Context:
Evaluate the integral \( \int_0^1 \ln(1-x^{1/n}) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For positive integer \( n \), consider the ordinary differential equation
\[ y'' + \frac{1}{x}y' + \left(1 - \frac{n^2}{x^2}\right)y = 0 \]
which has a solution in the form of the Bessel function of order \( n \), \( J_n(x) = x^n \sum_{k=0}^{\infty} a_kx^k \). Determine the coefficient \( a_1 \) by substituting the series representation of the Bessel function into the differential equation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( g: \mathbb{R}^3 \to \mathbb{R}^3 \) be defined by \( g(x,y,z) = (3x+4z, 2x-3z, x+3y) \) and let \( S = \{ (x,y,z) \in \mathbb{R}^3 : 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1 \} \). It is given that
\[ \iiint_{g(S)} (2x + y - 2z) \, dx \, dy \, dz = \alpha \iiint_S z \, dx \, dy \, dz. \]
Find the value of \( \alpha \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the interval on which the polynomial \( y = -\frac{3\pi}{2} + x - \frac{(2x-3\pi)^3}{48} \) best approximates the function \( y = \cos(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the box-counting dimension (Minkowski Dimension) of the set \( S = \left\{ \frac{1}{n^2} \mid n \in \mathbb{Z}^+ \right\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the quartic polynomial \(x^4 - 3x^3 + ... - 10 = 0\) with real coefficients, and two complex roots \(a + 2i\) and \(1 + bi\), where \(a\) and \(b\) are real numbers, determine the sum of the real roots of the polynomial.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Five bags, each of a distinct color (Blue, Green, Pink, Red, and Yellow), contain 5 balls of the same color as the bag. A ball is drawn from each bag and transferred to one of the other bags, ensuring that each bag still contains 5 balls at the end. Calculate the probability that, after all transfers, each bag contains 5 balls of its original color.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Convert the following limit to a Riemann sum and evaluate the resulting integral:
\[ \lim_{N \to \infty} \sum_{n=-N}^{N} \left( \frac{1}{N+in} + \frac{1}{N-in} \right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A fair $6$-sided die is repeatedly rolled until a $1, 4, 5$, or $6$ is rolled. What is the expected value of the product of all the rolls?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a unit segment [0, 1] that is colored randomly using two colors, white and black. What is the probability that a randomly chosen point on the segment is black after n steps?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the set \( A = \left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \in M(2, \mathbb{R}) \mid a, c \neq 0 \right\} \), given that \( A \) is an open subset of the differential manifold \( GL(2, \mathbb{R}) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute \(2^{2475} \bmod 9901\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $X$ and $Y$ are independent random variables, each following a normal distribution with mean $8$ and variance $10$, denoted as $X,Y\sim\mathcal{N}(8,10)$. Calculate the probability $\mathbb{P}(X\le 2Y)$. \(\boxed{?}\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a smooth projective variety $M$ of dimension $n$ over the complex numbers, where the cotangent bundle $T^*M$ forms a completely integrable system with $n$ independent regular functions $f_1, \ldots, f_n$ in involution. The moment map is given by $\mu: T^*M \to \mathbb{C}^n$, defined by $f_1, \ldots, f_n$. Let $Y \subset \mathbb{C}^n$ be the set of points $p \in \mathbb{C}^n$ such that the fiber $\mu^{-1}(p)$ is reducible. Determine the co-dimension of $Y$ and verify if it is always $\ge 2$. \( \text{Provide your answer in the form of a co-dimension value.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a line of $n$ bugs, each with a unique size, moving in the same direction. If a larger bug is behind a smaller bug, it will eat the smaller bug. What is the expected number of bugs remaining after a long enough time?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the angle of rotation that eliminates the cross term in the conic section defined by the equation \(5x^2 + 5y^2 - 6xy - 8 = 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n} \, dS \) where \( \mathbf{F}(x,y,z) = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), \( S \) is the plane surface whose boundary is the triangle with vertices at \((1,0,0), (0,1,0),\) and \((0,0,1)\), and \( \mathbf{n} \) is the unit normal to \( S \) having a nonnegative \( z \)-component. Use the vector representation \( \mathbf{r}(u,v) = (u+v)\mathbf{i} + (u-v)\mathbf{j} + (1-2u)\mathbf{k} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of integer solutions \((a, b)\) for the equation \( \frac{ab}{a+b} = 3600 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \frac{1}{2 \pi i} \int_{\partial B_1(0)} \frac{6z^{98}}{23z^{99}-2z^{81}+z^4-7}dz \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A server processes events from clients, where each client sends an event every second. The server processes an event in t seconds, where t < 1 second (e.g., 0.01 seconds). We want to serve 70,000 clients. The server can process up to n simultaneous requests, and a timeout occurs if this limit is exceeded. What should be the value of n so that a timeout occurs on less than 1% of requests?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Taylor Swift made the song $15$ in $2008$, and the song $22$ in $2012$. If she had continued at the same rate, in what year would her song $1989$ have come out?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the sum of distances $PX + PY$, where $P$ is a variable point on the line $2x + y + z = 1$. The fixed points are $X(1,1,1)$ and $Y(2,2,4)$. What is the minimum value of $PX + PY$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If Player 1 and Player 2 each independently sample from a Uniform(a, b) distribution, where a < b, what is the expected number of samples Player 2 must make to obtain a number higher than Player 1's sample?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a company with 100 shareholders, any group of 66 shareholders collectively owns at least 50% of the company's shares. What is the maximum percentage of shares that a single shareholder can own?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \iint_{D}\left(x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}\right)\mathrm{d}x\mathrm{d}y \) where \( f(x,y) \) is defined on \( D=\{(x,y)\mid x^2+y^2\le1\} \) and has continuous second-order partial derivatives in \( D \). Given that \( \frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}=\mathrm{e}^{-(x^2+y^2)} \), find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider three disjoint line segments of equal length labeled AB, CD, and EF. Determine the size of the group of symmetries of this configuration, where symmetries include exchanging line segments and flipping segments end for end.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Sidney Crosby is standing on the goal line, three feet outside one of the goal posts of a hockey goal that is six feet wide. How far should he move out perpendicular to the goal line to maximize the angle at which he can shoot at the goal?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Siva has the expression \( \frac{1}{2} \_ \frac{1}{4} \_ \frac{1}{8} \_ \frac{1}{16} \_ \frac{1}{32} \), where each blank is filled with either a plus or a minus based on a fair coin flip. He computes the value of the expression and repeats the process with new coin flips. What is the probability that the positive difference between his computed values is greater than \( \frac{1}{2} \)? If this probability can be expressed as \( \frac{a}{b} \) for relatively prime positive integers \( a \) and \( b \), find \( a + b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a group of 50 people, what is the minimum number of rotations required to seat them at tables of 5 so that each person meets every other person at least once?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit: \[ \lim_{n\to\infty}\left(\frac{\sin(2\sqrt{1})}{n\sqrt{1}\cos\sqrt{1}} + \cdots + \frac{\sin(2\sqrt{n})}{n\sqrt{n}\cos\sqrt{n}}\right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
设$x, y , z \in R ^ { + }$，求$\frac { 2 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } } { x y + \sqrt { 3 } y z }$的最小值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the isosceles trapezoid \(ABCD\), the bases \(AD\) and \(BC\) are related by the equation \(AD = (1 + \sqrt{15}) BC\). A circle is constructed with its center at point \(C\) and radius \(\frac{2}{3} BC\), which cuts a chord \(EF\) on the base \(AD\) of length \(\frac{\sqrt{7}}{3} BC\). Determine the ratio in which the circle divides side \(CD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of transitive $G$-sets of the symmetric group $S_3$ up to isomorphism.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Approximate the value of \( a = \frac{2.04^2}{\sqrt[3]{0.97 \sqrt[5]{1.02^2}}} \) using differential approximation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the rhombus $ABCD,$ the measure of $\angle{B}=40^{\circ}, E$ is the midpoint of $BC,$ and $F$ is the base of the perpendicular dropped from $A$ on $DE.$ Find the measure of $\angle{DFC}.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_A \frac{dz}{4z^2+1} \), where \( A = \{ z \in \mathbb{C} \mid |z-i|=1 \} \), in the negative direction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of the Ramsey number \( R(K_3, C_5) \), which is the smallest number of vertices \( n \) such that any graph on \( n \) vertices contains either a triangle \( K_3 \) or a cycle of length 5 \( C_5 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the dimension of the kernel of the linear transformation on the cyclic module \( M \), where \( M \cong \frac{D}{(p^k)} \) for a principal ideal domain \( D \), an irreducible element \( p \), and an integer \( k \geq 1 \). The transformation is defined as multiplication by \( p \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the value of the improper integral $ \int_0 ^ {\pi} \log (\sin (x)) dx$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
February 2, 2020, is an interesting "palindrome day" because the date written as 20200202 reads the same from left to right and right to left. What is the next "palindrome day" after this date?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an equilateral triangle with an area of 1, color parts of the triangle red, blue, and green such that each color forms exactly one connected region strictly inside the triangle, and no line parallel to any side contains points of multiple colors. Determine the maximum possible value of the minimum area among the red, blue, and green regions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest number of data points, \( n \), such that the kernel matrix \( K \) is guaranteed to be singular for any matrix \( X \in \{1, 2, \ldots, m\}^{d \times n} \) and any kernel function \( k : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define a sequence $\{a_{n}\}$ by
$$a_{0}=\frac{1}{2}, \quad a_{n+1}=a_{n}+\frac{a_{n}^{2}}{2012},(n=0, \quad 1, \quad 2, \quad \cdots),$$

find integer $k$ such that $a_{k}<1<a_{k+1}$.

(September 29, 2012, Hohhot)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of singular matrices of the form \(A=\begin{bmatrix} 1 & a & b\\ w&1 &c \\ w^2 &w & 1 \end{bmatrix}\) where \(a, b, c\) are elements of the set \(S=\{w, w^2\}\), and \(w\) is a complex cube root of unity with \(w \neq 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limiting value of \( \mathcal{P}_\infty \) as \( n \) approaches infinity, where \( \mathcal{P}_n \) is the ratio of twin primes \((p_1, p_2)\) with \( p_2 = p_1 + 2 \) and \( p_1 - 1 = a_1 \times b_1 \), \( p_2 + 1 = a_2 \times b_2 \), \( p_1 + 1 = p_2 - 1 = a \times b \) such that \( \min\{|b_1 - a_1|, |b - a|, |b_2 - a_2|\} \neq |b - a| \). Exclude the pair \((2, 3)\) from consideration.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the smallest non-abelian group that is not a complex reflection group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \(\iint_S (x, y, 3z) \, dS\) over the region \(S\), where \(S\) is the finite portion of the surface \(z = 4x^2 + y^2\) cut off by the plane \(z = 8x + 4y - 4\), and the normal to \(S\) points upwards.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the expression \(\frac{1}{(1-x)^4(1+2x^3+x^6)^4}\), simplify it and find the sum of the coefficients of \(x^{20}\) and \(x^{21}\) in the resulting expansion.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the degree of the minimal polynomial of $\cos \frac{\pi}{48}$ over $\mathbb{Q}$, given that it is a root of the $96$-th cyclotomic polynomial.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider 8 distinct boxes placed in a line, each containing between 1 and 8 red balls. The difference in the number of balls between any two adjacent boxes must be at most 1. Determine the number of distinct ways to distribute the balls among the boxes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of strongly monotonic increasing functions \(f:[n]\to \mathcal{P}([k])\) with respect to the subset relation, where \(n > k+1\). Use the pigeonhole principle to determine the value of \(a_{n,k}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many 13-card bridge hands, drawn from a standard 52-card deck, contain one or more quads (four cards of the same rank)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_R (y+2x^2)(y-x^2) \, dA \) where \( R \) is the region bounded by the curves \( xy=1 \), \( xy=2 \), \( y=x^2 \), and \( y=x^2-1 \) with \( x \) and \( y \) positive.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the simplest possible relation between the holomorphic functions $f$ and $g$ in a certain region of the complex plane, given that $$|f(z)|^2 + |f(z)| = |g(z)|^2 + |g(z)|.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the expression \( \sum_{k = n}^{5n} \binom{k-1}{n-1} \frac{1}{5^n} \left(\frac{4}{5}\right)^{k-n} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $X_1, X_2, \ldots, X_n$ are independent and identically distributed random variables, where $X_i > 0$ and $E[X_i] = \mu$, $\text{Var}(X_i) = \sigma^2$. If $m < n$, find $E\left(\frac{X_1 + X_2 + \cdots + X_m}{X_1 + X_2 + \cdots + X_n}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected number of picks required to end a game where you start by randomly selecting a number $K$ from the set $[0, N-1]$. The game ends if $K = 0$. If $K \neq 0$, you continue selecting numbers from the set $[0, K-1]$ until you eventually pick 0. What is this expected number of picks?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( k \) for which the matrix \( A = \begin{bmatrix} 1 & 1 & k \\ 1 & 1 & k \\ 1 & 1 & k \end{bmatrix} \) has an eigenbasis in \( \mathbb{R}^3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\gamma$ be a closed and continuously differentiable path in the upper half-plane $\{z \in \mathbb{C} : z = x + iy, x, y \in \mathbb{R}, y > 0\}$, excluding the point $i$. Determine the value of the contour integral:
\[ \frac{1}{2\pi i} \oint_\gamma \frac{2i}{z^2 + 1} \, dz \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A pyramid is inscribed in a cone such that the base of the pyramid is a quadrilateral with pairs of adjacent sides equal, and the angle between one pair of adjacent sides is \( \alpha \). Determine the ratio of the volume of the pyramid to the volume of the cone.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the conditions on \(m\) and \(n\) such that the integral \(\int_{0}^{\infty} \frac{x^{n}}{(1+x)^{m}}dx\) converges, where \(m,n \geq 0\) and \(m,n \in \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\rightarrow \infty}\frac{n!n^{x}}{x(x+1)...(x+n)} \] given that \( \text{Re}(x) > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
What is the minimum number of values needed to uniquely determine a line in three-dimensional space \(\mathbb{R}^3\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the expected value \( E[e^{vS}] \), where \( v \) and \( S \) are two independent random variables, each exponentially distributed with parameters \( \lambda_1 \) and \( \lambda_2 \), respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $n$ different wine bottles numbered $i=1, 2, \ldots, n$, where the $i$-th bottle is $i$ years old and is still good with a probability of $1/i$, find the expected value of the age of a randomly selected bottle that is still good.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the conditions under which the matrix $X = A^\top Y A$ is invertible, given that $A \in F^{m \times n}$ is full rank and $Y$ is positive definite.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \(\lim_{r \to \infty} \frac {r^C \int_0^{\frac{\pi}{2}} x^r \sin(x)\, dx}{\int_0^{\frac{\pi}{2}} x^r \cos(x)\, dx}\) where \(C\) is a real constant. Determine the value of \(\pi L - C\), given that \(L > 0\) is the limit's value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the intersection point of the projective lines corresponding to \(y = mx\) and \(y = mx + c\) (where \(c \neq 0\)) in the projective plane, given the embedding \((x,y) \to [1,x,y]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the sequence given by:

$$\lim_{n\to \infty} \frac{1}{n^4}\left(\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+\cdots+\frac{n}{a_n}\right)$$

where $a_n$ is defined recursively by $a_1=1$ and $a_{n+1}=\dfrac{a_n}{1+na_n}$. Find the value of this limit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of distinct labels that can be assigned to the vertices of a graph such that each vertex is adjacent to at least one vertex with a different label.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a gamma distribution with shape \(\alpha=2\) and rate \(\lambda=10\), find an expression for \(\mathbb{E}[X^{-2}]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum achievable accuracy for an algorithm $\psi$ that approximates an elementary function $f$ within a floating-point system characterized by $e$ bits for the exponent and $m$ bits for the mantissa, given that the domain $D(f) = [1,2)$ and the range $f(D(f)) = [1,2)$. Provide your answer in terms of $e$ and $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) using Stokes' Theorem, where \( S \) is the surface defined by the graph of \( f(x,y) = 4x - 8y + 30 \) over the rectangle \( R = \{ (x,y) \mid -2 < x < 3, 0 < y < 2 \} \). The vector field is given by \( \mathbf{F}(x,y,z) = -x^2 \mathbf{i} + xz \mathbf{j} + yx \mathbf{k} \). The curve \( C \) is the boundary of \( S \), oriented counterclockwise when viewed from above.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions to the equation \(x^{2} - 2x \cdot \sin(x \cdot y) + 1 = 0\) that lie within the circle defined by \(x^{2} + y^{2} \leq 100\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the interval of convergence for the series \( \sum_{n=1}^{\infty} \left(\frac{n - i}{i - 2n}\right)^{n} (z-2)^{n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the line integral \( \int_K \underline{u} \bullet \underline{t} \, ds \) for the vector field \( \underline{u}(x,y,z) = (x^3+2y+z,\,y^3+2x+z,\,1/2(x^2+y^2)+x+y) \), where \( K \) is the curve of intersection of the surfaces \( V: x+y-z=1 \) and \( S: z = 1-x^2-y^2, \ z \geq 0 \), along the path from \((1,0,0)\) to \((0,1,0)\). Use Stokes' theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a non-constant function \( x(t) \) that satisfies the differential equation \( (x')^2 + x^2 = 9 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set of positive real numbers \(x\) for which the series \(\sum_{n=1}^\infty \frac{x^{a(n)}}{n^3}\) converges, where \(a(n)\) is the number of zeroes in the base 3 representation of \(n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the ordinary differential equation $\dot{\vec{x}} = A\vec{x}$, where $A$ is an $n \times n$ matrix, and the first coordinate of the solution is $x_1(t) = 5t^2\cos(3t) + 2\sin(3t) - e^{2t}\sin(3t) + 4t$, determine the minimum dimension $n$ of the matrix $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P(x)$ be a polynomial of degree 3 such that $P(-1) = 10$, $P(1) = -6$, $P(x)$ has a local maximum at $x = -1$, and $P'(x)$ has a local minimum at $x = 1$. Determine the distance between the local maximum and local minimum of the curve $y = P(x)$. Express your answer as a positive number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the roots of the equation \( 3 \cos \frac{4 \pi x}{5} + \cos \frac{12 \pi x}{5} = 2 \cos \frac{4 \pi x}{5} \left(3 + \operatorname{tg}^{2} \frac{\pi x}{5} - 2 \operatorname{tg} \frac{\pi x}{5}\right) \) within the interval \([-11, 19]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest common divisor of all binomial coefficients \( \binom{n}{k} \) where \( k \) is odd, for a given integer \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( a, b, c \) are positive real numbers satisfying \( a + b + c = 1 \), evaluate the expression:
\[ f(a, b, c) = \frac{a^4}{a^3 + b^2 + c^2} + \frac{b^4}{b^3 + a^2 + c^2} + \frac{c^4}{c^3 + b^2 + a^2} \]
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Given column vectors $v_1, v_2, v_3 \in \mathbb{R}^2$, if for all distinct $i, j \in \{1, 2, 3\}$, the minimum eigenvalue of the matrix $v_iv_i^T + v_jv_j^T$ satisfies $\lambda_{\min}(v_iv_i^T+v_jv_j^T) \leq 1$, determine an upper bound for $\lambda_{\min}(v_1v_1^T+v_2v_2^T+v_3v_3^T)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the lower bound on the number of distinct angles formed by all possible 3-point subsets of a set of $m$ non-collinear points in the plane $\mathbb{R}^2$. Provide your answer as a function of $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three planets revolve about a star in coplanar circular orbits with the star at the center. All planets revolve in the same direction, each at a constant speed, and the periods of their orbits are 60, 84, and 140 years. The positions of the star and all three planets are currently collinear. They will next be collinear after $n$ years. Find $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer \( x \) such that \( 10^x \equiv 1 \pmod{1997} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a 29x29 table, the numbers 1 through 29 are each written 29 times. The sum of the numbers above the main diagonal is three times the sum of the numbers below this diagonal. Determine the number written in the central cell of the table.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest natural number \( m \) such that for all non-negative real numbers \( a_1 \ge a_2 \ge \ldots \ge a_{2014} \ge 0 \), the following inequality holds:
\[ \frac{a_1 + a_2 + \ldots + a_m}{m} \ge \sqrt{\frac{a_1^2 + a_2^2 + \ldots + a_{2014}^2}{2014}}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T \sim \mathcal{U}(0,1)$ and define the random variable $Y$ as follows: \[ Y = \begin{cases} -\Phi^{-1}(T+\frac{1}{2}) & \text{if} & 0 \leq T \leq \frac{1}{2} \\ -\Phi^{-1}(T-\frac{1}{2}) & \text{if} & \frac{1}{2} < T \leq 1, \end{cases} \] where $\Phi^{-1}$ is the inverse of the normal distribution function. Calculate the probability $P(Y \leq y)$ for a given $y \in \mathbb{R}$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
On line \( m \), three equilateral triangles are placed: \(\triangle ABC\), \(\triangle HFG\), and \(\triangle DCE\). It is given that \(BC = \frac{1}{3} CE\), and \(F\) and \(G\) are the midpoints of \(BC\) and \(CE\) respectively. Additionally, \(FM \parallel AC\) and \(GN \parallel DC\). Let the areas of the three parallelograms in the figure be \(S_{1}, S_{2}, S_{3}\), respectively. If \(S_{1} + S_{3} = 10\), find the value of \(S_{2}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements of order 9 in the group \(C_{135} \times C_6 \times C_9\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the cardinality of the set of all algebraic extensions of an infinite field $k$ that have the same cardinality as $k$. Provide your answer as a cardinal number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
ABC is an acute-angled triangle with area 1. A rectangle R has its vertices on the sides of the triangle, with two vertices on BC, one on AC, and one on AB. Another rectangle S has its vertices on the sides of the triangle formed by the points A, R3, and R4, with two vertices on R3R4 and one on each of the other two sides. What is the maximum total area of rectangles R and S over all possible choices of triangle and rectangles?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the Stieltjes integral \( \int_{0}^3 x^2 \, d\alpha \), where \( \alpha(x) = x+1 \) for \( 0 \le x \le 1 \) and \( \alpha(x) = x^2+2 \) for \( 1 < x \le 3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{C} f(z) \, dz \) over the upper half-circle \( C \) with \( 0 \leq \arg z \leq \pi \), starting from \( z=1 \), using the parametrization \( z = e^{ix} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a room with 3 bulbs, each bulb has a lifetime that is independently and identically distributed according to an exponential distribution with parameter \( \lambda = 1 \). Bulbs are replaced only when all bulbs burn out. What fraction of the time is the room only one-third lit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a = (123)(456) \) in \( S_{10} \). Determine the highest possible order of a permutation \( b \) in \( S_{10} \) such that \( b^k = a \) for some integer \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In trapezoid \(ABCD\), the bases \(AB\) and \(CD\) measure 41 and 24, respectively. The diagonals of the trapezoid are perpendicular to each other. Calculate the dot product of the vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a 2-player game, each player rolls k six-sided dice on their turn. The first player to accumulate a total of k or more sixes over all their turns wins the game. Compute the expected number of turns needed to complete the game.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the residue of the function \( \frac{e^{1/z}}{1-z} \) at the point \( z_0 = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral \( \int_0^{\infty} \frac{ \sin{x}}{x^2+1}dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic behavior of the sum \( \sum_{k = 1}^{\frac{n}{2} - \sqrt{n}} n \binom{n - 1}{k - 1} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A gun is located at the origin of an infinite number line and starts shooting bullets along the positive x-axis at a rate of one bullet per second. Each bullet's velocity is randomly chosen from a uniform distribution between 0 and 1 m/s. If two bullets collide, they explode and disappear. What is the probability that at least one bullet will travel infinitely without colliding with another bullet?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $S_0 = \{A, B\}$, where $A$ and $B$ are subsets of a set $M$ with a non-empty intersection, define $S_{i+1}$ recursively as the set of all subsets of $M$ that can be obtained by taking the union, intersection, or complement of any two elements in $S_i$. Determine the maximum possible cardinality of $\bigcup S_i$ as $i$ tends to infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the range of values of \(\lambda\) such that the quadratic equation \((1-\mathrm{i}) x^{2}+ (\lambda+\mathrm{i}) x+(1+\mathrm{i} \lambda)=0\) has two imaginary roots, where \(\lambda\) is a real number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{C} (z^2 - 3z + \text{Im} \, z)\,dz \) where \( C \) is the square with vertices \( z=0 \), \( z=1 \), \( z=1+i \), and \( z=i \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a function \( f(x, y) \) with continuous partial derivatives everywhere, and \( \nabla f(0, -18) = -2i + 3j \), define a new function \( g(x, y) = f(xy + x^2, xy - y^2) \). Calculate \( \nabla g(3, -3) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit using the Central Limit Theorem:
\[ \lim_{n\to\infty}p^{n}\sum_{k \geqslant{n(p^{-1}-1)}}^{\infty}\binom{n+k-1}{n-1}(1-p)^{k}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle ABC, three lines divide it into four congruent triangles and three pentagons. One of the triangles has all three sides along the new lines, while the other three triangles each have two sides along the new lines. If all four triangles are congruent, express the area of each triangle in terms of the area of triangle ABC.
Please reason step by step, and put your final answer within \boxed{}.


Context:
You are tossing an unbiased coin. The last 28 consecutive flips have all resulted in heads. Let \( x \) be the expected number of additional tosses you must make before you get 60 consecutive heads. Find the sum of all distinct prime factors of \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_1, X_2, \ldots,$ be independent random variables with distribution $P(X_k=2^k)=P(X_k=-2^k)=\frac{1}{2}$ for all $k$. Define $T_n = X_1 + X_2 + \cdots + X_n$. Determine if $\frac{T_n}{2^{n}}$ converges in distribution, and if so, find the limiting distribution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a square matrix $A$ of order $n$ where each element is either -1, 0, or 1, determine the maximum possible value of the determinant of $A$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \{ a_n \} \) be a sequence of non-negative real numbers such that the series \( \sum_{n=1}^{\infty} a_n \) is convergent. Determine the range of values of the real number \( p \) for which the series \( \sum_{n=1}^{\infty} \frac{\sqrt{a_n}}{n^p} \) diverges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rock contains $10^{20}$ atoms of a substance with an exponentially distributed lifetime and a half-life of 100 years. Determine the number of centuries that must pass until it is most probable that only 100 atoms remain.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a$, $b$, $c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ways in which the number 200 can be expressed as a sum of positive integers, denoted as \( p(200) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of terms required in the alternating series representation of the integral \( \int_{0}^{\frac{1}{2}} \frac{1}{1+x^6} \, dx \) to ensure the error is less than \( 10^{-5} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral: \[ \int_{-\infty}^{+\infty} \frac{x^2}{(x^2-8x+20)^2} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
An open can of oil is accidentally dropped into a lake, causing the oil to spread over the surface as a circular disc of uniform thickness. The radius of the disc increases at a steady rate of 10 cm/sec. At the moment when the radius is 1 m, the thickness of the oil slick is decreasing at a rate of 4 mm/sec. Given that the volume of oil remains constant, how fast is the thickness of the oil slick decreasing when the radius is 2 m?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer $n$, let $A_n$ and $B_n$ be the families of $n$-element subsets of $S_n=\{1,2,\ldots ,2n\}$ with respectively even and odd sums of elements. Compute $|A_n|-|B_n|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the integer value of \( \alpha = \angle BAC \) for which it is possible to find a right-angled triangle ABC with at least two integer-valued sides.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the series \( \sum_{n=1}^{\infty} \left( n\cdot \ln\frac{2n+1}{2n-1} - 1\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( I = \int_{-1}^{1}\int_{0}^{x^2} \sqrt{x^2-y}\,dy\,dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are $14$ players participating at a chess tournament, each playing one game with every other player. After the end of the tournament, the players were ranked in descending order based on their points. The sum of the points of the first three players is equal with the sum of the points of the last nine players. What is the highest possible number of draws in the tournament.(For a victory the player gets $1$ point, for a loss $0$ points, in a draw both players get $0,5$ points.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the line l with a slope of α is perpendicular to the line x+2y-λ=0, find the value of tan(π + α).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If 5% more is gained by selling an article for Rs. 600 than by selling it for a certain amount, the cost of the article is Rs. 400. What is the lower selling price?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设S={1,2,⋯,100}，求最小的正整数n，使得S的每一个n 元子集都含有4个两两互质的数．
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximal depth of an ideal in the polynomial ring $k[x,y]$ over an arbitrary field $k$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the conditions on the function $a(t)$ such that the stochastic process $X(t)$, defined by the 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.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The circle $C_1$ is tangent to the circle $C_2$ internally at the point $A$. Let $O$ be the center of $C_2$. The point $P$ is on $C_1$, and the tangent line at $P$ passes through $O$. Let the ray $OP$ intersect with $C_2$ at $Q$, and let the tangent line of $C_1$ passing through $A$ intersect with the line $OP$ at $R$. If the radius of $C_2$ is 9 and $PQ = QR$, find the length of the segment $OP$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A princess is equally likely to sleep on any number of mattresses from six to twelve each night. On half of the nights, a pea is placed under the lowest mattress. She never sleeps well if a pea is under six mattresses. If the pea is under seven mattresses, she sleeps well 10% of the time; under eight mattresses, 20% of the time; and so on, up to 60% of the time under twelve mattresses. One morning, she reports having slept incredibly well. What is the expected number of mattresses she slept on that night?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $AP_1, AP_2, AP_3$ be three arithmetic progressions with common terms $m, n, r$, respectively. If $AP_1 \cap AP_2 = AP(7, 15)$ and $AP_2 \cap AP_3 = AP(a, d)$, find the value of $a + d$. \( \text{Calculate } a + d. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an arithmetic-geometric sequence $\{ a_{n} \}$ that satisfies $a\_1 + a\_3 = 10$, $a\_2 + a\_4 = 5$, find the maximum value of the product $a\_1 a\_2 \ldots a\_n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( p > 0 \) for which the integral \( \int_{-1}^{0} x^{-p} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Six numbers are written in a row. It is known that among them there is one unit and any three consecutive numbers have the same arithmetic mean. Find the maximum value of the geometric mean of any three consecutive numbers in this row if the arithmetic mean of all 6 numbers is A.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit \(\displaystyle\lim_{n\to\infty}\sqrt[n]{|1-z^n|}\) for \(z\in\mathbb{C}\) with \(|z|\ne 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta, who did not attempt 300 points on the first day, had a positive integer score on each of the two days, and Beta's daily success ratio (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was $300/500 = 3/5$.

Find the largest possible two-day success ratio that Beta could have achieved.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The length of an open interval $\left(a,b\right)$ is defined as $b-a$. Through estimation, the zero point of the function $f(x)=\frac{1}{{2^x}}-{x^{\frac{1}{3}}}$ belongs to the open interval ____ $($just need to write one interval that meets the condition and does not exceed $\frac{1}{6}$ in length).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function f(x) = x^2 + 2x (x > 0), and the recursive definition f1(x) = f(x), f(n+1)(x) = f(f(n)(x)), n ∈ N*, find the maximum value of f2019(x) on the interval [1, 2]. Express your answer as a single mathematical expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( l^2= \{ (x_1,x_2,x_3,\dots):x_n \in \mathbb{R} \text{ for all } n \in \mathbb{N} \text{ and } \sum_{n=1}^{\infty} x_n^2 < \infty \}. \) For a sequence \((x_1,x_2,x_3,\dots) \in l^2,\) define \[\lvert \lvert (x_1,x_2,x_3,\dots)\rvert\rvert_2= \bigg( \sum_{n=1}^{\infty} x_n^2\bigg)^{\frac{1}{2}}.\] Consider the subspace \(M=\{ (x_1,x_2,x_3,\dots) \in l^2 :\sum_{n=1}^{\infty} \frac{x_n}{4^n}=0 \}.\) Let \(M^{\perp}\) denote the orthogonal complement of \(M\) in the Hilbert space \((l^2, \lvert \lvert \cdot \rvert \rvert_2).\) Consider \(\big(1,\frac{1}{2},\frac{1}{3},\frac{1}{4}, \dots \big) \in l^2.\) If the orthogonal projection of \(\big(1,\frac{1}{2},\frac{1}{3},\frac{1}{4}, \dots \big)\) onto \(M^{\perp}\) is given by \(\alpha\big( \sum_{n=1}^{\infty} \frac{1}{n4^n}\big)\big(\frac{1}{4},\frac{1}{4^2},\frac{1}{4^3}, \dots \big)\) for some \(\alpha \in \mathbb{R},\) find the value of \(\alpha\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points $K,A,L,C,I,T,E$ are such that triangles $CAT$ and $ELK$ are equilateral, share a center $I,$ and points $E,L,K$ lie on sides $CA, AT, TC$ respectively. If the area of triangle $CAT$ is double the area of triangle $ELK$ and $CI = 2,$ compute the minimum possible value of $CK.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The dispatcher Xiao Zhang from Factory A of a certain brand is responsible for transferring goods among five large supermarkets within the inner ring of the Second Ring Road. These five supermarkets are arranged in the order of A, B, C, D, and E within the ring-shaped area. Currently, the supermarkets have 15, 7, 11, 3, and 14 pieces of Product A, respectively. Now, to make the number of Product A pieces the same in each supermarket, how many pieces should Xiao Zhang transfer from each supermarket to its adjacent ones to minimize the total number of transfers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the angle between the vector \( \vec{a} \) and the vector \( \hat{i} - 2\hat{j} + 2\hat{k} \). The vector \( \vec{a} \) is parallel to the line of intersection of the plane determined by the vectors \( \hat{i} \) and \( \hat{i} + \hat{j} \), and the plane determined by the vectors \( \hat{i} - \hat{j} \) and \( \hat{i} + \hat{k} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum: $$ \sum_{k=1}^{+\infty}\frac{1}{k^{3}+k}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and  }x^3+bx+a \text{ have at least a common real zero .}\}$
How many elements are there in $S$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
有多少个正整数 $n$ 满足 $\sin (\sqrt{2})+\sin (2 \sqrt{2})+\cdots+\sin (n \sqrt{2})>2$? 填写满足条件的正整数个数。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum: $$\cos\left(\frac{2\pi}{2021}\right)+\cos\left(\frac{4\pi}{2021}\right)+\cdots+\cos\left(\frac{2020\pi}{2021}\right).$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the limit superior of the sequence \(|\sin(n)|^n\) as \(n\) approaches infinity, where \(n\) is an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Whole numbers whose decimal representation reads the same from left to right as from right to left are called symmetrical. For example, the number 5134315 is symmetrical, while 5134415 is not. How many seven-digit symmetrical numbers exist such that adding 1100 to them leaves them unchanged as symmetrical numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find \(g(2022)\) if for any real numbers \(x\) and \(y\) the following equation holds:
$$
g(x-y)=2022(g(x)+g(y))-2021 x y .
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 16 people standing in a circle: each of them is either truthful (always tells the truth) or a liar (always lies). Everyone said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x_1=\sqrt{10}$ and $y_1=\sqrt3$. For all $n\geq 2$, let
$$
\begin{align*}x_n&=x_{n-1}\sqrt{77}+15y_{n-1}\\y_n&=5x_{n-1}+y_{n-1}\sqrt{77}\end{align*}
$$
Find $x_5^6+2x_5^4-9x_5^4y_5^2-12x_5^2y_5^2+27x_5^2y_5^4+18y_5^4-27y_5^6.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum possible real number \( L \) such that the area of the intersection of the set \( I = \{(x,y) \mid x > (\frac{y^4}{9} + 2015)^{1/4}\} \) and the disk \( x^2 + y^2 \leq r^2 \) is less than \( Lr^2 \) for all \( r > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine a sequence of real numbers $(\alpha_n)_n$ such that the process $(Y_n)_{n \geq 0}$, defined by $Y_n := S_n^3 + \alpha_n \cdot S_n$, is a martingale with respect to the natural filtration $(F_n)_{n \geq 0}$ of a simple symmetric random walk $(S_n)$ on $\mathbb{Z}$. Provide the sequence $(\alpha_n)_n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the parametric equations for the folium of Descartes, \(x = \frac{3t}{1+t^3}\) and \(y= \frac{3t^2}{1+t^3}\), calculate the condition for a vertical tangent at the point \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the inner product \( \langle xx^T, A^TA \rangle \) given the definition \( \langle A, B \rangle = \sum_{i=1}^n\sum_{j=1}^n a_{ij}b_{ij} \). Express your answer in terms of \( \|Ax\|^2_2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
An archer has a probability of hitting a target of 0.4 with each shot. Determine the minimum number of attempts required for the archer to hit the target at least 80 times with a probability of 0.9.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X \sim \mathsf{Gamma}(2,3)$ and define $Y = 2X$. Find the probability density function (pdf) of $Y$ and evaluate it at $y = 13.5$. Provide your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p = 10007$ be a prime number. Determine the number of $2 \times 2$ matrices $X$ with elements from $\mathbb{Z}_p$ such that $X^2 \equiv I \pmod{p}$, where $I$ is the identity matrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the integral: $$\lim_{n \rightarrow \infty}\int_{0}^{\pi/4}{\frac {\sin^2(\frac{nx}{2})}{n\,\sin^2(\frac{x}{2})} }\,dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a difference triangle of order 8, where the total number of integers is 36, determine the closest possible total to 36 that can be achieved with either no repeated values or no missing values.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the density of the $xy$ coordinates in a 3-dimensional unit ball, given that $Z = 0$. Express your answer as a function of $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area of the domain D bounded by the loop of the curve \( (x+y)^5 = x^2y^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a_1, a_2, \ldots, a_9 \) be nine real numbers, not necessarily distinct, with an average of \( m \). Let \( A \) denote the number of triples \( 1 \le i < j < k \le 9 \) for which \( a_i + a_j + a_k \ge 3m \). Determine the minimum possible value of \( A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of $a + b$ given that $13 \mid (a + 11b)$ and $11 \mid (a + 13b)$, where $a$ and $b$ are positive integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\phi: c_0 \to \mathbb{R}$ be a linear functional defined by $\phi(x_n) = \sum_{n=1}^\infty (-2)^{1-n} x_n$, where $c_0$ is the space of sequences that converge to 0. Determine the norm of $\phi$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose you have $n$ identically and symmetrically distributed normal random variables. Determine the most negative common correlation coefficient possible between any two of them.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following limit of an integral: $$ \lim_{n\to\infty}\int_A\frac{\ln(n+y^3)-\ln n}{\sin(x/n)}d\lambda_2 $$ where $A=\{(x,y)\in\mathbb{R}^2_+:1<xy<4,1<y/x<4\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the matrix \( A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{pmatrix} \), find the value of \( E_3(A) \) such that the characteristic polynomial \( P_A(t) = t^4 - 4t^3 + 3t^2 - E_3(A)t - 1 \) is satisfied.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression \( \frac{\partial f'}{\partial f} \) given that \( f: \mathbb{R} \to \mathbb{R} \) and \( f' = \frac{df}{dt} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \to \infty} n \int_0^\pi \{ -x \}^n \, dx \] where \( \{ x \} \) denotes the fractional part of \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many distinct configurations exist for placing 1 to 8 red balls in each of 8 boxes such that the difference in the number of balls between any two adjacent boxes is at most 1?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the region of points $(x,y,z)$ such that
\[\left(x^{2}+y^{2}+z^{2}+8\right)^{2}\le 36\left(x^{2}+y^{2}\right). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A fair coin is tossed repeatedly, and the outcomes are recorded as a sequence of 'H' (heads) and 'T' (tails). Let $X$ be the number of tosses required for the sequence 'HTH' to appear for the first time. Calculate the expected value $\mathbb{E}(X)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the points where the partial derivatives of the function \( f(x,y) = |x+y|\sin|x+y| \) are discontinuous.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the conditional expectation \( E[\max(X,Y) | \min(X,Y)] \) when \( X \) and \( Y \) are two independent identically distributed random variables with an exponential distribution, where \( X,Y \sim \exp(1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A linear stationary dynamic system is described by the equation \(3 Y^{\prime}(t) + Y(t) = 4 X^{\prime}(t) + X(t)\). The system receives a stationary random function \(X(t)\) with a correlation function \(\mathbf{k}_{x}(\tau) = 6 e^{-2|\tau|}\). Determine the variance of the random function \(Y(t)\) at the output of the system in the steady-state regime.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \lim_{n \to \infty} U_n \) where \( U_n = \int_{n^2+n+1}^{n^2+1} \frac{\tan^{-1}x}{x^{0.5}} \, dx \) without performing the integration.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\rightarrow\infty}e^{\ln(1-\sin{\frac{1}{n}})^n} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
An investor needs to cover a liability of $1000 in two years. The investor can invest in a one-year pure discount bond and a three-year pure discount bond, both with a face value of $100. Assume the yield to maturity is 10%. How many units of the three-year pure discount bond must the investor purchase to ensure the liability is covered, even if the yield to maturity changes in the second year? Provide your answer as a decimal number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the number of ways there are to completely fill a $3\times 15$ rectangle with non-overlapping $1\times 3$ rectangles
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the outward flux \( \iint_S \mathbf{F} \cdot d\mathbf{S} \) where \( \mathbf{F}(x,y,z) = \frac{1}{(x^2+y^2+z^2)^{3/2}}\langle x,y,z\rangle \) and \( S \) is the outward oriented surface given by \( 9x^2+4y^2+16z^2=144 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the definite integral \( \int_0^3 x \, d[x] \), where \([x]\) is the greatest integer less than or equal to \(x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to\infty} \frac{1}{\sqrt{n}}\int_{1/{\sqrt{n}}}^{1}\frac{\ln(1+x)}{x^3}\,dx.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a positive integer \( n \), let \( \mathrm{rad}\; n \) be the product of \( n \)'s distinct prime factors and \( \mathrm{spf}\; n \) be the sum of \( n \)'s distinct prime factors. For any \( k \ge 1 \), let \( a_{n,k} \) be defined as \[ a_{n,k} = \begin{cases} 1 & \frac{\mathrm{rad}(n)}{\mathrm{rad}(n+k)} > 1 \\ 0 &\text{ otherwise} \end{cases} \] Find the limiting value of \[ \lim_{x \to \infty}\dfrac{1}{x}\sum_{n \le x}a_{n,k} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radii of circles drawn around the vertices of an equilateral triangle such that the circles do not intersect each other or the opposite side of the triangle, and together they cover the largest possible portion of the triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( M \) be the smallest positive multiple of 2012 that has 2012 divisors.
 Suppose \( M \) can be written as \( {\Pi }_{k = 1}^{n}{p}_{k}^{{a}_{k}} \) where the \( {p}_{k} \) ’s are distinct primes and the \( {a}_{k} \) ’s are positive integers.
 Find \( {\sum }_{k = 1}^{n}\left( {{p}_{k} + {a}_{k}}\right) \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest possible value of \(x^2 + y^2\) given that \(x\) and \(y\) are positive integers such that \(x(x+1)\) divides \(y(y+1)\), but neither \(x\) nor \(x+1\) divides either \(y\) or \(y+1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of positive integers less than $10^8$ whose digits sum to 24.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface area of the portion of the cylinder defined by $x^2 + y^2 = 8y$ that is located inside the sphere $x^2 + y^2 + z^2 = 64$. Provide your answer in terms of the appropriate units.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the number of ways to arrange the numbers $1, 2, 3, 4, 5, 6,$ and $7$ around a circle such that the product of every pair of adjacent numbers on the circle is at most $20.$ (Rotations and reflections count as different arrangements.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_\Gamma (z-y)\,dx-(x+z)\,dy-(x+y)\,dz \) using Stokes' Theorem, where \( \Gamma \) is the intersection of the sphere \( x^2+y^2+z^2=4 \) with the plane \( z=y \). Assume \( \Gamma \) is oriented anticlockwise when viewed from the positive \( z \)-axis. Use \( \nabla \times F = (0, 2, 0) \) to find the result.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the condition under which the expression \( n \binom{2n}{n} = \frac{(2n)!}{n!n!} \) is divisible by 4.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a_0 = 2$, $a_1 = 5$, and $a_2 = 8$, and for $n>2$ define $a_n$ recursively to be the remainder when $4(a_{n-1} + a_{n-2} + a_{n-3})$ is divided by $11$. Find $a_{2018}\cdot a_{2020}\cdot a_{2022}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the 2002th term of a binary sequence defined as follows: Start with $S_1 = 0$. For $k \ge 2$, obtain $S_k$ from $S_{k-1}$ by replacing every 0 with 01 and every 1 with 001. The sequence satisfies $S_k = S_{k-1}S_{k-2}S_{k-1}$ for all $k \ge 3$. What is the 2002th term of this sequence?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the greatest power of 2 that is a factor of $10^{1002}-4^{501}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest cardinality of a compact subset of a topological space that does not satisfy a given property.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \frac{1}{2\pi i}\oint_{|z|=R} \frac{z^2}{z-2} e^{\frac{1}{z}} \, dz \) using the residue theorem, where \( R \) is a sufficiently large number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the smallest infinite ordinal that is not order isomorphic to a reordering of the natural numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the interval of convergence for the series \( \sum_{n=1}^{\infty} \frac{n!x^n}{1\cdot 3\cdot 5\cdots (2n-1)} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The moisture content of freshly cut grass is $70\%$, while the moisture content of hay is $16\%. How much grass needs to be cut to obtain 1 ton of hay?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine if there exists a constant \( r \) such that with probability one, \( \limsup_{t \to \infty} \frac{M_t - m_t}{\sqrt{t \log \log t}} = r \), where \( M_t = \max\{B_s : 0 \le s \le t\} \) and \( m_t = \min\{B_s : 0 \le s \le t\} \) for a standard Brownian motion \( B_t \). If such an \( r \) exists, find its value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Students arrive at a help center according to a Poisson process with rate \( r \). When there are \( n \geq 1 \) students in the center, the first one to leave does so at a random \( \text{Exp}(2r) \) time. Suppose that there are currently no students in the center. What is the expected time until there are two students in the center?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the derivative of the function $F(x)$ is $\frac{d}{dx}F(x) = \frac{e^{\sin x}}{x}$ for $x > 0$, and if \( \int_1^4 \frac{2e^{\sin x^2}}{x} \, dx = F(k) - 1 \), find a possible value for $k$ by employing an appropriate substitution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
One hundred balls labelled $1$ to $100$ are to be put into two identical boxes so that each box contains at least one ball and the greatest common divisor of the product of the labels of all the balls in one box and the product of the labels of all the balls in the other box is $1$. Determine the number of ways that this can be done.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to \infty}\frac{1+2^2\sqrt{2}+\dots +n^2\sqrt[n]{n}}{n^2(n+1)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flux of the vector field \( \boldsymbol{\mathrm{F}} = x\boldsymbol{\mathrm{i}} + 4y \boldsymbol{\mathrm{j}} \) outwards across the triangle with vertices at \((0,0), (2,0), (0,2)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given 8 independent runners with finish times distributed as $X_1, X_2, \ldots, X_8 \sim \text{Exp}(1)$, find the expected value of the difference between the time of the first and second runners to finish, $E(T_2 - T_1)$, where $T_1$ is the time of the first runner to finish and $T_2$ is the time of the second runner to finish.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value of the points in the set \( P \) as \( N \to \infty \), where \( P = \{ x_i \mid x_i > 2 \} \) and \( x_i \) are independently generated from a normal distribution with mean 0 and standard deviation \( \sigma = 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For \(n \geq 2\), let \(G = (V, E)\) be the loop-free undirected graph where \(V\) is the set of binary \(n\)-tuples (of 0's and 1's) and \(E = \{ \{v, w\} : v, w \in V, \text{ and } v, w \text{ differ in exactly two positions} \}\). Determine the number of connected components, \(\kappa(G)\), of the graph \(G\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the set $X = \mathbb{N} \times \mathbb{Q}$ with the subspace topology induced by the usual topology on $\mathbb{R}^2$. Let $P = \{(n, \frac{1}{n}) : n \in \mathbb{N}\}$. Determine whether the set $P$ is open, closed, both, or neither in this topology.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum positive real number \( l \) such that the inequality \( 7\sqrt{a} + 17\sqrt{b} + l\sqrt{c} \geq 2019 \) holds, given that \( a + b + c = 1 \) and \( a, b, c > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given positive real numbers $a, b, c$ such that $a + b + c = 6$, determine the maximum value of the expression $A = a^2bc + a^2 + 2b^2 + 2c^2$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \iint_{x^2+y^2<\sqrt{x}}\sqrt{x^2+y^2}\,dxdy \) over the region where \( x^2 + y^2 < \sqrt{x} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit using improper integrals: \[ \lim _{ n\to \infty } \int _{ |x|<n }{ \int _{ |y|<n }{ \sin(x^2+y^2)\,dx\,dy } } \] Consider using a coordinate transformation to simplify the calculation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the sequence \(\left(\frac{N_{2r}}{N_r}\right)_{r\in\mathbb{N}}\), where \(N_r\) represents the number of unit squares with integer coordinates that intersect a circle of radius \(r\) centered at the origin.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral of the vector field \( F = (2x\sin(\pi y) - e^z, \pi x^2\cos(\pi y) - 3e^z, -xe^z) \) along the curve \( C \) from \((0,0,0)\) to \((1,1,\ln(2))\), where the curve is defined by the parametric equations \( z = \ln(1+x) \) and \( y = x \). Compute \( \int_{C} F \cdot dr \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit \( \lim_{n\to+\infty}n\int_{0}^{\frac{\pi}{2}}xf(x)\cos^n x\,dx \) for a continuous function \(f: [0, \frac{\pi}{2}] \to \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $a$, $b$, and $c$ as fixed positive integers, there are $a+b+c$ ducks sitting in a circle, each choosing either rock, paper, or scissors. Specifically, $a$ ducks choose rock, $b$ ducks choose paper, and $c$ ducks choose scissors. A move is defined as follows:

- A duck picking rock sitting behind a duck picking scissors switches places.
- A duck picking paper sitting behind a duck picking rock switches places.
- A duck picking scissors sitting behind a duck picking paper switches places.

Determine the maximum number of moves that can occur, in terms of $a$, $b$, and $c$, over all possible initial configurations.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be a random variable uniformly distributed over the interval $(0, 2)$, and $B$ be a random variable uniformly distributed over the interval $(1, 3)$. Define $W = \min(A, B)$ and $Z = \max(A, B)$. Find the covariance $\text{Cov}(W, Z)$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \oint_C \frac{\sin z}{z^2 + \pi^2} \, dz \) along the closed contour \( |z - 2i| = 2 \). Use Cauchy's integral formula to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the surface area of the 3D shape defined by the equation \((x^2+y^2+z^2+A^2-B^2)^2 = 4A^2(x^2+z^2)\), given that \(0 < B < A\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of $x^{20}$ in the expansion of $$(x^2 + x^3 + x^4)(x + x^2 + x^3 + \cdots + x^6)^4.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimal value of the function
\[\begin{array}{c}\ f(x) =\sqrt{15 - 12 \cos x} + \sqrt{4 -2 \sqrt 3 \sin x}+\sqrt{7-4\sqrt 3 \sin x} +\sqrt{10-4 \sqrt 3 \sin x - 6 \cos x}\end{array}\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $x$ for which the series \( \sum_{n=1}^{\infty} \frac{(2n)!\,x^n}{2^{2n}(n!)^2}\) is convergent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A group $G$ of order 275 (where 275 = $5^2 \times 11$) acts on a set of size 18. Determine the minimum number of orbits of length 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( z = f(x, y) \) and \( g(u, v) = uf(u^2, 2uv) \), calculate \( \frac{\partial^2 g}{\partial u \, \partial v}(1, 1) \). Use the following information: \( f(1, 2) = 4 \), \( \nabla f(1, 2) = (3, -1) \), \( \frac{\partial^2 f}{\partial x^2}(1, 2) = \frac{\partial^2 f}{\partial y^2}(1, 2) = 1 \), and \( \frac{\partial^2 f}{\partial x \, \partial y}(1, 2) = -1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the determinant of the following $6 \times 6$ matrix:
\[
\begin{pmatrix}
0 & 0 & 1 & 1 & 1 & 1\\
1 & 0 & 0 & 0 & 0 & 1\\
1 & 0 & 1 & 1 & 1 & 1\\
0 & 1 & 1 & 1 & 0 & 1\\
0 & 1 & 0 & 1 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0
\end{pmatrix}
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions to the equation \(\sqrt[a+bi]{1}\), where \(a\) and \(b\) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to\infty}\frac{{\left(\frac{1}{1^1} \cdot \frac{1}{2^2} \cdot \frac{1}{3^3}\cdots \frac{1}{n^n}\right)}^\frac{1}{n^2}}{\sqrt{n}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the dimension of the quotient space $C^{\infty}(\mathbb{R})/W$, where $W = \{f \in C^{\infty}(\mathbb{R}): f(0) = 0, f(1) = 0\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the theoretical maximum average compression rate of a reversible function applied to a random bit sequence, given that the function satisfies f'(f(s)) = s?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a regular 2016-gon, all vertices are initially white. Determine the minimum number of vertices that must be painted black so that no right or acute triangles can be formed using only the remaining white vertices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Assume that $(X,Y)$ are independent and identically distributed standard normal random variables. Compute the probability $\mathbb{P}(1<X^2+Y^2<2)$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The function \( g(x) \) is defined as \( \sqrt{\frac{x}{2}} \) for all positive \( x \). Evaluate the expression \( g\left(g\left(g\left(g\left(g\left(\frac{1}{2}\right)+1\right)+1\right)+1\right)+1\right) \) and express it as \( \cos(b) \) in degrees, where \( 0^\circ < b < 90^\circ \) and \( b = \frac{p}{q} \) for some relatively prime positive integers \( p, q \). Find \( p+q \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the subset \( \left\{ \begin{pmatrix} 1 & 2 & 3\\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{pmatrix}\begin{pmatrix}x \\y \\ z \end{pmatrix}:x^2+y^2+z^2 \leq 4\right\} \) of \( \mathbb{R}^3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A set S of distinct positive integers has the following property: for every integer x in S, the arithmetic mean of the set of values obtained by deleting x from S is an integer. Given that 1 belongs to S and that 2002 is the largest element of S, what is the greatest number of elements that S can have?
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Evaluate the limit: \[ \lim_{n\to \infty}\int_0^{\pi/2}\sqrt[n]{\sin^n{x}+\cos^n{x}}\,dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a,b>0)\) with left and right foci as \(F_{1}\) and \(F_{2}\), a line passing through \(F_{2}\) with an inclination angle of \(\frac{\pi}{4}\) intersects the hyperbola at a point \(A\). If the triangle \(\triangle F_{1}F_{2}A\) is an isosceles right triangle, calculate the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest lower bound, independent of \(\lambda\), for the expression:
\[
\frac{1+|\lambda|^2}{2} + \text{Re}\lambda - \frac{2\pi}{n+2}\text{Im}\lambda
\]
where \(|\lambda| < 1\), \(\text{Re}\lambda < 0\), \(\text{Im}\lambda > 0\), and \(n\) is a positive integer. You may choose \(n\) to be as large as necessary for the problem to make sense, but aim for the smallest possible lower bound on \(n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a straight stretch of one-way, two-lane highway, vehicles obey a safety rule: the distance from the back of one vehicle to the front of another is exactly one vehicle length for each 20 kilometers per hour of speed or fraction thereof. Suppose a sensor on the roadside counts the number of vehicles that pass in one hour. Each vehicle is 5 meters long and they can travel at any speed. Let \( N \) be the maximum whole number of vehicles that can pass the sensor in one hour. Find the quotient when \( N \) is divided by 10.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, the internal angles $A$, $B$, and $C$ satisfy the equation $$2(\tan B + \tan C) = \frac{\tan B}{\cos C} + \frac{\tan C}{\cos B}$$. Find the minimum value of $\cos A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sixth derivative of \( f(x) = e^x \left( 3\sin\frac{x}{\sqrt{3}} + 2\cos\sqrt{3}x \right) \) at \( x=0 \) without directly differentiating the function six times.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many square inches of paint are needed to fully paint a regular $6$-sided die with side length $2$ inches, except for the $\frac13$-inch diameter circular dots marking $1$ through $6$ (a different number per side)? The paint has negligible thickness, and the circular dots are non-overlapping.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the matrix representation of the operator \(|v_{j}\rangle \langle v_{k}|\) with respect to the orthonormal basis \(|v_{i}\rangle\) for an inner product space \(V\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the curvature 2-form \(\Omega(X,Y)\) for the Martinet distribution, where \(X=\partial x+\frac{1}{2}y^{2}\partial t\) and \(Y=\partial y\), given that \(\Omega=y\, \mathrm{d}x \wedge \mathrm{d}y\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the standard deviation of the insurance payment for an automobile with a damage deductible of $250, where repair costs are modeled by a uniform random variable on the interval $(0, 1500)$. Let $X$ denote the damage, and $Y$ the insurance payment, defined as:

$$Y = \begin{cases} 0 & \text{if } X < 250 \\ X - 250 & \text{if } 250 \le X \le 1500 \end{cases}$$

Calculate the standard deviation of $Y$. Use the formula $\text{var } Y = E(Y^2) - (EY)^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a nilpotent endomorphism $g: W \to W$ on a $12$-dimensional vector space $W$ over a field $K$, where the dimensions of $\operatorname{ker}g^i$ are provided in the following table:

$$\begin{array}{c|c} i & 12 & 11 & 10 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 \\ \hline \dim \operatorname{ker} g^i & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 11 &10 & 8 & 5 & 0 \end{array}$$

Determine the number of Jordan blocks of the nilpotent endomorphism $g$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of triangles required to cover the point set \( G_{2008} = \{(i, j) \mid i=1,2, \cdots, 2008 ; j=1,2, \cdots, 2008\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the joint probability density function for random variables $X$ and $Y$:
\[ f(x, y)=\begin{cases}
2(x+y)  & \text{if } 0<x<y<1 \\
0 &  \text{otherwise}
\end{cases}. \]
Find the real number $a$ such that the conditional expectation $E(X | Y=aX)=\frac{2}{9}$. \(a\) is a real number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the \(\sigma\)-algebra generated by the sets \(X_{i}:=\{(x_1,x_2,...,x_n):x_i > \frac{1}{2}\}\) for \(i=1,2,...,n\), where \(X=[0,1]^{n}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of black boxes required to pinpoint the location of the origin of a signal that caused spaceships to explode. Assume the signal spreads through space at a constant velocity uniformly in every direction, and each black box contains the spatial coordinates of the ship and the time it received the signal.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f: \mathbb{R}^2 \to \mathbb{R}^2 \) be defined by \( f(x,y) = (e^{2x+y}, e^{x+y}) \). Compute the density function \( \frac{df[\lambda_2]}{d\lambda_2} \) of the pushforward measure \( f[\lambda_2] \), where \( \lambda_2 \) is the Lebesgue measure on \( \mathcal{B}(\mathbb{R}^2) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sequence $(x_n)_{n\in N}$ satisfies $x_1 = 2015$ and $x_{n+1} =\sqrt[3]{13x_n - 18}$ for all $n \ge 1$.
Determine $\lim_{n\to \infty} x_n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$\int_0^\infty \int_{-x}^0 \frac{e^{-x^2/2}}{\sqrt {2\pi}} \frac{e^{-y^2/2}}{\sqrt {2\pi}}\, dy\, dx.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the function \( B = |x-1| + |2x-3| + |3x-5| + \cdots + |99x-197| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit of the sequence defined by \( a_n = \frac{1}{\sqrt{n}}\sum_{k=0}^n\sqrt{C_n^kp^k(1-p)^{n-k}} \), where \( C_n^k = \frac{n!}{k!(n-k)!} \) and \( p \in (0,1) \). Find \( \lim_{n \to \infty} a_n \).
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Find the volume of the solid in the first octant bounded by the surfaces $z = 1-y^2$, $y=2x$, and $x=3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \frac{1}{2\pi i}\oint_C \frac{z^2}{z^2+4}dz \), where \( C \) is the square with vertices \( \pm2 \) and \( \pm2+4i \), using Cauchy's integral formula.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $F$ be a field of order 32. Determine the number of solutions $(x,y) \in F \times F$ to the equation $x^2 + y^2 + xy = 0$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all positive integer solutions to the equation \( \log_2(x+1) = \lfloor \log_2(x) \rfloor + 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A single fair die is rolled repeatedly. What is the probability that two sixes in a row will appear before a five followed by a six?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) = 1 + \sum_{k=0}^{100}\frac{(-1)^{k+1}}{(k+1)!}x(x-1)\cdots (x-k) \). If \( S \) is the sum of all real roots of \( f \) and \( T \) is the sum of all real roots of \( f' \), find \( S-T \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Eliminate \(\alpha\) from the following equations: \(\cos(\theta-\alpha)=\lambda\sin2\theta\) and \(\sin(\theta-\alpha)=\lambda\cos2\theta\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the region bounded by the planes $6x+5y+6z = 6$, $y=x$, $x=0$, and $z=0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the symplectic group \(\operatorname{Sp}(2n,\mathbb{C})\) over the complex field.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a regular pentagon ABCDE, the sides are extended to their points of intersection, forming a pentagram. Let PQRST be another regular pentagon inscribed within the pentagram, such that the ratio of the length of a side of the pentagram to the length of a side of the inscribed pentagon is m:1. Determine the ratio of the area of the pentagram to the area of the inscribed pentagon.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many normal subgroups of the symmetric group $S_4$ are isomorphic to the Klein four-group?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $X_1, X_2, \ldots, X_n$ are independent and identically distributed (i.i.d.) random variables, find the expected value of the expression $\sum (X_i - \bar{X})^2$, where $\bar{X}$ is the sample mean. Calculate this without expanding the square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Assume $a_n > 0$ and $\lim\limits_{n \to \infty} \frac{a_1 + a_2 + \cdots + a_n}{n} = a < +\infty$. Determine $\lim\limits_{n \to \infty} \frac{a_1^p + a_2^p + \cdots + a_n^p}{n^p}$ where $p > 1$. Provide your answer as a single limit value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\mathbb{Z}$ denote the set of integers and $S \subset \mathbb{Z} $ be the set of integers that are at least $10^{100}$. Fix a positive integer $c$. Determine all functions $f: S \rightarrow \mathbb{Z} $ satisfying $f(xy+c)=f(x)+f(y)$, for all $x,y \in S$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an $n \times n$ bipartite graph $G$ with a minimum degree of at least $n - d$, consider the following process: Select vertices from the left side randomly with probability $p = \frac{1}{d+1}$ to form set $S$. Then, form set $T$ by including all vertices on the right side that are connected to every vertex in $S$. Determine the lower bound on the probability that the complete bipartite subgraph $S \times T$ contains a specific edge $uw$. Express this probability in terms of $p$, $n$, and $d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( X_1, X_2, \ldots \) be independent random variables such that \( P(X_n = 1) = P(X_n = -1) = \frac{1}{2} \). Define \( t = \inf \{ n : X_1 + X_2 + \cdots + X_n = 1 \} \). Find the expected value \( E\left(\left(\frac{1}{3}\right)^t\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a cyclic quadrilateral with $|AB|=26$, $|BC|=10$, $m(\widehat{ABD})=45^\circ$,$m(\widehat{ACB})=90^\circ$. What is the area of $\triangle DAC$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the divergence \( \nabla \cdot \mathbf{r} \) in cylindrical coordinates, where \( \mathbf{r} = \rho \hat{e}_\rho + z \hat{e}_z + \rho\theta \hat{e}_\theta \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The volume of a regular triangular pyramid, whose lateral face is inclined at an angle of $45^{\circ}$ to the base, is $9 \mathrm{~cm}^{3}$. Find the total surface area of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the determinant \( A = \begin{vmatrix} f(x) & g(x) & h(x) \\ i(x) & j(x) & k(x) \\ l(x) & m(x) & n(x) \end{vmatrix} \), where \( f(x), g(x), h(x), i(x), j(x), k(x), l(x), m(x), n(x) \) are functions of \( x \), determine the power of \( x-a \) in the factorization of \( A \) if substituting \( x = a \) makes exactly three rows identical.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given vectors \(a\), \(b\), and \(c\) such that \(|a| = |b| = 1\) and \(c = a \times b\), find the maximum value of the expression:
\[
|(a+2b+3c) \cdot ((2a+3b+c) \times (3a+b+2c))|
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that for each \(a \in (0,1)\), the limit \(\lim_{h \to 0^+} \int_{h}^{1-h} t^{-a} (1-t)^{a-1} dt\) exists and is denoted by \(g(a)\), and that \(g(a)\) is differentiable on \((0,1)\), find the value of \(g'\left(\frac{1}{2}\right)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( n \) for which the integral \( \int_0^\infty \frac{\sin(x+x^2)}{x^n} \, \mathrm{d}x \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $N$ sets, each containing between 1 and 100 elements, determine the number of ways to select one element from each set such that no two selected elements are the same. For example, if the sets are $\{1, 2, 3\}$, $\{4\}$, and $\{1, 2\}$, the answer is 4, corresponding to the arrangements $(1,4,2)$, $(2,4,1)$, $(3,4,1)$, and $(3,4,2)$. Find the total number of such arrangements for the given sets.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\xi_{1},\xi_{2},\ldots$ be a sequence of independent random variables such that $P(\xi_{i} = 2^i) = \frac{1}{2^i}$ and $P(\xi_{i} = 0) = 1 - \frac{1}{2^i}$ for $i \geq 1$. Find the almost sure value of the limit $\lim_{n\to \infty}(\xi_{1}+\cdots+\xi_{n})/n$. Apply the Law of Large Numbers to determine the limiting value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of rounds $N$ required such that you would accept to play the following game: You choose a finite sequence of natural numbers $(a_0, a_1, \ldots, a_n)$. In each round, Discriminant the cat gives you a natural number $\beta$, and you respond with the value of $\sum_{k=0}^{n} a_{k} \beta^{k}$. The game ends after $N$ rounds or when Discriminant guesses your sequence. For what values of $N$ would you accept to play this game?
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Find a $3 \times 3$ matrix $A$ that is invariant under any rotation, i.e., satisfies $AR = RA$ for any rotation matrix $R$. What is the form of such a matrix?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a $2024 \times 2024$ grid of unit squares. Two distinct unit squares are adjacent if they share a common side. Each unit square is to be coloured either black or white. Such a colouring is called $\textit{evenish}$ if every unit square in the grid is adjacent to an even number of black unit squares. Determine the number of $\textit{evenish}$ colourings.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \mathcal{H} = l^2(\mathbb{N}) \). If \( N \geq 1 \) and \( L: \mathcal{H} \rightarrow \mathbb{F} \) is defined by \( L(\{\alpha_n\}) = \alpha_N \), find a vector \( h_0 \in \mathcal{H} \) such that the linear functional \( L(h) = \langle h, h_0 \rangle \) holds for all \( h \in \mathcal{H} \), where \( \langle \cdot, \cdot \rangle \) denotes the inner product on \( l^2(\mathbb{N}) \) with \( \langle x, y \rangle = \sum_{n\geq1} x(n)\overline{y(n)} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a scalene triangle \( ABC \), one angle equals the difference between the other two angles, and one angle is twice another angle. The angle bisectors of angles \( A \), \( B \), and \( C \) intersect the circumcircle of the triangle at points \( L \), \( O \), and \( M \) respectively. Given that the area of triangle \( ABC \) is 32, find the area of triangle \( LOM \), rounding to the nearest whole number if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit: $$\lim_{x\to\infty} \left(\int_0^1 t^{-tx} dt\right)^{\frac{1}{x}}$$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Squares $ABCD$ and $EFGH$ have a common center and $\overline{AB}\parallel \overline{EF}$. The area of $ABCD$ is $2016$, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the difference between the largest and smallest possible integer values of the area of $IJKL$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the definite integral \( \int_0^{\pi} \cos(ie^{ix}) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Determine the lower bound for the index of the commutator subgroup of a non-abelian p-group using the conjugacy class equation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the integer closest to the integral \( \int_{0}^{6} x^{\{\sqrt{x}\}} \, dx \), where \( \{x\} \) denotes the fractional part of \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a circle, 101 nonzero integers are arranged such that each integer is greater than the product of the two integers that follow it in a clockwise direction. Determine the maximum number of negative integers that can be present among these 101 integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$A$ and $B$ move uniformly along two straight paths intersecting at right angles in point $O$. When $A$ is at $O$, $B$ is $500$ yards short of $O$. In two minutes they are equidistant from $O$, and in $8$ minutes more they are again equidistant from $O$. Then the ratio of $A$'s speed to $B$'s speed is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(p\) be a prime number and \((G,\cdot)\) be a group with \(p^3\) elements. We denote by \(\operatorname{C}(x)\) the centralizer of \(x\in G\). If \(|\operatorname{Z}(G)|=p\), determine the number of distinct centralizers \(\operatorname{C}(x)\) for \(x\in G\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate \( \lim_{R \rightarrow \infty} \int\int_{D_R} \frac{x^2 \cos y}{x^2+y^2+1} \, dx \, dy \), where \( D_R = \{(r\cos \theta, r\sin \theta):0\leq r\leq R,0\leq \theta\leq \frac{\pi}{R^2} \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the trigonometric equation for $x$ in degrees: $$\sin 84^\circ\cdot \sin(54^\circ-x)=\sin126^\circ\cdot \sin x.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the integral \( I = \int_0^\infty \frac{\sin^4 x}{x^3} \, dx \) using the Laplace transform method.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a tetrahedron \( ABCD \) with edge \( AB \) as the diagonal of the base of a quadrilateral pyramid, and edge \( CD \) parallel to the other diagonal of this base with its ends on the lateral edges of the pyramid, find the smallest possible volume of the pyramid if the volume of the tetrahedron is \( V \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a linear transformation \( T \) defined by \( T(f(t)) = f(3t - 2) \), find the determinant of \( T \) with respect to the basis \( \{1, t, t^2\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the closure of the set \( W = \{ f \in C[0,1] \mid f(0) = 0 \} \) with respect to the \( \|\cdot\|_1 \) norm.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p = 2027$ be the smallest prime greater than $2018$, and consider the polynomial $P(X) = X^{2031} + X^{2030} + X^{2029} - X^5 - 10X^4 - 10X^3 + 2018X^2$. Define $\mathrm{GF}(p)$ as the integers modulo $p$, and $\mathrm{GF}(p)(X)$ as the set of rational functions with coefficients in $\mathrm{GF}(p)$. Let $D: \mathrm{GF}(p)(X) \to \mathrm{GF}(p)(X)$ be a function such that for any $f, g \in \mathrm{GF}(p)(X)$ with $g \neq 0$, \[ D\left(\frac{f}{g}\right) = \frac{D(f) \cdot g - f \cdot D(g)}{g^2} \] and for any nonconstant polynomial $f$, $D(f)$ is a polynomial with degree less than that of $f$. If the number of possible values of $D(P(X))$ can be expressed as $a^b$, where $a$ and $b$ are positive integers with $a$ minimized, find $ab$. \(\text{Proposed by Brandon Wang}\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two players each have a randomly shuffled deck of 52 cards. They compare the cards in order, one at a time, checking if there is a match in rank (suit doesn't matter). What is the probability that there is no match in rank for any of the 52 card comparisons?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two pulleys with diameters of 80 mm and 200 mm are connected by a belt that is 1500 mm long. Calculate the distance between the axes of the pulleys, assuming the belt is tight. Provide your answer in millimeters.
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Jeffrey, being from Tennessee, sees 10s everywhere he looks. If he assigns to each of his 10000 lovers a unique integer ID number from 1 to 10000, how many of them will have the sequence "10" in their ID?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the bounded solution $u(x,t)$ of the heat equation $u_t = u_{xx}$ with the initial condition $u(x,0) = \frac{e^{2x}-1}{e^{2x}+1}$ as $t \to \infty$, specifically for $x = 1$. Express your answer as a single value or expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the conditional expectation \( \mathbb{E}\left[ \int_0^1 (W_t+1)^2 \, dt \big | \mathscr{F}_0 \right] \), where \( W_t \) is a standard Brownian motion and \( \mathscr{F}_0 \) is the sigma-algebra generated by \( W_0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \int_{|z|=2} z^n(1 - z)^m\ dz \) where \( m \) and \( n \) are negative integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct ways to fill a 3 × 3 grid with 0s and 1s, given that the center cell must contain a 0 and two grids are considered identical if one can be transformed into the other through a symmetry operation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of the real number $p$ for which the integral \[ I_p = \int_{0}^{1}\frac{r^{p+1}}{(1-r^{2})^{\frac{p}{2}-\frac{1}{2}}}dr \] converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence of the series \(\sum\limits_{n=0}^\infty 3^nz^{n!}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the size of the conjugacy class of the permutation \((123)(456)\) in the alternating group \(A_7\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum possible degree of a polynomial function $f(z)$ on the complex plane, given that $f(z) = \delta$ whenever $z^3 = \gamma$ and $z^3 = \gamma$ has exactly 3 distinct solutions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the real values of \( a \) (with \( a \neq -1 \)) for which the following limit holds:

\[
\lim_{n \to \infty} \frac{1^a + 2^a + \cdots + n^a}{(n+1)^{a-1} \left[\sum_{i=1}^n (na+i)\right]} = \frac{1}{60}
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the condition on a linear operator $A$ such that the operator $e^{tA}$ is norm-preserving for all $t \le 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral \(\int_{0}^{\infty} \left(e^{-\frac{1}{x^2}}-e^{-\frac{4}{x^2}}\right) dx\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit without using L'Hospital's rule: \[ \lim_{x \to 2} \frac{\sqrt{17 - 2x^{2}}\sqrt[3]{3x^{3} - 2x^{2} + 8x - 5} - 9}{(x - 2)^{2}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a Cartesian coordinate system, you can move from \((a,b)\) to \((a+1,b)\), \((a,b+1)\), or \((a+1,b+1)\), but the path must not form a right angle at any point. Determine the number of ways to move from \((0,0)\) to \((5,5)\) following these rules.
Please reason step by step, and put your final answer within \boxed{}.