Question: Determine all real numbers \(a\) such that the matrix \([A]_f = \begin{bmatrix} 1-a & 3-2a & 5-2a \\ a-3 & 2a-4 & 2a-3 \\ 3 & 4 & 4 \end{bmatrix}\) represents a Hermitian operator on \(\mathbb{R}^3\) with the standard scalar product, given the basis \(f = \{(1,0,0), (1,1,0), (1,1,1)\}\).

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left(1 + 2x\sqrt{x}\right)^{\frac{2}{\ln x}} \] without using L'Hôpital's rule.

Answer: e^3


Question: Given a prime number $p$, an integer $e$ such that $(p-1)$ and $e$ are relatively prime, and a value $m^e \mod p$, determine if it is possible to compute $m \mod p$. Provide a justification for your answer.

Answer: Yes


Question: Find the positive integer \( n \) such that \( \lfloor \log_2 1 \rfloor + \lfloor \log_2 2 \rfloor + \ldots + \lfloor \log_2 n \rfloor = 1994 \), where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \).

Answer: 312


Question: Using the power series of \( \sin x^3 \), the value of \( f^{(15)}(0) \) is equal to \( k\cdot11! \). Find the value of \( k \).

Answer: 273


Question: Two players play rock-paper-scissors until one player wins 3 times. What is the expected number of rounds played? Provide your answer as a decimal rounded to four decimal places.

Answer: 6.1875


Question: Evaluate the complex integral \( \int_{|z| = 1}\frac{z^2}{4e^z -z}\,\mathrm dz \) using residue calculus.

Answer: 0


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

Answer: 0


Question: Given that \( x^2 + y^2 + z^2 = 1 \) for \( x, y, z \in \mathbb{R} \), find the maximum value of the expression \( x^3 + y^3 + z^3 - 3xyz \).

Answer: 1


Question: Evaluate the integral \( \int_0 ^2\int_0^2 f(x+y)\, dx\, dy \), where \( f(t) \) denotes the greatest integer less than or equal to \( t \).

Answer: 6


Question: Calculate the area of the spherical triangle defined by the points (0, 0, 1), (0, 1, 0), and (1/√2, 0, 1/√2) on the unit sphere.

Answer: \dfrac{\pi}{4}


Question: Use Taylor's inequality to estimate the accuracy of the approximation \( f(x) \approx T_3(x) \) for \( f(x) = e^{-x^2} \cos(x) \) at \( a = 0 \), where \( T_3(x) = 1 - \frac{3x^2}{2} \). Estimate the error for \( |x| \leq 1 \).

Answer: \dfrac{25}{24}


Question: In a factory, 5% of screws are faulty, independently. Each shipment contains 10,000 screws and is eligible for a refund if more than \( r \) screws are faulty. Using Chebyshev's inequality, determine the minimum value of \( r \) such that no more than 1% of shipments will be eligible for a refund. Assume \( X \) is the number of faulty screws in a shipment, where \( X \sim B(10,000, 0.05) \).

Answer: 718


Question: Evaluate \( \lim_{x\rightarrow \infty} x\int_{0}^{x}e^{t^2-x^2}dt \).

Answer: \dfrac{1}{2}


Question: Determine the explicit value of $B_\alpha$ for $\alpha \in (\omega \cup \{\omega\}) \setminus \{0,1\}$. Specifically, find $B_\omega$. Here, $B_\alpha$ is the minimum cardinality of a set of fair functions $f: \omega \to \alpha$ that is equalising for $\alpha$, meaning for all $a,b \in \omega$, there exists $f$ in the set such that $f(a) = f(b)$. A function $f: \omega \to \alpha$ is fair if $|f^{-1}(\{j\})| = \aleph_0$ for all $j \in \alpha$. It is established that $B_2 \leq 3$. What is $B_\omega$?

Answer: \aleph_0


Question: Determine the order of the poles of the function \( \frac{z}{\cos z} \).

Answer: 1


Question: Let $p$ be a prime number. Determine whether there exists a non-commutative reduced ring of order $p^2$. Note that any ring of order $p^2$ with identity is commutative.

Answer: Yes


Question: Let $M$ be an $R$-module. If $M$ has finite rank, must it be finitely generated? Provide a justification for your answer.

Answer: No


Question: Consider the standard Brownian motion \(W(t), t \ge 0\). Find the probability \(p(W(1) \ge 0, W(2) \ge 0)\).

Answer: \dfrac{3}{8}


Question: A fair six-sided die is tossed \(2n+1\) times. What is the probability that an even-numbered face appears an odd number of times?

Answer: \dfrac{1}{2}


Question: If $a+bi$ is in the Motzkin set $E_k$, is $a-bi$ also in $E_k$? Justify your answer.

Answer: Yes


Question: Consider a list of polynomials $P_a(k_1,\ldots,k_n)$ for $a=1,\ldots,n$ in real variables $k_1,\ldots,k_n$ such that \( \sum_{a=1}^n k_a P_a(k_1,\ldots,k_n) = 0 \). Determine if there exists a list of polynomials $P_{ab}(k_1,\ldots,k_n)$ for $a,b=1,\ldots,n$ such that $P_{ab}+P_{ba}=0$ and \( P_a(k_1,\ldots,k_n) = \sum_{b=1}^n k_b P_{ab}(k_1,\ldots,k_n) \).

Answer: Yes


Question: Calculate the integral \( \int_A x^2z \, dV \) where \( A = \{(x,y,z) \in \mathbb{R}^3 : 0 \leq z \leq 1-x^2-y^2\} \) using Fubini's theorem and polar coordinates.

Answer: \dfrac{\pi}{48}


Question: In a triangle, consider the expression \( \sum \left(\frac{a}{b+c}\right)^{2} \). Determine if this expression implies that the triangle is equilateral.

Answer: Yes


Question: Let $G$ be a group and let $H$ and $K$ be two subgroups of $G$, each with 12 elements. Determine which of the following numbers cannot be the cardinality of the set $HK = \{hk: h \in H, k \in K\}$. Choose from the options: (a) 72, (b) 60, (c) 48, (d) 36.

Answer: B


Question: Maximize the function $f(x, y) = \sin(x) \cos(y)$ subject to the constraint $x^2 + y^2 = 1$ using Lagrange multipliers.

Answer: \sin(1)


Question: Let $A$ be a square right-stochastic matrix, meaning $A$ has nonnegative entries and each row sums to one. If $B$ is an invertible square matrix such that the product $AB$ is also right-stochastic, must $B$ itself be right-stochastic?

Answer: No


Question:
Find the limit of the sequence as \( n \to \infty \):
\[ a_n = \frac{n+1}{n}\ln(\sqrt{n}) - \frac{1}{n^2}\sum_{k=1}^{n}\ln(k+n)^k \]

Answer: -\dfrac{1}{4}


Question: Evaluate the limit: \[ \lim_{x\to0}\,3^{(1-\sec^2 20x)/(\sec^2 10x-1)} \]

Answer: \dfrac{1}{81}


Question: Determine whether there exists a non-central extension of a finite abelian p-group, which is not cyclic, by the alternating group $A_5$. Provide a justification for your answer.

Answer: Yes


Question: For which values of the natural number \( n > 2 \) is the graph \( G_n = (V, E) \), defined by \( V = \{A \subset \{1, 2, \ldots, n\} : |A| = 2\} \) and \( \{A, B\} \in E \iff A \cap B = \emptyset \), connected?

Answer: n \geq 5


Question: Given a vector \( \vec{x} \in \mathbb{R}^n \) (or \( \mathbb{C}^n \)), determine the maximum possible rank of a matrix \( A \in \mathbb{R}^{m \times n} \) (or \( \mathbb{C}^{m \times n} \)) such that \( A\vec{x} = \vec{0} \).

Answer: \min(m, n - 1)


Question: Evaluate the integral \( \int_{0}^{1} \frac{1}{1 + \left(1 - \frac{1}{x}\right)^{2015}} \, dx \).

Answer: \dfrac{1}{2}


Question: Let \( \psi \neq 1 \) be a non-trivial character of the finite field \( \mathbb{F}_p \). Define the Gauss sum \( G(\psi) = \sum_{x \in \mathbb{F}_p} \psi(x^2) \). Prove that \( |G(\psi)|^2 = p \).

Answer: p


Question: Let \( f:[1,\infty)\to \mathbb{R} \) be a differentiable function such that \( f(1)=1 \) and \( f'(x)=\frac{1}{1+f(x)^2} \). Determine \( \lim_{x \to \infty} f(x) \).

Answer: \infty


Question: The time it takes to service a car is an exponential random variable with rate 1. If A.J.'s car and M.J.'s car are both brought in at time 0, with work starting on M.J.'s car only when A.J.'s car has been completely serviced, what is the probability that M.J.'s car is ready before time 2?

Answer: 1 - \frac{3}{e^2}


Question: Let $R$ be a finite local ring with identity that has exactly one minimal left ideal. Determine if $R$ necessarily has exactly one minimal right ideal.

Answer: Yes


Question: Determine the largest integer \( n \) such that the function \( f: [-1,1] \to \mathbb{R} \), defined by \( f(x) = x^3 \sin(1/x) \) for \( x \neq 0 \) and \( f(0) = 0 \), is \( n \)-times continuously differentiable on the interval \([-1,1]\).

Answer: 1


Question: Determine the disc of convergence for the power series: $$\sum_{n=3}^\infty \left(1-\frac{1}{n^2}\right)^{-n^3}z^n$$

Answer: \{z \in \mathbb{C} \mid |z| < \frac{1}{e}\}


Question: Given that $a^{7!} + b^{8!} + c^{9!} + d^{10!} = x$ where $a$, $b$, $c$, and $d$ are natural numbers that are not multiples of 10, determine how many distinct values the unit's digit of $x$ can take.

Answer: 9


Question: Find the limit: \[ \lim_{x \to \infty} \left[\frac{1}{3} \left(3^{\frac{1}{x}} + 8^{\frac{1}{x}} + 9^{\frac{1}{x}}\right)\right]^x. \]

Answer: 6


Question: Find the smallest positive integer \( n \) such that the symmetric group \( S_n \) has a subgroup of order \( p^2 \), where \( p \) is a prime number.

Answer: 4


Question: Find the function \( F(x) \) such that \( [F(x)]^{100} = \int_{0}^{x} (F(t))^{100} \frac{dt}{1+\sin t} \).

Answer: 0


Question:
Consider a random variable $V$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that:
1) The support of $V$ is an open subset $\mathcal{V}$ of $\mathbb{R}^K$ with strictly positive Lebesgue measure.
2) The distribution of $V$ is absolutely continuous on $\mathcal{V}$ with respect to Lebesgue measure.

Which of these assumptions is sufficient to ensure that for all $v \in \mathcal{V}$, $\mathbb{P}(V=v)=0$?

Answer: 2


Question: Consider the symmetric group $G = S_5$. Let $L$ be the set of subgroups of $G$ that are isomorphic to the non-cyclic group of order 4. Two subgroups $H$ and $K$ in $L$ are considered equivalent if they are conjugate in $G$. Determine the number of equivalence classes in $L$.

Answer: 2


Question: Let $P \in \mathbb{R}^{n \times n}$ be an orthogonal matrix. Determine whether there exists a vector $x \in \mathbb{R}^n$ with no zero entries such that $Px$ also has no zero entries.

Answer: Yes


Question: Evaluate \( \lim_{n \to \infty} \int^{b}_{0} \frac{\sin nx}{nx} \, dx \) where \( b > 0 \).

Answer: 0


Question: Let $f: X \rightarrow S^1$ be a continuous map from a path-connected topological space $X$, and let $p: \mathbb{R} \rightarrow S^1$ be the universal covering map. Determine the condition under which there exists a continuous map $h: X \rightarrow \mathbb{R}$ such that $p \circ h = f$. Provide the condition in terms of the fundamental group of $X$ and the map $f$. \( \pi_1(X) \rightarrow \pi_1(S^1) \).

Answer: f_*(\pi_1(X)) = 0


Question: Find \(\lim_{x \to \infty} x^{(2-\sin(\frac{2}{x}))}(x\sin(\frac{2}{x})-2)\).

Answer: -\dfrac{4}{3}


Question: Calculate the limit: \[ \lim_{n\to\infty}{\dfrac{1}{n}\sum_{k=1}^{n}{\left(\dfrac{n}{k}-\left\lfloor\dfrac{n}{k}\right\rfloor\right)}} \]

Answer: 1 - \gamma


Question: Determine the Fourier transform of the derivative of the Dirac delta function, \(\delta'(t)\).

Answer: i\omega


Question: Determine the real value of \( \alpha \) for which the boundary value problem \( \Delta u(x,y) + 2u(x,y) = x - \alpha \) in the domain \( \Omega = (0, \pi) \times (0, \pi) \) with boundary condition \( u(x,y) = 0 \) on \( \partial \Omega \) has at least one weak solution. Here, \( \Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \). A function \( u \) is a weak solution if \( u \in H_0^1(\Omega) \) and the equation is satisfied in the sense of distributions.

Answer: \dfrac{\pi}{2}


Question:
Determine the value of \( p \) such that the system of planes given by the equations:
\[ a_1: x + py + pz = p \]
\[ a_2: x + p^2y = p^3 \]
\[ a_3: x + p^2y + pz = p \]
\[ a_4: x + py + 2pz = p \]
has exactly one intersection point.

Answer: -1


Question:
Evaluate the limit: \[ \lim_{n \to \infty}\left[(1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{n}{n})\right]^{\frac{1}{n}} \] Choose the correct answer from the following options:
(a) \(\frac{4}{e}\)
(b) \(\frac{3}{e}\)
(c) \(\frac{1}{e}\)
(d) \(e\)

Answer: A


Question: Find the area of the region bounded by the inequalities \(x^2 + y^2 \le 144\) and \(\sin(2x + 3y) \le 0\).

Answer: 72\pi


Question: If the conditions of the Existence and Uniqueness theorem are satisfied, does a unique solution exist for all $x \in (a, b)$?

Answer: No


Question: If \(\alpha\) is an algebraic number with degree 5555, what is the degree of \(\alpha^2\)?

Answer: 5555


Question:
Determine the values of parameters $a$ and $b$ such that the function \( f(x, y) = \frac{ax+by}{1+xy} \) is a binary operation on the interval \((-1, 1)\). Choose the correct option:

A. \( a = b = 2 \)
B. \( a + b \in (-1, 1) \)
C. \( a \in (-1, 1) \) and \( b \in (-1, 1) \)
D. \( a = b \in [-1, 1] \)
E. \( a + b = 1 \)

Answer: D


Question: Given that \([F(a):F]=n\) and there exists a polynomial \(p(x)\) of degree \(n\) with \(a\) as a root, is \(p(x)\) necessarily the minimal polynomial of \(a\) over \(F\)?

Answer: No


Question: Let $R$ be a principal ideal domain (PID) and $I$ a prime ideal of $R$ such that $0 \subset I \subset 1_R$. Suppose $I = \langle a \rangle$, where $a$ is a prime element of $R$. Is there any other prime ideal $J$ such that $0 \subset J \subset I$?

Answer: No


Question: Evaluate the surface integral \( \iint_S \vec{A} \cdot \hat{n} \, dS \) where \( \vec{A} = 6z\hat{i} + (2x+y)\hat{j} - x\hat{k} \) and \( S \) is the entire surface of the region bounded by the cylinder \( x^2 + z^2 = 9 \), \( x = 0 \), \( y = 0 \), \( z = 0 \), and \( y = 8 \). Here, \( \hat{n} \) is the unit normal vector to the surface \( S \).

Answer: 18\pi


Question: Determine whether the limit of the pseudospectrum of a linear operator $A$ as $\epsilon$ approaches zero is equal to the spectrum of $A$. Specifically, is it true that \( \lim_{\epsilon\rightarrow 0}\Lambda_{\epsilon}(A) = \Lambda(A) \), where the $\epsilon$-pseudospectrum is defined as \( \Lambda_{\epsilon}(A) = \{\lambda\in\mathbb{C}: ||(\lambda\mathbb{I}-A)^{-1}||\geqslant\epsilon^{-1}\} \) and the spectrum is \( \Lambda(A) = \{\lambda\in\mathbb{C}:(\lambda\mathbb{I}-A) \text{ is not invertible}\} \)?

Answer: Yes


Question: Does the floor topology on \( \mathbb{R} \), generated by the basis of half-open intervals \([a, b)\) where \(a < b\) are real numbers, have a countable basis?

Answer: No


Question: Calculate the value of \(3^{3^{3^{3^{3^{3^3}}}}} \mod 1000\).

Answer: 387


Question: Determine if there is a closed form for the limit: \[ \sqrt[2]{1+\sqrt[3]{1+\sqrt[4]{1+\cdots}}} \approx 1.5176001678777188... \]

Answer: 1.5176001678777188


Question: Given that $f \in C^2([0, 1])$ with $f(0) = 10$, $f(0.5) = 1$, and $f(1) = 7$, must the second derivative $f''$ attain some specific value? Justify your answer.

Answer: 60


Question: Given that the amount of wine in a bottle is normally distributed as \( N(1.05l, 0.004l^2) \) and the volume of a cask is normally distributed as \( N(22l, 0.16l^2) \), calculate the probability that the contents of 20 randomly chosen bottles will fit inside a randomly chosen cask.

Answer: 0.9794


Question: Evaluate the surface integral \( \iint_{S} (y^2 + 2yz) \, dS \) over the plane \( 2x + y + 2z = 6 \) in the first octant. Verify the setup of the integral and compute the correct value.

Answer: \dfrac{243}{2}


Question: Evaluate the improper integral: \[ \int_{0}^{1} \frac{dx}{(4-3x)\sqrt{x-x^2}} \]

Answer: \dfrac{\pi}{2}


Question: Find the sum of the last three digits of the number $89^{23}$. Express your answer as a single integer.

Answer: 22


Question:
Consider a ring $A$ that is smooth over the Noetherian henselian valuation ring $R = \mathbf{Z}_{(p)}^h$, with $I = pA$ being a principal ideal. The henselianization $A^h$ of $A$ along $I$ is given by the direct limit of étale ring maps:
\[A^h = \varinjlim_{s\in S} A_s\]
where $A \to A_s$ is étale and $A/I \to A_s/I$ is the identity. Can the index set $S$ be arranged to be a countable set?

Answer: Yes


Question: Consider a non-negative, smooth function \( f: [a,b] \rightarrow \mathbb{R}_{\geq 0} \). If there exists a sequence of pairwise disjoint points \( x_n \in [a,b] \) such that \( f(x_n) = 0 \) for all \( n \in \mathbb{N} \), does it follow that \( f(x) = 0 \) for all \( x \in [a,b] \)?

Answer: No


Question:
Let $Q(x)$ be a smooth symmetric matrix with constant signature $(p,q,k)$, where $x$ belongs to $\mathbb{R}^n$ and $p+q+k=m$. Locally around a point $x_0$, does there exist an invertible matrix $P(x)$ of size $m$ such that:
\[
P^T(x) Q(x) P(x) = \begin{pmatrix} Id_p &0&0\\0&-Id_q&0\\ 0&0&0_k\end{pmatrix}
\]

Answer: Yes


Question: Let \( x \in \mathbb{R}^k \). Prove that if there exists a constant \( c \ge 0 \) such that \( x \cdot y \le c\|y\| \) for all \( y \in \mathbb{R}^k \), then \( \|x\| \le c \).

Answer: \|x\| \le c


Question: Determine the largest integer $k$ such that for all sufficiently large $n \in \mathbb{N}$, less than 1% of all permutations of $[n]$ have at least $k$ fixed points.

Answer: 5


Question: Let $\mathscr{C}$ be an artinian, abelian category and $K(\mathscr{C})$ its Grothendieck group. If $[A]=0$ in $K(\mathscr{C})$ for an object $A$, can we conclude that $A=0$?

Answer: Yes


Question: Determine the number of zeros of the function \( f(z) = \log(z + 3) + z \) within the disk \( D_{1/4}(0) \) using Rouché's theorem.

Answer: 0


Question: Let $G$ be a group and for $x,y \in G$, define $[x,y] = x^{-1}y^{-1}xy$ to be the commutator of $x$ and $y$. If $y_1, \cdots, y_n \in G$, determine whether $[x, y_1 \cdots y_n]$ can be expressed as a product of conjugates of $[x, y_i]$ for $i = 1, \cdots, n$. Specifically, are there elements $g_1, \cdots, g_n \in G$ such that $$[x, y_1 \cdots y_n] = [x, y_{j_1}]^{g_1} \cdots [x, y_{j_n}]^{g_n},$$ where $\{j_1, \cdots, j_n\} = \{1, \cdots, n\}$ and $[x, y_{j_i}]^{g_i} = g_i^{-1}[x, y_{j_i}]g_i$?

Answer: Yes


Question: Find the product of all values of $x$ that satisfy the equation: \[ x^{\log_{10}(x)} = x\left(100 + 2^{\sqrt{\log_2(3)}} - 3^{\sqrt{\log_3(2)}}\right) \] Choose the correct answer from the following options: (a) 0.1 (b) 1 (c) 10 (d) 100.

Answer: C


Question: Let $X$ and $Y$ be Banach spaces, and let $T: X \rightarrow Y$ be a Fredholm operator. Consider a complement $V \subset X$ to $\ker T$ and a complement $W \subset Y$ to the image $T(X)$. Define the map $T': X \rightarrow Y/W$ as the composition of $T$ and the projection from $Y$ to $Y/W$. What is the kernel of $T'$? Is it equal to $\ker T$?

Answer: \ker T


Question: Let $F$ be a field of characteristic $0$ and let $D$ be the formal derivative map on $F[X]$. Determine the image of $F[X]$ under the map $D$. Justify your answer.

Answer: F[X]


Question: Suppose we have two local homeomorphisms $f: E \to X$ and $g: E' \to X$. If $S$ is a sheet of $E$, is $g^{-1}(f(E))$ homeomorphic to $f(E)$? Justify your answer.

Answer: No


Question: Is there a Riemann-integrable function $f: [a,b]\to\mathbb{R}$ such that $f$ has an antiderivative which is not equal to $t \mapsto \int_a^t f(x)dx +c$ for any $c\in\mathbb{R}$?

Answer: No


Question: Consider a triangle with vertices (-1,0), (1,0), and (0,1). Suppose $(X_1, X_2)$ is a random vector uniformly distributed over this triangle. Compute the expected value $E(X_1 + X_2)$. Provide your answer as a single number.

Answer: \dfrac{1}{3}


Question: When \((x^2 + x + 1)^{54}\) is expanded, how many coefficients of the resulting polynomial are divisible by 3?

Answer: 105


Question: Evaluate the limit: \[ \lim_{x\to 0}\left\lfloor \frac{x^2}{\sin x\cdot \tan x}\right\rfloor \] where \(\lfloor x \rfloor\) represents the floor function of \(x\).

Answer: 0


Question: Given that $a, b, c > 0$ and satisfy the equation $a^2 + b^2 + 4c^2 = 1$, find the maximum value of the expression $F(a, b, c) = ab + 2ac + 3\sqrt{2}bc$. \( \text{Provide your answer as a single number.} \)

Answer: \sqrt{2}


Question: Evaluate the limit: $$\lim_{x\rightarrow0}\left(\frac{a^x+b^x+c^x}{x}\right)^{\frac{2}{x}}$$ where $a, b, c > 0$.

Answer: \infty


Question: Let $X$ be a finite set, and let $F \langle X \rangle$ be the free group on the set $X$. Suppose there is a surjective homomorphism \( \varphi : F \langle X \rangle \longrightarrow G \). Can you conclude that the group $G$ is finitely generated?

Answer: Yes


Question: Let \(a_n\) be a sequence defined by \(a_1 = 3\) and \(a_{n+1} = 3^{a_n}\). Let \(b_n\) be the remainder when \(a_n\) is divided by 100. Find \(b_{2004}\).

Answer: 87


Question: Determine whether there exists a subset $A \subseteq \mathbb{R}$ that does not have the Baire property. A set $A \subseteq X$ has the Baire property if there exists a set $U \subseteq X$ such that the symmetric difference $A \bigtriangleup U = (A \setminus U) \cup (U \setminus A)$ is in the $\sigma$-ideal of meager sets $\mathcal{I}$ of the topological space $X$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to0}\left[1 + \left(\frac{\log \cos x}{\log \cos(x/2)}\right)^2 \right]^2$$

Answer: 289


Question: Find the number of ways to create a bunch of 10 balloons selected from white, gold, and blue balloons such that the bunch contains at least one white balloon, at least one gold balloon, and at most two blue balloons.

Answer: 24


Question: Let $M = A + iB$ be a complex $n \times n$ Hermitian matrix. We know that $$(\det M)^2 = \det \begin{pmatrix} A & -B \\ B & A \end{pmatrix}.$$ Also, $\det \begin{pmatrix} A & -B \\ B & A \end{pmatrix}$ is a polynomial in $n^2$ variables of degree $2n$. Is it true that $\det M$ is a polynomial of degree $n$ in these $n^2$ variables such that $D^2(M) = \det \begin{pmatrix} A & -B \\ B & A \end{pmatrix}$?

Answer: Yes


Question: In a free group $F(X)$, consider an element $z$ that can be expressed in two reduced forms: $z = x_1^{\epsilon_1} \cdots x_n^{\epsilon_n}$ and $z = y_1^{\delta_1} \cdots y_n^{\delta_n}$, where $\epsilon_i, \delta_i = \pm 1$ and $x_i, y_i \in X$. Can we conclude that $x_i = y_i$ for each $i$?

Answer: Yes


Question: Let \(a\) be a real number in the interval \([0,314]\) such that \(\displaystyle \int^{3\pi+a}_{\pi+a}|x-a-\pi|\sin \frac{x}{2}dx=-16.\) Determine the number of such real values of \(a.\)

Answer: 0


Question:
Calculate the expected value \( \mu_X(t) \) of the random process \( X(t) = \cos(2\pi f_o t + \phi) \), where \( f_o > 0 \) is a constant and \( \phi \) is a random variable with the probability density function:
\[ p_\phi (\varphi) = \frac{1}{4}[\delta (\varphi) + \delta (\varphi - \pi /2) + \delta(\varphi - \pi) + \delta(\varphi - 3\pi /2)] \]

Answer: 0


Question:
Solve the recurrence relation given by:
\[ T(n) = \frac{1}{n}(T(0) + T(1) + \ldots + T(n-1)) + 5n \]
with the initial condition \( T(0) = 0 \). Find a closed-form expression for \( T(n) \).

Answer: 10n - 5H_n


Question: Find an integer number \( a \) such that any integer \( b \) coprime with 34 is congruent to \( a^k \mod 34 \) for some non-negative integer \( k \).

Answer: 3


Question: Determine the number of pairs \((n, a)\) such that the congruence \(x^n \equiv a \pmod{31}\) has exactly 10 solutions.

Answer: 6


Question: Determine the values of $p$ for which the integral \( \int_0^\infty \frac{\log(x)}{1+x^p} \, dx \) converges.

Answer: p > 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left(n - \sqrt[2018]{n^{2018} - n^{2017}}\right) \]

Answer: \dfrac{1}{2018}


Question: Can the rational numbers in the interval $[0,1]$ be covered by disjoint open intervals such that the total length of these intervals is less than any given positive real number $\epsilon$? In other words, is it true that for every $\epsilon > 0$, there exists a sequence of disjoint open intervals $\langle I_n \rangle$ such that $\sum_{i \in I_n}|i| < \epsilon$ and $\Bbb Q \cap [0,1] \subset \bigcup I_n$?

Answer: Yes


Question: Evaluate the expression $a+b+c$ given that the integral \( \int_0^\pi \frac{x\sin x}{3+\cos^2 x}\mathrm{d}x = \frac{\pi^a}{b\sqrt c} \), where $a$ and $c$ are prime numbers.

Answer: 11


Question: Batteries A and B have lifetimes that are independent and exponentially distributed with a common mean of m years. Given that the probability that battery B outlasts battery A by more than one year is 0.33, calculate the value of m.

Answer: 2.41


Question: Let $V$ be a set and $V^V$ denote the set of all functions $f:V\to V$. Suppose $F\subseteq V^V$. Define $[V]^2 = \{\{x,y\}: x, y\in V \land x\neq y\}$. A set $E\subseteq [V]^2$ is $F$-compatible if all members of $F$ are graph homomorphisms from $(V,E)$ to itself. Is there always a largest $F$-compatible set that contains all other $F$-compatible sets?

Answer: Yes


Question: Let \( X \) and \( Y \) be independent and identically distributed random variables with probability mass function \( p(n) = \frac{1}{2^n} \). Find the probability \( P(X \geq 2Y) \).

Answer: \dfrac{2}{7}


Question: Solve for $x$ in the equation: \[ 4^x + 4^{x+1} + 4^{x+2} = 7^{x+1} - 7^{x-1} \] where the solution is in the interval $(-2, 2]$. Provide your answer as a single value of $x$.

Answer: 2


Question:
Find a polynomial \( f(x) \in \mathbb{Q}[x] \) such that:

1. \( f(x) \equiv 1 \pmod{(x-1)^2} \)
2. \( f(x) \equiv x \pmod{x^2} \)
3. \( \deg(f(x)) < 4 \)

Answer: -x^3 + x^2 + x


Question:
Let $B$ be the solution set of the equation:
\[ \left|x^2-3x+\sqrt{x^2+2x-3}+3-|-x+x^2+3|\right|+3=-x \]
Find the complement of $B$, denoted as $B^c$. Choose the correct answer from the following options:
\[
\begin{array}{ll}
1.&\varnothing\\
2.&\mathbb{R}\\
3.&[2,+\infty)\\
4.&[2,3]\\
\end{array}
\]

Answer: 2


Question: If X and Y are independent binomial random variables with parameters n and p, calculate the conditional expected value of X given that X + Y = m.

Answer: \dfrac{m}{2}


Question: Determine the points at which the complex function \( f(z) = |z| \bar{z} \) is differentiable.

Answer: 0


Question: For what real \( \alpha > 0 \) does the integral \( \int_{1}^{\infty}{\frac{1}{\sqrt{x^\alpha-1}}\, dx} \) converge?

Answer: \alpha > 2


Question: Given the differential equation \( \frac{d^2y(t)}{dt^2} + 2\frac{dy(t)}{dt} + 10y(t) = \frac{dx(t)}{dt} + x(t) \), where \( x(t) \) is the input and \( y(t) \) is the output, and the input is \( \frac{3u(t)}{e^t} \), find the zero-state output of the system.

Answer: e^{-t} \sin(3t) u(t)


Question: Consider the 2-by-2 matrix \( A = \begin{bmatrix} 0 & 1 \\ -a_0 & -a_1 \end{bmatrix} \). If \( \begin{bmatrix} x_{1,1}(t) \\ x_{1,2}(t) \end{bmatrix} \) and \( \begin{bmatrix} x_{2,1}(t) \\ x_{2,2}(t) \end{bmatrix} \) are linearly independent solutions to the matrix differential equation \( x' = Ax \), determine whether \( x_{1,1}(t) \) and \( x_{2,1}(t) \) are linearly independent solutions to the differential equation \( y'' + a_1 y' + a_0 y = 0 \). Answer true or false.

Answer: True


Question:
Find the number of permutations \(a_1, a_2, \ldots, a_{100}\) of the numbers \(1, 2, \ldots, 100\) such that:
\[|a_1 - 1| = |a_2 - 2| = \ldots = |a_{100} - 100|\]

Answer: 7


Question: Determine whether the following statement is true or false: "Any boundedly compact metric space is a countable union of its compact subsets."

Answer: A


Question: Calculate the value of the infinite series \( \sum_{n \geq 0} F_{3n} \cdot 2^{-3n} \), where \( F_n \) is the n-th Fibonacci number.

Answer: \dfrac{16}{31}


Question: Given the recurrence relation \( a_n = \ln(1+a_{n-1}) \) with \( a_n > 0 \), find the value of \( L \) such that \( \lim_{n\to\infty} na_n = L \).

Answer: 2


Question: Find the value of \(a\) such that the equation \(\frac{25}{\sqrt{x-1}} + \frac{4}{\sqrt{a-2}} = 14 - \sqrt{x-1} - \sqrt{a-2}\) has at least one solution.

Answer: 6


Question: Consider a subset \( A \subset \mathbb{R} \) defined as \( A = \{ k \} \), where \( k \in \mathbb{Z} \) and \( k = -N, -N+1, \ldots, 0, 1, 2, \ldots, N \). If we pick a number \( x \) from \( \mathbb{R} \) according to a normal distribution \( \mathcal{N}(0, \sigma) \) with \( \sigma > 0 \), what is the probability \( p \) that \( x \in A \) as \( N \to \infty \)?

Answer: 0


Question: Calculate the value of the expression \( Q = \frac{x + 1}{y} \) when the expression \( P = x + 2y + \frac{5x + 5y}{xy - 1} \) reaches its maximum value, given that \( x \) and \( y \) are positive numbers such that \( xy > 1 \).

Answer: 2


Question: Let $\Omega$ be a bounded open set in $\mathbb{R}^n$, and let $f$ be a function defined on $\Omega$ which is positive almost everywhere. Let $c$ be a fixed positive constant such that $c < m(\Omega)$, where $m(\Omega)$ denotes the Lebesgue measure of $\Omega$. Is it true that there exists a positive constant $M$ such that for any subset $V$ of $\Omega$ with $m(V) \geq c$, \[\int_V f(x)\,dx \geq M\]? Provide a justification for your answer.

Answer: Yes


Question:
Consider the fourth-degree polynomial \( acx^4 + b(a + c)x^3 + (a^2 + b^2 + c^2)x^2 + b(a + c)x + ac \), where \( a, b, \) and \( c \) are real numbers. Determine the nature of its roots:

(a) The polynomial has four complex (non-real) roots.
(b) The polynomial has either four real roots or four complex roots.
(c) The polynomial has two real roots and two complex roots.
(d) The polynomial has four real roots.

Choose the correct option.

Answer: B


Question: Consider the complex number $u = 2 + 2i$. Sketch an Argand diagram showing the points representing the complex numbers $1$, $i$, and $u$. Shade the region whose points represent the complex numbers $z$ which satisfy both the inequalities $|z-1| \le |z-i|$ and $|z-u| \le 1$. Determine the value of $|z|$ for the point in this region for which $\arg z$ is least.

Answer: \sqrt{7}


Question: From a point $P$ inside an ellipse, determine the number of points $Q_i$ on the ellipse such that the line segment $PQ_i$ is normal to the ellipse.

Answer: 4


Question:
Evaluate the following infinite series of integrals:
\[ \sum_{k=1}^\infty \int_{1/(k+\beta)}^{1/(k+\alpha)} \frac{1}{1+x}dx \]
where \(0 < \alpha < \beta < 1\). Choose the correct answer from the options below:
\[
\begin{align}
&(A)\ln \frac{\beta}{\alpha}\qquad\qquad (B)\ln\frac{1+\beta}{1+\alpha} \\
&(C)\ln\frac{1+ \alpha}{1+ \beta}\qquad\ (D)\infty
\end{align}
\]

Answer: B


Question: Find all functions \( f: \mathbb{Z} \to \mathbb{Z} \) such that for all integers \( k \), the equation \( f(f(k+1)+3) = k \) holds.

Answer: f(k) = k - 2


Question:
Given that vectors $\mathbf{\beta_1, \beta_2}$ are distinct solutions to the system of equations $\mathbf{AX}= \mathbf{b}$, and that $\mathbf{\alpha_1, \alpha_2}$ forms the basis to the corresponding linear homogeneous system $\mathbf{AX}=\mathbf{0}$, and that $k_1, k_2$ are arbitrary constants, which of the following is always a general solution to the system $\mathbf{AX}= \mathbf{b}$?

A) $k_1 \mathbf{\alpha_1} + k_2(\mathbf{\alpha_1 + \alpha_2}) + \frac{\mathbf{\beta_1 - \beta_2}}{2}$

B) $k_1 \mathbf{\alpha_1} + k_2(\mathbf{\alpha_1 - \alpha_2}) + \frac{\mathbf{\beta_1 + \beta_2}}{2}$

C) $k_1 \mathbf{\alpha_1} + k_2(\mathbf{\beta_1 - \beta_2}) + \frac{\mathbf{\beta_1 - \beta_2}}{2}$

D) $k_1 \mathbf{\alpha_1} + k_2(\mathbf{\beta_1 - \beta_2}) + \frac{\mathbf{\beta_1 + \beta_2}}{2}$

Select the correct option.

Answer: B


Question: Assume that \( \mathcal{H} \) is isomorphic to a Hilbert space \( \mathcal{G} \) via an isomorphism \( \Phi \). Define the inner product on \( \mathcal{H} \) as \( \langle \Phi(x), \Phi(y) \rangle_{\mathcal{H}} = \langle x, y \rangle_{\mathcal{G}} \). Can we conclude that \( \mathcal{H} \) is also a Hilbert space?

Answer: Yes


Question: Calculate the variance of the random variable $Y = XZ$, where $Z \sim \operatorname{Ber}(0.55)$ and $X \sim \exp(\frac{1}{8})$. Use the property $\operatorname{Var}[Y] = E[\operatorname{Var}[Y \mid Z]] + \operatorname{Var}[E[Y \mid Z]]$. Given that $E[Y \mid Z] = 8Z$ and $\operatorname{Var}[Y \mid Z] = 64Z$, find $\operatorname{Var}[XZ]$. Provide your answer as a single number.

Answer: 51.04


Question: Determine the domain of convergence for the series \( \sum^{\infty}_{n=1}\frac{n!x^{2n}}{n^n(1+x^{2n})} \).

Answer: (-\infty, \infty)


Question: Find the coefficient of \( a^8b^4c^9d^9 \) in the expansion of \((abc+abd+acd+bcd)^{10}\).

Answer: 2520


Question: Determine the number of continuous functions \( f \) that satisfy the equation \( xf(y) + yf(x) = (x+y)f(x)f(y) \). Choose from the following options: 1, 2, 3, or none of them.

Answer: 2


Question: Find the smallest positive integer solution to the indeterminate equation \(195y = 221x + 65\) using the Indian method of kuttaka.

Answer: 6


Question: Calculate \( \lim_{n\to\infty} \sum_{k=1}^{n}\frac{(-1)^{k-1}}{{k^2}\binom{n}{k}} \).

Answer: 0


Question: Evaluate the integral \( \int_0^{2\pi} e^{i\theta} e^{- i e^{i n \theta}} \, d\theta \) using contour integration techniques.

Answer: 0


Question: Evaluate the integral: \[ \int_0^\infty \frac{e^{-x}}{\sqrt{x}}\ dx \]

Answer: \sqrt{\pi}


Question: Evaluate the integral \( \int_0^1 \int_x^1 \frac{\cos(y)}{y} \, dy \, dx \). Determine if the integral converges or not.

Answer: \sin(1)


Question: Find the minimum value of \(x^2 + y^2 + z^2\) subject to the constraint \(yz + zx + xy = 3a^2\) using the Lagrange multiplier method.

Answer: 3a^2


Question: Calculate the contour integral \( \int_C \vec{F} \cdot \vec{dr} \) for the vector field \( \vec{F} = (y, x) \) along the closed path \( C: \vec{r}(t) = (\cos t, \sin t) \) for \( 0 \leq t \leq 2\pi \).

Answer: 0


Question: Let $X$ and $Y$ be independent and identically distributed random variables with $\mathbb{E}[X^4] = 1$. Determine the best upper bound for $\mathbb{E}[(X-Y)^4]$. Provide your answer as a single numerical value.

Answer: 8


Question:
A missile's guidance control system is modeled as a discrete-time Markov chain with states:

0: No correction required,
1: Minor correction required,
2: Major correction required,
3: Abort and self-destruct.

Given the one-step transition matrix and that the system is initially in state 1, determine the probability that the missile will eventually self-destruct.

Answer: 1


Question: Let $f$ be an entire function stable on $\mathbb{R}$, and let $E_{\mathbb{R}}$ be its real escaping set defined as: \[E_{\mathbb{R}} = \{ x \in \mathbb{R} : f^{(k)}(x) \rightarrow_{k \to \infty} \infty \} \] Given that $E_{\mathbb{R}}$ is a measurable null set with respect to the Lebesgue measure on $\mathbb{R}$, is $E_{\mathbb{R}}$ also empty?

Answer: Yes


Question: Let \( h \) be the function defined by \( h(x)=\int_{0}^{x^2}e^{x+t}dt \) for all real numbers \( x \). Find \( h'(1) \).

Answer: 3e^2 - e


Question: Compute the limit \( \lim_{n\rightarrow\infty}\sum_{k=0}^{n}\binom{n+1}{k}\left(\frac{x}{n}\right)^k \).

Answer: e^x


Question: Suppose \((X,Y) \sim BN(\mu_x=0, \mu_y=0, \sigma_x^2=1, \sigma_y^2=1, \rho=0.6)\). Find the value of \( c \) such that the random variables \( 2X + 3Y \) and \( 6X + cY \) are independent.

Answer: -\dfrac{38}{7}


Question: In \(\triangle ABC\), medians \(AD\) and \(BE\) are perpendicular. Given \(AC = 14\) and \(BC = 22\), find the length of \(AB\).

Answer: 2\sqrt{34}


Question: Given the polynomial \(0 + x + 12x^2 + 11x^3\) in the quotient ring \(\mathbb{Z}_{13} / (x^4 + 1)\), determine the smallest positive integer \(r\) such that the sequence of rows generated by multiplying the polynomial by \(x^i\) for \(i = 0, 1, 2, \ldots, r\) starts repeating. Assume the width of the matrix is 4. What is the value of \(r\)?

Answer: 8


Question: Find the number of integer solutions to the equation \(x + y + z + w = 10\) such that none of the variables \(x, y, z, w\) are equal to 2.

Answer: 144


Question: Let $(A_n)_{n\geq 0}$ be a collection of rings with compatible maps $f_{ij}:A_j\to A_i$ for $i\leq j$, and let $A=\varprojlim\limits_nA_n$ with the canonical projections $\pi_i:A\to A_i$. Suppose $B\subseteq A$ is a subring such that the maps $\pi_i|_{B}:B\to A_i$ are surjective for all $i$. Prove or disprove: $B=A$.

Answer: B \neq A


Question: Is it possible to extend a commutative ring $R$ to a commutative ring with unity $R'$ such that $R$ is a subring of $R'$?

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \sinh^{-1}(\sin(x)) \, dx \).

Answer: G


Question:
Find the period of the sinusoidal function given by:
\[ x(t) = \cos\left(\frac{4\pi t}{5}\right)\sin^2\left(\frac{8\pi t}{3}\right) \]

Answer: \dfrac{15}{2}


Question: Find the correlation coefficient between the minimum and maximum order statistics, \(X_{(1)}\) and \(X_{(n)}\), from a uniform distribution on the interval \([0,1]\).

Answer: \dfrac{1}{n}


Question: Evaluate the line integral \( \int_\ell \vec{F}(x) \cdot \vec{T}(x) \, ds \) where \( \vec{F} = (y^2, x, z^2) \) and the curve \( \ell \) is parameterized by \( \vec{v}(t) = (\cos t, \sin t, 1) \) for \( 0 \leq t \leq 2\pi \).

Answer: \pi


Question: Calculate the sum of the first 2017 Fibonacci numbers modulo 2017. Specifically, find \( F_{0} + F_{1} + F_{2} + \cdots + F_{2016} \equiv ? \pmod{2017} \), where the Fibonacci sequence \( \{F_n\} \) is defined by \( F_0 = 0, F_1 = 1, \) and \( F_{n+2} = F_{n+1} + F_n \).

Answer: 2016


Question: Find the radius of convergence for the series \( f(x) = \sum_{n=1}^\infty \left(1+\frac{(-1)^n}{n}\right)^{n^2} \cdot \frac{(2x+1)^n}{n} \).

Answer: \dfrac{1}{2e}


Question: Given that $2X^T = A^{-1}BA$, where $A$, $B$, and $X$ are $4 \times 4$ matrices, $\det(A) = 4$, and $\det(B^{-1}) = 3$, find $\det(X)$. Express your answer as a single number.

Answer: \dfrac{1}{48}


Question: Evaluate the double integral \( S = \int_{-1}^{1} \int_{0}^{2\pi} \sqrt{\frac{1}{16}v^{2}} \,du \,dv \). What is the value of \( S \)?

Answer: \dfrac{\pi}{2}


Question: Let $U = \cap_{n \in \mathbb{N}} U_n$ be a non-empty $G_{\delta}$-set, where each $U_n$ is open in a normal topological space $X$. If $\phi: X \rightarrow Y$ is a homeomorphism, is $\phi(U)$ a $G_{\delta}$-subset of $Y$? Provide a justification for your answer.

Answer: Yes


Question:
Consider the linear operator \( T \) from \((C[0,1], \|\cdot\|_1)\) to \((C[0,1], \|\cdot\|_2)\) defined by \[ T(f)(t) = \int_0^t f(s) \, ds \] where \( \|f\|_1 = \sup_{t \in [0,1]} |f(t)| \) and \( \|f\|_2 = \left( \int_0^1 |f(t)|^2 \, dt \right)^{1/2} \). Determine the norm of the operator \( T \). Choose the correct answer from the following options:

1. 1
2. \( \frac{1}{2} \)
3. \( \frac{1}{\sqrt{3}} \)
4. None of the above

Answer: 3


Question: Given the prior density function \((B+1)Bq(1-q)^{B-1}\) where \(q\) is the probability of death and is between 0 and 1, calculate the parameter \(B\) if the prior mean is 0.1.

Answer: 18


Question: Given the function \( f(x) = \prod_{r=0}^{n} \left( \binom{n}{r} x + r+1 \right) \), if \( f'(0) = \frac{3}{2} (7!) \), find the value of \( n \).

Answer: 5


Question: In triangle $ABC$, where $\angle A = 100^\circ$ and $\angle B = \angle C = 40^\circ$, the side $AB$ is extended to a point $D$ such that $AD = BC$. Determine the measure of $\angle BCD$ without using trigonometry.

Answer: 10^\circ


Question: Calculate the limit: $$\lim_{x\to0} \frac{\log{(1+x^3)}+\frac{1}{1+x^3}-1}{x^2-\arctan(x^2)}.$$

Answer: \dfrac{3}{2}


Question: Find the unit digit of the expression \((5+\sqrt{24})^{1234}+(5-\sqrt{24})^{1234}\).

Answer: 8


Question: Let $(X,d)$ be a metric space and $(A_n)_{n\in\mathbb{N}}$ be a sequence of non-empty closed subsets such that for every $\varepsilon > 0$, there exists $N\in\mathbb{N}$ such that for all $m,n\geq N$, the Hausdorff distance $d_H(A_m,A_n)<\varepsilon$. Is it possible to choose $x_n\in A_n$ for all $n\in\mathbb{N}$ such that $(x_n)_{n\in\mathbb{N}}$ is a Cauchy sequence?

Answer: Yes


Question: Compute the value of the fifth derivative at zero, \( f^{(5)}(0) \), for the function \( f(x) \) defined by the power series \( \sum_{n=0}^{\infty} (-1)^n \left(\frac{n+1}{n+3}\right)^{n^2} x^{2n} \).

Answer: 0


Question: Let $r \in \mathbb{Q}[\sqrt{2}]$. Define $\phi: \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$ by $\phi(a + b\sqrt{2}) = a - b\sqrt{2}$ for $a, b \in \mathbb{Q}$. Prove that $\phi(r) = r$ if and only if $r \in \mathbb{Q}$.

Answer: r \in \mathbb{Q}


Question: Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$. Let $N$ be the smallest positive integer such that $S(N) = 2013$. What is the value of $S(5N + 2013)$?

Answer: 18


Question: Find all natural number solutions \((x, y)\) to the equation \(3^x - 7^y = 2\).

Answer: (2, 1)


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

Answer: 0


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

Answer: True


Question: Determine the Lebesgue measure of the set of numbers in the interval (0,1) that contain 100 consecutive 4's in their decimal expansion.

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( \frac{\sin(x)-3\sinh(x)+2x}{x^2(\tanh(2x)+\sin(x))} \right) \]

Answer: -\dfrac{2}{9}


Question: Suppose $x$ and $y$ are elements in a group $G$ where the order of $y$ is not equal to $1$, $yx = xy^2$, and the order of $x$ is $3$. Determine the order of $y$.

Answer: 7


Question: Determine whether every bounded Borel function on a compact set is the pointwise limit of continuous functions. Provide a justification for your answer.

Answer: No


Question: Evaluate the integral \( \int_C \text{Im}(z) \, dz \) where \( C \) is the unit circle in the complex plane.

Answer: -\pi


Question: Evaluate the integral \( \int_{-\pi}^\pi \frac{\sin(13x)}{\sin x} \cdot \frac{1}{1+2^x} \, \mathrm{d}x \).

Answer: \pi


Question: Find an irreducible polynomial in \(\mathbb{Z}[x]\) that is reducible modulo 2, 3, and 5.

Answer: x^4 + 1


Question: Let $M$ and $N$ be $R$-modules. If $M \oplus N$ is a free module, is $M \oplus 0_N$ necessarily a free module? Provide a justification for your answer.

Answer: No


Question: Does the sequence \( \left(1 + \frac{z}{n}\right)^n \) converge to \( e^z \) as \( n \to \infty \) for a complex number \( z \)?

Answer: e^z


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

Answer: \dfrac{1}{e}


Question: Determine the side length of the largest cube that can fit inside a cone with a height of 12 units and a base radius of \(3\sqrt{2}\) units.

Answer: 4


Question: Determine the order of \( \wedge^2(\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}) \).

Answer: p


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left( \ln(e^{2x} - 1) - \frac{x^2 - 3}{x} \right) \]

Answer: \infty


Question: Let $A \subset \mathbb{R}^N \setminus \{0\}$ be a closed symmetric set such that if $x \in A$, then $-x \in A$. Suppose $A$ is homeomorphic to a sphere $S^n$, where $n \leq N$. Is it possible to construct an odd homeomorphism $F: A \to S^n$?

Answer: Yes


Question: Can a diffeomorphism exist between the entire spaces of $\Bbb R^n$ and $\Bbb R^m$ where $n \neq m$?

Answer: No


Question: Find the dimension of the product space \( X \times \mathbb{P}^{2} \), where \( X = \{ w_{0}^{3} = w_1(w_{1}^{2} - w_{2}^{2}) \} \subseteq \mathbb{P}_{w}^{2} \).

Answer: 3


Question: Solve the equation: \( \sqrt[4]{13x+1} + \sqrt[4]{4x-1} = 3\sqrt[4]{x} \) for \(x\).

Answer: \dfrac{1}{3}


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

Answer: \dfrac{3}{5}


Question: Let $F = GF(11)$ be the finite field of 11 elements. Let $G$ be the group of all non-singular $n \times n$ matrices over $F$, and let $H$ be the subgroup of those matrices whose determinant is 1. Find the index $[G:H]$. \( \boxed{?} \)

Answer: 10


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

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


Question:
Find the area of the portion of the sphere defined by $x^2 + y^2 + z^2 = 4z$ that lies inside the region defined by $x^2 + y^2 = z$ using double integrals. Express your answer in terms of $
abla$ and $dA$. The correct answer is known to be $4\pi$. Verify this result.

Answer: 4\pi


Question: Determine the maximal compact interval for which the series \( \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1} = \arctan(x) \) converges. Provide your answer as an interval.

Answer: [-1, 1]


Question: Determine whether the following statement is true or false: If $f:I\to\mathbb{R}$ is a one-to-one and continuous function, then $f$ is strictly monotone on $I$, where $I\subseteq\mathbb{R}$ is nonempty.

Answer: B


Question: Suppose $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, $g(x)=xf(x)-\int_0^xf(t)\ dt$, with $f(0)=0$ and $g(x)=O(x^2)$ as $x\to0$. Determine whether it is true that $f(x)=O(x)$ as $x\to0$. Provide a justification for your answer.

Answer: Yes


Question: Given three independent and identically distributed random variables $X, Y, Z$ following a Uniform $(0,1)$ distribution, find the probability $\Pr(X > YZ)$. Provide your answer as a single probability value.

Answer: \dfrac{3}{4}


Question: For the differential equation \( \frac{dy}{dt} = 2y + 3\cos 4t \), determine the initial value \( y(0) = y_0 \) such that the solution remains bounded for all \( t \).

Answer: -\dfrac{3}{10}


Question:
Find the maximum value of \(x\) that satisfies the equation:
\[ x(9^{\sqrt{x^{2}-3}} + 3^{\sqrt{x^{2}-3}}) = (3^{2\sqrt{x^{2}-3}+1} - 3^{\sqrt{x^{2}-3} + 1} - 18) \sqrt{x}+ 6x \]

Answer: 3


Question: Determine the number of positive integer solutions to the equation \(a^2 + b^2 = p_1p_2p_3\), where \(p_1, p_2,\) and \(p_3\) are distinct primes, each congruent to 1 modulo 4.

Answer: 8


Question: Determine the fundamental group of a 2-sphere with the identification $x = -x$. What is the fundamental group of this space?

Answer: \mathbb{Z}/2\mathbb{Z}


Question: Compute the limit: \( \lim_{n \to \infty} e^{\lambda n(e^{iu}-1)} \), where \( i = \sqrt{-1} \) and \( \lambda \) is the parameter for a Poisson random variable.

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}\sqrt[n] {2^n+3^n+\sin n} \]

Answer: 3


Question: For every integer $n > 1$, determine whether there exists a diagonal matrix $D \in M(n, \mathbb{R})$ such that for any matrix $A \in M(n, \mathbb{R})$, the condition $AD = DA$ implies that $A$ is also a diagonal matrix.

Answer: Yes


Question:
Find the limit as \( n \to \infty \) of the sum:
\[
\sum_{k=1}^{n} \frac{1}{\sqrt{n^2+k}}.
\]

Answer: 1


Question: If $B/K$ is a field extension of degree 1, does it imply that $B = K$? Justify your answer.

Answer: B = K


Question: Evaluate the integral \( \int (10x^2 + 3y^3 - 4z^2 + 8) \delta(2\vec{r}) \, d^3\vec{r} \), where \( \delta(2\vec{r}) \) is a three-dimensional delta function.

Answer: 1


Question: Determine whether there exists an element of order 30 in the group \(GL_{10}(\mathbb{C})\) that has integer coefficients.

Answer: Yes


Question: Determine whether the limit \( \lim_{(x,y)\to(0,0)}\frac{x^2+\sin^2y}{x^2+y^2} \) exists. If it does, find its value.

Answer: 1


Question: Determine the order of the automorphism \( \sigma \) in the Galois group \( \text{Gal} \left( \frac{\mathbb{Q}(\sqrt[3]{3},\sqrt{-3})}{\mathbb{Q}} \right) \), where \( \sigma(\sqrt[3]{3}) = \omega\sqrt[3]{3} \) and \( \sigma(\sqrt{-3}) = -\sqrt{-3} \).

Answer: 2


Question: Let $M$ be a Riemannian manifold, and let $x$ and $y$ be two points in $M$. Assume that $x$ is not in the cut locus of $y$. Does there exist a neighborhood $U$ of $x$ and a neighborhood $V$ of $y$ such that for every point $u$ in $U$ and for every point $v$ in $V$, $u$ is not in the cut locus of $v$?

Answer: Yes


Question: Given that \( \int \frac{f(x)}{x^2(x+1)^3}\,dx \) is a rational function and \( f(x) \) is a quadratic function with \( f(0) = 1 \), find \( f'(0) \).

Answer: 3


Question: Let $a$ and $b$ be two $2m$-dimensional vectors such that their entries are Rademacher random variables, and the sum of the variables for each vector is zero. That is, $$P(a_i=1)=P(a_i=-1)=P(b_i=1)=P(b_i=-1)=\frac{1}{2}$$ and $$\sum\limits_{i=1}^{2m}a_i=\sum\limits_{i=1}^{2m}b_i=0.$$ Find the expected value of the inner product $\langle a,b\rangle$. \(\text{Note: The index should start from 1 to } 2m.\)

Answer: 0


Question:
Given a continuous function \(f(x)\) such that \(f'(x) > 0\) for all \(x\) in \((-\infty, \infty)\) and \(f''(x) = 2\) for all \(x > 0\), find the limit:
\[ \lim_{x \to \infty} \frac{3x^2 - \frac{3}{x^2+1} - 4f'(x)}{f(x)}. \]

Answer: 3


Question: Let $f \in \mathscr{S}(\mathbb{R}^n)$, where $\mathscr{S}(\mathbb{R}^n)$ is the Schwartz space of all $C^\infty$ functions $f: \mathbb{R}^n \to \mathbb{C}$ such that $x \mapsto x^\alpha \partial^\beta f(x)$ is bounded for all multi-indices $\alpha, \beta$. Is it true that $\partial^\alpha f$ is uniformly continuous for all multi-indices $\alpha$? Provide a justification for your answer.

Answer: Yes


Question: Let $X$ be a locally compact Hausdorff space. Suppose $f: X \to \mathbb{R}$ is a continuous function and $g \in C_c(X)$, where $C_c(X)$ denotes the space of continuous functions with compact support. Is it true that the product $fg \in C_c(X)$? Justify your answer.

Answer: Yes


Question: Given a strictly decreasing sequence of non-negative integers \(\mu_{n},\mu_{n-1},\cdots,\mu_{1}\), determine if it is always possible to find \(x_{1}=2\) and \(x_{2},\cdots,x_{n}\) as odd integers such that the determinant \(\det\begin{pmatrix} x_{1}^{\mu_{1}} & x_{1}^{\mu_{2}} & \cdots & x_{1}^{\mu_{n}}\\ x_{2}^{\mu_{1}} & x_{2}^{\mu_{2}} & \cdots & x_{2}^{\mu_{n}}\\ \vdots & \vdots & \vdots & \vdots\\ x_{n}^{\mu_{1}} & x_{n}^{\mu_{2}} & \cdots & x_{n}^{\mu_{n}} \end{pmatrix}\not = 0\).

Answer: Yes


Question: Let \(T: \mathbb{R}^4 \to \mathbb{R}^4\) be a linear transformation such that the null space of \(T\) is \(\{(x,y,z,w) \in \mathbb{R}^4 : x+y+z+w=0\}\) and the rank of \((T-4I)\) is 3. If the minimal polynomial of \(T\) is \(x(x-4)^a\), find the value of \(a\).

Answer: 1


Question: Evaluate the double integral \( \iint_A xy^3e^{x^2-y^4} \,dx\,dy \) where \( A \) is the region bounded by \( x=4, \; y=0, \; x^2-y^4=9, \) and \( x^2-y^4=1 \). Use the substitution \( x^2-y^4=v, \; x^2=u. \)

Answer: e^9 - 2e


Question:
Determine the values of \( t \in \mathbb{R} \) for which the set of solutions to the following system of equations forms a subspace of \( \mathbb{R}^3 \):
\[
3x_{1}+(1-t^{2})x_{2}^{3}-x_{3}=0
\]
\[
x_{1}-5x_{2}+2(t+1)|x_{3}|=t^{3}-t
\]

Answer: -1


Question: Compute the expectation \( \mathbb{E} [Z_1 \mid Z_1 > Z_2] \) where \( Z_1 \) and \( Z_2 \) are independent and identically distributed standard normal random variables.

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


Question: A large tank is initially filled with 100 gallons of pure water. Brine containing 3 pounds of salt per gallon is pumped into the tank at a rate of 4 gallons per minute, while the well-mixed solution is pumped out at a rate of 5 gallons per minute. Determine the amount of salt in the tank after 30 minutes.

Answer: 159.58


Question:
Given two permutations of the set \( \{1, 2, \ldots, N\} \) where \( 3 \leq N \leq 1000 \), find the number of permutations \( X_1, X_2, \ldots, X_N \) such that when arranged in a 3xN matrix with the two given permutations, each column contains unique elements.

For example, for \( N = 4 \):

First permutation: \( \begin{pmatrix}3 & 1 & 2 & 4\end{pmatrix} \).
Second permutation: \( \begin{pmatrix}2 & 4 & 1 & 3\end{pmatrix} \).

Determine the number of possible permutations for the third row \( \begin{pmatrix}X_1 & X_2 & X_3 & X_4\end{pmatrix} \) such that:
\( X_1 \neq 3, 2 \),
\( X_2 \neq 1, 4 \),
\( X_3 \neq 2, 1 \),
\( X_4 \neq 4, 3 \).

The answer for this example is 2, with possible permutations \( \begin{pmatrix}1 & 3 & 4 & 2\end{pmatrix} \) and \( \begin{pmatrix}4 & 2 & 3 & 1\end{pmatrix} \).

Answer: 2


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a function satisfying the conditions \( f(10-x) = f(x) \) and \( f(2-x) = f(2+x) \) for all \( x \in \mathbb{R} \). Given that \( f(0) = 101 \), determine the minimum possible number of values of \( x \) in the interval \([0, 30]\) such that \( f(x) = 101 \).

Answer: 11


Question: Solve the differential equation \( \frac{d^2h}{dx^2} = 0 \) using the Galerkin method for \( 0 \le x \le 3 \), given the boundary conditions: \( h = 0 \) cm when \( x = 0 \) m and \( h = 10 \) cm when \( x = 3 \) m.

Answer: h(x) = \dfrac{10}{3}x


Question: Find the value of the limit: $$\lim _{ { x }\to { 0 } }{ \frac { \sin x-\arctan x }{  {x }^{ 2 }\log(1+x) } }$$ where the logarithm is the natural logarithm.

Answer: \dfrac{1}{6}


Question: Is there a group $G$ with a subgroup $H \subseteq G$ of finite index and an element $g \in G$ such that $H \cap g^{-1} H g$ has infinite index in $G$?

Answer: No


Question: Is it true that for every measurable cardinal \( \kappa \), there exists a normal, \( \kappa \)-complete, and non-principal ultrafilter on \( \kappa \)?

Answer: Yes


Question: Evaluate the limit: \( \lim_{x \to \infty} xe^{-x^2} \)

Answer: 0


Question: Let $X$ be a regular affine $\mathbb{C}$-scheme, and $A$ a finitely-generated $\mathbb{C}$-algebra. Consider a closed subscheme $Y \subset X \times \mathrm{Spec}(A)$ of codimension $1$ such that for each $t \in \mathrm{Spec}(A)$, the fiber $Y_t$ is an effective Cartier divisor of $X \times \{t\}$. Is $Y$ flat over $\mathrm{Spec}(A)$?

Answer: Yes


Question: Calculate the residue of \( \frac{e^{z}}{\sin(iz)} \) at \( -n\pi i \).

Answer: -i


Question: Evaluate the limit \( \lim_{x\rightarrow 0}\frac{\int_{0}^{x^{4}}e^{\sqrt t }\cdot\sin t \ dt}{x^{8}} \).

Answer: \dfrac{1}{2}


Question: Calculate \( \lim_{x \to \infty} \left(\frac{\sqrt[x]{1} + \sqrt[x]{2}}{2}\right)^x \).

Answer: \sqrt{2}


Question: Evaluate the line integral \( \oint \mathbf{F} \cdot \mathbf{n} \, ds \) where \( \mathbf{F} = y \mathbf{i} + x \mathbf{j} \) and the curve is a circle with radius one. Do not use Green's theorem in your solution.

Answer: 0


Question: Let \( f(x) = x^5 + x + 7 \). Find the value of the inverse function at the point 1035, i.e., compute \( f^{-1}(1035) \).

Answer: 4


Question: Determine the number of elements of order 15 in the symmetric group \( S_{13} \).

Answer: 79567488


Question: Find the probability that the quadratic equation \(x^2 + px + q = 0\) has both solutions positive, given that \(|p| \leq 1\) and \(|q| \leq 1\).

Answer: \dfrac{1}{48}


Question: Compute \( 5^{11469} \mod 1911 \).

Answer: 1448


Question: Evaluate the limit: \[ \lim_{x\rightarrow \infty}\left\{\left[(x+1)(x+2)(x+3)(x+4)(x+5)\right]^{\frac{1}{5}}-x\right\} \]

Answer: 3


Question: Compute the second-order mixed partial derivative \( \frac{\partial^2 f}{\partial y \partial x} \) at the point \((0,0)\) for the function \( f(x,y) = \frac{xy(x^2-y^2)}{x^2+y^2} \) when \((x,y) \neq (0,0)\), and \( f = 0 \) otherwise.

Answer: -1


Question: Given a continuously differentiable function $a(x)$ defined for $x > 1$, with $a'(x) < 0$ and $a'(x) > -\frac{a(x)}{x}$ for all $x > 1$, determine the limit $\lim_{x \to \infty} a(x)$. Provide a justification for your answer.

Answer: 0


Question: If $x, y, z$ are three distinct positive integers such that $x + y + z = 13$ and $xy, xz, yz$ form an increasing arithmetic sequence, find the value of $(x + y)^z$.

Answer: 117649


Question: Suppose $A(t)$ is a family of symmetric matrices smoothly depending on $t \in [0,1]$. Assume that $A(t)$ is null positive definite for all $t \in (0,t_0)$, and that $A(t_0)$ is null semi-positive definite but not null positive definite. Does there exist a null vector $v$ such that $A(t_0) v = 0$?

Answer: Yes


Question: Find the volume of the solid formed by revolving the region bounded by the graphs of \(x=0\), \(y=x(x-1)\), and \(y=0\) about the line \(x=-1\).

Answer: \dfrac{\pi}{2}


Question: Find the sum of all positive integers \( n \) for which \(|2^n + 5^n - 65|\) is a perfect square.

Answer: 6


Question: Let $C$ be a circumference of radius $a$ in the plane $2x+2y+z=4$, centered at the point (1,2,-2). Given the vector field $F(x,y,z)=(y-x,z-x,x-y)$, determine the value of $a$ such that the line integral $\oint_C F \cdot dr = -\frac{8\pi}{3}$. Use Stokes' Theorem to solve the problem.

Answer: 1


Question: Determine the number of subgroups of order 17 in the symmetric group \( S_{17} \).

Answer: 15!


Question: Find the remainder when \(2^{2019}\) is divided by 2019.

Answer: 8


Question: Prove that \( \lim_{x \to \infty} \left[1+\frac{a}{x}+o\left(\frac{1}{x}\right) \right]^x = e^a \).

Answer: e^a


Question: Find the value of $k \in \mathbb{R}$ such that \( \lim_{n\to\infty} \frac{\tan n}{n^k} = 0 \).

Answer: 1


Question: Given that $a, b, c, x, y, z \in \mathbb{R}^+$ and $ab + bc + ca = xy + yz + zx = 1$, find the minimum value of the expression $ay + az + bz + bx + cx + cy$. It is known that when $a = b = c = x = y = z = \frac{1}{\sqrt{3}}$, the value is 2, and as $a = b = x = y \to 1$ and $c = z \to 0$, the value also approaches 2. Determine the minimum value of the expression.

Answer: 2


Question: Suppose the function \( g: \mathbb{R} \rightarrow \mathbb{R} \) satisfies the inequality \[ \left| \sum_{r=1}^{n} 3^r \left( g(x+ry) - g(x-ry) \right) \right| \leq 1 \] for all positive integers \( n \) and all real values of \( x, y \). If \( g(13) = 17 \), find the value of \( g(2019) \).

Answer: 17


Question: Let $F:(0,\infty) \to [0,\infty)$ be a continuous function such that $F(1)=0$, $F$ is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$. Suppose $F$ is strictly convex on $[1-\epsilon,1+\epsilon]$ for some $\epsilon>0$. Does there exist a $\delta>0$ such that $F$ is convex at every point $y \in (1-\delta,1)$? Justify your answer.

Answer: Yes


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

Answer: 1


Question: Find the Cramer-Rao Lower Bound for the variance of any unbiased estimator of \( \log \sigma \), where \( X_1, X_2, \ldots, X_n \) form a random sample from a normal distribution with mean 0 and unknown standard deviation \( \sigma > 0 \).

Answer: \dfrac{1}{2n}


Question: Determine whether the Minkowski difference of two disjoint compact convex sets \(A\) and \(B\) is convex.

Answer: Yes


Question: Determine whether there exists a topological space $X$ such that $C_0(X) \neq \overline{C_c(X)}$, where $C_0(X)$ is the set of continuous functions on $X$ that vanish at infinity, and $C_c(X)$ is the set of continuous functions on $X$ with compact support.

Answer: Yes


Question: Determine the values of $t$ for which the integral \( \langle p, q \rangle = \int_t^2 p(x) q(x) \, dx \) defines an inner product on the vector space $V = P_2$, the space of polynomials of degree at most 2.

Answer: (-\infty, 2)


Question: Given distinct real numbers \(a\) and \(b\) that satisfy the equations \(|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|\), find the minimum value of \(|a-b|\).

Answer: 2


Question: In a graph G with diameter 2 and maximum degree \( \Delta < n-1 \), if a chordless 7-cycle exists, is there a vertex \( v \in V(G) \) such that the neighborhood \( N(v) \) contains at least three vertices of the 7-cycle?

Answer: Yes


Question: Given that \(\cos \left[\sqrt{(\sin x + \cos x)(1 - \sin x \cos x)}\right] = \sqrt{\cos (\sin x + \cos x) \cos (1 - \sin x \cos x)}\), find the value of \(\sin^5 x + \cos^5 x\).

Answer: 1


Question: Find the number of zeros of the complex polynomial \( P(z) = z^6 + (1 + i)z + 1 \) in the annulus \( \frac{1}{2} < |z| < \frac{5}{4} \) using Rouche's Theorem.

Answer: 6


Question: Given $N$ sets, each containing between 1 and 100 elements, determine the number of ways to select one element from each set such that no two selected elements are the same. For example, if the sets are $\{1, 2, 3\}$, $\{4\}$, and $\{1, 2\}$, the answer is 4, corresponding to the arrangements $(1,4,2)$, $(2,4,1)$, $(3,4,1)$, and $(3,4,2)$. Find the total number of such arrangements for the given sets.

Answer: 4


Question: Let $X$ be a compact Hausdorff space and $C(X)$ be the set of continuous functions on $X$. Suppose $F$ is a closed subspace of $X$. If $f \in C(X)$ satisfies $f|_{F} = 0$ implies $f = 0$, does it follow that $F = X$?

Answer: Yes


Question: Let $G$ be a non-abelian finite $p$-group and $H$ be a proper subgroup of $G$ with order $p^2$. Given that $C_G(H) = H$ and $|N_G(H)| = p^3$, determine the order of $Z(G)$. Assume $1 \neq Z(G) \leq H$. What is the order of $Z(G)$?

Answer: p


Question: Use von Neumann stability analysis to determine the stability condition for the mesh size $h$ in the discrete form of the equation $\frac{\partial c}{\partial x} = \frac{\partial^2 c}{\partial y^2}$. Apply the first-order forward finite difference for the first-order derivative and the central difference scheme for the second-order derivative. Assume $c_{i, j} = c(ih, jh)$ and the mesh size $h$ is the same in both $x$- and $y$-directions. What restriction arises for the mesh size $h$?

Answer: h \geq 2


Question: Find the smallest positive integer \( n \) such that \( n, n+1, \) and \( n+2 \) each have exactly four positive factors.

Answer: 33


Question:
Consider two real-valued sequences \(\{x_n\}\) and \(\{y_n\}\) satisfying the condition \(x_n^3 - y_n^3 \rightarrow 0\) as \(n \rightarrow \infty\). Determine which of the following statements is true:

(A) \(x_n - y_n \rightarrow 0\)

(B) \(x_n - y_n \rightarrow 0\) only if \(\{x_n\}\) converges

(C) \(x_n - y_n \rightarrow 0\) only if \(|x_n| - |y_n|\) converges

(D) \(x_n - y_n \rightarrow 0\) only if \(|x_n^2 + x_ny_n + y_n^2|\) converges

Choose the correct option.

Answer: A


Question: Find the solution to the functional differential equation \( f'(x) - f(\alpha x) = 0 \) with the initial condition \( f(0) = 0 \), where \( f: \mathbb{R}^+ \rightarrow \mathbb{R} \) and \( 0 < \alpha < 1 \).

Answer: 0


Question: Compute the covariance between $X_i$ and $Y = \sum_{k=-\infty}^{\infty} a_k X_k$, where $X_i$, $i \in \mathbb{Z}$, are independent and identically distributed Gaussian variables with mean $0$ and variance $\sigma_i^2$, $X_i \sim \mathcal{N}(0,\sigma_i^2)$. The random variable $Y$ is distributed as $\mathcal{N}(0,\sum_{i=-\infty}^{+\infty}a_i^2\sigma_i^2)$. Provide your answer in terms of $a_i$ and $\sigma_i^2$. \( \text{Cov}(X_i, Y) = ? \)

Answer: a_i \sigma_i^2


Question: Evaluate the summation: \[ \sum_{k=0}^{m}\frac{k^m}{k!(m-k)!}(-1)^{m-k} = 1 \]

Answer: 1


Question: Let $F$ be a field and $x$ be algebraic over $F$. Suppose the minimal polynomial of $x$ over $F$ is of odd degree. Prove that $F(x) = F(x^2)$. Justify your reasoning.

Answer: F(x) = F(x^2)


Question: Find all holomorphic functions \( f: \mathbb{C} \rightarrow \mathbb{C} \) such that \( f'(0) = 1 \) and \( f(x+iy) = e^{x}f(iy) \).

Answer: e^z


Question: How many permutations of the string "ABCDEF" exist such that no two adjacent characters in the original string are adjacent in the permutation?

Answer: 90


Question: Determine the radius of convergence of the series \( \sum_{n=1}^\infty \frac{nx^n}{n^2 + x^{2n}}. \)

Answer: 1


Question: Let $X$, $Y$, and $Z$ be independent random variables with probability density function $f(t) = 2t$ for $t \in [0,1]$, and $f(t) = 0$ otherwise. Define $W = \max(X, Y, Z)$. What is the expected value of $W$?

Answer: \dfrac{6}{7}


Question: Determine the number of solutions to the equation: \[ \sqrt{1-x} + \sqrt{1-2x} + \sqrt{1-4x} = x^2 + 2 \]

Answer: 2


Question: Determine if there exists a constant \( k > 1 \) such that for every natural number \( n \), if \( G \) is a weak \( n \)-Erdos graph, then the chromatic number \( \chi(G) \) satisfies \( \chi(G) \leq k \cdot n \). If such a \( k \) exists, find the smallest possible value of \( k \).

Answer: 2


Question: Suppose \(\gamma: \mathbb{S}^1 \to \mathbb{R}^2\) is a smooth, origin-symmetric, strictly convex curve. Determine whether there exists a special linear transformation \(A \in SL(2, \mathbb{R})\) such that the length of \(A\gamma\) is minimized.

Answer: Yes


Question: Find the real number \(x\) such that \(5^{2x-1} - 6e^{x-1} + 1 = 0\).

Answer: 1


Question: A planar rectangular shape with dimensions $N$ by $M$ is positioned in 3-space above a two-dimensional surface. Given a large number of random 3-space rotational orientations of the shape, determine the average surface area $A$ of its projection onto the two-dimensional surface.

Answer: \dfrac{NM}{2}


Question: Evaluate the integral \( \oint_{|z|=4} \frac{e^{1 /(z-1)}}{z-2} \, dz \) using Cauchy's Integral Formula.

Answer: 2\pi i


Question: Determine whether the following statement is true or false: Let $G$ be a 2-connected graph and $C$ be a smallest cycle in $G$. If $G$ is not a complete graph and not a cycle, then there exists a path from $x$ to $y$ for some $x, y \in V(C)$ such that each vertex in this path is not in $V(C) - \{x, y\}$.

Answer: True


Question: If $a>0$ and $(x+1)(x+2)(x+3)(x+4)+1=(ax^2+bx+c)^2$, find the value of $a+b+c$.

Answer: 11


Question: Provide an example of a finite nonabelian group whose exponent is equal to its order.

Answer: S_3


Question: Calculate the finite value of \(E(|X-Y|)\) where \(X\) and \(Y\) are independent standard uniform random variables on the interval \([0, 1]\).

Answer: \dfrac{1}{3}


Question: Let $f$ be an extended value function on $\mathbb{R}^n$ that is bounded from below, i.e., \( f(x) \geq \alpha \) for all \( x \in \mathbb{R}^n \). Prove or disprove that \( \liminf_{\|x\| \to +\infty} f(x) \geq \alpha \).

Answer: \liminf_{\|x\| \to +\infty} f(x) \geq \alpha


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{x^4 \, dx}{1+x^6} \) using the residue theorem.

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


Question: Let $U \subset \mathbb{R}^n$ be an open set, and let $f: U \to \mathbb{R}$ be a $C^1$ function. Suppose that for any $x_0 \in U$, there exists an $n$-variable polynomial $T_{x_0}$ of degree at most 2 such that $f(x + x_0) = T_{x_0} + o(\Vert x \Vert^2)$. Can we deduce that $f'$ is differentiable on $U$?

Answer: Yes


Question: What is the probability that there is no prime number between $n$ and $n + \ln(n)$?

Answer: \dfrac{1}{e}


Question: Given a quadrilateral with sides $a$, $b$, $c$, and $d$, find the minimum value of the expression \( \frac{a^2+b^2+c^2}{d^2} \).

Answer: \dfrac{1}{3}


Question: Find the function \( f(x) \) that satisfies the functional equation \( f(x+y) = 3^x f(y) + 9^y f(x) \) for all \( x, y \in \mathbb{R} \), given that \( f(1) = 6 \).

Answer: 9^x - 3^x


Question: If $H$ is a normal subgroup of $G$ and both $H$ and $G/H$ are abelian, is $G$ necessarily abelian? Provide a justification for your answer.

Answer: No


Question: Calculate the work done by the force field \(\bar{F}(x, y, z) = (x - yz)\bar{i} + (y - xz)\bar{j} + (x(1-y) + z^2)\bar{k}\) in moving a particle once around the triangle with vertices \((0, 0, 0), (1, 1, 1), (1, 1, 0)\) in that order.

Answer: -\dfrac{1}{2}


Question: Determine the values of $t > 1$ for which the set $F = \{a \in \mathbb{R} : a < 1\}$, together with the operations $a \boxplus b = a + b - ab$ and $a \boxdot b = 1 - t^{\log_t (1-a) \log_t (1-b)}$, forms a field.

Answer: t > 1


Question: Let $M$ be a finitely generated projective right module over a not-necessarily commutative ring $R$. For any non-zero $e \in M$, does there always exist a right $R$-module map $f: M \to R$ such that $f(e) \neq 0$?

Answer: Yes


Question: Consider the non-separable abstract $L_1$ space $L_1([0,1]^I)$, where $I$ is an uncountable set and $[0,1]^I$ is endowed with the product measure associated with the Lebesgue measure on $[0,1]$. Does this space contain subspaces isomorphic to $L_p([0,1]^I)$ for $1 < p \leq 2$?

Answer: No


Question: Suppose a function $f$ has a pole of order $m$ at $z = 0$, and satisfies the inequality \( \frac{1}{|z|^{3/2}} \le |f(z)| \le \frac{1}{|z|^{5/2}} \) for all \( z \) such that \( 0 < |z| \le 1 \). Determine the order \( m \) of the pole.

Answer: 2


Question: Given a fixed probability $p$ of winning a point in a game of tennis, determine the function $f(p)$ that represents the probability of winning the entire game. Assume the rules of Wii Tennis are the same as regular tennis. For $p = 0.5$, $f(p) = 0.5$ by symmetry. Show that $f(0.51) > 0.51$. Would using a Markov Chain be useful in solving this problem?

Answer: f(0.51) > 0.51


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \lim_{n \to \infty} \frac{1}{1 + n \sin^2 x} \]

Answer: 0


Question: Determine the rank of the matrix \(\begin{pmatrix} 4y+2z & 4x+4 & 2x-1 \\ y+2 & x & -1 \end{pmatrix}\) given that the variables \((x,y,z)\) satisfy the equations \(4xy + 2xz + 4y - z = 0\) and \(xy + 2x - z = 0\).

Answer: 2


Question: What is the maximum cardinality of a set \( Z \subset \mathbb{N} \) such that for any distinct \( a_1, a_2, a_3 \in Z \), the sum \( a_1 + a_2 + a_3 \) is a prime number?

Answer: 4


Question:
Consider the function $f(x) = |\log 2 - \sin x|$ for $x \in \mathbb{R}$. Define $g(x) = f(f(x))$. Determine whether $g$ is differentiable at $x = 0$, and if so, find $g'(0)$. Choose the correct statement:

- $g$ is differentiable at $x = 0$, and $g'(0) = -\sin(\log 2)$
- $g$ is not differentiable at $x = 0$
- $g'(0) = \cos(\log 2)$
- $g'(0) = -\cos(\log 2)$

Answer: C


Question: Consider a diffeomorphism $f: \mathbb{R}^2 \to \mathbb{R}^2$ acting on the unit disc $D = \{(x,y) : x^2 + y^2 \leq 1\}$. Let $D_xf$ denote the total derivative of $f$ at point $x$. Assume that the largest eigenvalue of $D_xf$ at $x$, denoted as $\lambda_x$, satisfies $|\lambda_x| > M$ for all $x \in D$ and some $M > 0$. Let $S$ be a square centered at $f(0)$ with side length 2. Determine if it is true that the image of the unit disc under $f$ will intersect $S$ for sufficiently large $M$. In other words, is $f(D) \cap S \neq \emptyset$ for large enough $M$?

Answer: Yes


Question: Let \( f(x) = \lim_{n \to \infty} \frac{3^n(\overbrace{\sin(\sin(\cdots\sin(x)))}^{n\text{ times}}+(\sqrt{2}\cos x+2)^n+2^n\cos x)}{3^n+\sin x(\sqrt{2}\cos x+2)^n} \). If \( l = \lim_{x \to \frac{\pi}{4}^+} f(x) \) and \( m = \lim_{x \to \frac{\pi}{4}^-} f(x) \), find the value of \( l^2 + m^2 \).

Answer: 2


Question: Calculate the integral \( \int_0^1 e^{x^2} \, dx \) using the Taylor expansion of \( e^{x^2} \) around \( x_0 = 0 \) to achieve an accuracy of less than 0.001. The Taylor expansion is given by \( e^{x^2} = 1 + x^2 + \frac{x^4}{3!} + \ldots \). Determine the number of terms required in the expansion such that the remainder term \( R_n(x) \) satisfies \( \left| \int_0^1 R_n(x) \, dx \right| < 0.001 \).

Answer: 5


Question: Let \( \kappa > \aleph_0 \) be a cardinal. Determine if there exists a connected topological space \((X, \tau)\) with \(|X| = \kappa\) such that every dense subset \(D \subseteq X\) satisfies \(|D| = |X|\).

Answer: Yes


Question: Suppose $\sum x_n$ is a convergent series and $(y_n)$ is a bounded sequence. Can the series $\sum x_n y_n$ diverge? If so, provide an example. If not, justify your answer.

Answer: Yes


Question: Prove that \( \lim_{x \to \infty} \frac{\tanh(x) - 1}{e^{-2x}} = -2 \) using the definition of a limit at infinity.

Answer: -2


Question: Let \( f(x) = x^3 + 2x^2 + 3x + 3 \) with roots \( a, b, c \). Find the value of \( \left(\frac{a}{a+1}\right)^{3} + \left(\frac{b}{b+1}\right)^{3} + \left(\frac{c}{c+1}\right)^{3} \).

Answer: 44


Question: Determine the exact value of the covering radius $R_v$ for a face-centered cubic (fcc) lattice, given that the maximum non-intersecting radius of each sphere is $r_c = \frac{1}{\sqrt{2}}$. Assume that sphere-sphere overlaps are allowed, and no point in the lattice is outside of a sphere. Express your answer in terms of $r_c$. \( R_v = \boxed{1} \)

Answer: 1


Question: Evaluate the line integral \( \int_\gamma y\,dx + z\,dy + x\,dz \) using Stokes' theorem, where \( \gamma = \{(x,y,z) : x^2 + y^2 + z^2 = 9, x + y + z = 0\} \) with positive orientation.

Answer: -9\sqrt{3}\pi


Question: Evaluate the complex contour integral \( \oint_C \frac{\cos(z-a)}{(z-a)}\mathrm{d}z \), where \( a \in \mathbb{R}^2 \) and \( C \) is the closed curve defined by \( |z-a|=\frac{|a|}{2} \).

Answer: 2\pi i


Question: Compute the size of the symmetry group of a configuration consisting of two congruent regular \(n\)-gons in the plane. Use the orbit-stabilizer theorem to find your answer.

Answer: 4n


Question: Evaluate the integral \( \int_C \frac{e^z}{z} \, dz \), where \( C \) consists of two circles centered at the origin with radii \( R = 2 \) and \( R = 1 \). Determine the value of the integral.

Answer: 4\pi i


Question:
Let $r_1, r_2, r_3, \ldots, r_n$ be the distinct real roots of the equation \(x^8 - 14x^4 - 8x^3 - x^2 + 1 = 0\). Find the value of \(r_1^2 + r_2^2 + r_3^2 + \cdots + r_n^2\). Choose the correct answer from the following options:

\((A)\,3\quad(B)\,14\quad(C)\,8\quad(D)\,16\)

Answer: C


Question: Let $Y = \operatorname{Pois}(n)$. Using the Normal approximation, also known as the Central Limit Theorem (CLT), estimate the probability \( p\Big[|Y-n| \geq 2\sqrt{n}\Big] \).

Answer: 0.0456


Question: Find the largest integer less than or equal to \((2+\sqrt{3})^4\).

Answer: 193


Question: Find all integer pairs \((j, k)\) such that \(2^j + 3 = 7^k\).

Answer: (2, 1)


Question: Let $a_{mn}$ be a double array of real numbers. Define \( A = \liminf_{n\rightarrow \infty} \limsup_{m\rightarrow\infty}a_{mn} \) and \( B = \limsup_{n\rightarrow \infty} \liminf_{m\rightarrow\infty}a_{mn} \). Determine which of the following is true: \( A \le B \) or \( A \ge B \).

Answer: A \geq B


Question: Evaluate the integral \( \int_{7}^{10}\sqrt{-40+14x-x^2}\,dx \).

Answer: \dfrac{9\pi}{4}


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\ln(\cos(4x))}{\ln(\cos(3x))} \]

Answer: \dfrac{16}{9}


Question: Determine whether there exists a bijection between the set $(0,1)^{r}$ and the set $ℂ^{r}$ for $r \in \mathbb{N}$.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to0}\left(\frac{\sin x}{x}\right)^{{6}/{x^{2}}}$$

Answer: \dfrac{1}{e}


Question: Find all nonnegative integer solutions \((a, b, c)\) to the equation \[ a!b! = a! + b! + c! \].

Answer: (3, 3, 4)


Question: Find the volume of the region that lies below the plane \( z = 3 - 2y \) and above the paraboloid \( z = x^2 + y^2 \).

Answer: 8\pi


Question: Let \( A = K[x,y,z] \). Find the radical of the ideal \( I = (xyz, xy+xz+yz, x+y+z) \).

Answer: (x, y, z)


Question: Evaluate the double integral \( \int_0^1 \int_x^{\sqrt{x}} \frac{e^y}{y} \, dy \, dx \).

Answer: e - 2


Question: In how many ways can you make $1.50 using exactly 50 coins? The coins may be pennies (1 cent), nickels (5 cents), dimes (10 cents), or quarters (25 cents).

Answer: 10


Question: Given the transformation $y = e^x$ and the probability density function $f_X(x) = e^{-x}$ for $0 \leq x < \infty$, determine the marginal density function $f_Y(y)$. Express your answer in terms of $y$.

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


Question: If \( \mathbb{E}[X | Y] = \mathbb{E}[X] \) for random variables \( X \) and \( Y \), does this imply that \( X \) and \( Y \) are independent?

Answer: No


Question: Let $f(x)$ be a real-valued function defined on the interval $(-5, 5)$ such that \[ e^{-x}f(x) = 2 + \int\limits_0^x (t^4 + 1)^{1/2} \ dt \] for all $x \in (-5, 5)$. If $f^{-1}(x)$ is the inverse function of $f(x)$, find \((f^{-1})^{'}(2)\).

Answer: \dfrac{1}{3}


Question: Evaluate the line integral \(\int_{c}(xy-y^2)dx-x^3dy\) over the region enclosed by the curves \(y = x^2\), \(y = x + 2\), and \(x = 0\) using Green's theorem.

Answer: \dfrac{4}{5}


Question: Find the smallest positive integer \( n \) such that \( 99^n + 100^n < 101^n \).

Answer: 49


Question: Evaluate the limit: \( \lim_{x \to -1} \frac{1}{x+1}(e^{2\pi i(x+1)}-1) \) without using l'Hopital's rule.

Answer: 2\pi i


Question: Solve the equation \( x = e^{\frac{1}{x}} \) for \( x \) without using the Lambert \( W \)-Function.

Answer: 1.763


Question: Determine the largest integer $n$ such that there exist functions $f_1, f_2, \ldots, f_n : [0,1] \to \mathbb{R}$ which are all bounded, non-decreasing, and mutually orthogonal in $L^2([0,1])$. Provide your answer as the maximal value of $n$. \( \boxed{n} \)

Answer: 2


Question:
Let $S = [0,1) \cup [2,3]$ and $f: S \to \mathbb{R}$ be a strictly increasing function such that $f(S)$ is connected. Determine which of the following statements is true:

- $f$ has exactly one discontinuity.
- $f$ has exactly two discontinuities.
- $f$ has infinitely many discontinuities.
- $f$ is continuous.

Answer: D


Question: Use the substitution \( z = xe^y \) to show that the differential equation \( x \frac{dy}{dx} = e^{-(x+y)} - 1 - x \) reduces to \( \frac{dz}{dx} + z = e^{-x} \). Then, find the particular solution to the original differential equation given that \( y(1) = -1 \).

Answer: y = -x


Question: Calculate the value of the product \( \sin(10^\circ) \sin(20^\circ) \sin(30^\circ) \sin(40^\circ) \sin(50^\circ) \sin(60^\circ) \sin(70^\circ) \sin(80^\circ) \). Use complex numbers in your solution.

Answer: \dfrac{3}{256}


Question: Find the value of the limit: $$\lim_{n\to\infty} \sum_{k=0}^n \frac{{k!}^{2} {2}^{k}}{(2k+1)!}$$

Answer: \dfrac{\pi}{2}


Question: Let \( |G| = p^n \) be a group of order \( p^n \). For each divisor \( d \) of \( p^n \), there are cyclic subgroups of order \( d \) in the group \( G \). These subgroups form a chain \( G_0 \subseteq G_1 \subseteq \ldots \subseteq G_n = G \), where \( |G_i| = p^i \) and each \( G_i \) is a cyclic \( p \)-group. If \( c_{p^i} \) represents the number of subgroups \( G_i \), determine the value of \( c_{p^i} \).

Answer: 1


Question: Evaluate the surface integral \( \iint_S x\;dy \times dz + y \; dz \times dx + z \; dx \times dy \) over the part of the plane \( x+y+z=2 \) in the first octant, where the normal vector \( n \) satisfies \( n \cdot (0,1,0) \geq 0 \).

Answer: 4


Question: Determine the number of positive integer values of \( n \) such that \( 2^{2^{2020}} - 1 \) is divisible by \( 2^{2^n} + 1 \).

Answer: 2019


Question: Apply one step of Richardson's Extrapolation to compute the improved value for $M$ given $N(3h) = 2.7172$ and $N(h) = 2.7182$. Assume $M = N(h) + Ah^{4/3} + Bh^{7/3}$ and use $h = 0.0001$ if needed.

Answer: 2.7185


Question: Let $P_k(x)$ denote the space of polynomials of at most degree $k$. Consider the differential operator $L: P_k \rightarrow P_k$ defined by \( L = \frac{1}{n!}D^n + \frac{1}{(n-1)!}D^{n-1} + \ldots + D + I \), where $D$ denotes differentiation with respect to $x$. If $k \leq n$, find the dimension of the kernel of $L - T$, where $T: P_k \rightarrow P_k$ is given by $T(p(x)) = p(x+1)$. Determine the dimension of the null space of this linear map.

Answer: k + 1


Question: In how many ways can 9 distinct marbles be distributed into 4 distinct bags such that each bag receives at least one marble?

Answer: 186480


Question: Let $I$ be an ideal in $A[x_1, \ldots, x_n]$, where $A$ is a Noetherian commutative ring, and suppose $I$ has a Groebner basis $G = \{g_1, \ldots, g_t\}$ with respect to some monomial order, where all the leading coefficients of each $g_i$ are equal to $1$. Given $S \subseteq \{x_1, \ldots, x_n\}$ and $f \in A[S] \cap I$, can we always find $f' \in A[S] \cap I$ such that the leading coefficient of $f'$ is $1$?

Answer: Yes


Question: Find the quadratic variation of the process \( Y_t = B_{2t} \), where \( (B_t)_{t \geq 0} \) is a standard Brownian motion.

Answer: 2t


Question: Determine the codimension of the set \( X = \{ A \in M_{m \times n}(\mathbb{R}) : \mathrm{rk}(A) = r \} \) as a submanifold of \( \mathbb{R}^{m \times n} \).

Answer: (m - r)(n - r)


Question: Evaluate \( \lim_{x\to \infty}x^p((x+1)^{1/3}+ (x-1)^{1/3} - 2x^{1/3}) \). If \( L = \lim_{x\to \infty}x^p((x+1)^{1/3}+ (x-1)^{1/3} - 2x^{1/3}) \) is a non-zero number, find \( \frac{p^2}{L} \).

Answer: -\dfrac{25}{2}


Question: Let \(a_r\) be the \(r^{th}\) term of a geometric progression (GP) such that \(\sqrt{a_1 \cdot a_3 \cdot a_5 \cdot a_7} = (18a_1 + a_1r^2)^2\) and \(\sum_{r=1}^{\infty} \frac{1}{a_r} = \frac{3}{4}\). Find the value of \(\sum_{r=1}^{10} \tan \left( \sin^{-1} \left( \frac{a_{r+1}}{\sqrt{a_r^2 + a_{r+1}^2}} \right) \right)\).

Answer: 30


Question: Determine the set of real numbers \(a\) for which the function \(f(x) = xe^{\frac{a}{x}}\), defined for \(x > 0\), has exactly two asymptotes.

Answer: (0, \infty)


Question: Find a composite number \( n \) such that \( n \mid a^n - a \) for every positive integer \( a \).

Answer: 561


Question: Is there a metric \(d\) on \(\mathbb{R}\) and \(a \in \mathbb{R}\) such that the function \(f:\mathbb{R} \to \mathbb{R}\), defined by \(f(x) = d(x, a)\), is differentiable?

Answer: Yes


Question: Determine the number of points at which the function \( f(x) = \lim_{n\to\infty}\left(4^n+x^{2n}+\frac{1}{x^{2n}}\right)^{\frac{1}{n}} \) is non-derivable.

Answer: 4


Question: A biased coin is tossed 100 times, resulting in 65 heads. Assuming a uniform prior distribution for the probability of heads, calculate the probability that the next toss will result in a head.

Answer: \dfrac{11}{17}


Question: Let $X$ be a non-negative integer-valued random variable such that $X \leq m$ and $E[X] = 2 m^{1-td/2}$. Prove that \( \Pr[X \geq m^{1-td/2}] \ge m^{-td/2} \).

Answer: m^{-td/2}


Question: Show that \( D^{-1/2}[D^{-1/2}t^2] = \frac{t^3}{3} \) using the integral operator \( D^{-1/2}(f(t)) = \frac{1}{\sqrt{\pi}} \int_0^t{(t-T)^{-1/2}f(T)dT} \).

Answer: \dfrac{t^3}{3}


Question: Let A be a square matrix such that its column space is equal to its null space. Prove that \( A^2 \) is the zero matrix.

Answer: A^2 = 0


Question: What is the value of the infimum when it is taken over an empty set?

Answer: \infty


Question:
Let $f: \Bbb R\rightarrow \Bbb R$ be a differentiable function such that $f '(0)=1$ and $f(x+y)=f(x)f(y)$ for all $x\in\Bbb R$. Determine which of the following statements is true:

- Both $f$ and $f '$ are decreasing functions.
- Both $f$ and $f '$ are increasing functions.
- $f$ is an increasing function but $f '$ is a decreasing function.
- $f$ is a decreasing function and $f '$ is a decreasing function.

Answer: B


Question: In triangle $ABC$, point $E$ is on $BC$ such that $BE : EC = 3:2$. Points $D$ and $F$ are on sides $AB$ and $AC$, respectively, such that $3AD = 2AF$. Let $G$ be the intersection of $AE$ and $DF$. Given $AB = 7$ and $AC = 9$, find the ratio $DG:GF$. Express your answer as a simplified ratio.

Answer: 9:7


Question: Determine the number of real solutions to the equation \( f(x) = 0 \), where \( f(x) = \sum_{i=1}^{2020} \frac{i^2}{x-i} \).

Answer: 2019


Question:
Consider \( \mathbb{R}^{2} \) with the \( \|\cdot\|_{1} \) norm and let \( M=\{(x,0):x\in\mathbb{R}\} \). Define \( g:M\rightarrow\mathbb{R} \) by \( g(x,0)=x \). Which of the following is a valid Hahn-Banach extension \( f \) of \( g \) to \( \mathbb{R}^2 \)?

A. \( f(x,y)=2x \)
B. \( f(x,y)=x+y \)
C. \( f(x,y)=x+2y \)
D. \( f(x,y)=x-2y \)

Answer: B


Question: Given two independent light bulbs with lifetimes modeled by the probability density functions \( f(x) = \frac{1}{100}e^{-x/100}, x>0 \) and \( f(y) = \frac{1}{50}e^{-y/50}, y>0 \), calculate the probability that the lifetime of the first light bulb (X) is longer than the lifetime of the second light bulb (Y).

Answer: \dfrac{2}{3}


Question: Determine the number of solutions to the quadratic equation \( \bar{x}^{2} = \bar{2} \) in the finite field \( \mathbb{Z}_{47} \).

Answer: 2


Question: In a right-angled triangle, medians are drawn from the acute angles to the opposite sides. If the maximum acute angle between these medians can be expressed as \( \tan^{-1}\left(\frac{p}{q}\right) \), where \( p \) and \( q \) are relatively prime positive integers, find \( p+q \).

Answer: 7


Question:
Prove that for any complex number $z \in \mathbb{C}$, the following inequality holds:
\[ \bigg\vert\frac{1-e^{iz}}{z^2}\bigg\vert \leq \frac{2}{\big\vert z\big\vert^2} \]

Answer: \frac{2}{|z|^2}


Question: The solutions to the equations \(z^2=4+4\sqrt{15}i\) and \(z^2=2+2\sqrt{3}i,\) where \(i=\sqrt{-1},\) form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form \(p\sqrt{q}-r\sqrt{s},\) where \(p,\) \(q,\) \(r,\) and \(s\) are positive integers and neither \(q\) nor \(s\) is divisible by the square of any prime number. What is \(p+q+r+s?\)

Answer: 20


Question: Calculate \( \lim_{n \to \infty} \left(\frac{n}{n^2 + 1} + \frac{n}{n^2 + 2} + \cdots + \frac{n}{n^2 + n}\right) \).

Answer: 1


Question: Evaluate the double integral \( \iint_D 3 \, dx \, dy \) where \( D = \{(x, y) : (x+y)^2 + (2x - y)^2 \le 4 \} \).

Answer: 4\pi


Question: Let \( F = \mathbb{F}_2 \) be the field with two elements. Consider the rings of polynomials and formal power series over \( F \) as \( F[t] \) and \( F[[t]] \) respectively. Suppose \( x \in F[[t]] \) is algebraic over \( F[t] \). Is it true that there exists a non-zero polynomial \( Q(y) = \sum\limits_{k = 0}^m q_k y^{2^k} \) with coefficients \( q_k \) from \( F[t] \) such that \( Q(x) = 0 \)?

Answer: Yes


Question: Compute the limit \( \lim_{x \to a} \frac{a^x-x^a}{x-a} \) using only fundamental limits, where \( a \) is a positive real number.

Answer: a^a (\ln a - 1)


Question: Let $G$ be a group of order 2021. Let $x$ and $y$ be elements in $G$ such that they are not equal to the identity element and have distinct orders. Let $H$ be the smallest subgroup of $G$ that contains both $x$ and $y$. Find the order of $H$.

Answer: 2021


Question:
Determine whether the following statement is true for all natural numbers \(x > 10\):

\[ P^{3}(2^{x}, 2^{x+1}) > P^{3}(1, 2^{x}) + 1 \]

where \(P^{3}(a, b)\) represents the sum of the frequency of the prime factor 3 of all integers in the interval \([a, b]\).

Answer: True


Question:
Implement a function `function [mpi, jmax] = mypi(tol)` that approximates \( \pi \) using the BBP formula:
\[ \pi = \sum_{k=0}^\infty \frac{1}{16^k} \left( \frac{4}{8k+1} - \frac{2}{8k+4} - \frac{1}{8k+5} - \frac{1}{8k+6} \right). \]
The function should continue iterating until the absolute difference between successive approximations is less than the specified tolerance `tol`. The output `mpi` is the approximation of \( \pi \), and `jmax` is the number of iterations required to achieve the specified tolerance. Calculate `jmax` for `tol = 10^{-10}`.

Answer: 8


Question:
Given the functional equation \( f(5x) = 8x - f(3x) - \sin^2(2x) \), find the value of the limit:
\[ \lim_{x \to 0} \frac{f(x) - x}{x^2} \]

Answer: -\dfrac{2}{17}


Question: Let $R$ be a commutative unital ring with identity $1$ such that the equation $nx = 1$ has a unique solution for each integer $n \ge 1$. Let $\xi$ be a nilpotent element of $R$ with nilpotency index $v$. Fix a positive integer $k$ and set $a = \sum_{i=0}^{v-1} \binom{1/k}{i} \xi^i$. Determine if it is true that $a^k = 1 + \xi$. Provide a proof or counterexample.

Answer: a^k = 1 + \xi


Question: Let \(z\) and \(w\) be two complex numbers satisfying the equations \(z + \bar{w} = z^2\) and \(w + \bar{z} = w^2\). Determine the number of ordered pairs \((z, w)\) that satisfy these conditions.

Answer: 2


Question:
Let $E$ be a Lebesgue measurable set in $\mathbf{R}^{n}$ with $0 < m(E) < \infty$, and let $f$ be a complex $L^{\infty}$ function on $\mathbf{R}^{n}$. For a given $0 < t < 1$, determine if there exists a Lebesgue measurable subset $A$ of $E$ such that:
\[
m(A) = t \cdot m(E), \quad \frac{1}{m(A)}\int_{A} f \,dm = \frac{1}{m(E)}\int_{E} f \, dm.
\]

Answer: Yes


Question: Calculate \(2^{947} \mod 1373\) without using a calculator. Express your answer as a single integer.

Answer: 177


Question: Evaluate the double integral: \[ \int_0^\pi \int_1^2 \delta(x^2 - y^2)\delta(\sin(x))\,\mathrm dy\,\mathrm dx \]

Answer: 0


Question: Let $X$ be a non-empty spectral space and $P$ be a closed subset of $X$. If $U_1$ and $U_2$ are two arbitrary quasi-compact open subsets such that $P \cap U_1 \neq \emptyset$ and $P \cap U_2 \neq \emptyset$, then $P \cap U_1 \cap U_2 \neq \emptyset$. Can we deduce that $P$ is irreducible from these conditions?

Answer: Yes


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

Answer: \dfrac{\pi}{4}


Question: Maximize \((1-a)(1-c)+(1-b)(1-d)\) subject to the constraints \(a^2+b^2=c^2+d^2=1\). Find the maximum value of the expression.

Answer: 3 + 2\sqrt{2}


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\frac{1}{n}\sum _{k=1}^{\lfloor\frac{n}{2}\rfloor} \cos\frac{k\pi}{n} \] where \(\lfloor\frac{n}{2}\rfloor\) denotes the largest integer not exceeding \(\frac{n}{2}\).

Answer: \dfrac{1}{\pi}


Question: Using Gauss' Theorem on Constructions, determine if a regular \( p^2 \) polygon is constructible. If it is, find the expression for \( \cos(2\pi/n) \).

Answer: 0


Question: In a connected electrical network with $n$ nodes, let nodes $a$ and $b$ have voltages set to 1 and 0, respectively. Is the maximum voltage difference between any two nodes in the network equal to 1, assuming the degree of each node is roughly the same?

Answer: 1


Question: Determine whether the sequence of functions \(f_n(t) = \frac{nt}{n+t}\) converges uniformly to \(f(t) = t\) on the interval \([0,1]\).

Answer: Yes


Question: Evaluate the expression \( \frac{\int_0^{\pi/2}\sin^{\sqrt{2}+1}(x) \, dx}{ \int_0^{\pi/2}\sin^{\sqrt{2}-1}(x) \, dx} \).

Answer: 2 - \sqrt{2}


Question: Suppose $^{2}B_{2}(q_1)$ and $^{2}B_{2}(q_2)$ are Suzuki simple groups where $q_1 = 2^{2n_1+1}$ and $q_2 = 2^{2n_2+1}$. If the order of $^{2}B_{2}(q_1)$ divides the order of $^{2}B_{2}(q_2)$, does it necessarily follow that $q_1 = q_2$?

Answer: No


Question: Given the polynomial \( f(x) = x^4 + ax^3 + bx^2 + cx + d \) with conditions \( f(1) = 10 \), \( f(2) = 20 \), and \( f(3) = 30 \), find the value of \( \frac{f(12) + f(-8)}{10} \).

Answer: 1984


Question:
Evaluate the limit by expressing it as the limit of integral sums of a properly chosen function:
\[ \lim_{n\to\infty} \frac{1}{n} \sqrt[n]{(n+1)(n+2)\cdots(n+n)}. \]

Answer: \dfrac{4}{e}


Question: Derive the moment generating function (mgf) for the standardized sum of random variables \( S_n^* = \frac{S_n - n\mu}{\sqrt{n} \sigma} = \frac{X_1 + \ldots + X_n - n\mu}{\sqrt{n}\sigma} \), where \( S_n = X_1 + \ldots + X_n \) and \( \mu \neq 0 \). Use L'Hopital's rule to show that the mgf of \( S_n^* \) converges to that of a standard normal variable, i.e., \( e^{t^2/2} \), as \( n \to \infty \).

Answer: e^{t^2/2}


Question: Evaluate \( \det(aI_n + bJ_n) \), where \( a, b \in \mathbb{R} \), \( I_n \) is the \( n \times n \) identity matrix, and \( J_n \in M_n(\mathbb{R}) \) is the matrix with every entry equal to 1.

Answer: a^{n-1}(a + bn)


Question: Does there exist a non-zero element $z \in A^2(\mathbb{R}^4)$ such that $z \wedge z \neq 0$?

Answer: Yes


Question: Solve the logarithmic equation analytically: \[ \log_2 (1 + \sqrt{x}) = \log_3 x \]

Answer: 9


Question: For a given integer \( n \), find the smallest base \( k \) such that \( n \) can be expressed as \( 1 + k^1 + k^2 + \ldots + k^{m-1} \) for some integer \( m \).

Answer: k


Question: Solve for \( x \) in degrees in the equation: \[ \sin 84^\circ \sin(54^\circ-x) = \sin 126^\circ \sin x. \]

Answer: 30


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

Answer: \dfrac{12}{13}


Question: Evaluate \( \lim\limits_{x\to\infty}(\sin\sqrt{x+1}-\sin\sqrt{x}) \).

Answer: 0


Question: Given the joint probability distribution of random variables X and Y as \( f(x,y) = Te^{-x-y} \) for \( 0 < y < x < \infty \), find the covariance \( \text{Cov}(X,Y) \).

Answer: \dfrac{1}{4}


Question: Calculate \( \lim_{x\to 0} \frac{x(\cosh x - \cos x)}{\sinh x - \sin x} \).

Answer: 3


Question: A circular birthday cake is divided into eight equal wedges. In how many distinct ways can red and green candles be placed on the cake, with one candle on each wedge, considering the symmetries of the circle?

Answer: 30


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^{1/n} \, \mathrm{d}x \]

Answer: \dfrac{3}{4}


Question: Find the sum of the solutions to the equation \(x^2 + 4x + 1 = \lfloor x \rfloor\), where \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\).

Answer: -2 + \sqrt{2}


Question: Find the integer $x$ such that $x \equiv 7 \pmod{37}$ and $x^2 \equiv 12 \pmod{37^2}$. Determine the value of $x$.

Answer: 1080


Question: In triangle $ABC$, the exradii $r_1, r_2, r_3$ are in harmonic progression. Given that the area of the triangle is 24 square centimeters and its perimeter is 24 centimeters, find the length of the smallest side of the triangle.

Answer: 6


Question: Let \( G \) be a connected, linear, semi-simple algebraic group over an algebraically closed field of characteristic zero, and let \( P \) be a maximal parabolic subgroup. Given that the quotient \( Z = G/P \) is a projective scheme, determine if the Picard group of \( Z \) is isomorphic to \( \mathbb{Z} \).

Answer: \mathbb{Z}


Question: A point $P$ is picked uniformly over a disc with radius $1$ and center at the origin. Subsequently, a new point $Q$ is picked uniformly over a disc with radius $|P|$ and center at the origin. Compute the expected value of $|Q|$. Note: A random point $(X,Y)$ uniformly distributed over a disc with radius $r > 0$ has density $f_{X,Y}(x,y) = \frac{1}{\pi r^{2}}$ for $x^{2} + y^{2} \leq r^{2}$.

Answer: \dfrac{4}{9}


Question: Let $A \in \mathbb{R}^{n \times p}$ and $B \in \mathbb{R}^{n \times d}$ be matrices with spectral norms (largest singular values) equal to 1. If we concatenate the columns of $A$ and $B$ to form the matrix $M = [A|B]$, what can be said about the spectral norm of $M$? Can it be bounded more tightly than 2?

Answer: \sqrt{2}


Question: Given that \( a + b + c = 0 \), evaluate the expression \( \frac{a^4 + b^4 + c^4}{a^2b^2 + c^2(a^2 + b^2)} \).

Answer: 2


Question: Assume \( f: [0, \infty) \to \mathbb{R} \) is a function such that for all \( 0 \leq a < b \in \mathbb{R} \), the restriction \( f_{|[a,b]}: [a,b] \to \mathbb{R} \) is integrable. If \( f(x) = \int_0^x f(t) \, dt \) for all \( x \in [0, \infty) \), prove that \( f \equiv 0 \).

Answer: 0


Question: For a given integer \( n \), determine the number of ways \( n! \) can be expressed as a product of factorials of smaller integers, \( a_1!a_2!\cdots a_n! \). Define this function as \( \Omega(n) \). Calculate \( \Omega(10) \).

Answer: 2


Question: Determine the number of polynomials \( f(x) \) of degree \( \geq 1 \) that satisfy the equation \( f(x^2) = f(x)^2 = f(f(x)) \).

Answer: 1


Question: Determine whether there exists a bounded function $f:\mathbb{R}\to\mathbb{R}$ of class $C^{\infty}$ such that for all $n = 0, 1, 2, \ldots$, the nth derivative of $f$ at 0, $f^{(n)}(0)$, equals the nth derivative of $g(x) = e^{-x} + x + 1$ at 0, $g^{(n)}(0)$. Provide a justification for your answer.

Answer: Yes


Question:
Let $k \in \mathbb{N}_0$, $A := (-1,1)$, and $f: A^2 \to \mathbb{R}$ be a function with the following properties:

- For every fixed $y \in A$, the function $f(\cdot, y)$ is $C^\infty$.
- For every fixed $x \in A \setminus \{0\}$, the function $f(x, \cdot)$ is $C^k$.

Is $f(0, \cdot)$ also $C^k$?

Answer: Yes


Question: Evaluate the infinite products \( \prod_{n=1}^\infty \cos\frac{1}{n} \) and \( \prod_{n=1}^\infty \sin\frac{1}{n} \).

Answer: 0


Question: Evaluate the limit of the sequence: \[ \lim_{n\to \infty} \frac{\sqrt[n]{e}+\sqrt[n]{e^2}+\sqrt[n]{e^3}+...+\sqrt[n]{e^n}}{n} \]

Answer: e - 1


Question: Can a metric be defined on a set $Y$, which contains more than one element, such that every continuous mapping $f: \mathbb{R} \rightarrow Y$ is constant? Justify your answer.

Answer: Yes


Question: Calculate the volume of the solid bounded by the surfaces $x = 1 - y^2$, $x = -1$, and $z^2 = 1 - x$. Determine the appropriate bounds for integration and find the volume.

Answer: 8


Question: If $A$ is a normal matrix with singular value decomposition (SVD) $A = USU'$ where $S$ is a diagonal matrix with non-negative real entries, is $A$ necessarily a diagonal matrix?

Answer: No


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

Answer: 1


Question: Find the fixed point through which all chords of a parabola with parameterization \((at^2, 2at)\) pass, given that the endpoints \(P(t_1)\) and \(Q(t_2)\) of the chord satisfy the relation \(t_1t_2 = k\), where \(k\) is a constant.

Answer: (-ak, 0)


Question: A farmer has a number of sheep and learns that he will die at the end of the year (day 365, where today is day 0). His utility is given by \( ab \), where \( a \) is the money he makes by selling his sheep (which have a fixed price) and \( b \) is the number of days he has left to enjoy the profit, i.e., \( 365-k \) where \( k \) is the day. Each day, his sheep breed and multiply their numbers by \( \frac{103}{101} \). On which day should he sell all his sheep to maximize his utility?

Answer: 314


Question: Show that the function \( f(x,y) = \frac{x^3 \sin(x-y)}{x^3-y^3} \) for \( x \neq y \), and \( f(x,y) = 0 \) for \( x = y \), is continuous at the point \((0,0)\).

Answer: 0


Question: Determine whether the sequence \( \frac{(2n-3)^{5}-25n^{4}}{(10\log_2(3n^{6}+5) + 8n^{2})(n^{3}-20)} \) converges. If it converges, find its limit.

Answer: 4


Question: Assume $z$ and $w$ are complex numbers satisfying $z^n = w^m$ for some integers $n, m \geq 2$. If $n$ and $m$ are coprime, does there always exist a complex number $t$ such that $t^m = z$ and $t^n = w$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_0^2 f(x) \, dx \), where \( t = f(x) \) is the solution to the equation \( t^5 + t = x \).

Answer: \dfrac{4}{3}


Question: Given a non-constant polynomial function \( f(x) \) such that \( f(x^3) - f(x^3 - 2) = f(x) \cdot f(x) + 12 \) for all \( x \), find \( f(5) \).

Answer: 744


Question: Evaluate the convolution integral of the functions \( f(x) = \frac{1}{6} \) for \( 0 \leq x \leq 6 \) and \( g(x) = x^2 - 3ix \). Find \( h(4) \), where \( h = f * g \) and \( * \) denotes the convolution operator.

Answer: 4 - 3i


Question: Find one solution to the differential equation \(x^2y'' - xy' + y = 0\) for \(x > 0\) using the characteristic equation method. The characteristic equation is \(x^2r^2 - xr + 1 = 0\).

Answer: x


Question: Find the smallest natural number \( n_0 \) such that \( 4 \log_2(n) \leq \frac{n}{2} \) for all \( n \geq n_0 \).

Answer: 44


Question: Let $A$ be a commutative ring and $U$ be an open subset of $\text{Spec}(A)$. Consider the ring of sections $B$ above $U$ of the affine scheme $\text{Spec}(A)$. For a prime ideal $p \in U$, the natural map $A \to B$ induces an embedding between the localizations $A_p \to B_p$. Determine whether this map is surjective. Equivalently, given $s \in A$ with $D(s) \subseteq U$ and $f \in B$, if the restriction of $f$ to $D(s)$ is 0, show that there exists an integer $N$ such that $s^N \cdot f = 0$ in $B$. Here, $D(s)$ is the set of all prime ideals of $A$ not containing $s$.

Answer: Yes