of 100 individuals.
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Context:
Determine the values of $s \in \mathbb{R}$ for which the series \[ I_s = \sum_{n=1}^{\infty} \left(\frac{a_1^{1/s} + a_2^{1/s} + \cdots + a_n^{1/s}}{n}\right)^s \] converges.
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Context:
Find an equivalent expression for the remainder \( R_n = \sum_{k=n+1}^{\infty}{\frac{1}{k^3}} \) of the converging series \( \sum_{n\geqslant1}{\frac{1}{n^3}} \).
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Context:
Determine the cardinality of the set \(E(a)\), where \(E(a) = \{(r_1, r_2, \ldots, r_n, \ldots) \mid r_n \in \mathbb{Q} \cap [0,1], r_n \leq r_{n+1}, \forall n \geq 1, \sum_{n=1}^\infty \frac{r_n}{2^n} = a\}\) for any \(a \in (0,1)\). What is \(|E(a)|\)?
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Context:
Determine the number of distinct squares that can be formed using 8 congruent triangular tiles, each of which can be colored either white or black. Two squares are considered the same if one can be transformed into the other through rotation or reflection.
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