Question: Compute the limit \( \lim_{p\rightarrow0^{+}}\int_{C_p}\frac{e^{3iz}}{z^{2}-1}dz \), where \( C_p: z = 1 + pe^{i\theta} \).

Answer: \pi i e^{3i}


Question: Prove that \( \sum_{k=3}^\infty k\left(\frac{5}{6}\right)^{k-3} = 48 \).

Answer: 48


Question: Given a vector field $v$ on $\mathbb{R}^3$ satisfying $\nabla \cdot v = 0$ and $\nabla \times v = \alpha v$ for some scalar function $\alpha: \mathbb{R}^3 \to \mathbb{R}$, compute $v \cdot \nabla \alpha$. Show that the result is $0$.

Answer: 0


Question:
Calculate the integral of the following function over the region \( I = [0,1] \times [0,1] \) using the Darboux sum convergence criterion:
\[
 f(x, y) = \begin{cases}
 1 & \text{if } y > x, \\
 0 & \text{if } y \le x
 \end{cases}
\]

Answer: \dfrac{1}{2}


Question: Let \(X_1, X_2, \ldots\) be i.i.d. Bernoulli random variables with parameter \(\frac{1}{4}\). Let \(Y_1, Y_2, \ldots\) be another sequence of i.i.d. Bernoulli random variables with parameter \(\frac{3}{4}\). Let \(N\) be a geometric random variable with parameter \(\frac{1}{2}\), i.e., \(P(N=k)=\frac{1}{2^k}\) for \(k=1,2,\ldots\). Assume the \(X_i\)'s, \(Y_j\)'s, and \(N\) are all independent. Compute \(\operatorname{cov}\left(\sum_{i=1}^{N}X_i,\sum_{i=1}^{N}Y_i\right)\).

Answer: \dfrac{3}{8}


Question: Evaluate the integral \( \int_0^1 \sqrt{1+x^4} \hspace{1mm} dx \) using series expansion, and provide the answer correct to two decimal places.

Answer: 1.09


Question: Evaluate the line integral \( \oint_C M\,dx + N\,dy \) for the vector field \( \vec{F} = (3x^2y^2)\hat{i} + (2x^2 + 2x^3y)\hat{j} \), where \( C \) is the counterclockwise circle centered at \((a,0)\) with radius \(a\). Use Green's Theorem to find the value of the integral.

Answer: 4\pi a^3


Question: Given the sequence defined by \(a_1 = 3\) and \(a_{n+1} = 3^{a_n}\), find the remainder when \(a_{2004}\) is divided by 100.

Answer: 87


Question: Find the minimum value of \( |f(z)| \) for the function \( f(z) = z^2 + 2 \) within the unit disc \( |z| < 1 \).

Answer: 1


Question: Evaluate the complex line integral \( \int_{\{|z|=4\}} \frac{z^2+2}{(z-3)^2} \, dz \).

Answer: 12\pi i


Question: Determine whether the exponential polynomial \( p(x) = -a e^{2x} + b e^{x} - cx + d \) always has a root for positive \( a \) and \( c \), regardless of the particular choice of \( b \) and \( d \).

Answer: Yes


Question: Find the area enclosed by the curve given by the equation \(x^4 + y^4 = 2xy\) using polar coordinates.

Answer: \dfrac{\pi}{2}


Question: Let $F=\mathbb{Z}_3$ and $V=F^4$. Define the subspaces $U=\text{span}\{(1,0,0,0),(1,0,1,0),(0,1,1,1)\}$ and $W=\text{span}\{(0,0,1,0),(-1,1,0,1),(1,1,1,1)\}$. Find the dimension of the intersection $U \cap W$. \[ \text{dim}(U \cap W) = ? \]

Answer: 3


Question: A 1-meter long stick is broken at a random point chosen uniformly along its length. The shorter part, denoted as \( X \), is used as the length of one side of a rectangle. The longer part of the stick is broken again at a random point chosen uniformly along its length, and the shorter part of this second breaking, denoted as \( Y \), is used as the length of the other side of the rectangle. Find the expected area of the rectangle.

Answer: \dfrac{1}{24}


Question: A function \(f\) is defined on the complex numbers by \(f(z) = (a+bi)z\), where \(a\) and \(b\) are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that \(|a+bi|=8\), find the value of \(b^2\).

Answer: \dfrac{255}{4}


Question: Use Green's theorem to calculate the area of the asteroid defined by the parametric equations \(x = \cos^3{t}\) and \(y = \sin^3{t}\) for \(0 \leq t \leq 2\pi\).

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


Question: Evaluate the line integral \( \int_C (1+y)\,dx - e^{\sin y}\, dy \), where \( C \) is the parabola \( y=x^2 \) from \((-1,1)\) to \((1,1)\).

Answer: \dfrac{8}{3}


Question: Let \( f : \mathbb{R}^n \to \mathbb{R} \) be a continuous function such that there exist points \( x, y \in \mathbb{R}^n \) with \( f(x)f(y) < 0 \). Determine if it is true that the Hausdorff dimension of the zero set of \( f \) is at least \( n-1 \).

Answer: True


Question: Find the smallest composite number \( n \) such that \( n \mid 2^n - 2 \).

Answer: 341


Question: Let \( f \in C^2(\mathbb{R}) \) be a function such that \((f(x))^2 \le 1\) and \((f'(x))^2 + (f''(x))^2 \le 1\) for all \(x \in \mathbb{R}\). Is it true that \((f(x))^2 + (f'(x))^2 \le 1\) for all \(x \in \mathbb{R}\)?

Answer: Yes


Question: How many surjections \( g: \mathbb{N} \rightarrow \mathbb{N} \) commute with the function \( t: \mathbb{N} \rightarrow \mathbb{N} \) defined by \( t(n) = n+1 \)?

Answer: 1


Question: Let $a$, $b$, and $c$ be non-negative real numbers such that $a+b+c = 1$. Find the maximum value of \[ \frac{1}{a^2-4a+9} + \frac{1}{b^2-4b+9} + \frac{1}{c^2-4c+9}. \]

Answer: \dfrac{7}{18}


Question: Determine the maximum value of \((x−1)^2+ (y−1)^2+ (z−1)^2\) subject to the constraints \(x^2+y^2+z^2 \leq 2\) and \(z \leq 1\).

Answer: 5 + 2\sqrt{6}


Question: Determine whether there always exists a projective transformation in $PGL_2(\mathbb{C})$ that maps the set of points $A = \{p_1, p_2, p_3\}$ to the set $B = \{q_1, q_2, q_3\}$ in the complex projective line $\mathbb{P}^1$. Provide a justification for your answer.

Answer: Yes


Question:
Find the limit of the following expression as \( n \to +\infty \):
\[ \lim_{n \to +\infty} \sqrt[n]{\frac{1}{3^n}+\frac{n^5}{5^n}+\frac{1}{2^{nn^{10^{10}}}}} \]

Answer: \dfrac{1}{3}


Question: Find the residue of \(2^{3^{2011}}\) modulo 17.

Answer: 8


Question: Find the maximum value of the expression \( \frac{x}{1+x^2} + \frac{y}{1+y^2} + \frac{z}{1+z^2} \) where \( x, y, z \) are real numbers satisfying \( x+y+z=1 \).

Answer: \dfrac{9}{10}


Question: Given that the volume of a regular tetrahedron, with side length \( t \), is \( V=\frac{1}{12}t^3\sqrt{2} \), find the rate of increase of the surface area when the volume is \( 144\sqrt{2} \text{ cm}^3 \) and is increasing at a rate of \( 12 \text{ cm}^3/\text{s} \).

Answer: 4\sqrt{6}


Question: Evaluate the integral \( \oint_C \frac{zf'(z)}{f(z)} \,ds \) using the argument principle, where \( f(z) = z^4 - 2z^3 + z^2 - 12z + 20 \) and \( C \) is the circle \( |z| = 5 \).

Answer: 40\pi


Question: Calculate the limit: \[ \lim _{n\to \infty }\frac{2^{4n+1} \ln(2n^3+1)+n^5\cdot 10^n}{15^n+4^{2n-1} \ln (5n)} \]

Answer: 24


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\frac{n!n^{x}}{x(x+1)...(x+n)} \] given that \( \text{Re}(x) > 0 \).

Answer: \Gamma(x)


Question: Find one triple of integers \((\ell, m, n)\) such that \(2 < \ell < m < n\) and the sums \(\sum_{k=1}^{\ell} k\), \(\sum_{k=\ell+1}^{m} k\), and \(\sum_{k=m+1}^{n} k\) form a geometric sequence.

Answer: (5, 9, 14)


Question: Let \( g:[0,1] \to \mathbb{R} \) be a function that is continuous on \([0,1]\) and differentiable on \((0,1)\). Suppose that for every \( x, y \in (0,1) \), if \( g(x) = g(y) \) then \( g'(x) = g'(y) \). Is \( g \) necessarily monotonic? Answer true or false.

Answer: True


Question: Find the sum of all positive integers \(\alpha\) less than 120 such that \(5\mid(2^\alpha+\alpha)(3^\alpha+\alpha)\).

Answer: 2100


Question: Determine the vector $abcd \in F_3^4$ such that the code $C = \{0000, 0010, 0020, abcd, 2110, 2100, 1220, 2120, 1210\}$ is linear. Consider the vectors $0010$ and $1220$ as a basis for $C$. Express $abcd$ as a linear combination of these basis vectors using coefficients $\alpha, \beta \in F_3 = \{0, 1, 2\}$. Find the vector $abcd$ that completes the code to make it linear.

Answer: 1200


Question: Let $E$ be a compact subset of a bounded open set $\mathcal{D},$ both of which are subsets of the real numbers. Let $f(x)$ be a differentiable function such that $f(x) = \mathcal{O}(1)$ for all $x \in \mathcal{D}$. Does there exist such a function where $f'(x) \neq \mathcal{O}(1)$ for some $x \in E$?

Answer: Yes


Question:
Let \( \{a_n\} \) be a sequence of real numbers. Determine which of the following conditions is equivalent to the statement that \( \lim_{n \to \infty} a_n \) exists:

(A) \(\lim_{n \to \infty} a_{2n}\) and \(\lim_{n \to \infty} a_{2n+2}\) exist.

(B) \(\lim_{n \to \infty} a_{2n}\) and \(\lim_{n \to \infty} a_{2n+1}\) exist.

(C) \(\lim_{n \to \infty} a_{2n}\), \(\lim_{n \to \infty} a_{2n+1}\), and \(\lim_{n \to \infty} a_{3n}\) exist.

(D) None of the above.

Answer: C


Question: Determine whether there exists a number field extension of \(\mathbb{Q}\) whose Galois group is isomorphic to \(\mathbb{Z}_3\).

Answer: Yes


Question: Calculate the variance of the function \( g(x) = (2x+1)^2 \) where \( x \) is a random variable with values 3, 6, and 9, having probabilities \( \frac{1}{6} \), \( \frac{1}{2} \), and \( \frac{1}{3} \) respectively. Use the formula \( \text{Var}[g(x)] = E[g(x)^2] - (E[g(x)])^2 \) to find the variance.

Answer: 12752


Question:
Find the determinant of the matrix:
\[
\begin{pmatrix}
\cos\left(a_{1}-b_{1}\right) & \cos\left(a_{1}-b_{2}\right) & \cos\left(a_{1}-b_{3}\right)\\
\cos\left(a_{2}-b_{1}\right) & \cos\left(a_{2}-b_{2}\right) & \cos\left(a_{2}-b_{3}\right)\\
\cos\left(a_{3}-b_{1}\right) & \cos\left(a_{3}-b_{2}\right) & \cos\left(a_{3}-b_{3}\right)
\end{pmatrix}
\]
where $a_1,\dots ,a_3,b_1,\dots, b_3\in\mathbb{R}$. Provide your answer in terms of $a_i$ and $b_j$.

Answer: 0


Question:
Calculate the energy of the signal \( y(t) = \frac{1}{3} x(2t) \), where the original signal \( x(t) \) is defined as:
\[
x(t)=\begin{cases}
t & 0 \leq t < 1 \\
0.5+0.5\cos(2 \pi t) & 1 \leq t < 2 \\
3-t & 2 \leq t < 3 \\
0 & \text{elsewhere}
\end{cases}
\]

Answer: \dfrac{25}{432}


Question:
Consider the joint distribution of a continuous variable \(X\) and a binary variable \(Y\):
\[
p(x,y;\beta,\rho_1,\rho_2,\phi_1,\phi_2)=\frac{1}{Z(\beta,\rho_1,\rho_2,\phi_1,\phi_2)}\exp(-0.5 \beta x^2+1_{y=0}\rho_1 x+1_{y=1} \rho_2 x+1_{y=0} \phi_1+1_{y=1} \phi_2)
\]
where \(Z\) is a normalizing constant. The conditional distribution of \(X\) given \(Y\) is Gaussian, and the conditional distribution of \(Y\) given \(X\) is Bernoulli with probability \(P(Y=1)\) depending on \(X\). Is it possible to use Gibbs sampling to sample from this joint distribution?

Answer: Yes


Question: Determine the number of distinct solutions of the congruence \((x^2-1)(x^2+1) \equiv 0 \pmod{4 \times 31^3}\).

Answer: 4


Question:
Evaluate the sum:
\[ \sum_{1\leq i < j \leq m} \sum_{\substack{1\leq k,l \leq n\\ k+l\leq n}} {n \choose k}{n-k \choose l}(j-i-1)^{n-k-l}. \]

Answer: m^n - m


Question: Evaluate the integral: \[ \int_{-1}^{1} \frac{1}{(e^x+1)(x^2+1)} \, dx \]

Answer: \dfrac{\pi}{4}


Question: Find the fifth derivative of the function \( y = \frac{x}{x^2-1} \) evaluated at \( x = 0 \). That is, compute \( y^{(V)}(0) \).

Answer: -120


Question: Determine whether the function \( g: C^{1}[0,2], ||*||_{C^1[0,2]} \rightarrow \mathbb{R} \) defined by \( g(f) = f'(1) \) satisfies a Lipschitz condition.

Answer: Yes


Question: Given the equation $x^3 + yz - xz^2 = 0$ which implicitly defines a function $z = f(x, y)$, and knowing that $f(-2, 1) = 2$, calculate the partial derivative $f_x(-2, 1)$. Use implicit differentiation to find your answer.

Answer: -\dfrac{8}{9}


Question: Determine the angle in degrees required to skew a circle so that it appears perfectly centered on each of the three visible sides of a cube from the viewpoint of a camera.

Answer: 54.74


Question: Evaluate the limit: \[ \lim_{x \to 2} \frac{\sqrt{2+\sqrt{2+\sqrt{2+\cdots+\sqrt{2+x}}}}-2}{x-2} \] where the expression consists of \( n \) nested radicals.

Answer: \dfrac{1}{4^n}


Question: Evaluate the integral \( I = \int_{x=0}^{\frac{1}{2}} \int_{y=x}^{1-x} \left( \frac{x-y}{x+y} \right)^{2} \, dy \, dx \) using the change of variables \( x = \frac{1}{2} (r-s) \) and \( y = \frac{1}{2} (r+s) \).

Answer: \dfrac{1}{12}


Question: Find all prime numbers $p$ such that $8p^4 - 3003$ is also a prime number.

Answer: 5


Question: The curves \(y=mx\) and \(y=\frac{x}{x^2+1}\) enclose a region for \(0<m<1\). Find the area of this region in terms of \(m\).

Answer: m - 1 - \ln m


Question:
A machine consisting of two components will fail if one of the components fails. Let $X$ be the time-to-failure random variable of the first component and $Y$ be the time-to-failure random variable of the second component, both measured in hours. The joint moment generating function of the time-to-failure of the two components is given by:
$$
M_{X,Y}(s,t)=\frac{1}{(1-3s-2t+6ts)}
$$
Calculate the probability that the machine will fail within the first hour.

Answer: 1 - e^{-5/6}


Question:
Given integers $n$ and $r_0$ with $r_0 < n$, define:
\[ r_1 = n \mod r_0 \]
\[ r_2 = n \mod r_1 \]
where $0 < r_1 < r_0$ and $0 < r_2 < r_1$. Determine if it is possible for $r_2$ to be greater than $\lceil \frac{1}{2}r_0 \rceil$. Provide a justification for your answer.

Answer: Yes


Question: Given the function \( F(x) = \int_0^{2x} \tan(t^2) \, dt \), find \( F''(1) \).

Answer: 16 \sec^2(4)


Question: Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x, y, z \), the equation \[ f(x-y) + f(y-z) + f(z-x) + 27 = 0 \] holds.

Answer: f(x) = ax - 9


Question: Evaluate the limit: \[ \lim _{x \rightarrow 0} \frac{12-6 x^{2}-12 \cos x}{x^{4}} \]

Answer: -\dfrac{1}{2}


Question:
Let \(A\) be a skew-symmetric matrix such that \(A^TA = I\). Determine the value of \(A^{4n-1}\) for \(n \in \mathbb{N}\). Choose from the following options:

(A) \(-A^T\)
(B) \(I\)
(C) \(-I\)
(D) \(A^T\)

Answer: D


Question: Evaluate the integral \( \int_0^{+\infty} \frac{\sin^4{x}}{x^4} \, dx \).

Answer: \dfrac{\pi}{3}


Question: Given a finite set of vectors $S_{n,k}$, where each vector has dimension $n$ with exactly $k$ positive ones and $n-k$ zeroes, and $k < \frac{n}{2}$, consider $n-1$ linearly independent vectors $u_1, \ldots, u_{n-1}$ from $S_{n,k}$ forming a matrix $A$. What is the minimal number of independent vectors remaining in $S_{n,k}$ that can be added to $A$ to increase its rank to $n$?

Answer: 1


Question: Let X be a random variable over the positive integers \( \{1, 2, 3, \ldots\} \) such that for each positive integer \( i \), \( \Pr[X = i] = 2^{-i} \). Write an analytical expression for the entropy \( H(X) \) and compute its value.

Answer: 2


Question: Is it consistent with ZFC that there exists a non-identity automorphism of $P(\omega_1)/\mathrm{NS}_{\omega_1}$?

Answer: Yes


Question: Evaluate the limit \( \lim\limits_{n \to \infty}\frac{2^{-n^2}}{\sum\limits_{k=n+1}^{\infty} 2^{-k^2}} \).

Answer: \infty


Question: Let $V$ be a finite-dimensional vector space, and let $X$ and $Y$ be subspaces of $V$. Suppose that for every subspace $W$ of $V$ such that $X + W = V$, it follows that $Y + W = V$. Can we conclude that $X \subseteq Y$?

Answer: Yes


Question: Suppose 55% of a large population of voters favor candidate A. Determine the minimum size of a random sample needed to ensure a 99% probability that the majority of voters in the sample will favor candidate A.

Answer: 556


Question: Frank alternates between flipping two weighted coins. The first coin has a \(\frac{2}{3}\) chance of landing heads and a \(\frac{1}{3}\) chance of landing tails. The second coin has a \(\frac{1}{4}\) chance of landing heads and a \(\frac{3}{4}\) chance of landing tails. Starting with the first coin, what is the probability that Frank sees two heads in a row before he sees two tails in a row?

Answer: \dfrac{13}{33}


Question: Consider a bit string $x = x(1), \\dots , x(n)$ containing exactly $m$ occurrences of 1. We perform the following operation on $x$: choose a random pair of indices $(i,j)$ and swap $x(i)$ and $x(j)$, resulting in a new string $x'$. If $i = j$, the string remains unchanged. Let $X_1 = x$, and let $X_2, \\dots, X_N$ be obtained by such a sequence of operations, where $X_{r+1}$ is derived from $X_r$. Show that for each index $i$, the probability $P(X_N(i) = 1)$ approaches $\frac{m}{n}$ as $N$ approaches infinity.

Answer: \dfrac{m}{n}


Question: Find the value of $k$ such that \( \csc \frac{\pi}{32} + \csc \frac{\pi}{16} + \csc \frac{\pi}{8} + \csc \frac{\pi}{4} + \csc \frac{\pi}{2} = \cot \frac{\pi}{k} \).

Answer: 64


Question: In triangle $ABC$, which is an acute triangle, points $D$, $E$, and $F$ are located on sides $BC$, $CA$, and $AB$, respectively. The line segment $AD$ is perpendicular to $BC$, $AE = EC$, and $CF$ bisects angle $ACB$ internally. The line $CF$ intersects $AD$ and $DE$ at points $M$ and $N$, respectively. Given that $FM = 2$ cm, $MN = 1$ cm, and $NC = 3$ cm, find the perimeter of triangle $ABC$. Provide your answer in centimeters.

Answer: 12\sqrt{3}


Question: Let $(X,d)$ be a metric space with an increasing sequence of subsets $A_n$ such that $A = \bigcup_n A_n$ is compact. Determine if it is possible for $A \setminus A_n$ to be compact for all finite $n$. Provide a justification for your answer.

Answer: Yes


Question: Is it true that the multivariate normal distribution \( N([s_1, s_2] | [0,0], [[\sigma, 0],[0, \sigma]]) \) can be decomposed as the product of its marginals, \( N(s_1 | 0, \sigma) \times N(s_2 | 0, \sigma) \)?

Answer: Yes


Question: Let \( P(x) = x^3 - ax^2 + x - 1 \), where \( a \) is a positive integer. If the sum of the cubes of the roots of \( P \) is 113, find the sum of the fourth powers of the roots of \( P \).

Answer: 547


Question: Evaluate the limit: $$ \lim_{(n,r) \rightarrow (\infty, \infty)} \frac{\prod\limits_{k=1}^{r} \left( \sum\limits_{i=1}^{n} i^{2k-1} \right)}{n^{r+1} \prod\limits_{k=1}^{r-1} \left( \sum\limits_{i=1}^{n} i^{2k} \right)} $$

Answer: 0


Question: Let $E$ be a complex Hilbert space. Let $A \in \mathcal{L}(\mathcal{H})$ and $B \in \mathcal{L}(E)^+$. Assume that $B^{1/2}A = AB^{1/2}$. Is it true that $AB = BA$?

Answer: Yes


Question: Evaluate the integral \( \oint_C \frac{\cos(z)}{e^{iz}+1} \mathrm{d}z \) using the Residue Theorem, where \( C:|z+3\pi|=2 \).

Answer: 2\pi


Question: Given a natural number \(b > 3\), determine the largest divisor of the set \(\{a^2 - 1 \mid a \ge b, \ 2 \nmid a, \ 3 \nmid a\}\).

Answer: 24


Question: Calculate the expected value of the square of the distance between two randomly chosen points in a unit square. Let $D$ denote the distance between the points. Find $E[D^2]$.

Answer: \dfrac{1}{3}


Question: Evaluate the integral \( \left| \int_{-9}^9 \lfloor \ln(x+\sqrt{1+x^2}) \rfloor \right| \), where \( |\cdot| \) represents the absolute value and \( \lfloor \cdot \rfloor \) represents the greatest integer function.

Answer: 9


Question: Determine the minimum number of distinct pairs \((x_i, y_i)\) needed to verify that a function \(f(x, y)\) is identical to the quadratic polynomial \(g(x, y) = Ax^2 + Bxy + Cy^2 + Dx + Ey + F\).

Answer: 6


Question: Find the integer value of $x$ that maximizes the function \( f(x) = a\left(1-\dfrac{a}{x}\right)\left(1-\dfrac{a}{bx}\right)^{n-2} \) given the constraints: \( a \in \{1, 1/4, 1/2, 3/4\} \), \( 50 \leq n \leq 500 \), \( 5 \leq b \leq 44 \), and \( 5 \leq x \leq 64 \). Use the derived formula \( x = \dfrac{a(n-1)}{n+b-2} \) as a starting point, and ensure that the final value of \( x \) is an integer within the specified range.

Answer: 64


Question: Consider a uniform random tournament with \(n\) vertices. For any two vertices \(x\) and \(y\), an edge is drawn from \(x\) to \(y\) with probability 0.5, and from \(y\) to \(x\) otherwise. Let \(S\) be the set of all out-degrees, with \(s_1\) as the largest element and \(s_2\) as the second largest (or equal to \(s_1\) if \(S\) is a singleton). Given a constant \(c\in (0,1)\), determine \(\lim_{n\rightarrow\infty}\text{Pr}[s_1-s_2<cn]\].

Answer: 1


Question: Find the limit: \( \lim_{x \to -\infty} y(x) \) where the function \( y(x) \) is defined by the equation \( y(x)e^{y(x)} = e^{x+1} \) for all \( x \in \mathbb{R} \).

Answer: 0


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^\infty 2^{n^2} \, x^{n!} \).

Answer: 1


Question: Find the limit of the sequence \( x_n = \prod_{j=2}^{n} \left(1 - \frac{2}{j(j+1)}\right)^2 \) as \( n \to \infty \).

Answer: \dfrac{1}{9}


Question: Given that $2f(x) + f(-x) = \frac{1}{x}\sin(x - \frac{1}{x})$, find the value of the integral: $$\int_{e^{-1}}^{e} f(x) \, \mathrm{d}x.$$

Answer: 0


Question: Let $p$ be a positive integer. Determine whether the following statement is true: If $2p+1$ divides $\left(\binom{2p}{p}+(-1)^{p-1}\right)$, then $2p+1$ is a prime number.

Answer: True


Question: Find the volume of the solid that lies under the surface \( z = x^2 + y^2 \), above the \( x \)-\( y \) plane, and inside the cylinder defined by \( x^2 + y^2 = 2x \).

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


Question: Let $U_1, U_2, \ldots, U_n$ be independent and identically distributed random variables, each following a uniform distribution $U(0, 1)$. Find $\lim_{n\to\infty} P(U_1 + U_2 + \cdots + U_n \le 3n/4)$. Provide your answer as a probability value.

Answer: 1


Question: Determine the values of \( p \) for which the limit \( \lim_{(x,y)\to (0,0)} \frac{\left|x\right|\cdot\left|y\right|^p}{x^2 + y^2} \) exists.

Answer: p > 1


Question: Calculate the value of \( S\left(S\left(2020^{2021}\right)\right) \), where \( S(n) \) denotes the sum of the digits of the integer \( n \).

Answer: 7


Question: Find the function $M(x, y)$ such that the differential equation $M(x, y) + (3xy^2 + 20x^2y^3)y' = 0$ is exact.

Answer: y^3 + 10xy^4


Question: Determine whether the following statement is true: For a smooth geometrically connected scheme $X$ over a field $k$ of characteristic 0, and a finite algebraic group $F$ over $k$, is $H_{et}^1(X,F) = H^1(\pi_1^{et}(X), F(\bar{k}))$?

Answer: True


Question: Let $g$ be a continuous function such that $g(1) = 1$ and for all real numbers $x$ and $y$, the equation $g(x + y) = 5g(x)g(y)$ holds. Determine the function $g(x)$.

Answer: 5^{x - 1}


Question: Evaluate the limit: \[ \lim _{x\to \infty }\frac{\left(x!\right)^3\left(3\left(x+1\right)\right)!}{\left(\left(x+1\right)!\right)^3\left(3x\right)!} \]

Answer: 27


Question: Calculate the second partial derivative \( \frac{\partial^2 f}{\partial x \partial y}(0,0) \) for the function \( f(x,y) = \exp(x^2 + xy + y^2) \).

Answer: 1


Question: Let \( C = \partial D_1(\mathbf{i}/2) \), which is the boundary of the disc with center \( \mathbf{i}/2 \) and radius \( 1 \). Compute the integral \( \int_C \frac{dz}{z^2+1} \).

Answer: \pi


Question: Determine the number of ways to arrange 6 red roses and 4 white roses in a garland such that no two white roses are adjacent.

Answer: 15


Question: Find the value of $2000! \mod 2003$. Justify each step in your solution.

Answer: 1001


Question: Evaluate the limit: \[ \lim_{(x,y)\to(0,0)} {(|x| + |y|) \ln{(x^2 + y^4)}} \]

Answer: 0


Question: Find the last digit of the number \(625703^{43898^{614961^{448629}}}\) using Euler's theorem.

Answer: 1


Question: Determine whether the extension \( \mathbb{Q}(e^{i2\pi/7})/\mathbb{Q} \) is a Galois extension.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{2\pi} f(z_0 + re^{i\theta}) e^{i\theta} \, d\theta \) where \( f \) is an entire function.

Answer: 0


Question: Compute the Euler characteristic \(\chi(\mathbb{C}\mathrm{P}^2)\) using techniques from differential topology.

Answer: 3


Question: Let $a, b, c, d$ be integers such that $\gcd(a, b, c, d) = 1$. Assume that the Diophantine equation $ax^2 + bxy + cxz + dyz - x = 0$ has a non-zero solution. Can we assert that it admits infinitely many solutions?

Answer: Yes


Question: Evaluate the integral \( \int_{-1}^1 \frac{e^x}{\sqrt{1-x^2}} \, dx \).

Answer: \pi I_0(1)


Question: Let \(X_1, X_2, \ldots, X_n, \ldots\) be independent random variables, where each \(X_n\) is uniformly distributed on \([0, n]\). Find a sequence \((a_n)\) such that \(\frac{1}{a_n} \sum_{k=1}^n X_k^2 \to 1\) in probability.

Answer: \dfrac{n^3}{9}


Question: Suppose $f_n: X \rightarrow \mathbb{R}$ are measurable functions such that every subsequence of $\{f_n\}$ has a subsequence that converges to a function $f$ almost everywhere. Does this imply that the sequence $f_n$ converges to $f$ almost everywhere?

Answer: Yes


Question: Compute the sum \( \sum_{n=0}^\infty \frac{n^2}{2^n} \).

Answer: 6


Question: Find all real roots of the polynomial equation: \[ (2x-5)^{2017}+(2x-5)^{2015}+(x-4)^{2017}+(x-4)^{2015}+3x-9 = 0. \]

Answer: 3


Question: Determine whether the sequence \( x_n = n(e^{\frac{2\pi i}{n}}-1) \) converges in \( \mathbb{C} \) with the usual norm. If it converges, find the limit.

Answer: 2\pi i


Question: Let $H_0^1$ be the space of all functions $g:[0,1]\to \mathbb{R}$ such that $g'$ is in $L^2[0,1]$ and $g(0)=g(1)=0$. Given that $H_0^1$ is a Hilbert space, and let $f\in C[0,1]$. Suppose $u\in C^{(2)}[0,1] \cap H_0^1$ satisfies the equation \( \int_0^1 u'v'\,dx = \int_0^1 fv\,dx \) for all $v\in H^1_0$. Prove that $-u'' = f$.

Answer: -u'' = f


Question:
Evaluate the Lebesgue integral of the function \( f(x) \) over the interval \([0,1]\), where \( f(x) \) is defined as follows:
\[
f(x) = \begin{cases} 9^n, & x \in \mathbb{Q}^c \\ 0, & x \in \mathbb{Q} \end{cases}
\]
Here, \( n \) is the number of zeroes immediately after the decimal point in the decimal representation of \( x \).

Answer: 9


Question: Find the product $ab$ if the equation of the line joining the complex numbers $-5 + 4i$ and $7 + 2i$ can be expressed in the form $az + b \overline{z} = 38$, where $a$ and $b$ are complex numbers.

Answer: 37


Question:
Find the radius of convergence for the power series:
\[ f(z) = \sum^{ \infty}_{j=0} 2^{j} z^{j^2} \]

Answer: 1


Question: Let \( f:[0,1] \to \mathbb{R} \) be a continuous function satisfying the equation \( \int_{0}^{1} f(x) \, dx = \frac{1}{3} + \int_{0}^{1} \left(f\left(x^{2}\right)\right)^{2} \, dx \). Determine the value of \( f\left(\frac{1}{4}\right) \).

Answer: \dfrac{1}{2}


Question: Evaluate the double integral \( \int \int_R xy \, dA \), where \( R \) is the square with vertices \((0,0), (1,1), (2,0), (1,-1)\). Use an appropriate change of variables to find the value of the integral.

Answer: 0


Question: Is the Sobolev space $H_0^1(\Omega)$, where $\Omega$ is a bounded 1D interval, a Hilbert space?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{z \to 0} \frac{z^2 - \overline{z}^2}{|z|} \] where \( z = x + iy \).

Answer: 0


Question: Determine whether the formula \( \sum_{k=0}^{2n}(-1)^k{2n\choose k}2^kF_{k+1}=5^n \) holds for all integer values of \( n \), where \( F_n \) is the \( n^{th} \) Fibonacci number.

Answer: 5^n


Question: Solve for $w$ in the equation: $2we^{w^2} - 2e^w = 0$.

Answer: 1


Question: Find the kernel of the homomorphism \( A[Y] \to A_a \) defined by sending \( Y \) to \( 1/a \), where \( a \) is an element of \( A \). Determine the ideal \( I \) in \( A[Y] \) that forms the kernel of this homomorphism.

Answer: (aY - 1)


Question: Find the smallest integer \( n > 1 \) such that \( \frac{1}{n}(1 + 2^2 + 3^2 + \ldots + n^2) \) is a perfect square.

Answer: 337


Question:
Compute the index of the unit circle with respect to the vector field given by the system of differential equations:
\[\begin {cases}
\frac{dx}{dt} = 2x^2-1\\
\frac{dy}{dt} = 2xy
\end{cases}\]

Answer: 2


Question: Evaluate the integral \( \int_0^\infty \frac{1}{(1+x^2)(1+x^{2020})} \, dx \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the integral \( \int_0^{\pi/2} \frac{1+2\cos x}{(2+\cos x)^2} \, dx \).

Answer: \dfrac{1}{2}


Question: Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and fix a Weierstrass equation for $E$ with coefficients in $\mathbb{Z}$. Are there infinitely many primes $p \in \mathbb{Z}$ such that the reduced curve $E/\mathbb{F}_p$ has Hasse invariant $1$?

Answer: Yes


Question: Calculate the value of \( 7^{7^{7^{7^{7^{7^7}}}}} \mod 100 \).

Answer: 43


Question:
Let \( \alpha(x) = \begin{cases} 0 & x=0, \\ 2 & 0<x\leq 1, \\ 1 & 1<x \leq 2 \end{cases} \).

Let \( f(x) \) be continuous on \([0,2]\). For the partition \( P = \{0,0.5,1,1.5,2\} \), find the sum \( S(P,t) \) approximating the integral \( \int_{0}^{2} f(x) \, d\alpha(x) \) for every \( t \) for which this sum is defined. What is the value of the integral? Justify your answer by referring to the definition of the integral.

Answer: 2f(0) - f(1)


Question: Determine whether the following statement is true or false: Let $f(x) = \lim f_n (x)$, where $f_n(x)$ is a sequence of functions. If each $f_n(x)$ is bounded on $[a, b]$ and $f(x)$ is also bounded on $[a, b]$, then $f_n \to f$ uniformly on $[a, b]$. Here, $a$ and $b$ are real numbers.

Answer: B


Question: Let $V$ be an infinite-dimensional vector space over the field $\mathbb{F}$, where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$. If the dimension of $V$ is $\alpha$, a cardinal number, determine the dimension of $V^*$, the algebraic dual of $V$. Express your answer in terms of $\alpha$.

Answer: 2^\alpha


Question: Let $X \geq 0$ be a random variable, $q>0$, and $\mathbb{E}[X^q]<1$. Prove that $y^q P(X>y) \to 0$ as $y \to \infty$.

Answer: 0


Question: Determine the number of solutions to the equation \((\tan x)\sin^2(2x) = \cos x\) within the interval \(-2\pi \leq x \leq 2\pi\).

Answer: 4


Question: Consider a sequence of subdivisions \(\{x_i^n\}_{i=0}^{m_n-1}\) of the interval \([0,1]\) such that \(\max_{i=1,...,m_n-1}\Delta x_i^n\to 0\). Determine whether the following limit always holds: \[ \lim_{n\to \infty }\sum_{i=0}^{m_n-1}o(\Delta x_i^n)=0. \]

Answer: 0


Question: In triangle $ABC$, points $D$, $E$, and $F$ are the first trisection points of sides $BC$, $CA$, and $AB$, respectively. Given that the area of triangle $ABC$ is 1, find the area of triangle $GHI$, where $G$, $H$, and $I$ are the points where cevians $AD$, $BE$, and $CF$ intersect. Assume that each cevian is divided in the ratio $1:3:3$. What is the area of triangle $GHI$?

Answer: \dfrac{1}{7}


Question: Evaluate the limit of the Riemann sum: \[ \lim_{n \to \infty} \frac{1}{n^3} \sum_{r=1}^n \frac{r^4}{(2r-1)(2r+1)} \]

Answer: \dfrac{1}{12}


Question: Determine the value to which the series \( \sum_{i = 1}^{\infty} \frac{1}{i(i-1)!} \) converges.

Answer: e - 1


Question: Given independent uniform random variables \(U_1, U_2, U_3, U_4\) where \(U_k \sim [-k\theta, k\theta]\), and the observed values \(u_1 = -3.8, u_2 = 8.2, u_3 = -12.6, u_4 = 15.6\), find the maximum likelihood estimate of \(\theta\).

Answer: 4.2


Question: Find the limiting distribution of \( \frac{Y_n - n}{\sqrt{2n}} \) as \( n \to \infty \), where \( Y_n \sim \chi^2(n) \), using moment generating functions.

Answer: N(0,1)


Question: Compute the limit: \[ \lim_{x \to 0} \frac{(e^x - 1 - x)^2 \cos x}{x(\sin x - x)} \] Choose the correct answer from the following options: (a) \( \frac{2}{3} \) (b) \( -\frac{3}{2} \) (c) \( \frac{3}{2} \) (d) \( -3 \)

Answer: B


Question: Evaluate the limit: $$\lim_{x\to \infty} \left(\cosh^{-1}(x^{3}) + \coth^{-1}(\sqrt{x^{2}+1}) - 3\sinh^{-1}(x)\right)$$

Answer: -2 \ln 2


Question: Find all real numbers \( x \) such that \( x[x[x[x]]] = 88 \), where \([\cdot]\) denotes the floor function.

Answer: \dfrac{22}{7}


Question: Calculate the probability that in a random selection of 13 cards from a standard 52-card deck, there is at least one denomination where all 4 cards of that denomination are present.

Answer: 0.0342


Question: In a group of 15 people, there will be five murders. Each murder involves one killer and one victim, except for the third murder, which involves one killer and two victims. After each murder, the killer and victim(s) are removed from the group. In the third murder, one of the victims can also be the killer, but the other victim must be different. The order of victims in the third murder matters. Calculate the total number of possible combinations of killers and victims for all five murders.

Answer: 76281004800


Question: Determine whether the limit \( \lim_{n \to \infty}{\frac{p_n}{p_{n+1}}} \) is less than 1, where \( p_n \) denotes the nth prime number.

Answer: No


Question: Evaluate the integral \( \int_0^1 f(t) \, dt \) where \( f(x) \) is an integrable function on \([0,1]\) that satisfies \( f(x) = x \) for \( x = \frac{n}{2^m} \), with \( n \) being an odd positive integer and \( m \) a positive integer. Calculate \( \int_0^1 f(t) \, dt \).

Answer: \dfrac{1}{2}


Question: Evaluate \( \lim_{n \to \infty} S_n \) where \( S_n = \sum_{i=1}^n \frac{1}{a_i} \) and \( a_k = \left( \prod_{i=1}^{k-1} a_i \right) + 1 \).

Answer: 1


Question: Consider the probability space \(([0,1]^2, \mathcal{B}[0,1], Leb_2)\). Let \(X: [0,1]^2 \rightarrow \mathbb{R}\) and \(Y: [0,1]^2 \rightarrow \mathbb{R}\) be random variables defined by \(X(\omega_1, \omega_2) = \omega_1 + \omega_2\) and \(Y(\omega_1, \omega_2) = \omega_2 - \omega_1\). Find the value of the joint cumulative distribution function \(F_{X,Y}(0.58,0.35)\).

Answer: 0.154975


Question: In a second-countable Hausdorff space \( \langle X, \mathcal{T} \rangle \), let \( \phi : 2^X \to [0,+\infty] \) be an outer regular outer measure. Determine whether all open subsets of \( X \) are Caratheodory-measurable by \( \phi \).

Answer: No


Question: Given two matrices $A$ and $B$ of size $n \times m$ with integer entries, determine the minimum number of moves required to make them identical. In each move, you can choose one of the matrices and either increment all elements of a selected row or column by 1. If it is impossible to make the matrices identical, state that it is impossible. For example, for $A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 2 \\ 0 & 0 \end{bmatrix}$, the matrices can be made identical in 2 moves.

Answer: 2


Question:
Consider the sequence \(T_n\) defined as follows:
\[ T_n = \frac{\sum\limits_{m=1}^{\binom{n}{2}}{\binom{u_{n,m}}{2}}}{\binom{g_n}{2}} \]
where \(G_n\) denotes the set of non-isomorphic simple graphs with \(n\) vertices, \(|G_n| = g_n\), and \(U_{n,m}\) denotes the set of non-isomorphic simple graphs with \(n\) vertices and \(m\) edges, \(|U_{n,m}| = u_{n,m}\). The sequence \(T_n\) represents the probability that two graphs among graphs with \(n\) vertices have the same number of edges. Determine whether the limit \(\lim_{n\rightarrow \infty} T_n\) exists and, if so, whether it is not equal to zero.

Answer: 0


Question: Find the 6th derivative of the function \( f(x) = \frac{\cos(5x^2) - 1}{x^2} \) at \( x = 0 \).

Answer: 18750


Question: Given two differentiable functions $f(x)$ and $g(x)$ on $\mathbb{R}^+$ such that $xf'(x) + g(x) = 0$ and $xg'(x) + f(x) = 0$ for all $x \in \mathbb{R}^+$, and $f(1) + g(1) = 4$, find the value of $f''(2) \cdot g''(2)$. \( \text{ } \)

Answer: \dfrac{1}{4}


Question: Evaluate the contour integral \(\oint_C \cos(z) \, dz\) where \(C\) is the unit circle \(|z| = 1\) in the complex plane.

Answer: 0


Question: Does every element \( M \) in \( \mathrm{SL}(2, \mathbb{R}) \) have a neighborhood that is diffeomorphic to \( \mathbb{R}^3 \)?

Answer: Yes


Question: Determine the values of \( p > 0 \) for which the vector field \( v_p(x,y) = \left(\frac{-y}{(x^2+y^2)^p}, \frac{x}{(x^2+y^2)^p}\right) \) has an anti-derivative on some open set \( U \subseteq \mathbb{R}^2 \setminus \{(0,0)\} \). For these values of \( p \), provide an example of such an open set \( U \) and an anti-derivative of \( v_p \).

Answer: 1


Question: Evaluate the integral \( \int_{-\infty}^{+\infty} \frac{\exp(1-ix)}{x^2+1} \, dx \) and determine if its value is \( \pi \).

Answer: \pi


Question:
Compute the operator norm of the linear functional $F$ defined on $C[-1,1]$ by:
$$
F(u) = \int_{0}^{1} u(x) \, dx - \int_{-1}^{0} u(x) \, dx
$$
where $||F|| = \sup_{ ||u|| = 1} \{ |F(u)| \}$ and $||u||_{\infty} = 1$ for $u \in C[-1,1]$. Determine the operator norm of $F$. \( \boxed{} \)

Answer: 2


Question: Given that $y = x$ is a solution to the differential equation \((x^2-x+1) \frac{d^2y}{dx^2} - (x^2+x)\frac{dy}{dx} + (x+1)y = 0\), find a second linearly independent solution.

Answer: (x - 1)e^x


Question: Let $k$ be a field and $k(Y)$ be the field of rational functions in the variable $Y$. If $f \in k[X,Y]$ is an irreducible polynomial, is it true that $f$ is also irreducible as a polynomial in $k(Y)[X]$? Provide a justification for your answer.

Answer: Yes


Question: Find the value of \(x\) that maximizes the probability function \(P_{\text{B}}(x) = \binom{n}{x}p^x q^{n-x}\), where \(p + q = 1\) and \(x \in \{0, 1, 2, \ldots, n\}\). Assume \(n\) is large and \(x\) is a continuous variable.

Answer: np


Question: Determine the value of \( c \in [3, \infty) \) for which there exists a measure space \((X, \mathcal{S}, \mu)\) such that the set of measures \( \{ \mu(E) : E \in \mathcal{S} \} = [0,1] \cup [3,c] \).

Answer: 4


Question: Determine whether there exists a non-zero vector \( w \in \mathbb{R}^4 \) such that \( Aw = 0 \) and \( Bw = 0 \), given that \( A, B \in M_{5 \times 4}(\mathbb{R}) \), \( Av = 0 \) and \( Au = 0 \) for \( v = \begin{pmatrix} 3 \\ -1 \\ 3 \\ 1 \end{pmatrix} \) and \( u = \begin{pmatrix} 2 \\ -1 \\ 1 \\ 1 \end{pmatrix} \), and \( \text{spanRow}(B) = \{(1,2,3,4),(5,6,7,8)\} \).

Answer: Yes


Question: Does the Banach space $l^1(\Gamma)$, where $\Gamma$ is an uncountable index set, have a separable pre-dual space?

Answer: No


Question: Let $V$ be a finite-dimensional real vector space, and let $f$ be a symmetric bilinear form on $V \times V$. The signature of $f$, denoted $s(f)$, is defined as $a-b$, where $a$ is the number of 1s and $b$ is the number of -1s on the diagonal when $f$ is represented by a diagonal matrix. Suppose $c, d \in V$ are such that $f(c, c) = 0$ and $f(c, d) = 1$. Prove that there exists a subspace $W \leq V$ such that $V$ is the direct sum of $\text{span}(c, d)$ and $W$, and that $s(f|_W) = s(f)$. \[ \text{Hint: Consider the orthogonal complement of $\text{span}(c, d)$ with respect to $f$.} \]

Answer: s(f|_W) = s(f)


Question: Consider the set of all square numbers of natural numbers. Using the axiom of specification, define the set \( A = \{ n \in \mathbb{N} \mid (\exists m)(m \in \mathbb{N}) \wedge n = m^2 \} \). Using the axiom of replacement, define the set \( B = \{ n^2 \mid n \in \mathbb{N} \} \). Determine whether \( A = B \).

Answer: A = B


Question: Calculate the surface integral \( \int \int_S (2z^2 - x^2 - y^2) \, dS \), where \( S \) is the surface defined by the intersection of the cone \( z = \sqrt{x^2 + y^2} \) and the cylinder \( (x-1)^2 + y^2 = 1 \).

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


Question: Determine whether the series \( S = \sum_{r=1}^\infty \tan^{-1}\left(\frac{2r}{2+r^2+r^4}\right) \) is finite. If it is finite, find the value of \( S \).

Answer: \dfrac{\pi}{4}


Question: Consider a game with 26 red cards and 26 black cards shuffled and placed face down on a table. A host turns up the cards one at a time. You can choose to stop the game at any point, even before any cards are revealed. Once you stop, the next card is turned up: if it is red, you win $1; if it is black, you lose $1. What is the expected payoff of this game?

Answer: 0


Question: Determine whether the following statement is true or false: Every positive rational number \( q \) can be expressed as \( q = \sum_{i=0}^{k} \frac{1}{n_i} \), where \( n_i \) and \( k \) are positive integers, and \( n_i \neq n_j \) if \( i \neq j \).

Answer: True


Question: Evaluate the limit: \[ \lim_{x\to e^+} (\ln x)^{\frac{1}{x-e}} \]

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


Question: Determine the sequence $s(k,n)$, which represents the number of sides of the intersection of a unit-radius regular polygon $P_k$ with $k$ sides and a rotating unit-radius regular polygon $P_n$ with $n \ge k$ sides, as the angle of rotation $\theta$ varies from $0$ to $2\pi$. Provide the sequence $s(k,n)$ for all $n \ge k$.

Answer: 2k


Question:
Compute the expectation \( \mathbb{E}(n^2) \) for a random walk given by the expression:
\[ \mathbb{E}(n^2) = \sum \limits_{n= -N}^{N} \left( \frac{1}{2} \right)^N \frac{N!}{(\frac{1}{2}(N+n))!(\frac{1}{2}(N-n))!} n^2 \]

Answer: N


Question: Determine the number of solutions to the equation \(e^x = x^3\).

Answer: 2


Question: Find the natural number \( n \) such that \( 2^{2^{2^{2n-1}}} \equiv 16 \pmod{127} \).

Answer: 1


Question: Approximate the integral \( \int_0^1 e^{x^2} \, dx \) using the Maclaurin series expansion of \( e^{x^2} \). Ensure that the approximation error is less than 0.0005 from the actual value.

Answer: 1.4625


Question: A polygon has interior angles such that the difference between any two consecutive angles is 5 degrees. If the smallest angle is 120 degrees, determine the number of sides of the polygon.

Answer: 9


Question: Solve for $x$ in the equation $3^x \cdot x^3 = 1$. Provide your answer to three decimal places.

Answer: 0.758


Question: Let $k > 1$ be an integer. Consider the vector $e \in \mathbb{Z}^k$ that has 1 in the first coordinate and 0 in every other coordinate. Determine the cardinality of the stabilizer of $e$ in the special linear group $\mathrm{SL}_k(\mathbb{Z})$ with respect to the natural action. In other words, how many elements of $\mathrm{SL}_k(\mathbb{Z})$ have $e$ as the first column?

Answer: \infty


Question: Calculate the area inside the loop defined by the equation \((x+y)^3 = axy\) using a double integral, where the parameter \(a\) is positive.

Answer: \dfrac{a^2}{60}


Question: Find an element in $\mathbb{Z}[\sqrt{-17}]$ that can be expressed both as a product of two irreducible elements and as a product of three irreducible elements.

Answer: 18


Question: Determine if the function \( f: \mathbb{C} \rightarrow \mathbb{C} \) given by \( f(z) = z^2 + z|z|^2 \) is differentiable at \( z = 0 \). If it is, find \( f'(0) \). Additionally, investigate whether \( f^{(n)}(0) \) exists for \( n \geq 2 \).

Answer: 0


Question: Find the function $f(x)$ such that for all $n$, the equation \( 2 f(n) + \frac{1}{3}f\left(\frac{1}{n}\right) = 12 \) holds.

Answer: \dfrac{36}{7}


Question: Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$. If $f(0)=0$, find the maximum value of $f(5)$. \( \boxed{0} \)

Answer: 0


Question: Evaluate the infinite series \( \sum_{k=1}^{\infty} k^2 \frac{2^{k-1}}{3^k} \).

Answer: 15


Question: Is it true that for functions $(f_i)$ on $L^p(\mathbb{R}^n)$, where $p \geq 1$, the following inequality holds: \[ \left\| \sum_{i=1}^{\infty} |f_i| \right\|_p \leq \sum_{i=1}^{\infty} \|f_i\|_p \]?

Answer: True


Question: Given a real symmetric matrix $A \in \mathbb{R}^{4 \times 4}$ with characteristic polynomial \( \chi_A(x) = x^4 - 9x^3 + cx^2 + dx + 37 \), and knowing that all eigenvalues of $A$ have the same sign, determine the signature of the matrix $A$. The signature is defined as the number of positive eigenvalues minus the number of negative eigenvalues.

Answer: 4


Question: In Geometric Algebra, if $A\tilde A=0$, does it necessarily follow that $\tilde AA=0$? Consider the properties of reversion, where $(AB)^\sim=\tilde B\tilde A$ for any multivectors $A$ and $B$, and reversion is linear, leaving scalars and vectors unchanged. Analyze whether this implication holds in both Euclidean and spacetime algebras.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x \rightarrow 0^+} \frac{\arctan(e^x+\arctan x)-\arctan(e^{\sin x}+\arctan(\sin x))}{x^3}$$

Answer: \dfrac{1}{6}


Question: Determine if there is a closed form for the integral \( I = \int_{-\infty}^{\infty} e^{-e^x-e^{-x}+x}~ dx \).

Answer: 2 K_1(2)


Question: Determine whether every W*-continuous linear functional belongs to $L^1(H)$. Is this statement true or false?

Answer: True


Question: In a [15,11] Hamming code, there are 2048 codewords. Calculate the number of binary words of length 15 that are at a Hamming distance of 1 from any of these codewords.

Answer: 30720


Question: Given two segments of lengths \(a\) and \(b\), construct a segment of length \(\sqrt{ab}\) using only a compass, without a segment of length 1 provided.

Answer: \sqrt{ab}


Question: Determine the values of $a$ for which the series \( \sum_{n=1}^{\infty} \frac{(\cos n)(\sin na)}{n} \) converges.

Answer: \mathbb{R}


Question: Evaluate the integral \( \int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \, dx \).

Answer: \dfrac{\pi}{2}


Question: Find a number \( a < 0 \) such that \( \int_a^0 \frac{e^{1/x}}{x^2} \, dx = \frac{e^{1/a}}{a^2} \).

Answer: -1


Question:
Let $A$ be an $n \times n$ self-adjoint matrix with eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$. Define the norm of a vector $x = (x_1, \ldots, x_n) \in \mathbb{C}^n$ as $\|x\| = \sqrt{|x_1|^2 + \cdots + |x_n|^2}$. Consider the polynomial $p(A) = a_0 I + a_1 A + \cdots + a_n A^n$. Determine the value of $\sup_{\|x\|=1} \|p(A)x\|$. Choose the correct option:

1. $\max \{a_0 + a_1 \lambda_j + \cdots + a_n \lambda_j^n : 1 \le j \le n\}$.
2. $\max \{|a_0 + a_1 \lambda_j + \cdots + a_n \lambda_j^n| : 1 \le j \le n\}$.
3. $\min \{a_0 + a_1 \lambda_j + \cdots + a_n \lambda_j^n : 1 \le j \le n\}$.
4. $\min \{|a_0 + a_1 \lambda_j + \cdots + a_n \lambda_j^n| : 1 \le j \le n\}$.

Answer: 2


Question: Compute the derivative of \( \text{trace}(U^T x y^T V) \) with respect to \( x \), where \( U \in \mathbb{R}^{d_x \times k} \), \( V \in \mathbb{R}^{d_y \times k} \), \( x \in \mathbb{R}^{d_x} \), and \( y \in \mathbb{R}^{d_y} \).

Answer: U V^\top y


Question: Evaluate the contour integral \( \oint_{C} \frac{e^z - 1}{z} \mathrm{d}z \) where \( C \) is the unit circle oriented counter-clockwise.

Answer: 0


Question: Calculate the limit: \[ \lim_{x\to{0^+}}\frac{\log_{\sin{x}}{\cos{x}}}{\log_{\sin{\frac{x}{2}}}\cos{\frac{x}{2}}} \]

Answer: 4


Question: Evaluate the line integral of \( \exp(2\overline{z}) \) over the unit circle \( C \) in the complex plane.

Answer: 4\pi i


Question: Find the closed form of the infinite product: \[ \prod_{k=0}^\infty \left(1+\frac{1}{2^{2^k}}\right) \]

Answer: 2


Question:
Find the product of all real roots of the polynomial equation:
\[ x^{101} - 3x^{100} + 2x^{99} + x^{98} - 3x^{97} + 2x^{96} + \cdots + x^2 - 3x + 2 = 0 \]

Answer: -2


Question: Determine the number of zeros of the polynomial \( f(z) = z^{4}+z^{3}+4z^{2}+2z+3 \) that lie in the first quadrant of the complex plane.

Answer: 0


Question: Determine whether the group $F^\times/N_{E/F}(E^\times)$ has order 2, where $F$ is a totally real number field and $E$ is a totally imaginary quadratic extension of $F$. Provide a justification for your answer.

Answer: 2


Question: Calculate the derivative with respect to \( y \) of the function \( g(y) = \int_{0}^{\frac{\pi}{y}} \frac{\tan^2(xy)}{2x} \, dx \), where \( y > 0 \).

Answer: 0


Question: Is it true that any proper submodule of a non-zero projective module is contained in a maximal submodule?

Answer: No


Question: Determine the number of non-isomorphic abelian groups of order 1764. Given that 1764 can be factored as \(2^2 \times 3^2 \times 7^2\), list the non-isomorphic abelian groups of this order.

Answer: 8


Question: A smaller cone is inscribed in a larger cone. Determine the radius of the smaller cone that maximizes its volume.

Answer: \dfrac{2}{3}R


Question: Let $M$ and $N$ be $d$-dimensional oriented Riemannian manifolds, and let $f: M \to N$ be a smooth map. Assume $M$ is compact with boundary. Does there exist a vector field $X \in \Gamma(TM)$ such that $\text{div} \, X = \det df$?

Answer: Yes


Question: Let $G$ be a compact and connected Lie group. Consider the adjoint representation of this Lie group, $Ad(g)$, with matrix components denoted by $(\alpha_{ij}(g))$. Prove that the determinant of the adjoint representation matrix is $+1$, i.e., $\det(Ad(g)) = +1$.

Answer: 1


Question: Let \( F: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) be a linear transformation such that \( F(0,0,1) = (2,-1,\alpha) \). The vectors \( (1,0,-1) \) and \( (0,1,0) \) are eigenvectors of \( F \) with associated eigenvalues \( 3 \) and \( -9 \) respectively. Determine the value of \( \alpha \) given that \( -2 \) is also an eigenvalue of \( F \).

Answer: -4


Question: Determine the Galois group of the polynomial \(x^4 - 4x^2 - 11\) over \(\mathbb{Q}\).

Answer: D_4


Question: Find all prime numbers \( p \) such that \( \sqrt{\frac{p+7}{9p-1}} \) is a rational number.

Answer: 11


Question: Evaluate the series \( \sum_{k=2}^{\infty} (k-1)k\left(\frac{1}{3}\right)^{k+1} \).

Answer: \dfrac{1}{4}


Question: Determine the number of matrices in \( GL(2, \mathbb{F}_5) \) that have a determinant of 3.

Answer: 120


Question: Evaluate the limit: $$\lim_{n\rightarrow\infty} \frac{(2n)!}{(n!)^2}$$

Answer: \infty


Question: Evaluate the limit: \[ \lim_{x \to 1} x^{\frac{1}{1-x}} \]

Answer: \dfrac{1}{e}


Question: Can every non-empty set satisfying the axioms of ZF be totally ordered?

Answer: No


Question: Determine whether the cardinality of every $T_2$, first countable, countably compact space is not more than $2^\omega$. Provide a justification for your answer.

Answer: Yes


Question: Consider a sequence of functions \((f_n(x))_n\) that converges uniformly to a function \(f(x)\) on a domain \(D\). Can there exist an \(N \in \mathbb{N}\) such that \(f_N(x)\) is unbounded? Provide a justification or an example if possible.

Answer: Yes


Question: Given the equations \( x = \min(\max(y,z+w), k-w) \) and \( z = \min(c, k-2w) \), where \( x, y, z, w, k \) are real numbers and \( w > 0 \), determine if the inequality \( z+w \le x \) is always true.

Answer: Yes


Question: Let \( f(x) \) be a non-negative continuous function on \( \mathbb{R} \) such that \( f(x) + f(x+\frac{1}{3}) = 5 \). Calculate the value of the integral \( \int_0^{1200} f(x) \, dx \).

Answer: 3000


Question: Find the integer $n > 2$ that satisfies the inequality $(\lg n)^{2^{100}} < \sqrt{n}$, where $\lg$ denotes the binary logarithm.

Answer: 2^{2^{108}}


Question: Calculate the flux of the vector field \( v = (yz, y^2z, yz^2) \) through the surface of the cylinder defined by \( x^2 + y^2 = 1 \) and \( 0 \leq z \leq 1 \). Use the divergence theorem to simplify your calculations.

Answer: 0


Question: Find the volume of the region bounded by the surfaces \( z = 5 - x^2 \), \( z = y \), and \( y = 1 \).

Answer: \dfrac{256}{15}


Question: Evaluate the infinite series: \[ \sum_{n=1}^{\infty} \cot^{-1}(2n^2) \]

Answer: \dfrac{\pi}{4}


Question: If $A_{N \times N} (x)$ is a complex $N \times N$ positive definite matrix whose components $a_{ij}(x)$ are $C^\infty$ functions, is the inverse matrix $A^{-1}$ also composed of $C^\infty$ functions?

Answer: Yes


Question: In an obtuse triangle $\triangle ABC$, let $I$ be the incenter. Among the triangles $\triangle IAB$, $\triangle IBC$, and $\triangle IAC$, one of them is similar to $\triangle ABC$. Determine the ratio of the angles of $\triangle ABC$.

Answer: 1:2:4


Question: Evaluate the limit \( \lim_{x\to0^{-}}\frac { f(-x){ x }^{ 2 } }{ \left\{ \frac { 1-\cos(x) }{ [f(x)] } \right\} } \). Given that \( f(x) \) is a real-valued continuous function except at \( x=0 \), with \( f'(x)<0 \) for negative \( x \) and \( f'(x)>0 \) for positive \( x \). Also, \( \lim_{x\to0^{+}} f(x)=2 \), \( \lim_{x\to0^{-}} f(x)=3 \), and \( f(0)=4 \). Here, \([\cdot]\) denotes the greatest integer function, and \(\{\cdot\}\) denotes the fractional part function.

Answer: 12


Question: Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point such that the distance from $P$ to the center of the circle is 3. Find the value of $PA_1^2 + PA_2^2 + \dots + PA_{11}^2.$

Answer: 143


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{\cos\left(\frac{\pi}{2\cos(x)}\right)}{\sin(\sin(x^2))} \]

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


Question: Evaluate the limit: \[ \lim_{x\to 1^+} (\ln x)^{\ln x} \]

Answer: 1


Question: Let \( p(x) \) be the polynomial of least degree with rational coefficients that has \( \sqrt[3]{7} + \sqrt[3]{49} \) as one of its roots. What is the product of all the roots of \( p(x) \)?

Answer: 56


Question: Given the function \( f(x+y) = f(x)f(y) \) for all \( x \) and \( y \), where \( f(5) = 2 \) and \( f'(0) = 3 \), find the value of \( f'(5) \).

Answer: 6


Question: If \( c = \binom{99}{0} - \binom{99}{2} + \binom{99}{4} - \binom{99}{6} + \cdots + \binom{99}{96} - \binom{99}{98} \), find \( \log_2{(-c)} \).

Answer: 49


Question: Evaluate the integral: $$ \int^{\infty}_{1} \left(3\mathrm{e}^{-\tfrac{x^2}{\sqrt{x-1}}}+x-\left|3\mathrm{e}^{-\tfrac{x^2}{\sqrt{x-1}}}-x\right|\right)\,\mathrm{d}x $$

Answer: 3


Question: Let \( X \) be a countably infinite (or larger) set with the cofinite topology. For every \( x \in X \), does there exist a family \( \xi \subset \tau \) such that \( \{ x \} = \bigcap \xi \)? If so, determine the cardinality of \( \xi \).

Answer: |X|


Question: Let $A$ and $B$ be real symmetric $n \times n$ matrices such that $AB = BA$. Is it true that $A$ and $B$ have a common eigenvector in $\mathbb{R}^n$?

Answer: Yes


Question: Calculate the limit \( \lim_{n \rightarrow \infty} \int_0^{2022\pi} \sin^n(x) \, dx \).

Answer: 0


Question:
Let \( C \) be the ellipse \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) traversed once in the positive direction. Define the function
\[
G(z) := \int_C \frac{\zeta^2-\zeta+2}{\zeta-z}\, d\zeta \quad \text{(for \( z \) inside \( C \))}
\]
Find \( G''(-i) \).

Answer: 4\pi i


Question: Determine the parameters of the subfield subcode of the Hamming code $Ham(3,4)$, which is a $[21,18,3]$ code, when intersected with $F_2^{21}$. What are the parameters for $Ham(3,4) \mid_{F_2} = Ham(3,4) \cap F_2^{21}$?

Answer: [21, 15, 3]


Question: Consider a function \( f: \mathbb{R} \to \mathbb{R} \) that is measurable with respect to the sigma-algebra \( \Sigma \), generated by the sets \([0, 1]\), \([2, 4]\), and \([3, \infty)\) in \( \mathbb{R} \). Does this function necessarily have to be measurable with respect to the Borel sigma-algebra \( \mathcal{B}(\mathbb{R}) \) of \( \mathbb{R} \)?

Answer: Yes


Question: Determine the total number of "magic" circles for a given integer \( n \). A "magic" circle is defined as a circle with \( 2^n \) items on its periphery, where each item holds either 0 or 1. For each item, consider the sequence of \( n \) items to the right. The set of these sequences must consist of all possible \( n \)-length combinations of 0 and 1. What is the number of such "magic" circles for a given \( n \)?

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


Question:
Find the closed form of the expression \( \sum\limits_{r=0}^n r^2 \binom{n}{r} p^{n-r}q^r \), where \( p > 0 \) and \( q = 1 - p \). Choose the correct option:

\( (A)\ np^2q^2 \\
(B)\ n^2p^2q^2 \\
(C)\ npq + n^2p^2 \\
(D)\ npq + n^2q^2 \)

Answer: D


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \frac{\cos^2(x)}{1+a^x} \, dx \) and show that it equals \( \frac{\pi}{2} \) for any non-zero \( a \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left[\dfrac{1}{2^{n/2}\Gamma(n/2)} \int_{n-\sqrt{2n}}^{\infty} t^{\frac{n}{2}-1}e^{-t/2}\,dt\right] \]

Answer: \Phi(1)


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty}4\sqrt{n}\sin (\pi\sqrt{4n^2+\sqrt{n}})$$

Answer: \pi


Question: Suppose that $G$ is a nonreduced affine group scheme of finite type over an algebraically closed field $k$. If $V$ is an irreducible representation of $G$, is $V$ absolutely irreducible?

Answer: Yes


Question: What is the dimension of the real numbers \(\mathbb{R}\) as a vector space over itself?

Answer: 1


Question: Determine the norm of the linear operator \( A: l_2 \rightarrow l_2 \) defined by \( A(x_1, x_2, \ldots) = (x_1, \frac{1}{2}x_2, \ldots, \frac{1}{2^{n-1}}x_n, \ldots) \).

Answer: 1


Question: Consider the sequence $a_n=\left(\underbrace{\frac{1}{n}, \ldots, \frac{1}{n}}_{n \text{ elements}}, 0, 0, 0, 0, \ldots \right)$ for any $n\in \mathbb{N}$. Is there a norm on a sequence space such that $a_n$ converges to the zero sequence?

Answer: Yes


Question: Determine all polynomials \(P(x)\) with real coefficients such that there exists a positive integer \(n\) for which the equation \[P(x + \frac{1}{n}) + P(x - \frac{1}{n}) = 2P(x)\] holds for all \(x\).

Answer: P(x) = ax + b


Question: Evaluate the limit: \[ \lim _{x\to 0}\left(\frac{\sqrt[5]{1+3x^4}-\sqrt{1-2x}}{\sqrt[3]{1+x}-\sqrt{1+x}}\right) \] without using L'Hospital's rule.

Answer: -6


Question:
Calculate the following limit superior:
\[ \limsup_{n\to\infty}\left(\frac{A^n}{n}+\frac{B^n}{n^2}\right)^{\frac{1}{n}} \]
where \( A, B > 0 \).

Answer: \max\{A, B\}


Question: Are there infinitely many positive integers \(N\) such that \(N\) is relatively prime to two given integers \(p > 1\) and \(q > 1\), and there exist positive integers \(k\) and \(l\) such that \(p^k \equiv q \pmod{N}\) and \(q^l \equiv p \pmod{N}\)?

Answer: Yes


Question:
An airline has collected an i.i.d. sample of 10,000 flight reservations and found that 5% of passengers did not show up for their flights. The airline decides to sell 100 tickets for a flight that can hold only 99 passengers. Consider the following sampling process:

- Sample 10,100 passenger show-up events independently at random according to an unknown distribution.
- Split them into 10,000 passengers in the collected sample and 100 passengers booked for the 99-seat flight.

Bound the probability of observing a sample of 10,000 with 95% show-ups and a 99-seat flight with all 100 passengers showing up, following the above sampling protocol. Provide a bound close to 0.0062.

Answer: 0.0062


Question: Evaluate the integral \( \int_0^1 \frac{x^{m-1} + x^{n-1}}{(1+x)^{m+n}} \, dx \) in terms of the Beta function.

Answer: B(m, n)


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int_{\mathbb{R}} e^{x-nx^2} \,dx \]

Answer: 0


Question: Let $f(z) = z^3 e^{z^2}$ for $z \in \mathbb{C}$, and let $\Gamma$ be the circle $z = e^{i\theta}$, where $0 \leq \theta \leq 4\pi$. Using the argument principle, determine the value of \( \frac{1}{2\pi i} \oint_{\Gamma} \frac{f'(z)}{f(z)} \, dz \).

Answer: 6


Question: Evaluate the limit: \[ \lim_{x \rightarrow 0} -2\int_{0}^{x^{2}} \cos t^{2} \, dt \] and express the result in terms of a Maclaurin series expansion up to the order of $x^2$. Ignore terms of order $o(x^4)$.

Answer: -2x^2


Question: Is a closed subset of isolated points in a compact set necessarily finite? Provide a proof or counterexample to support your answer.

Answer: Yes


Question:
Calculate the marginal probability \( P(X_1=27, X_2=20) \) given the joint probability distribution:
\[
f(X_1, X_2 \mid p_1, p_2) = p_1^{x_1} (1 - p_1)^{(n_1-x_1)}p_2^{x_2}(1-p_2)^{(n_2-x_2)},
\]
where \( p_1 \sim \text{Unif}(0,1) \) independently \( p_2 \sim \text{Unif}(0,1) \), \( n_1=34 \), and \( n_2 = 56 \).

Answer: \dfrac{1}{1995}


Question: Suppose \( f:D(0,1) \longrightarrow \mathbb{C} \) is a holomorphic function, where \( D(0,1) = \{z \in \mathbb{C} \mid |z|<1\} \), and assume the maximum \(|f(z)| \le 2\). Estimate \(|f^{(3)}(i/3)|\) using Cauchy's estimate.

Answer: \dfrac{81}{2}


Question: Evaluate the limit: \[ \lim_{n\to\infty} n \int_0^1 nx^{n-1}\left(\frac{1}{1+x}-\frac{1}{2}\right)\mathrm{d}x. \]

Answer: \dfrac{1}{4}


Question: Find the remainder when \((a+b+c)^{333} - a^{333} - b^{333} - c^{333}\) is divided by \((a+b+c)^3 - a^3 - b^3 - c^3\).

Answer: 0


Question: Determine the number of ways to divide 90 workers into 6 groups such that 5 of the groups contain exactly 30 workers each, and each group has at least one worker. Each group will be assigned a different task. Provide your answer as a single number.

Answer: 0


Question: Consider the function \( f(x) = x^4 + x^3 + x^2 + x + 1 \), where \( x \) is an integer and \( x > 1 \). Determine the remainder when \( f(x^5) \) is divided by \( f(x) \).

Answer: 5


Question: Let $N$ be an $n \times n$ nilpotent matrix such that $N^k = 0$ for some integer $k$. Determine whether $(DN)^k = 0$ for any diagonal matrix $D$. Provide a proof or counterexample.

Answer: (DN)^k = 0


Question: You are at location \((0,0)\) and know the treasure is within 100 meters of you. Person A claims that they walked 108 meters to the treasure, with their distance calculated by \(|x| + |y|\). Person B claims that they walked 105 meters to the treasure, with their distance calculated by \(\frac{|x|}{\sqrt{3}} + \max\left\{\frac{|x|}{\sqrt{3}}, |y|\right\}\). Determine the number of possible locations for the treasure.

Answer: 4


Question: Solve for real $x$ in the equation: \[ 2^{\left(\log_2\sqrt{x^2-6x+9}\right)} = 3^{\log_\sqrt{x}(x-1)}. \]

Answer: 2


Question: Find the volume of the set \( \Omega = \bigcup_{t \in [0,1]} T(t) \), where \( T(t) \) is the triangle formed by the points \( A(t) = (t,t^3,t) \), \( B(t) = (t,t,t) \), and \( C(t) = (0,2t,t) \). Assume \( T(t) \) is empty if the points are collinear.

Answer: \dfrac{1}{15}


Question: Determine whether there exists a non-abelian nilpotent Lie algebra \( \mathfrak{n} \) over \( \mathbb{R} \) and an automorphism \( \alpha: \mathfrak{n} \to \mathfrak{n} \) such that \( \alpha \) is periodic, the fixed subspace of \( \alpha \) is the origin, and there is an \( \alpha \)-invariant lattice \( L \subset \mathfrak{n} \).

Answer: Yes


Question: Prove that if a Dirichlet character \( \chi \) modulo \( N \) is defined modulo \( N' \), where \( N' \mid N \), then the conductor \( c(\chi) \) divides \( N' \).

Answer: c(\chi) \mid N'


Question: Evaluate the limit of the improper integral: \[ \lim_{r \to 0^+} \int_{A} \frac{1}{|u|} \, du \] where \( A = \{ u \in \mathbb{R}^3 : r \leq |u| \leq 1 \} \) and \( r \in (0,1) \). Determine if the limit exists, and if not, explain why.

Answer: 2\pi


Question: Determine whether the ratio \( \frac{\pi_g(n)}{\pi(n)^2} \) has a limit as \( n \to \infty \), where \( \pi(n) \) is the number of primes \( p \) for \( p \le n \) and \( \pi_g(n) \) is the number of Gaussian primes \( z = x + yi \) for \( |x| \) and \( |y| \le n \).

Answer: \infty


Question: Determine the number of solutions to the equation \(x_1 + x_2 + \dots + x_8 = 30\) subject to the constraints \(2 \leq x_i \leq 5\) for \(i = 1, 2, \dots, 6\), and \(x_7\) and \(x_8\) must be either 5 or 10.

Answer: 658


Question: Find the minimum value of the function \( F(x,y,z,w) = \frac{1}{x+y+z} + \frac{1}{x+y+w} + \frac{1}{x+z+w} - \frac{2}{x+y+z+w} \) subject to the constraints \( x, y, z, w \geq 0 \) and \( 0 \leq x+y, y+z, z+x, x+w, y+w, z+w \leq 1 \).

Answer: 1


Question: Let $K$ be a number field, and let $A$ and $B$ be abelian varieties over $K$. Assume that $A$ and $B$ are isogenous over the algebraic closure of $K$, and that the endomorphism ring of $A_{\bar{K}}$ is $\mathbb{Z}$. Determine whether there exists a quadratic extension $L$ over $K$ such that $A$ and $B$ are isogenous over $L$.

Answer: Yes


Question: Evaluate the limit \( \lim\limits_{x\to a}\left(\frac{x^x-a^x}{x^a-a^a}\right) \) using L'Hospital's rule.

Answer: 1


Question: Determine the number of continuous functions \( f: [0,1] \rightarrow \mathbb{R} \) that satisfy the equation \[ \int_{0}^{1} f(x) \, dx = \frac{1}{3} + \int_{0}^{1} f^2(x^2) \, dx. \]

Answer: 1


Question: In how many distinct ways can 5 identical red beads, 3 identical green beads, and 2 identical blue beads be arranged in a circular necklace?

Answer: 252


Question: Evaluate the limit: $$\lim_{x \to 0^{+}} \sqrt{|\sin x - \tan x | } \int_{\cos x}^{1+ \sin x} e^y \, \, \mathrm{d}y$$

Answer: 0


Question: Determine the dimension of the unitary group $U(n)$. Provide a correct calculation and explanation for your answer.

Answer: n^2


Question: Calculate the directional derivative of the function \( u = \arcsin\left(\frac{z}{\sqrt{x^2+y^2}}\right) \) at the point \( A = (1, 1, 1) \) in the direction of the vector \( \mathbf{AN} \), where \( N = (3, 2, 3) \).

Answer: \dfrac{1}{6}


Question: Consider an analytic function $f$ of a complex variable $z$: $f(x + i y) = f(z)$. At any point $z$ in the domain, are the Gaussian curvatures of the surfaces $(x,y, \operatorname{Re} f(z))$ and $(x,y, \operatorname{Im} f(z))$ the same?

Answer: Yes


Question: X deposits $1 at time $t=0$ with a force of interest given by \( \delta_t = \frac{t^2}{k} \). Y deposits the same amount at the same time with a nominal interest rate of 8% per annum, convertible semiannually. At \( t=5 \), their accumulated values are equal. Determine the value of \( k \).

Answer: 106.24


Question:
For every \( \alpha \ge 0 \), determine if the following integral evaluates to zero:
\[ \int_0^{\pi/4} (\sin(t)\cos(t))^\alpha(1-(\alpha+2)\cos^2(2t)) \, dt = 0 \]

Answer: 0


Question: Is there always a partition $P$ such that the upper sum $U(f, P)$ equals the infimum of all upper sums $U(f)$ for a bounded function $f$ on a compact set $A$? Consider the definition of the Riemann Integral where $U(f) = \inf\{U(f,P) \mid P \in \mathcal{P}\}$. Determine if such a partition $P$ always exists.

Answer: No


Question: Find the area enclosed between an arc of the cycloid given by the parametric equations \(x=a(\theta - \sin (\theta))\) and \(y=a(1-\cos (\theta))\) and its base.

Answer: 3\pi a^2


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

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


Question: In the finite field \(\mathbb{F}_{101}\), where the discrete logarithms are given as \(L_2(3)=69\) and \(L_2(5)=24\), compute the discrete logarithm \(L_2(60)\).

Answer: 95


Question: Determine the region of convergence for the series \( \sum_{n=1}^{\infty} \frac{z^n}{n (\log n)^2} \).

Answer: |z| \leq 1


Question: On Christmas Eve, Santa crashed on an island inhabited by elves. Among these elves, having blue eyes is considered shameful, and any elf who discovers they have blue eyes will leave the island the next midnight. The elves cannot see their own eye color, and they never mention each other's eye color. After Santa's visit, he inadvertently revealed that there are both blue and brown-eyed elves on the island. On the 43rd midnight after Santa's visit, some elves left the island. How many blue-eyed elves were there on the island when Santa crashed?

Answer: 43


Question: Evaluate the integral \( \int_0^a \arcsin\sqrt{\frac{a}{a+x}}\,dx \) using the substitution \( x = a\tan^2\theta \).

Answer: a


Question: A frog makes 3 jumps, each exactly 1 meter long, with the directions of the jumps chosen independently at random. What is the probability that the frog's final position is no more than 1 meter from its starting position?

Answer: \dfrac{1}{4}


Question: Let \( A = \begin{pmatrix} \cos \frac{2 \pi}{5} & -\sin \frac{2 \pi}{5} \\ \sin \frac{2 \pi}{5} & \cos \frac{2 \pi}{5} \end{pmatrix} \) and \( B = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \). Let \( S \) be the set of all matrices that can be generated from taking products of \( A \) and \( B \), in any order. For example, the matrix \( A^2 B^4 B \) is in the set \( S \). Find the number of distinct elements in \( S \).

Answer: 10


Question: Determine all real numbers \(a\) such that the inequality \(\ln(1+x) < x - \frac{x^2}{2} + ax^3\) holds for all \(x > -1\).

Answer: \dfrac{1}{3}


Question: Find the covariance of $X$ and $|X|$, where $X \sim \mathcal{N}(0, \sigma)$ and $|X|$ follows a half-normal distribution.

Answer: 0


Question: In a normed algebra with an involution \(\ast\), does the inequality \(\|a\|^2 \le \|a^\ast a\|\) imply \(\|a^\ast\| \|a\| \le \|a^2\|\) for any element \(a\)? Determine whether this implication is true or false.

Answer: False


Question: What is the minimal possible value of the Shannon entropy \(H(X_1 X_2 X_3 X_4)\) for 4 pairwise independent unbiased Bernoulli random variables \(X_1, X_2, X_3, X_4\)?

Answer: 3


Question: Given an $n \times n$ full rank matrix $A$ with condition number $\kappa = \frac{\sigma_{\max}}{\sigma_{\min}}$, where $\sigma$ are the non-zero singular values of $A$, determine the condition number of $A^TA$.

Answer: \kappa^2


Question: Consider a sequence of operators \(\{A_n\}_{n=1}^{\infty}\) where each \(A_n: \ell_1 \to \ell_1\) and \(\|A_n\|_1 \leq 1\). Given that the unit ball \(\{T : \ell_1 \to \ell_1 \text{ such that } \|T\|_1 \leq 1\}\) is compact in the weak operator topology, does the sequence \(A_1 \leq A_2 \leq A_3 \leq \ldots\) converge in the weak operator topology to some operator \(A : \ell_1 \to \ell_1\)?

Answer: Yes


Question: Evaluate the double integral \( \int_0^1 \int_0^1 \frac{x-y}{(x+y)^3} \, dy \, dx \).

Answer: \dfrac{1}{2}


Question:
Determine which of the following expressions can represent the first fundamental form of a surface:

1. \( ds^2 = du^2 + 4 dudv + dv^2 \)
2. \( ds^2 = du^2 + 4 dudv + 4 dv^2 \)
3. \( ds^2 = du^2 - 4 dudv + 6 dv^2 \)
4. \( ds^2 = du^2 + 4 dudv - 2 dv^2 \)

Select the correct expression.

Answer: 3


Question: Evaluate the limit: \[ \lim_{t \to 1^+} \lim_{x \to \infty} \left( \frac{t^x - 1}{x^t - 1} \right)^{1/x} \]

Answer: 1


Question: Let $F$ be a field of order $32$. Determine the number of non-zero solutions $(a,b) \in F \times F$ to the equation $x^2 + xy + y^2 = 0$. Provide your answer as a single integer.

Answer: 0


Question: Assume we have 100 different balls numbered from 1 to 100, each placed in a separate bin. Calculate the variance of the number of balls located between ball #1 and ball #2.

Answer: \dfrac{4949}{9}


Question: Find the limit of the sequence defined by the recurrence relation \( q_{n+1} = q_n + \frac{2}{q_n} \) with initial condition \( q_0 = 1 \).

Answer: \infty


Question: Determine whether the inclusion map \(i: X \times \{0,1\} \cup A \times I \rightarrow X \times I\) is a cofibration, given that \((X,A)\) is a relative CW-complex.

Answer: Yes


Question: Given $f$ as the density function of a random variable, determine whether the following statement is true: $$\lim_{x\rightarrow \pm\infty}xf(x)=0.$$

Answer: False


Question: Find a polynomial \( p(x) \) such that \( p(-1)=p'(-1)=p''(-1)=p(1)=p'(1)=p''(1)=0 \).

Answer: (x^2 - 1)^3


Question: Is the function \( f: \mathbb{N} \rightarrow \mathbb{N} \) well-defined, where \( f(n) \) is the smallest \( m \in \mathbb{N} \) such that the decimal expansion of \( 2^m \) contains \( n \) consecutive zeros?

Answer: Yes


Question: Evaluate the integral \( \int_0^{2\pi}\frac{\cos 3x}{5-4\cos x}dx \) using complex integration techniques.

Answer: \dfrac{\pi}{12}


Question: Let \( f(z) \in \mathbb{C}[z] \) be a monic polynomial of degree \( n \). Determine if the following inequality holds: \[ \max\{\,|f(x)|: \, -1\leq x\leq 1\}\geq 2^{1-n} \,\, ? \]

Answer: Yes


Question:
Find the cokernel of the map $A: \mathbb{Z}^2 \to \mathbb{Z}^7$ given by the matrix:
\[
\begin{pmatrix}
*&1&1&0&-1&-1&-1&0\\
&0&2&1&3&0&-1&-1
\end{pmatrix}^T
\]

Answer: \mathbb{Z}^5


Question: Suppose that vectors $x_1, x_2, \ldots, x_n$ in a linear space have the property that for each $i$, the sum of all vectors except $x_i$ is parallel to $x_i$. If at least two of the vectors $x_1, x_2, \ldots, x_n$ are not parallel, find the value of the sum $x_1 + x_2 + \cdots + x_n$. \( \boxed{0} \)

Answer: 0


Question: Find all functions \( f:(0,1)\rightarrow \mathbb{R} \) such that \( f(xy)=f(x(1-y)) \) for all \( x,y\in (0,1) \).

Answer: f(x) = c


Question: Derive the asymptotic distribution of \( \sqrt{n} (\hat{\theta}_{MLE} - \theta) \) as \( n \to \infty \), where \( \hat{\theta}_{MLE} = \frac{n}{\sum_{i=1}^n{X_i}} \) is the maximum likelihood estimator for the parameter \( \theta \) of an exponential distribution. Show that it is asymptotically \( N(0, \theta^2) \).

Answer: N(0, \theta^2)


Question: Given natural numbers \(A\) and \(B\) such that \(A \le B\), you repeatedly toss a fair coin to decide the next step. If the result is heads, decrement \(A\) by 1. If the result is tails and \(A < B\), increment \(A\) by 1; otherwise, do nothing. Determine the expected number of coin flips required for \(A\) to reach 1.

Answer: (2B - A)(A - 1)


Question: Determine the asymptotic distribution of the statistic \(8nB = - 8n\operatorname{ln}(\sqrt{0.2\hat{p}} + \sqrt{0.8(1-\hat{p})} )\), where \(\hat{p} = (1/n)\sum_i X_i = \bar{X}_n\) is the estimator of \(p\) from a random sample \(X_1, ..., X_n\) taken from a Bernoulli\((p)\) distribution. Assume \(H_0: p=0.2\) and \(H_1: p\neq 0.2\).

Answer: \chi^2_1


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{\cos 2 + \cos 4 + \cos 6 + \dots + \cos{2n}}{n} \).

Answer: 0


Question: Evaluate the infinite series: \[ \sum_{n=1}^\infty \frac{n^2 \cdot (n+1)^2}{n!} \]

Answer: 27e


Question: Solve for \(x\) in the equation: \[ \tan^{-1}(x+1) + \tan^{-1}x + \tan^{-1}(x-1) = \tan^{-1}3. \]

Answer: 5 - \sqrt{19}


Question: Let \( A \subset \mathbb{R}^n \) be a connected set that is not a singleton. Is it true that the Jordan measure of \( A \) is always non-zero?

Answer: No


Question:
Determine the value of \( f(\pi/6) \) such that the function \( f(x) = (\sin{3x})^{\tan^2(3x)} \) is continuous at \( x = \pi/6 \). Choose the correct option:

A) \( e \)

B) \( e^{-1/2} \)

C) \( e^{1/2} \)

D) \( e^2 \)

Answer: B


Question: Find all pairs of natural numbers \((a, b)\) such that \((b!)^{a!} - (a!)^{b!} = 28\).

Answer: (3, 2)


Question: Determine the region in the complex plane where the series \( \sum_{n=1}^\infty \frac{1}{1+z^n} \) converges for \( z \in \mathbb{C} \).

Answer: |z| > 1


Question: What is the cardinality of a single equivalence class of Cauchy sequences in the rational numbers, corresponding to a single real number?

Answer: 2^{\aleph_0}


Question: Does an infinite dimensional operator $T$ having an eigenvalue of $0$ imply that $T^{-1}$ does not exist?

Answer: Yes


Question: Find the pointwise limit of the sequence of functions \( f_n(x) = \frac{nx}{1+n^2x^2} \) as \( n \to \infty \) on the interval \( I = [0,1] \).

Answer: 0


Question: Evaluate the double integral \( \iint_D (1+xy)\,dA \) where \( D=\{(x,y):1\leq x^2 +y^2\leq 2, y\geq0\} \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the integral: \[ \int\limits_{-\infty}^\infty \left[ e^{-at^3} \delta(t-10) + \sin (5 \pi t) \delta(t) \right] \mathrm{d}t \]

Answer: e^{-1000a}


Question: Find the minimum value of \( \int_0^\pi f^2(x) \, dx \) given that \( \int_0^\pi f(x) \sin x \, dx = 1 \) and \( \int_0^\pi f(x) \cos x \, dx = 1 \), where \( f(x) \) is continuous on \([0, \pi]\).

Answer: \dfrac{4}{\pi}


Question: Find the minimum value of the product \( ab \) if the roots of the cubic equation \( x^3 - ax^2 + bx - 2 = 0 \) are all positive.

Answer: 18


Question: Cars arrive according to a Poisson process with a rate of 2 per hour, and trucks arrive according to a Poisson process with a rate of 1 per hour. These processes are independent. What is the probability that at least 3 cars arrive before a truck arrives?

Answer: \dfrac{8}{27}


Question: Compute the contraction of the 1-form $\alpha = zdy - ydz$ with the vector field $\mathbb{X} = (0, y, -z)$ on $\mathbb{R}^3$. What is the result of $i_\mathbb{X}\alpha$?

Answer: 2yz


Question: Evaluate the infinite series: \[ \sum_{n=1}^{\infty} \frac{n}{e^{2\pi n}-1} = \frac{1}{e^{2\pi}-1} + \frac{2}{e^{4\pi}-1} + \frac{3}{e^{6\pi}-1} + \cdots \]

Answer: \dfrac{1}{24} - \dfrac{1}{8\pi}


Question: Given a subgroup \( H \) of \( \mathbb{Z}^2 \) generated by the vectors \((1, 6), (3, 5), (7, 11)\), how many distinct cosets are there in the quotient group \( \mathbb{Z}^2/H \)?

Answer: 1


Question: Determine an upper bound on the number of iterations required for the recurrence relation \( R(0) = n \) and \( R(t) = R(t-1) - \max\left(1, \left\lfloor\sqrt{R(t-1)}\right\rfloor\right) \) to terminate, where the recurrence stops when \( R(\cdot)=0 \).

Answer: 2\sqrt{n}


Question: Find the mass of the region $V = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \le z \}$, where the mass density is given by $g(x,y,z) = \sqrt{x^2 + y^2 + z^2}$. Use spherical coordinates to evaluate the triple integral and find the mass of $V$. Express your answer as a single number.

Answer: \dfrac{\pi}{10}


Question: Determine whether $z^k + z^{-k}$ belongs to the ring $\mathbb{Z}[z + z^{-1}]$ for every positive integer $k$, where $z$ is a non-zero element of $\mathbb{C}$.

Answer: Yes


Question: Evaluate the limit: \[ \lim\limits_{n\to\infty} \frac{1}{n^2}\sum_{2\leq i < j \leq n} \log_i j \]

Answer: \dfrac{1}{2}


Question: If $R$ is a non-commutative ring, will $R[x]$, the polynomial ring over $R$, still be a ring?

Answer: Yes


Question: Determine the top de Rham cohomology group $H^m_{\mathrm{dR}}(M;\mathbb{R})$ for a noncompact smooth, connected, oriented $m$-manifold $M$ with nonempty boundary.

Answer: 0


Question: Given a regular planar graph $G$ whose line graph $L(G)$ is also planar and not isomorphic to $G$, determine the number of edges in $L(G)$. Provide your answer as a single integer.

Answer: 12


Question: Given a set of 1000 positive integers with a mean of 10, determine the maximum possible value of the standard deviation of the set.

Answer: 27\sqrt{111}


Question: Bob gives Alice a weighted coin that lands on tails with some probability \( p \). The probability distribution is given by \( P(p \le z) = z^4 \). Alice flips the coin 5 times, and each time it lands on tails. What is the probability that Alice's next flip is also tails?

Answer: \dfrac{9}{10}


Question: Evaluate the limit: \[ \lim _{n \to \infty} \int_{0}^1 \arcsin(\sin(nx)) \, dx \]

Answer: 0


Question: Calculate the line integral \( I = \int \limits_\Gamma x^2 \, dy - 2xy \, dx \), where \( \Gamma \) is the part of the curve \( x^4 - y^4 = 6x^2y \) from point \( A(-4\sqrt{2}, 4) \) to point \( B(0,0) \).

Answer: \dfrac{44}{3}


Question: Evaluate the triple integral \( \int_0^1 \int_0^1 \int_{x^2}^1 12xz \, e^{y^2z} \, dy \, dx \, dz \) by changing the order of integration to \( dxdydz \). Verify that the final result is \( 3e - 6 \).

Answer: 3e - 6


Question: Find the value of the limit: \[ \lim_{n\rightarrow \infty}\frac{1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n^3}}{\ln(n)} \]

Answer: 3


Question: Compute \( \lim_{k \to \infty} \int_{\mathbb{R}} f_k \, \mathrm{d}\lambda \), where \( f_k(x) = \frac{k}{\lfloor \sqrt{k^2x} \rfloor} \chi_{[4/k^2,1]}(x) \).

Answer: 2


Question: Suppose $\mathcal{G}$ is an étale groupoid with a locally compact, Hausdorff unit space $\mathcal{G}^{(0)}$. If $f \in C_c(\mathcal{G})$ vanishes outside $\mathcal{G}^{(0)}$, is $f$ necessarily continuous on $\mathcal{G}^{(0)}$?

Answer: Yes


Question: Let $f$ be a positive continuous function on a compact interval $[a,b]$. Determine the limit \( \lim_{n \to \infty} \left[ \int_{a}^{b} f(x)^n \, dx \right]^{1/n} \).

Answer: M


Question: Find the singular point \(x = a\) of the solution to the differential equation \(y' - \left(\frac{6}{17}\sin x\right)y = \frac{5}{32} x^4 y^2\) with the initial condition \(y(0) = 1\). (Hint: Consider the behavior of \(y(x)\) when \(y \gg 1\).)

Answer: 2


Question:
Let $X$ be a metric space and $A$ and $B$ be two dense subsets in $X$. Determine which of the following statements is correct:

1. If $A$ is open, $A \cap B$ is dense in $X$.
2. If $A$ is closed in $X$, $A \cap B = \emptyset$.
3. $(A-B) \cup (B-A)$ is dense in $X$.
4. $A^\circ \cup B^\circ$ is dense in $X$.

Select the correct statement.

Answer: 1


Question: Let $G$ be a finite group with normal subgroups $H$ and $K$ of orders 25 and 14, respectively. Determine whether $G$ has an element of order 35.

Answer: Yes


Question: Evaluate the limit \( \lim_{x \to 0} \frac{\sin^{200}(x)}{x^{199} \sin(4x)} \).

Answer: \dfrac{1}{4}


Question: Evaluate the line integral \( \oint_C (xy \, dx + xy^2 \, dy) \) using Stokes' Theorem, where \( C \) is the square in the xy-plane with vertices \((1,0), (-1,0), (0,1), (0,-1)\).

Answer: \dfrac{1}{3}


Question:
Let $U$ be a random variable with a uniform distribution on the interval $(0,1)$. Define $X = \sin(2\pi U)$ and $Y = \cos(2\pi U)$. Calculate the correlation coefficient between $X$ and $Y$, denoted as:
\[ \text{Corr}(X,Y) = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \]

Answer: 0


Question: Consider the sequence of fractions \(\left\{\frac{1}{a} : a \in \mathbb{N}\right\}\) (e.g., \(\frac{1}{2}, \frac{1}{3}, \ldots\)). Starting with 1, subtract each fraction from the sequence as long as the result remains positive. For example, subtract \(\frac{1}{2}\) and \(\frac{1}{3}\), but skip \(\frac{1}{4}\) to \(\frac{1}{6}\) if the result would be zero or negative. Continue with fractions like \(\frac{1}{7}\) and \(\frac{1}{43}\). Determine if there is a lower limit to how small the result can be, or if it can be made arbitrarily close to zero.

Answer: 0


Question: Does there exist a function $z: \mathbb{N} \to [0,1)$ such that the infinite product \( \prod_{k=1}^\infty z(k) \) converges to a nonzero value? Assume that \( z(k) \to 1 \) as \( k \to \infty \).

Answer: Yes


Question: Determine the cokernel of the map $O_S \to F_\infty/O_\infty$, where $k$ is a field of characteristic not equal to 2, $F$ is the quadratic extension of $k(T)$ generated by $\sqrt{T^3 + 1}$, $X$ is the set of all places of $F$, $S = \{\infty\} \subset X$ with $\infty$ being the unique place of $F$ such that $\text{ord}_\infty(T) < 0$, and $O_S = k[T, \sqrt{T^3 + 1}]$. What is the cokernel of $O_S \to F_\infty/O_\infty$?

Answer: k


Question: Determine the cardinality of the set S, where S is the set of all infinite increasing sequences of binary strings. An infinite increasing sequence of binary strings is a series $w_1, w_2, \ldots$ of finite binary strings such that for every number $i$, the string $w_i$ is a prefix of $w_{i+1}$. For example, a sequence could be "101", "10100", "101001", "1010010111", and so on.

Answer: 2^{\aleph_0}


Question: A particle traces a smooth closed curve \(\mathcal{C}\) in \(\mathbb{R}^2\) with length \(L\). Another particle traces a curve \(\mathcal{C'}\) at a constant distance \(d > 0\) from \(\mathcal{C}\), remaining orthogonal to \(\mathcal{C}\). What is the length of the curve \(\mathcal{C'}\)?

Answer: L + 2\pi d


Question:
Given the moment generating function of a multivariate normal random variable \((X, Y, Z)\):
\[ M(t_1, t_2, t_3) = \exp(t_1 - 2t_3 + t_1^2 + t_1t_2 + \frac{3}{2}t_2^2 - t_1t_3 + \frac{5}{2}t_3^2) \]
Find the value of \(a\) such that \(X\) and \(Y + aX\) are independent.

Answer: -\dfrac{1}{2}


Question: Given a two-dimensional absolutely continuous random variable with the joint density function \( f_{X,Y}(x,y) = \frac{1}{2} \) for \( 0 < x < 1 \) and \( 0 < y < 4x \), find the probability that \( P(X + Y > 1) \).

Answer: \dfrac{4}{5}


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

Answer: e^{-2}


Question: Evaluate the limit: $$\lim_{(x,y)\to(0,0)} (x^2+y^2)^{x^2y^2}.$$ Determine whether the limit is 0 or 1.

Answer: 1


Question: Determine if there are $\omega_1$-many limit ordinals that are also limits of limit ordinals less than $\omega_1$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the limit: $$\lim_{n \to \infty} \left(n - \sum_{k=1}^n \cos{\frac{\sqrt{k}}{n}}\right)$$

Answer: \dfrac{1}{4}


Question: Find all integer solutions to the equation \(x^2 + y^2 + z^2 = 2xyz\).

Answer: (0, 0, 0)


Question: Let \((R, \mathfrak{m})\) be a Noetherian local ring containing a field. Does the \(\mathfrak{m}\)-adic completion \(\widehat{R}\) of \(R\) also contain a field? Provide a justification for your answer.

Answer: Yes


Question: Determine if it is sufficient for an Egyptian fraction to be greedy if each term \(a_{n+1}\) is greater than or equal to \(a_n \times (a_{n}-1)+1\).

Answer: Yes


Question: Calculate the line integral \(\oint_C (x - y^3)dx + x^3dy\) where \(C\) is the circle \(x^2 + y^2 = 1\) traced counterclockwise.

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


Question: Let $X$ and $Y$ be vector spaces with dimensions $p$ and $q$, respectively. Suppose $T$ is a subspace of $X$ with dimension $r$. Determine the dimension of the space of linear transformations from $X$ to $Y$ that vanish on $T$. Express your answer as a single number.

Answer: q(p - r)


Question: Suppose $R$ is a principal ideal domain (PID) and $I$ and $J$ are ideals of $R$ such that $0 \subset J \subseteq I$. If $I$ is maximal and $J$ is principal, determine whether $I = J$.

Answer: No


Question: Find the sum of the non-real solutions of the equation \((x+1)^4 + (x+3)^4 = 4\).

Answer: -4


Question: Evaluate the definite integral \( \int_{0}^{\frac{\pi}{2}} \sin|2x - \alpha| \, dx \) where \( \alpha \) is a real number in the interval \([0, \pi]\).

Answer: 1


Question: Calculate the limit: $$\lim_{x\rightarrow 0} \frac{ \sin^2(\sqrt x) - \sin^2(x^2) - x} {x^3}$$

Answer: -\infty


Question: Suppose $\lambda_1, \lambda_2, \dots, \lambda_k$ are complex numbers such that for all $n \in \mathbb{N}$, the equation $$\lambda_1^n + \lambda_2^n + \dots + \lambda_k^n = 0$$ holds. Determine whether it must be true that $\lambda_1 = \lambda_2 = \dots = \lambda_k = 0$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether a non-Hermitian matrix of complex numbers can have real eigenvalues.

Answer: Yes


Question: Find the minimum value of \( \frac{x_1 + x_2 + \cdots + x_n}{x_1 x_2 \cdots x_n} \) given that \( x_i > 1 \) for all \( i \).

Answer: 0


Question: Evaluate the integral by changing the order of integration: \[ \int_{0}^{9}\int_{0}^{3}\int_{2y}^{6} \frac{2\cos x^2}{3\sqrt{z}}\,dx\,dy\,dz \]

Answer: \sin 36


Question: Evaluate the integral: \[ \int_0^{2\pi}\sum_{i=1}^{M}b_i \cos (it)\sin(i\xi) \sum_{j=1}^{M} b_j\cos(j\xi)\sin(j\tau) \, d\xi \]

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: Evaluate the improper integral: $$\int_{\mathbb{R}} \frac{\sin^{4}(x)}{x^2}dx$$.

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit: $$\lim_{x\to 1}\ln(1-x)\cot\frac{\pi x}2$$

Answer: 0


Question: Compute the outward flux of the vector field \( F = \langle x, y, 1 \rangle \) through the closed surface formed by the paraboloid \( z = x^2 + y^2 \) and the plane \( z = 1 \). The flux through the plane \( z = 1 \) is given as \( \pi \). Determine the flux through the paraboloid \( z = x^2 + y^2 \).

Answer: 0


Question: Determine the fundamental group of a topological space created by attaching four spheres along four distinct arcs, with each arc connecting two different spheres.

Answer: \mathbb{Z}


Question: Determine whether it is possible for a divergent series \( S'_n = \sum \limits _{k=0} ^{n} x_k \) to satisfy \( \lim _{n \rightarrow \infty} \dfrac{S'_n}{S_n} = 0 \), where \( S_n = \sum \limits _{k=0} ^{n} \dfrac{1}{k} \).

Answer: Yes


Question: Determine the asymptotic order of the probability \(\mathbb{P}_\lambda(Z^2 \leq X)\) as \(\lambda \to 0^+\), where \(Z \sim N(\mu, 1)\) with \(\mu \sim N(0, 1)\), and \(X \sim \exp(\lambda)\) is independent of \(Z\). Here, \(Z^2\) is a non-central chi-squared distribution with 1 degree of freedom.

Answer: 1 - 2\lambda


Question: Let $p$ and $q$ be integers such that $q < 0 < p$. Prove that $M_q < G < M_p$ for positive real numbers $x_1, x_2, ..., x_n$ that are not all equal, where $G$ is the geometric mean.

Answer: M_q < G < M_p


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{\cos(at)}{t}\ dt \) for \( a \in \mathbb{R} \).

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^{-1}\frac{e^x-e^{-x}}{1-e^{-2x}}\,dx \).

Answer: \dfrac{1}{e}


Question: Evaluate the limit: $$\lim_{x\to\infty}\left(\frac{3x^2-x+1}{2x^2+x+1}\right)^\left(\frac{x^3}{1-x}\right)$$

Answer: 0


Question: Find the maximum value of the function \( f(x, y, z) = 4x^2 + yz + 15 \) subject to the constraint \( x^2 + y^2 + z^2 = 1 \). Determine the coordinates of the points where this maximum occurs.

Answer: 19


Question:
Determine which subspace $X \subseteq \mathbb{R}$ with the usual topology and containing $\{0,1\}$ allows for a continuous function $f: X \to \{0,1\}$ such that $f(0) = 0$ and $f(1) = 1$. Choose from the following options:
1. $X = [0,1]$
2. $X = [-1,1]$
3. $X = \mathbb{R}$
4. $[0,1] \not\subset X$

Answer: D


Question: Determine whether the following statement is true or false: For a number field $K$ and a discrete valuation $v$ on $K$, the completion of the localization of the ring of integers $\mathcal{O}_{K}$ at $v$, denoted $\widehat{\mathcal{O}_{K,v}}$, is isomorphic to the inverse limit of the sequence $\{\mathcal{O}_K/\mathfrak{p}^n\}$, where $\mathfrak{p}$ is the prime ideal corresponding to $v$. Provide a proof or counterexample.

Answer: True


Question: Let $\kappa$ be a regular uncountable ordinal, and consider the field of surreal numbers $No(\kappa)$ with birthdate $< \kappa$. Define $S$ as the subset of $\mathbb{R}((x))^{No(\kappa)}_{<\kappa^+}$ containing Hahn series of either length $<\kappa$ or length $\kappa$ with a cofinal $\kappa$-sequence of exponents in $No(\kappa)$. Is $S$ a subfield of $\mathbb{R}((x))^{No(\kappa)}$, thus serving as a completion of $No(\kappa)$?

Answer: Yes


Question:
Determine the limit of the sequence \((x_n)_{n \geq 0}\) defined by the recurrence relations:
\[\begin{cases}
x_0 = 2017 \\
x_{n+1} = x_n + e^{-2018x_n}
\end{cases}\]
for every \(n \ge 0\).

Answer: \infty


Question: Evaluate the definite integral: $$\int_0^{\pi/2} \sqrt{1 - \sqrt{3}\sin 2x + 2\cos^2x} \, dx$$

Answer: 3 - \sqrt{3}


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

Answer: \dfrac{4}{e}


Question: Evaluate the integral: $$\int_{1}^{2} \int_{1}^{2} \int_{1}^{2} \int_{1}^{2} \frac{x_{1}+x_{2}-x_{3}-x_{4}}{x_{1}+x_{2}+x_{3}+x_{4}} d x_{1} d x_{2} d x_{3} d x_{4}$$

Answer: 0


Question: Calculate the flux of the vector field $\nabla \times G$ across the surface of the hemisphere $x^2+y^2+z^2=4$, $z\ge 0$, where $G=(x^2z, y^2z, z^3)$. Provide your answer as a single numerical value.

Answer: 0


Question:
Determine the limiting temperature distribution \( \lim\limits_{t\to\infty} u (x, t) \) for a 1-dimensional rod of length \(L\), given the temperature distribution at time \(t\):

\[u(x,t)= \frac {a_0}2 + \sum_{n=1}^{\infty} a_n \cos\left(\frac {n\pi x} L\right) e^{\frac {-4\pi^2n^2t}{L^2}}\]

Answer: \dfrac{a_0}{2}


Question: Calculate the determinant of the matrix \(A=\begin{pmatrix} \sin\alpha & \cos\alpha & a\sin\alpha & b\cos\alpha & ab \\ -\cos\alpha & \sin\alpha & -a^2\sin\alpha & b^2\cos\alpha & a^2b^2 \\ 0 & 0 & 1 & a^2 & b^2 \\ 0 & 0 & 0 & a & b \\ 0 & 0 & 0 & -b & a \end{pmatrix}\) where \(\alpha,a,b \in \mathbb{R}\). Given that the matrix is block upper triangular, use this property to determine its determinant.

Answer: a^2 + b^2


Question: Evaluate the limit \( \lim_{x \to 0}\frac {P_n}{x} \) where the sequence \( P_n \) is defined recursively by \( P_n = a^{P_{n-1}} - 1 \) for all \( n = 2, 3, \ldots \) and \( P_1 = a^x - 1 \), with \( a \) being a positive real number.

Answer: (\ln a)^n


Question: Identify a function from the real numbers to the real numbers that cannot be expressed as the pointwise limit of a sequence of measurable functions.

Answer: \chi_V


Question: Determine whether the following statement is true or false: If every proper subgroup of a finite group $G$ is cyclic, and $G$ has at least two distinct prime divisors, then $G$ is cyclic.

Answer: False


Question: Determine the number of 3-digit prime numbers \( \overline{abc} \) that satisfy the equation: \( (b+3)(b-3)=4ac \).

Answer: 0


Question: Simplify the expression: $$(A^T A)^{-1}A^T(B^{-1}A^T)^T B^T B^2 B^{-1}$$ assuming all matrices inverted in the expression exist.

Answer: B


Question: Evaluate the limit: \[ \lim_{x \to 0^+} ((x+9)^x - 9^x)^x \]

Answer: 1


Question: Compute the limit: $$\lim_{x\to 0}x^2 e^{\sin(1/x)}$$

Answer: 0


Question: What is the characteristic of a finite product of fields, each having characteristic 0? For instance, if \( \mathbb{C} \) has characteristic 0, determine the characteristic of \( \mathbb{C}^n \) for \( n > 1 \).

Answer: 0


Question: Determine the number of permutations in the symmetric group $S_8$ whose cube is a 5-cycle.

Answer: 4032


Question: Determine whether any open set \( A \) can always be expressed as the union of (potentially infinite) balls.

Answer: Yes


Question: Calculate the limit: $$\lim_{n \to \infty} \frac{\frac{\sin(1)}{1} + \frac{\sin(2)}{2} + \frac{\sin(3)}{3} + \cdots + \frac{\sin(n)}{n}}{n}$$

Answer: 0


Question: Evaluate the limit: $$\lim_{x \to 2} \frac{e^x - e^2 - (x^2 - 4x + 4)\sin (x - 2)}{\sin[(x - 2)^2] + \sin (x-2)}$$

Answer: e^2


Question: Determine the number of integer solutions to the equation \(\frac{xy}{z}+\frac{xz}{y}+\frac{yz}{x} = 6\), where \(x, y, z\) are non-zero integers.

Answer: 16


Question: Determine whether the expression \( \frac{1}{\sqrt{2\pi \varepsilon}}\exp\left( -\frac{x^2}{2\varepsilon}\right) \) converges to the Dirac delta distribution as \( \varepsilon \to 0 \).

Answer: Yes


Question: Determine whether there exist functions that are weakly convex, as defined by Artin, but are not continuous.

Answer: Yes


Question: Evaluate the limit: $$ \lim _{x \rightarrow 0}\left(x^{2}\left(1+2+3+\dots+\left[\frac{1}{|x|}\right]\right)\right) $$ where $[a]$ denotes the greatest integer less than or equal to $a$.

Answer: \dfrac{1}{2}


Question: Find the limit as \( n \to \infty \) of the expression: \[ \frac{(7n^5+2^n)^n}{2^{n^2}+(-7)^n\log(n)} \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{y \rightarrow -\infty} (-2y+5)\ln(\sqrt{y^2 + 3} - y + 1). \]

Answer: \infty


Question: Determine the order of the group of $3 \times 3$ matrices over $\mathbb{F}_2$ with determinant 1.

Answer: 168


Question: Evaluate the limit \( \lim _{x\to 0^+}\left(\left(\left(1+x\right)^x-1\right)^x\right) \) as \( x \) approaches 0 from the positive side.

Answer: 1


Question: A graph has 20 vertices and does not contain a $K_7$. If each edge is colored either green or yellow such that there are no monochromatic triangles, determine the maximum number of edges this graph can have.

Answer: 160


Question: Find the minimum value of the function \( \fermi(k) = \int_{0}^{1} |x^{2} - k^{2}| \, dx \) for \( k \in [-1, 1] \).

Answer: \dfrac{1}{4}


Question: Consider the function \( f(z) = \sum_{n=1}^{\infty} z^{n!} \). Determine whether this function is analytic within the open unit disk \(|z| < 1\), and if so, find the radius of convergence \(R\) for the power series representation.

Answer: 1


Question: In a topological vector space $(A, \tau)$, consider a subset $B$ of $A$ that is metrizable and complete with a translation-invariant metric. If the metric topology on $B$ coincides with the induced subspace topology, is $B$ necessarily closed in $\tau$?

Answer: Yes


Question: Let $E$ be a space and $d$ and $d'$ be two metrics on $E$. Suppose that for any $x, x_0 \in E$, we have $d'(x, x_0) < d(x, x_0)$. If a sequence $x_n$ converges to $x$ with respect to $d$, does it necessarily converge to $x$ with respect to $d'$? Provide a justification for your answer.

Answer: Yes


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

Answer: \pi \ln 2


Question: Calculate the limit: $$ \lim_{x \rightarrow \infty } \left(-6e^{11x} + 9\sin(x) + 3e^{8x} \right)^{\frac{8}{4x}} $$

Answer: e^{22}


Question:
Find the value of \( \lambda \) that satisfies the equation:
\[
\begin{vmatrix} a^2+\lambda^2 & ab+c\lambda & ca-b\lambda \\
 ab-c\lambda & b^2+\lambda^2& bc+a\lambda\\
 ca+b\lambda & bc-a\lambda & c^2+\lambda^2 \end{vmatrix} \cdot \begin{vmatrix} \lambda & c & -b\\ -c& \lambda & a\\ b & -a & \lambda \end{vmatrix}=(1+a^2+b^2+c^2)^3.
\]

Answer: 1


Question:
Find the function $y(x)$ such that the arc length from $0$ to $x$ is given by the expression:

$$s = \frac{1}{2} x \sqrt{1+4x^2} + \frac{1}{4} \ln(2x + \sqrt{1+4x^2}).$$

Answer: x^2


Question: Find the angle between the parabolas \(y^2=2px-p^2\) and \(y^2=p^2-2px\) at their intersection points.

Answer: 0


Question: Find the points where the function \( f(z) = z + 2iz^2 \operatorname{Im}(z) \) is differentiable using the definition of a derivative.

Answer: 0


Question: Find the tangent space to the surface defined by the equation $x^2 + y^2 + z^2 + w^2 = 1$ at the point $P = (0, 0, 0, 1)$. Provide the equations that describe this tangent space.

Answer: w = 1


Question: Determine the values of \(c > 0\) for which the series \(\sum a_n\) converges, given that \(\frac{a_{n+1}}{a_n} < \frac{n^c}{(n+1)^c}\) and \(a_n > 0\) for all \(n\).

Answer: c > 1


Question: Find the value of \( n \) such that the limit \[ \lim_{x\to 0}\frac{\sin^2x-\sin x-e^x\sin x+e^x-\frac{x^3}{3}}{x^n} \] is a non-zero finite number.

Answer: 0


Question: Let $G$ be a finite group of isometries acting properly discontinuously on a Riemannian manifold $\overline{M}$. If $\pi : \overline{M} \to \overline{M}/G$ denotes the quotient map and $M$ is a connected hypersurface in the quotient space $\overline{M}/G$, determine whether the number of connected components of the preimage $\pi^{-1}(M)$ is a divisor of the order of $G$. Provide a justification for your answer.

Answer: Yes


Question: In a commutative ring with unity $R$, if two elements $a$ and $b$ are associates (i.e., $a \mid b$ and $b \mid a$), must $a$ be expressible as $ub$ for some unit $u \in R$? Provide a justification for your answer.

Answer: Yes


Question: In triangle $\Delta ABC$, points $P$ and $Q$ are located on sides $AB$ and $AC$, respectively. Given that the ratio $|AP|:|PB|=1:5$ and $|AQ|:|QC|=2:1$, lines $BQ$ and $CP$ intersect at point $F$. Using vector methods, find the ratio $|CF|:|FP|$.

Answer: 3:5


Question: Determine whether there is a canonical homeomorphism between the sphere $S^n \subset \mathbb{R}^{n+1}$ and the reduced suspension $SS^{n-1}$. Provide a justification for your answer.

Answer: Yes


Question: If the Hessian matrix of a function $f(\textbf{x})$ is positive semidefinite on the entire domain, are all stationary points of $f(\textbf{x})$ minima?

Answer: Yes


Question: If $\operatorname{Gal}(K/F)\cong S_3$, is $K$ necessarily the splitting field of some irreducible cubic polynomial in $F[x]$?

Answer: Yes


Question: Calculate the value of \( \left\lfloor\sum_{n=1}^{9999} \frac {1} {n^{\frac{1}{4}}}\right\rfloor \).

Answer: 1332


Question: What is the cardinality of the set $B$, where $B$ is the set of all subsets of the natural numbers $\mathbb{N}$ that contain exactly 2 elements?

Answer: \aleph_0


Question: Given a graph with N vertices (3 ≤ N ≤ 1000) and M edges, determine the minimum number of additional vertices that need to be added to ensure that every vertex is either included in the new set of vertices or is adjacent to a vertex in the new set.

Answer: 1


Question: Calculate the partial derivative of the function \( f(x,y) = \begin{cases} x \arctan\left[\frac{x}{y}\right] & \text{if } y \neq 0 \\ 0 & \text{if } y = 0 \end{cases} \) with respect to \( x \) at the point \( (0,0) \).

Answer: 0