Question: Let $x = [1/n, 1/n, \dots, 1/n]$ and $y = [y_1, y_2, \dots, y_n]$ be probability vectors on $n$ elements, where $\sum_{i=1}^{n} x_i = \sum_{i=1}^{n} y_i = 1$ and $x_i, y_i \geq 0$. Is it always true that $x$ is majorized by $y$? Provide a justification for your answer.

Answer: Yes


Question: If $f:X\to Y$ is a surjective function and $g:Y \to Z$ is a function such that the composition $g \circ f$ is injective (1-1), must $g$ be injective (1-1) as well? Provide a justification for your answer.

Answer: Yes


Question: Suppose $X$ and $Y$ are bivariate normal random variables, both with mean 0 and standard deviation 1. Given the correlation coefficient $\rho$, determine the value of $E[X|X,Y]$. Provide your answer as an expression in terms of $X$ and $Y$.

Answer: X


Question: Determine the number of sets \( A \) of natural numbers, where each element \( A_i \) is less than \( N \), no numbers in \( A \) repeat, and the sum of the numbers in \( A \) is exactly \( N \).

Answer: Q(N) - 1


Question: Let $M: \Omega \times \mathbb{R}^+ \longrightarrow \mathbb{R}$ be a martingale such that \( \sup_{t \in \mathbb{R}^+} \mathbb{E}[M_t^2] < \infty \). Determine if $M$ is uniformly integrable, i.e., is it true that \( \lim_{K \rightarrow \infty} \left[ \sup_{t \in \mathbb{R}^+} \mathbb{E}\left[ M_t^2 \mathbb{I}_{\left\{M_t^2 > K\right\}}\right]\right] = 0 \)?

Answer: Yes


Question:
Given two matrices \( A \) and \( C \) of order \( n \times n \) and \( m \times n \) respectively, consider the block matrix:

\[ D=\left( \begin{matrix} C\\ CA\\ :\\ :\\ CA^{q-1} \end{matrix}\right) \]

where \( q \in \mathbb{N}^* \). Determine the smallest value of \( q \) such that \( \text{rank}(D) = n \).

Answer: n


Question:
Consider a simple random walk on the set $\{0, 1, \ldots, k\}$ with reflecting boundaries at $0$ and $k$. The random walker earns $k$ dollars upon reaching $0$ or $k$ and loses $1$ dollar at each intermediate state. Define the reward function $r(x)$ as follows:

$$r(x)=\begin{cases} k, & \text{if } x = 0 \text{ or } x = k, \\ -1, & \text{if } x = 1, \ldots, k-1. \end{cases}$$

The transition matrix $P$ for this random walk is given by:

$$P=\begin{pmatrix}0 & 1 & 0 & \dots & 0 &0 \\ 1/2 & 0 & 1/2 & \dots & 0 &0 \\ 0 & 1/2 & 0 & \dots & 0&0 \\ \vdots & \vdots & \vdots & \ddots & 1/2&0 \\ 0 & 0 & 0 & 1/2 &0&1/2 \\ 0&0&0&0&1&0\end{pmatrix}$$

Find the stationary distribution $\pi_x$ for the transition matrix $P$. Then, compute the expected long-term reward $\mathbb{E}[r(x)]=\sum_x r(x)\pi_x$. What is the expected average earnings after $10,000$ steps?

Answer: \dfrac{1}{k}


Question: Evaluate the integral \( \int_{0}^{\infty}\frac{dx}{(x^2+1)(x^2+4)} \).

Answer: \dfrac{\pi}{12}


Question: Evaluate the limit: $$\lim_{n\to \infty}\frac{1+2^2\sqrt{2}+\dots +n^2\sqrt[n]{n}}{n^2(n+1)}$$

Answer: \dfrac{1}{3}


Question: Given the function \( f(x) = (x+1)^{1/3} + x \), find the value of \( (f^{-1})'(9) \).

Answer: \dfrac{12}{13}


Question: Let \( f(z) = \sin\left(z + \mathrm{e}^{3z}\right) \). Find \( \frac{\partial f}{\partial \bar{z}}(z) \).

Answer: 0


Question: Find an element $x$ in a group $G$ such that $xabx^{-1} = ba$, where $a$ and $b$ are elements of $G$. What is $x$?

Answer: a^{-1}


Question: Evaluate the limit: $$\lim \limits_{x\to 0} \frac{\tan\sqrt[3]{x}\ln(1+3x)}{(\tan^{-1}\sqrt{x})^2(e^{5\sqrt[3]{x}}-1)}$$

Answer: \dfrac{3}{5}


Question: If $a, b, c$ are integers determined by rolling a single die three times, find the probability that the following inequality holds: $$ \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \leq \frac{a+b}{b+c} + \frac{b+c}{c+a} + \frac{c+a}{a+b}. $$

Answer: \dfrac{1}{36}


Question: Find the limit: \( \lim_{x \to \infty} x^a a^x \) for \( 0 < a < 1 \).

Answer: 0


Question: Find a function $f(x)$ such that \( \int_0^1 \left(4x^3 - f(x)\right)f(x)\,dx = \frac{4}{7} \).

Answer: 2x^3


Question: Evaluate the limit using L'Hopital's rule: $$\lim_{x\to\infty}\frac{\int_0^x te^{-2t}\,dt+\frac{1}{4}}{\frac 1x}$$

Answer: \infty


Question:
Consider the sequence defined by:

\[ x_{1} = 1 \]
\[ x_{n+1} = x_{n} + \cos x_{n} \]

Determine whether the limit \( \lim_{n \rightarrow \infty} x_{n} \) exists. If it does, find its value.

Answer: \dfrac{\pi}{2}


Question: Determine the limit of the sequence of functions \( f_n(x) = \begin{cases} n-n^2x, & 0 < x < \dfrac{1}{n}\\[0.3cm] 0, & x = 0 \text{ or } x > \dfrac{1}{n} \end{cases} \) as \( n \to \infty \).

Answer: 0


Question: Find all functions \( f: \mathbb{R}^+ \rightarrow \mathbb{R}^+ \) satisfying the functional equation \( f(1+xf(y)) = yf(x+y) \) for all \( x, y \in \mathbb{R}^+ \).

Answer: \dfrac{1}{x}


Question: Determine the limit of the function \( \frac{1 - \cos(xy)}{xy} \) as \((x,y)\) approaches \((0,0)\).

Answer: 0


Question: Suppose the random variable $X$ has a moment-generating function \( M(t) = \frac{e^{at}}{1-bt^2} \) for \(-1 < t < 1\). Given that the mean \( \mu = E[X] = 3 \) and variance \( \sigma^2 = \text{Var}[X] = 2 \), determine the value of \( a + b \).

Answer: 4


Question: Let $A$ be a square matrix in $M^{\Bbb C}_{n \times n}$ with a minimal polynomial of degree $k$. Find the dimension of the subspace $W$ spanned by the set $\{I, A, A^2, A^3, \ldots, A^m, \ldots\}$, where $I$ is the identity matrix.

Answer: k


Question: Evaluate the limit: $$ \lim_{x \to \frac{1}{\sqrt{2}}^-} \frac{\arccos \left(2x\sqrt{1-x^2} \right)}{x-\frac{1}{\sqrt{2}}} $$

Answer: -2\sqrt{2}


Question: Determine the number of solutions to the equation \(2 - \frac{x^3}{3} = \sin x\) by analyzing the behavior of the functions involved using derivatives.

Answer: 1


Question: Prove that for every positive integer $a$, the inequality $F(a) < 0$ holds, where $F(a) = \sum_{i=1}^{a} (-1)^{\mu(i)}$ and $\mu(i)$ is the Möbius function.

Answer: F(a) < 0


Question: Determine the accumulation points of the sequence \(a: \mathbb{N} \rightarrow \mathbb{R}\) defined by \(a_1 = 0\) and \(a_{n+1} = \sqrt{2a_n + 3} + 2\).

Answer: 3 + 2\sqrt{2}


Question: In triangle ABC, given that \( \tan\left(\frac{A}{2}\right) = \frac{5}{6} \) and \( \tan\left(\frac{C}{2}\right) = \frac{2}{5} \), find the ratio of the side lengths \( BC:CA:AB \).

Answer: 87:74:61


Question: Evaluate the limit: $$\lim_{x \to \infty}\frac{1}{\sqrt{x}}\log\left(\sinh \sqrt{x+2}-\sinh{\sqrt{x+1}} \right)$$

Answer: 1


Question: Determine the number of triplets \((m, n, p)\) where \(p\) is a prime number that satisfy the equation \(m^2 - 3mn + p^2n^2 = 12p\).

Answer: 6


Question: Determine the smallest non-abelian group $G$ such that the power map $f(x) = x^n$ is a homomorphism for some integer $n > 2$. What is this group?

Answer: S_3


Question: Two circles \(C_1\) and \(C_2\) have radii of 10 cm and 8 cm, respectively, and are tangent to each other internally at point \(A\). Let \(AD\) be the diameter of \(C_1\). Points \(P\) and \(M\) lie on \(C_1\) and \(C_2\), respectively, such that \(PM\) is tangent to \(C_2\). Given that \(PM = \sqrt{20}\) and \(\angle PAD = x^\circ\), find the value of \(x\).

Answer: 60


Question: Consider a polynomial of degree \( k \) in one variable. If the polynomial vanishes at \( x = a \), and \( k-2 \) derivatives vanish at \( x = a \) and 2 derivatives vanish at \( x = b \), does the polynomial vanish identically? Answer with "yes" or "no".

Answer: yes


Question: Let \( A \) be a \( 7 \times 7 \) matrix with rank \( 3 \), and \( a \) be a \( 7 \times 1 \) column vector. Determine the smallest possible rank of the matrix \( A + (aa^T) \).

Answer: 2


Question: Determine the values of \( n \) for which the polynomial \( x^n + y^n \) is irreducible over the integers \( \mathbb{Z} \).

Answer: 2^k


Question: Determine the values of $m$ for which the function $f(x) = e^x(x^2 + x + m)$ has exactly one local extremum in the interval $(-1, 1)$. Provide your answer in terms of $m$.

Answer: (-5, 1)


Question: Calculate the integral \( \int_{C}{xy^{2}dy-yx^{2}dx} \) where \( C \) is the circle \( x^{2}+(y-1)^{2}=1 \) using polar coordinates and Green's theorem.

Answer: \dfrac{3\pi}{2}


Question: Use Green's theorem to find the area of the closed loop of the Lemniscate of Gerono described by the equation $x^4 = x^2 - y^2$ in the region $x \ge 0$. The curve can be parametrized as $x(t) = \sin t$, $y(t) = \sin t \cos t$, for $0 \le t \le 2\pi$. Determine the area of this region.

Answer: \dfrac{2}{3}


Question: Is the Ramsey number $R(0,a)$ defined, and if so, what is its value?

Answer: 0


Question: Find the coefficient of $x^{25}$ in the expansion of $(1+x^3 + x^8)^{10}$.

Answer: 2520


Question: Evaluate the integral: $$\int_{- \infty}^\infty \int_{-\infty}^y \frac{1}{2 \pi} e^{-(1/2) ( x^2+y^2 )}\mathrm {d}x\,\mathrm{d}y$$

Answer: \dfrac{1}{2}


Question: Given the vector space $P_3$ consisting of polynomials with a degree of 3 or less, and a basis $\{1, (1-x), (1-x)^2, (1-x)^3\}$. Consider the linear functional (evaluation function) $ev_3: P_3 \rightarrow \mathbb{R}$ that evaluates a polynomial at $x=3$. If $\{f_1, f_2, f_3, f_4\}$ is the dual basis of $P_3$, compute $f_3(p)$ for the polynomial $p = x^3 + 5x^2 + 3x + 4$.

Answer: 8


Question: Prove that for scalar fields \( \phi \) and \( \lambda \), the following identity holds: \[ \operatorname{div}(\vec{\nabla} \phi \times \vec{\nabla} \lambda) = 0 \] Use index notation to demonstrate this result.

Answer: 0


Question: Determine the values of \(\alpha\) for which the improper integral \( \int_{1}^{\infty}\frac{\sin (x^\alpha)}{x} \, \mathrm{d}x \) converges.

Answer: \alpha \neq 0


Question: Evaluate the double integral \(\int_{v=0}^{\infty}\int_{u=0}^{\infty}\frac{\sqrt{uv}}{(u+v)^2}e^{-\frac{(u+v)}{2}}dudv\) using a change of variables.

Answer: \dfrac{\pi}{4}


Question:
Find a general formula for the sum:
\[ \sum_{k=1}^{50} \frac{k^2\binom{n}{k}}{\binom{n}{k-1}} \]
where $n$ is a given integer. Verify the formula for $n = 50$ and show that it simplifies to $22100$. The proposed general formula is $425(3n - 98)$. Confirm or derive this formula.

Answer: 425(3n - 98)


Question: Find the limit: $$\lim_{x\to \infty} \frac{(x+3)^x}{\sum_{n=3}^{x+2}n^x}.$$

Answer: e - 1


Question: Given that $x$ satisfies the equation $\sqrt{40-9x} - 2\sqrt{7-x} = \sqrt{-x}$, find the value of $2x+5$.

Answer: -13


Question: Determine the smallest possible value of the expression \((a-b)^2+(b-c)^2+(c-d)^2+(d-a)^2\) for real numbers \(a\), \(b\), \(c\), and \(d\) that satisfy the conditions \(a+b+c+d=0\) and \(a^2+b^2+c^2+d^2=1\). Choose the correct interval for this value: i) \((0, 1.5)\) ii) \((1.5, 2.5)\) iii) \((2.5, 3)\) iv) \((3.5, 4)\).

Answer: ii


Question: Evaluate the limit \( \lim_{n\to\infty} \frac{1}{n!}\int_0^r (ax - bx^2)^n \sin(x) \, dx \), where \( a \) and \( b \) are constants.

Answer: 0


Question: Calculate the determinant of the matrix \( D_n = \begin{vmatrix} n & -1 & -3 & 0 & 0 & \cdots & 0 & 0 & 0 \\ n & 1 & 2 & -3 & 0 & \cdots & 0 & 0 & 0 \\ n & 0 & 1 & 2 & -3 & \cdots & 0 & 0 & 0 \\ n & 0 & 0 & 1 & 2 & \cdots & 0 & 0 & 0 \\ n & 0 & 0 & 0 & 1 & \cdots & 0 & 0 & 0 \\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots \\ n & 0 & 0 & 0 & 0 & \cdots & 1 & 2 & -3 \\ n & 0 & 0 & 0 & 0 & \cdots & 0 & 1 & 2 \\ n & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 1 \\ \end{vmatrix} \)

Answer: n^2


Question: Find the limit of the sequence \(a_n = \frac{n^{2001}}{1.001^n}\) as \(n \to \infty\).

Answer: 0


Question: Suppose we have two power series \(0\leq\sum_{n=0}^{\infty}a_{n}(x-\alpha)^{n}\leq \sum_{n=0}^{\infty}b_{n}(x-\alpha)^{n}\) for all \(|x-x_{0}|<r\) for some \(r>0\). The left-hand series converges for \(|x-x_{0}|<r\). Moreover, suppose that the right-hand series converges for \(|x-x_{0}|<R\), where \(R>r\). Can we conclude that the left-hand series also converges for \(|x-x_{0}|<R\)? Provide a justification for your answer.

Answer: Yes


Question: Determine whether every finite ring $R$ with identity, where $a + a = 0$ for all $a \in R$, is a subring of $Mat_{n \times n}(\mathbb{F}_2)$ for some $n$. Provide a proof or counterexample.

Answer: Yes


Question: Prove that the numerator $p$ of the fraction representing the alternating harmonic series up to 1331, given by \( 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots + \frac{1}{1331} = \frac{p}{q} \), is divisible by 1997.

Answer: 1997


Question: Determine whether the following statement is true or false: If a function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ is differentiable at a point $x_0$, then all its partial derivatives exist and are continuous in a neighborhood of $x_0$ when $m \geq 2$. Provide a justification for your answer.

Answer: B


Question: Consider an odd perfect number $N = q^k n^2$ in Eulerian form. Determine whether it is true that if $n < q^{k + 1}$, then $k \neq 1$. Provide a justification for your answer.

Answer: True


Question: Find the coefficient of \(x^{14}\) in the expansion of \((1-x^4)^8(1-x)^{-8}(1+2x^5+x^{10})\).

Answer: 17706


Question: Determine the necessary and sufficient condition for the expression \(x^3 + y^3 + z^3 + kxyz\) to be divisible by \(x + y + z\) for all integers \(x, y, z\).

Answer: -3


Question: Alice picks a number from the set \( \{1, 2, \ldots, 1000000\} \). Bob can ask yes/no questions to determine Alice's number, but Alice is allowed to lie once. What is the minimum number of questions Bob must ask to guarantee he correctly identifies the number?

Answer: 25


Question: Determine whether there exists a continuous, monotonically increasing function \( f: [0,1)\rightarrow [0,\infty) \) such that \( \lim_{x\to 1^-}f(x)=\infty \) and \( f'=0 \) almost everywhere on \([0,1)\).

Answer: Yes


Question: Calculate the limit \( \lim_{n\to \infty}\left(\dfrac{f\left(a+\frac{1}{n}\right)}{f(a)}\right)^{1/n} \), given that the function \( f:\mathbb{R}\to \mathbb{R} \) is differentiable at \( a\in \mathbb{R} \) and satisfies \( f(a) > 0 \).

Answer: 1


Question: Determine the number of positive integer solutions to the equation \(\frac{1}{x}+\frac{1}{y}=\frac{1}{pq}\), where \(p\) and \(q\) are distinct prime numbers.

Answer: 9


Question: Suppose $G$ is a group and $H$ and $K$ are two distinct normal subgroups of $G$ such that both the quotient groups $G/H$ and $G/K$ are simple. Can such subgroups exist with the property that the order of $H$ is not equal to the order of $K$? Provide a justification for your answer.

Answer: Yes


Question: Let $\rho$ and $\sigma$ be linear operators on a Hilbert space $A$. Determine whether the following statement is true: For all positive semi-definite operators $\sigma \geqslant 0$, we have \(Tr(\rho \sigma) \geqslant 0\) if and only if \(\rho \geqslant 0\). Here, $Tr$ denotes the trace and $\rho \geqslant 0$ means that $\rho$ is a positive semi-definite operator.

Answer: True


Question: What is the name of the operation $U^TRU$, where $U$ is an isometric embedding from $\mathbb{R}^d$ to $\mathbb{R}^n$ with $d<n$?

Answer: compression


Question: Given a set of positive integers $(a,b,c)$ with $a<b<c$, and a function $f(n) = a^n + b^n + c^n$, you are provided with the values of $f(n)$ for $n = p, q, r$ where $p < q < r$. Determine the value of $f(s)$ modulo $M$, where $M$ is a given integer constant and $s$ is an integer such that $p < s < r$. Provide your answer as an integer in the range $0$ to $M-1$. \( \text{(Assume all necessary values are given.)} \)

Answer: f(s) \mod M


Question: Evaluate the integral \( \int_{0}^{100}\left\{ \sqrt{x}\right\} \,dx \), where \( \{.\} \) denotes the fractional part of \( x \).

Answer: \dfrac{155}{3}


Question: Determine the limit of the sequence \(\{x_n\}_{n \in \mathbb{N}}\) defined by \(x_n = \sqrt[n]{n \sqrt[n]{n \sqrt[n]{n \ldots}}}\) as \(n\) approaches infinity. Prove whether the limit exists and, if so, find its value.

Answer: 1


Question: Calculate the limit \(\lim_{x \to 0} \left(\frac{1 + \sin x}{1 - \cos x}\right)^{\frac{1}{x}}\).

Answer: \infty


Question: Calculate the limit: $$\lim_{x\to\infty} \left(x-\sqrt[3]{2-x^3}\right).$$

Answer: \infty


Question: Evaluate the integral \(\int_{-\infty}^{\infty} \frac{x\sin x}{x^2 +4} \, dx\) using contour integration and Jordan's lemma.

Answer: \pi e^{-2}


Question: Determine the largest possible value for the minimum distance among all pairs of 30 points placed inside a $5\times5$ square.

Answer: 1


Question: Find the multiplicative inverse of $n+1$ modulo $n^2$, where $n$ is an integer greater than 1.

Answer: n^2 - n + 1


Question: Let $N \unlhd K$ be a normal subgroup of a group $K$, and let $q:K\to K/N$ be the natural quotient map. Consider a subset $A\subseteq K$ and its conjugate closure in $K$, denoted by $\langle A^K\rangle$. If $\langle A^K\rangle$ is an Abelian group, is $\langle(q(A))^{K/N}\rangle$ also Abelian?

Answer: Yes


Question: Evaluate the limit \( \lim_{x\rightarrow 0^+}\exp\left(-\frac{1}{x}\right)\ln\left(\frac{1}{x^2}\right) \).

Answer: 0


Question:
Find the variance of the random variable $X$ given its joint density with $Y$:

$$ f_{X,Y}(x,y) = \begin{cases} 2-x-y, & 0 < x, y < 1 \\ 0, & \text{otherwise}. \end{cases} $$

Answer: \dfrac{11}{144}


Question: Evaluate the limit: $$\lim_{x\to 0} \left( \frac{1+x\cdot4^x}{1+x\cdot5^x} \right)^{\frac{1}{x^2}}$$

Answer: \dfrac{4}{5}


Question: Determine the value of n such that in a lottery where 5 balls are drawn without replacement from a set of balls numbered 1 through n, the probability of matching exactly 0 balls is as close as possible to the probability of matching exactly 1 ball. The goal is to create a scenario where matching at least one ball is slightly favored.

Answer: 34


Question: Evaluate the limit: $$ \lim_{n \to \infty} \left( \int_1^{1+\frac{1}{n}} \left\{ [(x+1)\ln x +1] e^{x (e^x \ln x +1)} + n \right\} \, dx \right)^n $$

Answer: e^e


Question: Consider a matrix $A$ in the real n-dimensional general linear group $G$. Define the map $l: G \to G$ by $l(B) = AB$, which represents left multiplication by $A$. Examine the differential of $l$ at the identity matrix $I$ of $G$, denoted as $$l_{\{*,I\}}: T_I(G) \to T_A(G).$$ Is this differential $l_{*,I}$ still equivalent to left multiplication by $A$?

Answer: Yes


Question: In a connected graph $G=(V,E)$ with $n$ vertices, where all edges form $3$-cycles, the graph has exactly $n-2$ $3$-cycles and a minimum degree of $2$. Every vertex is part of at least one $3$-cycle. Determine the maximum number of vertices with a degree of $2$ in $G$ as a function of $n$. Provide your answer as a function of $n$.

Answer: n - 2


Question: Let $a, b, c$ be integers such that $\gcd(a, b, c) = 1$. Determine whether the equation $a(xy) + c(zw) + b(zx) = 1$ always has an integer solution for $x, y, z, w$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the covariance of $X_2$ and $\min(X_1, X_2)$, where $X_1$ and $X_2$ are independent random variables uniformly distributed on $(0,1)$. Provide your answer as a single numerical value.

Answer: \dfrac{1}{24}


Question: Is the function \( f(z) = \frac{1}{z} \) considered an entire function when its domain is extended to include the point at infinity, \( \mathbb{C} \cup \{\infty\} \), with \( f(\infty) = 0 \) and \( f(0) = \infty \)?

Answer: No


Question: Evaluate the limit \( \lim\limits_{x \to 0}\dfrac{\int_0^{x^2}f(t){\rm d}t}{x^2\int_0^x f(t){\rm d}t} \), given that \( f(x) \) is a continuously differentiable function satisfying \( f(0)=0 \) and \( f'(0) \neq 0 \).

Answer: 1


Question: Let $M$ be a Riemannian manifold with submanifolds $S$ and $T$. Suppose $	heta$ is a differential $k$-form on $M$ such that \( \oint_{[S]} \theta = r \neq 0 \). If \( \eta \) is the Hodge star of \( \theta \) and the \( \mathbb{Z}\pi \)-intersection number of \( S \) and \( T \) is \( \pm 1 \), determine the value of \( \oint_{[T]} \eta \).

Answer: \pm r


Question:
Compute the expected value of $\sin(\pi XY)$ given the joint density function of continuous random variables $X$ and $Y$:

$$f_{\small X,Y}(x,y) = \begin{cases} \frac{1}{3}(x+y), & \text{for } 0 < x < 1 \text{ and } 0 < y < 2 \\
0, & \text{otherwise} \end{cases}$$

Evaluate the integral:

$$\mathbb{E}[\sin(\pi XY)] = \iint_{\mathbb{R}^2} \sin(\pi XY) \cdot f_{\small X,Y}(x,y) \,dx\,dy$$

Answer: \dfrac{1}{\pi}


Question: Evaluate the sum: \[ 2\cos\frac{2\pi}{11} + 2\cos\frac{4\pi}{11} + 2\cos\frac{6\pi}{11} + 2\cos\frac{8\pi}{11} + 2\cos\frac{10\pi}{11} \]

Answer: -1


Question: Let $X(t) = \varepsilon_1 \cos(t) + \varepsilon_2 \sin(t)$, where $\varepsilon_1$ and $\varepsilon_2$ are independent standard normal random variables. Find the covariance function $C_X(s, t)$ of the process $X(t)$. Express your answer in terms of $s$ and $t$.

Answer: \cos(s - t)


Question:
Determine which of the following series is divergent:

(a) $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}\sin^2\left(\frac{1}{n}\right)$

(b) $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2}\sin\left(\frac{1}{n}\right)$

(c) $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}\log n$

(d) $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}\tan\left(\frac{1}{n}\right)$

Answer: C


Question:
Determine the limit of the sequence $a_n$ defined by $a_1 = 0$, $a_2 = 100$, and for $n \geq 2$:

$$a_{n+1} = a_n + \frac{a_n - 1}{(n)^2 - 1}$$

Answer: 199


Question: Determine the number of integer solutions to the Diophantine equation \(x^4 - 4y^2 = 2\).

Answer: 0


Question:
Determine whether the following statements are equivalent:

1. There exists an element $z$ in the ring $\mathbb{Z}/n\mathbb{Z}$ such that $z^t = 0$ for some positive integer $t$ and $z$ is not the zero element.
2. The integer $n$ is divisible by the square of some prime number.

Answer: Yes


Question: Let $R$ be a topological ring and $q \in R$ be an element such that $1-q$ is invertible. Determine whether the infinite geometric series $\sum_{n=0}^{\infty} q^n$ converges, and if so, find its limit.

Answer: (1 - q)^{-1}


Question: Evaluate the line integral \( \int_C e^y \, dx + x \sin y \, dy \) over the curve \( C \), where \( C \) is the intersection of the plane \( z=1 \) and the surface \( \frac{x^2}{9} - \frac{y^2}{16}=1 \), oriented counterclockwise.

Answer: 0


Question: Find the area of the region $R$ bounded by $y=f(x)>0$, $x=1$, and $x=2$, given that the volume of the solid formed by rotating $R$ about the $y$-axis is $\pi$ cubic units, and the volume formed by rotating $R$ about the line $x=-3$ is $6\pi$ cubic units. Determine the area using the washer or shell method.

Answer: \dfrac{5}{6}


Question: Let $M \subset \mathbb{C}$ be a sub-field that is not contained in $\mathbb{R}$ and is closed under complex conjugation. Define $L(M)$ as the set of all lines that intersect at two points in $M$. Prove that if $z$ is the intersection of two lines from $L(M)$, then $z \in M$.

Answer: z \in M


Question: You have a Lego block of length 10 and three additional Lego blocks of lengths 3, 4, and 5. These shorter blocks are randomly placed on the block of length 10 such that their boundaries are within the boundaries of the block of length 10. The position of a short block is defined by the left-most knob it covers on the 10-block. The block of length 3 can be placed in positions 1 to 8, the block of length 4 in positions 1 to 7, and the block of length 5 in positions 1 to 6, all with equal probability. The blocks are placed independently. Let $X$ be the number of knobs on the large block that are covered by all three shorter blocks. Calculate the expected value $E[X]$. \(X \in \{0, 1, 2, 3\}\).

Answer: \dfrac{6}{7}


Question:
Find the value of $z$ in the following equation involving inverse matrices:

$$ \begin{pmatrix} a & b & c \\ 1 & 2 & 3 \\ d & e & f \\ \end{pmatrix} ^{-1}= \begin{pmatrix} 1 & 2 & 3 \\ x & y & z \\ 4 & 5 & 6 \\ \end{pmatrix} $$

Given that $A^{-1} = B$, and $AB = I_3$, where $I_3$ is the 3x3 identity matrix.

Answer: -\dfrac{21}{2}


Question: Find \( \lim_{x\to 1^-} \left\lfloor \dfrac{\ln (1-x)}{x^2-2x+2}\right\rfloor \).

Answer: -\infty


Question: Does there exist a function whose inverse maps Borel sets to measurable sets, but does not map all measurable sets to measurable sets?

Answer: Yes


Question: Evaluate the infinite series: $$\sum_{k=0}^{\infty} \sum_{j=0, j\neq k}^{\infty} \sum_{i=0, i\neq j, i\neq k}^{\infty} \frac{1}{3^i3^j3^k}$$

Answer: \dfrac{81}{208}


Question: Determine the values of \(x\) for which the series \(\sum_{n=1}^{\infty}\frac{x^n}{n}\) converges absolutely. Express your answer in interval notation.

Answer: (-1, 1)


Question: Find the smallest positive integer \( n \) such that \( n! \) ends in exactly 150 zeroes and is divisible by 16.

Answer: 610


Question: Evaluate the limit: $$ \lim_{x \to 0^+} \frac{\log x}{\frac{1}{x+x^{x+1}}} $$

Answer: 0


Question: Determine whether the number given by the sum \( N = \sum_{n=0}^{\infty} b^{-n!} \), where \( b \geq 2 \) is an integer, is a Liouville number.

Answer: Yes


Question: Given $x > 0$ and $y > 0$, if \( \frac{3}{2x^2 + 3xy} + \frac{5}{3xy + 4y^2} = 2 \), find the supremum of $xy$.

Answer: \dfrac{5}{6}


Question:
Calculate the limit of the expression as \( n \) approaches infinity:
\[ \lim_{n\to \infty} e^{-n} \sum_{i=0}^{n} \frac{n^i}{i!} \]

Answer: \dfrac{1}{2}


Question: Given the functional $f(x_1, x_2, x_3) = x_1 - 3x_2 + x_3$ and the dual basis $\mathcal{A}^* = \{g_1, g_2, g_3\}$ where $g_1(x_1, x_2, x_3) = x_1 - x_3$ and $g_2(x_1, x_2, x_3) = x_1 - 2x_2$, find the functional $g_3$ such that the third coordinate of $f$ in the basis $\mathcal{A}^*$ is 1.

Answer: -x_1 + x_2 + x_3


Question: Find the remainder when $5^{100}$ is divided by $630$ using the Chinese remainder theorem.

Answer: 625


Question: Evaluate the limit: $$\lim_{x \to 0} \ln\left(\frac{\cos x}{1+\sin x}\right)^{\frac{1}{x}}$$.

Answer: -1


Question: Evaluate the double integral \( \int_0^\frac{\pi}{2} \int_0^\frac{\pi}{2} \sqrt{\cos^2(x-y)} \,dx\,dy \).

Answer: 2


Question: Determine the cardinality of the quotient ring of Gaussian integers when factored by a non-zero element with relatively prime components.

Answer: a^2 + b^2


Question: Calculate the limit: $$ \lim_{n \to \infty} \frac{1\cdot3\cdot5\cdot\cdot\cdot(2n-1)}{2\cdot4\cdot6\cdot\cdot\cdot(2n)} $$

Answer: 0


Question: If $U_n$ is an approximate unit for a $C^*$-algebra $A$, determine whether $U_n^2$ is also an approximate unit for $A$. Provide a justification for your answer.

Answer: Yes


Question:
Consider the system of equations:
1. $(2x)^{\ln 2} = (3y)^{\ln 3}$
2. $3^{\ln x} = 2^{\ln y}$

Find the value of $x_0$ for the solution $(x_0, y_0)$. \( \boxed{} \)

Answer: \dfrac{1}{2}


Question: Find the minimum value of the expression \( \cos{a} + \cos{b} + \cos(a+b) \).

Answer: -\dfrac{3}{2}


Question: Determine whether the expression \(2^n5^{2n+1}+3^{2n+1}7^{n+1}\) is divisible by 13 for all integers \(n\).

Answer: Yes


Question: Determine the maximum value of the integral \( \int_{0}^{y}\sqrt{x^4 + (y - y^2)^2}\,dx \) for \( y \in [0, 1] \).

Answer: \dfrac{1}{3}


Question: Let $g(x)$ be a differentiable function that satisfies the equation \(g(x+y) = g(x) + g(y) + 3x^2y + 3xy^2 - 1\) for all real values of $x$ and $y$. Given that \(\lim_{h \to 0} \frac{g(h)+1}{h} = 5\), find the value of \(g(3)\).

Answer: 43


Question: Find the limit as \( n \to \infty \) of the expression \( \frac{2^{n^k}}{2^{2^{\iddots^2}}} \), where the denominator is a tower of \( n \) twos.

Answer: 0


Question: Given the differential equation $y''' - ay'' + a^2y' - a^3y = 0$ with initial conditions $f(0) = 1$, $f'(0) = 0$, and $f''(0) = a^2$, determine how many positive values of $x$ satisfy $f(x) = 0$.

Answer: 0


Question: Let \( f \) be a differentiable function on \(\mathbb{R}\) that satisfies \( f(x + y) = f(x) \cdot f(y) \) for all \( x, y \in \mathbb{R} \). Given that \( f(1) = 2 \), determine the function \( f(x) \).

Answer: 2^x


Question: Calculate the variance of the sum of two random variables, $X$ and $Y$, where $X$ is a binomial random variable with parameters $n$ and $p$, and $Y$ is a Bernoulli random variable with parameter $p$. Given that $X$ and $Y$ are dependent and $E(Y|X=x) = \frac{x}{n}$, find $\text{Var}(X + Y)$. \( \text{Var}(X + Y) = ? \)

Answer: (n + 3)p(1 - p)


Question: Determine the values of $m \in \mathbb{R}$ for which the system of inequalities \(x^2-y+m \le 0\) and \(y^2-x+m \le 0\) has a unique solution $(x,y)$. Provide your answer in terms of $m$.

Answer: \dfrac{1}{4}


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2} \frac{\sec\theta}{2\tan^2\theta+1}\,\mathrm{d}\theta \).

Answer: \dfrac{\pi}{2}


Question: Let \( \{ Y_t \}_{t \ge 0} \) be defined by \( Y_t := 5+2t+2B_t \), where \( \{ B_t: t \ge 0 \} \) is a standard Brownian Motion. Let \( \{ T_t \}_{t \ge 0} \) be defined by \( T_t := e^{Y_t} \). Calculate the probability \( P(T_2 > T_4, T_2 >5) \).

Answer: 0.0783


Question: 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.

Answer: (-\infty, 1)


Question: Consider the sequence defined by $x_1 = 1$ and $x_{n+1} = x_n + \cos(x_n)$ for all $n \in \mathbb{N}$. Determine whether the sequence $(x_n)$ converges, and if it does, find its limit.

Answer: \dfrac{\pi}{2}


Question: Determine whether the strong duality theorem states that both the primal and dual linear programs having feasible solutions are equivalent to them both having optimal solutions, and that for any optimal primal solution $x$ and optimal dual solution $y$, the equality $c^Tx = b^Ty$ holds.

Answer: Yes


Question: Determine whether the sequence \( y_n = \frac{\sqrt[n]{\prod\limits ^{n}_{k=1}( 2n+k)}}{2n} \) converges, using the logarithmic version of the Cesàro-Stolz theorem.

Answer: \dfrac{27}{8e}


Question: Evaluate the limit \( \lim_{n\to\infty}\int_{-1}^1 \frac{nx^2}{1+n^2x^4} \, dx \) using the Lebesgue Dominated Convergence Theorem.

Answer: 0


Question: Find the maximum value of the function \( f(x, y) = 2x + y \) subject to the constraints \( 2x^2 + y^2 \leq 1 \) and \( x \leq 0 \). Use the method of Lagrange multipliers to solve this problem.

Answer: 1


Question: Determine an upper bound for the integral \( \int_\gamma \frac{z^2}{(z-2)^3} \, dz \) using the inequality \( \Big|\int_\gamma f(z) \, dz \Big|\leq \|f\|_{\infty,\gamma}\cdot L(\gamma) \), where \( \gamma \) is the circle \( |z+1|=2 \).

Answer: 4\pi


Question:
Given non-negative real numbers $a, b, c$ such that $ab + bc + ca = 1$, determine the minimum value of the expression:
\[ P = \sqrt{a + 2} + \sqrt{b + 2} + \sqrt{c + 2} - \sqrt{2 - abc}. \]

Answer: 2\sqrt{3}


Question: Evaluate the limit: \[ \lim_{n\to \infty}\int_0^1 \left(5\sqrt[n]{(1-x^2)^n}+10\sqrt[n]{(1-x^3)^n}\right)\,dx. \]

Answer: \dfrac{65}{6}


Question: Let $\Omega \subset \mathbb{R}^N$ be an open and bounded domain, and consider a continuous operator $T: L^2(\Omega) \to L^2(\Omega)$ with a nullspace consisting only of the zero function. Prove or disprove that there exists a constant $C > 0$ such that $\|Tx\|_{L^2(\Omega)} \geq C\, \|x\|_{L^2(\Omega)}$ for all $x \in L^2(\Omega)$, with $x \neq 0$.

Answer: False


Question: Determine the norm of the linear operator \( T: L_p[0,1] \to L_p[0,1] \) defined by \( Tf(x) = xf(x) \) for \( p \in [1, \infty) \).

Answer: 1


Question: Determine whether the following statement is true or false: For $a>0$, the region of integration of the triple iterated integral $$\int_{0}^{a}\int_{0}^{a}\int_{0}^{x}f(x,y) dzdydx$$ lies above a square in the xy-plane and below a plane.

Answer: A


Question: Given that a series $\sum_{n \geq 1}a_{n}$ converges, determine if it is always possible to find a subsequence $a_{n_k}$ such that $a_{n_{k+1}} \geq a_{n_k}$ for all $k \in \mathbb{N}$. Answer with 'Yes' or 'No'.

Answer: No


Question: Determine the number of positive continuous functions $f(x)$ on the interval $[0,1]$ that satisfy the following conditions: \( \int_{0}^{1} f(x) \, dx = 1 \), \( \int_{0}^{1} x f(x) \, dx = \alpha \), and \( \int_{0}^{1} x^2 f(x) \, dx = \alpha^2 \), where \( \alpha \) is a given real number.

Answer: 0


Question: Determine if the set \( C = \{f \in \mathcal{C}^1 [0,1] \mid |f(x)| \leq 2 \text{ and } |f'(x)| \leq 2 \text{ for all } x \in [0,1] \} \) is equicontinuous.

Answer: Yes


Question: Consider an abelian group \(A\) of order \(p^4\), where \(p\) is a prime number. If there exists a subgroup \(N\) such that \(A/N \cong \mathbb{Z}/p^3\mathbb{Z}\), determine how many distinct groups \(A\) satisfy this condition. Use the classification of abelian groups of order \(p^4\): \(\mathbb{Z}/p^4\mathbb{Z}\), \(\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}\), \(\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}\), \(\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^3\mathbb{Z}\), and \(\mathbb{Z}/p^2\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}\).

Answer: 2


Question: Evaluate the sum \( \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n(n+2)} \).

Answer: \dfrac{1}{4}


Question: Given an AR(1) process with correlation parameter \(\rho\), determine the maximum possible correlation between its elements.

Answer: 1


Question: Calculate the area of the surface defined by \( z = x^2 + y^2 + 2 \) that lies above the disc \( x^2 + y^2 \le 1 \). Provide your answer correct to four decimal places.

Answer: 5.3304


Question: In a metric space, determine whether an unbounded set necessarily implies that the set is infinite. Provide a justification for your answer.

Answer: Yes


Question: Find the minimum page size $s \in \Bbb N$ such that the inequality \( \frac{i + l}{1 + \lfloor \frac{i}{s} \rfloor} \leq s \) holds, given that $i > l$ and $\{i, l\} \in \Bbb N$. What is the minimum value of $s$?

Answer: 2


Question: Determine the minimum number of vertices for a polyhedron defined by the constraints given by the identity matrix and a vector with all entries equal to 0.5.

Answer: 1


Question: Determine the maximum possible order of an element in the multiplicative group \( \mathbb{Z}_{2^k}^* \) for \( k \ge 3 \).

Answer: 2^{k-2}


Question: Find the coefficient of \(x^{10}\) in the function composition \(f(f(x))\), where \(f(x) = x + x^2 + x^4 + x^8 + x^{16} + x^{32} + \ldots\).

Answer: 40


Question: In how many ways can 12 identical red balls, 6 identical green balls, and 4 identical blue balls be arranged in a row such that no two balls of the same color are adjacent?

Answer: 0


Question: Evaluate the limit: $$\lim_{x\rightarrow \infty}\frac{\ln \left( \sqrt[x]{b_1^x+b_2^x+...+b_m^x}-b_2 \right)}{x},\ b_1\ge b_2\ge ...\ge b_m>0$$

Answer: 0


Question: Let $X$ be an exponentially distributed random variable with parameter $\lambda = 3$. Find the probability density function (PDF) of $Y = e^{\frac{1}{2}X}$. Express your answer in terms of $y$.

Answer: \dfrac{6}{y^7}


Question:
Consider an $n \times n$ matrix $A$ over the complex numbers $\mathbb{C}$ such that every non-zero vector in ${\mathbb{C}}^n$ is an eigenvector of $A$. Determine which of the following statements must be true:

1. All eigenvalues of $A$ are equal.
2. All eigenvalues of $A$ are distinct.
3. $A = \lambda I$ for some $\lambda \in \mathbb{C}$, where $I$ is the $n \times n$ identity matrix.
4. The minimal polynomial and characteristic polynomial of $A$ are equal.

Select the correct statement.

Answer: 3


Question: Determine whether every tree-ordered set that is also connected is a semi-lattice. Provide a justification for your answer.

Answer: Yes


Question: Given that $X_1, X_2, \ldots, X_n$ are correlated normal random variables with $E(X_i) = 0$, $Cov(X_i, X_j) = 0.2 \times 2$ for $i \neq j$, and $Var(X_i) = 2$, where $n = 100$, find the covariance matrix of the multivariate normal distribution for $X = (X_1, X_2, \ldots, X_n)'$. Then, calculate the variance of $\sum_{i=1}^n X_i$. Provide your answer for the variance in a boxed form.

Answer: 4160


Question: Let $F$ be a closed set in $\mathbb{R}^m$ with $m\geq2$. For $r>0$, define $F_r$ as the set of points $x$ in $\mathbb{R}^m$ such that \( \text{dist}(x,F)=r \). Suppose for some $r>0$, the set $F_r$ has a bounded component, meaning there exists $R>0$ such that a component of $F_r$ is contained within the ball centered at the origin with radius $R$. Is it true that for all $0<\rho<r$, the set $F_\rho$ also has a bounded component?

Answer: Yes


Question: Determine the number of solutions \((x, y)\) in natural numbers to the equation \(3 \times 2^x + 2 = 7^y\).

Answer: 0


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

$$\lim_{x \to \infty} \left[-x^{n+1} e^{-x}\right]_{x=0}^{x=\infty}$$

Answer: 0


Question: Find all natural number solutions $(m, n)$ to the equation $5^m = 2 + 3^n$.

Answer: (1, 1)


Question: How many distinct necklaces can be formed using 20 beads, each of a different color?

Answer: \dfrac{19!}{2}


Question: If a bounded operator $T$ on a Banach space $X$ is Fredholm with index zero and has a trivial null space, does it follow that $T$ is surjective?

Answer: Yes


Question: Calculate the flux of the vector field \( G = 3x\hat{i} - y\hat{j} - z\hat{k} \) through the surface of the solid enclosed by the cylinder \( y^2 + z^2 = 9 \) and the planes \( x = 0 \) and \( x = 5 \), oriented outward.

Answer: 45\pi


Question: In a queueing system with $n$ customers and $m$ servers, where the service time for each customer is exponentially distributed with mean $\mu$, calculate the expected time until the next customer begins service, given that there are currently $n-m$ customers waiting in the queue.

Answer: \dfrac{\mu}{m}


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{f_{n+1} \cdot f_{n-1}}{f_n^2} \), where \( f_n \) is the \( n \)th Fibonacci number, using Cassini's identity: \( f_{n-1} \cdot f_{n+1} - f_n^2 = (-1)^n \).

Answer: 1


Question: Find the value of the third derivative \( y''' \) at \( x = 1 \) for the function defined by the equation \( x^2 + xy + y^3 = 1 \).

Answer: 42


Question: Let $T$ be a linear operator from $l_1$ space to itself defined as \(T(x_1,x_2,\cdots , x_n,\cdots )=(x_2+x_1, x_3+x_2,\cdots, x_{n+1}+x_n, \cdots).\) Find the norm of $T$. \(\boxed{\text{Answer}}\)

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \rightarrow \infty} \left(x^{\frac{2}{3}} - (x^2+1)^{\frac{1}{3}}\right) \cdot \left(x^{\frac{4}{3}} + x^{\frac{2}{3}}(x^2+1)^{\frac{1}{3}} + (x^2+1)^{\frac{2}{3}}\right) \]

Answer: -1


Question: Let $A$ be an abelian group and $D$ be a divisible abelian group. Determine if any homomorphism $f: H \rightarrow D$, where $H \leq A$, can be extended to a homomorphism $g: A \rightarrow D$. Provide a justification for your answer.

Answer: Yes


Question: Given a positive integer $k$, let $S_{k}$ denote the set of all numbers in the interval $[0,1]$ whose decimal expansion has a prime number at the $k$th decimal place. Determine the Lebesgue measure of $S_{k}$.

Answer: \dfrac{2}{5}


Question: Find the limit of \( \frac{|\text{Tr}(W^3)|}{\|W\|^2} \) as \( W \) approaches the zero matrix, where \( W \) is an \( n \times n \) real matrix and \( \|\cdot\| \) represents any matrix norm.

Answer: 0


Question: Find a prime number $p$ such that the Legendre symbol $(2/p) = -1$, meaning 2 is a non-quadratic residue modulo $p$. Do not use the Legendre symbol formula in your solution.

Answer: 3


Question: Find an elegant relation between the intersection points of the sphere with equation \((x-a)^2 + (y+r)^2 = r^2\) and the cubic function \(y = x^3 - 3x^2 + r\). The intersection points are \((x_1, y_1)\) and \((x_2, y_2)\), where \(x_1^3 - 3x_1^2 + r = y_1\) and \(x_2^3 - 3x_2^2 + r = y_2\).

Answer: x_1 + x_2 = 2a


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{1}{n}\sum_{i=\left\lfloor \sqrt{n}\right\rfloor }^{n}\ln\left(\frac{n+1}{i}\right) \]

Answer: 1


Question: Let $f: M \to N$ be a smooth map between finite-dimensional real smooth manifolds. If $f$ is a local diffeomorphism at $p \in M$, is the induced map $f_{* p}$ surjective? Provide a justification for your answer.

Answer: Yes


Question: Determine the minimal polynomial of $b = \bar{y}$ over $\mathbb{Q}$, where $L = \mathbb{Q}[y]/(y^2 + 3)$ and $L \simeq \mathbb{Q}(\sqrt{-3})$. Given that $[L:\mathbb{Q}]=2$, find the minimal polynomial of $b$.

Answer: x^2 + 3


Question: Consider a linear transformation $T : \mathbb{R}^4 \rightarrow \mathbb{R}^4$. Suppose there exists a vector $v \in \mathbb{R}^4$ such that the set $\{v, T(v), T(T(v))\}$ is linearly independent, and $T(T(T(v))) = 0$. Determine the rank of $T$. \(T(T(v))\) is in both the null space and the range of $T$. What is the rank of $T$?

Answer: 2


Question: Determine whether the function \( f(x, y) = \frac{y^2 \sin x}{x^2 + y^2} \) is continuous at the point (0,0).

Answer: Yes


Question:
In a unique number system, the following relationships hold:

1. $a^2 = a$
2. $a + a = b$
3. $y^2 = y$
4. $y + y = y$
5. $z^2 = bb$
6. $z + z = aa$

Given these conditions, determine how the decimal number $99$ can be represented in this number system.

Answer: bya


Question: Determine the number of integer solutions to the equation \((x^3-x)+(y^3-y)=z^3-z\) where \((x,y,z) > 1\) and not all three variables are equal.

Answer: 1


Question: Evaluate the integral \( \int_{-1}^{1} (3^x + 3^{-x}) \tan{x} \, dx \).

Answer: 0


Question: Given a function $f$ that satisfies the equation $f\left(\frac{x+y}{2}\right) = \frac{f(x) + f(y)}{2}$ for all real numbers $x$ and $y$, and knowing that $f'(0) = -1$, determine the value of $f(2)$. \( \boxed{} \)

Answer: -2


Question: Let $R$ be a Noetherian local domain that is not a Unique Factorization Domain (UFD). Consider a height one prime ideal $P$ in $R$. Determine if there exists an element $x \in P$ such that $P$ is the only minimal prime ideal containing $x$. Provide a justification for your answer.

Answer: Yes


Question: Find the radius of convergence of the power series \( \sum_{n=1}^\infty b_n z^n \) derived from the power series \( \sum_{n=0}^\infty a_n z^n \), given that \( \lim_{n \to \infty} \frac{a_n}{a_{n+1}} = z_0 \) exists and \( z_0 \) is a pole on the unit circle.

Answer: 1


Question: Can the set \( A := \left\{ f \in \mathbb{Q}[X] \mid f(0)=f'(0)=0 \right\} \) be generated by a single element?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to \infty}\left(\sum_{k=n+1}^{2n}\left(2(2k)^{\frac{1}{2k}}-k^{\frac{1}{k}}\right)-n\right)$$

Answer: (\ln 2)^2


Question: Calculate the integral \( \int_E 1 \, d\lambda \) for the set \( E := \{(x,y,z) \in \mathbb{R}^3 : (x-z)^2 + (y-e^z)^2 \leq 3\sin(\pi z), \; z \in [0,1]\} \).

Answer: 6


Question: Compute the limit: $$\lim_{x \to 0^+} \frac{\sqrt{e^x-1}-e^x\sqrt{x}}{x}.$$

Answer: 0


Question: Determine whether the function \( f: \mathbb{C} - \{0\} \rightarrow \mathbb{C} \) defined by \( z \rightarrow \frac{1}{z} \) can be extended continuously to the Riemann sphere.

Answer: Yes


Question: Find the roots of the function \( f(x) = \frac{3}{4} x - \frac{1}{2}\sin(2x) + \frac{1}{16} \sin(4x) \) within the range \([0, \pi]\) without using numerical approximation methods.

Answer: 0


Question: Determine the interval of uniform convergence of the series \( \sum_{n=0}^\infty \frac{xe^{-nx}}{e^x-1} \) for \( x \in (0, \infty) \).

Answer: [a, \infty)


Question: Find the remainder when $6^{936}$ is divided by $2014$. You may use Euler's theorem or any other method to solve the problem.

Answer: 1008


Question: Let $E \subset \mathbb{C}$ be a set with the following property: for any sequence of distinct elements $(e_n)_{n \in \mathbb{N}}$ from $E$, $e_n \to 0$. Is the set $E$ necessarily countable? Answer yes or no.

Answer: Yes


Question: Calculate the Generalized Directional Derivative (Clarke Derivative) of the function \( f(x) = -|x| \) at the point \( x = 0 \).

Answer: |v|


Question: Let $Q(x)$ be a quadratic form with integer coefficients. If $Q(x)$ is positive-semidefinite over $\mathbb{R}$ and positive-definite over $\mathbb{Q}$, is $Q(x)$ positive-definite over $\mathbb{Z}$? Answer yes or no.

Answer: Yes


Question: Determine whether a real-valued function of a real variable, defined explicitly by a formula using elementary algebraic operations (addition, subtraction, multiplication, division, roots), can have infinitely many zeros.

Answer: Yes


Question: Two spheres with radii \( R \) and \( r \) (where \( R > r \)) are packed inside a cylindrical pipe with a diameter of 100 units. The spheres touch each other and the walls of the pipe. Derive an expression for the distance \( x \) between the centers of the spheres along the axis of the pipe in terms of \( R \) and \( r \).

Answer: 2\sqrt{Rr}


Question:
Given the following information:

\[ \mathbf{E}(Y)=2, \quad \mathbf{Var}(X)=2, \quad \mathbf{E}(Y|X)= 2+X, \quad \mathbf{E}(\mathbf{Var}(Y|X))=3, \]

determine the value of \(\mathbf{Cov}(X,Y)\).

Answer: 2


Question: Determine whether the following statement is true or false: If $f$ is Riemann integrable on $[a, b]$, then the function $F(x) = \int_a^x f(t) \, dt$ is differentiable on $(a, b)$ with $F'(x) = f(x)$ for all $x \in (a, b)$. Provide a justification for your answer.

Answer: B


Question: Evaluate the integral \( \int^{2021}_{-1}g(x)\,dx \) given that the function \( g: \mathbb{R}-\{0\} \rightarrow \mathbb{R} \) satisfies the functional equation \( g(x)\cdot g(y)=2g(xy)-g\left(\frac{2020}{x}\right)\cdot g\left(\frac{2020}{y}\right) \) for all \( x,y \in \mathbb{R}-\{0\} \), with \( g(2020)=1 \) and \( g(-3)=-1 \).

Answer: 2020


Question: Calculate the surface integral of the curl of the vector field \( F = \langle 2y, xz, 1 \rangle \) over the surface defined by \( z = f(x,y) = 3 - 2x - \frac{3}{2}y \). Use a normal vector obtained from the cross product of two vectors on the surface.

Answer: -3


Question: Evaluate the limit of the sequence \(b_n=\frac{n\sin(n\pi/2+\pi/4)+n(-1)^{n+1}}{n^2-1}\) as \(n\) approaches infinity.

Answer: 0


Question: Determine the number of ordered pairs \((\mathcal{A}, \mathcal{B})\) of nonempty sets such that \(\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}\), \(\mathcal{A} \cap \mathcal{B} = \emptyset\), the number of elements in \(\mathcal{A}\) is not an element of \(\mathcal{A}\), and the number of elements in \(\mathcal{B}\) is not an element of \(\mathcal{B}\).

Answer: 772


Question: Calculate the definite integral \( \int_0^{4\pi}\frac{dx}{5+4\cos x} \).

Answer: \dfrac{4\pi}{3}


Question: Given a black and white matrix and a stamp matrix, determine if it is possible to cover certain parts of the black and white matrix using copies of the stamp matrix. Answer 'yes' or 'no'.

Answer: no


Question: Solve for $x$ in the equation $$\sqrt{\log_x\left(\sqrt{3x}\right)} \cdot \log_3 x = -1$$ where $0 < x < 1$. \( \boxed{\text{Answer}} \)

Answer: \dfrac{1}{9}


Question: Evaluate the integral \( \displaystyle \int_{|C|=2} \frac{1}{(z-(2-i))(z-(2+i))} \, dz \) using contour splitting, where \(|C|=2\) denotes a circle of radius 2 centered at the origin.

Answer: 0


Question: Determine the convergence and find the limit of the sequence \( x_n = \left( \frac{n^n}{n!} \right)^{\frac{1}{n}} \).

Answer: e


Question: Given the equation \( \frac{\sqrt{1+f(x)^2}}{1+f(x)} = \frac{1}{(1+y)^1} \), where \( y=\frac{\sqrt{1+f(x)^2}}{1+f(x)} \), find the derivative \( y' \) using logarithmic differentiation or another appropriate method.

Answer: 0


Question: In a right triangle $\triangle ABC$ with the right angle at $A$, the angle bisectors $BD$ and $CE$ divide $\angle ABC$ and $\angle ACB$, respectively. Given that $AE = 8$ and $AD = 9$, determine the perimeter of $\triangle ABC$. \(\text{(Note: Consider alternative methods to solve without directly solving a system involving non-linear terms.)}\)

Answer: 72


Question: Construct a 9-degree angle using geometric methods.

Answer: 9^\circ


Question: Determine the group of symmetries for the elements of the quaternion group, excluding 1 and -1.

Answer: S_4


Question: Calculate the sum of the series \( \sum f_n \), where \( f_n(x) = \frac{nx}{1+n^2x^2} - \frac{(n-1)x}{1+(n-1)^2x^2} \) for \( x \in [0,1] \), using the method of telescoping series.

Answer: 0


Question: Determine all integer values of \( n \) such that the equation \( 2\sin{nx} = \tan{x} + \cot{x} \) has real solutions.

Answer: n \equiv 2 \pmod{8}


Question: Find all entire functions $f(z)$ such that $f\left(\frac{1}{n}\right) = \frac{1}{n^2}$ for all positive integers $n$.

Answer: z^2


Question: Determine the value of $r$ that minimizes the probability of a draw in a poll where $k$ people each vote for $r$ out of $n$ options with uniform probability. Express $r$ as a function of $k$ and $n$, i.e., $r = f(k, n)$. Assume the highest scoring option wins unless there is a draw.

Answer: \dfrac{n}{2}


Question: Evaluate the limit: $$\lim_{x \rightarrow \infty} \frac{\log{(e^{\pi x} -1)}}{x}$$ without using L'Hôpital's rule.

Answer: \pi


Question: Determine whether the converse of the statement "If a matrix $G$ has full rank, then $A = G'G$ is non-singular" is true.

Answer: True


Question: Determine whether the quotient map \( \pi\colon \mathbb{C}^{n+1}\backslash \{0\} \to \mathbb{CP}^n \) is a submersion.

Answer: Yes


Question: Evaluate the limit: $$\lim _{x\rightarrow - \infty}\{x+|x|^{\frac{2}{3}}(6-x)^\frac{1}{3}\}.$$

Answer: 2


Question: Determine the number of unique paths of length 16 from the origin (0,0) to the point (7,9) on a 2D grid, where each step is either to the right (from (a,b) to (a+1,b)) or upward (from (a,b) to (a,b+1)), and no path intersects the line y = x - 3.

Answer: 9620


Question: Determine whether the proposition "If \(\sum{a_n^2}\) converges absolutely, then \(\sum{a_n}\) also converges absolutely" holds true or false. Provide a proof or counterexample.

Answer: False


Question: Evaluate the limit: \[\lim\limits_{n\to\infty}\frac{\zeta(3,n)}{\frac{1}{n}+\frac{1}{3n^3}}\] where \(\zeta(3,n)\) is the generalized Riemann zeta function.

Answer: 0


Question: Determine the Galois group of the polynomial \( p(x) = (x^2 - 2)(x^2 - 3) \) over \( \mathbb{Q} \).

Answer: \mathbb{Z}_2 \times \mathbb{Z}_2


Question: Compute the limit of the function \( \frac{1}{A} \int_{1}^{A} A^{1/x}\, dx \) as \( A \) approaches infinity.

Answer: 1


Question: Find the smallest positive integer $N$ such that $N = 3a^2 - ab^2 - 2b - 4$ for positive integers $a$ and $b$. Provide your answer as a single integer.

Answer: 2


Question: Find the adjoint of the operator $A: L^2 \to L^2$ defined by $Ax = (\lambda_1 x_1, \lambda_2 x_2, \ldots)$, where $|\lambda_n| \leq 1$ and $\lambda_n \in \mathbb{R}$.

Answer: A^* x = (\lambda_1 x_1, \lambda_2 x_2, \ldots)


Question: Minimize the expression \( (3+2a^2)(3+2b^2)(3+2c^2) \) subject to the constraints \( a+b+c=3 \) and \( a, b, c > 0 \). Find the minimum value and the values of \( a, b, c \) that achieve this minimum.

Answer: 125


Question: Evaluate the sum \( \sum_{b=0}^{N-1}\left(\frac{b}{p}\right)\zeta_M^{-kb} \) for an odd prime \( p \), where \( p \mid N \), \( M \mid N \), \( (k,M)=1 \), and \( p \not\mid M \). Here, \( \left(\frac{b}{p}\right) \) denotes the Legendre symbol.

Answer: 0


Question: Suppose $\varphi : \mathbb{C} \to \mathbb{C}$ is a continuous function. Are the roots of $\varphi$ necessarily discrete? Answer yes or no and justify your answer.

Answer: No


Question: Determine the maximal possible degree of the field extension \( \mathbb{Q}(\sqrt{5},i)/ \mathbb{Q}(\sqrt{5}) \).

Answer: 2


Question: Let $a,b \in \mathbb{R}$ and let $\Gamma$ denote the curve with parameterization $\gamma(t) = a\cos(t) + ib\sin(t), 0 \leq t \leq 2\pi$. Compute \( \int_{\Gamma} |z|^2 \, dz \).

Answer: 0


Question: Suppose a sequence of distinct elements $\{x_n\}$ converges to $x$, but $x$ is not in the sequence. Given an integer $N$, determine if there exists a positive number $\epsilon$ such that there is an integer $M > N$ with the property that for all $n \geq M$, $|x_n - x| < \epsilon$ and the set $\{x_n : n \geq M\}$ is equal to the set $\{x_n : |x_n - x| < \epsilon\}$. Is it possible to find such an $\epsilon$?

Answer: Yes


Question:
Consider a sequence $\{a_n\}_n$ with $a_n \in \mathbb{R}^k$ for all $n$ and $k>1$. We know that the following expressions are equivalent when $a_n \in \mathbb{R}$:

$$\lim_{n \rightarrow \infty} a_n=a \tag{$*$}$$

$$\lim_{n \rightarrow \infty} a_n-a=0 \tag{$**$}$$

$$\lim_{n \rightarrow \infty} |a_n-a|=0 \tag{$***$} $$

Are these expressions still equivalent when $a_n \in \mathbb{R}^k$ for all $n$ and $k>1$, replacing the absolute value in $(***)$ with the Euclidean distance?

Answer: Yes


Question: Find the pedal equation of the curve \(y^2 = 4a(x+a)\) with respect to the origin \((0,0)\). Express your answer in terms of the distance \(p\) from the origin to the tangent, and the distance \(r\) from the origin to a point on the curve.

Answer: p^2 = a r


Question: Find the values of $a$ and $b$ such that the set $H = \{ z \in \mathbb{C} \ | \ bz^{n} = a \}, a \in \mathbb{C^{*}}, b \in \mathbb{C}$ forms a subgroup of $(\mathbb{C^{*}},\cdot)$.

Answer: a = b


Question: Let $U = Z(x^2 - y)$ in $k[x,y]$, where $k$ is a field. Show that $I(U) = (x^2 - y)$. Recall that $Z(I) = \{(x,y) \in k^2 \mid \forall f \in I, f(x,y) = 0\}$ and $I(U) = \{f \in k[x,y] \mid \forall (x,y) \in U, f(x,y) = 0\}$. Prove that $f \in I(U)$ if and only if $f(x, x^2) = 0$ for all $x \in k$.

Answer: (x^2 - y)


Question: Is the extension \( \mathbb{C}[x,y,z]/(z^2 - xy, y^2 - xz) \) integral over \( \mathbb{C}[x] \)?

Answer: Yes


Question: Find the limit \( \lim_{x \to \infty} (a^x+b^x-c^x)^{\frac{1}{x}} \) given that \( a > b > c > 0 \).

Answer: a


Question: Determine the locus of the orthocenter of the triangle formed by the tangents to the parabola \((x-y)^2-2(x+y)+1=0\).

Answer: x + y = 0


Question: Can a Turing machine with a limited tape length output every bit of \(\pi\) one after another correctly?

Answer: No


Question: Calculate the mass of the surface defined by \( z = \sqrt{2xy} \) over the region where \( 1 \leq x \leq 5 \) and \( 1 \leq y \leq 4 \), given the density function \( \sigma(x,y,z) = 6z \).

Answer: 396


Question: Determine the values of $x \in \mathbb{R}$ for which the series \( \sum_{n=2}^\infty \frac{1}{n\sqrt{\ln (n+x^2)}} \) converges.

Answer: \emptyset


Question: Find the limit of the sequence \(\lim_{n\to\infty} \left(1+\frac{x^4}{n^4}\right)^{\frac{n^2}{2}}\) for any real number \(x\).

Answer: 1


Question: Determine the number of permutations possible for an array of length \( n \) when any number of adjacent elements can swap places simultaneously.

Answer: n!


Question: Evaluate the limit: \[ \lim_{x\to0^+} \left(\frac{x^{\sin(x)}-1}{x}\right) \]

Answer: -\infty


Question:
Find the limit of the following sum as \( n \) approaches infinity:
\[
\lim_{n\to\infty}\left[\frac{1}{\sqrt{n^2+2}}+\frac{1}{\sqrt{n^2+4}}+\frac{1}{\sqrt{n^2+6}} + \dots + \frac{1}{\sqrt{n^2+2n}}\right]
\]

Answer: 1


Question: In a non-commutative monoid $M$, if $a \in M$ and there exists an element $b \in M$ such that $a * b = n$, where $n$ is the neutral element, does it necessarily follow that $b * a = n$? Provide a justification for your answer.

Answer: No


Question: Given a sequence of independent and identically distributed (iid) random variables, \(\{X_n\}_{n \geq 1}\), with a common probability density function (pdf) \(f_X(x) = xe^{-x}I_{x>0}\), where \(I_{x>0}\) is the indicator function for \(x > 0\). Define the sample mean as \(\overline{X_n} = \frac{1}{n}\sum_{i=1}^{n}{X_i}\) for \(n = 1, 2, \ldots\). What is the limit of \(P(\overline{X_n} = 2)\) as \(n\) approaches infinity? Choose from the following options: \(0\), \(1\), \(\frac{1}{2}\), or \(\frac{1}{4}\).

Answer: 0


Question: Evaluate the integral \( \int_4^6 \sin^4 (\pi x) \cdot \cos^2 ( 2 \pi x) \,\mathrm{d}x \) using trigonometric identities.

Answer: \dfrac{7}{16}


Question: Given a linear operator $L: X \rightarrow Y$ between two Banach spaces $X$ and $Y$, if $L$ is injective and has a closed range, is $L$ boundedly invertible on its range?

Answer: Yes


Question:
Let $A, B \in M_n(\mathbb{R})$ with $\det(A) > 0$ and $\det(B) < 0$. For $t \in [0, 1]$, consider $C(t) = tA + (1-t)B$. Determine which of the following statements is true:

(a) $C(t)$ is invertible for all $t \in [0, 1]$.
(b) There exists $t_0 \in (0, 1)$ such that $C(t_0)$ is not invertible.
(c) $C(t)$ is not invertible for each $t \in [0, 1]$.
(d) $C(t)$ is invertible for only finitely many $t \in [0, 1]$.

Answer: B


Question: Solve the equation \( \sqrt[x]{\frac{2}{(x+1)}} = {(x+1)}^{x+2} \) for real numbers \( x \).

Answer: \sqrt{2} - 1


Question: What is the maximum number of moves required to transform one position of a Rubik's cube into another?

Answer: 20


Question: Derive the distribution of \( \sum_{i=1}^k \frac{X_i^2}{\sigma_i^2} \) where \( X_i \) are independent normal random variables with mean 0 and variances \( \sigma_i^2 \).

Answer: \chi^2(k)


Question: Given \((2x^2+3x+4)^{10}=\sum_{r=0}^{20}a_rx^r\), find the value of \(\frac{a_7}{a_{13}}\).

Answer: 8


Question: Let \( f(x) = \int_0^x \frac{dt}{\sqrt{1+t^4}} \) and \( g = f^{-1} \). Find \( g'(0) \).

Answer: 1


Question: Calculate the sum of the infinite series \( \sum_{n=1}^{\infty}{\dfrac{n^2}{3^n}} \).

Answer: \dfrac{3}{2}


Question: In a commutative ring with unity $A$, let $q$ be a $p$-primary ideal such that $\sqrt{q}=p$. Suppose for $x \in A$, if $x \not\in q$, then $(q:x)$ is $p$-primary, and if $x \not\in p$, it holds that $(q:x)=q$. Determine if the converse is true: if $(q:x)=q$, can we conclude that $x \not\in p$?

Answer: Yes


Question: Determine the number of unique natural numbers that are products of two distinct primes and satisfy the condition that the sum of their positive divisors is divisible by the number itself.

Answer: 1


Question: Determine whether the following statement is true or false: If a Lipschitz map $u:\mathbb{R}^d\to\mathbb{R}$ and a measure $\mu$ on $\mathbb{R}^d$ satisfy the condition $\mathcal{H}^{d-1}(A)<+\infty\implies \mu(A)=0$ (where $\mathcal{H}^{d-1}$ is the $d-1$-dimensional Hausdorff measure), then the set on which $u$ is not differentiable has $\mu$-measure $0$. Provide a justification for your answer.

Answer: False


Question: Determine if the limit \( L = \lim_{n \rightarrow \infty} \int_{0}^{n} \frac{f(x/n)}{(1+x)^{4}} \, dx \) exists for a continuous function \( f : [0, \infty) \rightarrow \mathbb{R} \) such that \( |f(x)| \leq 1 + x^{2} \) for all \( x \geq 0 \), and find its value.

Answer: \dfrac{f(0)}{3}


Question: Compute \( \lim\limits_{n \to \infty} \frac{X(n)}{n} \) where \( X(t), t \ge 0 \) is a standard Brownian motion and \( n = 1, 2, 3, \ldots \).

Answer: 0


Question: Determine the minimum value of the function \( f(x,y,z) = x^{z}+y^{z}-(xy)^{\frac{z}{4}} \) for all positive real numbers \( x, y, \) and \( z \).

Answer: -\dfrac{1}{8}


Question: Let $f:\mathbb{C} \to \mathbb{C}$ be an entire function satisfying $|f(z)| \leq 100 \ln |z|$ for $|z| \geq 2$, and assume that $f(i) = 2i$. Determine the value of $f(1)$. \( \boxed{?} \)

Answer: 2i


Question: How many natural numbers between 2 and 60 are there that have no primitive roots and require at least four squares to be expressed as a sum of squares?

Answer: 5


Question: Solve the equation for $x$ in terms of $a$: $$ x (\mathrm{exp}({\frac{2x}{a^2}}) + 1) = \mathrm{exp}({\frac{2x}{a^2}}) - 1 $$

Answer: 0


Question: Using Stokes' Theorem, evaluate the surface integral \(\iint_S \nabla \times F \cdot dS\) where \(F = yz\hat{i} - xy\hat{j} + \hat{k}\) and \(S\) is the upper hemisphere defined by the equation \(x^2 + y^2 + (z - 1)^2 = 4\) with \(z \geq 1\). Determine if the result is \(-4\pi\).

Answer: -4\pi


Question: Calculate the surface integral \(\iint_{S}\text{curl}(\vec F)dS\) for the vector field \(\vec{F}(x,y,z)=(y^2z, xz,x^2y^2)\) over the part of the paraboloid \(z=x^2+y^2\) that lies within the cylinder \(x^2+y^2=1\), considering outward-facing normal vectors. Determine the value of the integral.

Answer: \pi


Question: Evaluate the limit: \[ L = \lim_{n\to\infty} \left(\sqrt[n+1]{(n+1)!} - \sqrt[n]{n!}\right) \]

Answer: \dfrac{1}{e}


Question: Evaluate the limit of the expression as $k$ approaches infinity using a definite integral: $$\lim_{k\to\infty}\sum_{i=1}^{k}{\frac{k}{k+2i}}$$

Answer: \infty


Question: Suppose $M$ is a correlation matrix with correlation $\rho$ between any pair of random variables. Determine the range of possible values for the determinant of $M$. \( \text{Provide your answer as an interval.} \)

Answer: [0, 1]


Question: Determine the limit: \[ \lim_{n\to\infty} \left(\frac{n^2 + 2n \cos(n)}{n^2 + 1}\right)^{1/n} \]

Answer: 1


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

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

where $a$ is a constant and $h$ approaches zero.

Answer: f(x)


Question: Suppose $E$ is an extension field of $\mathbb{Z}_5$ and $E$ has exactly $78125$ elements. Determine the degree $\deg(a/\mathbb{Z}_5)$ for every $a \in E - \mathbb{Z}_5$. Provide your answer as a single integer.

Answer: 7


Question: Evaluate the triple integral \( \iiint xyz\,dx\,dy\,dz \) over the volume of the tetrahedron with vertices \((0,0,0)\), \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\).

Answer: \dfrac{1}{720}


Question: For a sequence of functions $f_n:[0,1]\to \mathbb{R}$, where each $f_n(x)$ takes only the values 0 and 1, and the intervals where $f_n(x)=1$ shrink and cover the entire interval $[0,1]$ infinitely many times as $n$ increases, determine the value of $\limsup_{n \to \infty} f_n(x)$ for a fixed $x \in [0,1]$. \(\text{Provide your answer as either 0 or 1.}\)

Answer: 1


Question: Determine the values of $p$ for which the sequence $\{a_n\}$, defined by the inequality $0 < a_{n+1} \leq a_n + \frac{1}{n^p}$, converges. Here, $p$ is a real number and $n \geq 1$. Provide your answer as a set of values for $p$.

Answer: (1, \infty)


Question: Determine whether there exists a subspace of the space of 2x3 matrices over the real numbers, denoted as $M^R_{2x3}$, that is isomorphic to the space of polynomials of degree at most 4, denoted as $R_4[x]$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the integral of the function \( \operatorname{sgn}(\sin(\pi/x)) \) over the interval \((0, 1)\).

Answer: 1 - 2 \ln 2


Question: Evaluate the limit of the sequence \( Y_n = \frac{500^n}{\sqrt{n!}} - 1 \).

Answer: -1


Question: Calculate the limit: $$\lim_{x\to -\infty} \left(\frac{1}{x}\arctan \frac{1}{x}-\frac{1}{x}\frac{\pi}{2}\right)$$ without using a calculator.

Answer: 0


Question: Calculate the limit \(\lim_{n\rightarrow \infty}\int_0^1 e^{x^n} \, dx\) and justify the existence of this limit.

Answer: 1


Question:
Suppose $X$ and $Y$ are independent random variables such that $X\sim\mathsf{Bin}\left({k},\frac{2}{5}\right)$ and $Y\sim\mathsf{Bin}\left({m},\frac{2}{5}\right)$. Determine the limit:

$$\displaystyle{ \lim_{m\to\infty}\mathbb{P}\left[X\geq t+Y\right] }$$

where $t=\frac{2m}{5}$.

Answer: 0


Question: Evaluate the limit: $$ \lim_{n \to \infty} \frac{\int_{0}^{1} \frac{e^{-x^2} dx}{(1+x^{2})^{n}}}{\frac{1}{n^2}} $$

Answer: \infty


Question: Evaluate the line integral \(\oint_C {\bf B}\cdot d{\bf x}\) for the vector field \({\bf B}({\bf x})={\rho}^{-1}e_{\phi}\), where \(C\) is the ellipse defined by \(z=0, \rho=2\cos\phi\) and \(0\le \phi \le \pi\).

Answer: \pi


Question: Use Stokes' theorem to evaluate the integral \( \int_C (x+ 2y)\,dx + (2z+ 2x)\,dy + (z+y)\,dz \), where \( C \) is the intersection of the sphere \( x^2 + y^2 + z^2 = 1 \) and the plane \( y = z \). Assume \( C \) is oriented counterclockwise when viewed from above.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{\sin(\sin(\sin x)) - x}{x^3}.$$

Answer: -\dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{n\to \infty} \left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n}\right). \]

Answer: \ln 2


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty }n\left [ \widetilde{H_{n}}-H_{2n}+H_n \right ] \] where \( \widetilde{H_{n}} \) is the alternating harmonic number defined as \[ \widetilde{H_{n}}=\sum_{j=1}^{n}\frac{\left ( -1 \right )^{j-1}}{j} \]

Answer: \dfrac{1}{4}


Question: Suppose that \(x\) and \(y\) satisfy the equations \(3\sin x + 4\cos y = 5\) and \(4\sin y + 3\cos x = 2\). Find the value of \(\sin(x+y)\).

Answer: \dfrac{1}{6}


Question: Let $f_n:[0,1]\rightarrow \mathbb{R}$ be functions for $n\in\mathbb{N}$ such that (1) for some $M>0$, $\vert f_n\vert_{L_\infty([0,1])}\leq M$ for all $n\in\mathbb{N}$ and (2) $(f_n)$ weakly converges to some $f$ in $L_2([0,1])$. Determine whether it is true that for almost all $x\in[0,1]$, $f(x)\in[\liminf \, f_n(x),\limsup \,f_n(x)]$. Provide a justification for your answer.

Answer: True


Question: In the ring $\mathbb{Z}_5[x]/(x^3 + 3x + 2)$, determine the inverse of the element $(x+2) + (x^3 + 3x + 2)$, if it exists.

Answer: 3x^2 + 4x + 1


Question: Consider a joint probability density function $f: \mathbb{N}_1 \times \mathbb{N}_2 \rightarrow \mathbb{R}$ and probability measures $\mu$ and $v$. If the double integrals \( \int_{\mathbb{N}_1}\int_{\mathbb{N}_2}f(w_1,w_2)\mu(dx)v(dx) \) and \( \int_{\mathbb{N}_2}\int_{\mathbb{N}_1}f(w_1,w_2)v(dx)\mu(dx) \) yield different results, can we conclude that Fubini's theorem is not applicable to these integrals? Provide a justification for your answer.

Answer: Yes


Question: Calculate the limit: $$\lim_{x\rightarrow 0}\frac{\cos(2x)-\sqrt{1+x^4}\sin(x)}{\tan^6(x)}$$

Answer: \infty


Question: Find the value of \(a \in \mathbb{R}\) such that the solutions of the system \(\dot{x} = 3x + y + 1; \dot{y} = 6x + 2y + a\) are bounded for \(-\infty < t < \infty\). Determine whether these solutions are stable.

Answer: 2


Question: Evaluate the limit \( \lim _{z \rightarrow \infty} z e^{-\frac{1}{2} z^{2}} \) using a method other than L'Hospital's Rule.

Answer: 0


Question: Given a Poisson distribution with a double mode at $X=1$ and $X=2$, calculate the probability $P(X=0)$. Express your answer as a decimal or a fraction.

Answer: 0.1353


Question:
Given the following information:

- \(\nabla(r^2) = 2 \vec r\)
- \(\hat n = \frac{1}{\sqrt{x^2 + y^2}}( x \hat i + y \hat j)\)
- \(a = \int_C y^2 \, ds\) and \(b = \int_C x^2 \, ds\)

where \(C\) is a closed plane curve, \(a\) and \(b\) are the moments of inertia about the \(x\) and \(y\) axes, \(\hat n\) is the unit outward vector, and \(r = \left|x \hat i + y \hat j\right|\), find the value of the integral \(\int_C (x^2 + y^2) \, ds\).

Answer: a + b


Question: In a non-abelian group $G$, is it possible to find two distinct elements $a$ and $b$ such that their centralizers are equal and not equal to the entire group?

Answer: Yes


Question: Find the residue of \( \frac{(z+1)^3}{(z-2)^2} \) at \( z=2 \).

Answer: 27


Question: Given that \(\lim_{x\to 0} \frac{ae^x - b\cos x + ce^{-x}}{x\sin x} = 2\), find the value of \(a + b + c\).

Answer: 4


Question: Evaluate the limit: $$\lim_{n \to \infty} \frac{1}{n^4}\left(\left \lfloor{1^3 x}\right \rfloor + \left \lfloor{2^3 x}\right \rfloor + \cdots+ \left \lfloor{n^3 x}\right \rfloor\right)$$ where $x$ is a real number.

Answer: \dfrac{x}{4}


Question: Expand the function \( f(z) = \frac{1}{1+e^z} \) using a Laurent series and find the residue at the singularity points \( z_0 = (2k+1)\pi i \) for integer \( k \).

Answer: -1


Question: Find the maximum value of the expression \( \sqrt{\sin^2x + 2a^2} - \sqrt{-1 - \cos^2x + 2a^2} \) for real numbers \(a\) and \(x\).

Answer: \sqrt{2}


Question: Determine whether the free group on four letters is torsion-free.

Answer: Yes


Question: Determine the limit \( \lim_{n \to \infty} \frac{\sqrt[n]{n!}}{n} \).

Answer: \dfrac{1}{e}


Question: Evaluate the line integral \(\int_\gamma x^2 \, dy\) where \(\gamma\) is the path connecting the points \((0,0)\), \((1,1)\), and \((2,0)\) in that order.

Answer: -2


Question: Calculate the perimeter of the evolute of the ellipse given by the equation \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \).

Answer: \dfrac{38}{3}


Question: Evaluate the line integral \( \int_C Pdx + Qdy \) where \( P(x,y) = xe^{-y^2} \) and \( Q(x,y) = -x^2ye^{-y^2} + \frac{1}{x^2+y^2+1} \). The curve \( C \) is the boundary of the square defined by \(-a \leq x \leq a\) and \(-a \leq y \leq a\), oriented positively.

Answer: 0


Question: Given the equations $x^2 - 2y = -13$, $y^2 - 4z = 14$, and $z^2 + 6x = -15$, find the value of $xy + xz + 6yz$. Provide your answer as a single number.

Answer: 3


Question: Determine whether a continuous function $g:\mathbb{R}\rightarrow \mathbb{R}$, which is smooth on the open set $\mathbb{R}-\{0\}$, must be smooth everywhere on $\mathbb{R}$. Provide a justification for your answer.

Answer: No


Question: Given the recursive sequence \( u_0 = 2 \) and \( u_{n+1} = \frac{2}{3}u_n - n - \frac{8}{3} \), and another sequence defined as \( v_n = u_n + x \cdot n - 1 \), find the value of \( x \) such that \( v_n \) forms a geometric sequence.

Answer: 3


Question:
Given real numbers \(x, y, z\) that are not all zero, determine all possible values of the expression:
\[ A = \frac{|x+y|}{|x|+|y|} + \frac{|y+z|}{|y|+|z|} + \frac{|z+x|}{|z|+|x|}. \]

Answer: [1, 3]


Question:
Gary can be Cheerful (1), So-so (2), or Glum (3). The following transition matrix describes the Markov chain that records Gary's mood changes:

$$P=\begin{bmatrix}0.5 & 0.4 & 0.1\\0.3 & 0.4 & 0.3\\0.2 & 0.3 & 0.5\end{bmatrix}$$

Given the stationary distribution vector:

$$ \pi = (0.33871,0.370968,0.290323)$$

If Gary is cheerful today, calculate the expected number of days until he is glum three days in a row.

Answer: \dfrac{236}{9}


Question: Let \( \{A_n\} \) be a sequence of events such that \( \operatorname{Cov}(I_{A_i},I_{A_j}) = E[I_{A_i}I_{A_j}] - E[I_{A_i}]E[I_{A_j}] \leq 0 \) for all \( i \neq j \). Prove that if \( \sum P(A_i) = \infty \), then \( P[\lim \sup A_n] = 1 \).

Answer: 1


Question: Given the 5-adic number \(x = 2 + 1 \cdot 5 + 3 \cdot 5^2 + 2 \cdot 5^3 + \ldots \), find the first four digits of \(\frac{1}{x}\) in its 5-adic expansion.

Answer: 3334


Question: Evaluate the limit: $$\lim_{n\to\infty} \left(\frac{1}{\sqrt{n^2+1}} + \cdots + \frac{1}{\sqrt{n^2+n}}\right).$$

Answer: 1


Question: In triangle $ABC$, where $AC = BC = 24$, a circle with center $J$ is inscribed. Given that $CH$ is the altitude from $C$ to $AB$ (with $CH \perp AB$ and $H \in AB$), and the ratio $CJ:CH = 12:17$, determine the length of $AB$.

Answer: 20


Question: Calculate the sum \(3 \binom{n}{3} + 5 \binom{n}{5} + 7 \binom{n}{7} + \cdots\).

Answer: n(2^{n-2} - 1)


Question: Given the joint probability density function \( f_{xy}(x,y) = 1 \) for \( 0 < x < 1 \) and \( x < y < x+1 \), and \( 0 \) elsewhere, derive the marginal probability density function of \( Y \), \( f(y) \), and calculate the expected value \( E[Y] \).

Answer: 1


Question: Evaluate the limit \( \lim_{x \to \infty} \frac{x^x}{x!} \).

Answer: \infty


Question: Determine the number of distinct continuous functions that can be defined from a separable Hausdorff space \((X, \tau)\) to the interval \([0, 1]\).

Answer: \mathfrak{c}


Question: Determine whether the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(0) = 0 \) and \[ f(x) = \int_0^x \cos \frac{1}{t} \cos \frac{3}{t} \cos \frac{5}{t} \cos \frac{7}{t} \, dt, \quad \forall x \neq 0, \] is differentiable at \( x = 0 \).

Answer: \dfrac{1}{8}


Question: Determine a pointwise convergent subsequence of the sequence of functions \( f_n(x) = (-1)^n x^n \).

Answer: f_{2k}(x) = x^{2k}


Question: Let $\{x_n\}_{n=1}^{\infty}$ be a convergent sequence in a normed space $(S, \|\cdot\|)$. Find the limit of the sequence \[\left\{\frac{x_1 + 2x_2 + \cdots + nx_n}{n^2}\right\}_{n=1}^{\infty}.\]

Answer: \dfrac{L}{2}


Question: Calculate the expectation $E[(B_t - B_s)^5]$ for a Standard Brownian Motion $(B_t)_{t\ge0}$, where $s < t$. Provide your answer in terms of $t$ and $s$.

Answer: 0


Question: For odd integers A and x greater than 1, determine if it is always true that \(A^x - A\) is divisible by both \((A+1)/2\) and \((A-1)/2 + 1\).

Answer: Yes


Question: Determine whether the following statement is true or false: If $\lim_{x \rightarrow c} f(x)$ does not exist, then there exists a sequence $(x_n) \subset A$, where $x_n \neq c$ and $(x_n) \rightarrow c$, but $\lim_{n \rightarrow \infty} f(x_n)$ does not exist.

Answer: A


Question: Evaluate the integral \( \oint_{|z+2i|=1} \frac{\sin(z)}{z^3+16z} \, dz \) using the residue theorem. Factor the denominator to identify the poles and their respective orders within the contour.

Answer: 0


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty} n \int_{-n}^{n} \cos^2(\pi x) \left(\frac{1}{(x-3)(x-4)} + \frac{1}{(x+3)(x+4)}\right) dx$$

Answer: -2


Question:
In the ring $\mathbb{Z}/2\mathbb{Z}$, determine which polynomial is NOT in the ideal generated by $1+x^2$ and $1+x^3$.

A. $1+ x^4$
B. $x^5+x+1$
C. $1+x^6$

Answer: B


Question: Consider normed linear spaces X and Y, and a finite-rank continuous linear transformation T from a subspace W of X to Y. Is it possible to extend T to a continuous linear transformation from X to Y that maps X onto the range of T?

Answer: Yes


Question: Evaluate \( \lim_{n \to \infty} \int_1^2 \frac{x^n}{x^n+1} \, dx \).

Answer: 1


Question: Determine the degree of the field extension \(|\mathbb{Q}(\sqrt[m]{p}, \sqrt[n]{q}) : \mathbb{Q}|\), given primes \(p\) and \(q\), and positive integers \(m\) and \(n\).

Answer: mn


Question: Calculate the mass of the tile in the first quadrant defined by the inequalities \(1 < xy < 3\) and \(1 < x^2 - y^2 < 3\) using the density function \(p(x, y) = (x^4 + y^4)(x^2 + y^2)\).

Answer: 26


Question:
Evaluate the limit as \( n \to \infty \):
\[
\left(\frac{n^4-3n^3-n^2+2n-1}{n^5+n^4-n^3-3n^2-3n+1}\right)^{\frac{6n^5-2n^4-2n^3+n^2-2n}{9n^4-2n^3+n^2+3n}}
\]

Answer: 0


Question: For two $C^\infty$-compatible charts on a topological manifold, determine if the inverse transition maps are also $C^\infty$. Provide a justification for your answer.

Answer: Yes


Question: Find the limit of the sequence $\{b_n\}$ as $n$ approaches infinity, given that $\lim\limits_{n\to\infty}\dfrac{b_n^{2n+1}}{2n+1}=1$. Express your answer as a single value.

Answer: 1


Question: Determine whether the sequence \(b_n = \frac{1}{\sqrt{n^2+2n+2}} + \frac{1}{\sqrt{n^2+2n+3}} + \ldots + \frac{1}{\sqrt{n^2+3n+1}}\) converges.

Answer: 1


Question: Let $1 < p < 2$ and $u \in W_0^{1,p}(\Omega)$, where $\Omega$ is a smooth bounded open set of $\mathbb{R}^N$, $N \geq 2$. Suppose that \( \int_{\Omega} \nabla u \nabla \phi \, dx = 0, \forall \, \phi \in C_0^{\infty}(\Omega). \) Determine whether it is true that $u = 0$. Provide a justification for your answer.

Answer: u = 0


Question: Given the sequence \((x_n)_{n \ge 1}\) defined as follows: \[x_1 > 0, \quad \sum_{k=1}^{n}x_k = \frac{1}{\sqrt{x_{n+1}}}\] for all \(n \ge 1\), determine the limit \[\lim_{n \to \infty} n^2x_n^3.\]

Answer: \dfrac{1}{9}


Question: Given a convex but non-smooth function $f : \mathbb R^2 \to \mathbb R$, if at a particular point in $\mathbb R^2$ the function increases in the directions of $(1,0)$, $(-1,0)$, $(0,1)$, and $(0,-1)$, can we conclude that this point is a local (and thus global) optimum due to the function's convexity? Answer with 'Yes' or 'No' and justify your answer.

Answer: Yes


Question: Let $p(x)$ and $q(x)$ be two monic polynomials with real coefficients of even degree $n \geq 4$. If the coefficient of $x^{n-1}$ in $p(x)$ is less than that in $q(x)$, determine whether the polynomial $p(x) - q(x)$ must have a real root.

Answer: Yes


Question: Let $X$, $Y$, and $Z$ be independent exponentially distributed random variables with means $1/\lambda_X$, $1/\lambda_Y$, and $1/\lambda_Z$, respectively. Calculate the probability $P(X > \max\{Y, Z\})$. Additionally, consider a queueing system with three cash boxes, A, B, and C, each with independent exponential service times with a mean of 5. Determine the probability that cash box D, which is waiting, is the last one to leave the system. Express your answer as a single probability value.

Answer: \dfrac{1}{3}


Question: Let \( N \) be the positive integer with 2008 decimal digits, all of them 1. That is, \( N = 111\ldots111 \), with 2008 occurrences of the digit 1. Determine the 1005th digit after the decimal point in the expansion of \( \sqrt{N} \).

Answer: 2


Question: Given sequences of positive numbers $a_n$ and $b_n$ such that \( \lim_{n\to\infty}a_n^n = a \) and \( \lim_{n\to\infty}b_n^n = b \), where \( a, b \in (0, \infty) \), determine the limit \( \lim_{n\to\infty}(pa_n + qb_n)^n \) for nonnegative numbers \( p \) and \( q \) with \( p + q = 1 \).

Answer: a^p b^q


Question: Evaluate the integral \( \int_{0}^{\pi} \cos^4(x + \sin 3x) \, dx \).

Answer: \dfrac{3\pi}{8}


Question: Compute the limit: \[ \lim_{n \to \infty} \frac{(2n)!}{(2^n \cdot n!)^2} \] using bounding techniques or simplification methods.

Answer: 0


Question: Evaluate the limit \( \lim\limits_{n \rightarrow +\infty} \int_{0}^{1} nxe^{-\sqrt{n}x}e^{-x^2}dx \).

Answer: 1


Question: Determine whether the converse of the statement "If $d$ divides $p-1$ for a prime $p$, then $x^d \equiv 1 \pmod{p}$ has exactly $d$ roots" is true. Specifically, is it true that if $x^d \equiv 1 \pmod{p}$ has $d$ roots, then $d$ must divide $p-1$?

Answer: True


Question: Evaluate the limit \( \lim_{k \to \infty} \int_0^k x^n \left(1 - \frac{x}{k} \right)^k \mathrm{d} x \) for a fixed integer \( n \geq 0 \).

Answer: n!


Question: Consider the function \( g(x) = ax^2 + bx + c \) with parameters \( a \), \( b \), and \( c \). Given that \( \frac{dc}{du} = 2 \) and \( c = 1 \) when \( u = 0 \), determine the minimum number of integrals needed to express \( B(3) = \int_0^3 \{g(x)\} \, dx \), where \( \{\cdot\} \) denotes the fractional part function.

Answer: 1


Question: Let $f: \mathbb{N} \to \{0,1\}$ be a function. Define the set $A$ such that a function $h: \mathbb{N} \to \{0,1\}$ belongs to $A$ if there exists a natural number $k$ for which $f(i) = h(i)$ for all $i \ge k$. Determine the cardinality of the set $A$. \( \boxed{} \)

Answer: \aleph_0


Question: Determine whether a finite group $G$ has a subgroup of index 2 if and only if its abelianization $G/[G,G]$ has even order.

Answer: Yes


Question: Solve the equation $x^x = 2^2$ for the real number $x$.

Answer: 2


Question: Find the limit of the expression \( \frac{\ln(\frac{\cos\alpha}{\cos\beta})}{\beta-\alpha} \) as \( \beta \) approaches \( \alpha \), given that \( 0<\alpha<\beta<\frac{\pi}{2} \).

Answer: \tan \alpha


Question: Determine the number of elements in the symmetric group $S_8$ that have a cycle structure of $4^2$, meaning permutations composed of two 4-cycles.

Answer: 1260


Question: Find the limit of the sequence \(\frac{n}{\sqrt[n]{n!}}\) as \(n\) approaches infinity, and explain its relation to the number \(e\).

Answer: e


Question: Given two sequences of events $\{A_n\}$ and $\{B_n\}$ with $\mathbb{P}(\liminf_n A_n) = \mathbb{P}(\limsup_n B_n) = 1$, determine if $\mathbb{P}(\limsup_n A_n \cap B_n) = 1$.

Answer: 1


Question: For any prime number \( q \) greater than 3, determine if there exists an integer \( s \) such that \[ q = \sqrt{\frac{\sigma_2(t_s) - \sigma_1(t_s)}{2} + 1} \] where \( t_s = (2, 2, \ldots, 2) \) is an \( s \)-tuple of the integer 2, and \( \sigma_1(t_s) \) and \( \sigma_2(t_s) \) are the first and second elementary symmetric polynomials on \( t_s \), respectively.

Answer: Yes


Question: Calculate the flux of the vector field \(\mathbf{F}(x, y, z) = \langle e^{z^2}, 2y + \sin(x^2z), 4z + \sqrt{x^2 + 9y^2} \rangle\) through the surface \(S\) defined by the inequalities \(x^2 + y^2 \leq z \leq 8 - x^2 - y^2\). The surface \(S\) consists of two coaxial half-cones. Use the Divergence Theorem to find the flux.

Answer: 96\pi


Question: Calculate the volume measure \( \lambda_3(A) \) for the set \( A \) defined as \[ A = \left\{(x, y, z) \in \mathbb{R}^3 \middle| z^2 < \frac{x^2}{4} + \frac{y^2}{9} < 2z\right\} \]. Use Fubini's Theorem to determine the limits of integration and evaluate the integral \( \int_A 1 \, d\lambda_3 \).

Answer: 8\pi


Question: Calculate the limit: $$\lim_{x\to 0}(\cos x)^{\dfrac{1}{\ln(\sin^2(x))}}$$ without using L'Hospital's rule.

Answer: 1


Question: Determine the number of distinct ways to color the faces of a cube using 6 different colors, where two colorings are considered equivalent if one can be obtained from the other by a rotation of the cube.

Answer: 30


Question: Given positive integers $n$ and sequences of positive numbers $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ such that \(\sum_{i=1}^{n}a^2_{i} = \sum_{i=1}^{n}b^2_{i} = 1,\) find the maximum value of the minimum ratio \(\min_{1\le i\le n}\left(\frac{a_{i}}{b_{i}}\right).\)

Answer: 1


Question: Determine if the order of the commutator subgroup $G'$ is greater than $p^2$ for any group $G$ of order $p^3$, where $p$ is a prime.

Answer: No


Question: Determine whether the limit \( \lim_{(x,y) \to (0,0)} f(x,y) \) exists for the function \( f(x,y) = x\sin (1/y)+y\sin (1/x) \), where the function is defined as 0 when \( x=0 \) or \( y=0 \). If the limit exists, find its value.

Answer: 0


Question: Given the equation \(x^2 + xy + y^3 = 1\), find the value of \(y'''\) at the point where \(x = 1\).

Answer: 42


Question: Find the minimum value of the function \( f(x,y) = \sin^2(x) + \sin^2(y) - \cos(\pi - x - y) \) over all possible values of \( x \) and \( y \).

Answer: -1


Question:
Let $N$ be the number of sequences $a_1, a_2, a_3, \ldots, a_{24}$ that satisfy the following conditions:
1. For all $1 \le i \le 24$, $1 \le a_i \le 24$.
2. If $i$ is even, then the sum of the first $i$ terms, $a_1 + a_2 + \cdots + a_i$, is congruent to $1$ modulo $2$.
3. If $i$ is divisible by $3$, then the sum of the first $i$ terms is congruent to $2$ modulo $3$.

Given that $N = p_1^{a_1}p_2^{a_2}$ where $p_1$ and $p_2$ are prime numbers, find $p_1 + a_1 + p_2 + a_2$. Use the Chinese Remainder Theorem to assist in solving this problem.

Answer: 81


Question: Determine if it is possible to express the right-hand side of the identity \( (1+6x^3+9x^4)^3+(1-6x^3+3x-9x^4)^3+(1-9x^3-6x^2)^3=9x^2+9x+3 \) as a perfect cube by adding a suitable term to both sides. If possible, find the term.

Answer: 3x^3


Question: Determine the greatest value of \( p \) such that the series \[ \sum_{n=1}^{\infty}(-1)^n \cdot \tan \left(\frac{1}{\sqrt{n^p}}\right) \cdot \ln \left(1+\frac{1}{n^{2 p}}\right) \] converges conditionally.

Answer: \dfrac{2}{5}


Question: Find a subspace $W$ of the vector space $V$ with maximal dimension such that for all vectors $w$ in $W$, the quadratic form $$ q(x,y,z) = 2x^2 - \frac{1}{2}y^2 + 4z^2 $$ satisfies $q(w) \geq 0$. Determine the dimension of this subspace.

Answer: 2


Question: In the 100 Prisoners Problem, a single prisoner is allowed to open 50 boxes. If the prisoner uses the random method and gains information about the incorrect boxes after each selection, what is the probability of finding the correct box?

Answer: \dfrac{1}{2}


Question: Prove that the expression \( \sum_{i=1}^{2^{n-1}}(-1)^i(2i-1)^2 \) equals \( 2^{2n-1} \).

Answer: 2^{2n-1}


Question: Evaluate the integral \( \int_0^{+\infty}\cos 2x\prod_{n=1}^{\infty}\cos\frac{x}{n}dx \).

Answer: \dfrac{\pi}{4}


Question: Let $G$ be a finite group with normal subgroups $H$ and $K$. Given that $|G:H|=4$ and $|G:K|=5$, find the possible values of $|G:H \cap K|$.

Answer: 20


Question: Determine whether the level surface \(N_k = \{(x_1,...,x_n): z(x_1,...,x_n) = k, k \in \mathbb{R}\}\) of the tangent plane \(z(x_1,...,x_n)\) of a function \(f(x_1,...,x_n)\) at a point \((a_1,...,a_n) \in N_k\) is the same as the tangent plane \(p(x_1,...,x_n)\) at \((a_1,...,a_n)\) of the level surface \(N_k = \{(x_1,...,x_n): f(x_1,...,x_n) = k, k \in \mathbb{R}\}\) of the function.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to +\infty} \frac{n}{\ln(n)}(\sqrt[n]{n}-1) \]

Answer: 1


Question: Evaluate the limit: $$\lim_{x\to 0}\left(\frac{e^{-x^{-2}}}{x^{100}}\right)$$

Answer: 0


Question: Determine the quasi components of the set of rational numbers \(\mathbb{Q}\) when it is equipped with the topology induced by the real numbers \(\mathbb{R}\).

Answer: \{q\}


Question: Evaluate the limit: $$\lim_{x \to 1^-} \left(\ln\left(\left(\frac{x-1}{x+1}\right)^2\right) \cdot x - 2 \cdot \ln|x^2-1|\right)$$

Answer: -4 \ln 2


Question: Evaluate the limit: $$\lim_{n\to\infty} \int_{-\infty}^{\infty} \frac{ne^{\cos x}}{1+n^2x^2}\ dx$$

Answer: e\pi


Question: A square pyramid has equilateral triangles as its sides. A cuboid with dimensions 25 (height) by 50 (width) by 50 (length) has a volume of 62500. Determine the length of the square base of the pyramid such that the cuboid can be fully contained within the pyramid, with the bottom of the cuboid positioned 26 units above the base of the pyramid.

Answer: 50 + 51\sqrt{2}


Question: Given two dependent random variables $X$ and $Y$, with $E(Y|X=x)=5x$ and $\text{Var}(Y|X=x)=2x+x^2$, and $E(X)=12$ and $\text{Var}(X)=4$, compute the variance of $Y$. Assume the expectation of $Y$ has been computed as $E(Y)=5E(X)$. What is the variance of $Y$?

Answer: 272


Question: Consider the group $G = SL2(p)$, consisting of $2 \times 2$ matrices with determinant $1$ and entries in the field $F(p)$ of integers modulo $p$. Define the action of $G$ on the set $X = F(p) \cup \{ \infty \}$ using Möbius transformations. Determine the orbit of $\infty$ and the stabilizer of $\infty$. Use this information to calculate the order of $SL2(p)$. Provide your answer as a single integer.

Answer: p(p^2 - 1)


Question:
Find the point of intersection between the plane $x + 2y + z = 10$ and the paraboloid $z = x^2 + y^2$ that is closest to the origin. Use the method of Lagrange multipliers to minimize the distance squared from the origin, given by $d^2(x, y, z) = x^2 + y^2 + z^2$, subject to the constraints:
$$g(x, y, z) = x + 2y + z - 10 = 0$$
$$h(x, y, z) = x^2 + y^2 - z = 0$$
Solve the system of equations to find the solution.

Answer: (1, 2, 5)


Question: Determine the maximum number of inflection points that a cubic planar curve, given by the equation $F(x,y)=0$ where $\deg F=3$, can have. Consider the real roots of the determinant $F_{xx}F_y^2-2F_{xy}F_xF_y+F_{yy}F^2_x$ in your analysis.

Answer: 3


Question: Determine the function $f(x)$ that satisfies the relation $f(x+y) = f(x) + f(y) - 2xy + (e^x - 1)(e^y - 1)$ for all $x, y \in \mathbb{R}$ and the condition $f'(0) = 1$.

Answer: e^x - x^2 - 1


Question: Calculate the expected exit time of a random walk on the interval $(-a, b)$ starting at position $x$ using a recurrence relation.

Answer: (a + x)(b - x)


Question: Determine whether there exists an infinitely differentiable function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that matches the absolute value function \( |x| \) at every integer point \( x \in \mathbb{Z} \).

Answer: Yes


Question: Find the smallest real number \( c \) such that the inequality \( x + y - (xy)^c \geq 0 \) holds for all \( x, y \in [0,1] \).

Answer: \dfrac{1}{2}


Question: Evaluate the improper integral \(\int_{0}^{\infty} \frac{1-x^2}{x^4+3x^2+1}\ dx\) and simplify the result.

Answer: 0


Question: Calculate the sum of the series: \[ \sum_{k=1}^{n+1} \binom{n+1}{k}(-1)^{k-1}\frac{1}{k} \]

Answer: H_{n+1}


Question: Given two independent continuous random variables $X$ and $Y$ with supports $[a,b]$ and $[c,d]$ respectively, determine the support of their sum $X+Y$. Provide a rigorous proof.

Answer: [a + c, b + d]


Question: In a pirate coin distribution game, determine the minimum number of coins the third youngest pirate must be offered to vote in favor of a proposal.

Answer: 1


Question: Determine the maximum number of functions in the set \( A = \{ f \in C^1[0,1] : f(0) = 0, f(1) = 1, |f'(t)| \le 1 \text{ for all } t \in [0,1] \} \) that can have different slopes at a given point \( c \in (0,1) \).

Answer: 1


Question: Transform the second-order differential equation \(x'' - c(x')^2 + d(x + 2) = 0\) into a first-order differential equation in the form \((x,v)' = G(x,v)\) where \(v = x'\). Find the zeros of \(G(x,v)\).

Answer: (-2, 0)


Question: Find the value of the fractional part \( x = \left\{\frac{3^{1001}}{82}\right\} \), where \( \{\} \) denotes the fractional part.

Answer: \dfrac{3}{82}


Question: Determine whether the following statement is true or false: If $f \in L^2(0,1)$ and $\int_0^1 x^n f(x)\,dx = 0$ for all positive integers $n$, then $f(x) = 0$ almost everywhere on $(0,1)$.

Answer: True


Question: Evaluate the limit: $$\lim_{{x \to \frac{\pi}{2}}} \left(1+\cos x\right)^{\tan x}$$

Answer: e


Question: Find the minimum value of the expression \(2\csc(2x)+\sec(x)+\csc(x)\) for \(x \in (0,\frac{\pi}{2})\).

Answer: 2 + 2\sqrt{2}


Question: Let $T: L^2([0,1]) \rightarrow L^2([0,1])$ be a linear operator defined by $T(f)(x) = x \cdot f(x)$. Determine a bound for the operator norm $\| T \|$. Justify your answer.

Answer: 1


Question: Determine the number of non-isomorphic quotient groups of the symmetric group $S_4$.

Answer: 4


Question: Determine whether the statement "If \(\sum_{n=1}^\infty a_n^2\) converges, and \(a_n>0\) for all \(n\), then \(\sum_{n=1}^\infty a_n\) also converges" is true or false. Justify your answer with a proof or a counterexample.

Answer: False


Question: Find the sum of all positive integers that cannot be expressed as the sum of two abundant numbers.

Answer: 4179871


Question: For distinct primes $p$ and $q$, find the number of non-isomorphic semidirect products of the group $\mathbb{Z}_p$ by the group $\mathbb{Z}_q$, given that $(p-1)/q$ is not an integer.

Answer: 1


Question: Determine whether there exists a function $g(x)$ and its inverse $g^{-1}(x)$ such that the second derivative of $g(x)$ is positive everywhere and the second derivative of $g^{-1}(x)$ is negative everywhere.

Answer: Yes


Question: Krishna is preparing a pizza with 8 slices and has 10 different toppings available. Each slice can have one topping, and the same topping cannot be used on adjacent slices. How many different ways can Krishna arrange the toppings on the pizza slices?

Answer: 43046730


Question: Evaluate the integral \( \int_0^1 \ln^n(x) \, dx \).

Answer: (-1)^n n!


Question: Determine whether the expression \(\big[\log(n) + \sum_{j=1}^{n-1} (\log(j) - (j+1)(\log(j+1)) + j \log(j) +1)\big]\) is in \(O(\log(n))\).

Answer: O(\log(n))


Question: Calculate the residue of the function \( f(z) = \frac{e^{-z}}{(z+2)^3} \) at \( z = -2 \) using the Residue Theorem.

Answer: \dfrac{e^2}{2}


Question:
Find the closed-form expression for the limit:

$$\lim_{n\to \infty}\left(\frac{1}{ \Gamma\left(\displaystyle \frac{1}{1}\right)}+ \frac{1}{ \Gamma\left( \displaystyle \frac{1}{2}\right)}+ \cdots + \frac{ 1}{ \Gamma \left( \displaystyle \frac{1}{ n}\right) }- \log\left( \Gamma\left(\displaystyle\frac{1}{n}\right)\right)\right),$$

where the sequence is known as the Gammaharmonic series.

Answer: \gamma


Question: Find the minimum value of \( P = 3x^2 + xy - 2y^2 \) subject to the constraint \( x^2 - xy + 2y^2 = 3 \).

Answer: -3


Question: Does there exist a function $f(x)$ such that for some integer $a$, $f(a) = 1$, $f(x) = 0$ for all integers $x$ between $1$ and $a$, and $\lim_{x \to \infty} f(x) = 1$?

Answer: Yes


Question: Evaluate the limit: $$\lim_{(x,y)\rightarrow (0,0)}\frac{e^{xy}}{x+1}$$ Determine if the limit exists, and if it does, find its value.

Answer: 1


Question: Determine whether a continuous function $f: \mathbb{R} \to \mathbb{R}$ that satisfies $f(f(x)) = x$ for all $x \in \mathbb{R}$ must have a fixed point, i.e., a point $c$ such that $f(c) = c$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: For any positive number \( \epsilon \), there exists a positive integer \( N \) such that the difference between the tangent of \( N \) and the nearest integer to the tangent of \( N \) is less than \( \epsilon \).

Answer: True


Question:
Let $P$ and $Q$ be two non-empty subsets of a normed linear space $X$ and define $P+Q=\{x+y \in X: x \in P \text{ and } y \in Q\}$. Determine which of the following statements is FALSE:

(A) If $P$ or $Q$ are convex, then $P+Q$ is convex.
(B) If $P$ or $Q$ is open, then $P+Q$ is open.
(C) $P+Q$ must be closed if $P$ and $Q$ are closed.
(D) If $P$ is closed and $Q$ is compact, then $P+Q$ is closed.

Answer: C


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^{\infty} n!x^{n^2} \).

Answer: 1


Question: Evaluate the limit: $$\lim\limits_{x \to \infty}\frac{\ln^{1000} x}{x^5}$$

Answer: 0


Question: Find all positive integer solutions \((a, b, c)\) to the equation \(2^a + 3^b = 5^c\), excluding the solution \((1, 1, 1)\).

Answer: (4, 2, 2)


Question: Let \( \mathcal{S_n}(\mathbb{R}) \) be the set of all real \( n \times n \) symmetric, positive semi-definite matrices with at least one entry strictly greater than 1. Determine whether it is true that for all \( A \in \mathcal{S_n}(\mathbb{R}) \), the spectral radius \( \rho(A) \ge 1 \).

Answer: True


Question: Calculate the area of the spherical triangle defined by the points \((0, 0, 1)\), \((0, 1, 0)\), and \(\left(\dfrac{1}{\sqrt{2}}, 0, \dfrac{1}{\sqrt{2}}\right)\) using the spherical Gauss-Bonnet formula.

Answer: \dfrac{\pi}{4}


Question: Determine if the limit of the function \( f: \mathbb{R}^2 \to \mathbb{R} \) defined by \[ f(\mathbf{x}) = \frac{x_1x_2^2}{x_1^4+x_2^2} \quad \text{if } \mathbf{x} \neq \mathbf{0} \] \[ f(\mathbf{0}) = 0 \] is 0 as \( \mathbf{x} \to \mathbf{0} \) using the epsilon-delta definition.

Answer: 0


Question: Determine whether there exists a power series \( \sum_{n=0}^{\infty}c_{n}x^{n} \) that converges for all \( x \in [0, R) \) for some \( R > 0 \), but diverges for all \( x \geq R \).

Answer: Yes


Question: Given a real \(2\times2\) matrix \(A\) such that \(A^6 = I\), where \(I\) is the identity matrix, determine the number of distinct possibilities for the characteristic polynomial of \(A\). The characteristic polynomial must divide the annihilating polynomial \(x^6-1 = (x-1)(x^2+x+1)(x+1)(x^2-x+1)\).

Answer: 5


Question: In the ring $\mathbb{Z}_7[x]/\langle x^3 + 2x + 1 \rangle$, find the multiplicative inverse of the element $3x^2 + 4 + \langle x^3 + 2x + 1 \rangle$. Provide your answer as a polynomial in $\mathbb{Z}_7[x]$.

Answer: 6x^2 + 5x + 4


Question: Given two polynomials $p(x)$ of degree $m$ and $q(x)$ of degree $k$, where the equation $p(x) = q(x)$ has at least $n+1$ distinct solutions, and $n \geq m$ and $n \geq k$, determine the relationship between $p(x)$ and $q(x)$. Provide a justification for your conclusion.

Answer: p(x) = q(x)


Question: Given a complex Banach space $X$ and an element $A \in \mathfrak L(X)$, where $\mathfrak L(X)$ is the set of bounded linear operators on $X$, denote the resolvent set of $A$ by $\rho(A)$ and the spectral radius of $A$ by $r(A)$. If $\lambda \in \mathbb{C} \setminus \overline{B}_{r(A)}(0)$, where $\overline{B}_{r(A)}(0)$ is the closed ball of radius $r(A)$ centered at the origin, the series $\sum_{n=0}^\infty \frac{A^n}{\lambda^n}$ converges absolutely and satisfies $\lambda R_\lambda(A) = \sum_{n=0}^\infty \frac{A^n}{\lambda^n}$. Can it be directly deduced from this equation and the absolute convergence that the resolvent function $f(\lambda) = R_\lambda(A) = (\lambda - A)^{-1}$ is analytic?

Answer: Yes


Question: Suppose $\langle \cdot, \cdot \rangle_{1}$ and $\langle \cdot, \cdot \rangle_{2}$ are inner products on a real vector space $V$. If $\langle v, v \rangle_{1} = \langle v, v \rangle_{2}$ for any $v \in V$, does it follow that $\langle v, w \rangle_{1} = \langle v, w \rangle_{2}$ for all $v, w \in V$? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: Evaluate the integral \(\int \frac{2z^3 + 5z - 1}{(z+2)^4} \, dz\) over the circle centered at \(-2\) with radius 1 using Cauchy's Integral Formula.

Answer: 4\pi i


Question: Determine the values of \(\lambda \in (-1, 1)\) for which the matrix \(I - \lambda P\) is positive definite, given that \(P\) is a matrix with entries in \([0,1]\) and all row sums equal to 1.

Answer: (-1, 1)


Question: Calculate the integral \( \int_{B'} y\sqrt{x^2+y^2}e^{-(x^2+y^2)}\,dx\,dy \) over the region \( B' \), where \( B' = B_1 \cup B_3 \), and \( B_3 = \{(x,y) \in \mathbb{R}^2: 1 \le x \le 2, |y| \le (x-1)^2\} \).

Answer: 0


Question: Given a set $X$ with at most $\kappa$ operations, each having arity less than $\kappa$, and a subset $S$ of $X$ with cardinality less than or equal to $\kappa$, determine the maximum cardinality of the substructure of $X$ generated by $S$.

Answer: \kappa^{<\kappa}


Question: Consider the set $X = \{A = (a_{ij}) \in M_{22} (\mathbb{R}) \mid \operatorname{tr}(A) = 0 \text{ and } |a_{ij}| \leq 2 \text{ for all } 1 \leq i,j \leq 2\}$. Let $Y = \{\det(A) \mid A \in X\} \subset \mathbb{R}$. Determine whether there exist $\alpha < 0$ and $\beta > 0$ such that $Y = [\alpha, \beta]$. Is this statement true or false?

Answer: True


Question: Determine the value of \(\alpha\) for which the series \(\sum_{n=0}^{\infty} (\sqrt{n^\alpha + 1} - \sqrt{n^\alpha})\) converges.

Answer: \alpha > 2


Question: For a continuously differentiable function $f(x)$ on $[1,\infty)$, if $f(x) \to a$ as $x \to \infty$ for some constant $a$, does it necessarily follow that $f'(x) \to 0$ as $x \to \infty$? Provide a justification for your answer.

Answer: No


Question: Find a non-zero function $f(x)$ such that $\dfrac{d}{dx}f(x)=f(x-1)$. If no such function exists, explain why.

Answer: e^{W(1) x}


Question: Evaluate the integral: $$\int_{0}^{\pi/2}{\frac{dx}{(1+\tan x)^6}}$$

Answer: \dfrac{11}{60}


Question: Determine whether there exist any transcendental numbers that can be expressed in the form $u + v\sqrt{2}$, where $u$ and $v$ are rational numbers.

Answer: No


Question: Given a lattice $L$ in $\mathbb{C}$, determine the number of distinct points in the complex torus $X=\mathbb{C}/L$ that satisfy the equation $n*q=0$, where $n$ is a positive integer.

Answer: n^2


Question: Find the smallest integer \( n > 1 \) such that \( (n+1)^2 \) divides \( n^{2017} + 1 \).

Answer: 2016


Question: Alice and Bob start on a coordinate plane at time \( t = 0 \) at points \( A(1, 0) \) and \( B(-1, 0) \), respectively. They each choose a direction uniformly at random and travel at a speed of 1 unit per minute in their chosen directions. Both have a 1-unit radius field of view, meaning they can see each other if the distance between them is at most 1 unit. What is the probability that Alice and Bob will ever see each other?

Answer: \dfrac{1}{3}


Question: Calculate the area of the region enclosed by the intersection of the curves \( y = \frac{2}{x^2+1} \) and \( y = |x| \).

Answer: \pi - 1


Question: Evaluate the limit: $$\lim_{x\to0}\frac{e^{\cos x \ln \sin x}-1}{x^2}$$

Answer: -\infty