Question: Let $A$ be an $n \times n$ symmetric positive-definite matrix such that both $A$ and its inverse $A^{-1}$ have no zero entries. Determine whether it is true or false that the Schur's complement $S$ of the block $A_{n,n}$ of matrix $A$ also has no zero entries, where $S = A_{1:n-1,1:n-1} - A_{1:n-1,n}A_{n,n}^{-1} A_{n,1:n-1}$. Provide your answer as "True" or "False".

Answer: True


Question: Determine whether there exists a non-constant entire function whose image is not the entire complex plane $\mathbb{C}$. Provide an example if such a function exists.

Answer: e^z


Question: Is there a $2 \times 2$ symmetric matrix with eigenvectors $u$ and $v$ such that the angle between $u$ and $v$ is not $\pi/2$? In other words, can the dot product between $u$ and $v$ be non-zero?

Answer: Yes


Question:
Evaluate the triple integral \( \iiint_D z \, dxdydz \) where \( D = \{(x,y,z) \in \mathbb{R}^3: y \geq 0, z \geq 0, x^2+y^2+z^2 \leq 2, x^2+y^2 \leq 1\} \). Use cylindrical coordinates to find the value of the integral. Choose the correct answer from the following options:

(A) \( \frac{3\pi}{2} \) \qquad (B) \( \frac{\pi}{2} \) \qquad (C) \( \frac{3\pi}{8} \) \qquad (D) \( \frac{5\pi}{6} \) \qquad (E) \( \frac{3\pi}{4} \)

Answer: C


Question:
Consider the equation \( \frac{2P^2}{x} + \frac{3Q^2}{x - 1} = 6 \), where \( P, Q \in \mathbb{R}_0 \) (non-zero real numbers). Let \( \alpha \) and \( \beta \) be the roots of this equation. Determine the nature of the roots:

(A) \( \alpha, \beta \in (0, 1) \)
(B) \( \alpha, \beta \in (1, \infty) \)
(C) One root lies in \((0, 1)\) and the other root lies in \((1, \infty)\)
(D) \( \alpha, \beta \) are imaginary roots.

Answer: C


Question: Find the area enclosed by the parametric curve \(\gamma(t) = (at - a\sin t, a - a\cos t)\) for \(t \in [0, 2\pi]\) and \(a > 0\). Use Green's theorem to compute the area.

Answer: 3\pi a^2


Question: Find the density function of the random variable \( W = X \cdot Y \), where \( X \) and \( Y \) are independent and uniformly distributed on the interval \([0,1]\).

Answer: -\ln w


Question: Let $H$ be a finite subgroup of a group $G$. Is the number of left cosets of $H$ in $G$ always finite?

Answer: No


Question: Consider a system of polynomials $f_1, \ldots, f_m \in \mathbb{F}[x_1, \ldots, x_n]$ each of degree $d$, where $n = O(1)$. If this system has a solution in the algebraic closure $\bar{\mathbb{F}}$, what is the minimum degree of extension of $\mathbb{F}$ required to guarantee a solution? Express your answer in terms of $m$ and $d$.

Answer: d^m


Question:
Let $X_1, X_2, \ldots, X_n$ be iid random variables with mean $\mu$, and let $N$ be a random variable independent of all the $X_i$'s. Show that the conditional expectation of the sum $\sum_{i=1}^N X_i$ given $N=n$ is $n\mu$, i.e., prove that:
\[
E\left(\sum_{i=1}^N X_i \mid N=n\right) = n\mu.
\]

Answer: n\mu


Question: Determine the number of different $2n$-block sequences $A_i$ that can be constructed such that $|A_i \cap A_j| = n$ for all $i \neq j$. Each sequence consists of $n$ occurrences of the symbol $\mathsf{a}$ and $n$ occurrences of the symbol $\mathsf{b}$. For example, for $n=2$, possible sequences include $A_0 = \mathsf{aabb}$, $A_1 = \mathsf{abab}$, and $A_2 = \mathsf{abba}$. Find the total number of such sequences for a given $n$. \(n\) is a positive integer.

Answer: 2n - 1


Question:
Consider the quadratic equation \((a+c-b)x^2 + 2cx + b+c-a = 0\), where \(a, b, c\) are distinct real numbers and \(a+c-b \neq 0\). Suppose that both roots of the equation are rational. Which of the following must be true?

(a) \(a, b,\) and \(c\) are rational
(b) \(\frac{c}{a-b}\) is rational
(c) \(\frac{b}{c-a}\) is rational
(d) \(\frac{a}{b-c}\) is rational

Select the correct option.

Answer: B


Question: Solve the initial value problem for the partial differential equation \( uu_x + yu_y = x \) with the initial conditions \( x = s \), \( y = s \), and \( u = 2s \).

Answer: u = 3y - x


Question:
Find all integer values of $k$ that satisfy the equation:
\[
\frac{k-15}{2000} + \frac{k - 12}{2003} + \frac{k - 9}{2006} + \frac{k - 6}{2009} + \frac{k - 3}{2012} = \frac{k - 2000}{15} + \frac{k - 2003}{12} + \frac{k - 2006}{9} + \frac{k - 2009}{6} + \frac{k - 2012}{3}.
\]

Answer: 2015


Question: Determine the Hausdorff dimension of \(\mathbb{R}^n\).

Answer: n


Question: Let \( L \) be a field extension of \( \mathbb{F}_2 \) with \([L : \mathbb{F}_2]=2\). Determine the minimal polynomial of an element \( \beta \in L \setminus \mathbb{F}_2 \).

Answer: x^2 + x + 1


Question: Let \( S \) be a hyperbolic surface and \( p: M \to S \) be a covering map. Determine if there exists a metric on \( M \) such that \( M \) is a hyperbolic surface and \( p \) is a local isometry.

Answer: Yes


Question: Find a primitive root of 71.

Answer: 7


Question: Calculate the value of \( \frac{J}{I} \), where \( I = \int_0^\infty x^{1/2}e^{-x^3}\,dx \) and \( J = \int_0^1 x^{-2/3}(1-x)^{-1/3}\,dx \).

Answer: 2\sqrt{3\pi}


Question: Evaluate the limit: $$\lim_{n \to \infty}\frac{1}{n}\int_{0}^{n}xg(x)\mathrm{d}x$$ where $g:\mathbb{R}\to [0,\infty]$ is continuous and the Riemann integral $\int_{0}^{\infty}g(x)\mathrm{d}x$ is convergent.

Answer: 0


Question: Consider a matrix \( A \) that is positive semi-definite. Let \( B \) be a square matrix composed of replicas of \( A \) as sub-blocks, such that \( B = \begin{pmatrix} A & A \\ A & A \end{pmatrix} \) or \( B = \begin{pmatrix} A & A & A \\ A & A & A \\ A & A & A \end{pmatrix} \), etc. Determine whether \( B \) is positive semi-definite.

Answer: B


Question: Evaluate the integral \( \int_{0}^{2\pi} \sqrt{2+2\sin x} \, dx \).

Answer: 8


Question: Determine the integer values of \(a\) and \(b\) for which the quotient rings \(\mathbb{Z}[x]/(x^2-a)\) and \(\mathbb{Z}[x]/(x^2-b)\) are isomorphic.

Answer: a = b


Question: Is every simply-connected nilpotent Lie group algebraic?

Answer: Yes


Question: Determine the values of the real number $a$ for which the series \( \sum_{n=1}^{\infty} \left\{ e-\left(1+\frac{1}{n}\right)^{n+a} \right\} \) converges.

Answer: \dfrac{1}{2}


Question: Find the smallest integer value of $n$ such that when a fair coin is tossed $n$ times, the probability that the proportion of heads is between 0.49 and 0.51 is greater than 0.95.

Answer: 9604


Question: Find the value of \( \sum_{n=1}^{89} \sin^6(n) \) expressed as a fraction \( \frac{m}{n} \), where \( m \) and \( n \) are integers.

Answer: \dfrac{221}{8}


Question: Find the limit of the expression \( \sqrt[n]{2^n-n^2} \) as \( n \to \infty \).

Answer: 2


Question: Let $C[0,1]$ be the real vector space of all continuous real-valued functions on $[0,1]$. Define the linear operator $T: C[0,1] \to C[0,1]$ by \[(Tf)(x) = \int_0^1 \sin(x+y)f(y) \, dy, \quad x \in [0,1].\] Find the dimension of the range space of $T$. \( \text{Provide your answer as a single integer.} \)

Answer: 2


Question: Compute the limit: \[ \lim_{n \rightarrow \infty} \sqrt{n} \cdot \left[\left(1+\dfrac{1}{n+1}\right)^{n+1}-\left(1+\dfrac{1}{n}\right)^{n}\right] \]

Answer: 0


Question: Calculate the value of \((1+i\sqrt{3})^{2017} + (1-i\sqrt{3})^{2017}\).

Answer: 2^{2017}


Question: Let $b$ be an integer greater than 2, and let $N_b = 1_b + 2_b + \cdots + 100_b$ (the sum contains all valid base $b$ numbers up to $100_b$). Compute the number of values of $b$ for which the sum of the squares of the base $b$ digits of $N_b$ is at most 512.

Answer: 30


Question:
For non-negative values of \(a\), find the maximum value of the expression:
\[
P = 3x\sqrt{a-y^2} - 3y\sqrt{a-x^2} + 4xy + 4\sqrt{a^2 - ax^2 - ay^2 + x^2y^2}
\]

Answer: 5a


Question: Evaluate the contour integral \( I = \oint_C \frac{\sin(\pi z)}{(z-1)^5} \, dz \), where \( C \) is the circle \( |z|=2 \).

Answer: 0


Question: For a graph $G$ with $n \geq 1$ vertices, determine the maximum number of vertices in its block-cutpoint graph $BC(G)$. Provide your answer as a single integer.

Answer: 2n - 3


Question: Determine the limit \( \lim_{N \rightarrow \infty} \frac{f(N)}{N^2} \), where \( f(N) \) is the number of positive integer solutions \((x, y, z)\) to the equation \( x + y + z = N \) with the constraint \( x \leq y \leq z \).

Answer: \dfrac{1}{12}


Question:
Let $A$ and $B$ be real invertible matrices such that $AB = -BA$. Determine the correct statement about the traces of $A$ and $B$ from the following options:
1. $\text{Trace}(A) = \text{Trace}(B) = 0$
2. $\text{Trace}(A) = \text{Trace}(B) = 1$
3. $\text{Trace}(A) = 0, \text{Trace}(B) = 1$
4. $\text{Trace}(A) = 1, \text{Trace}(B) = 0$

Answer: 1


Question: In a triangle $ABC$, the angles are in the ratio $A:B:C = 1:2:4$. Find the value of $\tan A \tan B + \tan B \tan C + \tan C \tan A$. Use the identity $\tan A \tan B + \tan B \tan C + \tan C \tan A = 1 + \sec A \sec B \sec C$ to assist in your calculation.

Answer: -7


Question: Consider the set of non-negative integers \(\mathbb{N}\) with the function \(f : \mathbb{N} \to \mathbb{R}\) defined by \(f(0) = 0\) and \(f(n) = \frac{1}{n}\) for all \(n > 0\). Define the metric \(d : \mathbb{N} \times \mathbb{N} \to \mathbb{R}\) by \(d(x, y) = |f(x) - f(y)|\). Is the metric space \((\mathbb{N}, d)\) compact?

Answer: Yes


Question: Determine whether the following statement is true or false: If there exists an invertible matrix $P_{n \times n}$ such that $P$ diagonalizes both matrices $A_{n \times n}$ and $B_{n \times n}$, then $P$ also diagonalizes $A + B$. Provide a justification for your answer.

Answer: True


Question: Is the graph of the Conway base 13 function connected?

Answer: Yes


Question: Compute the first cohomology group $H^1(S^1, \underline{\mathbb{Z}})$ of the circle $S^1$ with constant coefficients $\underline{\mathbb{Z}}$. Use the Mayer-Vietoris sequence and the given open cover $U = S^1 \setminus \{(1,0)\}$ and $V = S^1 \setminus \{(-1,0)\}$ to determine the group.

Answer: \mathbb{Z}


Question: Given an integer $k \geq 1$, determine the minimal integer $n$ such that there exist matrices $A_1, A_2, \ldots, A_k \in M_n(\mathbb{C})$ satisfying the conditions $A_i A_j = -A_j A_i$ for all $i \neq j$, and $A_1 A_2 \cdots A_k \neq 0$. Additionally, each $A_i^2$ must be zero. Find this minimal $n$.

Answer: 2^k


Question: Let $j: W \to X$ be a closed immersion of schemes and $f: Y \to X$ a morphism. The base change of $j$ along $f$ defines a closed immersion $W \times_X Y \to Y$. The inclusion $k: X \setminus W \to X$ and the inclusion $l: Y \setminus (W \times_X Y) \to Y$ are both open immersions. Is the fiber product $(X \setminus W) \times_X Y$ isomorphic to $Y \setminus (W \times_X Y)$ in the canonical way, i.e., is $l$ the base change of $k$ along $f$?

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{e^{iax}}{x^2+1} \, dx \) for \( a \in \mathbb{R} \).

Answer: \pi e^{-|a|}


Question: Find the roots of the equation \( f(x) = g(x) \) for \( x \neq \frac{\pi}{2} (2k+1) \), where \( f(x) = e^{\tan{x}} + 3^{\sin{x}} \) and \( g(x) = 2 - 2010x - \arctan{x} \).

Answer: 0


Question: Use Parseval's identity to find the length \(|f|\) of the function \(f(x) = 2 \cos(14x) + 4 \cos(11x) + 2 \sin(27x) - \cos(19x) + 5 \cos(140x)\).

Answer: 5\sqrt{2\pi}


Question:
Determine the value(s) of \( L \geq 0 \) for which the volumes of the solids of revolution obtained by revolving the region bounded by the parabola \( y = L \cdot x \cdot (2-x) \), the line \( y = 0 \), and the line \( x = 1 \) about the lines \( y = 0 \) and \( x = 1 \) are equal. Choose the correct option:

(a) \( L = 0 \)
(b) \( L = 0, L = \frac{15}{16} \)
(c) \( L = 0, L = \frac{15}{14} \)
(d) \( L = 0, L = \frac{13}{12} \)
(e) None of the above.

Answer: B


Question: Let \( Z \) be a standard normal random variable and \( f(x) \) be a real-analytic function such that \( |f(x)| \le x^4 \). Define the function \( g(t) = E[f(Z-t)] \). Is it true that \( g(t) \) is also a real analytic function? If so, can this be shown without using complex analysis tools?

Answer: Yes


Question: Does there exist a Lipschitz map on a locally compact second countable group $G$ that is 1 on a bounded set $A \subset G$ and 0 on $G \setminus N_{\varepsilon}(A)$, where the Lipschitz constant depends only on $\varepsilon$ and not on the size of $A$?

Answer: Yes


Question:
Given that $t$ is a real and positive number, and matrices $A$ and $B$ are defined as follows:

\[ A = \begin{pmatrix} \dfrac{t^2+1}{t} & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & 25 \end{pmatrix} \quad \text{and} \quad B=\begin{pmatrix} \dfrac{2t}{t^2+1} & 0 & 0 \\ 0 & \dfrac{3}{t} & 0 \\ 0 & 0 & \dfrac{1}{5} \end{pmatrix} \]

Calculate the infinite series of matrix products:

\[ X=(AB)^{-1} + (AB)^{-2} + (AB)^{-3} + \cdots \]

Let \( Y = X^{-1} \). Determine which of the following options is correct:

1. $\text{det}(Y)=10$
2. $X\cdot \text{adj(adj}(Y))=8I$
3. $\text{det}(Y)=20$
4. $X\cdot \text{adj(adj}(Y))=5I$

Answer: 2


Question: Consider a firm with quality \( \theta \in \{0, 1, \ldots, 100\} \), each equally likely. With probability \( \frac{1}{10} \), the firm can only send the message \( m = \varnothing \), and with probability \( \frac{9}{10} \), it can send either \( m = \varnothing \) or \( m = \theta \). A consumer observes the firm's message and forms a belief about the firm's expected quality \( b = E[\theta|m] \). The firm aims to maximize \( b \). Determine if there exists an equilibrium cutoff \( \theta^* \) such that: (i) every firm with \( \theta < \theta^* \) sends \( m = \varnothing \), and (ii) every firm with \( \theta \ge \theta^* \) sends \( m = \theta \) if possible. If such an equilibrium exists, find the value of \( \theta^* \).

Answer: 24


Question: Let \( \mathfrak{g} \) be a real semisimple Lie algebra that admits an invariant inner-product. Is every connected Lie group with Lie algebra \( \mathfrak{g} \) compact?

Answer: Yes


Question: Evaluate the definite integral \( \int_2^{10} g(y) \, dy \), where \( g(y) = f^{-1}(y) \) is the inverse function of the polynomial \( f(x) = x + x^3 \).

Answer: \dfrac{51}{4}


Question: A particle starts at position 0 on the integer number line. Every second, it moves one unit to the left or right with equal probability of \( \frac{1}{2} \). What is the probability that the particle reaches position +1?

Answer: 1


Question: Let \(\langle x_n\rangle\) be a sequence defined by the recursive relation \(x_{n+1} = x_n + a + \sqrt{b^2 + 4ax_n}\) for \(n \geq 0\), with \(x_0 = 0\), where \(a\) and \(b\) are fixed positive integers. Find \(\lim_{n\to\infty}\frac{x_n}{n^2}.\)

Answer: a


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

Answer: e^{-1/4}


Question: In a revolving restaurant, there are four round tables, each with three seats. How many different ways can 12 people be seated in this restaurant?

Answer: 5913600


Question: Determine whether there exists a function \( f: [-1,1] \to \mathbb{R} \) such that \( f^{(n)}(-1) = f^{(n)}(1) = 0 \) for all integers \( n \).

Answer: Yes


Question: Given $e_{60}=17$, $_{10}P_{50}=0.8$, and the $10$-year curtate temporary life expectancy at age $50$ is $9.2$, find $e_{50}$ using the recursive formula $e_x = P_x(1 + e_{x+1})$. Calculate $e_{50}$.

Answer: 22.8


Question: Determine whether the subgroup $H$ of $GL_n(\mathbb{R})$, consisting of all matrices with positive determinant, is a normal subgroup of $GL_n(\mathbb{R})$. Answer Yes or No.

Answer: Yes


Question: Evaluate the complex integral \( \int_{|z|=3} \frac{z}{z+2} \, dz \).

Answer: -4\pi i


Question:
Find the number of positive integer solutions \((a, b, c)\) that satisfy the equation:
\[ a^{b^c}b^{c^a}c^{a^b} = 5abc \]

Answer: 3


Question: Is it possible to construct a norm \( N \) on \( \mathbb{R}^2 \) such that the ratio of the circumference to the diameter of a standard Euclidean circle is 42?

Answer: Yes


Question: Consider a group \( G \) defined by the generators \( a, b \) and the relations \( a^2 = b^3 = a^{-1}b^{-1}ab = e \). Prove that \( G \) is isomorphic to the cyclic group \( \mathbb{Z}_6 \).

Answer: \mathbb{Z}_6


Question: Let \( n \) and \( m \) be fixed natural numbers. Consider independent random variables \( N, X_1, X_2, \ldots \) such that \( N \sim \text{Poisson}(n) \) and \( X_k \sim \text{Poisson}(m) \) for all \( k \). Define \( S_N = \sum_{k=1}^N X_k \) with \( S_0 = 0 \). Find \( E(S_N) \).

Answer: mn


Question: Determine a lower bound for the convergence radius of the $q$-expansion of the modular lambda function, given by \( \lambda = \sum_{j=0}^\infty a_j q^j \). Find a real number \( R \) such that for all \( \tau \in \mathfrak{h} \) with \( |q(\tau)| \leq R \), the series \( \lambda(\tau) = \sum_{j=0}^\infty a_j q^j(\tau) \) converges.

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{3^{3n}(n!)^3}{(3n)!}\right)^{1/n} \]

Answer: 1


Question: In a rating system with $R$ rating classes, each class has a probability of default $p_{i0}$ for $i=1,2,...,R$. After classifying $N$ companies into these classes, with $N_i$ companies in each class, you observe $d_i$ defaults in each class after one year. You want to test if the observed defaults align with the expected defaults using the test statistic $X^2=\sum_{i=1}^R \frac{(d_i-N_ip_{i0})^2}{N_i p_{i0}}$. Assuming the expected counts $N_ip_{i0}$ are adequate and defaults are independent, determine the degrees of freedom for this chi-squared test. Is it $R$ or $R-1$? Provide a justification for your answer.

Answer: R


Question: Let $G$ be a finite supersolvable group. Is it true that $G$ has a minimal normal subgroup for each prime divisor of its order?

Answer: No


Question: Compute the residue of the function \( \frac{z}{e^z - 1} \) at its poles.

Answer: 2\pi i k


Question: Let $A$ and $B$ be compact metric spaces with Borel probability measures $m_A$ and $m_B$, respectively. Suppose $f: A \to B$ is a continuous surjection. If $m_A(K) = 1$ for a Borel set $K \subseteq A$, is it true that $m_B(f(K)) = 1$? Assume $A$ is second-countable.

Answer: No


Question: Determine whether the map \( \phi : L^3([0,1], \mathbb{C}) \to \mathbb{C} \) defined by \( \phi(f) = \int_{0}^{1} f(x^2) \, dx \) is a bounded linear functional on the Banach space \( L^3([0,1], \mathbb{C}) \). If it is bounded, find its norm.

Answer: 2^{1/3}


Question: Given real numbers $a_1$ and $a_{100}$, and a sequence defined by $a_i = a_{i-1}a_{i+1}$ for $2 \leq i \leq 99$, where the product of the first 50 terms is 27 and the product of all 100 terms is also 27, find the value of $a_1 + a_2$.

Answer: 12


Question: For the mean \(\bar{X}_n\) of \(n\) independent and identically distributed normal observations with expected value \(E[X] = \mu\) and variance \(\text{Var}(X) = \sigma^2\), consider the statistic \(T_n = \frac{\sqrt{n}}{\bar{X}_n}\). Show that \(\text{Var}(T_n) = \infty\).

Answer: \infty


Question: Is it true that the description of automorphisms of $SL(n, \mathbb{Z})$ given by principal congruence subgroups holds for even $n$ as well? Refer to Hau-Reiner's paper for generators of the group of automorphisms of $SL(n, \mathbb{Z})$ induced by automorphisms of $GL(n, \mathbb{Z})$.

Answer: Yes


Question: Let $K/F$ be a field extension and $A, B \in M_n(F)$. Suppose $A$ and $B$ are conjugate in $GL_n(K)$, meaning there exists a matrix $P \in GL_n(K)$ such that $PAP^{-1} = B$. Is it true that $A$ and $B$ are also conjugate in $GL_n(F)$? In other words, can we find a matrix $C \in GL_n(F)$ such that $CAC^{-1} = B$?

Answer: Yes


Question: Determine the rank of the $n \times n$ matrix $A$ where each entry $a_{i,j}$ is given by $a_{i,j} = \frac{1}{i+j}$. What is the rank of this matrix?

Answer: n


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(\dfrac{1^p+2^p+\cdots+n^p}{n^{p}}-\dfrac{n}{p+1}\right) \] where \( p \in \mathbb{N} \) is fixed.

Answer: \dfrac{1}{2}


Question: Let $A$ be a matrix 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 $A$. Consider the differential of $l$ at the identity matrix $I$ of $G$, denoted as $l_{*,I}: T_I(G) \to T_A(G)$. Is the differential $l_{*,I}$ of $l$, left multiplication by $A$, still left multiplication by $A$? Justify your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \to \infty} \dfrac{3((n+1)!)(n-1)}{3^n + (n!)n^2}$$

Answer: 3


Question: Evaluate the limit of the sequence \( s_n = \int_{0}^{1} \frac{nx^{n-1}}{1+x} \, dx \) for \( n \geq 1 \). Determine where the sequence \( (s_n) \) converges. Choose from the following options: (A) 0, (B) \( \frac{1}{2} \), (C) 1, (D) \infty.

Answer: B


Question: Let \( T : \mathbb{R}^7 \to \mathbb{R}^7 \) be a linear transformation with nullity \( N(T) = 2 \). Determine the minimum possible rank of \( T^2 \).

Answer: 3


Question:
Given the second-order linear differential equation \(x^2y''+xy'+(-1+x^2)y=f(x)\), where \(J(x)\) and \(N(x)\) are linearly independent solutions, the general solution is given by:
\[y=-J(x)\int N(x) xe^{-x^2} dx +N(x)\int J(x) xe^{-x^2} dx\]
Determine the correct form of \( f(x) \) from the following options:

A) \( f(x)=x^2e^{-x^2} \)
B) \( f(x)=xe^{-x^2} \)
C) \( f(x)=e^{-x^2} \)
D) \( f(x)=x^{-1}e^{-x^2} \)
E) None of the above

Select the correct option.

Answer: E


Question: Determine the order of the quotient group \((\mathbb{Z}_{12} \times \mathbb{Z}_4)/\langle ([3], [2]) \rangle\).

Answer: 12


Question: Calculate the volume of a paperweight with a slanted top described by the plane equation \( x + y + z = 2 \). The edges of the paperweight are orthogonal to the \(xy\)-plane, and the base is a triangle with vertices \((1, 0, 0)\), \((0, -1, 0)\), and \((0, 1, 0)\). Use a triple integral to find the volume of the paperweight.

Answer: \dfrac{5}{3}


Question: Consider two universal graph homomorphisms, \( u: G \to H \) and \( v: H \to K \). Is the composition \( v \circ u: G \to K \) also a universal graph homomorphism?

Answer: Yes


Question: Let $d \in \mathbb{N}$ and $B_1 = \{x \in \mathbb{R}^d; |x| \leq 1\}$. Suppose $f \in L^1(B_1)$ is a radial function, meaning $f(x) = f(|x|)$, and satisfies $\|f - f_n\|_{L^1(B_1)} \to 0$ as $n \to \infty$ for some sequence $(f_n)_{n \in \mathbb{N}} \subset C(B_1)$, where $f_n(x) = 0$ if $|x| = 1$. Does there exist a sequence $(g_n)_{n \in \mathbb{N}} \subset C(B_1)$ such that $g_n(x) = 0$ if $|x| = 1$, $\|f - g_n\|_{L^1(B_1)} \to 0$ as $n \to \infty$, and each $g_n$ is radial?

Answer: Yes


Question: Calculate the expected straight line distance through the Earth (in kilometers) between two points chosen uniformly on the surface of a perfect sphere with a radius of 6378 kilometers.

Answer: 8504


Question: Evaluate the path integral of the vector field \( F = \langle \arccos x, xy - e^y \rangle \) over the triangular path from \((0,0)\) to \((2,3)\) to \((2,0)\) and back to \((0,0)\).

Answer: 3


Question: Find the particular solution to the recurrence relation \(a_{n+2} = 4a_{n+1} - 4a_n + n^2 - 5n + 2\). The homogeneous solution is given by \(c_1 2^n + c_2 n 2^n\).

Answer: n^2 - n


Question: Determine the minimum cardinality of subset $A$ in a set of $n^2$ bins, partitioned into two non-empty subsets $A$ and $B$, such that when balls are thrown independently and uniformly at random into the bins until $\\log(n)$ balls are placed in bins belonging to $A$, the total number of balls thrown $m$ satisfies $$\frac{m}{\log n}=\Theta\left(\frac{n^2}{|A|}\right)$$ with high probability as $n$ approaches infinity.

Answer: 1


Question: Consider the system \( \Sigma: \dot{x} = f(x) \), where \( x \in \mathbb{R}^n \) and \( f: \mathbb{R}^n \to \mathbb{R}^n \) is a smooth function vanishing at the origin. Assume \( \Sigma \) has a globally exponentially stable equilibrium at \( x=0 \). Apply a global coordinate transformation \( z = T(x) \), where \( T: \mathbb{R}^n \to \mathbb{R}^n \) is a global \( \mathcal{C}^1 \) diffeomorphism with \( T(0) = 0 \). The transformed system is \( \Sigma': \dot{z} = g(z) = \frac{\partial T(x)}{\partial x}f(x)\big|_{x=T^{-1}(z)} \). Is \( \Sigma' \) globally exponentially stable at \( z=0 \)?

Answer: Yes


Question: Given the partial differential equation $u_{xx} - u_{yy} = -1$ with the conditions $u(x,0) = 0$ and $u_y(x,0) = 0$ for $x \in \mathbb{R}$, find the value of $u\left(\frac{1}{2}, \frac{1}{2}\right)$. Assume $u(x,y)$ is a solution for $x \in \mathbb{R}$ and $y > 0$. Provide your answer as a single value.

Answer: \dfrac{1}{8}


Question:
Suppose you have measurable sets $P, Q \subset \mathbb{R}^2$ such that there exist sets $P_1, \ldots, P_n, Q_1, \ldots, Q_n$ satisfying:

- $\bigsqcup\limits_{i=1}^n P_i = P$ and $\bigsqcup\limits_{i=1}^n Q_i = Q$.
- For each $i \in \{1, \ldots, n\}$, $P_i$ can be transformed to $Q_i$ via a finite sequence of translations, rotations, and reflections.

Does it follow that $\mu(P) = \mu(Q)$?

Answer: \mu(P) = \mu(Q)


Question: Evaluate the limit: \[ L = \lim_{x\to0^+}\;x^x \]

Answer: 1


Question: Find the remainder when the number consisting of 124 repeated '1's is divided by 271.

Answer: 27


Question: Find the residue of the function \( f(z) = \frac{\cot(z)\coth(z)}{z^3} \) at \( z = 0 \).

Answer: -\dfrac{7}{45}


Question: Determine the radius of convergence for the series \( \sum_{n=1}^\infty \frac{(4-x)^n}{\sqrt{n^4+5}} \).

Answer: 1


Question: Determine the value of the Ramsey number $R(K_3, C_5)$. Use the hint that $R(3,4) = 9$ to assist in your calculation.

Answer: 9


Question: Is there a function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that is differentiable everywhere, but has a derivative that is nowhere differentiable?

Answer: Yes


Question: Determine whether the following statement is true or false: "The dimension of every minimal sufficient statistic is less than any sufficient statistic (minimal or not)."

Answer: B


Question: Determine whether there exists a function \( h(x): \mathbb{R} \rightarrow \mathbb{R} \) such that \( h(0) = 0 \) and \( h(2\pi) = 0 \) for which the integral \( \int_0^{2\pi} (h')^2 - h^2 \, dx \) is negative.

Answer: Yes


Question: Determine whether there exists a function \( f: \mathbb{R} \to \mathbb{R} \) that is continuous only at the points \( 1, 2, \ldots, 100 \).

Answer: Yes


Question: Consider an ellipse with major and minor axes of lengths 10 and 8, respectively. Determine the radius of the largest circle that can be inscribed in this ellipse, given that the center of the circle is at one of the foci of the ellipse.

Answer: 2


Question: Evaluate the limit \( \lim_{x\rightarrow 0}\frac{\exp{(ax)}-\exp{(bx)}+(b-a)x}{1-\cos{(x)}} \) using L'Hopital's Rule.

Answer: a^2 - b^2


Question: Find the dimension of the linear space \( V = \{ P \in \mathbb{R}_4[x] : xP'''(x) + P''(x) = P'(-1) + P(0) = 0 \} \).

Answer: 1


Question: In an isosceles right triangle $ABC$ with $AB = BC = 4$ cm, point $P$ is the midpoint of $BC$. Points $Q$ and $S$ lie on $AC$ and $AB$, respectively, such that triangle $PQS$ is equilateral. What is the area of triangle $PQS$?

Answer: 22\sqrt{3} - 36


Question: Let \( X_1, X_2, \ldots, X_n \) be i.i.d. random variables, each uniformly distributed over the interval (0, 1). Define \( S_n = \sum_{k=1}^n X_k \). Determine \( P(S_n \leq 1) \).

Answer: \dfrac{1}{n!}


Question:
Determine whether the following statement is true or false:

For any normed vector space $A$ and any linear subspace $B \subsetneq A$ that is not necessarily closed, and for any $\epsilon > 0$, there exists an $a \in A$ such that $||a|| = 1$ and $d(a, B) = \inf_{b \in B} ||a - b|| > 1 - \epsilon$. Provide a proof or a counterexample.

Answer: False


Question: Suppose $m$ and $n$ are positive integers such that for all integers $a$ and $b$ with $a \neq b$, the condition $a-b \mid a^m-b^n$ holds. Is it true that $m = n$?

Answer: Yes


Question: Determine whether the sum \( \sum\limits_{k\ge1}\frac{\log(t+k)}{k^2} \) is \( O(\log(t)) \) for \( t > 0 \).

Answer: O(\log(t))


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: Given that $f(f(f(x))) + f(x) = 2$ for all $0 \leq x \leq 2$, where $f(x)$ is a continuous function, find the value of $\int_0^2 f(x) \, dx$. Provide a detailed solution.

Answer: 2


Question: Find all polynomials $p \in \mathbb{C}[x]$ such that $p(0) = 1$ and $|p(z)| \leq 1$ for all $z$ on the unit circle $|z| = 1$. Provide a complete characterization of such polynomials.

Answer: 1


Question: Evaluate the limit \( \lim_{n\to\infty} (x_n\sqrt{n})^{\sqrt{n^2-1}} \), where the sequence \( x_n \) is defined recursively by \( x_{n+1} = \frac{x_n}{\sqrt{1+x_n^2}} \) with the initial condition \( x_1 = 2 \).

Answer: e^{\frac{3}{8}}


Question: Determine the set of values of $x$ for which the inequality \((1-a^2)x^2+(2a-3)x+1<0\) holds for all values of $a$.

Answer: (1, 2)


Question: Let \( f:D(0, 2)\to\,\mathbb{C} \) be an analytic function such that \( f\left(\frac{2}{n}\right)=\frac{1}{n+1} \) for \( n = 2, 3,\ldots \). Calculate \( f\left(\frac{3}{5}\right) \).

Answer: \dfrac{3}{13}


Question:
Compute the limit:
\[ \lim_{m \to \infty} \frac{g(x_m) - g(0)}{x_m - 0} = \lim_{m \to \infty} \frac{g(x_m)}{x_m} \]
where \( x_m = 2^{-m} \) and \( g(x) = \sum_{n=0}^\infty \frac{1}{2^n} h(2^n x) \) with \( h(x) = |x| \) on \([-1,1]\) and extended to \(\mathbb{R}\) by defining \( h(2+x) = h(x) \).

Answer: \infty


Question: Let \( f : \mathbb{C} \rightarrow \mathbb{C} \) be a holomorphic function such that for \( x, y \in \mathbb{R} \), \( \text{Im}f(x + iy) = x^2 - y^2. \) Find the function \( f. \)

Answer: i z^2


Question: Find the number of ways to distribute 10 identical gift boxes to 6 people (A, B, C, D, E, F) such that the total number of boxes given to A and B together does not exceed 4.

Answer: 2121


Question: Let \( S \) be the part of the sphere \( x^2+y^2+z^2=25 \) which lies above the plane \( z=4 \). Orient \( S \) by upward pointing normal vectors. Evaluate \( \int\!\!\!\int_{S}\text{curl}F\,dS \) where \( F=yze^xi+(x+ze^x)j+yzk \). (Hint: Consider a different surface.)

Answer: 9\pi


Question: Evaluate the integral \( \int_{-\infty}^0 e^x\sqrt{1-e^x} \, dx \).

Answer: \dfrac{2}{3}


Question: Evaluate the limit: \[ \lim_{m\to\infty} \frac{\int_0^{\pi/2}(\sin x)^{2m}dx}{\int_0^{\pi/2}(\sin x)^{2m+1}dx} \]

Answer: 1


Question: Calculate \( \lim_{n \to \infty} \cos\left(\frac{a}{n}\right)^{n^2} \) where \( a \in \mathbb{R} \setminus \{0\} \).

Answer: e^{-\frac{a^2}{2}}


Question: Determine whether the limit \( \lim_{n\to \infty} n(en! - \lfloor e n! \rfloor) \) exists. If it does, find its value.

Answer: 1


Question: Find the limit: \[ L = \lim_{n \to \infty} \frac{1}{2n}\sqrt[n]{\frac{(2n+1)!}{n!}}. \]

Answer: \dfrac{2}{e}


Question: Let $\omega: \mathbb{Z} \rightarrow \mathbb{Z}$ be an arbitrary permutation of the integers, and let $\varepsilon > 0$. Does there exist a syndetic set $A \subset \mathbb{Z}$ such that the upper uniform density $D^+(\omega(A)) < \varepsilon$?

Answer: Yes


Question: Consider the system of differential equations given by \(x' = y - p(x)\) and \(y' = x - q(y)\), where \(p\) and \(q\) are polynomials of degree \(n\). Determine the maximum number of critical points this system can have and justify why this number is achievable.

Answer: n^2


Question: Let $f,g:|K|\to|L|$ be continuous maps such that the sets $f^{-1}(\text{star}(v,L))\cap g^{-1}(\text{star}(v,L))$ with $v$ a vertex of $L$ form an open covering of $|K|$. Determine an integer $m$ and a simplicial map $s:|K^m|\to|L|$ that simplicially approximates both $f$ and $g$. Provide the construction of such a map and justify its properties.

Answer: m


Question: Given the sequence defined by the recurrence relation \(A_{n+1} + (-1)^n A_n = 2n - 1\) for all \(n\), find the value of \(\sum_{k=1}^{60} A_k\).

Answer: 1830


Question: Find the volume of the solid defined by the region \( S = \{(x,y,z) \in \mathbb{R}^3 : |x| + 2|y| \le 1-z^2\} \).

Answer: \dfrac{16}{15}


Question: Calculate the integral \( \int_{\gamma}\omega \) on the curve \( \gamma(t)=(t\cos(2\pi t), t\sin(2\pi t), t^2) \), where \( t \in [0,1] \), and the curve is contained in the paraboloid \( S \) defined by \( S = \{(x,y,z) \mid z = x^2 + y^2\} \). The differential form is \( \omega = ydx + xdy + (x^2 + y^2)dz \). Evaluate the integral and find the value of \( \int_{\gamma} \omega \).

Answer: \dfrac{1}{2}


Question: Find the extremals of the functional \( \int_0^1 (y^2 + x^2 y') \, dx \) subject to the boundary conditions \( y(0) = 0 \) and \( y(1) = A \).

Answer: y(x) = x


Question:
Let \(f\) be an entire function such that for each \(a \in \mathbb{R}\), there exists at least one coefficient \(c_n\) in the expansion \(f(z) = \sum\limits_{n=0}^{\infty} c_n (z-a)^n\) that is zero. Determine which of the following statements is true:

1. \(f^{(n)}(0) = 0\) for infinitely many \(n \geq 0\).
2. \(f^{(n)}(0) = 0\) for every \(n \geq 0\).
3. \(f^{(2n+1)}(0) = 0\) for every \(n \geq 0\).
4. There exists \(k \geq 0\) such that \(f^{(n)}(0) = 0\) for all \(n \geq k\).

Answer: 4


Question: Given a differentiable function $f: \mathbb{R}^2 \to \mathbb{R}$ with $f(2,1) = 3$ and $\nabla f(2,1) = (4,3)$, find $\nabla G(1,1)$ where $G(x,y) = x^2yf(x^2 + y^2, xy)$. Compute the gradient of $G$ at the point $(1,1)$. \( \nabla G(1,1) = \boxed{?} \)

Answer: (17, 14)


Question:
Determine whether the following proof is correct:

Let \((X,d)\) be a metric space and \(A\) a subset of \(X\). If \(r > 0\), then the set \(V_r(A) = \{x \in X : d(x,A) \le r\}\) is a closed set. The proof is as follows:

"If \(x \in X \setminus V_r(A)\), then \(x \in \{x \in X : d(x,A) > r\}\). Therefore, there exists \(\epsilon > 0\) such that \(B(x,\epsilon) \subset X \setminus V_r(A)\). Thus, \(X \setminus V_r(A)\) is open, and consequently, \(V_r(A)\) is closed."

Is this proof correct?

Answer: Yes


Question:
Let \( S_n=\sum_{k=1}^n \frac{n}{n^2+kn+k^2} \) and \( T_n=\sum_{k=0}^{n-1}\frac{n}{n^2+kn+k^2} \) for \( n=1,2,3,\dotsc \). Determine which of the following statements are true:

(A) \( S_n < \frac{\pi}{3\sqrt3} \)
(B) \( S_n > \frac{\pi}{3\sqrt3} \)
(C) \( T_n < \frac{\pi}{3\sqrt3} \)
(D) \( T_n > \frac{\pi}{3\sqrt3} \)

Provide the correct choices.

Answer: AD


Question: Let $G$ be a finite group, and let $\phi$ and $\psi$ be two automorphisms of $G$ of order $p$, where $p$ is a prime number. Suppose that for every $g \in G$, the sets $\{\phi(g), \phi^2(g), \ldots, \phi^p(g)\}$ and $\{\psi(g), \psi^2(g), \ldots, \psi^p(g)\}$ are equal. Is the subgroup generated by $\phi$ equal to the subgroup generated by $\psi$?

Answer: Yes


Question: Determine the smallest integer \( C_k \) such that \( \frac{C_k}{n+k+1}\binom{2n}{n+k} \) is an integer for all \( n \geq k \), where \( k \) is a given positive integer.

Answer: 2k + 1


Question: Let \( \{X_n\}_n \) be a sequence of i.i.d. simple random variables with mean 0 and variance 1. Define \( S_n = X_1 + \cdots + X_n \). Given that \( P\left(\limsup_{n} \frac{S_n}{\sqrt{2n\log\log(n)}} = 1\right) = 1 \) and \( P\left(\liminf_{n} \frac{S_n}{\sqrt{2n\log\log(n)}} = -1\right) = 1 \), deduce that with probability 1, the set of limit points of the sequence \( \left\{\frac{S_n}{\sqrt{2n\log\log(n)}}\right\} \) is the closed interval from \(-1\) to \(+1\).

Answer: [-1, 1]


Question: Verify whether the statement \( E[y - \text{sign}(s - 0.5)] = 0 \) is true, given that \( s \sim \text{Uniform}(0,1) \) and \( y \in \{-1, 1\} \) with \( \text{Prob}(y=1)=s \).

Answer: True


Question: Calculate the 2-D Laplacian \( \Delta^2 G \) of the function \( G(x,y) = \ln(x^2+y^2)/2 \). Then, for the interior \( D \) of the circle \( C \) with radius \( a \), evaluate the integral \( \int\int_D \Delta^2 G \, dA \).

Answer: 2\pi


Question: Determine whether for every \( \epsilon > 0 \), there exists a non-zero function \( f \in L^1(\mathbb{R}) \) such that \( \|\hat{f}\|_\infty \leq \epsilon \|f\|_1 \).

Answer: Yes


Question: Evaluate the triple integral \( \iiint_V (15x+30z)\,dx\,dy\,dz \) over the region \( V \) defined by the conditions: \( z = x^2+3y^2, \ z = 0, \ y=x, \ y=0, \ x=1 \).

Answer: 18


Question: Let \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) be a function such that \( f(f(x+y)) = f(x^2) + f(y^2) \) for all integers \( x \) and \( y \). Given that \( f(f(2020)) = 1010 \), find \( f(2025) \).

Answer: 505


Question: Evaluate the expression: \[ 10^{10^{10}} \sin\left( \frac{109}{10^{10^{10}}} \right) - 9^{9^{9}} \sin\left( \frac{101}{9^{9^{9}}} \right) - 8^{8^{8}} \sin\left( \frac{17}{8^{8^{8}}} \right) + 7^{7^{7}} \sin\left( \frac{76}{7^{7^{7}}} \right) + 6^{6^{6}} \sin\left( \frac{113}{6^{6^{6}}} \right) \]

Answer: 180


Question: Let $X$ and $Y$ be Banach spaces, and let $B(X,Y)$ and $B(Y,X)$ be the Banach spaces of bounded linear operators from $X$ to $Y$ and from $Y$ to $X$, respectively. Suppose $T,T_0,T_1,\dots \in B(X,Y)$ with $T_n \rightarrow T$ as $n \rightarrow \infty$ under the operator norm. Assume that for all $n \in \mathbb{N}$, $T_n$ is invertible, and the sequence $T_0^{-1},T_1^{-1},\dots$ is a Cauchy sequence in $B(Y,X)$. Since $B(Y,X)$ is complete, there exists $S \in B(Y,X)$ such that $T_n^{-1} \rightarrow S$ as $n \rightarrow \infty$. Is $T$ necessarily invertible with $T^{-1} = S$?

Answer: Yes


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x \) and \( y \), the following equation holds: \[ f(xf(x) + 2y) = f(x^2) + f(y) + x + y - 1. \]

Answer: f(x) = x + 1


Question: Evaluate the integral: \[ \int_{-\infty}^{\infty} \frac{\cos(\pi (x+1))}{x}\left[\sum_{n=1}^{\infty}\delta (x-n)\right]dx \]

Answer: \ln 2


Question: Evaluate the triple integral \( \iiint\limits_{V} xe^{-z} \, dV \) where \( V \) is the portion of the cone \( z=\sqrt{x^2+y^2} \) for \( x \geq 0 \) using cylindrical polar coordinates.

Answer: 4


Question: In a circle with radius $R$, an equilateral triangle $ABC$ is inscribed. A line is drawn through point $C$ that intersects $AB$ at point $M$ and the circle again at point $N$. Determine the product $CM \cdot CN$.

Answer: 3R^2


Question:
Let \( \langle p(x) \rangle \) denote the ideal generated by the polynomial in \( \mathbb{Q}[x] \). Let \( f(x) = x^3 + x^2 + x + 1 \) and \( g(x) = x^3 - x^2 + x - 1 \). Let \( I = \langle f(x) \rangle + \langle g(x) \rangle \). Determine which of the following statements is true:

(A) \( I = \langle x^3 + x \rangle \)
(B) \( I = \langle f(x)g(x) \rangle \)
(C) \( I = \langle x^2 + 1 \rangle \)
(D) \( I = \langle x^2 - 1 \rangle \)

Provide a justification for your answer.

Answer: C


Question: Let \( f \) be a non-vanishing holomorphic function on the open unit disc \( D \) such that \( |f(z)| \to 1 \) as \( |z| \to 1 \). Is \( f \) a constant function?

Answer: Yes


Question: Find the points $(x,y)$ on the circle $x^2+y^2=1$ and the directions $(u_1,u_2)$ such that the directional derivative of $f(x,y)=3x^2+y^2$ is maximized. The directional derivative is given by $6xu_1+2yu_2$, with the constraint $u_1^2+u_2^2=1$. Determine the maximum value of the directional derivative and the corresponding points and directions.

Answer: 6


Question: Does there exist a complex non-diagonalizable orthogonal matrix?

Answer: Yes


Question: Evaluate the complex integral \( \int_{|z|=2} \frac{1}{z^2+1} \, dz \).

Answer: 0


Question: What is the smallest number of vertices, $n$, in a polygon such that there exists an interior point $p$ where the distance from any vertex $b$ to $p$ is greater than the distance from $b$ to either of its adjacent vertices?

Answer: 7


Question: Evaluate the limit: \[ \lim_{x \to 0}\frac{(x+4)^\frac{3}{2}+e^{x}-9}{x} \] without using L'Hôpital's rule.

Answer: 4


Question: Given an infinite sequence of integers \(<a_0;a_1,a_2,\dots>\) such that for all positive integers \(n\), \(a_n > 0\), does the simple continued fraction always converge to a real number?

Answer: Yes


Question: Determine the radius of convergence for the function \( f(z) = \frac{\sin z}{(z-\pi)^2} \).

Answer: \pi


Question: Given a matroid $M$ with a ground set $E$ of size $2n$, where there exists a subset $A \subseteq E$ of size $n$ such that both $A$ and $E \setminus A$ are independent, determine the minimum number of subsets $B \subseteq E$ such that both $B$ and $E \setminus B$ are independent.

Answer: 2^n


Question: Given 11 chess players and 5 distinct tables, in how many ways can the players be paired to play, considering that the order and position (color) of the players matter?

Answer: 39916800


Question: Prove the identity: \[ \binom{n}{m} m \sum_{i=0}^{m-1} \frac{\binom{m-1}{i}(-1)^i}{n-m+1+i} = 1 \] for fixed integers \( n \) and \( m \).

Answer: 1


Question: In trapezium $ABCD$, sides $AB$ and $CD$ are parallel. A circle $S$ with diameter $AB$ touches side $CD$ and passes through the midpoints of diagonals $AC$ and $BD$. Determine the measure of the smallest angle of the trapezium.

Answer: 30^\circ


Question: Evaluate the limit: \[ \lim_{n\to \infty}\left(\sum_{k=1}^{2n} \frac{1}{\binom{2n}{k}}\right)^n \]

Answer: e


Question: Evaluate the limit: \[ \lim_{x \to 1} \Gamma(1-x) \cos\left(\frac{\pi}{2}x\right) \] where \( \Gamma(x) \) is the Gamma function.

Answer: \dfrac{\pi}{2}


Question: Let $G$ be a free abelian group of rank $k$. If $S$ is a subset of $G$ that is linearly independent over $\mathbb{Z}$, is it true that $|S| \le k$? Answer 'True' or 'False'.

Answer: A


Question: Given that \(\lim_{n \rightarrow \infty} \int_{0}^{a_{n}} x^{n} \, dx = 2\) and \(a_n > 0\), find \(\lim_{n \rightarrow \infty} a_n\).

Answer: 1


Question: Given that \( X \) and \( Y \) have a bivariate normal distribution with parameters \( \mu_X=5, \mu_Y=10, \sigma^2_X=1, \sigma^2_Y=25, \rho >0 \), and the condition \( P(4 < Y < 16|X=5)=0.9544 \), find the correlation coefficient \( \rho \).

Answer: 0.8


Question: Evaluate the line integral \( \int_{C} \mathbf{F} \cdot d\mathbf{r} \) where \( \mathbf{F}(x,y) = (2x + 3y, 3x + 4y) \) and \( C \) is the upper part of the circle of radius \( \sqrt{5} \) centered at \( (2,0) \), running from the point \( (1,2) \) to \( (3,2) \).

Answer: 20


Question: Determine the range of values for the parameter \( \alpha \in \mathbb{R} \) such that the inequality \[ 1 + \log_5(x^2 + 1) \geq \log_5(\alpha x^2 + 4x + \alpha) \] holds for all \( x \in \mathbb{R} \). Express the range in the form \((A, B]\).

Answer: (2, 3]


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function satisfying \( f\left(\frac{x+y}{3}\right) = \frac{2 + f(x) + f(y)}{3} \) for all real \( x \) and \( y \), and \( f'(2) = 2 \). Define \( g(x) = |f(|x|) - 3| \) for all \( x \in \mathbb{R} \). Determine the number of non-differentiable points of \( g(x) \).

Answer: 3


Question: Given \( \sum_{r=1}^{13}\frac{1}{r} = \frac{x}{13!}\), find the remainder when \(x\) is divided by 11.

Answer: 9


Question: Find the area of the region defined by the inequality \( \left( \frac{x^2}{9}+\frac{y^2}{25} \right)^2 \le x^2 + y^2 \).

Answer: 255\pi


Question: In triangle $\triangle PQR$, where $PQ=PR=5$ and $QR=6$, the triangle is inscribed in circle $\omega$. Determine the radius of a circle centered on $QR$ that is tangent to both circle $\omega$ and line $PQ$. Provide your answer as a single numerical value.

Answer: \dfrac{20}{9}


Question: Determine the values of \((a, b)\) for which the series \(\sum_{n=1}^{\infty} \left( \ln(n) + a\ln(n+2) + b\ln(n+3) \right)\) converges.

Answer: (-3, 2)


Question: Find the absolute minimum of the function \( f(x,y,z) = xy + yz + zx \) subject to the constraint \( x^2 + y^2 + z^2 = 12 \). Use the method of Lagrange multipliers to solve the problem.

Answer: -6


Question: Consider the group $G=\mathrm{SL}(2,\mathbb{R})$ and the subgroup $K=\mathrm{SO}(2)$. Let $\pi$ be a continuous irreducible representation of $G$ in a Banach space $H$. Determine whether there always exists a nonzero vector $v \in H$ such that the span $\langle \pi(k)v : k \in K \rangle$ is finite-dimensional.

Answer: Yes


Question: For what values of \( p \) does the series \( \sum_{n \ge 2} \frac{1}{n(\log n)^{p}} \) converge?

Answer: p > 1


Question:
Find the maximum value of \( t \) such that the following integral equals zero:
\[
\int_{-1}^{t}\left[\sin\left(x^{3} - 9x^{2} + 18x\right) + x^{2} -6x + \frac{11}{3}\right] \, \mathrm{d}x = 0
\]

Answer: 3


Question: Simplify the expression: $$\sqrt[3]{a+\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}+\sqrt[3]{a-\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}$$

Answer: 1


Question: Evaluate the limit: \[ L = \lim_{x\to \infty} \int_0^x \frac{(1+\cos t)^2}{x} \, dt \]

Answer: \dfrac{3}{2}


Question: Find the natural nonzero number \(a\) such that there exists an integer \(n\) for which \(a^{n+1} + 2^{n+1} + 1\) is divisible by \(a^n + 2^n + 1\).

Answer: 4


Question: Let $f \in L^1(\mathbb{R}^d)$ be a uniformly continuous function, where $d > 1$. Determine whether the following limit holds: $$\lim_{R\to\infty} \int_{|x|=R} |f(x)| \ dS(x) = 0,$$ where $dS$ is the surface measure on the sphere of radius $R$. Provide a proof or a counterexample.

Answer: 0


Question: Evaluate the integral \( I(a,b) = \int_{a}^{b} \frac{\left(e^{\frac{x}{a}}-e^{\frac{b}{x}}\right)dx}{x} \) for \( a, b \in \mathbb{R^+} \).

Answer: 0


Question: Determine the radius of convergence of the power series: \[ \sum \frac{1}{n^n \cdot 2^{2n}}(x+2)^{n^2} \]

Answer: 1


Question: In triangle \( ABC \), the measure of angle \( \angle B \) is \( 16^\circ \). Let \( I \) be the incenter and \( E \) be the excenter relative to side \( BC \). If \( IE = 2 \times AC \), calculate the measure of angle \( \angle ACB \).

Answer: 132^\circ


Question:
Calculate the flow integral \( \int_\gamma \vec{v} \cdot \vec{n} \,ds \) from inside to outside for the vector field \( \vec{v}=(x-y^2, y-x^2) \) along the path defined by the following parametrizations:

1. \( \gamma_1: [\pi/2, -\pi/2] \to \mathbb{R}^2, \quad t \mapsto \begin{pmatrix}t\\ \cos(t) \end{pmatrix} \)
   - Normal vector: \( \vec{n_1} = \begin{pmatrix}\sin(t) \\ 1 \end{pmatrix} \)

2. \( \gamma_2: [-\pi/2, \pi/2] \to \mathbb{R}^2, \quad t \mapsto \begin{pmatrix}t\\ 0 \end{pmatrix} \)
   - Normal vector: \( \vec{n_2} = \begin{pmatrix}0 \\ -1 \end{pmatrix} \)

Evaluate the integrals:

\[ \int_{\gamma_1} \vec{v} \cdot \vec{n_1} \,ds = \int_{\pi/2}^{-\pi/2}\big(t\sin(t)-\cos^2(t)\sin(t)+\cos(t)-t^2 \big)\,dt \]

\[ \int_{\gamma_2} \vec{v} \cdot \vec{n_2} \,ds = \int_{-\pi/2}^{\pi/2}t^2\,dt \]

Combine the results to find the total flow integral. Verify that the result is \( 4 \). Identify any errors in the calculation if the result does not match.

Answer: 4


Question: Find the invariant factor decomposition of the tensor product \( \mathbb{Z}_6 \otimes_{\mathbb{Z}} \mathbb{Z}_9 \).

Answer: \mathbb{Z}_3


Question: Evaluate the series \((0.5)^2 + (1)^2 + (1.5)^2 + (2)^2 + \ldots + (25)^2\) and verify if the result is equal to 11381.25.

Answer: 10731.25


Question: Does every group of order $2k$, where $k$ is an odd positive integer, have a subgroup of order $k$? Answer yes or no.

Answer: Yes


Question: Evaluate the expression \( \frac{\int_0^{\pi}x^3\log(\sin x)\,dx}{\int_0^{\pi} x^2 \log(\sqrt{2} \sin x)\,dx} \).

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


Question: Evaluate the integral: \[ \int_0^{\frac{\pi}{4}} \left( \frac{\sin^2{(5x)}}{\sin^2{x}} -\frac{\cos^2{(5x)}}{\cos^2{x}} \right)\mathop{dx} \]

Answer: \dfrac{20}{3}


Question: Determine whether a complex number that is both a Gaussian integer and an Eisenstein integer must be an ordinary integer. Justify your answer.

Answer: Yes


Question: Let $S = \{z \in \mathbb{C} : |z| \leq c\}$ be the closed ball of radius $c$ in the complex plane. Determine the orthogonal complement $S^{\perp} \subseteq \mathbb{C}$.

Answer: \{0\}


Question: Determine the number of orbits of the action of $GL_3(\mathbb{R})$ on $\mathbb{R}^3$ by left multiplication.

Answer: 2


Question: Provide an example of an algebraic number that is not constructible using Euclidean methods.

Answer: \sqrt[3]{2}


Question: Find the limit: \[ \lim_{n \to \infty} \frac{n^{\sqrt{n}}}{n!} \]

Answer: 0


Question: In a particular water sample, ten bacteria are found, of which three are of type A. What is the probability of obtaining exactly six type A bacteria in a second independent water sample containing 12 bacteria in total?

Answer: 0.0792


Question: Determine whether a collection of subsets of \( \mathbb{N} \) such that no subset contains another can be uncountable.

Answer: Yes


Question: Let $f$ be a continuous function such that \( \int_0^1 f(1-x^2) \, dx = \sqrt{3} \). Find the value of \( \int_0^2 x f(2x-x^2) \, dx \).

Answer: 2\sqrt{3}


Question: Find the limit: \[ \lim_{ n\to \infty }\sqrt[n]{\prod_{i=1}^n \frac{1}{\cos\frac{1}{i}}} \]

Answer: 1


Question: Consider an infinite sequence of flips of a fair coin. Determine whether the event "For every natural number \(n\), there exists a sequence of \(n\) consecutive heads between flip number \(3^n\) and flip number \(3^{n+1}\)" has a positive probability.

Answer: Yes


Question: Calculate the size of the stabilizer of the element \((1\ 2\ 3)(4\ 5\ 6)\) in the symmetric group \(S_7\) under the group action defined by \(x \cdot y = xyx^{-1}\).

Answer: 18


Question: Determine the dimension of the moduli space of lines on a general quadric hypersurface in \(\mathbb{P}^n\).

Answer: 2n - 5


Question: Find the function \( g(x) \) that is implicitly defined by the relation: \[ \sqrt{x^2 - y^2} + \arccos\left(\frac{x}{y}\right) = 0, \quad y \neq 0. \]

Answer: x


Question:
Determine the value of $11^A$, where $A$ is given by the limit:
\[ A = \lim_{N \to \infty} \left| \frac{\sum\limits_{\alpha=0}^N \frac{(1 + i\pi)^\alpha}{\alpha!}}{\sum\limits_{\alpha=0}^N \frac{1}{\alpha!}} \right|. \]

Answer: 11


Question: Determine whether there is always at least one prime number between two numbers $x$ and $y$ if $\operatorname{li}(y) - \operatorname{li}(x) = 1$ and $x < y$. Provide a justification for your answer.

Answer: No


Question: Evaluate the integral \( \int_\gamma \frac{dz}{z} \), where \( \gamma = -i + 5e^{2it} \) for \( t \in [0, 2\pi] \).

Answer: 4\pi i


Question: Compute the limit: \[ \lim_{M\to\infty}M\left(\frac{1}{s} - \frac{e^{-s/M}}{s}\right) \]

Answer: 1


Question: Let \( R = k[x_{1}, \ldots, x_{n}] \) where \( k \) is a field, and let \( I \) be a homogeneous ideal. Consider the graded minimal free resolution \( \cdots \to R_{1} \to R_{0} \to R/I \to 0 \) of \( R/I \). Is it always true that \( R_{0} = R \)?

Answer: Yes


Question: Determine whether there exists a singular cardinal \( \lambda \) such that for every cardinal \( \kappa < \lambda \), \( 2^\kappa < \lambda \). Provide a justification for your answer.

Answer: Yes


Question: For any natural number \( k \), determine the degree of the polynomial obtained by eliminating the parameter from the parametric equations \( x(t) = \cos(t) \) and \( y(t) = \cos(2^k t) \).

Answer: 2^k


Question: Evaluate the integral: \[ \int_{-\infty }^{\infty } \frac{\sin^2x}{x^2} \, dx \]

Answer: \pi


Question: Let $H: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be an injective linear map. Prove that there exists a constant $c > 0$ such that $\lvert Hx \rvert_{\text{sup}} \geq c \lvert x \rvert_{\text{sup}}$ for all $x \in \mathbb{R}^{n}$, where $\lvert x \rvert_{\text{sup}} = \max\{\lvert x_i \rvert : i = 1, \ldots, n\}$.

Answer: c


Question: Find the maximum value of \( p \) such that the expression \( 2x^4y^2 + 9y^4z^2 + 12z^4x^2 - px^2y^2z^2 \) is always nonnegative for all real numbers \( x, y, \) and \( z \).

Answer: 18


Question: If a sequence of functions \(f_n\) converges to \(f\) in measure and satisfies \(\|f_n\|_{L^p(\mathbb{R})} = \|f\|_{L^p(\mathbb{R})}\), does it imply that \(f_n\) converges to \(f\) in \(L^p\)?

Answer: Yes


Question: Let \(\Omega\) be a bounded domain with a sufficiently smooth boundary. Suppose \(u \in W^{1, 2}_{0}(\Omega)\) and \(F \in C^{\infty}(\mathbb{R} \rightarrow \mathbb{R})\) such that \(F(u(x)) = 0\) for almost every \(x \in \Omega\). Must it be true that \(F(0) = 0\)? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to\infty} \frac{\sum_{k=0}^{x/2}\binom{x}{2k}2k(x-2k)}{\sum_{k=0}^{x/2}\binom{x}{2k}(x-2k)^{2}} \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to 0} \left[ \frac{1}{x^2} \int_0^x \frac{t + t^2}{1 + \sin t}\, \mathrm{d} t \right] \]

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true or false: If a function \( f: [a,b] \to \mathbb{R} \) is such that its restrictions to each subinterval \([c_{i-1},c_i]\) are Riemann integrable, where \( a = c_0 < c_1 < \cdots < c_m = b \), then \( f \) is Riemann integrable over \([a,b]\).

Answer: A


Question: Let $E$ be a subset of a metric space $X$. For $r > 0$, define the set of $r$-far interior points as \( E_r = \{ x \in E : d(x,E^c) > r \} \). Is the set $E_r$ always open? Equivalently, is the function $x \mapsto d(x,E^c)$ continuous?

Answer: Yes


Question: Find the last three digits of the number formed by the tetration: $7^{7^{7^{...}}}$, where there are 1001 sevens.

Answer: 343


Question: Let $L$ be an operator. If $L$ is injective and has a closed range, does this imply that $L$ is boundedly invertible on its range?

Answer: Yes


Question: Given the function $f(x+y) = f(x)f(y)$, with $f'(0) = 11$ and $f(3) = 3$, find the value of $f'(3)$. Provide your answer as a single number.

Answer: 33


Question: Let \( A = \begin{pmatrix} B & C \\ C' & D \end{pmatrix} \) be a block matrix, where \( B \) and \( D \) are skew-symmetric matrices. If \( A \) is an odd order matrix, prove that \( \det A = 0 \).

Answer: 0


Question: Suppose we have two $n \times m$ matrices $A$ and $B$ with coefficients in an algebraically closed field of characteristic $0$, where $n > m$ and both matrices have rank $m$. Additionally, assume that the $m \times m$ minors of both matrices are identical. Determine whether there exists an $m \times m$ matrix $C$ with determinant $1$ such that $A \cdot C = B$. Justify your answer.

Answer: Yes


Question: Calculate the limit: \[ \lim_{x\rightarrow +\infty}\left[x\left(4\arctan\left(\frac{x+1}{x}\right)-\pi\right)\right] \] without using L'Hospital's rule or Taylor expansions.

Answer: 2


Question: Let $f(x)$ be a Lipschitz real-valued function defined on a closed interval $I$. Is the derivative $f'(x)$ necessarily square integrable, i.e., in $L^2(I)$?

Answer: Yes


Question: Determine whether the following statement is true or false: If \( R[x] \) is a unique factorization domain (UFD), then \( R \) is a UFD.

Answer: A


Question:
Find all functions \(f:\mathbb{Z}\to\mathbb{N}\) such that, for every \(x,y\in\mathbb{Z}\), the following condition holds:
\[\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)\]

Answer: f(x) = 1


Question: Assume we have a function \( f \in W^{1, \infty}(\mathbb{R}^n \times \mathbb{R}) \). Is the mapping \( x \mapsto f(x, y) \) weakly differentiable for almost every \( y \in \mathbb{R} \)? Provide a justification for your answer.

Answer: Yes


Question: Suppose \( A \in Mat_{n,n}(\mathbb{C}) \) is Hermitian. Does every unitary diagonalization of \( A \), given by \( D = U^H A U \), imply that \( D \) has only real entries?

Answer: Yes


Question: Evaluate the line integral \( \int_C \sin(x)\sin(y)dx - \cos(x)\cos(y)dy \) where \( C \) is the line segment from \((0,-\pi)\) to \(\left(\frac{3\pi}{2},\frac{\pi}{2}\right)\).

Answer: 0


Question: Determine the values of \( m \) for which the integral \( \iint_{\mathbb{R^2}} \frac{1}{(1+x^2+y^2)^m} \, dx \, dy \) converges.

Answer: m > 1


Question: Calculate the homology group of the quotient space $X$ given by the labelling scheme $aabcb^{-1}c^{-1}$. Determine to which of the following spaces $X$ is homeomorphic: $S^2$, $P_1$, $P_2$, $\dots$, $T_1$, $T_2$, $\dots$, where $P_m$ is the $m$-fold connected sum of projective planes and $T_m$ is the $m$-fold connected sum of tori.

Answer: P_3


Question: Given that \(|x| \leq 1\) and \(|ax^2 + bx + c| \leq 1\) for all \(x\) in the interval \([-1, 1]\), where \(a, b, c \in \mathbb{R}\), find the maximum possible value of \(|2ax + b|\).

Answer: 4


Question: Let \(\omega\) be a primitive root modulo a prime \(p\). Determine the value of the Legendre symbol \(\left(\frac{\omega}{p}\right)\).

Answer: -1


Question:
Consider the set \( A = \left\{ \sum_{i=0}^\infty \frac{a_i}{5^i} : a_i = 0, 1, 2, 3, 4 \right\} \) contained in \( \mathbb{R} \). Determine which of the following statements is true about the set \( A \):

1) \( A \) is a finite set.
2) \( A \) is countably infinite.
3) \( A \) is uncountable but does not contain an open interval.
4) \( A \) contains an open interval.

Which statement holds?

Answer: 4


Question: Determine whether it is possible to find uncountably many infinite sets of natural numbers such that any two of these sets have only finitely many common elements. Answer with 'Yes' or 'No'.

Answer: Yes


Question: There are 15 passengers, each with an assigned seat on a flight. Each passenger can either take their correct seat or a different one. Calculate the probability that at least one passenger sits in their assigned seat.

Answer: 1 - \frac{1}{e}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{n} \int_0^\infty e^{-x/n}f(x) \, dx, \] where $f$ is Riemann integrable on $[0,A]$ for all $A > 0$ and $f(x) \to -1$ as $x \to \infty$.

Answer: -1


Question: Evaluate the limit: \[ \lim_{x\to \infty} x^2 \int_{0}^{x} e^{t^3 - x^3} \, dt \]

Answer: \dfrac{1}{3}


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{1}{\ln n}\sum_{k=1}^n \frac{k}{k^2+1} \]

Answer: 1


Question: Is it true that every abelian group $G$ is the quotient of a free abelian group $F$? Provide a justification for your answer.

Answer: Yes


Question: Let $W(t)$ be a Wiener process. Define $X(t) = W^2(t)$ for $t \geq 0$. Calculate the covariance $\operatorname{Cov}(X(s), X(t))$ for $s, t \geq 0$. \( \operatorname{Cov}(X(s), X(t)) = \boxed{?} \)

Answer: 2 (\min(s, t))^2


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^\infty\frac{\tanh^{(n)}(0)}{n!} z^n \). Justify your answer.

Answer: \dfrac{\pi}{2}


Question: Determine the degree and a basis for the field extension \(\mathbb{Q}(\sqrt{2+\sqrt{5}})\).

Answer: 4


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( x \left\lfloor \frac{1}{x} \right\rfloor \right) \]

Answer: 1


Question: Find the largest integer $k$ for which $3^k$ divides the binomial coefficient $\binom{400}{200}$.

Answer: 2


Question: Using Taylor's inequality, find an upper bound on the error of the approximation \( f(x) = x\ln(x) \approx T_3(x) \) on the interval \([1/2, 3/2]\), where \( T_3(x) \) is the third-degree Taylor polynomial about \( x=1 \).

Answer: \dfrac{1}{24}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{e^n} \sum_{k=0}^n \frac{n^k}{k!} \]

Answer: \dfrac{1}{2}


Question: Find the value of \( \sum \frac{1}{a_n} \), where \( a_n \) denotes the \( n \)-th composite squarefree number.

Answer: \infty


Question: Determine the set of positive real numbers \( b \) for which the series \( \sum_{k=1}^\infty \frac{2^{\sin k}}{k^{b}} \) converges.

Answer: (1, \infty)


Question: Determine the number of divisors of \(2^2 \cdot 3^3 \cdot 5^3 \cdot 7^5\) that are of the form \(4n+1\).

Answer: 48


Question: Determine whether there exists a metric on the interval $X = (-1,1)$ such that the completion of $X$ results in the addition of exactly one point.

Answer: Yes


Question: Determine whether the function \( g(x) = E[Y \mid Y < x] \), where \( Y = \max(Z_1, \ldots, Z_n) \) and each \( Z_i \) is i.i.d. with density function \( f(z) > 0 \) for \( z \in (0, a) \), is increasing in \( x \) for \( x \in (0, a) \).

Answer: Yes


Question: In a triangle $\triangle ABC$, if $\cos A \cos B \cos C = \frac{1}{3}$, find the value of $\tan A \tan B + \tan B \tan C + \tan C \tan A$. \(\text{???}\)

Answer: 4


Question:
Let \( \{a_n\}_{n \geq 1} \) be a sequence of numbers in \((0,1)\) such that \( a_n \to 0 \) and \( \sum_{n \geq 1} a_n = \infty \). Determine if the following limit is true:
\[
\lim_{n \to \infty} \sum_{i=1}^n \prod_{j=n-i+1}^n a_j = 0.
\]

Answer: 0


Question: Determine whether every subset of zero Lebesgue measure in \(\mathbb{R}\) is a \(K_{\sigma}\) set, where a \(K_{\sigma}\) set is defined as a countable union of compact sets. Provide a justification for your answer.

Answer: No


Question: Determine the number of real solutions to the equation \( \sin x + 2\sin 2x - \sin 3x = 3 \) for \( x \in (0, \pi) \).

Answer: 0


Question: Is it possible to deduce an integer between 0 and 15 by asking seven true or false questions, if the respondent may lie at most once?

Answer: Yes


Question: Find the mass of the surface of the ellipsoid \(E=\{\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\}\) if the density at a point \((x, y, z)\) on the surface is given by \(\rho=\frac{r}{4\pi abc}\), where \(r\) is the distance from the origin to the tangent plane at \((x, y, z)\).

Answer: 1


Question: Compute \( \lim\limits_{n \to \infty} {\frac{1 \cdot 3 \cdot 5 \cdots(2n - 1)}{2 \cdot 4 \cdot 6 \cdots (2n)}} \).

Answer: 0


Question: Let \( f(x) \) be a polynomial of at least the second degree with rational coefficients, which is irreducible over the rationals. If \( a+bi \) is a root of \( f(x) \), does the smallest subfield of the rationals containing \( a+bi \) also contain \( a-bi \)? Answer yes or no.

Answer: No


Question: Determine whether the following statement is true: For an invertible matrix $A$ of size $n \times n$ with no eigenvalue of magnitude 1, i.e., $|\lambda_i| \neq 1$ for all $i$, the rank of the matrix $A - A^{-1}$ is $n$. Justify your answer.

Answer: True


Question: Let \( A \in \mathbb{R}^{n \times n} \) be an invertible symmetric matrix, and let \( X \in \mathbb{R}^{n \times n} \) be any matrix. Show that the matrix \( B = A^{-1}XA - X \) is anti-symmetric, i.e., \( B^T = -B \).

Answer: B^T = -B


Question: Evaluate the limit \( \lim_{n \to \infty} \left\| e^{\frac{A+B}{n}} - e^{\frac{A}{n}} e^{\frac{B}{n}} \right\| \), where \( ||.|| \) denotes the operator norm, and \( A \) and \( B \) are non-commuting linear operators. Show that this limit is bounded above by a term that is \( o\left(\frac{1}{n}\right) \).

Answer: 0


Question: Find \(\lim_{x\rightarrow\infty}x\left(\frac{1}{e}-\left(\frac{x}{x+1}\right)^x\right).\) Use the expansion \(\ln{(1+t)}=t-\frac{t^2}{2}+O(t^3)\) and \(e^t=1+t+O(t^2)\) to simplify the expression and determine the limit.

Answer: -\dfrac{1}{2e}


Question: Evaluate the integral \( \int_{C(0,2)} \frac{e^z}{z-1} \, dz \) using the Cauchy Integral Theorem.

Answer: 2\pi i e


Question: Find a concrete embedding of the group $M_2(\mathbb{Z}_2)$, under matrix addition, into the symmetric group $S_n$ for some $n$. Provide the value of $n$ and describe the embedding map.

Answer: 8


Question: Find the envelope of the family of straight lines given by the equation \( x \cos \alpha + y \sin \alpha = a \sec \alpha \), where \( \alpha \) is a parameter.

Answer: y^2 = 4a(a - x)


Question: Find \( \lim _{t \rightarrow \infty} \mathbb{E}\{|N(t)-\lambda t|\} \) for a Poisson process \( \boldsymbol{N}(\boldsymbol{t}) \) with rate \( \lambda \).

Answer: \infty


Question: You toss a fair coin one million times. What is the probability of obtaining at least one sequence of six consecutive heads followed by six consecutive tails?

Answer: 1


Question: Let \( f:[a,b]\to\mathbb{R} \) be a differentiable function on \([a,b]\), with \( \lim\limits_{x\to a}f(x)=\lim\limits_{x\to a}f'(x)=0 \), and \( f'(x)\ne 0 \) in a neighborhood of \( a \). Determine whether it is necessarily true that \( \lim_{x\to a}\frac{f(x)}{f'(x)}=0 \).

Answer: 0


Question: Given $a > 0$ and $a, b, x \in \mathbb{R}^n$, if for any $x > 0$, whenever $a^T x \leq 1$ we have $b^T x \leq 1$, does it imply that $a \geq b$?

Answer: a \geq b


Question: In a semisimple monoidal category with duals, if $V$ is a simple object, can $V \otimes V$ contain $V^*$ as a summand?

Answer: Yes


Question: Determine the number of \(3 \times 3\) permutation matrices \(A\) such that \(AA^t = I\). What is the cardinality of this set?

Answer: 6


Question:
Determine if the following identity is true:
\[ \lim_{n\to \infty}\int_0^{\pi/2} \frac{x^{\frac 1n}+\sin x}{\tan x+x^{\frac 1n}} \, dx = \int_0^{\pi/2} \frac{1+\sin x}{\tan x+1} \, dx = 1.62\ldots \]

Answer: False


Question: Given the equations \(x+y=2010\) and \((\sqrt[3]{x}+\sqrt[3]{y})(\sqrt[3]{x}+11)(\sqrt[3]{y}+11)=2942\), find the value of \(\sqrt[3]{x}+\sqrt[3]{y}\).

Answer: 12


Question: Find the volume of the solid tetrahedron with vertices at \((0,0,0)\), \((0,0,1)\), \((0,2,0)\), and \((2,2,0)\) using only double integrals.

Answer: \dfrac{2}{3}


Question: Find the radius of convergence of the power series \( \sum a_n z^n \), where \( a_n \) is the number of divisors of \( n^{50} \).

Answer: 1


Question: Minimize the expression \( \frac{(x^2+1)(y^2+1)(z^2+1)}{(x+y+z)^2} \) for positive values of \( x, y, \) and \( z \).

Answer: \dfrac{3}{4}


Question: Let \( q \) be a prime power. Determine whether any prime divisor of the expression \( 1 + q + q^2 + q^3 + q^4 \) is congruent to 1 modulo 10.

Answer: Yes


Question: Evaluate the limit \( \lim_{x \rightarrow 2k\pi}\frac{2(1-\cos x)+x\sin x}{2\sqrt{1-\cos x}} \) to find \( f'(2k\pi) \) for the function \( f(x)=x \sqrt{1-\cos(x)} \).

Answer: 0


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{\ln(n)}{n} \sum_{k=1}^n \frac{a_k}{k} \) for a given sequence \( a_1, a_2, a_3, \ldots, a_n \), if \( \lim_{n \to \infty} a_n = a \).

Answer: 0


Question: Determine whether the function \( f(x,y) = \begin{cases} x & \text{if } y=0 \\ 0 & \text{otherwise} \end{cases} \) can have all of its partial derivatives exist at a point but not be continuous at that point. Justify your answer.

Answer: No


Question: Compute the integral \(\int_{\mathbb{C}P^1} \omega_{FS}\), where \(\omega_{FS}\) is the Fubini-Study form on \(\mathbb{C}P^1\). Given that \(\mathbb{C}P^1\) is covered by two open sets \(U_1\) and \(U_2\), with \(U_1 = \{[Z_1,Z_2]\in \mathbb{C}P^1 : Z_1 \neq 0\}\) and the coordinate map \(z: \mathbb{C}P^1 \rightarrow \mathbb{C}\) defined by \([Z_1, Z_2]\mapsto Z_2/Z_1\), show that \(\int_{\mathbb{C}P^1} \omega_{FS} = \pi\).

Answer: \pi


Question: Evaluate the double integral: $$\int_1^\infty \int_1^\infty \frac{1}{m^2+n^2} \, dm \, dn.$$

Answer: \infty


Question:
Let $Z_1, Z_2, Z_3$ be independent standard normal random variables. Determine which of the following expressions has a Chi-Square distribution with 1 degree of freedom:
\[
\begin{align}
A) & \quad \frac{Z_1^2 + Z_2^2}{2} \\
B) & \quad (Z_1+Z_2)^2-Z_1^2 \\
C) & \quad Z_1^2+Z_2^2-Z_3^2 \\
D) & \quad \frac{(Z_1+Z_2-Z_3)^2}{3} \\
E) & \quad (Z_1+Z_2-Z_3)^2
\end{align}
\]

Answer: D


Question: Given that \( \lim _{ x\rightarrow 0 }{ \frac { f(x) }{ { x }^{ 2 } } =a } \) and \( \lim _{ x\rightarrow 0 }{ \frac { f(1-\cos x) }{ g(x)\sin^2x } = b }\), where \(b \neq 0\), find \( \lim _{ x\rightarrow 0 }{ \frac { g(1-\cos2x) }{ x^4 } }\).

Answer: \dfrac{a}{b}


Question:
Determine which of the following methods correctly calculates the predictive hazard function in a Bayesian context:

1. \(h_p(t) = \frac{f_p(t)}{1-F_p(t)}\).

2. \(h_p(t) =  \int h(t\mid\theta)\pi(\theta \mid \text{Data})d\theta\).

Answer: 1


Question: Calculate the sum \[ \sum_{n=1}^{\infty}\left((2n-1)\left(\sum_{k=n}^{\infty}\frac{1}{k^2}\right)-2\right). \]

Answer: -\dfrac{1}{2}


Question: Calculate the area inside the circle given by the polar equation \( r = 3a\cos(\theta) \) and outside the cardioid given by \( r = a(1+\cos(\theta)) \).

Answer: \pi a^2


Question: Find the least positive residue of \(3^{83} \pmod{3600}\).

Answer: 27


Question: Find the mean cross-sectional area of a sphere with radius 1.

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


Question: Does there exist a semigroup such that every element can be expressed as a product in the form \( y = xy \) for some \( x \), but the semigroup lacks a left identity element?

Answer: Yes


Question: Is the principal Dirichlet character \(\chi_0\bmod{1}\) considered primitive?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{x \tan(2x) - 2x \tan(x)}{(1 - \cos(2x))^2} \]

Answer: \dfrac{1}{2}


Question: Let \( f: S^1 \to S^1 \) be defined by \( f(z) = z^n \), where \( n \in \mathbb{N} \). Determine the image of the generator \( g \) of \( H_1(S^1) \) under the induced map \( f_*: H_1(S^1) \to H_1(S^1) \).

Answer: n \cdot g


Question: Find the limit of the series: \[ \sum_{k=0}^\infty \left( \sum_{j=0}^k \binom{k}{j} \left(-\frac{1}{3}\right)^j \right) \]

Answer: 3


Question: Determine the number of solutions of the equation \(x^3 + 2x^2 + 5x + 2\cos x = 0\) within the interval \([0, 2\pi]\) without using a graph.

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \frac{\int_{0}^{x}\sqrt{t}\cos t\ dt}{x^2} \]

Answer: \infty


Question: Let \( X \sim \mathrm{Geometric}(1/11) \). Use Jensen's inequality to determine a lower bound on \( \mathrm{E}(X^4) \) by applying Jensen's inequality to \( X^2 \) with \( f(x) = x^2 \).

Answer: 53361


Question: Consider a number field $K$ equipped with a total order $\le$. Is there always a field homomorphism $\phi:K \rightarrow \Bbb R$ that respects the total order, i.e., with $\phi(x) > \phi(y)$ for $x > y$?

Answer: Yes


Question: Let $S$ be a regular oriented surface such that for any simple, closed, and positively oriented curve in $S$, the integral of the geodesic curvature along this curve is always the same. What can be concluded about the Gaussian curvature of $S$?

Answer: 0


Question: Determine whether the tensor product \( \mathbb{Q} \otimes_\mathbb{Z} \mathbb{Z}/n \) is equal to zero. Justify your answer.

Answer: 0


Question: Compute the residue of the function \( f(z) = \frac{\exp(4z) - 1}{\sin^2(z)} \) at \( z_0 = 0 \).

Answer: 4


Question: Consider the attaching space $D^1 \cup_f D^1$ where $f: S^0 \to S^0$ is a map. Determine if this space is homeomorphic to $S^1$. Assume $S^0 = \{1, -1\}$ and $f$ is either the identity map or the map that swaps the points.

Answer: S^1


Question: Find the value of \( \limsup_{n \to \infty} \frac{\log(\mathrm{lcm}(1,2, \dots, n))}{n} \).

Answer: 1


Question: Calculate the maximum value of \[ \frac{1}{\sqrt{a^2 - ab + 3b^2 + 1}} + \frac{1}{\sqrt{b^2 - bc + 3c^2 + 1}} + \frac{1}{\sqrt{c^2 - ca + 3a^2 + 1}} \] given that \( a, b, c > 0 \) and \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \le 3 \).

Answer: \dfrac{3}{2}


Question: Determine whether an odd perfect number can be divisible by $5313$. Provide a justification for your answer.

Answer: No


Question: Evaluate the integral \(\iint_{x^2+y^2 \leq a^2} |xy| \, dx \, dy\), where the region of integration is the area inside the circle \(x^2 + y^2 = a^2\).

Answer: \dfrac{a^4}{2}


Question: A sequence is defined by \(a_1 = 0\), \(a_2 = 4\), and \(a_n = 4(a_{n-1}-a_{n-2})\) for \(n > 2\). Determine the greatest value of \(n\) such that \(n < 100\) and \(a_n\) is a power of 2.

Answer: 65


Question: Calculate the surface area of the sphere defined by \(x^2 + y^2 + z^2 = 1\) that lies above the disc \((x-1/2)^2 + y^2 \le 1/4\) in the xy-plane.

Answer: \pi - 2


Question: Determine if \((-2-i)\) is a root of the polynomial \(p(x) = x^5 + (1+2i)x^4 - (1+3i)x^2 + 8 + 44i\) using the Horner scheme.

Answer: Yes


Question: Find an element of order 3 in the multiplicative group \( \mathbb{Z}_{67}^* \).

Answer: 29


Question: Determine whether the function $f: \mathbb{R} \to \mathbb{R}$, which satisfies the condition \(|f(x)-f(y)| \le 7{|x-y|}^{201}\) for all real numbers $x$ and $y$, is a constant function.

Answer: Yes


Question: Compute the homology group $H_2(\mathbb{S}^3/G; \mathbb{Z})$ for a finite subgroup $G$ of $SO(4)$ acting freely on $\mathbb{S}^3$. Is this homology group zero?

Answer: 0


Question: For the function \( f(x) = \frac{x^{n-x}}{(n-x)!} \), find the constant \( n \) such that \( f(x) \) has a maximum at the point \((n, 1)\).

Answer: e^{-\gamma}


Question: Evaluate the surface integral \(\int_S \vec{F} \cdot d\vec{s}\) where \(\vec{F} = x \vec{i} - y \vec{j} + z \vec{k}\) and the surface \(S\) is the closed cylinder defined by \(x^2 + y^2 \le 4\) and \(0 \le z \le 1\). Verify your answer using the Divergence Theorem.

Answer: 4\pi


Question:
Let \(\alpha, \beta\) (\(\alpha > \beta\)) be the roots of the quadratic equation \(x^2 - x - 4 = 0\). Define \(P_n = \alpha^n - \beta^n\) for \(n \in \mathbb{N}\). Find the value of the expression:
\[
\frac{P_{15}P_{16} - P_{14}P_{16} - P_{15}^2 + P_{14}P_{15}}{P_{13}P_{14}}
\]

Answer: 16


Question: Solve for \( x \) in the equation: \[ e^{-\frac{x}{a}} \cdot \sin(b + cx) = \sin(b) \]

Answer: 0


Question:
Consider a probability space $(\Omega, \mathcal{F}, \mu)$. Define the essential supremum of a function $f : \Omega \to \mathbb{R}$ as
\[ \esssup f = \inf_{A : \mu(A) = 0} \sup_{x \notin A} f(x). \]
Is the set of equivalence classes of almost everywhere equal, bounded $\mathbb{R}$-valued functions on $\Omega$ a Banach space when equipped with the norm $\|f\| = \esssup |f|$?

Answer: Yes


Question: The equation $g(x) = 0$ has a simple root in the interval $(1,2)$. The function $g(x)$ satisfies the conditions $|g'(x)| \ge 4$ and $|g''(x)| \le 3$. Assuming the Newton-Raphson method converges for all initial approximations in $(1,2)$, determine the maximum number of iterations required to obtain the root correct to 6 decimal places after rounding.

Answer: 4


Question:
Find a solution $X$ to the following system of congruences:
\[ X \equiv 1 \pmod{5} \]
\[ X \equiv 3 \pmod{6} \]
\[ X \equiv 5 \pmod{11} \]
\[ X \equiv 10 \pmod{13} \]

Answer: 621


Question: Evaluate the integral \( \int_{0}^{\frac{\pi}{6}} \cos x \sqrt{1-2\sin x} \, dx \) using the substitution \( u = \cos x \).

Answer: \dfrac{1}{3}


Question: If an entire function has uncountable zeros, what can you conclude about the function?

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{2^{n/\log_2 n} (\log_2 n)^4}{n^3} \]

Answer: \infty


Question: Determine the degree of the field extension \([\Bbb Q(\alpha):\Bbb Q(\alpha^2+3)]\), where \(\alpha\) is a root of the polynomial \(f(t) = t^4 + 9t^2 + 15\).

Answer: 2


Question: Calculate the integral \( \int_{\lambda} \frac{dz}{(z^2-1)^2} \), where \( \lambda \) is a path in \( \mathbb{R}^2 - \{1,-1\} \).

Answer: 0


Question: Let \( f: \mathbb{R}^+ \to \mathbb{R}^+ \) be a twice-differentiable function such that \( xf''(x) + f'(x) + f(x) \leq 0 \) for all \( x > 0 \). Find \( \lim\limits_{x \to \infty} f(x) \).

Answer: 0


Question: Let $\bar{a}, \bar{b} \in \mathbb{Z}/k$ such that $\gcd((a,k),(b,k))=1$. Does Bezout's lemma imply the existence of integers $x, y \in \mathbb{Z}$ such that $x\bar{a} + y\bar{b}$ is a generator of $\mathbb{Z}/k$?

Answer: Yes


Question: How many 4 × 4 matrices with entries from \{0, 1\} have an odd determinant?

Answer: 20160


Question: Evaluate the integral \( \iint_{\mathbb{R}^2} \frac{1}{(1+4x^2+9y^2)^2} \, dx \, dy \).

Answer: \dfrac{\pi}{6}


Question: In triangle $ABC$, points $D$ and $E$ lie on sides $BC$ and $AC$, respectively, such that $\frac{BD}{DC} = \frac{CE}{EA} = 2$. Let $AD$ and $BE$ intersect at point $P$. Given that the area of $\triangle BPD$ is 252 more than the area of $\triangle APE$, find the sum of the areas of $\triangle APB$ and $\triangle CEPD$. Provide your answer as a single numerical value.

Answer: 432


Question: Let $X_i$ for $i \in \mathbb{N}$ be independent and identically distributed random variables, each following a Binomial distribution $\text{Bin}(10, p)$ for some $p \in [0, 1]$. Define $Y_n = \sum_{i=1}^n X_i$ and $Y_m = \sum_{i=1}^m X_i$ for $n, m \in \mathbb{N}$ where $n \leq m$. Compute the covariance $t_{n,m} = \text{Cov}(Y_n, Y_m)$. Specifically, calculate $E(Y_n Y_m)$. Provide your answer in terms of $n$, $m$, and $p$. \(n \leq m\)

Answer: 10np(1 - p)


Question: Let $(X,A,\nu)$ be a probability space and $T:X\to X$ a measure-preserving transformation with respect to $\nu$. Consider a measurable partition $P=\{P_0,...,P_{k-1}\}$. Define the set of all possible itineraries as $I=\{(i_1,...,i_n,...)\in k^{\mathbb{N}}; \text{ there exists } x\in X \text{ such that } T^n(x)\in P_{i_n} \text{ for all } n\in\mathbb{N}\}$. Suppose that $I$ is countably infinite. Is it true that the entropy of $T$ with respect to $P$ is $0$, i.e., $h(T,P)=0$?

Answer: 0


Question: Let \( A \) be an \( n \times n \) matrix with integer entries such that \( \det(A) = 1 \). If the entries of \( A \) are reduced modulo a positive integer \( k \), what is the determinant of the resulting matrix? Justify your answer.

Answer: 1


Question: Consider a set of integers: 7, 9, 11, 13. Define y as the highest common factor (HCF) found so far, initially set to 1. For each nth generation, add 1 to each member of the set, calculate the HCF of the new set, and update y if the new HCF is greater than the current y. For example, when n is 9, the set becomes 16, 18, 20, 22, and the HCF is 2. Determine if there is a limit to y as n approaches infinity.

Answer: 2


Question: A committee consists of 12 members. Each pair of members either mutually like or dislike each other, and each member likes exactly 6 other members. Is it possible to arrange all 12 members around a round table so that any two adjacent members like each other?

Answer: Yes


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

Answer: 0


Question: Find the inverse Z-transform of the function \( \frac{1}{(1-z^{-1})^{2}} \).

Answer: (n + 1) u[n]


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty } \int_0^1 \sqrt{ \frac 1 x + n^2 x^{2n} } \ \mathrm dx \]

Answer: 3


Question: Evaluate the integral \( \int_{0.5\pi}^{\pi} \frac{dx}{1-\cos x} \).

Answer: 1


Question: Find the limit \( \lim_{n \to \infty} \{en!\} \), where \( \{x\} \) denotes the fractional part of \( x \).

Answer: 0


Question: Determine the quadratic character of 293 modulo 379.

Answer: -1


Question: Find the greatest common divisor (gcd) of the complex numbers $6-7i$ and $2-9i$.

Answer: 2 + i


Question: Evaluate the integral \( \int_{|z| = \frac{1}{2}} \frac{e^{1/z}}{1-z} \, dz \).

Answer: 2\pi i (e - 1)


Question: Evaluate the limit using the Riemann sum approach: \[ \lim_{n \rightarrow +\infty} \sum_{k=1}^{n}\frac{(3k-2n)^{5}}{n^{6}}. \]

Answer: -\dfrac{7}{2}


Question: Consider the set $E = \{1/n : n \in \mathbb{N}\}$ in $\mathbb{R}^1$ with the standard metric $\rho(x,y) = |x-y|$. Let $E'$ be the set of all limit points of $E$. Determine if $E$ and $E'$ have the same limit points. Justify your answer.

Answer: No


Question: Evaluate the limit: \[ \lim_{n \to \infty} \inf_{x \in \mathbb{R}} (e^x - nx) \]

Answer: -\infty


Question: What is the minimum number of values needed to uniquely determine a line in three-dimensional space \(\mathbb{R}^3\)?

Answer: 4


Question: If a cubic fourfold $X$ contains a quartic scroll, is it necessarily a pfaffian?

Answer: Yes


Question:
Determine the interval of convergence for the power series:
\[ \sum_{n=2}^\infty(-1)^n\frac{(x-3)^n}{(\sqrt[n]{n}-1)n} \]

Answer: (2, 4]


Question: Determine the order of the entire function \( f(z) = \sin(z) \).

Answer: 1


Question:
Find the value of \(a\) such that for \(a \in \mathbb{R}\) and \(a \neq 1\), the following limit holds:
\[ \lim_{n \to \infty} \frac{1^a + 2^a + \cdots + n^a}{(n+1)^{a-1} \left[ (na+1) + (na+2) + \cdots + (na+n) \right]} = \frac{1}{60}. \]

Answer: 7


Question: Determine whether there exists a non-holomorphic but real differentiable function $f : D^2 \setminus \{0\} \to \mathbb{R}$ such that $xf(x) \to 0$ as $x \to 0$, but $f(x)$ is not bounded near 0.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to \infty} \frac{J_0(2\sqrt{(n+1)x})}{J_0(2\sqrt{nx})} \] where \( J_0 \) is the Bessel function of the first kind.

Answer: 1


Question: Determine the number of distinct characteristic polynomials in the set of 2x2 matrices over \( \mathbb{Z}_2 \).

Answer: 4


Question: Given that \( p \) is a prime number, and the sum of all positive divisors of \( p^4 \) is a perfect square, determine the number of possible primes \( p \).

Answer: 1


Question: Compute the flux integral \( \int \int_{M} \text{curl}(\mathbf{u}) \cdot \mathbf{N} \, dS \) for the surface \( M: z = e^{-x^{2}-y^{2}}, x^{2}+y^{2} \leq 1 \), where \( \mathbf{N} \) is the upwards unit normal on \( M \) and the vector field \( \mathbf{u} = (x^{4}-y^{3}, \cos(x), \sin(z)) \). The answer is \( \frac{3\pi}{4} \).

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


Question: Calculate \(333^{333^{333}} \mod 17\).

Answer: 11


Question: Let $u_1, u_2, u_3, u_4, u_5$ be linearly independent vectors in $\mathbb{R}^m$. Consider the matrix $A = \sum_{i=1}^{5} u_i u_i^T$. What is the rank of $A$?

Answer: 5


Question: Determine the fundamental group of a punctured cylinder. Let $X$ be the standard open cylinder, defined as the quotient of $\mathbb{C}$ by the action of $z \mapsto z+1$. Let $X^*$ be $X$ with a point removed. What is $\pi_1(X^*)$?

Answer: \mathbb{Z} * \mathbb{Z}


Question:
Consider the generalized sieve method described, which uses the following cross-out forms for integers starting from 81:
\[
6ij + i + j, \\
6ij + i + 5j, \\
6ij + 5i + 5j, \\
\]
where \(i, j \geq 1\). Can we reduce the number of these cross-out forms to less than three while still achieving the same filtering effect? Provide a justification for your answer.

Answer: No


Question: Suppose we have a curve $C$ defined by $F(X,Y) = 0$, where $F$ is a polynomial, and a line $l$ that is tangent to $C$ at some point $P$. Let $M$ be an invertible linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$. Define a new curve $C'$ as the image of $C$ under $M$ and a new line $l'$ as the image of $l$ under $M$. Is $l'$ necessarily tangent to $C'$?

Answer: Yes


Question: Determine whether the statement "If $A$ is an invertible square matrix and $\det(ABA) = 0$, then $\det(B) = 0$" is true or false. Justify your answer.

Answer: True


Question: Evaluate the integral \( \int_\gamma \frac{z^4+z^2+1}{z^3-1} \, dz \) where \( \gamma \) is the circle \( |z-i|=1 \), using Cauchy's integral formula. Determine if the solution is correct.

Answer: 0


Question: Given the data set $x_1 = 0.2$, $x_2 = 0.6$, $x_3 = 0.7$, compute the Kolmogorov-Smirnov statistic $D_N = \max\{D^{+},D^{-}\}$, where $F$ is the cumulative distribution function of the uniform distribution on $(0,1)$. Calculate $D_N$ by hand.

Answer: 0.3


Question: Evaluate the integral \( \int_{-\infty}^\infty e^{-(x-ti)^2} \, dx \), where \( t \in \mathbb{R} \) and \( i = \sqrt{-1} \).

Answer: \sqrt{\pi}


Question: Given triangle ABC with its inscribed circle centered at O, where AO = 3, BO = 4, and CO = 5, find the perimeter of triangle ABC.

Answer: 24


Question:
Given the joint distribution of $(X,Y)$:
\[ f(x,y) = \frac{y}{(\alpha e^{-\beta x} + \sigma)^2} \exp\left(-\frac{y}{(\alpha e^{-\beta x}) + \sigma} - x\right), \quad x,y > 0 \]

Calculate the marginal distribution of $X$, $f_X(x)$, by evaluating the integral:
\[ f_X(x) = \int_{0}^{\infty} f(x,y) \, dy \]

Answer: e^{-x}


Question: Determine the values of \( n \) for which the following limit exists, ensuring the function is continuous at \((0,0)\): \[ \lim_{(x,y)\to(0,0)}\frac{|x||y|^n}{x^2 + y^2} \]

Answer: n > 1


Question: Let $E$ be a regular hexagon centered at the origin in the plane $\mathbb{R}^2$. Consider a harmonic function $f$ defined in $E$ such that $f$ takes the value 1 on one side of the hexagon and 0 on the other five sides. Determine the value of $f$ at the origin.

Answer: \dfrac{1}{6}


Question: Calculate the flux of the vector field \( F = (\cos(xyz), \sin(xyz), xyz) \) through a square that is parallel to the \(XY\) plane, with its center at \((0,0,a)\) and side length \(a\).

Answer: 0


Question: Calculate \( \lfloor \frac{1}{L} \rfloor \), where \( L = \lim_{n\to\infty}\sum_{r=1}^n \frac{rn}{n^3+r} \).

Answer: 2


Question: Two people enter adjacent lines in a grocery store. The service time for person A is exponentially distributed with a mean of 4 minutes, and the service time for person B is exponentially distributed with a mean of 6 minutes. What is the probability that person A finishes their service before person B?

Answer: \dfrac{3}{5}


Question: Evaluate the integral \( \int_{\gamma}\frac{dz}{z} \) where \( \gamma \) is a path from \(-i\) to \(i\) such that the image of \( \gamma \) is contained in \( \mathbb{C} \setminus \mathbb{R}^- \).

Answer: i\pi


Question: Is it possible to construct a discontinuous function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(x + y) = f(x) + f(y) \) for all real numbers \( x \) and \( y \)?

Answer: Yes


Question: Let $D \subset \mathbb{R}^2$ be the closed unit disk with boundary $\mathbb{S}^1$. Suppose $a, b \in \mathbb{S}^1$ are two distinct points, and $A, B \subset D$ are two disjoint closed sets such that $A \cap \mathbb{S}^1 = \{a\}$ and $B \cap \mathbb{S}^1 = \{b\}$. Determine whether any distinct points $x, y \in \mathbb{S}^1 \setminus \{a, b\}$ can be connected by a continuous path lying entirely in $D \setminus (A \cup B)$. In other words, does there exist a continuous function $f: [0, 1] \to D \setminus (A \cup B)$ such that $f(0) = x$ and $f(1) = y$?

Answer: Yes


Question: Let $M$ be the set of all holomorphic functions $f$ from the unit disk to itself such that $f(0)=\frac{2}{3}$. Determine the supremum of $|f'(0)|$ over all functions in $M$.

Answer: \dfrac{5}{9}


Question: Evaluate the limit: \[ \lim _{x\rightarrow0}\frac{1-\cos\left(\frac{1-\cos x \cos 2x}{x^2}-\frac {5}{2}\right)\cos2x}{x^2} \]

Answer: 2


Question: Evaluate the triple integral \( \iiint \frac{dx\,dy\,dz}{\sqrt{1 - \left(\frac{x^2}{9}+\frac{y^2}{16}+\frac{z^2}{25}\right)}} \) over the region defined by the ellipsoid \( \frac{x^2}{9}+\frac{y^2}{16}+\frac{z^2}{25}=1 \).

Answer: 60\pi^2


Question: Determine the pushout of the diagram \( \mathbf{1} \stackrel{f}{\longleftarrow} \mathbf{2} \stackrel{g}{\longrightarrow} \mathbf{1} \), where \( \mathbf{1} \) and \( \mathbf{2} \) denote the discrete finite categories of one and two objects, respectively, and both \( f \) and \( g \) are the trivial constant functions.

Answer: \mathbf{1}


Question:
Express the following limit of an infinite series as a single equation in terms of $x$:
\[ \lim_{k\to \infty} \left(\sum_{n = 0}^\infty \frac{x^{2n}(k-n)!}{(k+n)!}\right) \]

Answer: 1


Question:
Determine the condition on the parameter $h$ that ensures the convergence of the series:
\[
\sum_{n=0}^{\infty} \left(\frac{h}{4}\right)^n \frac{(2n)!\times (2n)!}{n! \times n!\times n! }.
\]

Answer: 0


Question: Compute the limit: \[ \lim_{n\to\infty} \sum_{k=1}^{n} \left(\frac{k}{n^2}\right)^{\frac{k}{n^2} + 1} \]

Answer: \dfrac{1}{2}


Question: Let $h$ be a positive odd integer and $n = 4h^2$. Consider the polynomial $R(t) = r_0 + r_1 t + \cdots + r_{n-1} t^{n-1}$ with integer coefficients $r_i \in \{-1, 1\}$. Suppose $R(1) = 2h$ and for every complex number $\omega$ such that $\omega^n = 1$ and $\omega \notin \{-1, 1\}$, $R(\omega)$ is a nonzero integer. Determine whether all these integers $R(\omega)$ have the same sign.

Answer: Yes


Question:
Determine the appropriate way to write the following inequality involving norms and mappings:

Given a continuous and bounded function $x : \mathbf{R} \to \mathbf{C}$, where $x(t) \neq 0$ only if $a \leq t \leq b$ and $|x(t)| \leq 1$ for all $t$, consider the mappings:
\[ Aux_{\tau} : L^{2}(\mathbf{R}) \times L^{2}(\mathbf{R}) \rightarrow L^{2}(\mathbf{R} \times \mathbf{R}) \]
and
\[ A : L^{2}(\mathbf{R}) \rightarrow L^{2}(\mathbf{R} \times \mathbf{R}) \]
where
\[ A(x)(t,f) = \int Aux_{\tau} (x)(t,f) d\tau. \]

Choosing $x=y$ for $Aux_{\tau}$, the norm is given by the Schwartz inequality:

1. \[ \lVert A(x) \rVert_{L^{2}( \mathbf{R} \times \mathbf{R} )} \leq \int \lVert Aux_{\tau}(x) \rVert_{L^{2}( \mathbf{R} \times \mathbf{R} )} d\tau. \]
2. \[ \lVert A(x) \rVert_{L^{2}(\mathbf R) \to L^2(\mathbf R \times \mathbf R)} \leq \int \lVert Aux_{\tau}(x) \rVert_{L^{2}(\mathbf R) \to L^2(\mathbf R \times \mathbf R)} d\tau. \]
3. \[ \lVert A(x) \rVert_{L^{2}(\mathbf R) \to L^2(\mathbf R \times \mathbf R) } \leq \int \lVert Aux_{\tau}(x) \rVert_{L^{2}(\mathbf R) \times L^2(\mathbf R) \to L^2(\mathbf R \times \mathbf R)} d\tau. \]
4. \[ \lVert A(x) \rVert_{L^2} \leq \int \lVert Aux_{\tau}(x) \rVert_{L^2} d\tau. \]

Which is the correct way to write the inequality?

Answer: 1


Question: Determine whether the function \( f(x) = (x-1)\sin\left(\frac{1}{x-1}\right) \) is differentiable at \( x = 0 \) given that \( f(0) = \sin 1 \). Provide a proof using first principles.

Answer: \cos 1 - \sin 1


Question: Solve the recurrence relation \( T(n) = T(\sqrt{n}) + T(n - \sqrt{n}) + n \) with the base case \( T(N) = O(1) \) for \( N < 2 \).

Answer: O(n^{3/2})


Question: Let $(X, \mu)$ be a measure space with a positive, finite measure $\mu$. Consider a decreasing sequence $\{ f_{j} \in L^{\infty}(X) \}$ that converges pointwise to $f$, denoted as $f_{j} \searrow f$. Assume that \( \int_{X} f_{j} \, d\mu \geq -1 \) for each $j$. Can we conclude that \( \int_{X} f \, d\mu \geq -1 \)?

Answer: Yes


Question: Find the expected duration of a randomly selected long-distance telephone call, given that the duration $Y$ has the following properties: $P(Y=3)=0.2$, $P(Y=6)=0.1$, and otherwise $Y$ has a continuous density function given by \( f(y) = \frac{1}{4}ye^{-y/2} \) for \( y > 0 \).

Answer: 4


Question:
Find the value of the infinite power tower:
\[ \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{.^{.^{.^{.^{.}}}}}}}} \]

Answer: 2


Question: Using triple integrals and Cartesian coordinates, find the volume of the solid bounded by the surfaces \(z = 3x^2 + 3y^2 - 7\) and \(z = 9 - x^2 - y^2\).

Answer: 32\pi


Question: Calculate the sum of the series: \[ \sum^{10}_{i=1} i \left( \frac{1^2}{1+i} + \frac{2^2}{2+i} + \cdots + \frac{10^2}{10+i} \right) \]

Answer: \dfrac{3025}{2}


Question: Find the limit: \[ \lim_{n\to\infty} \frac{n^{\log(n)}}{(\log n)^n} \]

Answer: 0


Question: Determine the value of \( \Omega \) such that the integrating factor \( \mu = (x^2+y^2)^\Omega \) makes the differential equation \( xdy -(x^2+y^2+y)dx = 0 \) exact.

Answer: -1


Question: Find the average distance from points within a unit ball in \( \mathbb{R}^3 \) to the origin.

Answer: \dfrac{3}{4}


Question: Use Green's theorem to evaluate the line integral \( \int_{\gamma} y\,dx + x^2\,dy \), where \( \gamma \) is the circle parameterized by \( g(t) = (\cos(t), \sin(t)) \) for \( 0 \le t \le 2\pi \).

Answer: -\pi


Question: Evaluate the limit: \[ \lim_{n\to \infty} \int_0^n (1+x/n)^{-n}\log(2+\cos(x/n))\,dx \] Justify your answer.

Answer: \log 3


Question: Determine the number of positive solutions to the equation \( \cos(97x) = x \).

Answer: 31


Question: Find \( \lim_{x \to \infty}\int_0^x{e^{-x^2+t^2}}\,dt \).

Answer: 0


Question: Determine whether the inequality \( 2\varphi(n) > n \) holds for all integers \( n = 2^k - 1 \) where \( k \) is an odd integer.

Answer: Yes


Question: Determine the limit of the function \( f(x) = \sum_{n=1}^{\infty} \frac{1}{n^6 + x^4} \) as \( x \to \infty \).

Answer: 0


Question: Find the volume of the solid bounded by the surfaces \( x=0 \), \( y^{2}+z^{2}=4 \), and \( x+z=4 \).

Answer: 16\pi


Question: Evaluate the integral \( \int_{-\infty}^{+\infty} \frac{1}{\sqrt{2\pi}} e^{iaz-z^{2}/2} \, dz \).

Answer: e^{-a^2/2}


Question: Find the maximum likelihood estimate (MLE) of $m$ in the linear model $y = mx + e$, where $m$, $x$, and $e$ are independent and identically distributed as $N(0,1)$, given a single observation $(x, y)$. Assume $x = 1$ and $y = 3$. Provide the analytical expression for the MLE of $m$.

Answer: 3


Question: Evaluate the limit \( \lim_{n \to \infty} I_n^n \), where \( I_n = \int_1^{1+\frac{1}{n}}\left\{[(x+1)\ln x +1] e^{x (e^x \ln x +1)}+n \right\}\, dx \) for \( n=1,2,\ldots \).

Answer: e^e


Question: Determine the norm of the functional $l: C[0,\pi] \rightarrow \mathbb{R}$ defined by \[ l(x) = x(0) - x(\pi/4) + \int_{0}^{\pi} x(s) \sin s \, ds. \]

Answer: 4


Question: Determine the splitting field of the polynomial \(x^8 + x^6 + x^5 + x^4 + x^3 + x^2 + 1\) over the finite field \(\mathbb{F}_2\).

Answer: \mathbb{F}_{2^6}


Question: A fair coin is flipped repeatedly until three consecutive tails (TTT) appear. What is the expected number of heads that will be observed before this sequence occurs?

Answer: 7


Question: Evaluate the integral \( \int_{-2}^{2} \frac{3x^2}{1+e^x} \mathrm{d}x \).

Answer: 8


Question: Find the gradient of the expression \( \mathrm{trace}(A'(X\circ X-1)) \) with respect to the matrix \( X \), where \( \circ \) denotes the Hadamard product, \( A \) is a matrix of positive real values, and \( 1 \) is a matrix of all ones.

Answer: 2A \circ X


Question: Is there a group $G$ with a finite index subgroup $H$ such that for every subgroup $N$ of $H$ which is normal in $G$, the index $[G:N]$ is at least $n!$?

Answer: Yes


Question:
Show that the two antiderivatives of the function \( f(x) = \frac{1}{\sqrt{e^{2x} - 9}} \) are equivalent, meaning they differ by a constant. The antiderivatives are given by:

1. \( \frac{2}{3} \arctan \left(\frac{e^x + \sqrt{e^{2x}-9}}{3}\right) + C \)
2. \( \frac{1}{3}\arctan \frac{\sqrt{e^{2x} - 9}}{3} + C \)

Verify that these expressions are equivalent.

Answer: \frac{\pi}{6}


Question: Given the equations \(p = q(4-q)\), \(q = r(4-r)\), and \(r = p(4-p)\) where \(p, q, r \in \mathbb{R}\), find the maximum value of \(p + q + r\).

Answer: 9


Question: Compute the exterior derivative \(d\beta\) of the two-form \(\beta = z(y^2-x^2)\,dx\,dy - x^2y\,dx\,dz - xy^2\,dy\,dz\) in three-dimensional real space.

Answer: 0


Question: Evaluate the infinite series: \[ \sum_{n=0}^{\infty} \frac{16n^2 + 20n + 7}{(4n+2)!} \]

Answer: e + \sin 1


Question: Find the splitting field of the polynomial $x^4 - 4$ over $\mathbb{Q}$ and determine its degree over $\mathbb{Q}$.

Answer: 4


Question: Suppose every eigenvalue \( \lambda \) of a matrix \( X \in M_n(\mathbb{C}) \) satisfies \(|\lambda| < 2\pi\). Prove that if \( \exp(X)v = v \) for some vector \( v \in \mathbb{C}^n \), then \( Xv = 0 \).

Answer: Xv = 0


Question: Solve the differential equation: \((x^2-y^2)dx+2xydy=0\).

Answer: x^2 + y^2 = Cx


Question: In a fair coin game, you win $1 if a head shows up and lose $1 if a tail shows up. What is the expected number of rounds you will play before you lose the first dollar from your own pocket?

Answer: \infty


Question: Given that 3 is a primitive root of 17, solve the congruence equation \(7^x \equiv 6 \pmod{17}\).

Answer: 13


Question: Suppose $A$ is a real matrix such that every term in the determinant sum \( \det(A) = \sum \pm a_{1,i_1} a_{2,i_2} \cdots a_{n,i_n} \) is zero. Is it true that $A$ must have a zero row or zero column? Answer "Yes" or "No" and justify your answer.

Answer: No


Question: Let $K$ be an extension of a field $F$ such that $[K:F] = 13$. Suppose $a \in K - F$. Determine the value of $[F(a):F]$. \( \text{(Note: 13 is a prime number.)} \)

Answer: 13


Question:
Consider the determinant \( D(x) = \begin{vmatrix} x & (1+x^2) & x^3 \\ \log(1+x^2) & e^x & \sin(x) \\ \cos(x) & \tan(x) & \sin^2(x) \end{vmatrix} \). Determine which of the following statements is true:

(a) \( D(x) \) is divisible by \( x \)
(b) \( D(x) = 0 \)
(c) \( \frac{d}{dx}D(x) = 0 \)
(d) None of these

Find the correct option.

Answer: a


Question: Determine the degree of the image \( \nu_{n,d}(C) \) under the Veronese map, where \( C \subset \mathbb{P}^n \) is a curve of degree \( a \). Is this degree simply \( ad^n \)?

Answer: a d


Question: Determine whether the sequence \( a_n = \frac{1}{n} + \frac{1}{n+1} + \cdots + \frac{1}{2n} \) is convergent for \( n \) in natural numbers. If it is convergent, find its limit.

Answer: \ln 2


Question: Given a tree graph where the root node has a value of \( m \), each node at the second level has a value of \( k = \frac{m}{n+1} \), and each subsequent level has nodes with values \( k = \frac{k}{n+1} \), continuing this process until the leaf nodes have a value of 1. If the number of leaf nodes (nodes with value 1) is given as 9 and the root value \( m \) is 16, determine the branching factor \( n \) of the tree.

Answer: 3


Question: Evaluate the line integral \( \int_{\gamma} \mathbf{x} \cdot \mathbf{n}(\mathbf{x}) \, ds(\mathbf{x}) \), where \( \mathbf{x} = (x, y) \in \mathbb{R}^2 \) and \( \mathbf{n}(\mathbf{x}) \) is the unit outward normal to the ellipse \( \gamma \) given by \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \).

Answer: 12\pi


Question: Find all positive integers \( n \) such that \( 3^{n-1} + 5^{n-1} \) divides \( 3^n + 5^n \).

Answer: 1


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty} \frac{n^n}{n!} \cdot (z-1)^n \).

Answer: \dfrac{1}{e}


Question:
Determine the set of all values of \(a\) such that the equation \((a-1)(x^4+x^2+1)+(a+1)(x^2+x+1)^2 = 0\) has real and distinct roots. Choose from the following options:

(a) \(\left(0,\frac{1}{2}\right)\)
(b) \(\left(-\frac{1}{2},0\right)\cup \left(0,\frac{1}{2}\right)\)
(c) \(\left(-\infty, -2 \right)\cup (2,\infty)\)
(d) \(\left(-\frac{1}{2},0\right)\)

Answer: B


Question: Determine whether the sequence of operators \( \{T_n:H\rightarrow H\}_n \) with \( H=l^2(\mathbb{N};\mathbb{R}) \) defined by \( T_n(x_0,x_1,x_2,...)=(x_n,x_{n-1},...,x_0,0,0,...) \) converges weakly to zero.

Answer: Yes


Question: Given that \( Y_n \) converges to \( Y \) in distribution and that \( P(Y \le t) \) is continuous, prove that \( \frac{Y_n}{\sqrt{n}} \) converges to 0 in probability.

Answer: 0


Question: Evaluate the integral \( \int_{-1}^{1}\frac{\cos x}{e^{\frac{1}{x}}+1}dx \) using the hint: \( f(x)=\frac{f(x)+f(-x)}{2}+ \frac{f(x)-f(-x)}{2} \).

Answer: \sin 1


Question: Determine the cardinality of the set \( A = \{ X \subseteq \mathbb{N} : |X| = \aleph_0 \land |\bar{X}| < \aleph_0 \} \).

Answer: \aleph_0


Question: Determine whether the following statement is true or false: If a commutative ring with $1$ does not have a unit of additive order $2$, then it does not have any element of additive order $2$. Justify your answer.

Answer: False


Question: Determine the expected value of the quadratic variation process \( \langle S \rangle \) for a simple symmetric random walk \( S = (S_n)_{n \in \mathbb{N}_0} \), where \( S_n = \sum_{k=1}^n Z_k \) and \( (Z_k)_{k \in \mathbb{N}} \) are i.i.d. random variables with \( \mathbb{P}[Z_1 = 1] = \mathbb{P}[Z_1 = -1] = \frac{1}{2} \).

Answer: n