Context:
Given the joint probability density function:
\[
f_{X,Y}(x,y) = \begin{cases}
ye^{-xy-y} & \text{if } x > 0, y > 0 \\
0 & \text{otherwise}
\end{cases}
\]
Calculate the distribution of the product $XY$. Assume $X$ and $Y$ are dependent random variables.
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Context:
Evaluate the sum \( \sum_{n=1}^{2019} \lambda(n) \left\lfloor \frac{2019}{n} \right\rfloor \), where \( \lambda(n) = (-1)^{\Omega(n)} \) and \( \Omega(n) \) is the number of prime factors of \( n \) counted with multiplicity. Use the properties of \( \lambda(n) \) to find the value of the sum.
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Context:
In a 4D spacetime, a signal spreads uniformly in all directions at an unknown velocity. What is the minimum number of spacetime points required to determine the origin of the signal?
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Context:
Determine the number of possible Beta-reductions for the lambda calculus expression \((\lambda x.\lambda y.y)(\lambda x.x)((\lambda x.x)(\lambda y.y))\). Consider the order of evaluation in your calculation.
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Context:
Simplify the infinite series: $$\sum_{i=0}^\infty \frac{4^i}{(i!)^2}$$
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Context:
Evaluate the integral: $$\int_{-\pi/2}^{\pi/2}\sin(\tan(\tan(x)))\tan(x)dx$$
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Context:
Suppose $X$ and $Y$ are independent random variables, each uniformly distributed over the interval $(0,1)$. Calculate the expected value $\mathbb{E}[X | \max \{X, Y\}]$. Provide your answer as a single number.
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Context:
Determine the order of the smallest subgroup of the symmetric group $S_4$ that contains the set of permutations \( \{ (12)(34), (14)(32), (24), (31) \} \).
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Context:
Let \(u(x,y)=x^3+ax^2y+bxy^2+2y^3\) be a harmonic function and \(v(x,y)\) be its harmonic conjugate. If \(v(0,0)=1\), find \(|a+b+v(1,1)|\).
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Context:
How many ordered pairs of sets(A,B)have the properties:
\( A \subseteq  \{ 1,2,3,4,5,6\} \)
 \( B \subseteq  \{ 2,3,4,5,6,7,8\} \)
 \( A \cap  B \) has exactly3elements.
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Context:
Determine the supremum of \(|q(z_1,z_2,z_3)|\) over the set \(\{|z_i| = 1, i = 1,2,3\}\), where \(q(z_1,z_2,z_3) = z_1^2z_2^2z_3^2 - z_1^6 - z_2^6 - z_3^6\).
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Context:
Evaluate \( \frac{\partial^6 f}{\partial x^4 \partial y^2}(0,0) \) for the function \( f(x,y) = \frac{1}{1-x^2y} \).
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Context:
Calculate the surface integral \( \iint_S F \cdot dS \) for the vector field \( F(x, y, z) = xi + yj + zk \), where \( S \) is the triangular surface with vertices \((0, 1, 1), (1, 0, 1), (1, 1, 0)\) and \( n \) is the unit normal to \( S \) pointing towards the origin. Use the parameterization \( r(u, v) = (1-v)i + (u+v)j + (1-u)k \).
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Context:
Determine the closure of the subspace $X = \mathrm{lin}\{e_n : n \in \mathbb{Z}^+\}$ in $l^\infty$, where $e_j$ is the sequence with a 1 in the $j$-th position and 0 elsewhere.
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Context:
Determine the number of linearly independent matrices in a commuting family of $4 \times 4$ complex matrices.
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Context:
Calculate the degree of the field extension \([\mathbb{Q}(\sqrt[n]{m}):\mathbb{Q}]\), where \(m\) is an integer with a prime factorization of the form \(m = p \prod p_i^{n_i}\), and \(p \neq p_i\). Assume \(n\) is a natural number.
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Context:
Consider a Markov chain where each day there is a 0.2 probability of rain in the morning. The probability of rain in the afternoon given it rained in the morning is 0.6, and the probability of rain in the afternoon given it did not rain in the morning is 0.3. John goes to the office in the morning and leaves in the afternoon, and he has a total of 3 umbrellas. Let \( X_n \) be the number of umbrellas in his office at the \( n \)-th night. Given the transition probability matrix:
\[
P_{transition} = \begin{pmatrix}0.92 & 0.08 & 0 & 0 \\ 0.24 & 0.68 & 0.08 & 0 \\ 0 & 0.24 & 0.68 & 0.08 \\ 0 & 0 & 0.24 & 0.76 \end{pmatrix}
\]
What is the proportion of time John has 0 umbrellas in his office at night?
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Context:
Square \( ABCD \) has center \( O \). Points \( P \) and \( Q \) are on \( AB \), \( R \) and \( S \) are on \( BC \), \( T \) and \( U \) are on \( CD \), and \( V \) and \( W \) are on \( AD \), so that \( \triangle APW \), \( \triangle BRQ \), \( \triangle CTS \), and \( \triangle DVU \) are isosceles and \( \triangle POW \), \( \triangle ROQ \), \( \triangle TOS \), and \( \triangle VOU \) are equilateral. What is the ratio of the area of \( \triangle PQO \) to that of \( \triangle BRQ \)?
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Context:
What linear algebra operation transforms a matrix $V$ such that $V'^T M V' = I$, where $I$ is the identity matrix, given that $V^T M V = E$?
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Context:
Given two distributions $p$ and $q$ over a common finite set $X$ with a total variation distance $TV(p, q) = \epsilon$, and a function $f: X \to \mathbb{R}$ such that $0 \leq \sum_{x \in X} f(x) p(x) \leq 1$ and $0 \leq \sum_{x \in X} f(x) q(x) \leq 1$, determine the tightest upper bound on the expression \( \Big|\sum_x f(x)(p(x)-q(x))\Big| \) in terms of $\epsilon$. Consider how the condition $TV(p, q) = \epsilon$ can be used to refine the bound beyond the trivial upper bound of 2.
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Context:
Determine the number of positive solutions to the equation \( \cos(97x) = x \).
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Context:
In the binomial expansion of \((1+\sqrt{7})^{207}\), determine the value of \(k\) for which the term \(C_{207}^{k}(\sqrt{7})^{k}\) is maximized.
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Context:
Estimate the value of $\sqrt{90} - \sqrt{88}$.
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Context:
Given a list of prime numbers under 100,000, determine the last prime number that appears in the decimal expansion of pi.
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Context:
Determine the smallest integer $k$ such that the semigroup $C_n$ is equal to the union of the $k$-th powers of the set $A$, where $A = \{\begin{bmatrix}2\\1  \end{bmatrix},\begin{bmatrix}3\\2  \end{bmatrix},\cdots, \begin{bmatrix}n\\n-1  \end{bmatrix}\}$ and $C_n$ is the semigroup of all order-preserving and decreasing transformations in the full transformations semigroup $T_n$. \[ C_n = \bigcup\limits_{i=1}^{k} A^k \]
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Context:
Determine the 4-torsion points of the elliptic curve $E: y^2 = x^3 - 5$ over the field of rational numbers $\mathbb{Q}$.
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Context:
Determine whether the proportion of finite topologies of size \( n \) that are \( T_{0} \) approaches 1 or 0 as \( n \) approaches infinity.
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Context:
Given a Markov chain \( X_n \) with state space \( S = \{1, 2, 3, 4\} \) and transition matrix \( P \) as follows:
\[
P = \begin{pmatrix}
  \frac{1}{4} & 0 & \frac{1}{2} & \frac{1}{4} \\
  \frac{1}{4} & \frac{1}{4} & \frac{1}{2} & 0 \\
  0 & \frac{2}{3} & \frac{1}{3} & 0 \\
  0 & \frac{1}{2} & 0 & \frac{1}{2}
\end{pmatrix}
\]
Calculate \( EX_{\tau} \), where \( \tau = \inf \{ n : X_n = 2 \text{ or } X_n = 3 \} \), given that the initial state is \( X_0 = 1 \).
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Context:
Calculate the entropy for a random variable with a delta function probability density function given by $\Pr(X=x) = \delta(x-x_0)$. What is the entropy?
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Context:
Given the monthly demand for the Instant Pie Maker is \(D(x,y)= \frac{1}{100}ye^{xy/800} \text{ units}\), where \(x\) dollars are spent on online advertisements and \(y\) dollars are spent on social media marketing, and the spending is defined as \(x=15+t^{1/2}\) and \(y=t\cos(t)\) dollars, find the rate at which the demand is changing with respect to time 16 months from now. Round your answer to 3 decimal places.
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Context:
Let \( \omega \) be the incircle of triangle \( {ABC} \) . Line \( {l}_{b} \) is parallel to side \( {AC} \) and
 tangent to \( \omega \) . Line \( {l}_{c} \) is parallel to side \( {AB} \) and tangent to \( \omega \) . It turned ou
 that the intersection point of \( {l}_{b} \) and \( {l}_{c} \) lies on circumcircle of \( {ABC} \)
 Find all possible values of \( \frac{{AB} + {AC}}{BC} \)
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Context:
Two players, A and B, alternately draw from the set \( S = \{0, 1\} \) with replacement. A player wins by drawing two consecutive zeros. Player A draws first. What is the probability that player A wins?
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Context:
Determine the number of elements in the quotient ring \( \Bbb Z_5[i]/\langle 1+i\rangle \).
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Context:
Let $ABC$ be a triangle with $AB=20, AC=21,$ and $\angle BAC = 90^{\circ}.$ Suppose $\Gamma_1$ is the unique circle centered at $B$ and passing through $A,$ and $\Gamma_2$ is the unique circle centered at $C$ and passing through $A.$ Points $E$ and $F$ are selected on $\Gamma_1$ and $\Gamma_2,$ respectively, such that $E, A, F$ are collinear in that order. The tangent to $\Gamma_1$ at $E$ and the tangent to $\Gamma_2$ at $F$ intersect at $P$. Given that $PA \bot BC$, compute the area of $PBC$.
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Context:
A positive integer is written on each vertex of a pentagon, with a different integer on each vertex. On each side of the pentagon, the least common multiple (LCM) of the numbers at the vertices forming that side is written. If the same integer $n$ is written on each side, determine the minimum possible value of $n$.
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Context:
Find the area of the intersection between the sphere \(x^2 + y^2 + z^2 = 1\) and the cylinder \(x^2 + y^2 - y = 0\).
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Context:
Compute the Lefschetz number \( L(f) \) for the map \( f : S^n \to S^n \) defined by \( f(x_0, x_1, \dots, x_n) = (x_0, -x_1, \dots, -x_n) \), where \( S^n \) is the unit sphere in \( \mathbb{R}^{n+1} \).
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Context:
Evaluate the limit: \[ \lim_{n\to\infty} \int_{0}^{\infty} \frac{x^n}{1+x^n} \, dx \]
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Context:
In the space \(\mathbb{R}^3\), let \(S\) be the set of all spheres and \(C\) be the set of all circles. Consider the topologies \(\tau_3\) and \(\tau_4\) on \(\mathbb{R}^3\) generated by the subbases \(S\) and \(C\), respectively. Determine the relationship between the topologies \(\tau_3\) and \(\tau_4\).
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Context:
A chain consists of $2k$ white beads and $2m$ black beads. Determine the minimal number of cuts required to divide the chain into two parts, each containing exactly $k$ white beads and $m$ black beads.
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Context:
For a positive integer \( k \), define the set \( R_{n} \) as follows:
\[
R_{n} =
\begin{cases}
\{-k, -(k-1), \cdots, -1, 1, \cdots, k-1, k\}, & \text{if } n = 2k; \\
\{-k, -(k-1), \cdots, -1, 0, 1, \cdots, k-1, k\}, & \text{if } n = 2k+1.
\end{cases}
\]

A device contains balls connected by red or white strings. A "marking" is a coloring of each ball with elements from \( R_n \). A marking is "good" if any two balls connected by a string have different colors. A marking is "insightful" if any two balls connected by a white string have different colors and any two balls connected by a red string have colors whose sum is not zero.

For a fixed positive integer \( n \geq 3 \), assume any device has a "good" marking regarding \( R_n \) and an "insightful" marking regarding \( R_m \). Find the smallest positive integer \( m = m(n) \).
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Context:
For the differential equation \( y' = a - 2b \sin{y} \), where \( a \) and \( b \) are constants, determine the approximate value of \( \frac{y(2T) - y(T)}{T} \) for sufficiently large \( T \), given that \( 2b < a \).
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Context:
In a group $G$, given that $a^5 = e$ and $aba^{-1} = b^2$ for elements $a$ and $b$, find the order of the element $b$, denoted by $o(b)$. Provide your answer as a single integer.
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Context:
Given the linear regression model with 100 observations:
\[ y_i = b_0 + b_1 x_i + \epsilon_i \]
The estimated coefficients are:
\[ \hat{\beta} = \begin{pmatrix} 9 \\ 1 \end{pmatrix} \]
And the variance-covariance matrix of the estimated coefficients is:
\[ V(\hat{\beta}) = \hat{\sigma}^2 (X'X)^{-1} = \begin{pmatrix} 3 & 0.12 \\ 0.12 & 0.064 \end{pmatrix} \]
Calculate the sum of squared residuals, \( e'e \).
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Context:
Find the Lebesgue measure of the interval \([n, n + \frac{1}{n^2}]\) for \(n \in \mathbb{N}\) using the definition of the n-dimensional Lebesgue measure: \( \lambda^{n}([[a,b))) = \prod_{j=1}^{n}(b_{j}-a_{j}) \).
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Context:
Find the particular solution to the non-homogeneous differential equation \(y'' + \cos(t) y' = \cos(t)\) given that the homogeneous solution is \(y_h (t) = A + Be^{- t \cos (t)}\) with \(A,B \in \mathbb{R}\).
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Context:
The sum \( 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{45} \) is expressed as a fraction with the denominator \( 45! = 1 \cdot 2 \cdot \ldots \cdot 45 \). Determine how many zeros (in decimal notation) the numerator of this fraction ends with.
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Context:
Given a real periodic signal with period $T_0=2$ and Fourier coefficients $$X_k=\left [\frac{2}{3}, \ \frac{1}{3}e^{j\pi/4}, \ \frac{1}{3}e^{-i\pi/3}, \ \frac{1}{4}e^{j\pi/12}, \ e^{-j\pi/8}\right ]$$ for $k=0,1,2,3,4$, calculate the integral $\int_0^{T_0}x^2(t)\, dt$ using the Fourier coefficients.
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Context:
Find the measure of the angle \( \delta = \arccos \left( \left( \sin 2195^\circ + \sin 2196^\circ + \cdots + \sin 5795^\circ \right)^{\cos 2160^\circ} + \cos 2161^\circ + \cdots + \cos 5760^\circ \right) \).
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Context:
Determine the smallest cardinality of a basis for the topology \( \tau \) on \( \mathbb{R}^2 \), where \( \tau \) consists of the empty set, \( \mathbb{R}^2 \), and all open disks \( \{ (x, y) \in \mathbb{R}^2 \mid x^2 + y^2 < r \} \) for all \( r > 0 \).
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Context:
In triangle \( ABC \), \( \angle BAC = 75^\circ \), \( AB = 1 \), and \( AC = \sqrt{6} \). A point \( M \) is chosen on side \( BC \) such that \( \angle BAM = 30^\circ \). The line \( AM \) intersects the circumcircle of triangle \( ABC \) at a point \( N \) different from \( A \). Find the length of \( AN \).
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Context:
Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6. The distance between the centers of the spheres is 13. A plane is tangent to these two spheres and intersects the surface of the cylinder, forming an ellipse. Find the sum of the lengths of the major axis and the minor axis of this ellipse.
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Context:
Let \( R \) be the set of real numbers and \( C \) be the set of all functions \( f : [0, 1] \to R \) that have a continuous derivative and satisfy \( f(0) = 0 \) and \( f(1) = 1 \). Determine \( \inf_{f \in C} \int_{0}^{1} | f'(x) - f(x) | \, dx \).
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Context:
Determine the values of $s$ for which the integral \[ \int\limits_{e_k}^{\infty} \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x \cdots (\ln^{(k)} x)^s } \] converges, where $\ln^{(k)}$ denotes the $k$-fold iterated logarithm.
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Context:
Evaluate the limit: $$\lim_{x\to 0}\left(\frac{1}{\sin(x)\arctan(x)}-\frac{1}{\tan(x)\arcsin(x)}\right)$$
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Context:
Evaluate the integral \( \int_{-\infty}^{\infty} dk\left[\Im\left( \frac{1}{(\omega+i\eta+k)^2+(\omega+i\eta+k+q)^2}\right) \right] \) for an infinitesimal \( \eta \).
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Context:
Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be an entire function satisfying $|f(z)| \leq C|\cos(z)|$ for all $z \in \mathbb{C}$, where $C \in \mathbb{R}$ is a constant. Determine the possible form of $f(z)$. Provide your answer in the form of a function or expression.
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Context:
Calculate the flux of the vector field \( \mathbf{F} = (x^2+y^2+2+z^2)\mathbf{i} + (e^{x^2} + y^2)\mathbf{j} + (3+x)\mathbf{k} \) outward across the part of the spherical surface \( x^2 + y^2 + z^2 = 2az + 3a^2 \) that is above the \( xy \)-plane, where \( a > 0 \).
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Context:
Evaluate the line integral \( \int_\mathcal{E} F \cdot dr \) where \( F(x,y,z) = x\,i+xy\,j+x^2yz\,k \) and \( \mathcal{E} \) is the ellipse \( x^2 + 4y^2 - 8y + 3 = 0 \) in the xy-plane, without using parameterization.
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Context:
Evaluate the limit: $$\lim_{x\to\infty}\frac{\sum_{r=1} ^x re^\frac{r}{x}}{x^2}$$
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Context:
How many points at infinity are there in the complex plane?
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Context:
Given that \( \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = -4 \), find the determinant of the matrix \( \begin{vmatrix} 2a_3 & 2a_2 & 2a_1 \\ b_3-a_3 & b_2-a_2 & b_1-a_1 \\ c_3+3b_3 & c_2+3b_2 & c_1+3b_1 \end{vmatrix} \).
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Context:
Find the largest integer \( n \) such that \( 7^n \) divides \( \binom{2016}{1003} \).
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Context:
Alan, Barb, Cory, and Doug are on the golf team, Doug, Emma, Fran, and Greg are on the swim team, and Greg, Hope, Inga, and Alan are on the tennis team. These nine people sit in a circle in random order. The probability that no two people from the same team sit next to each other is $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
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Context:
Solve the equation $x^x = 10$ for $x$.
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Context:
Calculate the total work required to pump all the water out of a cylindrical tank with a radius of 2 meters and a length of 10 meters. Assume the tank is filled with water, the acceleration due to gravity is 9.8 m/s², and the density of water is 1000 kg/m³. Express your answer in terms of Joules.
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Context:
Under ideal conditions of uniform gravitational acceleration and no friction, what shape would the top ends of a set of straight ramps form if they were arranged such that they all ended at the same base point and took an equal amount of time for a point-like body to slide down from the top to the base?
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Context:
Determine the values of $z$ for which the generating function $\sum_{n=1}^{\infty} g_{n} z^{n}$, where $g_{n}$ is the number of isomorphism classes of groups of order $n$, converges.
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Context:
Find the minimum value of the function \(f(x)=12x^2\int_0^1yf(y)dy+ 20x\int_0^1y^2f(y)dy+4x\).
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Context:
Determine the time complexity of the recursive function given by:

$$T(n) =\left\{\begin{aligned} &2T((n-3)/3) + k_3n^3&&\text{ when }n > 1\\ &k_1 &&\text{ when } n = 1 \end{aligned}\right.$$
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Context:
Determine the value of $n$ for which a cyclic $n$-gon can be divided by non-intersecting diagonals into $n-2$ triangles, such that each triangle is similar to at least one other triangle.
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Context:
Determine the number of integer solutions for \( n \) such that \( 0 \leq n < 23^2 \) and \( n^5 + 2n^4 + n^3 - 3n + 2 \equiv 0 \pmod{23^2} \).
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Context:
Ana, Bob, Cho, Dan, and Eve want to use a microwave. In order to be fair, they choose a random order to heat their food in (all orders have equal probability). Ana's food needs $5$ minutes to cook, Bob's food needs $7$ minutes, Cho's needs $1$ minute, Dan's needs $12$ minutes, and Eve's needs $5$ minutes. What is the expected number of minutes Bob has to wait for his food to be done?
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Context:
Determine the values of \( \lambda \) for which the function \( N_\lambda(x,y) = \sqrt{x^2 + y^2 + 2\lambda xy} \) defines a norm on \( \mathbb{R}^2 \).
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Context:
Given a composite system in a pure state with its Schmidt decomposition, determine the rank of the reduced density matrix.
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Context:
Determine the maximum number of students who can participate in a math competition where each student solves 3 problems, each scored from 0 to 7, such that no student scores equal to or higher than another student on all three problems.
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Context:
Find the limit of \( \frac{{{k-1}\choose n}}{{{n+k-1}\choose n}} \) as \( k/n^2 \to a > 0 \) and both \( k, n \to \infty \).
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Context:
Let $a$, $b$, and $c$ be nonzero real numbers such that $a + \frac{1}{b} = 5$, $b + \frac{1}{c} = 12$, and $c + \frac{1}{a} = 13$. Find $abc + \frac{1}{abc}$.
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Context:
The lateral edge length of a regular n-sided prism is equal to the side length of its base. Find the maximum possible value of n. Express your answer as a whole number.
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Context:
Determine the range of values for $r$ such that the function $f$ is continuous on the interval $[0, \infty)$. Provide your answer as an interval or union of intervals.
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Context:
Calculate the integral \( \int_{0}^{3}\sqrt{4-x^2}\,dx \) using a Riemann sum.
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Context:
Calculate the sum \[ \sum_{n=1}^{\infty}\left((2n-1)\left(\sum_{k=n}^{\infty}\frac{1}{k^2}\right)-2\right). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set \( S \) given that it consists of all positive real numbers, is closed under addition, and for any interval \([a, b] (a>0)\), there exists an interval \([c, d] \subseteq [a, b]\) such that \([c, d] \subseteq S\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of roots of the polynomial \(z^4 + z^3 - 4z + 1 = 0\) within the annular region \(1 < |z| < 2\) using Rouché's theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the function $f(u)$ in the system of equations:
\[ (u_x)^2 - (u_t)^2 = 1 \\
   u_{xx} - u_{tt} = f(u) \]
given that $u = \sqrt{x-t}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the intersection number of the divisors \(\iota^*\mathcal{O}(3,2)\) and \(\iota^*\mathcal{O}(1,4)\) on the smooth subvariety \(\iota:X=V_+(f)\subset\mathbb{P}^2\times\mathbb{P}^1\), where \(f\in H^0(\mathcal{O}(1,2))\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a second degree polynomial $t(x)$ such that for every second degree polynomial $r(x)$, there exist constants $a$, $b$, and $c$ satisfying the equation $r(x) = a \cdot p(x) + b \cdot q(x) + c \cdot t(x)$, where $p(x) = x^2 + 5x - 3$ and $q(x) = 4x^2 + 18x + 4$. Determine $t(x)$. \(t(x)\) should be a second degree polynomial.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limiting distribution of the sequence of random variables $Y_n = \frac{X_n - n}{n^2}$, where $X_n$ is uniformly distributed over the interval $[n, n^2]$ for $n = 1, 2, 3, \dots$. What does $Y_n$ converge to in distribution as $n \to \infty$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim _{x \to 0} \left[\frac{3}{x^4}(\cos^{-1}x - \sin^{-1}x )\right]^{\frac{3}{x^3}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( \int_E z(x^2+y^2)\ dx\ dy\ dz \), where \( E \) is the region in \( \mathbb{R}^3 \) defined by the inequalities \( x^2+y^2+z^2 \le 4 \), \( \frac{1}{3}(x^2+y^2) \le z^2 \le 3(x^2+y^2) \), and \( z\ge 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_1 \sim N(3,\sqrt{5})$ and $X_2 \sim \chi^2(15)$. Assume $X_1$ and $X_2$ are independent. Determine the distribution of \[ Z := \sqrt{\frac{3}{X_2}}(X_1-3). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether there exists a natural number $n$ such that $n^n + (n+1)^n$ is divisible by 1987.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of zeros of the polynomial \( g(z) = z^4 + iz^3 + 1 \) that lie in the region \( \{ z \in \mathbb{C} : \text{Re}(z) > 0, \text{Im}(z) > 0 \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the smallest number of subsets, $n$, into which the set of all integers can be partitioned such that none of the subsets contain an infinite arithmetic progression?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{|z|=2} \frac{dz}{z\sqrt{1-\frac{1}{z^2}}} \) over the circle \(|z|=2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $p$ for which the function \[ f(x,y) = \begin{cases} \frac{(x+y)^{p}}{x^2-y^2} & \text{if $(x,y) \neq (0,0)$} \\ 0 & \text{if $(x,y)= (0,0)$} \end{cases} \] is differentiable at the point $(0,0)$. Provide your answer as a set of values for $p$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the size of the symmetry group of a configuration consisting of two congruent regular \(n\)-gons in the plane. Use the orbit-stabilizer theorem to find your answer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a function $g(t)$ such that its Fourier transform satisfies \( \int_{-\infty}^{\infty} g(t)e^{-j\omega t}dt=\omega e^{-2\omega ^2} \) for any real \( \omega \), find the value of \( \int_{-\infty}^{\infty} y(t)dt \), where \( y(t)=\int_{-\infty}^{t}g(\tau)d\tau \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the general solution to the Clairaut differential equation: \( y = xy' - \ln y' \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\alpha$ be the radian measure of the smallest angle in a $3{-}4{-}5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7{-}24{-}25$ right triangle. In terms of $\alpha$, what is $\beta$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the largest number of starting clues in a Sudoku puzzle that does not guarantee a unique solution?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the time complexity of the conjugate gradient method when solving the system of linear equations $Ax = b$, where $A$ is a sparse, symmetric, and positive definite matrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Five points, no three of which are collinear, are given. What is the least possible value of the numbers of convex polygons whose some corners are from these five points?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(f(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n\), where \(a_i\) are non-negative integers. Given that \(f(1) = 21\) and \(f(25) = 78357\), find the value of \(\frac{f(10) + 3}{100}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A positive integer $x$ is randomly generated with equal probability from the set $\{1, 2, \cdots, 2016\}$. What is the probability that the sum of the digits of $x$ in binary representation does not exceed 8?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a projective curve $X$ over a field $k$, determine if there exists a positive integer $N$ such that the equation $h^0(\mathcal{O}_X(D)) = \deg D + 1 - p_a(X)$ holds for any Cartier divisor $D$ with $\deg D > N$. Provide the value of $N$ if it exists.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a single server queue, the arrival rate is 2 when the queue is empty and 1 otherwise, while the departure rate is a constant 3. Using Little's Law and the invariant distribution, calculate the average waiting time in the queue.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of arrangements of the letters in the word INSTRUCTOR such that: (a) the vowels appear in alphabetical order (IOU), (b) there are at least 2 consonants between each vowel, and (c) the arrangement begins or ends with the two consecutive Ts.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the expression as \( n \) approaches infinity: $$\lim_{n\to \infty}\frac{1}{2n}\log\left(\binom{2n}{n}\right)$$
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \[ f(x + y) + f(z) = f(x) + f(y + z) \] for all real numbers \( x, y, z \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of complex roots of the equation $x^n = a$, where $a \ge 0$ and $n$ is not an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a smooth closed curve $\mathcal{C}$ in the plane $\mathbb{R}^2$, with a curve $\mathcal{C'}$ located at a constant distance $d > 0$ from $\mathcal{C}$ and orthogonal to it at all points, if $\mathcal{C}$ has length $L$, find the length $L'$ of $\mathcal{C'}$. Express your answer in terms of $L$ and $d$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of 4-digit numbers such that if any single digit is removed, the resulting 3-digit number divides the original 4-digit number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In class 12th grade (12), there are 6 students, with 2 wearing red clothes, 2 in blue, and the other two each wearing black and yellow, respectively. They need to stand in a line for a photo, with the requirement that students wearing the same color clothes do not stand next to each other, and the students wearing black and yellow must stand next to each other. There are a total of ___ different ways to arrange this.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\pi_1$, $\pi_2$, $\pi_3$ be three planes in a projective space such that $\pi_1 \cap \pi_2 \cap \pi_3 = \emptyset$, $\dim(\pi_1 \cap \pi_2) = 1$, and $\dim(\pi_1 \cap \pi_3) = \dim(\pi_2 \cap \pi_3) = 0$. Determine the dimension of $\pi_1 \lor \pi_2 \lor \pi_3$. \(\lor\) denotes the span of the planes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values for the parameter \(\lambda\) such that the transformation \(T\) defined by \((Tf)(x) = x + \lambda \int_0^x (x-t)f(t)dt\) is a contraction mapping with respect to the supremum norm on the space of continuous functions \(C([0,1])\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a constant symmetric tensor $V_{ij}$, calculate the product $V_{ij}V^{jk}$ by contracting the $j$ index. Express your answer as a matrix multiplication result.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the average number of steps required for a knight, starting at a corner of an 8x8 chessboard, to return to its starting corner when making random but legal moves.
Please reason step by step, and put your final answer within \boxed{}.


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


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


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


Context:
Find the radius of convergence, \( R \), of the series \( \sum\limits_{n=0}^{\infty} \dfrac{(n!)^{k+2}\cdot x^n}{((k+2)n)!} \), where \( k \) is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the total differential of the function \( g(x,y) = f(f(x,y), f(y,x)) \) at the point \((1,1)\), given that \( f(1,1) = 1 \), \( \partial_x f(1,1) = 1 \), and \( \partial_y f(1,1) = 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Squares of a chessboard are painted in 3 colors - white, gray, and black - in such a way that adjacent squares sharing a side differ in color; however, a sharp change in color (i.e., adjacency of white and black squares) is prohibited. Find the number of such colorings of the chessboard (colorings that are identical upon rotating the board by 90 and 180 degrees are considered different).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two trains are 80 kilometers apart and are traveling toward each other on the same track, each at a speed of 25 kilometers per hour. A bee starts at the first train and flies back and forth between the two trains until they are 100 meters apart. The bee completes 36 rounds, landing 18 times on each train. Assuming the bee has no acceleration or deceleration and takes no extra time when turning, calculate the average speed of the bee.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Through the right focus \( F_{2} \) of the ellipse \(\frac{x}{2} + y^{2} = 1\), draw a chord \(\mathrm{AB}\) with an inclination angle of \(45^\circ\). Then, what is the length \(|AB|\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the complex number $z$ that maximizes the expression $|z-w| - |z-w^2|$, where $z$ lies on the ellipse defined by $|z-4| + |z+4| = 10$, and $w$ is a non-real cube root of unity. Assume $z = x + iy$, and the ellipse's equation is $\frac{x^2}{25} + \frac{y^2}{9} = 1$. Determine the value of $z$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit for any integer $n > 1$:

$$\lim_{x \to \infty} \left[-x^{n+1} e^{-x}\right]_{x=0}^{x=\infty}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the probability that at 10:00 AM on December 31, the Absent-Minded Scientist has exactly two packs of tablets in the cabinet, given that he starts with one pack of 10 tablets and orders a new pack whenever he sees only one tablet left?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of minutes before Jack arrives at the park that Jill arrives at the park, given that they are 2 miles apart, Jill cycles at a constant speed of 12 miles per hour, and Jack jogs at a constant speed of 5 miles per hour.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose X, Y, and Z are independent normal random variables. X has a mean of 5 and variance of 16, Y has a mean of 7 and variance of 25, and Z has a mean of 10 and variance of 9. Find the probability that the average of X and Z is greater than Y.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( \lim_{x \to 0} f(x) \) if the function \( f(x)=\lim_{n\to\infty} \frac{(1-\cos(1-\tan(\frac{\pi}{4}-x)))(1+x)^n+\lambda\sin((n-\sqrt{n^2-8n})x)}{x^2(1+x)^n+x} \) is continuous at \( x=0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Estimate the number of planes in an airline's fleet, given that on his fifteenth flight, the Absent-Minded Scientist flew on a plane named "Siméon Denis Poisson" for the second time, with no repetitions on previous flights.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume enclosed by the hyperboloid \(-x^2-y^2+z^2=1\) and the plane \(z=2\) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the rank of the first homology group of the double branched cover of a knot $K$ in $S^3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the norm of the bounded linear functional \( f \) defined by \( f(x) = x'(t_0) \) for all \( x \) in the space \( C'[a,b] \), where \( C'[a,b] \) is the space of all continuously differentiable functions on \([a,b]\) with the norm \( ||x|| = \max_{a \leq t \leq b} |x(t)| + \max_{a \leq t \leq b} |x'(t)| \). Here, \( t_0 \) is a fixed point in \([a,b]\). What is the norm of \( f \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a random variable $X$ with differential entropy $h(X)$, determine the change in differential entropy when $X$ is multiplied by a negative constant $a$. Express your answer in terms of $h(X)$ and $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $x_0=x_1=1$, and for $n\geq1$

$x_{n+1}=\frac{x_n^2}{x_{n-1}+2x_n}$,

find a formula for $x_n$ as a function of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In space, there are four pairwise skew lines \( l_{1}, l_{2}, l_{3}, l_{4} \), such that no three of them are parallel to the same plane. A plane \( P \) intersects these lines at points \( A_{1}, A_{2}, A_{3}, A_{4} \), forming a parallelogram. Determine the number of lines that the centers of such parallelograms trace out.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Estimate the number of iterations of Newton's method needed to find a root of \( f(x) = \cos(x) - x \) to within \( 10^{-100} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If three lines from the family of lines given by \( C: x \cos t + (y + 1) \sin t = 2 \) enclose an equilateral triangle \( D \), what is the area of the region \( D \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the line integral \( \oint_C P\,dx + Q\,dy \) over the ellipse \( \frac{x^2}{25} + \frac{y^2}{36} = 1 \), where the vector fields are given by:
\[
P = \frac{-y}{(x-1)^2 + y^2}, \quad Q = \frac{x-1}{(x-1)^2 + y^2}
\]
Determine the value of the integral, considering that the vector field is undefined at the point \((1,0)\) inside the ellipse.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the solid formed by the intersection of the paraboloid given by the function \( f(x,y) = 3-x^2-y^2 \) and the plane \( z=1-2x-2y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer $m$ such that in any distribution of 2011 stickers among 33 friends, if every pair of friends is missing at least $m$ stickers, then there is at least one sticker that none of the friends have.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \lim_{x\to \infty} \frac{ \cot^{-1} (\sqrt {x+1} -\sqrt x)}{\sec^{-1} \left(\frac{2x+1}{x-1}\right)^x} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the optimal asymptotic behavior of the ratio \( \frac{\|(-\Delta)^{\frac{s}{2}}F_d\|_{L^2}^2}{\|(-\Delta)^{\frac{s}{2}}f\|_{L^2}^2} \) as \( d \to \infty \), where \( F_d = f^{\otimes d} \) and \( f \in H^s(\mathbb{R}) \) with \( \|f\|_{L^2} = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of labeled trees on 9 vertices where exactly three vertices have a degree of 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the least ordinal \( \alpha \) such that \( V_{\alpha} \) satisfies the axioms of power, pairing, and infinity simultaneously, given that \( V_{\omega} \) does not satisfy the axiom of infinity, \( V_{\omega+1} \) does, and \( V_{\omega+\omega} \) satisfies all three axioms.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum constant \( c \) such that for any positive integer \( m \) and a \( kn \)-regular graph \( G \) with \( m \) vertices, the vertices of \( G \) can be colored with \( n \) different colors so that the number of monochrome edges is at most \( cm \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $v_1, v_2, \ldots, v_n$ are $n$ i.i.d. continuous random variables with the range $[\underline{v}, \bar{v}]$. Determine the limit of the expectation of the minimum of these random variables, $E[\min(v_1, v_2, \ldots, v_n)]$, as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a_1,a_2,\ldots,a_{2017}$ be the $2017$ distinct complex numbers which satisfy $a_i^{2017}=a_i+1$ for $i=1,2,\ldots,2017$. Compute$$\displaystyle\sum_{i=1}^{2017}\frac{a_i}{a_i^2+1}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \oint_C \boldsymbol{F} \cdot d\boldsymbol{l} \) for the vector field \( \boldsymbol{F} = x\hat{e}_x + z\hat{e}_y + x\hat{e}_z \), where the contour \( C \) consists of the following segments: a straight line from the origin to \((0, 0, b)\); a straight line from \((0, 0, b)\) to \((0, b, 0)\); and a straight line from \((0, b, 0)\) back to the origin.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{R\rightarrow \infty} \int_{C_R} \frac{6z^6 + 5z^5}{z^6 + z^5 + 10}dz \), where \( C_R=\{z \in \mathbb{C} : |z|=R \} \) for \( R>0 \), using the Residue theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the splitting field of the polynomial \(x^8 + x^6 + x^5 + x^4 + x^3 + x^2 + 1\) over the finite field \(\mathbb{F}_2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A car is equipped with one spare tire. The lifetimes of the four tires at the start of a long-distance journey are independent identically distributed exponential random variables with a mean of 5000 miles. The spare tire has an exponential lifetime with a mean of 1000 miles. Calculate the expected number of miles that can be covered without needing to visit a tire shop.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the derivative of \( \frac{\partial{A_{ij}\vec{x}}}{\partial{A_{ij}}} \), where \( A_{ij}\vec{x} \) is a vector whose \( i^{th} \) entry is given by \( A_{i1}x_1 + A_{i2}x_2 + \ldots \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_D \frac{e^{1/w^2}}{w-2}dw \), where \( D \) is the annulus centered at the origin with inner radius 1 and outer radius 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the derivative of \( \text{Tr}(S^T SUU^T) \) with respect to the matrix \( U \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of the parameter \( \alpha \in \mathbb{R} \) for which the function \( f: (0,1) \rightarrow \mathbb{R} \), defined by \( f(x) = \frac{x-1}{x^{\alpha} \ln(x)} \), is integrable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the residue of the function \( \frac{e^{2/z}}{(z+2)^3} \) at the point \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the residue of the function \( \frac{(z^6-1)^2}{z^5(2z^4-5z^2+2)} \) at \( z=0 \) without directly computing the 5th derivative using the residue formula.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum possible length of a path that includes all vertices in an Eulerian semi-graceful graph with $n$ vertices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T: P_2(\mathbb{R}) \to P_2(\mathbb{R})$ be a linear transformation defined by the following extensions: $T(x) = x-2$, $T(x^2-x) = x+2x^2$, and $T(x-2+x^2) = -1-x^2$. Compute $T(2+x^2)$. Provide your answer as a polynomial.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the characteristic function of the Dirac delta function?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two circles \( \Omega_1 \) and \( \Omega_2 \) have centers \( O_1 \) and \( O_2 \) with radii 3 and 4, respectively. Points \( A \) and \( B \) lie on circles \( \Omega_1 \) and \( \Omega_2 \), such that segment \( AB \) is tangent to both circles and intersects \( O_1O_2 \). Given that \( O_1O_2 = 25 \), find the area of quadrilateral \( O_1AO_2B \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x\rightarrow0} \frac{(4^{\arcsin(x^2)} - 1)(\sqrt[10]{1 - \arctan(3x^2)} - 1)}{(1-\cos\tan6x)\ln(1-\sqrt{\sin x^2})} \]
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Evaluate the infinite series: $$\sum_{n=1}^{\infty}\frac{g(n)}{n(n+1)}$$ where the function $g(n)$ is defined as follows: $$ g(n) = \begin{cases}0&n=1\\g(z)& n = 2z\\g(z) - 1&n = 2z + 1\end{cases} $$ Find the value of this summation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For all positive integers $n$, denote by $\sigma(n)$ the sum of the positive divisors of $n$ and $\nu_p(n)$ the largest power of $p$ which divides $n$. Compute the largest positive integer $k$ such that $5^k$ divides\[\sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)},\]where $N=6^{1999}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangular playing field of size \( 13 \times 2 \) is to be completely covered using an unlimited number of dominoes of sizes \( 2 \times 1 \) and \( 3 \times 1 \). The dominoes must cover the field without gaps or overlaps, and no domino should extend beyond the playing field. Additionally, all dominoes must be oriented the same way, with their long sides parallel to each other. Determine the number of such coverings possible.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $B$ be an orthogonal matrix with $\det(B) < 0$. Determine the range of possible values for $\det(B-I)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
8. You have been kidnapped by a witch and are stuck in the *Terrifying Tower*, which has an infinite number of floors, starting with floor 1, each initially having 0 boxes. The witch allows you to do the following two things:

- For a floor  $i$ , put 2 boxes on floor  $i+5$ , 6 on floor  $i+4$ , 13 on floor  $i+3$ , 12 on floor  $i+2$ , 8 on floor  $i+1$ , and 1 on floor  $i$ , or remove the corresponding number of boxes from each floor if possible.
- For a floor  $i$ , put 1 box on floor  $i+4$ , put 3 boxes on floor  $i+3$ , 6 on floor  $i+2$ , 5 on floor  $i+1$ , and 3 on floor  $i$ , or remove the corresponding number of boxes from each floor if possible.

At the end, suppose the witch wishes to have exactly  $n$  boxes in the tower. Specifically, she wants them to be on the first 10 floors. Let  $T(n)$  be the number of distinct distributions of these  $n$  boxes that you can make. Find  $\displaystyle\sum_{n=1}^{15} T(n)$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an acute triangle \(ABC\), the altitudes from \(A\), \(B\), and \(C\) intersect at the orthocenter \(H\). Let \(S\) be the midpoint of the circular arc \(BHC\) on the circumcircle of triangle \(BCH\). If the segments \(AS\) and \(AH\) are equal in length, determine the measure of angle \(\angle BAC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of \( \frac{m}{m^2} \) over \( k \), where \( m = m_P(V) \) and \( V = V(x^2-y^3, y^2-z^3) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the equation \( xf(x) - yf(y) = (x-y)f(x+y) \) for all real numbers \( x \) and \( y \).
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Context:
Find an irrational number $x$ such that $x^x$ is rational. Provide a specific example of such a number.
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Context:
Determine the largest interval around \(x=0\) where the function \(f(x) = x^5 - 5x + 3\) is invertible.
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Context:
There are twenty individuals numbered $1,2,...,20$. Each individual randomly and independently chooses 10 others from this group to call. Let $X$ be the total number of calls (incoming and outgoing) involving Individual 1, and let $Y$ be the number of calls between Individual 1 and Individual 2. Calculate the expected value of $X$ given that $Y=1$, denoted as $E(X|Y=1)$.
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Context:
An equilateral triangle with side length 3 is divided into nine smaller triangular cells. Initially, zeros are written in these cells. In one move, you can choose two numbers located in neighboring cells (sharing a side) and either add one to both or subtract one from both. Determine the value of \( n \) such that after several moves, the cells contain consecutive natural numbers \( n, n+1, \ldots, n+8 \) in some order.
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Context:
Determine the local order of convergence of the iteration procedure given by \( x_{n+1} = x_{n} - p \cdot \frac{f(x_{n})}{f'(x_{n})} \) for a function \( f \in C^{p+1}(\mathbb{R}) \) with a \( p \)-fold root at \( x^* \), where \( p \in \mathbb{N} \).
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Context:
Determine the dimension of $H^0(T, \mathcal{L})$, where $T$ is a complex torus of dimension 2 and $\mathcal{L}$ induces a $(m,n)$-polarization on $T$. Provide your answer as a single integer.
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Context:
The integers $0 \leq a < 2^{2008}$ and $0 \leq b < 8$ satisfy the equivalence $7(a+2^{2008}b) \equiv 1 \pmod{2^{2011}}$. Then $b$ is
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Context:
Three balls are randomly and independently thrown into buckets labeled with positive integers such that the probability of a ball being thrown into bucket \(i\) (for \(i = 1, 2, \ldots\)) is \(2^{-i}\). It is not allowed for a bucket to contain more than one ball. The probability that the three balls fall into buckets labeled in an arithmetic sequence is given by \(\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. Find \(p+q\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the \( \mathbb{C} \)-dimension of the vector space \( \mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \), where \( \mathbb{R} \) is the field of real numbers and \( \mathbb{C} \) is the field of complex numbers. Consider \( \mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \) as a \( \mathbb{C} \)-vector space via \( a(b\otimes c) := ab \otimes c, \) for \( a, b, c \in \mathbb{C} \).
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Context:
Semicircle \( \Gamma \) has diameter \( \overline{AB} \) of length14. Circle \( \Omega \) lies tangent to \( \overline{AB} \) at a point \( P \) and intersects \( \Gamma \) at points \( Q \) and \( R \) . If
 \( {QR} = 3\sqrt{3} \) and \( \angle {QPR} = {60}^{ \circ  } \) ,then the area of \( \bigtriangleup {PQR} \) is \( \frac{a\sqrt{b}}{c} \) ,where \( a \) and \( c \) are relatively prime positive integers,and
 is a positive integer not divisible by the square of any prime. What is \( a + b + c \) ?
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Context:
Evaluate the contour integral \( \oint_{|z|=1} z(z+1) \cosh\left(\frac{1}{z}\right) \, dz \) where the contour is the unit circle \(|z|=1\) oriented counterclockwise.
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Context:
Determine the maximum possible rank of a set such that the ranks of all its finite subsets are bounded by a given ordinal \( \beta \).
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Context:
In the 22nd century (the years 2101 through 2200, inclusive), find the expected number of years that have either a record low or a record high rainfall, assuming that annual rainfalls are i.i.d. draws from a continuous distribution.
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Context:
Determine the asymptotic behavior of the expression \( n^3 \left ( 1 - \left ( \frac{c \log n}{n} \right )^3 \right )^{n^3} \) as \( n \rightarrow \infty \), where \( c > 0 \).
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Context:
Determine whether the function \( \frac{1}{|x|^\alpha} \) is integrable at infinity in dimensions \( d \geq 2 \) for any \( \alpha \).
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Context:
Evaluate the integral: $$\int_0^1 A\sin(L+r\cos(2\pi n x +\phi))\cos(2\pi nx + \phi)\ \text{d}x$$ where $n$ is a natural number, and $A$, $L$, $r$, and $\phi$ are real numbers.
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Context:
Find the derivative of the trace of the product of three matrices, \( \text{tr}(ABC) \), with respect to the matrix \( B \), where \( A \) is an \( m \times n \) matrix, \( B \) is an \( n \times p \) matrix, and \( C \) is a \( p \times q \) matrix.
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Context:
Given the force of interest \( \delta_t = \frac{2t}{10+t^2} \) for \( t \geq 0 \), calculate \( \ddot{s}_{\bar{2}|} \).
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Context:
Determine a generator of the multiplicative group \( \mathbb{Z}^{*}_{242} \).
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Context:
Determine the number of spheres that are tangent to all the faces of a given triangular pyramid $T$. Provide your answer as a single integer.
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Context:
Let $\mathbb{k}$ be a field, and consider the vector space $\mathbb{k}_{10}[x]$ of polynomials in $x$ with degree at most 10. Define the shift operator $T: \mathbb{k}_{10}[x] \to \mathbb{k}_{10}[x]$ such that $T(f(x)) = f(x+1)$. Determine the minimal polynomial $\mu(t)$ of $T$. \( \boxed{\mu(t)} \)
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Context:
Find the smallest integer \(x\) such that the absolute difference between \(\pi(x)\) and \(\operatorname{li}(x) - \frac{1}{2}\operatorname{li}(\sqrt{x})\) is greater than the absolute difference between \(\pi(x)\) and \(\operatorname{li}(x)\).
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Context:
Determine the number of elements in the conjugacy class of the permutation $(123)(456)$ in the alternating group $A_7$.
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Context:
Construct a polynomial over the finite field \( \mathbb{Z}_p \) with a degree less than \( p^3(p-1) \) that evaluates to 0 at the origin and 1 at all other points.
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Context:
A point $P$ lies inside equilateral triangle $ABC$. Let $A'$, $B'$, $C'$ be the feet of the perpendiculars from $P$ to $BC, AC, AB$, respectively. Suppose that $PA = 13$, $PB = 14$, and $PC = 15$. Find the area of $A'B'C'$.
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Context:
Given the function \( f(x) = |1 - 2x| \) for \( x \in [0, 1] \), determine the number of solutions to the equation \( f(f(f(x))) = \frac{1}{2} x \).
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Context:
Determine the constant $c_0$ in the power series representation of the shifted exponential function $y = e^{-a(gx-b)^{2}}$. The power series is given by:

$$e^{-a(gx-b)^{2}} = c_{0}\sum_{k=1}^{\infty}\frac{(-2ag)^{k}}{(2k)!!}(gx-b)^{k}$$

where the coefficients $c_{2k}$ are defined by the recurrence relation $c_{2k} = \frac{(-2ag)^{k}}{(2k)!!}c_{0}$ for $k = 1, 2, 3, \dots$. What is the appropriate value for $c_0$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the periodicity of the entire function $g(z)$, given that $g(iy)$ and $g(2+iy)$ are real for all $y \in \mathbb{R}$. Provide your answer as the period of the function.
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Context:
Determine the value of \( t \) such that the test \( \delta = \chi[\bar{X}< a - t\sqrt{\frac{\sigma^2}{n}}] + \chi[\bar{X}> b + t\sqrt{\frac{\sigma^2}{n}}] \) for testing \( H_0: \theta \in [a, b] \) versus \( H_1: \theta \not\in [a, b] \) has a size of 0.05.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the isomorphism group of the directed graph that becomes isomorphic to the complete bipartite graph $K_{3,3}$ when all its edges are made undirected.
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Context:
Determine the convergence of the series \( \sum\limits_{n=1}^{\infty} \frac{2n+1}{3n^2+1} x^n \) in terms of \( x \).
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Context:
Let \( \xi = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5} \) be a complex fifth root of unity. Define \( a = 20\xi^2 + 13 \xi \), \( b = 20\xi^4 + 13\xi^2 \), \( c = 20\xi^3 + 13\xi^4 \), and \( d = 20\xi + 13\xi^3 \). Calculate \( a^3 + b^3 + c^3 + d^3 \).
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Context:
Given the function $f: \mathbb{N}^+ \rightarrow \mathbb{N}$ defined by the rules $f(2^n) = n^2$ and $f(3n) = f(2n) + 5$, find the value of $f(3^{100})$. Provide your answer as a single integer.
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Context:
Given the equations:
\[ \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} = 9, \]
\[ \frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} = 32, \]
\[ \frac{a^3}{b+c} + \frac{b^3}{c+a} + \frac{c^3}{a+b} = 122, \]
find the value of \(abc\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the series \( \sum_{k=0}^{n/2} C(n-k, k) \), where \( C(n, k) \) denotes the number of combinations of \( n \) items taken \( k \) at a time.
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Context:
Determine the rank of the first homology group of the double branched cover of a knot in $S^3$.
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Context:
Determine the metric $d(A,B)$ for the sets $A = \{ x \mid x^2 \leq x + 1 \}$ and $B = [-1, 1]$ in the measure space $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)$. The metric should be a real-valued non-negative function, symmetric, and satisfy the triangle inequality. What is the value of $d(A,B)$?
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Context:
Calculate the area of the sphere defined by \(x^2 + y^2 + z^2 = a^2\) that lies outside the regions defined by the cylinders \(x^2 + y^2 - ax = 0\) and \(x^2 + y^2 + ax = 0\).
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Context:
Given a polynomial $f$ of degree $1007$ such that $f(k) = 2^k$ for $0 \le k \le 1007$, determine the value of $f(2015)$. \( \boxed{} \)
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Context:
Compute the expected value \( E[e^{X^2Y}] \) where \( X \sim \text{Exp}(2) \) and \( Y \sim \text{Uniform}(-1, 1) \) are independent random variables.
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Context:
Given a linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}_2[x] \) defined by \( T(1,0) = 1+x \) and \( T(1,1) = 1-x \), determine \( T^{-1}(2x) \).
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Context:
In how many distinct ways can 5 identical red beads, 3 identical green beads, and 2 identical blue beads be arranged in a circular necklace?
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Context:
Find the radius of the circle formed by the intersection of the plane $x + 2y - z = 4$ and the sphere $x^2 + y^2 + z^2 - x + z - 2 = 0$.
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Context:
Evaluate the expression \( \frac{J_{6,3}}{J_{5,2}} \), where \( J_{m, n} = \int_{0}^{\pi/2} \cos^m x \cos nx \, dx \).
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Context:
Evaluate the improper integral expression: \[ \int_{-\infty}^{e} \ln x \,dx - \int_{-\infty}^{e} \frac{x}{e} \, dx. \]
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Context:
 Cindy cuts regular hexagon $ABCDEF$ out of a sheet of paper. She folds $B$ over $AC$, resulting in a pentagon. Then, she folds $A$ over $CF$, resulting in a quadrilateral. The area of $ABCDEF$ is $k$ times the area of the resulting folded shape. Find $k$.
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Context:
Jefferson Middle School has the same number of boys and girls. Three-fourths of the girls and two-thirds of the boys went on a field trip. What fraction of the students were girls?
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Context:
Evaluate the integral: \[ \int_{0}^{\infty}\text{sech}^2(x+\tan(x))\,dx \]
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Context:
Given real numbers \(a\), \(b\), and \(c\) such that:
\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 4 \left( \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a} \right) = \frac{c}{a+b} + \frac{a}{b+c} + \frac{b}{c+a} = 4 \]
Determine the value of \(abc\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral: $$I=\int_{-1}^1 \left(\frac{1}{x}+1-\frac{\sqrt{1-x^2}}{x}\right )\arctan\left(\frac{2}{x^2}\right)dx.$$
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Context:
Calculate \( \lim_{x \rightarrow 0}{\frac{\int_{0}^{x} \frac{\sin(t)}{t}-\tan(x)}{2x(1-\cos(x))}} \) using Maclaurin polynomials.
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Context:
Given an ellipse with a known semi-major axis and a circle with a known radius, calculate the semi-minor axis of the ellipse if it is tangent to the circle at one point.
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Context:
How many of the rearrangements of the digits $123456$ have the property that for each digit, no more than two digits smaller than that digit appear to the right of that digit? For example, the rearrangement $315426$ has this property because digits $1$ and $2$ are the only digits smaller than $3$ which follow $3,$ digits $2$ and $4$ are the only digits smaller than $5$ which follow $5,$ and digit $2$ is the only digit smaller than $4$ which follows $4.$
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Context:
Determine the number of possible values of \(\theta\), where \(0 < \theta < \pi\), for which the system of equations
\[ (y + z)\cos 3\theta = xyz\sin 3\theta \]
\[ x\sin 3\theta = \frac{2\cos 3\theta}{y} + \frac{2\sin 3\theta}{z} \]
\[ xyz\sin 3\theta = (y + 2z)\cos 3\theta + y\sin 3\theta \]
has a solution \((x_0, y_0, z_0)\) with \(y_0z_0 \neq 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( \triangle ABC \), the sides are \( AB = 30 \), \( BC = 40 \), and \( CA = 50 \). Squares \( A_1A_2BC \), \( B_1B_2AC \), and \( C_1C_2AB \) are constructed externally on the sides of \( \triangle ABC \). The lines \( A_1A_2 \), \( B_1B_2 \), and \( C_1C_2 \) intersect pairwise at points \( P \), \( Q \), and \( R \). Determine the length of the shortest altitude of triangle \( \triangle PQR \).
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Context:
What is the proportion of prime numbers \(p\) such that the Legendre symbols \(\left(\frac{2}{p}\right) = 1\) and \(\left(\frac{-5}{p}\right) = -1\)?
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Context:
In a right triangle \(ABC\) with hypotenuse \(AB\), a point \(P\) is marked on leg \(AC\). Let \(D\) be the foot of the perpendicular from \(A\) to line \(BP\), and \(E\) be the foot of the perpendicular from \(P\) to side \(AB\). A point \(T\) is chosen such that lines \(TA\) and \(TP\) are tangent to the circumcircle of triangle \(PAB\), with \(O\) as the center of this circumcircle. The perpendicular from \(T\) to line \(DE\) intersects line \(BC\) at point \(Q\). The line through \(C\) parallel to \(OQ\) intersects segment \(BO\) at point \(K\). Determine the measure of angle \(\angle OKE\).
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Context:
Find the product of all possible values of $a$ such that the system of equations \(|2x| - y = 5\) and \(x - |2y + 2| = a\) has an odd number of distinct solutions.
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Context:
Determine the number of distinct compact Hausdorff spaces, up to homeomorphism, that exist with a given uncountable cardinality \( \kappa \).
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Context:
Find the value of \( a \) such that the point on the curve \( f(x) = e^x \) closest to the line \( y = \frac{x}{a} - 1 \) has a \( y \)-coordinate of \( \frac{1}{e} \).
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Context:
Determine the smallest integer $j$ such that it is possible to fill the cells of a $10 \times 10$ table with numbers from 1 to 100, ensuring that every set of 10 consecutive numbers lies within some $j \times j$ sub-square of the table.
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Context:
Let \( \text{Revolution} (x) = x^3 + Ux^2 + Sx + A \), where \( U \), \( S \), and \( A \) are integers such that \( U + S + A + 1 = 1773 \). Given that \( \text{Revolution} \) has exactly two distinct nonzero integer roots \( G \) and \( B \), find the minimum value of \( |GB| \).
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Context:
In a group \( G \) of individuals, each person can be a friend of another. Determine the smallest number of individuals that must be chosen as subgroup leaders so that every person in the group has at least one friend who is a leader. Express your answer in terms of graph theory, where this problem involves finding the minimum number of radial-subgraphs.
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Context:
Find the sum of the two positive integers $n$ such that when $1^3+2^3+3^3+\cdots +n^3$ is divided by $n+5$, the remainder is $17$.
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Context:
Find the order of the group given by the presentation: $$\langle a, b\mid a^3, b^3, [a, [b, a^{-1}]], [b, [b, a^{-1}]]\rangle.$$
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Context:
Given a projective curve \( C \) in \( \mathbb{P}_\mathbb{C}^2 \), if a point \( p \in C \) has multiplicity \( m \), and \( \mathcal{O}_p(C) \) is the local ring of \( C \) at \( p \) with its maximal ideal \( \mathfrak{m} \), find the dimension of \( \mathfrak{m}^k/\mathfrak{m}^{k+1} \) over \( \mathbb{C} \) for \( k < m \).
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Context:
Determine the sign of the permutation represented by the matrix: \[ \begin{pmatrix} 1 & \dots & p & p+1 & \dots & p+q \\ q+1 & \dots & p+q & 1 & \dots & q \end{pmatrix} \]
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Context:
In a triangulation of a polygonal domain $\Omega$ in $\mathbb{R}^3$, suppose $z$ is an interior vertex shared by $m$ tetrahedra. Determine the number of interior faces adjacent to $z$ in terms of $m$. Express your answer using $m$.
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Context:
Determine the number of zeros of the function \( y = \tan(2015x) - \tan(2016x) + \tan(2017x) \) in the interval \([0, \pi]\).
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Context:
Determine the cardinality of the set $G$ of all entire functions $g$ such that $g(x) = x^k$ for all $x \in (-1, 1)$, where $k$ is an odd integer $\geq 1$. \( K \) is given as an odd integer $\geq 1$. \( G \) is the set of all entire functions $g$ such that $g(x) = x^k$ for all $x \in (-1, 1)$. What is the cardinality of $G$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $G$ be a group of order $p^3$, where $p$ is a prime number. If $|Z(G)|=p$, determine the number of conjugacy classes of $G$.
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Context:
Consider a 4 × 4 grid of squares, where each square can be colored either black or white. Two colorings are considered identical if one can be transformed into the other by any combination of flipping, rotating, or swapping the two colors (changing all black squares to white and vice versa). Determine the number of non-identical colorings of the grid.
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Context:
Determine the minimum degree $n$ of the Taylor polynomial $T_n(x)$ for the function \( f(x) = \int_{\frac{\pi}{2}}^{x} \frac{\cos(t)}{t - \frac{\pi}{2}}dt \) around \( a=\frac{\pi}{2} \) such that \( \mid f(\pi) - T_n(\pi) \mid \leq \frac{1}{100} \).
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Context:
Determine the number of connected components of the set \( L \cap \Delta \), where \( L = \alpha b \) is a line in \( \mathbb{R}^n \) with \( \alpha \in \mathbb{R} \) and \( b \in \mathbb{R}^n \) is fixed, and \( \Delta = \{ x \in \mathbb{R}^n : \pi(x) \text{ has roots in the open unit disk of } \mathbb{C} \} \). Assume \( n > 2 \).
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Context:
Find the reproducing kernel Hilbert space (RKHS) induced by the kernel $k: X \times X \to \{0, 1\}$, where $X$ is an uncountably infinite set such as $X = [0, 1]$ or $X = \mathbb{R}$. The kernel is defined as $k(x, y) = \delta_{x, y}$, where $\delta_{x, y} = 1$ if $x = y$ and $0$ otherwise.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sum of ages of some children born at the intervals of 3 years each is 60 years. The age of the youngest child is 6 years. How many children are there?
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Context:
An $m\times n\times p$ rectangular box has half the volume of an $(m+2)\times(n+2)\times(p+2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\le n\le p.$ What is the largest possible value of $p$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the operator norm of the linear operator $A$ defined by \( Ax=(x_1+x_2, x_2+x_3, \ldots) \) for all $x \in \ell_2$. Given that the operator is bounded with \( \sum_{i=0}^\infty|x_i+x_{i+1}|^2 \leq 6\sum_{i=0}^\infty|x_i|^2 \), find the norm of $A$ by evaluating the supremum of $\|Ax\|$ for $\|x\|=1$. \( \| \cdot \| \) denotes the $\ell_2$ norm.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a regular triangular prism \(P-ABC\), a hemisphere is inscribed such that its base coincides with the base of the prism, and all three lateral faces of the prism are tangent to the hemisphere. Given that the radius of the hemisphere is 1, determine the height of the prism that minimizes its volume.
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Context:
Calculate the integral \( \int_{[0,1]} x^3 \, d\mu_F \), where \( F \) is the Cantor function.
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Context:
Find all complex numbers $z$ such that $iz + \sqrt{1-z^2} \le 0$, where the square root is defined as follows: for $z = re^{i\theta}$ with $-\pi < \theta \le \pi$, $\sqrt{z} = \sqrt{r}e^{\frac{1}{2}i\theta}$ and $\sqrt{r} \ge 0$. Determine the set of all such $z \in \mathbb{C}$.
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Context:
Determine the minimum diameter that can be ensured when a planar set with a unit diameter is divided into $k$ pieces.
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Context:
Determine the number of roots of the equation \( \overbrace{f(f(\ldots f}^{10 \text{ times } f}(x) \ldots)) + \frac{1}{2} = 0 \), where \( f(x) = |x| - 1 \).
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Context:
Determine the real rank of the $C^*$-algebra $C_0(\mathbb{R})$. Provide your answer as a single value.
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Context:
Determine the number of real solutions for the equation \(3^{x+2}=\frac{1}{x^2}\).
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Context:
Find the arithmetic mean of all positive rational numbers less than 1 that can be expressed with a 10-digit period in decimal representation, where all 10 digits are distinct.
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Context:
In a standard deck of 52 cards numbered from 1 to 52, 26 cards are selected at random without replacement. Determine the number of ways to select these 26 cards such that the sum of the reciprocals of their ranks equals exactly 2.
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Context:
Determine the order of subgroups that appears most frequently in the symmetric group $S_n$. Specifically, find the order $k$ for which the number of subgroups of $S_n$ with order $k$ is maximized.
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Context:
Determine the degree of accuracy for an $n$-point Gauss-Legendre quadrature when applied to the multidimensional integral \( \int_{a}^{b}{\int_{p}^{q}{f(x,y) \,dx \,dy}} \).
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Context:
Determine the values of $x$ in the interval $(-3, 17)$ for which the series \( \sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n[\log (n+1)]^2} \) converges.
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Context:
Given a list of numbers \([a_1, a_2, a_3, \ldots, a_n]\) and a target value \(t\), determine the number of non-negative integer solutions to the equation \(x_1a_1 + x_2a_2 + \ldots + x_na_n = t\), where each \(x_i\) is a non-negative integer.
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Context:
Let \( X = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 = 1\} \cup ([-1,1] \times \{0\}) \cup (\{0\} \times [-1,1]) \). Determine the maximum number of distinct points \( n_0 \) that can be removed from \( X \) such that the remaining set \( X - \{p_1, p_2, \ldots, p_{n_0}\} \) is still connected. What is the value of \( n_0 \)?
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Context:
Find the infimum of the integral \( \int_{\mathbb{R}^5} u^2 - \ln(1+u^2) \, \text{dx} \) subject to the constraint \( \int_{\mathbb{R}^5} u^{14} = 1 \).
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Context:
Suppose \( f:D(0,1) \longrightarrow \mathbb{C} \) is a holomorphic function, where \( D(0,1) = \{z \in \mathbb{C} \mid |z|<1\} \), and assume the maximum \(|f(z)| \le 2\). Estimate \(|f^{(3)}(i/3)|\) using Cauchy's estimate.
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Context:
A group of order 15 acts on a set of order 22 with no fixed points. Determine the number of orbits in this group action.
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Context:
Find the smallest positive integer \( n \) such that all its prime factors are greater than 18, and it can be expressed as \( n = a^3 + b^3 \) for positive integers \( a \) and \( b \).
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Context:
Determine the intervals on which the series \( \sum_{n=1}^\infty ne^{-nx} \) converges uniformly.
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Context:
Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is $50\%$ more than the regular. After both consume $\tfrac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and $2$ additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?
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Context:
Let \( A \) be a non-singular matrix that satisfies \( A^2B = A \) for some matrix \( B \). Given the set \( S = \{-1, 0, 1\} \), and \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) where \( a, b, c, d \in S \), determine the probability that \( \det(A + B) = \det A + \det B \).
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Context:
Determine the interval of convergence for the series: \[ \sum_{n=1}^{\infty}\frac{((n+1)x)^n}{n^{n+1}} \]
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Context:
Evaluate the limit: \[ \lim_{n \rightarrow \infty} \left(\left(\frac{f(n)}{g(n)}\right)^q \left(\frac{1-f(n)}{1-g(n)}\right)^{(1-q)}\right)^n \] where \( n \in \mathbb{N} \), \( f(n),g(n) \in [0,1] \), \( q\in (0.5,1) \), \( f(n) \leq g(n) \), and both \( f(n) \) and \( g(n) \) converge to some real number in \([0,1]\).
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Context:
In a game, three players, $A$, $B$, and $C$, compete to earn the highest score. The score of each player is represented by a Poisson distributed random variable: $A$ with parameter $\lambda_A = 2$, $B$ with $\lambda_B = 3$, and $C$ with $\lambda_C = 4$. The scores are independent. Given that the total combined score of the players is $A + B + C = 6$, with the scores being $1, 2$, and $3$, respectively, but not knowing which player got which score, calculate the conditional probability that player $C$ had the highest score, i.e., $C = 3$. \( \boxed{?} \)
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Context:
Find all positive integers \( n > 2 \) such that the only functions \( f: \mathbb{Z} \to \mathbb{Z} \) satisfying the following property are linear functions of the form \( f(x) = bx + c \), where \( b \) and \( c \) are integers: If \( a_1, a_2, \dots, a_n \) form an arithmetic progression, then the numbers \( f(a_1), f(a_2), \dots, f(a_n) \) also form an arithmetic progression, possibly in a different order.
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Context:
Calculate the flux of the vector field \( \boldsymbol{F}(x,y,z) = \nabla \phi + \nabla \times \boldsymbol{G} \) out of the cylindrical surface of the region \( D = \{(x,y,z) : x^2 + y^2 \leq 1, 0 \leq z \leq 1\} \), where \( \phi(x,y,z) = x^2 - y^2 + \frac{3}{2\pi}z^2 \) and \( \boldsymbol{G}(x,y,z) = \frac{3}{\pi}zy\boldsymbol{i} + \arctan(x^2)\boldsymbol{j} + x^5 y \sqrt{z^2 +1}\boldsymbol{k} \). Use the divergence theorem and the identity \( \nabla \cdot(\nabla \times \boldsymbol{F}) = 0 \).
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Context:
Find the value of $a$ such that the following limit holds true:

$$\lim_{x\to\frac{\pi}{6}}\left(\sqrt3 \tan x\right)^{\frac{(a+1)\tan(3x)}{\sin x}} = e^{-\frac{8\sqrt3}{3}}$$

After applying $\lim_{x\to 0}(1 + x)^{\frac{1}{x}}$, we get:

$$e^{\lim_{x\to\frac{\pi}{6}}\left(\frac{\sqrt3 \tan x - 1}{\sin x}\right) (a+1)\tan(3x)} = e^{-\frac{8\sqrt3}{3}}$$

Determine the value of $a$.
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Context:
Determine the value of \( \alpha \) that results in the fastest convergence of the given iteration formula.
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Context:
Evaluate the infinite series: \[ \sum_{n=1}^\infty \left(1 - \left(1+\frac{1}{n}\right)^{-n}\right). \]
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Context:
Given the sphere with the parametric representation \(X(\theta,\phi) = (r \sin\phi \cos\theta, r \sin\phi \sin\theta, r \cos \phi)\), find the Christoffel symbol \(\Gamma^1_{22}\).
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Context:
Let proposition $p$: $(x-2)^2 \leq 1$, and proposition $q$: $x^2 + (2a+1)x + a(a+1) \geq 0$. If $p$ is a sufficient but not necessary condition for $q$, find the range of real numbers $a$.
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\boxed{(-\infty, -4] \cup [-1, +\infty)}


Context:
Find a non-zero solution to the equation \( K(Q,m,F_\text{x})=\frac{{F_\text{x}}^2\cdot\left(m-1\right)}{\sqrt{{\left(m\cdot{F_\text{x}}^2-1\right)}^2+{F_\text{x}}^2\cdot{\left({F_\text{x}}^2-1\right)}^2\cdot{\left(m-1\right)}^2 \cdot Q^2}} \) for any values of \( m \) and \( Q \).
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Context:
Find a basis for a generalized eigenspace of the linear operator $T$ on $P_2(\mathbb{R})$, where $T(f(x))=2f(x)-f'(x)$, consisting of a union of disjoint cycles of generalized eigenvectors.
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Context:
In a game played inside the unit disc D, Player A chooses a point p0 ∈ D. At each step n, Player B draws a line ℓn through pn−1, and Player A chooses a point pn ∈ D ∩ ℓn. Player B wins if they can force the sequence (pn) to converge. If Players A and B play randomly, what is the probability that Player B wins?
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Context:
Let $|\alpha| < \frac{\pi}{2}$. Evaluate the following definite integral.
$$\int_{0}^{\frac{\pi}{2}} \frac{dx}{\{\sin (a+x)+\cos x\}^{2}}$$
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Context:
Evaluate the definite integral \( \int_0^1 f(x) \, dx \), where \( f(x) = \begin{cases} 2x\sin\left(\frac{1}{x}\right) - \cos\left(\frac{1}{x}\right) & : x \in (0,1] \\ 0 & : x = 0 \end{cases} \). Determine if the integral is defined, and if so, find its value.
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Context:
Suppose 10 people consisting of 5 married couples are randomly seated around a round table. Let \( X \) be the number of wives who are seated next to their husbands. Calculate the variance \( \text{Var}(X) \).
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Context:
Evaluate the limit: \[ \lim_{n \to \infty} \int_0^n \frac{dx}{1+n^2\cos^2x} \]
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Context:
Let \( z = f(x,y) \) be a differentiable function such that \( f(3,3) = 1, \) \( f_x(3,3) = -2, \) \( f_y(3,3) = 11, \) \( f(2,5) = 1, \) \( f_x(2,5) = 7, \) and \( f_y(2,5) = -3. \) Suppose \( w \) is a differentiable function of \( u \) and \( v \) satisfying the equation \( f(w,w) = f(uv, u^2+v^2) \) for all \((u,v)\). Find \( \frac{\partial w}{\partial u} \) at \((u,v,w) = (1,2,3)\).
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Context:
Evaluate the infinite sum: \[ \sum_{n=1}^{\infty}\frac{2H_{n}}{(n+1)^2} \] where \( H_n \) is the \( n \)th harmonic number.
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Context:
Evaluate the product \( \mathop{\prod }\limits_{{k = 0}}^{{2}^{1999}}\left( {4{\sin }^{2}\frac{k\pi }{{2}^{2000}} - 3}\right) \)
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Context:
Evaluate the exact value of the alternating series \( \displaystyle \sum_{n=0}^\infty \frac{(-1)^{n+1}n}{(2n+1)!} \cdot \pi^{2n} \).
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Context:
Consider the transfer function matrix \(G(s)\) of a continuous-time system given by:
\[G(s) = \begin{bmatrix}\frac{1}{s^2+2s}&\frac{s+1}{s} \\ -\frac{1}{s+1} & \frac{1}{s^2+4s+3} \end{bmatrix}\]
Suppose that \((A,B,C,D)\) is a state-space model that forms a minimal realization. What is the dynamic order of this system (i.e., the dimension of the state-space)?
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Context:
Determine if there exists a natural number $n$ such that the equation $n = x^2 + y^2$ has more than three solutions in natural numbers $x$ and $y$. If such an $n$ exists, provide an example.
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Context:
Determine the dimension of the quotient ring $\mathbb{R}[x]/J$, where $\mathbb{R}[x]$ is the polynomial ring over $\mathbb{R}$, and $J=\{f(x)\in \mathbb{R}[x]\mid f(0)=f'(0)=f''(0)=0\}$.
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Context:
Determine the frequency \( \omega(x) \) of the function \( x(t) = \sin(t) + \sin(\pi t) \), where the frequency is defined by \( \omega(x) = \limsup_{T \to \infty} \frac{\pi}{T} \nu(x,T) \), and \( \nu(x,T) \) represents the number of zeros of \( x(t) \), counting with their multiplicities, on the interval \((0,T]\).
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Context:
Compute the limit: \[ \lim \limits_{x \rightarrow 0} \dfrac{\sin(\sin(\sin(x)))}{\cos\left(\frac{\pi}{2}\cos(x)\right)}\cdot x^k \]
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Context:
Find a function that is not bounded below and whose second derivative is bounded above.
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Context:
Determine the maximum possible shortest distance between any two points among 8 points placed either on the boundary or inside an equilateral triangle with a side length of 1.
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Context:
Let $k$ be a field, $f(x) \in k[x]$ be an irreducible polynomial over $k$, and $\alpha$ be a root of $f$. If $L$ is a field extension of $k$, determine what $k(\alpha) \otimes_k L$ is isomorphic to.
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Context:
Determine the values of $p$ for which the Fourier transform exists for functions in the $L^p$ space.
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Context:
设实数 $a$ 满足 $a<9 a^{3}-11 a<|a|$, 则 $a$ 的最小值的$\sqrt{3}$是 $\qquad$.
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Context:
For $ a\in\mathbb{R}$, let $ M(a)$ be the maximum value of the function $ f(x)=\int_{0}^{\pi}\sin (x-t)\sin (2t-a)\ dt$.

Evaluate $ \int_{0}^{\frac{\pi}{2}}M(a)\sin (2a)\ da$.
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Context:
Evaluate the limit: \[ \lim_{x\to\infty} \frac{\sum_{k=0}^{x/2}\binom{x}{2k}2k(x-2k)}{\sum_{k=0}^{x/2}\binom{x}{2k}(x-2k)^{2}} \]
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Context:
Find the smallest number of mutually visible pairs of birds among 155 birds sitting on the boundary of a circle, where two birds are mutually visible if the angle at the center between their positions is at most 10 degrees.
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Context:
Determine the order of the group of all 2x2 matrices with entries from the integers modulo a prime number \( p \).
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Context:
Sumith lists out the positive integer factors of $12$ in a line, writing them out in increasing order as $1$, $2$, $3$, $4$, $6$, $12$. Luke, being the mischievious person he is, writes down a permutation of those factors and lists it right under Sumith's as $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$. Luke then calculates$$gcd(a_1, 2a_2, 3a_3, 4a_4, 6a_5, 12a_6).$$Given that Luke's result is greater than $1$, how many possible permutations could he have written?
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Context:
Determine the necessary and sufficient condition that the coefficients of the cubic equation \(x^{3}+a x^{2}+b x+c=0\) must satisfy if its roots form a geometric progression.
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Context:
Evaluate the limit: $$\lim_{a\to\infty}\left[\int_{t}^{t+\frac{1}{2}}\psi(x)\,dx -\frac{\psi\left(t+\frac{1}{4}\right)+\psi\left(t+\frac{1}{2}\right)}{4}\right]$$
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Context:
Given the finite difference approximations:
- \( \frac{\partial}{\partial x} \rightarrow O(\Delta^4) \)
- \( \frac{\partial^2}{\partial x^2} \rightarrow O(\Delta^2) \)

Determine the resulting order of accuracy for the approximation of \( \frac{\partial^3 f}{\partial x^3} \) using the approach:
\[ \frac{\partial}{\partial x} \left(\frac{\partial^2 f}{\partial x^2}\right) \rightarrow O(\Delta^?) \]
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Context:
Let \( S \) be a nonempty closed bounded convex set in the plane. Let \( K \) be a line and \( t \) a positive number. Let \( L_1 \) and \( L_2 \) be support lines for \( S \) parallel to \( K \), and let \( \overline{L} \) be the line parallel to \( K \) and midway between \( L_1 \) and \( L_2 \). Let \( B_S(K,t) \) be the band of points whose distance from \( \overline{L} \) is at most \( \left( \frac {t}{2} \right) w \), where \( w \) is the distance between \( L_1 \) and \( L_2 \). Determine the smallest \( t \) such that \[ S \cap \bigcap_K B_S (K, t) \ne \emptyset \] for all \( S \). Here, \( K \) runs over all lines in the plane.
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Context:
Find the maximum value of $p$, where $m$, $n$, and $p$ are natural numbers satisfying the equation $2mnp = (m + 2)(n + 2)(p + 2)$, with the condition $m \leq n \leq p$. Provide your answer as a single integer.
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Context:
5. The sphere touches all the edges of a tetrahedron, in which there is a vertex from which the edges emanate pairwise perpendicularly. Find the radius of this sphere if the radius of the sphere circumscribed around this tetrahedron is $3 \sqrt{3}$.
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Context:
Given the summation \( S_n=\sum^{n-1}_{i=0}\sum^{i-1}_{j=0}\sum^{j-1}_{k=0} (i+j+k) \), there exist positive integers \( A \) and \( B \) such that \( \frac{1}{S_3}+\frac{1}{S_4}+\frac{1}{S_5}+\dots=A-\frac{2\pi^2}{B} \). Find the value of \( A+B \).
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Context:
Find the product of all quadratic nonresidues in $\mathbb{Z}_{103}$.
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Context:
Evaluate the contour integral \( \int_\gamma \frac{1}{z+1} \, \mathrm{d}z \), where \( \gamma(t) \) is a path in the complex plane such that \( \gamma(0) = -i \) and \( \gamma(1) = i \), with \( \gamma(t) \in \mathbb{C} \setminus (-\infty,-1] \) for all \( t \in [0,1] \).
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Context:
Calculate the Jacobian determinant for the transformation from variables \((u, v, w)\) to \((x, y, z)\) given by the equations:

\[ u = x + y + z, \quad v = x + 2z, \quad w = x + 2y + 3z. \]
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Context:
In a double elimination tournament with a varying skill levels and a few byes assigned to the top players, a total of 63 matches are conducted. How many players participated in this tournament?
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Context:
Given the harmonic function \(u(x,y) = x^3 + ax^2y + bxy^2 + 2y^3\) and its harmonic conjugate \(v(x,y)\), where \(v(0,0) = 1\), find the value of \(|a + b + v(1,1)|\).
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Context:
In a $101 \times 101$ square grid, each unit square contains one chocolate, and there are doorways between adjacent squares. It costs $\$1$ to pass through a doorway. Jake can start from any square and must collect all $5101$ chocolates. What is the minimum amount of money Jake needs to ensure he can collect all the chocolates?
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Context:
Determine the positive integer $k$ such that the non-constant complex analytic function $f$ on $\mathbb{C}$ satisfies the functional equation:
\[ f(s) = (-1)^{k} f(2-s) \]
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Context:
Let triangle $\triangle ABC$ have side lengths $AB = 7, BC = 8,$ and $CA = 9$, and let $M$ and $D$ be the midpoint of $\overline{BC}$ and the foot of the altitude from $A$ to $\overline{BC}$, respectively. Let $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, such that $m\angle{AEM} = m\angle{AFM} = 90^{\circ}$. Let $P$ be the intersection of the angle bisectors of $\angle{AED}$ and $\angle{AFD}$. If $MP$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b,$ and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, then find $a + b + c$.
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Context:
At least how many weighings of a balanced scale are needed to order four stones with distinct weights from the lightest to the heaviest?
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Context:
Evaluate the integral \( I = \int_0^1 \frac{dx}{\sqrt{(1-x^2)(1-x^2k^2)}} \) for \( 0 < k < 1 \).
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