Question: Find a closed formula, not involving summation notation, for the expression \( \sum_{k=0}^{n-1}\binom{2n}{2k+1} \).

Answer: 2^{2n-1}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{r=0}^n(-1)^r\binom{n}{2r}\left(\frac{x}{n}\right)^{2r} \]

Answer: \cos x


Question: Find the Cauchy principal value of the integral \( \int_{-\infty}^{\infty}\frac{x}{(x^2+4)(x^2-2x+5)}\,dx \) using the method of residues.

Answer: \dfrac{\pi}{34}


Question: The event that a man arrives at a bank follows a Poisson process with parameter \( \lambda \). Given that two men visited the bank between 9:30 AM and 10:30 AM, what is the probability that both entered before 10:00 AM?

Answer: \dfrac{1}{4}


Question: Evaluate the surface integral \( \iint_{X} y \, dS \) over the parametrized surface \( X(s,t) = (2s, s+t, 2t) \), where \( D \) is the triangular region with vertices at \((0,0)\), \((0,1)\), and \((1,1)\).

Answer: \sqrt{6}


Question: Find the closed form of the infinite series: \[ \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{\left\lfloor \sqrt{k} \right\rfloor} \]

Answer: \ln 2


Question: Determine whether the limit \( \lim_{n\rightarrow\infty}\sum_{k=1}^n\ln\left(\frac{k(1+\alpha)}{n-k+1+(n-k)\alpha}\right) \) exists.

Answer: 0


Question: Evaluate the integral \( \int_0^1 \int_0^{2\sqrt{1-x}} \sqrt{x^2+y^2} \, \mathrm{d}y\,\mathrm{d}x \) using the transformation \( x=u^2-v^2 \) and \( y=2uv \).

Answer: \dfrac{56}{45}


Question: Given the function \( F(x, y) = 2x^4 + 3y^3 + 5xy \) and the condition \( F(x, y) = 10 \), find the derivative \( y'(x) \) at the point \( x = 1 \) and \( y = 1 \). Assume a small change in \( x \) from 1 to 0.999. Use the Implicit Function Theorem to determine how \( y \) changes with respect to \( x \).

Answer: -\dfrac{13}{14}


Question: Let \( A = \{1, 2, 3, 4, 5, 6, 7\} \). Determine the number of onto functions \( f: A \to A \) such that at least 3 elements of \( A \) are mapped to themselves.

Answer: 407


Question: Find a sufficient statistic for \( \theta \) given the distribution \( f_x(x) = \frac{e^{-|x-\theta|}}{2} \).

Answer: (X_{(1)}, X_{(2)}, \ldots, X_{(n)})


Question: Given an open covering \(\{U_\lambda\}_{\lambda\in\Lambda}\) of \(\mathbb{R}^n\) and a family of functions \(f_\lambda:U_\lambda\rightarrow \mathbb{R}\) for each \(\lambda\in\Lambda\), such that \(f_\lambda-f_\mu: U_\lambda\cap U_\mu\rightarrow\mathbb{R}\) is constant for any \(\lambda,\mu\in\Lambda\), determine whether there exists a function \(f:\mathbb{R}^n\rightarrow \mathbb{R}\) such that for each \(\lambda\in\Lambda\), the function \(f-f_\lambda\) is constant on \(U_\lambda\).

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty}^{\infty} e^{-(t-k)^2} \, dt \), where \( k \) is a constant.

Answer: \sqrt{\pi}


Question: Let $G$ be a connected compact Lie group. For any finite subset $A \subset G$, consider the centralizer $Z_G(A) = \{g \in G \mid ag = ga \text{ for all } a \in A\}$. Is $Z_G(A)$ a connected subgroup of $G$?

Answer: No


Question: Let \( S = \{ \tan(k) : k = 1, 2, \dots \} \). Find the set of limit points of \( S \) on the real line.

Answer: \mathbb{R}


Question: Given the function \( f(x,y,z) = \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \), find the value of \( xf_x + yf_y + zf_z \) without calculating \( f_x, f_y, \) and \( f_z \) individually. Use the concept of Euler's homogeneous function theorem to solve the problem.

Answer: 0


Question:
Determine the fundamental period of the continuous time signal given by:
\[ x(t) = \sum_{n=-\infty}^{\infty}e^{-(2t-n)}u(2t-n) \] where \( u(t) \) is the unit step function.

Answer: \dfrac{1}{2}


Question: Let $A$ be a finitely generated algebra over a field $k$. Suppose $x$ is a zero divisor in $A$. Is it true that the height of the principal ideal $(x)$ is zero?

Answer: No


Question:
Given a configuration of points in $X \in \mathbb{R}^{n \times 2}$ with zero column means, the variance of each axis $x$ and $y$ is given by:
\[ \mathrm{var}(x) = \frac{1}{n} x^T x \quad \text{and} \quad \mathrm{var}(y) = \frac{1}{n} y^T y. \]
If $\mathrm{var}(x) > \mathrm{var}(y)$, does this imply that the configuration has a larger spread along the $x$-axis?

Answer: Yes


Question:
Calculate the total variation \( V_0^{2\pi}(f) \) of the function \( f(x) \) defined as follows:
\[
f(x) = \begin{cases} 1 & \text{for } x = 0 \\ x & \text{for } x \in (0, \pi] \\ \sin x & \text{for } x \in (\pi, 2\pi] \end{cases}
\]

Answer: 3 + 2\pi


Question: Evaluate the limit: \[ \lim _{x \to a}\left(\frac{\sin(x)}{\sin(a)}\right)^{\frac{1}{x-a}} \] where \( a \) is any number.

Answer: e^{\cot(a)}


Question:
Consider the symmetric group $S_n$ on $n$ symbols and the alternating group $A_n$, which is the subgroup of even permutations. Determine which of the following statements is true:

(a) There exists a finite group which is not a subgroup of $S_n$ for any $n \geq 1$.
(b) Every finite group is a subgroup of $A_n$ for some $n \geq 1$.
(c) Every finite group is a quotient group of $A_n$ for some $n \geq 1$.
(d) No finite abelian group is a quotient of $S_n$ for $n > 3$.

Select the correct statement.

Answer: B


Question: Given the function \( f(x) = \frac{x^3 + 1}{3} \) with fixed points \( \alpha, \beta, \gamma \) in the intervals \((-2,-1)\), \((0, 1)\), and \((1, 2)\) respectively, a sequence is defined as \(x_1 = \gamma - 0.01\) and \(x_{n+1} = f(x_n)\) for all \(n = 1, 2, 3, \ldots\). Given that the sequence converges, find \( \lim_{n\rightarrow\infty}x_n \).

Answer: \beta


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(x+y) = f(x) + f(y) \) for all real numbers \( x \) and \( y \), and \( f(1) = 0 \).

Answer: 0


Question: Given a compact, complete, separable metric space $(X, d)$, and a sequence of constant speed geodesics $g_n : [0,1] \rightarrow X$ with the same endpoints, where $g_n(0) = g_{n+1}(0)$ and $g_n(1) = g_{n+1}(1)$ for every $n \in \mathbb{N}$, can a subsequence of $g_n$ be extracted that converges to a constant speed geodesic? Assume the space of geodesics on $X$ is compact, complete, and separable with the metric $\delta(f,g) = \sup_{x \in [0,1]} d(f(x),g(x))$. Provide a justification for your answer.

Answer: Yes


Question: Let $A$ be an $F$-algebra that is finite-dimensional over a field $F$. Is $A$ necessarily commutative?

Answer: No


Question: Given a stationary stochastic process with a spectral density \( S_{XX}(\omega) = 1 - \frac{|\omega|}{8 \pi} \), find the mean square value of the process.

Answer: 4


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left( \tan^{-1}\left(\frac{1+x}{4+x}\right)-\frac{\pi}{4}\right)x \]

Answer: -\dfrac{3}{2}


Question: Evaluate the limit: \[ \lim_{n\to\infty} \int_0^\infty \frac{1}{1+x^n} \, dx \]

Answer: 1


Question: Find all positive integer pairs \((m, n)\) such that \(6^m + 2^n + 2\) is a perfect square.

Answer: (1, 3)


Question: Find the real number \(x\) such that \( \int_0^x 2020^{(x^2 + 2020)} \, dt = x \).

Answer: 0


Question: Find the value of \( l \) for which the integral \( I(l) = \int_{0}^{\infty} \frac{x^l}{2x^6 + 4x^5 + 3x^4 + 5x^3 + 3x^2 + 4x + 2} \, dx \) is minimized, where \( 0 < l < 5 \).

Answer: 2


Question: Determine the number of graphs with 5 vertices labeled a, b, c, d, and e, such that exactly one vertex has degree 4.

Answer: 205


Question: Let $(E,d)$ and $(F,d')$ be separable metric spaces with their Borel algebras. Suppose $f: E \rightarrow F$ is a continuous surjective function, and $Q$ is a probability measure on $F$ with separable support. Determine whether there exists a probability measure $P$ on $E$ such that $Q$ is the pushforward measure of $P$ by $f$. Provide a justification for your answer.

Answer: Yes


Question: Solve the equation \(x^2 + \ln(x) = 0\) for \(x\).

Answer: 0.6529


Question: Find the classical solution $u = u(x_1, x_2)$ to the partial differential equation $$(x_2^2 + u)u_{x_1} + x_2 u_{x_2} = 0$$ with the initial condition $u(x_2^2/2, x_2) = 0$. Provide the solution $u(x_1, x_2)$.

Answer: 0


Question: Determine the order of convergence of the Newton-Raphson method when applied to the equation \(2x^2 + 1 - e^{x^2} = 0\) with an initial approximation \(x_0\) sufficiently close to zero, for the root \(x = 0\). Fill in the blank: The order of convergence is \(\boxed{\ }\).

Answer: 1


Question: Determine the maximum value of $x$ such that $\mathbb{Z}$, when made into a compact space, satisfies the $T_x$ separation axiom. Can this space be metric or completely metric?

Answer: 1


Question: Evaluate the expression: \[ \binom{5}{0}\binom{50}{5} - \binom{5}{1}\binom{40}{5} + \binom{5}{2}\binom{30}{5} - \binom{5}{3}\binom{20}{5} + \binom{5}{4}\binom{10}{5} \] where \( \binom{n}{r} \) denotes the binomial coefficient.

Answer: 100000


Question: Find three natural numbers \(n_1, n_2, n_3\) such that \(n_1 < n_2 < n_3\), \(n_1 + n_2 + n_3 = 784\), and \(n_1^2 + n_2^2 = n_3^2\). Calculate the product \(n_1 \times n_2 \times n_3\).

Answer: 11524800


Question: Evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \frac{1}{4\cos^{2}x + 9\sin^{2}x} \, dx \).

Answer: \dfrac{\pi}{12}


Question: Given that $X \sim \operatorname{Uniform}(\theta, \theta+1)$, we are testing the hypotheses $H_0: \theta \le 0$ versus $H_1: \theta > 0$. The test rejects $H_0$ if $X > 0.95$. Compute the probability of a type I error for this test.

Answer: 0.05


Question: Calculate the double integral \( \iint\limits_D (x+y)^3 \cdot (x-y)^2 \,\mathrm{d}x\,\mathrm{d}y \) over the region \( D \) defined by the lines \( x+y=1 \), \( x-y=1 \), \( x+y=3 \), and \( x-y=-1 \).

Answer: \dfrac{20}{3}


Question: Determine the smallest non-negative integer \( n \) such that the ring \( S_n = C^n([0,1], \mathbb{R}) \), consisting of all continuously \( n \)-times differentiable functions on the interval \([0,1]\), is not an integral domain.

Answer: 0


Question: Prove that for a smooth function \( f \in \mathcal{C}^{\infty}_{K}(\mathbb{R}) \) with compact support and \( t \geq 0 \), the function \( g_{t}(x) = \frac{1}{t}\int_{\mathbb{R}}(f(x+\sqrt{t}z)-f(x)-\frac{t}{2}f^{''}(x))e^{-\frac{1}{2}z^2}dz \) converges uniformly to 0 as \( t \rightarrow 0 \).

Answer: 0


Question: Solve the equation \(\frac{x(\log_{2}(x))^2}{2} = x^{3/2}\) for \(x\).

Answer: 4


Question: Compute the length of the curve defined by the equations \(z^2 = 2x\) and \(3y = xz\) between the points \((0,0,0)\) and \(\left(\frac{1}{2},\frac{1}{6},1\right)\).

Answer: \dfrac{7}{6}


Question: Let \( f_1, f_2, \ldots \) be continuous functions on \([0,1]\) such that \( f_1 \geq f_2 \geq \cdots \) and \( \lim_{n\to\infty} f_n(x) = 0 \) for each \( x \in [0,1] \). Does the sequence \( \{f_n\} \) converge uniformly to 0 on \([0,1]\)?

Answer: Yes


Question: A car requires repair after traveling 5000 km. The driver travels an average of 40 km per day with a standard deviation of 7 km per day. Determine the maximum number of days the driver can travel such that the probability of exceeding 5000 km is less than 0.01.

Answer: 120


Question: Find an expression for the function \( h(z) = \sum_{n=1}^{\infty}\frac{(z-2)^n}{n} \) as an elementary function.

Answer: -\ln(3 - z)


Question: Show that \( \int_0^t 1_{B_s=0} \, dB_s = 0 \), where \( B_s \) is a Brownian motion process.

Answer: 0


Question: In a connected graph, if the heaviest edge of each cycle is removed, does the resulting graph form a spanning tree? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_0^\infty \frac{\sin^2 x}{x^2} \, dx \). Given that \( \int_0^\infty \frac{\sin x \cos x}{x} \, dx = \frac{\pi}{4} \).

Answer: \dfrac{\pi}{2}


Question:
Let \( f(x,y) \) be a polynomial of \( x \) and \( y \) satisfying the following conditions:

1. \( f(1,2) = 2 \)
2. \( yf(x,f(x,y)) = xf(f(x,y), y) = (f(x,y))^2 \).

Determine all such functions \( f(x,y) \).

Answer: xy


Question: Evaluate the line integral \( \int_C \frac{2x^3+2xy^2-2y}{ x^2+y^2} \, dx+\frac{2y^3+2x^2y+2x}{ x^2+y^2} \, dy \) where \( C \) is any simple closed loop that contains the origin.

Answer: 4\pi


Question: Determine whether the function $f(x) = x$ satisfies the integral equation \( \big(\,f(x)\big)^2=2 \int_0^x f(t)\,dt \) for all $x \geq 0$, given that $f:[0, \infty)\longrightarrow \mathbb{R}$ is continuous and $f(x) \neq 0$ for all $x>0$. Is it true that $f(x) = x$ for all $x \geq 0$?

Answer: f(x) = x


Question: Given that \( M^2 \equiv 51 \pmod{59} \), find the value of \( M^{12} \pmod{59} \).

Answer: 7


Question: Calculate the integral \( \int x \, dy - y \, dx \) over the curve parameterized by \( x = a (t - \sin t) \) and \( y = a (1 - \cos t) \).

Answer: -6\pi a^2


Question:
Evaluate the sum of the series \( \sum_{n \in \Bbb{Z}} |c_n|^2 \), where for each integer \( n \), \( c_n \) is defined as:
\[ c_n = \frac{1}{\sqrt{2\pi}} \int_{-\pi}^{\pi} e^{i(n-i)x} \, dx \]
What is the value of \( \sum_{n \in \Bbb{Z}} |c_n|^2 \)?

(a) \( \cosh(\pi) \) \hspace{2cm} (b) \( \sinh(\pi) \) \hspace{2cm} (c) \( \cosh(2\pi) \) \hspace{2cm} (d) \( \sinh(2\pi) \)

Answer: D


Question: Let $\phi(n)$ be the Euler phi-function. For an integer $a > 1$, determine the remainder when $\phi(a^n - 1)$ is divided by $n$ using the Euclidean algorithm.

Answer: 0


Question: Let $R$ be a Noetherian commutative, infinite ring with unity such that distinct ideals have distinct index, meaning if $I$ and $J$ are ideals of $R$ and $I \neq J$, then $R/I$ and $R/J$ are not bijective as sets. Determine whether it is necessarily true that $R$ is Artinian, i.e., every prime ideal of $R$ is maximal.

Answer: No


Question: Determine the value of \(a\) such that the matrix \(A\) is not diagonalizable, where \(A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & a & a-2 \\ 0 & -2 & 0 \end{pmatrix}\).

Answer: 4


Question: Evaluate the integral \( \int_{-1}^{1} \frac{d}{dx}\left( \arctan\left(\frac{1}{x}\right)\right)dx \).

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


Question: Express the operation \( C_{ij} = B_{ik} B_{jl} A_{kl} \) in terms of standard matrix multiplication using Einstein's notation.

Answer: C = B A B^\top


Question: Calculate the surface area of the hemisphere defined by the equation \(x^2 + y^2 + z^2 = 4\) that is bounded below by the plane \(z = 1\).

Answer: 4\pi


Question: Evaluate the limit: \[ \lim_{n\to \infty} \frac{1 + \frac{1}{2} + \cdots + \frac{1}{n}}{(\pi^n + e^n)^{1/n} \log_e n}. \]

Answer: \dfrac{1}{\pi}


Question: Find a closed form expression for the limit \( \lim_{n\to\infty}\frac{1}{2^n}\left(1 + x^{1/n}\right)^n \) where \( x < 1 \).

Answer: \sqrt{x}


Question: Find the limit of the sequence defined by the recurrence relation \( a_{n+1} = \frac{a_n + n a_{n-1}}{n+1} \) with initial conditions \( a_0 = 0 \) and \( a_1 = 1 \).

Answer: \ln 2


Question: Given that there is at least one solution to the equation \(a^{2} + 2b^{2} = p^{11}q^{13}\), where \(a, b \in \mathbb{Z}\) and \(p, q\) are odd prime numbers, determine the total number of integer solutions.

Answer: 336


Question: Calculate \(7^{77} \mod 221\) without using a calculator. Show your work and any tricks or methods you use to simplify the computation.

Answer: 193


Question: Find all bounded harmonic functions \( h \) in the upper half-plane \( H = \{(x,y) \in \mathbb{R}^2 : y > 0\} \) such that \( h(x,0) = 1 \) for all \( x \in \mathbb{R} \).

Answer: 1


Question: Find the value of \(a\) such that \(g(a)\), the maximum of \(|x^2 - ax|\) over the interval \([0, 1]\), attains its minimum.

Answer: 2\sqrt{2} - 2


Question: Find the real number $b > 1$ such that the graphs of $y = b^x$ and $y = \log_b(x)$ intersect at exactly one point.

Answer: e^{1/e}


Question: Find the value of $y$ such that $11^{112111} \equiv y \pmod{113}$. Use Fermat's Little Theorem to simplify the computation.

Answer: 72


Question: Find the largest even number \( n \) such that there exist positive integers \( a \) and \( b \) satisfying the equation \( n \cdot \text{lcm}(a,b,2016) = ab - 2016 \).

Answer: 62


Question: Suppose $V_0$ and $V_1$ are models of ZFC, and let $\varphi$ be a first-order sentence in a finite language $L$. Is it true that $V_0 \models$ "The deductive closure of $\varphi$ is a complete $L$-theory" if and only if $V_1 \models$ "The deductive closure of $\varphi$ is a complete $L$-theory"?

Answer: No


Question: Let \( f: \mathbb{N}^{\star} \to \mathbb{N}^{\star} \) be a bijective function such that \( \lim_{n \to \infty} \frac{f(n)}{n} \) exists. Determine the value of this limit.

Answer: 1


Question: Let $G$ be a connected reductive linear algebraic group over an algebraically closed field $k$ of characteristic $0$. The coordinate ring of $G$ is denoted by $k[G]$. Determine the conditions under which $k[G]^* \neq k^*$, where $k^*$ is the unit group of the field $k$. Specifically, if $G$ is semisimple and simply connected, is it true that $k[G]^* = k^*$?

Answer: k[G]^* = k^*


Question: Is the analytic fundamental group of a smooth complex algebraic curve, considered as a Riemann surface, residually finite?

Answer: Yes


Question: In how many ways can you draw 6 cards from a standard 52-card deck such that you have at least one card from each suit (hearts, spades, diamonds, clubs)?

Answer: 8682544


Question: Determine the number of planes that contain the points $A(2, 5, 7)$ and $B(4, 11, 16)$ and are at a distance of 2 units from the origin.

Answer: 0


Question: Given a free abelian group $G$ of rank 2 with $\\mathbb{Z}$-basis $x, y$, and a subgroup $H$ with $\\mathbb{Z}$-basis $2x+y, 2x-3y$, determine the elements of the quotient group $G/H$. What is the structure of $G/H$?

Answer: \mathbb{Z}_8


Question: Find \(\lim_{x \to 0^{+}}(\ln{(\cot{x})})^{\tan x}\).

Answer: 1


Question: Calculate the limit \( \lim_{x\rightarrow 0}\dfrac{e^{-\frac{1}{x}}}{x^n} \) where \( n > 0 \) is an integer.

Answer: 0


Question: Find the constant-coefficient linear differential equation of the smallest possible order that has \( y(x) = x^3 \) as a solution.

Answer: y'''' = 0


Question: Compute the line integral of the vector field \((2x-3y)\,dx + (5x+6y)\,dy\) over the curve \(C\) defined by the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) in the first quadrant, using Green's theorem. The curve \(C\) is traversed in a direction such that the \(y\)-coordinate increases. Provide the value of the line integral.

Answer: 3 - 12\pi


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

Answer: e^{t - 1}


Question:
For a positive real number \(a\), let \(S(a)\) denote the set of points \((x, y)\) satisfying \(|x|^a + |y|^a = 1\). A positive number \(a\) is said to be good if the points in \(S(a)\) that are closest to the origin lie only on the coordinate axes. Determine which of the following statements is true:

(A) All \(a\) in \((0, 1)\) are good and others are not good.
(B) All \(a\) in \((1,2)\) are good and others are not good.
(C) All \(a > 2\) are good and others are not good.
(D) All \(a > 1\) are good and others are not good.

Answer: C


Question: Determine the number of positive integers \( n \) such that the expression \( n + 2n^2 + 3n^3 + \cdots + 2005n^{2005} \) is divisible by \( n - 1 \).

Answer: 16


Question: Calculate the limit: $$\lim_{n\to\infty}\frac{5n^n}{3n!+3^n}$$

Answer: \infty


Question: Determine the number of homomorphisms from the group \(C_5 \times C_4 \times C_4\) onto \(C_{10}\).

Answer: 12


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(x) = f(x/2) + \frac{x}{2}f'(x) \). Find the value of \( f''(20) - f''(10) + 4f''(8) \).

Answer: 0


Question: Evaluate \( \lim_{n\to \infty} \int_n^\infty \left(1+\frac{x}{n} \right)^n e^{-x} \, dx = 0. \)

Answer: 0


Question: Determine the cardinality of the set of sequences of rational numbers that converge to 1.

Answer: 2^{\aleph_0}


Question: Identify the familiar space that is isometric to the metric identification of the pseudometric space $(C(\mathbb{I}), \mu)$, where the pseudometric $\mu$ is defined by $\mu(f, g) = |f(x_0) - g(x_0)|$ for functions $f, g \in C(\mathbb{I})$.

Answer: \mathbb{R}


Question: Determine the degree of the field extension $[\mathbb{Q}(\sqrt[3]{2}+\sqrt{5}):\mathbb{Q}]$. Provide your answer as a single integer.

Answer: 6


Question: Evaluate the limit: $$\lim_{x\to 0^+} \frac{\sqrt{\log^2(x)-\sin(x)}}{\log(x) + e^x}$$

Answer: -1


Question: Evaluate the integral \( \int_{-\infty}^{a} \delta(x-a) \, dx \), where \( \delta(x-a) \) is the Dirac Delta function.

Answer: 1


Question: Find all positive integer solutions $(a, b)$ to the equation $a^3 - 5a + 7 = 3^b$.

Answer: (1, 1)


Question: Let $(a_n)_{n \in \mathbb{N}}$ be a sequence such that all non-trivial subsequences converge. A subsequence $(a_{\phi(n)})$ is trivial if there exist $N$, $\alpha$, and $\beta$ such that for all $n \geq N$, $\phi(n) = \alpha n + \beta$. Does the sequence $(a_n)$ converge?

Answer: Yes


Question: Determine the minimum number of replications $M$ required for $Par(3)$ distributed random variables $X_1, X_2, \ldots, X_M$ such that $P(\mu - 0.05 \leq \bar{X}_M \leq \mu + 0.05) = 0.9$, where $\bar{X}_M$ is the sample mean and $Var(X_i) = 3/4$. Use Chebyshev's inequality to find $M$. Assume the mean $\mu$ is unknown.

Answer: 3000


Question: Let $X$ be a continuous non-negative local martingale with $X_0=1$ and $X_t \to 0$ almost surely as $t \to \infty$. For $a > 1$, define $\tau_a = \inf\{t \geq 0 : X_t > a\}$. Show that $\mathbb{P}(\tau_a < \infty) = \mathbb{P}(\sup_{t \geq 0} X_t > a) = 1/a$. Use the expected value of $X_{t \wedge \tau_a} = a \mathbf{1}_{\tau_a \leq t} + X_t \mathbf{1}_{\tau_a > t}$ to assist in your proof.

Answer: \frac{1}{a}


Question: Determine whether the function \( f(z) = \frac{\sin(2z)}{e^z - 1} \) is holomorphic at \( z = 0 \).

Answer: Yes


Question: Assume that $X$ has five path components. If $CX$ is the cone on $X$, determine the relative homology group $H_1(CX,X)$. Provide your answer in terms of an isomorphism to a known group.

Answer: \mathbb{Z}^4


Question: Let $a, b, c$ be the side lengths of an acute triangle $\triangle ABC$. Given that \( \frac{b}{a} + \frac{a}{b} = 6 \cos C \), find the value of \( \frac{\tan C}{\tan A} + \frac{\tan C}{\tan B} \).

Answer: 4


Question: Determine the values of \( \alpha \) for which the integral \( \int_{\mathbb{R}^n}\dfrac{1}{\Vert x\Vert^{\alpha}}\,dx \) is finite.

Answer: \emptyset


Question: Find the volume of the region bounded by the planes $4x+2y+4z=6$, $y=x$, $x=0$, and $z=0$.

Answer: \dfrac{3}{4}


Question:
Calculate the determinant of the symmetric matrix of order 2018 given by:
\[\begin{bmatrix}1^{2016} & 2^{2016} & \cdots & 2018^{2016}\\2^{2016} & 3^{2016} & \cdots & 2019^{2016}\\\vdots & \vdots & \ddots & \vdots\\2018^{2016} & 2019^{2016} & \cdots & 4035^{2016}\end{bmatrix}\]

Answer: 0


Question: Find the order of the group of symmetries of a dodecahedron using the formula: Order of G = (Order of stabilizer of s) x (Order of Orbit of s), where G is a group acting on a set S and s is an element of S.

Answer: 120


Question: Solve the linear recurrence relation \(a_n = \binom{9}{1}a_{n-1} - \binom{9}{2}a_{n-2} + \binom{9}{3}a_{n-3} - \dots + \binom{9}{9}a_{n-9}\) with the initial conditions \(a_0 = a_1 = a_2 = \dots = a_7 = 0\) and \(a_8 = 1\) for \(n \ge 9\).

Answer: \dbinom{n}{8}


Question: Consider a bipartite graph $G$ with partition $V = X \cup Y$. Suppose $G$ is not Hamiltonian but has a Hamiltonian path. Is it possible for $|X| = |Y|$? Provide a justification for your answer.

Answer: Yes


Question: Prove that any nonabelian group of order 6 is isomorphic to \( S_3 \).

Answer: S_3


Question: What is the maximum number of distinct prime factors that a number less than $2^{64}$ can have? In other words, find the largest value of $\omega(n)$ for $n < 2^{64}$.

Answer: 15


Question: Consider the field of rational numbers \( \mathbb{Q} \). Let \( K_1 \subseteq \mathbb{C} \) be the minimal splitting field of the family \( \{x^n-a : a \in \mathbb{Q}, n \geq 1\} \). Define \( K_2 \subseteq \mathbb{C} \) as the splitting field of the family \( \{x^n-a : a \in K_1, n \geq 1\} \). Continuing this process, we obtain an ascending chain \( \mathbb{Q} \subseteq K_1 \subseteq K_2 \subseteq \cdots \) of subfields of \( \mathbb{C} \). Since \( K_1, K_2, \ldots \) are algebraic extensions of \( \mathbb{Q} \), they are subfields of \( \overline{\mathbb{Q}} \), the algebraic closure of \( \mathbb{Q} \) in \( \mathbb{C} \). Let \( K = \cup_i K_i \). Is \( K \) a proper subfield of \( \overline{\mathbb{Q}} \)?

Answer: Yes


Question: Evaluate the limit: \( \lim_{z\to i\pi} \left( z-i\pi \right)\frac{e^z}{1+e^z} \)

Answer: 1


Question: Let \( x_1, x_2, x_3, x_4, x_5 \) be the roots of the polynomial equation \( x^5 + x + 1 = 0 \). Calculate the value of \( \prod_{k=1}^{5} (2+x_k^2) \).

Answer: 51


Question: Determine whether the Mobius transformation \( f_{a}(z) = \frac{z-a}{1-\overline{a}z} \), with \( a = -0.5 \), maps the unit circle to itself.

Answer: Yes


Question: Find the residue of the function \( f(z) = \frac{1}{z - \sin z} \) at \( z = 0 \).

Answer: \dfrac{3}{10}


Question: Prove by induction that for any well-formed formula \( \alpha \), the number of sentence symbols \( s \) is equal to the number of binary connective symbols \( c \) plus one, i.e., \( s = c + 1 \).

Answer: s = c + 1


Question: Determine the number of values of $x$ that satisfy the equation \((\sqrt{2})^x + (\sqrt{3})^x = (\sqrt{13})^{\frac{x}{2}}\).

Answer: 1


Question: In a javelin throwing competition, each athlete throws independently, and the distances are exponentially distributed with parameter \( \lambda \). The first competitor will lead the competition until the \( N \)-th opponent throws the javelin a greater distance and takes the lead. Calculate the expected value \( E(N) \).

Answer: \infty


Question: Let \(x, y, z\) be real numbers such that \(x^2 + y^2 + z^2 = 1\). Determine the maximum value of \(\sqrt{6}xy + 4yz\) and find \(2A^2 - 4\), where \(A\) is this maximum value.

Answer: 7


Question: Consider a set \( A \) with six elements. Determine the number of functions \( f: A \to A \) such that the image of \( f \) contains exactly three elements.

Answer: 10800


Question: Let $U$ be a uniform random variable on the interval $[-\pi, +\pi]$. Consider a sequence of random variables $X_k = \cos(kU)$ for $k = 1, 2, \ldots$. Define $S_n = X_1 + X_2 + \ldots + X_n$. Determine $\lim_{n \rightarrow \infty} P\left(\left|\frac{S_n}{n}\right| < \epsilon\right)$ for a given $\epsilon > 0$.

Answer: 1


Question: Let \( A \subseteq \mathbb{R}^n \) be such that for every continuous function \( f : A \to \mathbb{R} \), the image \( f(A) \) is closed in \( \mathbb{R} \). Is the set \( A \) closed in \( \mathbb{R}^n \)?

Answer: Yes


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

Answer: \infty


Question: Calculate the expected value of the squared distance from the origin for a simple symmetric random walk in \(\mathbb{Z}^2\) at time \(n\).

Answer: n


Question: Evaluate the limit: \[ \lim_{n\to\infty} \left( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{2n} - \ln n \right) \]

Answer: \gamma + \ln 2


Question: Four machines, M1, M2, M3, and M4, process a binary code sequentially. M1 produces a binary code 0 with probability \( \frac{1}{2} \) and code 1 with probability \( \frac{1}{2} \), then passes it to M2. Machines M2, M3, and M4 each change the code to its complement with probability \( \frac{1}{3} \) or leave it unchanged with probability \( \frac{2}{3} \). Given that the final output is 0, what is the probability that M1 originally produced a 0?

Answer: \dfrac{14}{27}


Question: Compute the greatest common divisor (GCD) of the polynomials \(1+x+x^2+x^3\) and \(1+2x+3x^2\) over \(\mathbb{Z}[x]\).

Answer: 1


Question: Express the module $M=\mathbb{Z}^3/N$ as a direct sum of cyclic modules, where $N=\{(x,y,z)\in \mathbb{Z}^3\;|\; 2x+3y-5z=0\} \subset \mathbb{Z}^3$. Provide the decomposition.

Answer: \mathbb{Z}


Question: Given a triple of digits \((a, b, c)\), where \(a, b, c \in \{0, 1, \dots, 9\}\), you can guess a triple of digits \((m, n, p)\) each turn. You will be informed whether the statement \(a = m\) or \(b = n\) or \(c = p\) is true, but not which specific equality holds. What is the minimum number of turns required to guarantee determining the triple \((a, b, c)\)?

Answer: 10


Question: The number of workplace injuries, $N$, occurring in a factory on any given day is Poisson distributed with mean $\lambda$. The parameter $\lambda$ is a random variable uniformly distributed on the interval $[0,3]$. Calculate $\text{Var}[N]$. Provide your answer as a single number.

Answer: \dfrac{9}{4}


Question: Let $f(x) = a_nx^n + a_{n−1}x^{n−1} + \\ldots + a_0$ be a polynomial of degree $n > 0$ with integer coefficients and $a_n \neq 0$. Let $p$ be a prime number and $r$ an integer. Prove that if $f(r) \equiv 0 \pmod{n}$, then there exists a polynomial $g(x)$ of degree $n − 1$ such that $(x−r)g(x) = a_nx^n + a_{n−1}x^{n−1} + \\ldots + a_1x + b_0$ where $a_0 \equiv b_0 \pmod{p}$. Determine whether this statement is true or false.

Answer: False


Question: Evaluate the limit \( \lambda = \lim\limits_{n \to \infty}{n\int_0^{\frac{\pi}{2}}(\sin x)^{2n} dx} \) without using Wallis' integral formula.

Answer: \infty


Question: Find a $C^∞$ map $f: X \to Y$ where $X \subseteq \mathbb{R}^m$ and $Y \subseteq \mathbb{R}^n$, such that $f$ is one-to-one and onto, its inverse is continuous, but $f$ is not a diffeomorphism.

Answer: f(x) = x^3


Question: Two teams play a series of games where the first team to win 4 games is declared the overall winner. Each team has an equal probability of 0.5 of winning each game. What is the expected number of games played in the series?

Answer: \dfrac{93}{16}


Question: Find the function \( g(x) \) such that \( g: \mathbb{R}^{+} \rightarrow \mathbb{R} \) is differentiable and satisfies the functional equation \( 2g(x) = g(xy) + g\left(\frac{x}{y}\right) \) for all \( x, y \in \mathbb{R}^{+} \), with the conditions \( g(1) = 0 \) and \( g'(1) = 1 \).

Answer: \ln x


Question: Given a power series \( \sum_{n=0}^{\infty} a_n x^n \) with a radius of convergence \( R \), determine the radius of convergence of the series \( \sum_{n=0}^{\infty} a_n^3 x^n \).

Answer: R^3


Question: Given an element \( v = (v_1, \dots, v_d) \in \mathbb{Z}^d \) with \( \gcd(v_1, \dots, v_d) = 1 \), there exists a regular integer matrix \( A \in \mathop{GL}_d(\mathbb{Z}) \) such that \( A(v) = (1, 0, \dots, 0) \). Assume \( |v_1| + \dots + |v_d| = n \). Determine the smallest \( \ell \in \mathbb{N} \) such that the image \( A\left([-1, 1]^d\right) \) is contained in the cube \( [-\ell, \ell]^d \). Find the asymptotic behavior of \( \ell(n) \) up to a multiplicative constant.

Answer: \Theta(n)


Question: Evaluate the integral: \[ \int_{\mathbb{R}^3} \frac{e^{-k |\vec{x}|}}{|\vec{x}|^2} \, \mathrm{d} \vec{x} \] where \( k > 0 \).

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


Question: Calculate \( \mathbb{E}[X^4] \) for a random variable \( X \sim \mathcal{N}(0,1) \).

Answer: 3


Question: Does there exist an infinite locally finite group of finite rank and bounded exponent?

Answer: No


Question: Find the last digit of $AD^{AD}$ in hexadecimal when expressed in octal.

Answer: 5


Question: Find an integer $x$ such that $x^2 \equiv 7 \pmod{2579}$. If no such integer exists, state that 7 is not a quadratic residue modulo 2579.

Answer: 88


Question: Prove that if a matrix is both Hermitian and unitary, then all its eigenvalues are ±1.

Answer: \pm 1


Question: Given a $3 \times 3$ matrix $A$ with rational entries such that $A^3 = 2A$ and $A$ is not the zero matrix, calculate the trace of $A^2$, denoted as $\text{tr}(A^2)$. It is known that the determinant of $A$, $\det(A)$, is zero.

Answer: 4


Question:
Find the equilibrium position of an object that satisfies the equation of motion given by:
\[ 4\frac{d^{2}x}{dt^{2}} = -x^{3} + x^{2} - x + 1 \]

Answer: 1


Question: Evaluate the integral \( \int_{\gamma} \frac{z^2 + 3\cos(z)}{z^2 - 4} \, dz \), where \( \gamma \) is the contour of an equilateral triangle inside \( |z| = 2 \), with \(-2i\) as one of its vertices.

Answer: 0


Question: Calculate the value of \( \left\lfloor \sum_{n=2}^{10000} \frac{1}{\sqrt{n}} \right\rfloor \).

Answer: 197


Question: Given a random variable \(X\) with probability density function \(f_X(x;\theta) = \frac{1}{2\theta}\) for \(-\theta < x < \theta\), and zero otherwise, determine whether \(|X|\) is a sufficient statistic for \(\theta\).

Answer: Yes


Question: Let the complex function \( f(z) = u(z) + iv(z) \) be analytic on \( |z| < 1 \) with \( f(0) = 2 + i \). If \( r \) is a real number satisfying \( 0 < r < 1 \), find the value of the integral \( \int_{0}^{2\pi} (u^2(re^{i\theta}) - v^2(re^{i\theta})) \, d\theta \).

Answer: 6\pi


Question: Evaluate the line integral \( \oint_C (4y^3 + \cos x^2) \, dx - (4x^3 + \sin y^2) \, dy \), where \( C \) is the boundary of the disk centered at the origin with radius 2, oriented clockwise.

Answer: 96\pi


Question: A fair die with \( n \) sides is rolled. If the \( r \)-th face is shown, \( r \) balls are removed from an urn containing \( 2n \) balls (\( n \) tangerine and \( n \) heliotrope) and placed in a bag. What is the probability that a ball removed at random from the bag is tangerine?

Answer: \dfrac{1}{2}


Question: Evaluate the integral: \[ \frac{2}{\pi}\int_{-\pi}^\pi\frac{\sin\frac{9x}{2}}{\sin\frac{x}{2}}dx \]

Answer: 4


Question: Determine whether the infinite series \( \sum_{n=0}^{\infty} \frac{(n+1)(n+2)}{2^n} \) converges to 16.

Answer: 16


Question: Evaluate the integral \( \int_0^{2\pi} x \sin^6 x \cos^5 x \, dx \).

Answer: 0


Question: Simplify the function \( f(x) = \lim\limits_{n \to \infty}\frac{\ln(e^n+x^n)}{n} \) for \( x > e \).

Answer: \ln x


Question: Solve for \( x \) in the equation: \[ \frac{(1+x)^{36} -1}{x} = \frac{20142.9}{420} \]

Answer: 0.0158


Question: Find the integer \(k\) such that \[ 1 + \sum_{r=0}^{18} (r(r+2)+1)r! = k! \]

Answer: 20


Question: For certain positive integers \(a, b, c,\) and \(d\), the set of complex numbers \(z\) that satisfy \(|z-5\sqrt{3}-5i|=5\) is equivalent to the set of complex numbers \(z\) that satisfy \(\left| \frac{1}{z} - \frac{1}{a \sqrt{b}} + \frac{i}{c} \right| = \frac{1}{d}\), where \(b\) is not divisible by the square of a prime. Find \(a+b+c+d\).

Answer: 38


Question: Given $n \equiv 0 \,(\mathrm{mod}\,4)$, determine the minimum value of $m$ such that there exists an $m \times n$ partial Hadamard matrix that cannot be extended to an $(m+1) \times n$ partial Hadamard matrix.

Answer: \frac{n}{2}


Question: Given an integer \( x \) and an odd integer \( y \), determine the number of consecutive odd numbers starting from \( y \) that need to be added to \( x \) to result in a perfect square. For example, if \( x = 20 \) and \( y = 39 \), the perfect square is 100, achieved by adding 39 and 41 to 20. Find the number of consecutive odd numbers needed.

Answer: 2


Question: Compute the line integral \( \int_{\gamma} F \cdot ds \), where \( F(x,y) = (x,y^2) \) and \( \gamma \) is the curve of the parabola \( x=3y^2 \) from \((0,0)\) to \((3,1)\).

Answer: \dfrac{29}{6}


Question:
Consider the sequence \(\langle x_n\rangle\), \(n\ge 0\), defined by the recurrence relation \(x_{n+1} = c \cdot x_n^2 - 2\), where \(c > 0\). Suppose there exists a non-empty, open interval \((a,b)\) such that for all \(x_0\) satisfying \(a < x_0 < b\), the sequence converges to a limit. Determine the value to which the sequence converges.

A. \((1+\sqrt{1+8c})/2c\)
B. \((1-\sqrt{1+8c})/2c\)
C. \(2\)
D. \(2/(2c - 1)\)

Answer: B


Question: Consider a symmetric random walk starting at the origin where the positions are non-negative integers. Each move goes right or left with probability \( \frac{1}{2} \), except if you are at the origin, you always go right. Let \( X \) be the number of steps to reach position 10. Does \( X^2 \) have a mean?

Answer: Yes


Question: Calculate the expected number of consecutive random number inputs required to unlock a lock with the combination 5556, given that the lock checks for the sequence without needing a reset between inputs.

Answer: 10000


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 $x^2 + xy + y^2 = 0$.

Answer: 0


Question: Solve the partial differential equation \( u_x - 6u_y = u \) using the method of characteristic curves, given the Cauchy data \( u(x,y) = e^x \) on the line \( y = -6x + 2 \).

Answer: e^x


Question: For integers $a, b > 0$ where the ratio $\frac{a}{b}$ is not a power of 10, determine if the set of integers $n \ge 0$ such that the digit sum $s(an) = s(bn)$ has zero density.

Answer: Yes


Question: Calculate the volume of the region bounded by the hyperboloid \(x^2 + y^2 - 3z^2 = 1\) and the planes \(z = 1\) and \(z = -1\).

Answer: 4\pi


Question: Given the 0-form \(\omega = xy \, dx + 3 \, dy - yz \, dz\), directly compute \(d(d\omega)\) and verify that it equals zero.

Answer: 0


Question: Determine if there exists a value \( \alpha^* \in (0,1) \) such that \( \lim_{n \to \infty} T_{\alpha^*}(p_n) = k \) for some real number \( k \), where \( T_{\alpha}(p_n) = p_n^{\alpha} \prod_{i=1}^n \left( 1 - \frac{1}{p_i^{\alpha}} \right) \). If such an \( \alpha^* \) exists, find the value of \( k \).

Answer: 0


Question: Consider the sequence of integers $a_1, a_2, \ldots$ defined recursively by $a_1 = 2013$ and for $n \ge 1$, $a_{n+1}$ is the sum of the 2013th power of the digits of $a_n$. Determine whether there exist distinct positive integers $i$ and $j$ such that $a_i = a_j$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \oint_C \frac{1}{z - 10 - 12i} \, dz \) around the rectangle with vertices \(7+7i, 12+7i, 12+15i,\) and \(7+15i\) in the anti-clockwise direction. Is the value of the integral \( \frac{\pi i}{2} \)?

Answer: 2\pi i


Question: Given a $4\times4$ matrix with rows $v_1$, $v_2$, $v_3$, and $v_4$ and a determinant of $-1$, determine the determinant of the matrix \( \begin{pmatrix} 2v_1 + 3v_4 \\ v_2 \\ v_3 \\ 9v_1 + 2v_4 \end{pmatrix} \).

Answer: 23


Question: An icosahedron is a regular polyhedron with 20 equilateral triangular faces. If the faces are colored either blue or white, what is the maximum number of blue faces that can be colored such that no two blue faces are adjacent?

Answer: 8


Question:
Given the following probabilities:

- \( P[W|T] = 0.8 \)
- \( P[W|T \cap G] = 0.65 \)
- \( P[W | G' \cap T] = 1 \)

Find \( P[G'|T] \).

Answer: \dfrac{3}{7}


Question: Evaluate the integral \( \int_0^1 \log\left(\frac{1+ax}{1-ax}\right)\,\frac{{\rm d}x}{x\sqrt{1-x^2}} \) and show that it equals \( \pi\arcsin(a) \) for \( |a| \leq 1 \).

Answer: \pi \arcsin(a)


Question:
Find the relation between $x$ and $y$ by eliminating the parameter $t$ from the equations:
\[ x = \frac{3 + 3t^2}{t^2 - 1} \quad \text{and} \quad y = \frac{3t}{t^2 - 1} \]

Answer: x^2 - 4y^2 = 9


Question:
Determine all real numbers \(a\) for which the limit \(\lim_{n\to \infty}a^nA^n\) exists and is non-zero, where \(A\) is the \(3 \times 3\) matrix:
\[
A=\begin{pmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{pmatrix}.
\]

Answer: \dfrac{1}{3}


Question: Let \( \kappa \geq \aleph_0 \) be a cardinal, and suppose that \( \mathcal{U} \) is a non-principal ultrafilter on \( \kappa \). Consider \( \mathcal{U} \) as a poset \((\mathcal{U}, \subseteq)\). If there exist posets \( P \) and \( Q \) such that \( \mathcal{U} \cong P \times Q \), does this imply that one of \( P \) or \( Q \) consists of only one point?

Answer: Yes


Question: Determine the dimension of the ideal \( I = \langle x+1 \rangle / \langle x^9+1 \rangle \) in the ring \( R = \mathbb{Z}_2[x]/\langle x^9+1 \rangle \).

Answer: 8


Question: Find the unique root of the equation \( F(x) = 0 \) for a continuously differentiable function \( F(x) \) defined on the interval \([a,b]\) with \( F(a)<0 \) and \( F(b)>0 \), given that \( 0<K_1\leq F'(x)\leq K_2 \) for \( a\leq x\leq b \). Use the contraction mapping theorem by defining \( f(x) = x - \lambda F(x) \) and show that \( f(x) \) is a contraction mapping.

Answer: c


Question: Evaluate the integral \( \int_{0}^{\infty} \Phi(x) \phi(x) \, dx \), where \( \Phi(x) \) is the cumulative distribution function and \( \phi(x) \) is the density function of a standard normal variable \( N(0,1) \).

Answer: \dfrac{3}{8}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^n \frac{1}{n+n^2 \sin \left( \frac{x}{n^2} \right)} \, \mathrm{d}x. \]

Answer: \ln 2


Question: Evaluate the surface integral \(\iint_{S} \frac{\mathbf{x}}{||\mathbf{x}||} \cdot d\mathbf{S}\), where \(S\) is the surface defined by \(x^3 + y^3 + z^3 = a^3\).

Answer: 4\pi a^2


Question:
Let $A$ be an $n \times n$ invertible matrix. Define a function $F: \mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ by $F(x,y) = \langle Ax, y \rangle$. Let $DF(x,y)$ denote the derivative of $F$ at $(x,y)$, which is a linear transformation from $\mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$. Determine which of the following statements is true:

1) If $x \neq 0$, then $DF(x,0) \neq 0$.
2) If $y \neq 0$, then $DF(0,y) \neq 0$.
3) If $(x,y) \neq (0,0)$, then $DF(x,y) \neq 0$.
4) If $x = 0$ or $y = 0$, then $DF(x,y) = 0$.

Choose the correct statement.

Answer: 3


Question: Calculate the integral \( \int_{-\infty}^{\infty} \frac{dx}{(x^2 + 1)^3} \) using the residue theorem.

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


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x \) and \( y \), the equation \( f(x^2 + f(y)) = y + f^2(x) \) holds, where \( f^2(x) = f(x) \cdot f(x) \).

Answer: f(x) = x


Question: Find the remainder when \((34! + 75^{37})^{39}\) is divided by 37.

Answer: 14


Question: Find the residue of the function \( \sin \left(\frac{z}{z-1}\right) \) at the point \( z = 1 \).

Answer: \cos(1)


Question: Let \( f(x) \) be a strictly increasing and differentiable function. Evaluate the limit: \[ \lim_{x \to 0} \frac{f(x^2) - f(x)}{f(x) - f(0)}. \]

Answer: -1


Question: Let $A$ and $B$ be subsets of the real numbers such that $|A| = |B| = n$. Determine the best-known upper bound on the number of four-tuples in $A \times B$ where the four points are co-circular.

Answer: O(n^{8/3})


Question: Let $A$ be an $n \times n$ nonsingular complex matrix with all real eigenvalues. Suppose $A$ satisfies the condition that \( \text{trace}(A^2) = \text{trace}(A^3) = \text{trace}(A^4) \). Determine the trace of the matrix $A$. \( \text{trace}(A) \).

Answer: n


Question: Find the function \( f(x) \) that satisfies the functional equation \( f(x+1) = x f(x) \).

Answer: \Gamma(x)


Question: Determine the number of roots of the equation \( \frac{1}{2}e^z + z^4 + 1 = 0 \) that lie in the left half-plane \( \text{Re}(z) < 0 \).

Answer: 2


Question: Determine the 100th digit to the right of the decimal point in the decimal representation of \((1 + \sqrt{2})^{3000}\).

Answer: 9


Question: Determine whether the following statement is true or false: Suppose that $A_1,A_2,\ldots ,A_m$ are $m$ linearly independent real $n\times n$ matrices such that $A_iA_j=0$ for all $i\neq j$. Then $m\le n$. Provide a proof or a counterexample.

Answer: True


Question: Determine whether the following statement is true or false: "In solving a linear program by the simplex method, starting with a feasible tableau, a different feasible point is generated after every pivot step."

Answer: False


Question: A fair die is thrown repeatedly. Let $X_n$ denote the sum of the first $n$ throws. Determine $\lim_{n\rightarrow \infty}P(X_n \text{ is a multiple of 13})$. Provide your answer as a single probability value.

Answer: \dfrac{1}{13}


Question: Compute the derivative of the functional \( J[f] = \int_{0}^{1} f'(x)^2 \, \mathrm{d}x \) with respect to the function \( f \).

Answer: -2 f''(x)


Question: Is it true that for any joint probability distribution function $f(x,y)$ of two continuous random variables, the integral \( \int_{-\infty}^{\infty} f(t,t) \, dt \) is necessarily upper bounded by 1? Provide a justification for your answer.

Answer: No


Question: Find the maximum value of the function \( f(x,y,z) = x^2y^2z \) subject to the constraints \( x + y + z \leq 5 \) and \( x, y, z \geq 0 \).

Answer: 16


Question: Find a real number \(a\) such that the function \(g_a(x) = \begin{cases} x^a \sin \left(\frac{1}{x}\right) & x \neq 0 \\ 0 & x = 0 \end{cases}\) is differentiable on \(\mathbb{R}\) and its derivative \(g_a'(x)\) is unbounded on the interval \([0,1]\).

Answer: \dfrac{3}{2}


Question: Let $p$ be a prime number. Prove that if $p \mid a^p - b^p$, then $p^2 \mid a^p - b^p$.

Answer: p^2 \mid a^p - b^p


Question: Evaluate the definite integral \( \int_{-\infty}^{+\infty} \frac{x^2 - 7x + 1}{x^4+1} \, \mathrm{d}x \).

Answer: \pi \sqrt{2}


Question: Given the polynomial \(7X^4 - 14X^3 - 7X + 2 = 0\), find the sum of the cubes of its roots. Let \(x_1, x_2, x_3, x_4\) be the roots of the polynomial. Use Viete's formulas to determine \(x_1^3 + x_2^3 + x_3^3 + x_4^3\).

Answer: 11


Question: How many elements of the set of all $2 \times 2$ matrices with entries in the field with 7 elements are similar to the matrix $\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$?

Answer: 56


Question: Let $J$ be the all-one square matrix. Determine if it is true that over any field, for all square matrices $A$, the inequality \(|\operatorname{rank}(J \pm A) - \operatorname{rank}(A)| \leq c\) holds for some positive constant $c$.

Answer: 1


Question: Let $\Gamma$ be a graph such that for any vertex $v$, $\deg v \geq 50$, and there exist two vertices $a$ and $b$ such that $d(a,b) = 8$. Prove that the number of vertices $|V| \geq 200$. Provide a detailed explanation of your reasoning.

Answer: 200


Question: Solve the equation $x^2 + y^2 = pz^2$ for $x, y, z \in \mathbb{N}$, where $p$ is a prime such that $p \equiv 3 \pmod{4}$. Prove that the only solution is $x = y = z = 0$.

Answer: x = y = z = 0


Question: Given the polynomial \( W(x) = -7x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 \), where \( W(1) = -2 \), \( W(2) = -4 \), \( W(3) = -6 \), \( W(4) = -8 \), and \( W(5) = -10 \), find the constant term \( a_0 \).

Answer: 840


Question: Consider the real-valued one-dimensional Schrödinger operator with potential $V(x)$, defined as $L: H^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$ by the expression $$L(u)=-u''+V(x)u$$ where $V(x)$ is bounded. Is this operator self-adjoint?

Answer: Yes


Question: Find the limit of the sequence defined by the recurrence relation \(x_{n+1} = x_n(2 - ax_n)\), where \(a\) is a positive real number.

Answer: \dfrac{1}{a}


Question: Calculate the line integral \( \int_C \frac{z\arctan(z)}{\sqrt{1+z^2}}\,dz + (y-z^3)\,dx - (2x+z^3)\,dy \), where the contour \( C \) is defined by the equations \( \sqrt{1-x^2-y^2}=z \) and \( 4x^2+9y^2 = 1 \).

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


Question: In the finite field \( \mathbb{F}_{p^n} \), where \( p \) is a prime number and \( n > 1 \), does there exist a subgroup of the additive group \( (\mathbb{F}_{p^n}, +) \) of order \( p^k \) for any integer \( k \) such that \( 1 < k < n \)?

Answer: Yes


Question: Is a centerless Lie group necessarily connected? Consider a Lie group $G$ with $Z(G) = \{g \in G : hg = gh \text{ for all } h \in G\} = \{e\}$. Determine if $G$ must be connected.

Answer: No


Question: In a parallelogram $ABCD$, points $E$ and $F$ are located on segments $AD$ and $CD$, respectively, such that \( \frac{|AE|}{|ED|} = \frac{|DF|}{|FC|} = \frac{1}{2} \). Determine the ratio in which the line segment $\overline{EF}$ divides the diagonal $\overline{BD}$. Use the theorem about the composition of two homotheties: If $\mathcal{H}_{M,k_1}$ and $\mathcal{H}_{N,k_2}$ are homotheties, then their composition $\mathcal{H}_{M,k_1} \circ \mathcal{H}_{N,k_2}$ is again a homothety $\mathcal{H}_{S,k}$ with $k = k_1k_2$ (if $k \neq 1$), and its center $S$ lies on the line $MN$. What is the ratio?

Answer: \dfrac{7}{2}


Question: Let $F$ be a field and $(K,*)$ be an associative $F$-algebra which, as a vector space, is finitely generated over $F$. Given an element $a \in K$, do there necessarily exist elements $a_1, a_2 \in K$ such that $a_1 * a_2 = a$?

Answer: No


Question: Evaluate the limit \( \lim_{x\rightarrow{\infty}} \frac{\int_{0}^{1} (1+(xy)^{n})^{\frac{1}{n}}dy}{x} \) where \( n\geq 2 \).

Answer: \dfrac{1}{2}


Question:
Let \( a, b, c \) be real numbers with \( a \neq 0 \). Suppose \( \alpha \) is a root of the equation \( a^2x^2 + bx + c = 0 \) and \( \beta \) is a root of the equation \( a^2x^2 - bx - c = 0 \), with \( 0 < \alpha < \beta \). Determine which of the following is always true for a root \( \gamma \) of the equation \( a^2x^2 + 2bx + 2c = 0 \):

A) \( \gamma = \frac{\alpha + \beta}{2} \)

B) \( \gamma = \alpha + \frac{\beta}{2} \)

C) \( \gamma = \alpha \)

D) \( \alpha < \gamma < \beta \)

Answer: D


Question: Show that the equation $\tan^{-1}x = kx$, where $k$ is a constant, has only one real root when $k < 0$. State the value of this root.

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to\infty}\left(\frac{\ln(x-2)}{\ln(x-1)}\right)^{x\ln x}. \]

Answer: \dfrac{1}{e}


Question: Determine the number of zeros of the polynomial \( h(z) = z^5 + 3z - 2 \) that lie within the unit disk \( D_1(0) \) using Rouche's Theorem.

Answer: 1


Question: Calculate the number of integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 = 30\) where each \(x_i\) is in the range \(-10 \leq x_i \leq 20\).

Answer: 17266


Question: Is the size of the Galois orbit of an algebraic number \( \alpha \) under the action of \( \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \) bounded above by the degree of the splitting field of \( \alpha \)?

Answer: Yes


Question: Calculate the volume of the region \( \Omega = \{(x,y,z) \in \mathbb{R}^3 \mid (\frac{x}{2})^2 + y^2 + z^2 \le 1\} \).

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


Question:
Consider the following sets of functions on \( \mathbb{R} \):

- \( W \): The set of all constant functions on \( \mathbb{R} \)
- \( X \): The set of polynomial functions on \( \mathbb{R} \)
- \( Y \): The set of continuous functions on \( \mathbb{R} \)
- \( Z \): The set of all functions on \( \mathbb{R} \)

Which of the sets has the same cardinality as that of \( \mathbb{R} \)?

A) Only \( W \)
B) Only \( W \) and \( X \)
C) Only \( W, X \) and \( Y \)
D) All of \( W, X, Y \) and \( Z \)

Answer: C


Question: Given the joint probability density function (PDF) \( f_{XY}(x,y) = 8xy \) for \( 0 < x < 1 \) and \( 0 < y < x \), find the probability density function \( f_Z(z) \) where \( Z = \frac{Y}{X} \). Determine the bounds of integration needed to calculate the cumulative distribution function \( F_Z(z) = P(Z < z) \).

Answer: 2z


Question: Determine the number of 4-tuples \((a,b,c,d) \in \mathbb{N}^{4}\) such that \(\mathrm{lcm}(a,b,c) = \mathrm{lcm}(b,c,d) = \mathrm{lcm}(c,d,a) = \mathrm{lcm}(d,a,b) = 2^{5}3^{4}5^{3}7^{2}\).

Answer: 42723153


Question: Suppose that $f(x)$ is bounded on the interval $[0,1]$, and for $0 < x < 1/a$, we have $f(ax) = bf(x)$, where $a, b > 1$. Calculate $$\lim_{x \to 0^+} f(x).$$

Answer: 0


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

Answer: 2\sqrt{2}


Question:
Given the real numbers $x$, $y$, and $z$ that satisfy the following system of equations:
\[\begin{cases} x + y + z = 3 \\
x^2 + y^2 + z^2 = 9 \\
xyz = -2 \end{cases}\]
Find the value of $x^4 + y^4 + z^4$. Provide your answer as a single number.

Answer: 57


Question:
In a projective space $\mathbb{P}^{k+2}$, consider two complementary subspaces $\mathbb{P}^1$ and $\mathbb{P}^k$, and let $C\subset\mathbb{P}^k$ be a degree $k$ rational normal curve. Given an isomorphism $\phi:\mathbb{P}^1\rightarrow C$, consider the rational normal scroll
$$S_{(1,k)} = \bigcup_{p\in \mathbb{P}^1}\left\langle p, \phi(p)\right\rangle\subset \mathbb{P}^{k+2}$$
where $\left\langle p, \phi(p)\right\rangle$ is the line through $p$ and $\phi(p)$. Let $\Sigma_{(1,k)}$ be the Segre embedding of $\mathbb{P}^1\times \mathbb{P}^k$ in $\mathbb{P}^{2k+1}$. Can $S_{(1,k)}$ be obtained by intersecting $\Sigma_{(1,k)}$ with a linear subspace of dimension $k+2$ in $\mathbb{P}^{2k+1}$?

Answer: Yes


Question: Determine if 47 is the largest number that can be uniquely partitioned into five parts (15, 10, 10, 6, 6), such that no two parts are relatively prime.

Answer: 47


Question: Let $p$ be a prime number other than $2$ or $5$. Let $n$ be the smallest positive integer such that $10^n \equiv 1 \pmod{p}$. Does $10^n - 1$ divide $10^{p-1} - 1$?

Answer: Yes


Question: Determine whether the sequence \(a_n = \frac{(n+1)^{100}}{e^{\sqrt{n}}}\) for \(n \geq 1\) is convergent.

Answer: 0


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

Answer: [-1, 1)


Question: Given the equations $x + y + z = 0$, $x^3 + y^3 + z^3 = 9$, and $x^5 + y^5 + z^5 = 30$, find the value of $xy + yz + zx$. Express your answer in terms of $k$. \( \boxed{k} \)

Answer: -2


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left[ \int_{-\infty}^{x \ln n} \frac{1}{\sqrt{2\pi}} e^{-\frac{y^2}{2}} \, dy \right]^n \] where \( x > 0 \).

Answer: 1


Question: Evaluate the integral: \[ \int_{-\pi/4}^{\pi/4}{ (\cos{t} + \sqrt{1 + t^2}\sin^3{t}\cos^3 {t})\,dt} \]

Answer: \sqrt{2}


Question: Let $f:[0,1]\rightarrow \mathbb{R}$ be a bounded function such that $f$ is integrable on $[\delta,1]$ for every $0<\delta<1$. Determine whether $f$ is integrable on $[0,1]$. Answer true or false.

Answer: A


Question: Given a set of points within a time range from 0 to N, select 4 points such that the distance between each selected point is maximized. For example, if the range is 0-10 and the points are {1, 2, 4, 5, 7, 9, 10}, determine the 4 points that maximize the distance between them.

Answer: \{1, 4, 7, 10\}


Question: Find the smallest integer \( n \) such that \( \sum_{i=0}^{n}\frac{e^{-1500}1500^i}{i!} \ge 0.95 \).

Answer: 1564


Question: Find the norm of the vector \( v \) given the following conditions: \( \angle uv = \frac{\pi}{4} \), \( ||u|| = 3 \), and \( (u-v) \perp u \).

Answer: 3\sqrt{2}


Question: Let $f$ be a $C^\infty(\mathbb{R},\mathbb{R})$ function such that $\lim_{x \to \infty} f(x) = L$ and $f''$ is bounded. Prove that $\lim_{x \to \infty} f'(x) = 0$. Provide a detailed proof of this statement.

Answer: 0


Question: Is the set of discontinuity points, which are all jumps, of a càdlàg function necessarily countable?

Answer: Yes


Question: In triangle \(ABC\), point \(P\) is located inside such that \(\angle ABP = 20^{\circ}\), \(\angle PBC = 10^{\circ}\), \(\angle ACP = 20^{\circ}\), and \(\angle PCB = 30^{\circ}\). Determine the measure of \(\angle CAP\) in degrees.

Answer: 20


Question: Determine the value of \( c \) that makes the Riemannian metric \( g = (1-x^2-y^2-z^2)^{2c}(dx \otimes dx + dy \otimes dy + dz \otimes dz) \) geodesically complete on the open unit ball \( B = \{ (x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 <1 \} \).

Answer: -1


Question:
Given some $4$-tuples of positive real numbers $(a_1,b_1,c_1,d_1),\dots,(a_n,b_n,c_n,d_n)$ with all $a_i,b_i,c_i,d_i\leq 1$, determine the best possible value $k$ between $1$ and $3$ such that we can always partition $\{1,2,\dots,n\}$ into two subsets $X,Y$ satisfying:
\[1+\sum_{i \in X}a_i\geq \sum_{i \in Y}a_i, \quad 1+\sum_{i \in X}b_i\geq \sum_{i \in Y}b_i,\]
\[\sum_{i \in X}c_i\leq 1+\sum_{i \in Y}c_i, \quad \sum_{i \in X}d_i\leq 1+\sum_{i \in Y}d_i.\]

Answer: 2


Question:
Let \( A \) be a \( 3 \times 3 \) real-valued matrix such that \( A^3 = I \) and \( A \neq I \). What is the trace of \( A \)?

(a) 0
(b) 1
(c) -1
(d) 3

Answer: A


Question: Find the volume of the solid obtained by revolving the region bounded by \( y = \sin\left(\frac{\pi x}{12}\right) \) and the x-axis about the line \( x = -1 \) using the method of cylindrical shells.

Answer: 336


Question: Given a linear transformation $T$ on a finite vector space $V$ such that $Ker(T) = Im(T)$, determine the minimal polynomial of $T$.

Answer: x^2


Question:
Evaluate the limit:
\[ L = \lim_{x \to 0^+} x \left( \left[ \frac{1}{x} \right] + \left[ \frac{2}{x} \right] + \left[ \frac{3}{x} \right] + \cdots + \left[ \frac{12}{x} \right] \right) \]
where \([t]\) denotes the greatest integer less than or equal to \(t\).

Answer: 78


Question: Evaluate the limit \( \lim_{n\to\infty}\frac{x_n}{\sqrt n} \), where \( x_1 > 0 \) and \( x_{n+1} = x_n + \frac{1}{x_n} \) for \( n \in \mathbb{N} \).

Answer: \sqrt{2}


Question: Determine the limit of the sequence \( \left(\frac{1}{n^2}+\frac{2}{n^2-1}+\frac{3}{n^2-2}+ \cdots + \frac{n}{n^2-n+1}\right) \) as \( n \to +\infty \).

Answer: \dfrac{1}{2}


Question: Find the distribution of the ratio \( \frac{\bar{X}}{\bar{Y}} \), where \( \bar{X} \) and \( \bar{Y} \) are the sample means of two independent random samples from the same exponential distribution with parameter \( \lambda > 0 \).

Answer: F(2n, 2m)


Question: Is it possible to find a sequence of Littlewood polynomials \( f_n \in L_n \) such that \( M(f_n, 1) \leq C \sqrt{n+1} \) for all \( n \geq 1 \), where \( C > 0 \) is a constant depending on the sequence?

Answer: Yes


Question: A camera is located 100 feet from a straight road along which a car is traveling at 120 ft/sec. The camera turns so that it is always pointed at the car. In radians per second, how fast is the camera turning as the car passes closest to the camera?

Answer: \dfrac{6}{5}


Question: Suppose $H, G, K$ are abelian Hausdorff topological groups and $0 \to H \overset{\alpha}{\to} G \overset{\beta}{\to} K \to 0$ is an exact sequence of continuous homomorphisms. If $H$ and $K$ are compact, is $G$ necessarily compact as well?

Answer: Yes


Question: Let $M$ be a $3 \times 3$ singular matrix with eigenvalues $2$ and $3$. Determine the number of linearly independent eigenvectors of the matrix $M^3 + 2M + I_3$.

Answer: 3


Question: Evaluate the limit \( \lim_{x\to+\infty}x^{\alpha} B(\alpha, x) \) where \( \alpha > 0 \) and \( \alpha, x \in \mathbb{R} \).

Answer: \Gamma(\alpha)


Question: Find the volume of the solid in \(\mathbb{R}^3\) that lies above the \(xy\)-plane, inside the cylinder \(x^2+y^2= 2\), and below the paraboloid \(z=x^2+y^2\).

Answer: 2\pi


Question:
The roots of the polynomial equation \( x^3 - 3x - 1 = 0 \) are \( a, b, \) and \( c \). Calculate the value of the expression:
\[ L = \frac{1}{a^2 - 2a + 2} + \frac{1}{b^2 - 2b + 2} + \frac{1}{c^2 - 2c + 2} \]

Answer: \dfrac{39}{37}


Question: In an undirected, weighted, connected graph $G$ with no negative weights and all weights distinct, it is known that the shortest path between any two vertices is on the minimum spanning tree (MST). Verify whether the shortest path between any two vertices $u$ and $v$ is unique. Is this statement true or false?

Answer: True


Question: Let \( \Omega \) be a connected open subset of \( \mathbb{C} \). Determine whether it is necessarily true that any two points of \( \Omega \) can be joined by a non-selfintersecting curve, that is, an injective continuous map \( \gamma:[0,1]\to \Omega \).

Answer: Yes


Question:
Consider the triangular array where $A_n \stackrel{iid}{\sim} \text{Bern}(\pi)$ and $Y_n \stackrel{iid}{\sim} N(\mu, \sigma^2)$, with $\pi_n=\dfrac{A_1+A_2+...+A_n}{n}$. The array is structured as follows:

\[\begin{align*}
\frac{A_1Y_1}{\pi_1} \\
\frac{A_1Y_1}{\pi_2}, & \frac{A_2Y_2}{\pi_2} \\
\frac{A_1Y_1}{\pi_3}, & \frac{A_2Y_2}{\pi_3}, & \frac{A_3Y_3}{\pi_3} \\
& \vdots \\
\frac{A_1Y_1}{\pi_n}, & \frac{A_2Y_2}{\pi_n}, & \ldots, & \frac{A_nY_n}{\pi_n}
\end{align*}\]

Let $Z_n$ be the average of the $n$-th row. Determine whether the sequence $Z_n$ converges almost surely, in probability, or in distribution.

Answer: \mu


Question: Given that \( \int_{0}^{\pi}(f(x)+f''(x))\sin(x)~dx=-1 \) and \( f(\pi)=1 \), where \( f(x) \) has a continuous second derivative, compute \( f(0) \).

Answer: -2


Question: Let \( A(-1,0) \), \( B(3,0) \), and \( P(4,1) \) be points in the plane. Consider a line \( PQ \) passing through \( P \). Determine the range of slopes \( m \) for which there are two points on \( PQ \) such that the line segment \( AB \) subtends a right angle at these points. If the range of slopes is \((\lambda_1, \lambda_2)\), find \( \lambda_1 + \lambda_2 \).

Answer: \dfrac{6}{5}


Question: A coffee express machine has the following probabilities for each request: it brews coffee with probability \( p_c = 0.9 \), pours hot water with probability \( p_w = 0.09 \), and gets out of order with probability \( p_f = 0.01 \). Let \( X \) denote the number of coffees the machine prepares before it gets out of order. Compute \( \operatorname{E}[X] \).

Answer: 90


Question: Given the function \( F(x, y) = x + y + x^5 - y^5 \), which defines a smooth function \( \phi: U \rightarrow \mathbb{R} \) such that \( F(x, \phi(x)) = 0 \) for all \( x \in U \), find the value of \( \phi^{(2004)}(0) \).

Answer: 0


Question: Compute the line integral \(\int_C (y^2z^3\,dx + 2xyz^3\,dy + 3xy^2z^2\,dz)\) where \(C\) is the straight line path from \(a = (1,1,1)\) to \(b = (2,2,2)\).

Answer: 63


Question: Find all values of \(x\) in the interval \(0 \le x \le 2\pi\) for which the function \(f(x) = \cot(2x)\) is equal to its second derivative \(f''(x)\).

Answer: \dfrac{\pi}{4}, \dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4}


Question: Find the angle between the vector \( x = (2,2,1,1) \) and the subspace formed by the linear combination of the vectors \( a_1 = (3,4,-4,-1) \) and \( a_2 = (0,1,-1,2) \).

Answer: \dfrac{\pi}{3}


Question: Find the sum of the series \( \sum_{1 \leq x < y < z}^\infty \frac{1}{3^x4^y5^z} \).

Answer: \dfrac{1}{4484}


Question: Given the matrix \( A = \begin{bmatrix} \sin (\pi/18) & -\sin (4\pi/9) \\\ \sin (4\pi/9) & \sin (\pi/18) \end{bmatrix} \), find the smallest natural number \( n \) such that \( A^n = I \), where \( I \) is the identity matrix.

Answer: 9


Question: Find all pairs of natural numbers \((x, y)\) such that \(2^x + 1 = y^2\).

Answer: (3, 3)


Question:
Let \( S, T_n : (C[0,1], || \cdot ||_\infty) \to (\mathbb{R}, | \cdot |) \) be linear operators defined as follows:

\[ S(f) = \int_{0}^{1} f(x) \, dx \]
\[ T_n(f) = \frac{1}{n} \left( \frac{1}{2}f(0) + \sum_{k=1}^{n-1} f\left(\frac{k}{n}\right) + \frac{1}{2}f(1) \right) \]

Show that \( S \) and \( T_n \) are bounded operators and that \( ||S - T_n|| = 2 \).

Answer: 2


Question:
Calculate the coefficient of \(x\) in the determinant of the matrix function:
\[
f(x) = \det\begin{pmatrix}
x & (1+\sin x)^3 & \cos x \\
1 & \ln(1+x) & 2 \\
x^2 & 1+x^2 & 0
\end{pmatrix}
\]

Answer: -2


Question: Determine the number of non-zero coefficients in the Zhegalkin polynomial representation of the logical expression \(x_1 \lor x_2 \lor x_3 \lor \ldots \lor x_n\).

Answer: 2^n - 1


Question: The plane given by the equation \(x + 2y + 3z = 7\) is rotated about the line where it intersects the yz-plane by an angle \(m\). In its new position, the plane contains the point \((-1, 0, 2)\). Find \(\cos m\).

Answer: \dfrac{6}{7}


Question: Find the limit of the sequence \(a_n = \int_n^{n+\sqrt{n}} \frac{\sin x}{x} \, dx\).

Answer: 0


Question: Find \( \lim_{n \to \infty} a_n^n \) where \( a_n = \frac{(n!)^2}{(n+k)!(n-k)!} \) for each \( k \in \mathbb{N} \).

Answer: e^{-k^2}


Question: Evaluate the integral \( \int_{0}^{\pi/2}\sin^{5}x\cos^{7}x \, dx \) using the Beta function.

Answer: \dfrac{1}{120}


Question: Determine whether there exist integer functions \(f(n)\) and \(g(n)\) such that \(\lim_{n \to \infty} f(n)\zeta(2) + g(n) = 0\) and \(f(n)\zeta(2) + g(n) \neq 0\) for all positive integers \(n\).

Answer: Yes


Question: Given that $(X|Y=y) \sim N(y, y^2)$ and $Y \sim U[3,9]$, where $N(y, y^2)$ is a normal distribution with mean $y$ and variance $y^2$, and $U[3,9]$ is a uniform distribution on $[3,9]$, find $\text{Var}(X)$. Use the law of total variance: $\text{Var}(X) = E(\text{Var}(X|Y)) + \text{Var}(E(X|Y))$. Compute the necessary expectations and variances to determine $\text{Var}(X)$. \( \boxed{42} \)

Answer: 42


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

Answer: 0


Question: Let \( \{X(t), t \geq 0\} \) be a Brownian motion with drift parameter \( \mu = 3 \) and variance parameter \( \sigma^2 = 9 \). If \( X(0) = 10 \), find \( P(X(0.5) > 10) \).

Answer: 0.7603


Question: Compute the limit \( \lim_{n\to\infty}\frac{\sum_{k=1}^n|\cos(k^2)|}{n} \).

Answer: \dfrac{2}{\pi}


Question: Determine the range of the expression \( \sin \alpha \cdot \sin \beta + \sin \beta \cdot \sin \gamma + \sin \gamma \cdot \sin \alpha \), where \( \alpha, \beta, \gamma \) are the angles a line makes with the positive \( x, y, \) and \( z \) axes, respectively.

Answer: [1, 2]


Question: Determine the radius of convergence of the series \( \sum_{n=1}^{\infty}\frac{(1+nx)^n}{n!} \).

Answer: \dfrac{1}{e}


Question: Let $P(x)$ be a polynomial of least degree with a local maximum of 6 at $x=1$ and a local minimum of 2 at $x=3$. If $P(1)=6$ and $P(3)=2$, find $P'(0)$. Assume $P(x)$ is a cubic polynomial.

Answer: 9


Question:
Let \( f(x) \) be defined as follows:
\[
f(x) = \begin{cases} e^{-1/x^2}, & x \neq 0 \\ 0, & x = 0 \end{cases}
\]
Calculate \( f'''(0) \).

Answer: 0


Question:
Consider an infinite sequence of non-empty subsets \( A_1 \supset A_2 \supset \cdots \supset A_n \supset A_{n+1} \supset \cdots \) in \( \mathbb{R}^3 \). Which of the following conditions ensures that the intersection \( \cap_{i=1}^{\infty} A_i \) is non-empty?

A. Each \( A_i \) is uncountable.
B. Each \( A_i \) is open.
C. Each \( A_i \) is connected.
D. Each \( A_i \) is compact.

Answer: D


Question:
A particle moves in a straight line, and its velocity $v$ depends on the distance $d$ from its starting point according to the equation:
\[ v = \frac{3d+4}{2d+1} \]
Calculate the particle's acceleration when $d = 2$. (Note: The derivative of velocity with respect to time gives the acceleration.)

Answer: -\dfrac{2}{5}


Question: Determine whether the set of all injective functions from \( \mathbb{N} \) to \( \mathbb{N} \) is equinumerous with \( \mathbb{R} \).

Answer: Yes


Question:
Find the limit as \( n \to \infty \) of the series:
\[ \sin^4 x + \frac{1}{4} \sin^4(2x) + \frac{1}{16} \sin^4(4x) + \cdots + \frac{1}{4^n} \sin^4(2^n x) \]
Choose the correct option:
(A) \( \sin^4 x \)
(B) \( \sin^2 x \)
(C) \( \cos^2 x \)
(D) Does not exist

Answer: B


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{\log(1^1 +2^2 +\cdots+ n^n)}{\sqrt{n^4 + 2n^3\log(n)}-\sqrt{n^4-n^3}}. \]

Answer: 1


Question: Calculate \(10^{10^{10}} \pmod{7}\).

Answer: 4


Question: Let \( f(n) = \left(\frac{n+3\ln(n)}{n}\right)^n \) and \( g(n) = 27^{\ln(n)} \). Determine the correct asymptotic notation to fill in the blank: \( f(n) = \Box g(n) \), where the box represents one of the notations \( o \), \( \omega \), or \( \Theta \).

Answer: o


Question: Consider two decks, each containing 4 different cards labeled 1, 2, 3, and 4. You draw one card from each deck simultaneously, without replacement, for a total of 4 draws. What is the probability that in each draw, the two cards are different?

Answer: \dfrac{3}{8}


Question: Given that $a + b + c + abc + ab + bc + ac = 1000$, find the value of $a + b + c$.

Answer: 28


Question: Evaluate the complex integral \( \int_{|z|=1} e^{\frac{1}{z}} \cos{\frac{1}{z}} \, dz \) using the residue theorem.

Answer: 2\pi i


Question: Given the sequence defined by \( a_1 = \cos 1 \) and \( a_n = \max(a_{n-1}, \cos n) \) for natural numbers \( n \), find the value of \( \lim_{n \to \infty} a_n \).

Answer: 1


Question: Let $k$ be a $p$-adic field with ring of integers $\mathcal{O}_K$ and residue field $\mathbb{F}$. Consider a projective quadric $Q$ that is smooth over $\mathcal{O}_K$. If the reduction $\bar{Q}$, which is smooth over $\mathbb{F}$, contains a line isomorphic to $\mathbb{P}^1$ defined over $\mathbb{F}$, does it follow that $Q$ contains a line isomorphic to $\mathbb{P}^1$ defined over $\mathcal{O}_K$? Justify your answer.

Answer: Yes


Question: Let $k$ be a field and $A$ a finite-dimensional commutative $k$-algebra. Suppose a finite group $G$ acts on $A$ by automorphisms, and $A^G = \{a \in A \mid ga = g \text{ for any } g \in G \}$ is isomorphic to $k$. Is the action of $G$ on $\operatorname{Spec}(A)$ transitive under these assumptions?

Answer: Yes


Question: Evaluate the integral \( I = \int_{0}^{2\pi} \left(\frac{1}{3-2\cos x +\sin x}\right)\,dx \) using the residue theorem.

Answer: \pi


Question: Calculate the probability that the sum of 100 independent random variables, each following an exponential distribution with rate parameter 0.5, is less than or equal to 215. The probability density function of each variable is given by \( f(x) = 0.5e^{-0.5x} \) for \( x \ge 0 \). Find \( P\left(\sum_{i=1}^{100}\xi_i \le 215\right) \).

Answer: 0.7734


Question:
Determine the radius of convergence for the power series:
\[ \sum_{n=1}^{\infty }(-1)^n x^{3n} \frac{1\cdot 4\cdot 7\cdot ...\cdot (3n-2)}{n!\cdot 3^{n}} \]

Answer: 1


Question:
Consider the following time series processes:
\begin{align*}
\epsilon_t &\sim i.i.d.(0,1) \\
x_t &= \alpha x_{t-1}+\epsilon_t \\
y_t &= y_{t-1}+x_t \\
z_t &= z_{t-1}+y_t
\end{align*}
Assume $\alpha = 0.8$. Determine the order of integration $d$ for the process $z_t$, such that $z_t$ is an $I(d)$-process.

Answer: 2


Question: Let $X$ and $Z$ be independent standard normal random variables, i.e., $X, Z \sim N(0,1)$. Calculate the probability $P(X > 0, Z > -X)$. Provide your answer as a single probability value.

Answer: \dfrac{3}{8}


Question: Let \( f(x) = \cos(x/2) \cdot \cos(x/4) \cdots \cos(x/2^n) \). If \( \lim_{n \to \infty} f(x) = g(x) \) and \( \lim_{x \to 0} g(x) = k \), find the value of \( \lim_{k \to 1} \frac{1-k^{2011}}{1-k} \).

Answer: 2011


Question:
Determine the order of the generalized quaternion group \( \Bbb H_n \) and list all its elements, given the matrices:
\[ R=\begin{pmatrix}\omega & 0\\0&\omega^{-1}\end{pmatrix} \] and \[ S=\begin{pmatrix}0&-1\\1&0\end{pmatrix} \]
where \( \omega \) is a primitive \( 2^n \)-th root of unity. The group \( \Bbb H_n \) is defined as \( \langle S,R\rangle \) in \( \rm{SL}(2,\Bbb C) \). Use the relations:
\[ R^{2^n}=1, \quad S^2=-1, \quad SR=R^{-1}S, \quad S^{-1}R=R^{-1}S^{-1} \]
to find the order and list all elements of \( \Bbb H_n \).

Answer: 2^{n+1}


Question: Compute the limit: $$ \lim_{z \to 0} \frac{(1-\cos z)^2}{(e^z-1-z)\sin^2z} $$

Answer: \dfrac{1}{2}


Question: Find the limit of \( \frac{{{k-1}\choose n}}{{{n+k-1}\choose n}} \) as \( k/n^2 \to a > 0 \) and both \( k, n \to \infty \).

Answer: e^{-1/a}


Question: A special fair die has 10 faces numbered from 1 to 10. Let \( r_i \) be the result of the \( i \)-th throw and \( a \) be the imaginary fifth root of unity. Calculate the probability that \( \sum_{i=1}^5 a^{r_i} = 0 \).

Answer: \dfrac{24}{625}


Question: Evaluate the limit: $$\lim_{x\to 0}\left(\frac{1}{x^5} \int_0^x e^{-t^2} \, dt - \frac{1}{x^4} + \frac{1}{3x^2}\right)$$

Answer: \dfrac{1}{10}


Question: Let $X$ be a Banach space over $\mathbb{C}$, and let $T: X \to X$ be a conjugate linear map. If the graph $G(T) = \{ (x, Tx) \mid x \in X \}$ of $T$ is closed in $X \times X$, is $T$ continuous?

Answer: Yes


Question: Calculate the absolute value of the line integral \( \int \vec{F} \cdot d\vec{r} \) for the vector field \( \vec{F} \) such that \( \nabla \times \vec{F} = x\hat{i} - y\hat{j} \), where \( C \) is the closed curve forming the boundary of the surface \( S \) of the cone \( z = \sqrt{x^2 + y^2} \) bounded by the planes \( z = 0 \) and \( z = 3 \). Use Stokes' theorem to find the result.

Answer: 0


Question: Compute the limit: \[ \lim_{n \to \infty} n \left( \frac{\arccos\left(\frac{3}{n^3}\right)}{\arccos\left(\frac{3}{(n+1)^3}\right)} - 1 \right) \]

Answer: 0


Question: Let $I$ be a finite nonempty set, and $D$ a nonvoid compact subset of an N-dimensional Euclidean space. Consider continuous functions $V:D\rightarrow\mathbb{R}$ and $u_i:D\rightarrow\mathbb{R}$ for each $i \in I$. These functions satisfy the condition: if $u_i(p) = u_i(q)$ for all $i\in I$, then $V(p) = V(q)$. Define a mapping $W:L\rightarrow\mathbb{R}$, where $L = \{ (u_1(p),...,u_I(p)): p \in D\}$, by $W(u) = V(p)$ for any $p \in D$ such that $u = (u_1(p),...,u_I(p))$. Is the function $W$ continuous?

Answer: Yes


Question: Find the value of \( k \) such that \( \int_0^k x^x \, dx = 1 \).

Answer: 1.194


Question: Determine an upper bound for the function \( f(n) \) that satisfies the recurrence relation \( f(n) = 4f(n/2) + n \) for real numbers \( n \).

Answer: O(n^2)


Question: Find the limit of the sequence \( x_n = n^{n^{0.5}} - 2^n \).

Answer: -\infty


Question: Calculate the volume of the solid in $xyz$-space bounded by the surfaces \( z=\frac{1}{x^2+y^2+1} \) and \( z=\frac{1}{x^2+y^2+4} \).

Answer: 2\pi \ln 2


Question:
Determine the function whose Taylor series expansion is given by the expression:
\[ f(x) = \sum\limits_{N=0}^\infty (N-x)^2 \frac{x^N}{N!} \]

Answer: x e^x


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

Answer: 1


Question: Find the radius of convergence, \( R \), of the series \( \sum\limits_{n=0}^{\infty} \dfrac{(n!)^{k+2}\cdot x^n}{((k+2)n)!} \), where \( k \) is a positive integer.

Answer: (k+2)^{k+2}


Question: Evaluate the integral \( \int_0^n \frac{f(x)}{f(x) + f(n-x)} \, dx \) given that \( f(x) \) is continuous on \([0,n]\) and \( f(x) + f(n-x) \neq 0 \) for all \( x \) in \([0,n]\).

Answer: \dfrac{n}{2}


Question: Compute \( \int_{\gamma} y \, dx \) where \( \gamma \) is a curve in \( \mathbb{R}^2 \) with self-intersecting points at \((0,0)\), \((2,1)\), and \((4,2)\).

Answer: 0


Question: Evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \left[ e^{\ln\left(\cos x \cdot \frac{d(\cos x)}{dx}\right)} \right]dx \).

Answer: -\dfrac{1}{2}


Question: Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous and strictly convex function. Determine whether $f(x) = 0$ can have more than two solutions.

Answer: No


Question: Verify whether the transpose of the projection matrix \( P = A(A^TA)^{-1}A^T \) is equal to \( P \) itself, given that \( A \) is a matrix with more rows than columns \( (m > n) \).

Answer: P


Question: Determine whether the following inequality holds for the Lebesgue integral: \[ \left|\int_A f(x) \, d\mu(x) \right| \leq \left(\sup_{x \in A} |f(x)|\right) \cdot \left|\int_A \, d\mu(x)\right|. \]

Answer: Yes


Question: Let $f$ be a function in $L^{1}(\mathbb{R})$ that is infinitely differentiable with uniformly bounded derivatives. Determine whether it is true or false that the derivatives of $f$ are also in $L^{1}(\mathbb{R})$. Provide a justification for your answer.

Answer: True


Question: Determine whether the following statement is true: For points $x$ and $y$ on a Riemannian manifold $M$, with the logarithmic map $\log_xy$ and the parallel transport operator $P^{x\rightarrow y}$, is it true that $P^{x\rightarrow y}(\log_xy) = -\log_yx$?

Answer: True


Question: Find the value of \( a \) such that the chord of the parabola \( y = -a^2x^2 + 5ax - 4 \) is tangent to the curve \( y = \frac{1}{1-x} \) at the point \( x = 2 \) and is bisected by that point.

Answer: 1


Question: Compute the limit: \[ \lim_{x\to 0}\frac{\sin(\tan(x))-\sin(\sin(x))}{\tan(\tan(x))-\tan(\sin(x))} \]

Answer: 1


Question: Determine the number of group homomorphisms from the symmetric group $S_3$ to the cyclic group $\mathbb{Z}/6\mathbb{Z}$.

Answer: 2


Question: Determine the range of values for \( p \) such that the interval \((1,2)\) lies between the roots of the equation \(2^x + 2^{2-x} + p = 0\).

Answer: (-\infty, -5)


Question: Solve the equation \(2^x \cdot 3^y = 1 + 5^z\) in positive integers \(x, y,\) and \(z\).

Answer: (1, 1, 1)


Question: Evaluate the integral \( \int_0^1 t^{\frac{1}{z}}(1-\ln t^{\frac{1}{z}})\,dz \).

Answer: t


Question: Evaluate the infinite product: \[ \prod_{n=2}^\infty \frac{n^3 - 1}{n^3 + 1} \]

Answer: \dfrac{2}{3}


Question: Find the largest positive integer \( n \) such that for all real numbers \( a_1, a_2, \ldots, a_n, a_{n+1} \), the equation \[ a_{n+1}x^2 - 2x\sqrt{a_1^2 + a_2^2 + \cdots + a_n^2 + a_{n+1}^2} + (a_1 + a_2 + \cdots + a_n) = 0 \] has real roots.

Answer: 4


Question: Find all points of continuity for the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) defined by \[ f(x) = \begin{cases} 2x & \text{if } x \in \mathbb{Q} \\ x + 3 & \text{if } x \in \mathbb{R} - \mathbb{Q} \end{cases} \]

Answer: 3


Question: A function \( f: \mathbb{R} \to \mathbb{R} \) satisfies the equation \( f(x+y) = f(x)f(y) \) for all \( x, y \in \mathbb{R} \) and \( f(x) \neq 0 \) for all \( x \in \mathbb{R} \). Given that \( f'(0) = 2 \), find \( f'(x) \) in terms of \( f(x) \).

Answer: 2f(x)


Question: Find the remainder when the polynomial \( P(x) = x^{100} + x^{50} - 2x^4 - x^3 + x + 1 \) is divided by \( x^3 + x \).

Answer: 2x^2 + 2x + 1


Question: Determine the radius of convergence of the series \( \sum a_n x^{n^2} \), given that the series \( \sum a_n x^n \) has a radius of convergence \( R \).

Answer: 1


Question: Let $f(x)$ be a polynomial such that $f(x)f(1/x) + 3f(x) + 3f(1/x) = 0$ and $f(3) = 24$. Find the value of $f(2) + f(-2)$.

Answer: 18


Question: Evaluate the integral \( \int_0^1 \frac{3x^4 + 4x^3 + 3x^2}{(4x^3 + 3x^2 + 2x + 1)^2} \, dx \).

Answer: \dfrac{1}{10}


Question: Determine the Jacobson radical of the module $m = {}_B B$, where $B = k[r,s]/(r^2,s^2,rs)$ is a polynomial ring and $k$ is a field.

Answer: (r, s)


Question: Assume $G$ is a connected planar graph with 100 edges. The edges can be split into two sets, $S_1$ and $S_2$, such that $|S_1|=60$ and $|S_2|=40$. For all $e$ in $S_1$, the face on one side of $e$ has 3 edges, and the face on the other side has 10 edges. For all $e$ in $S_2$, the two faces on each side of $e$ are distinct from each other and both have 10 edges. How many vertices does $G$ have?

Answer: 68


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

Answer: e^{\frac{1}{6}}


Question: Determine the radius of convergence of the power series \( \sum_{n\geq2}\frac{z^{n}}{\ln(n)} \).

Answer: 1


Question: Suppose that \(X\) and \(Y\) are distributed uniformly over the triangle with vertices at \((-2,0)\), \((0,2)\), and \((2,0)\). Calculate the conditional expectation \(E[Y^2|X=1]\).

Answer: \dfrac{1}{3}


Question: Find the product of all solutions to the equation \(x^{1+\log_{10}x} = 100000x\). Choose the correct answer from the following options: \((A)~ 10 \quad (B)~ 10^5 \quad (C)~ 10^{-5} \quad (D)~1\).

Answer: D


Question:
Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a smooth function with a positive definite Hessian at every point. Let $(a,b) \in \mathbb{R}^{2}$ be a critical point of $f$. Determine the nature of the critical point at $(a,b)$:

A. $f$ has a global minimum at $(a,b)$.
B. $f$ has a local, but not global minimum at $(a,b)$.
C. $f$ has a local, but not global maximum at $(a,b)$.
D. $f$ has a global maximum at $(a,b)$.

Choose the correct option.

Answer: A


Question: Find a pair of coprime integers \(x\) and \(y\) such that \((p^y + 1) \bmod (p^x + 1) = 0\) for a given prime number \(p\). If such a pair exists, provide an example.

Answer: (1, 3)


Question:
Evaluate the following limit and find the greatest integer value of the reciprocal of the function at a specific point:

If
$$
\lim_{n \to\infty} n^k \int_0^{1/n} x^{x+k-1} dx = f(k)
$$
for $k \in \mathbb{N}$, determine
$$
\left[\frac{1}{f(5)}\right],
$$
where the square brackets denote the greatest integer function (i.e., ceil).

Answer: 5


Question: Let \( H \) be a Hilbert space and \( f: H \rightarrow \mathbb{C}^n \) be a linear function such that \( \overline{N(f)} = N(f) \), where \( N(f) \) is the annihilator of \( f \). Is \( f \) continuous when \( n > 1 \)?

Answer: Yes


Question: Let $X$ be a locally compact Hausdorff space and $A_1, A_2 \subset X$. Suppose that $\overline{A_1}^{X} \cap \overline{A_2}^{X} = \emptyset$ and $\overline{A_1}^{X}$ is compact. Let $X_\infty$ be the one-point compactification of $X$. Does this imply that $\overline{A_1}^{X_\infty} \cap \overline{A_2}^{X_\infty} = \emptyset$?

Answer: Yes


Question: Let a set \( X \) be the union of two locally closed subsets \( U \) and \( V \) such that \( U \) does not lie in the closure of \( V \). Let the restriction of a complex \( R \) of constructible sheaves on \( X \) to a smooth open subset \( A \) of \( U \) be an irreducible local system \( S \) on \( A \). Determine if the intermediate extension \( IC(U, S) \) is one of the irreducible perverse subquotients in one of the perverse cohomologies of \( R \).

Answer: Yes


Question: Evaluate the surface integral \( \int_S \mathbf{A} \cdot \mathbf{n} \, dS \) over the entire surface of the region above the \(x\)-\(y\) plane bounded by the cone \(x^2+y^2=z^2\) and the plane \(z=4\), where \(\mathbf{A} = 4xz\mathbf{i} + xyz^2 \mathbf{j} + 3z \mathbf{k}\).

Answer: 320\pi


Question: Find the length of the path \( \gamma(t) = t - ie^{-it} \) for \( 0 \leq t \leq 2 \).

Answer: 4(1 - \cos 1)


Question: Given two strictly concave, strictly increasing, and everywhere differentiable functions $f, g: \mathbb{R}^+_0 \to [0,1]$ with $f(0) = g(0) = 0$ and $\lim_{x \to \infty} f(x) = \lim_{x \to \infty} g(x) = 1$, determine the maximum number of intersections between $f$ and $g$ for $x > 0$. Exclude the intersection at $x = 0$.

Answer: 1


Question: Evaluate the integral \( \int_4^8 \frac {\ln(9-x)}{\ln(9-x)+\ln(x-3)}\,dx \).

Answer: 2


Question: Evaluate the infinite sum \( \sum\limits_{n=0}^{\infty}\frac{(-1)^n}{2n+1} \).

Answer: \dfrac{\pi}{4}


Question:
Find the limit of the following nested square root expression as \( n \) approaches infinity:

\[ \lim_{n \to \infty}\left( 1 + \sqrt{2 + \sqrt{3+ \ldots + \sqrt {n + \sqrt{n+1}}}}\right) \]

Answer: 3


Question: Determine the minimum number of times you need to roll a fair six-sided die to be at least 99% certain that the sample mean of the rolls is between 3 and 4. Use the central limit theorem to find your answer.

Answer: 78


Question: Verify the approximation \( \sqrt{\frac{z}{z-2h}} - \sqrt{\frac{z}{z+2h}} \approx \frac{2h}{z} \) given that \( h>0 \) and \( |z| \gg h \).

Answer: \dfrac{2h}{z}


Question: If $f(z)$ is an analytic function, is the function $f(-z)$ also analytic?

Answer: Yes


Question: Suppose we have a box (parallelepiped) A completely contained within another box B. Is the surface area of box A necessarily less than the surface area of box B? Note that the sides of A are not necessarily parallel to the sides of B.

Answer: Yes


Question: Determine the Lebesgue measure of the set \( A = \left \{x \in [0,1] \cap (\mathbb{R} - \mathbb{Q}) \mid \text{in the decimal expansion of } x, \text{ the first 1 (if it exists) is followed by 2} \right \} \).

Answer: \dfrac{1}{10}


Question: Let $a$, $b$, and $c$ be positive real numbers such that \[ \frac{a^3+2a}{a^2+1} + \frac{b^3+2b}{b^2+1} + \frac{c^3+2c}{c^2+1} = \frac{9}{2}. \] Determine whether the inequality \[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \ge 3 \] holds true.

Answer: True


Question: Evaluate the limit of the series using Faulhaber's formula: \[ S = \lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^{99}}{n^{100}}. \]

Answer: \dfrac{1}{100}


Question: Evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \), where \( S \) is the surface composed of the paraboloid \( y = x^2 + z^2 \) for \( y \in [0,1] \) and the disk \( x^2 + z^2 \leq 1 \) at \( y = 1 \), oriented with outward normal vectors. The vector field is given by \( \mathbf{F} = (0, y, -z) \).

Answer: 0


Question: A right circular cylindrical container with a closed top is to be constructed with a fixed surface area. Determine the ratio of the height to the radius that maximizes the volume of the cylinder.

Answer: 2


Question: Determine the maximum integer value of $n$ for which the equation $x = n^x$ has a solution.

Answer: 1


Question: Let $A$ be a non-singular square matrix of order $2$ such that \(|A + |A|\operatorname{adj}A| = 0\), where \(\operatorname{adj}A\) represents the adjugate of matrix \(A\), and \(|A|\) represents \(\det(A)\). Evaluate \(|A – |A|\operatorname{adj}A|.\)

Answer: 4


Question: Determine the primes $p$ for which there exists an integer $q$ such that the congruence \(X^2 + (pq-1)Y^2 \equiv 0 \pmod{p^2}\) has no solution for co-prime integers $X$ and $Y$. Provide a list of such primes or a condition that these primes satisfy.

Answer: 2


Question: Calculate the line integral of the vector field \( F(x,y,z) = \left(4x\left(1+\frac{z^2}{8}\right),\frac{\pi}{3}\cos\left(\frac{\pi}{3}y\right),\frac{x^2z}{2}-e^{\frac{z}{3}}\right) \) along the curve \( C \) parametrized by \( \gamma(t) = (e^{t(t-3)}, t, 2t) \). Evaluate \( \int_C F \cdot ds \).

Answer: 12 - 3e^2


Question: Evaluate the double integral \( \int _0^1\int _0^{\frac{1}{x}}\frac{x}{1+y^2}\,dx\,dy \).

Answer: \dfrac{1}{2}


Question: Calculate the partial derivative: $$\frac{\partial}{\partial a}\int_{-\infty}^a(a-x)f(x)\text{d}x$$ where $f(x)$ is a probability density function (pdf). Is the result equal to $a$?

Answer: F(a)


Question: Evaluate the complex integral \( I = \frac{1}{2\pi i} \int_C |1+z+z^2|^2 \, dz \), where \( C \) is the unit circle centered at the origin, and the integral is taken in the anticlockwise direction.

Answer: 2


Question: Evaluate the improper integral: $$\int_{0}^{\infty} \left(e^{-\frac{1}{x^2}}-e^{-\frac{4}{x^2}}\right) dx$$

Answer: \sqrt{\pi}


Question: Find the Lebesgue measure of the interval \([n, n + \frac{1}{n^2}]\) for \(n \in \mathbb{N}\) using the definition of the n-dimensional Lebesgue measure: \( \lambda^{n}([[a,b))) = \prod_{j=1}^{n}(b_{j}-a_{j}) \).

Answer: \dfrac{1}{n^2}


Question: Verify Stokes' Theorem for the surface \( S = \{(x,y,z) \mid x^2+y^2+z^2=1, z\ge 0\} \) and the vector field \( F = (z^2, x, y^2) \). Compute both \( \iint_S (\nabla \times F) \cdot dS \) and \( \oint_{\partial S} F \cdot ds \) and show that they are equal.

Answer: \pi


Question: Find the sum of the series \( \sum_{k=1}^{\infty} \frac{k^3+6k^2+11k+5}{(k+3)!} \).

Answer: \dfrac{5}{3}


Question: Calculate the volume of the region bounded by the paraboloid \( x^2 + y^2 = z \) and the plane \( z = 4y \) using triple integrals.

Answer: 8\pi


Question: Find the last two digits of \(139^{139^{100}}\).

Answer: 39


Question: Find the maximum value of \(72\int\limits_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx\) for \(y\in[0,1].\)

Answer: 24


Question: Determine whether there exist any Gaussian composite numbers that are 'viewable', where a Gaussian integer $a+bi$ is 'viewable' if $(a,b)=1$.

Answer: Yes


Question: Determine the order of the element \(4U_5(105)\) in the quotient group \(U(105)/U_5(105)\), where \(U(105)\) is the group of units modulo 105 and \(U_5(105)\) is the subgroup of elements congruent to 1 modulo 5.

Answer: 2


Question: Consider a matrix $A$ that is not a scalar multiple of the identity matrix. Is it always possible to find a matrix $B$ such that the first diagonal element of the matrix $BAB^{-1}$ is zero?

Answer: Yes


Question: Let \( L: \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation defined by \( L([x,y]) = [x+2y, -x+2y] \). Consider the set \( A = \{[x,y] : x \ge 0, 0 \le y \le 2-x\} \), which represents a triangle in the plane. Find the measure of the image \( L(A) \).

Answer: 8


Question: A system consists of three components. Component 1 is in series with two parallel components, 2 and 3. Each component operates independently and has a lifetime that follows an exponential distribution with rate parameter \( \lambda = 0.002 \). Calculate the probability that the system will last more than 500 hours.

Answer: 0.2209


Question: Let $\{B_t\}_{t\ge 0}$ be a standard Brownian motion, and let $\mu$ and $a$ be given real numbers with $\mu \neq 0$. Define $X_t = \mu t + B_t$. Consider the set $L = \{\tau \in \Re_+ \mid X_\tau = a\}$. Is it true that for every $T \in \Re_+$, the set $[0,T] \cap L$ is almost surely finite?

Answer: Yes


Question: Find the function $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfies the equation \[ f(x^2 + x) + 2f(x^2 - 3x + 2) = 9x^2 - 15x \] for all $x \in \mathbb{R}$. Then, calculate $f(2016)$. \( \text{Provide your answer in a boxed format: } \boxed{} \)

Answer: 6044


Question: Let $X$ and $Y$ be exponential random variables with parameters $\alpha$ and $\beta$, respectively. Find the expected value $E\left[\frac{X}{Y}\right]$ using the probability density function $f_{\frac{X}{Y}}(u) = \frac{\alpha\beta}{(\alpha u + \beta)^2}$. Evaluate the integral to determine the expected value.

Answer: \infty


Question: If $f$ is a holomorphic function and $g$ is a non-holomorphic function, and their product $f \times g$ is holomorphic, must $f$ be identically zero? Justify your answer.

Answer: Yes


Question: Evaluate the surface integral \( \iint_S \nabla \times \vec{v} \cdot \vec{N} \, dS \) where \( \vec{v} = (xz, -y, x^2y) \) and \( S \) consists of the five faces of the cube \([0,2] \times [0,2] \times [0,2]\) that are not on the xy-plane. The vector \( \vec{N} \) points outward from the surface. Use Stokes' theorem to find the value of the integral.

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to \infty}\dfrac{\sum\limits_{k=2}^n k\cos\dfrac{\pi}{k}}{n^2} \]

Answer: \dfrac{1}{2}


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^{\infty} a_n z^n \), where the sequence \( (a_n) \) is defined recursively by \( a_{n+3} = a_{n+2} + a_{n+1} - a_n \) for \( n \in \mathbb{N}_0 \), with initial terms \( a_0, a_1, a_2 \in \mathbb{C} \).

Answer: 1


Question: Find a closed form expression for the sum \( \sum_{k=0}^n \binom{2k}{k}\binom{2n-2k}{n-k} \) for non-negative integers \( n \).

Answer: 4^n


Question: Determine the value of \( \sup \{ c \mid c > 0, n! \in O((\frac{n}{c})^n) \} \).

Answer: e


Question: Find the radius of convergence for the series \( \sum \frac{(3n)!}{(n!)^2}x^n \).

Answer: 0


Question:
Given a matrix \( A \in \mathbb{R}^{5 \times 5} \) and a parameter \( \alpha \in \mathbb{R} \), consider the linear system:
\[ Ax = \begin{pmatrix} 1 \\ -1 \\ \alpha^2 + \alpha \\ 1 \\ -1 \end{pmatrix} \]
The general solution to this system is:
\[ x = \begin{pmatrix} t+u+1 \\ s+t-u \\ s-t+1 \\ -s-t+u \\ -s-u-1 \end{pmatrix} \]
Additionally, the following vectors are in the column space of \( A \):
\[ \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \\ \alpha^2-3 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \\ -1 \\ \alpha^2-2 \end{pmatrix} \]
Find the value of \( \alpha \) such that \( A^2 = 0 \).

Answer: -1


Question:
Let $A$ be the set of all 2017-digit numbers that satisfy the following property: If $a$ is in $A$ and $a_1, a_2, \ldots, a_{2017}$ are the digits of $a$ (in order), then there exists a positive integer $S$ such that:
\[ S = \sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \ldots + \sqrt{a_{2016} + \sqrt{a_{2017} + S}}}}} \]
Let $m$ and $n$ be the least and greatest elements of $A$, respectively. If $s_m = \sum_{i=1}^{2017} m_i$ and $s_n = \sum_{i=1}^{2017} n_i$, compute $s_m + s_n$. \( \boxed{} \)

Answer: 16136


Question: Evaluate the limit \( \lim_{z \to 0} \frac{(e^{\sin(z)} - 1)^3}{\sin(z) - z} \).

Answer: -6


Question: Determine if the signal \( y = \cos(2\pi t) + \sum_{n} \delta(t - n) \) is periodic. If it is periodic, find its period.

Answer: 1


Question: Compute the expectation \( \mathbb{E}[X^2Y^2] \) where \( X \) and \( Y \) are standard Gaussian random variables with correlation \( \rho \).

Answer: 1 + 2\rho^2


Question: Determine whether the following statement is true or false: If $a$ is a quadratic non-residue modulo $p$, then $a$ is a primitive root modulo $p$. Justify your answer.

Answer: False


Question: If $q \equiv k \equiv 1 \pmod{4}$, is it necessarily true that $\gcd\bigg(\sigma(q^k),\sigma(q^{(k-1)/2})\bigg)=1$? Here, $q$ is a prime number and $\sigma$ denotes the classical sum-of-divisors function.

Answer: No


Question: Evaluate the integral \( \int_{-\infty}^\infty |f\ast f'|^2(x)\,dx \) given that \( \mathcal{F}[f](\omega)=\frac{1}{1+|\omega|^3} \) using Plancherel’s theorem.

Answer: \dfrac{1}{9\pi}


Question: Determine whether the following statement is true or false: "The free group generated by two generators is isomorphic to the direct product of two infinite cyclic groups."

Answer: B


Question:
Let \( x_n = \min\{x \in \mathbb{R} : (1+x)^n \geq 1+nx \} \), where \( n \) is a natural and odd number. Determine whether the following limit is true:
\[ \lim_{n \to \infty} x_n = -2. \]

Answer: -2


Question:
Determine the value of \( A \) such that the limit \( \lim_{(x,y)\to (0,0) } f(x,y) \) exists, where
\[
f(x,y) = \begin{cases}
\frac{x^4+y^4}{y(x^2+y^2 ) } , & y \neq 0 \\
A , & y=0
\end{cases}
\]

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{1}{x} \int_x^{2x} e^{-t^2} \, dt \]

Answer: 1


Question: Given the limit expression \( \lim_{x\to\infty}\left[x.\ln\left(\dfrac{a}{x^3}+b-\dfrac{c}{x}\right)\right]=-5 \), find the value of \( b+c \).

Answer: 6


Question: What is the remainder when \(41!\) is divided by \(83\)?

Answer: 82


Question: In a von Neumann algebra, if $A_{\alpha}$ converges to $0$ in the $\sigma$-weak topology, do the positive parts $(A_{\alpha})_{+}$ necessarily converge to $0$ in the $\sigma$-weak topology?

Answer: No


Question: Does every finite digraph have a unique (up to isomorphism) irreducible reduction?

Answer: Yes


Question: Evaluate the infinite series: $$\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}\sum_{n=0}^\infty \frac{1}{k2^n+1}.$$

Answer: 1


Question: Determine the slope of the function \( xe^{-(\frac{1}{x}+ \frac{1}{|x|})} \) at \( x = 0 \) using the right-hand differential approach.

Answer: 0


Question: Determine the value of the limit \( L = \lim \limits_{x \to 0}\frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4} \) such that \( L \) is finite. What is the value of \( a \) that satisfies this condition?

Answer: 2


Question: Find the inverse of the polynomial $x^2 + x + 1$ in the finite field $\mathbb{F}_2[x]$ modulo $x^4 + x + 1$. Provide your answer as a polynomial in $\mathbb{F}_2[x]$.

Answer: x^2 + x


Question: Demonstrate that \( \left\lfloor \sum_{k=1}^{10000} \frac{1}{\sqrt{k}} \right\rfloor = 198 \).

Answer: 198


Question: A soccer player takes 19 consecutive penalty shots. The probability of failing the $n$-th shot is $\frac{1}{2n+1}$. What is the probability that the player succeeds in an even number of these shots?

Answer: \dfrac{19}{39}


Question: Determine the value of \( c \) such that the integral \( \int_0^\infty xe^{-cx^2} \, dx = 1 \).

Answer: \dfrac{1}{2}


Question: Given a symmetric positive semi-definite matrix $A = \{a_{ij}\}$, where $0 < a_{ij} < 1$, determine if the matrix $B = \{\frac{1}{1 - a_{ij}}\}$ is also a symmetric positive semi-definite matrix.

Answer: B


Question: Find a function \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for every \( t > 0 \), the Armijo criterion \( f(x_k + td_k) \leq f(x_k) + t \nabla f(x_k)^T d_k \) is not satisfied when \( \alpha = 1 \). Provide an example of such a function.

Answer: f(x) = x^2


Question: Evaluate the limit \( \lim\limits_{x \to a} \frac{x^a - a^x}{x^a - a^a} \) where \( a \) is a positive real number different from 1, without using L'Hopital's rule.

Answer: 1 - \ln a


Question: Find the smallest natural number $x$ of the form $3qp$, where $q$ and $p$ are prime numbers, such that $x$ divides $(n^x - n)$ for all positive integers $n$. Provide a solution using elementary number theory.

Answer: 561


Question: Let $A$ and $B$ be non-empty subsets of $\mathbb{R}$, where $A$ is bounded above and $B$ is bounded below. Suppose that for every $\epsilon > 0$, there exist $x \in A$ and $y \in B$ such that $0 < y - x < \epsilon$. Prove or disprove that $\sup A = \inf B$.

Answer: \sup A = \inf B


Question: Given that $\alpha$, $\beta$, and $\gamma$ are the roots of the polynomial $x^3 - x^2 + px - 1 = 0$, and that $(\alpha^3 + 1)(\beta^3 + 1)(\gamma^3 + 1) = 2019$, find the product of all possible values of $p$. Note that $p$ may be a complex number.

Answer: 2010


Question: Let \(a_n\) be a sequence, and define \(b_n = \frac{a_1 + a_2 + \, ... \, + a_n}{n}\). If \(b_n\) converges, does \(\frac{a_n}{n}\) necessarily converge to 0? Provide a justification for your answer.

Answer: Yes


Question: Determine whether there exists a subset $A \subseteq (-1, 1)$ such that $|A \cap \mathbb{Q}| = \varnothing$ and $|A| = \aleph_0$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim\limits_{x \to 1^-} \left(\ln\left(\left(\frac{x-1}{x+1}\right)^2\right) \cdot x - 2 \cdot \ln|x^2-1|\right) \]

Answer: -4 \ln 2


Question: Determine the largest constant $c$ such that the inequality \[ p\ln\left(\frac{p}{q}\right)+(1-p)\ln\left(\frac{1-p}{1-q}\right)-4c(p-q)^2\ge 0 \] holds for all $0 < p, q < 1$. Find the largest $c$.

Answer: \dfrac{1}{2}


Question: Let $A \to B \to B/I$ be ring homomorphisms where $B$ and $B/I$ are both flat over $A$, and $I$ is generated by a regular sequence of length $d$. Given any other homomorphism $A \to C$, determine if the ideal $C \otimes_A I$ of $C \otimes_A B$ is generated by a regular sequence of length $d$.

Answer: Yes


Question: Consider the parametric curve given by \( c(t) = (a\cos(t), b\sin(t)) \) for \( 0 \leq t \), where \( a = 1 > b > 0 \) in \( \mathbb{R}^2 \). Fix \( 0 < t_1 < \pi/2 \) such that \( x_1 = c(t_1) \). If \( x_i = c(t_i) \) and \( x_i \) is a positive multiple of \( c'(t_{i-1}) \), determine the number of limit points for subsequences in \( \{x_i\} \). Prove that \( x_n \) has exactly 4 limit points.

Answer: 4


Question: Evaluate the flux of the curl of the vector field \( F = (z-y, 0, y) \) through the surface \( S' \), which is the portion of the sphere with radius \( \sqrt{2} \) centered at the origin that lies above the curve \( C \) (in the region \( z \geq 1 \)) and has \( C \) as a boundary. Specify the orientation used for \( S' \). The curve \( C \) is defined by \( x^2 + y^2 = 1 \) in the plane \( z = 1 \).

Answer: \pi


Question: Find the projection of $x^3$ onto the subspace $M$ of $L^2[-1,1]$, where $M$ is generated by the functions $p_0(x) = 1$, $p_1(x) = x$, and $p_2(x) = x^2$. Express the projection $P_M(x^3)$ in terms of $p_0$, $p_1$, and $p_2$. You may use the Gram-Schmidt process if needed.

Answer: \dfrac{3}{5} p_1(x)