Question: Determine the polynomials $g(x)$ for which $g(S_n)$ is a martingale, where $S_n$ is the position of a simple symmetric random walk after $n$ steps, defined as $S_n = X_1 + \cdots + X_n$ with $X_i$ i.i.d. and $P(X_i = 1) = P(X_i = -1) = 1/2$. The coefficients of $g$ cannot depend on $n$.

Answer: g(x) = ax + b


Question: If $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous, open surjection, is it necessarily a homeomorphism?

Answer: Yes


Question: Given the functional equation $f(f(n)) = 3n$ for each positive integer $n$, where $f(n+1) > f(n)$, find the value of $f(10)$.

Answer: 19


Question:
Determine the limit of the following integral as \( \varepsilon \to 0 \):
\[
\lim_{\varepsilon\rightarrow 0}\frac{1}{\varepsilon}\int_{D(\varepsilon)}\frac{1 + \sin(x) + \sin(y)}{\sqrt{x^2 + y^2}} \, d(x,y)
\]
where \( D(\varepsilon) = \{ (x,y) \in \mathbb{R}^2 \mid 0 < x^2 + y^2 < \varepsilon^2 \} \).

Answer: 2\pi


Question: Evaluate the series \( \sum\limits_{n \geq 0} \left(S_{n + 2} + S_{n + 1}\right)^2 (-1)^n \), given the recurrence relation \( S_{n + 1} = T_n - S_n \) for \( n \geq 1 \), with initial conditions \( S_1 = \frac{3}{5} \) and \( T_1 = 1 \). The sequences \( T_n \) and \( S_n \) satisfy the condition \( \frac{T_n}{S_n} = \frac{T_{n + 1}}{S_{n + 1}} = \frac{T_{n + 2}}{S_{n + 2}} = \dots \).

Answer: \dfrac{9}{13}


Question: Find the lowest-degree polynomial function that passes through the following points: \((1, 0), (2, 3), (3, 3), (4, 6), (5, 1), (6, 4), (7, 6), (8, 2), (9, 5), (10, 0), (11, 3), (12, 5)\).

Answer: 11


Question: Let $F$ be a field and $E$ an extension field of $F$. Suppose $\alpha \in E$ is transcendental over $F$. Determine if the elements of $F(\alpha)$ can be expressed as \( \left\{ \frac{f(\alpha)}{g(\alpha)} : f, g \in F[x] \right\} \). Is this statement correct?

Answer: Yes


Question: Let \( \phi: \mathbb{R}[X] \to \mathbb{C} \) be a homomorphism such that \( \phi(X) = 1 + i \). Determine the kernel of \( \phi \), denoted as \( \ker \phi \).

Answer: (X^2 - 2X + 2)


Question: Given the set $A = \{0, 1, 4, 16, r\}$, which is a subset of $\mathbb{Z}_{21}$, determine the value of $r$ such that $A$ is a $\lambda$-difference set for some $\lambda$. Provide the value of $r$.

Answer: 14


Question: Calculate the value of \( 718^{143} \mod 1260 \).

Answer: 1192


Question: Consider all labeled trees with nodes \(\{1, 2, 3, \ldots, n\}\). How many of these trees contain the edge \(\{1, 2\}\)?

Answer: 2n^{n-3}


Question: If $R[X]$ is a Noetherian ring, can we conclude that $R$ is also Noetherian?

Answer: Yes


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

Answer: 0


Question: Solve the equation \(\sqrt{\frac{\sqrt{x!}\times \sqrt{x!!}\times \sqrt{x!!!}}{\sqrt{\left( x-2 \right)!+x}}}=12\) for \(x\).

Answer: 6


Question: Let $X$ be a projective Noetherian scheme over $\mathbb{C}$. Determine whether $H^0(\mathcal{O}_X(-t))=0$ for any $t>0$.

Answer: 0


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

Answer: -\pi i


Question: Suppose \( \int_{0}^{a}f(x)dx = 1 \). Determine if it is true that for every \( b>0 \), the inequality \( \int_{0}^{a}\frac{e^{bx}}{b}\left(f(x)\right)^2dx \ge 1 \) holds.

Answer: True


Question: Find the limit of the expression \( \sum_{k = n}^{5n} \binom{k-1}{n-1} \frac{1}{5^n} \left(\frac{4}{5}\right)^{k-n} \) as \( n \to \infty \).

Answer: \dfrac{1}{2}


Question: Given a Hilbert manifold $M$, does there exist a smooth map into some very large Hilbert space that takes geodesics to straight lines?

Answer: No


Question: Let $G$ be a group in which every proper subgroup is contained in a maximal subgroup of $G$. Determine whether $G$ must be finitely generated.

Answer: No


Question: Given that \( \Delta u = 0 \) in the unit ball and \( u(1, \varphi, \theta) = \sin^2 \varphi \), find the value of \( u \) at the origin.

Answer: \dfrac{2}{3}


Question: Consider a 4002-digit number formed by randomly arranging the digits 4 and 8. What is the probability that this number is exactly divisible by 19?

Answer: \dfrac{1}{19}


Question: Find a real-valued function \( f \) that satisfies the equation \( f(x) + f(y) + 2xyf(xy) = \frac{f(xy)}{f(x+y)} \) for all positive real numbers \( x \) and \( y \).

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


Question: Find the value of \( \overline{mnp} \) such that \( \overline{mnp} = 4p(m+n)^2 \).

Answer: 216


Question: Does the dihedral group $D_{15}$ have a normal subgroup of order 6? Justify your answer.

Answer: No


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

Answer: 1


Question: Let \(f(z)\) be an entire function such that \(|f(z)| \leq |e^{z}|\) on the contour \(|z| = 1\). Determine the maximum possible value of \(|f(\ln(2))|\).

Answer: 2


Question: Determine whether the matrix \( \begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \) is in Jordan form.

Answer: Yes


Question: Minimize the expression \( \{tr^Trt^T\}_{ii} \) subject to the constraint \( r^Tt = 1 \), where \( r, t \in \mathbb{R}^{n \times 1} \). Assume that the diagonal elements of \( t \) are ordered in nondecreasing order.

Answer: 0


Question: Determine the values of \( \beta \) for which the series \( \sum_{n=1}^{\infty} \left( \sqrt{1+\frac{(-1)^n}{n^\beta}} - 1 \right) \) converges.

Answer: \beta > \dfrac{1}{2}


Question: Solve the linear first-order partial differential equation \( u_x + u_t = u \) for \( x \in \mathbb{R}, t > 0 \) with the initial condition \( u(x, 0) = \cos x \).

Answer: e^{t} \cos(x - t)


Question: Find the sum of all possible non-negative integer values of $x$ such that $(xy-7)^2 = x^2 + y^2$.

Answer: 14


Question: Find the minimum value of the integral \( \int_0^2 [g(x)]^2 \, dx \), where \( g(x) = -\sqrt{3}x + (1+\sqrt{3}) + C \) and \( C \) is a constant that can vary.

Answer: 2


Question: Determine the quotient group $D_{10}/\langle \sigma \rangle$, where $D_{10}$ is the dihedral group of a regular pentagon, given by $D_{10} = \{1, \sigma, \sigma^2, \sigma^3, \sigma^4, \tau, \sigma\tau, \sigma^2\tau, \sigma^3\tau, \sigma^4\tau\}$ with $\sigma = (12345)$ and $\tau = (13)(45)$. What is the structure of this quotient group?

Answer: \mathbb{Z}_2


Question: Is there a pair of random variables \((X, Y)\) such that the expected value of \(X\) goes to infinity, the expected value of \(Y\) goes to minus infinity, but the expected value of \(X+Y\) goes to infinity?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\to+\infty}\sum_{k=0}^{n}(-1)^{k}\sqrt{\binom{n}{k}} \).

Answer: 0


Question: Evaluate the integral \( \int_{\partial R_r}\frac{dz}{(z-2)^2} \), where \( R_r = \{ z \in \mathbb{C} :  |\text{Re}(z)| \leq r \text{ and } |\text{Im}(z)|\leq 1\} \) for \( r>2 \), without using the residue theorem.

Answer: 0


Question: Evaluate the integral \( \int_{0}^{\infty}\frac{2t}{e^{t\pi}+1} \,dt \).

Answer: \dfrac{1}{6}


Question: Find the limit \( \lim_{n \to \infty} \frac{(2n^{\frac{1}{n}}-1)^n}{n^2} \).

Answer: 1


Question: Is it possible to have a coordinate system \( f: M \to \mathbb{R}^n \) such that \( df_1 \wedge \cdots \wedge df_n(p) = 0 \)?

Answer: No


Question: Given a square matrix $A$ where all eigenvalues satisfy $|\lambda| < 1$, determine whether there exists a matrix $T$ such that $\Vert T^{-1}AT\Vert < 1$.

Answer: Yes


Question: Evaluate the integral: \[ \int_{-\infty}^{+\infty}\frac{1}{\sqrt{4\pi}}e^{x}e^{-\frac{x^2}{4}}dx \]

Answer: e


Question: Find the points where the function \( f(x,y) = x^2 + y^3 + \frac{768}{x+y} \) attains its extrema.

Answer: (6, 2)


Question: Evaluate the limit \( \lim_{n \to \infty} \sum_{k=0}^{2n+1} \frac{n}{n^2 + k} \).

Answer: 2


Question: Given the function \( f(x,y) = \frac{xy \sin x}{x^2+y^2} \) for \((x,y) \neq (0,0)\) and \( f(0,0) = 0 \), find the directional derivative of \( f \) at the origin in the direction of the unit vector \( \phi = (\phi_1, \phi_2) \). Calculate \( \frac{\partial f(0,0)}{\partial \phi} \).

Answer: \phi_1^2 \phi_2


Question: How many times must a fair die be rolled so that there is less than a 1% chance that the mean of the scores differs from 3.5 by more than 0.1?

Answer: 1936


Question: What is the outer measure of a Vitali set constructed from the interval [0, 1]?

Answer: 1


Question: Evaluate the limit: $$\lim_{x\to\infty} x\left[\left(1+\frac{1}{2x}+\frac{1}{4x^2}+\frac{1}{8x^3}\right)^{1/3}-1\right].$$

Answer: \dfrac{1}{6}


Question: Let \( f: [a, b] \to [0, \infty) \) be a continuous function such that \( f\left(\frac{a+x}{2}\right) + f\left(\frac{2b+a-x}{2}\right) = f(x) \) for all \( x \in [a, b] \). Prove that the only function that satisfies these conditions is \( f \equiv 0 \).

Answer: 0


Question: Find the limit: \[ \lim_{x \to 0} \left( \csc^2 x - \frac{1}{x^2} \right) \]

Answer: \dfrac{1}{3}


Question: Find the maximal value of \( \left| \int_{0}^{1} (f(x))^2 - f(x) \, dx \right| \), where \( f \) is a continuous function from \([0, 1]\) to \([-1, 1]\) and satisfies \(|f(x)| \leq x\) for all \(x \in [0, 1]\).

Answer: \dfrac{5}{6}


Question: Find a polynomial in \( \mathbb{Q}[x] \) such that the ideal generated by this polynomial in \( \mathbb{Q}[x] \) makes the quotient ring \( \mathbb{Q}[x]/I \) isomorphic to \( \mathbb{Q}[\sqrt{2} + \sqrt{3}] \).

Answer: x^4 - 10x^2 + 1


Question: Find the second derivative of the function \( f(x) = \sec(3x)\sqrt{324\cos^2(3x) + 396 + 121\sec^2(3x)} \). Then, evaluate the second derivative at \( x = \frac{\pi}{12} \).

Answer: 1584


Question: Evaluate the limit \( \lim_{n\to \infty}\left(\cos{\frac{x}{n}}\right)^{n^2} \) where \( x \in \mathbb{R} \).

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


Question: Find the length of the longest line segment that can be enclosed inside the region defined by \( A = \{(x,y): x = u + v, y = v, u^2 + v^2 \leq 1 \} \).

Answer: 1 + \sqrt{5}


Question: Consider the set \( A = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \). How many onto functions \( f: A \to A \) exist such that for all even \( x \), \( f(x) \neq x \)?

Answer: 229080


Question: Is it true that for each real number $w$, there exist relatively prime positive integers $a$ and $d$ such that in the arithmetic sequence with first term $a$ and common difference $d$, the first prime appears at a position greater than $w$?

Answer: Yes


Question: Find the value of the infinite series \( \sum_{n=1}^{\infty} \frac{n^3}{3^n} \).

Answer: \dfrac{33}{8}


Question: Given a right triangle ABC with \( A = 90° \), \( b + c = 18 \), and \( \cot(B) = 4\sin^2(C) \), determine if these conditions can be satisfied. Assume \( 0 < B, C < 90° \).

Answer: Yes


Question: Mr. Porter has 12 students in his combinatorics class. In the first week, the students form 4 groups of 3 people each. In the second week, they need to form 6 groups of 2 people each, ensuring that no two students who were in the same group in the first week are in the same group in the second week. In how many ways can the students form the groups in the second week? (A) 1296 (B) 2546 (C) 2995 (D) 3348 (E) 10395

Answer: D


Question: Determine the Hausdorff dimension of the set $A$ of non-recurrent walks on $\mathbb{Z}$, where walks are identified with infinite binary sequences in the Cantor space $2^\mathbb{N}$. A non-recurrent walk is defined as a sequence $x$ for which $\frac{\sum_{k=1}^n x(k)}{n} = \frac{1}{2}$ for only finitely many $n$. What is the Hausdorff dimension of this set?

Answer: 1


Question:
Given that $F(x)$ is a primitive of the function $f(x) = e^{x^2}$ with $F(-1) = 0$, determine the value of $F(1)$. Choose the correct option:

A) $F(1) < 0$
B) $F(1) = 0$
C) $F(1) > 2$
D) $F(1) = 2$

Answer: C


Question: Evaluate the limit of the term $b^n V(c^n x)$ as $n \to \infty$ in the functional equation $V(x) = ax + bV(cx)$, where $a = \frac{1 + pg}{r + p}$, $b = \frac{p}{r + p}$, and $c = 1 + g$, with $d_0, g > 0$, $r \ge 0$, and $0 \le p \le 1$. Assume $V(x)$ is a function that satisfies this equation.

Answer: 0


Question: Let $X_1, X_2, \ldots, X_n$ be independent, identically distributed nonnegative random variables. Given $k \le n$, compute the expected value: \[ E\left[\frac{\sum_{i=1}^k X_i}{\sum_{i=1}^n X_i}\right]. \]

Answer: \dfrac{k}{n}


Question: Approximate the area of the surface on the hemisphere $x^2+y^2+z^2=9$, $z\geq 0$, that lies above the region $R = \{(x,y)|0\leq x \leq 1, 0 \leq y \leq 1\}$ in the plane, using the Trapezoidal rule in both directions. The area is given by the integral \( \iint_R\sqrt{1 +[f_x(x,y)]^2+[f_y(x,y)]^2}dA \).

Answer: 1.05


Question: Consider the graded $R$-algebra $A = \bigoplus_{n=0}^\infty A_n$, where $R = \mathbb{C}[q^{\pm 1}]$, $A_0 = R$, and each $A_n$ is a free $R$-module. The algebra $A' = A/(q-1)$ is a graded algebra over $R/(q-1) = \mathbb{C}$. Assume $A'$ is commutative. Is it possible for $A'$ to be finitely generated as an algebra over $\mathbb{C}$ while $A$ is not finitely generated as an algebra over $R$?

Answer: Yes


Question: Let $\mathbb{P}$ be the set of all prime numbers, and let $p_i$ denote the $i$th prime number, with $p_1 = 2$. Define $m = \prod_{i=1}^n p_i$. Determine the number of solutions to the congruence $x^2 + x \equiv 0 \pmod{m}$. Conjecture: $2^n$. Provide the number of solutions.

Answer: 2^n


Question:
Calculate the limit of the following sum as \( n \to \infty \):
\[ \lim_{n\to\infty}\left(\frac{3^1}{\sqrt{9^n+1}}+\frac{3^2}{\sqrt{9^n+2}}+\cdots+\frac{3^n}{\sqrt{9^n+n}}\right) \]

Answer: \dfrac{3}{2}


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

Answer: e


Question: Evaluate the limit: \[ \lim_{n \to +\infty}\int_{n}^{n+7}\frac{\sin x}{x}\,\mathrm{d}x. \]

Answer: 0


Question: True or False: If the Wronskian of n functions vanishes at all points on the real line, then these functions must be linearly dependent in \( \mathbb{R} \).

Answer: B


Question: Calculate the integral \( \int_{-\infty}^\infty |f(x)|^2 \mathrm{d}x \) using Plancherel's formula, where \( f(\xi) = \int_0^1 \sqrt{x} \sin(\xi x) \mathrm{d}x \).

Answer: \dfrac{\pi}{2}


Question: Find the limit of the sequence: \[ \lim_{n\to \infty} \sqrt[n]{1^{2022}+2^{2022}+\ldots+n^{2022}} \]

Answer: 1


Question: Find the minimum value of the function \( f(t) = 2\cos(100t) + 18\cos(2000\pi t) \).

Answer: -20


Question: Prove the identity \( \sum_{l=0}^{n-1} \cos\left(\frac{2\pi kl}{n}\right)^2 = \frac{n}{2} \) for integers \( n > 2 \) and \( 1 \leq k \leq \left\lfloor \frac{n-1}{2} \right\rfloor \) without using character theory.

Answer: \dfrac{n}{2}


Question: Calculate the line integral of the vector field \( \mathbf{F} = x^2y\mathbf{i} + z\mathbf{j} + (2x-y)\mathbf{k} \) along the path from \((1,1,1)\) to \((2,-3,3)\). The path is parameterized by \( \mathbf{r}(t) = (t+1, -4t+1, 2t+1) \) for \( t \) in \([0, 1]\).

Answer: -\dfrac{10}{3}


Question:
Consider two convex sets $C$ and $D$ with $C \cap D \neq \emptyset$. Let $x^*$ be the solution to the optimization problem:
\[ \min_{x \in C} \max_{y \in D} |x-y|^2 \]
Determine if it is true that $x^* \in D$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty} \prod_{k=1}^\infty \left(\frac{p_n}{p_k}\right)^{\frac{p_n}{n(p_n+p_k)}} \] where \( p_n \) is the \( n \)th prime number.

Answer: 1


Question: Determine the dimension of the quotient space \( \frac{C_{0}}{M} \), where \( C_{0} = \{(x_{n}): x_{n} \in \mathbb{R}, x_{n} \rightarrow 0\} \) and \( M = \{(x_{n}) \in C_{0}: x_{1} + x_{2} + \cdots + x_{10} = 0\} \).

Answer: 1


Question: Find the limit: \[ \lim_{n\to +\infty}\left[\frac{\frac{a_n}{s_n}}{-\ln(1-\frac{a_n}{s_n})}\right] \] where \(a_n=\sqrt{n}\) and \(s_n=a_1+a_2+\ldots+a_n\).

Answer: 1


Question:
Solve the equation involving the continued fraction:
\[
x = 1 + \frac{1}{x + \frac{1}{x + \frac{1}{x + \frac{1}{x + \ldots}}}}
\]
Find the value of \(x\).

Answer: \dfrac{3}{2}


Question: Find the residue of the function $f(z^2)$ at $z = 0$, given that $f(z)$ is analytic in the annulus $0 < |z| < n$.

Answer: 0


Question: If $A_t$ and $B_t$ are independent Brownian motions and $a$ is a constant, determine whether $(\sin a)A_t + (\cos a)B_t$ is a Brownian motion.

Answer: Yes


Question: Find an ideal $I$ of $\mathbb{Z}/2\mathbb{Z}[x]$ such that the quotient ring $\mathbb{Z}/2\mathbb{Z}[x]/I$ is a field with 8 elements.

Answer: (x^3 + x + 1)


Question: Compute the integral \( \int_\gamma \left(\frac{1}{z}+\frac{1}{z^2}\right)dz \) along the path \( \gamma \), which is the unit circle parameterized by \( z(t) = e^{it} \) for \( 0 \leq t \leq 2\pi \).

Answer: 2\pi i


Question: How many different six-digit positive integers are there, where each digit is between 0 and 7 (inclusive), and the sum of the digits equals 20?

Answer: 15666


Question: Evaluate the integral \( \int_{\mathbb{R}^n} \exp(-||x||^2) \, d\mu \).

Answer: \pi^{n/2}


Question: A $7 \times 7$ board is filled with $49$ distinct numbers, one in each cell. You can ask questions where each question reveals the set of numbers in any $3$ chosen cells. What is the minimal number of questions required to determine the number in each cell of the board?

Answer: 33


Question: Consider a sequence \((x_1, x_2, \ldots)\) generated by sampling uniformly and independently from the integers \(\{0, 1, 2, \ldots, 9\}\). From this sequence, construct the real number \(x = 0.\overline{x_1x_2x_3\ldots}\). What is the probability that \(x\) is a rational number?

Answer: 0


Question: Maximize \(z-x\) subject to the constraint \(x^2 + y^2 + z^2 = 1\).

Answer: \sqrt{2}


Question: Two people agree to meet between 12:00 and 13:00. Each person's arrival time is independently and identically distributed, following a uniform distribution over this interval. Given that one person has already arrived, calculate the probability that the waiting time for the other person is at least 30 minutes.

Answer: \dfrac{1}{4}


Question: Find the minimum value of \( n \) such that the Taylor polynomial \( T_1^n\left(\frac{1}{x}\right) \) approximates \( \frac{1}{x} \) with an error less than \( 10^{-3} \) on the interval \( x \in [0.9, 1.1] \) using the Lagrange remainder.

Answer: 3


Question: Let ${\bf x}=(x(u,v),y(u,v),z(u,v))$ be a $C^1$ vector-valued function on an open neighborhood $U$ of $(u_0,v_0)$ such that $({\bf x}_u\times{\bf x}_v)\cdot {\bf k}\neq 0$ at $(u_0,v_0)$. Determine if it is true that $z$ can be locally solved in terms of $x$ and $y$. Specifically, does there exist a $C^1$ function $f$ defined in a neighborhood $O$ of $(x(u_0,v_0),y(u_0,v_0))$ such that $$z(u,v)=f(x(u,v),y(u,v))$$ for all $(x(u,v),y(u,v))\in O$?

Answer: Yes


Question: Compute the first cohomology group $H^1(G, \mathbb{Z}_2G)$, where $G$ is the group of integers $\mathbb{Z}$. Provide your answer with justification.

Answer: \mathbb{Z}_2


Question: Let \( E \) be the set of all continuous functions \( f: [0,1] \to \mathbb{R} \) such that \( f(x) + f(y) \ge |x-y| \) for all \( x, y \in [0,1] \). Find the minimum value of \( \int_0^1 f(x) \, dx \) for \( f \in E \).

Answer: \dfrac{1}{4}


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

Answer: \dfrac{1}{2}


Question: Determine the value of \(a\) such that the quadrilateral \(ABCD\) is cyclic, where the line \(x + 2y + a = 0\) intersects the circle \(x^2 + y^2 = 4\) at points \(A\) and \(B\), and the line \(12x - 6y - 41 = 0\) intersects the circle \(x^2 + y^2 - 4x - 2y + 1 = 0\) at points \(C\) and \(D\).

Answer: 2


Question: Consider a collection of $k$-variate Gaussian distributions $\{\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda})\}$, where $\lambda$ is a label. For each distribution, $\mu \in [-1,1]^k$, the variance of each coordinate $X_i$ is $1$, and all covariances $\text{Cov}[X_i, X_j] \in [-1,1]$. Let $\Lambda$ be a distribution over labels $\lambda$. Sample a vector $X \in \mathbb{R}^d$ by first sampling $\lambda \sim \Lambda$, then $X \sim \mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda})$. Now, sample $Y$ from $\mathcal{N}(\mathbb{E}_{\lambda \sim \Lambda}[\mu_\lambda], \mathbb{E}_{\lambda \sim \Lambda}[\Sigma_\lambda])$. Determine if $X$ and $Y$ can ever be identically distributed.

Answer: Yes


Question: Transform the differential equation \( x^3 y''(x) - x y'(x) + 2y(x) = 0 \) into the Sturm-Liouville form \( (p(x) y'(x))' + q(x) y(x) = 0 \) by finding the function \( p(x) \).

Answer: e^{1/x}


Question: Find the limit: $$\lim_{n \to\infty} \dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$$

Answer: -1


Question:
Evaluate the integral \( \int^{n+1}_1 \frac{(\{x\})^{[x]}}{[x]} \, dx \) where \( n \in \mathbb{N} \), \( [x] \) is the integer part of \( x \), and \( \{ x \} = x - [x] \). Choose the correct answer from the following options:

(a) \( \log_e(n) \)
(b) \( \frac{1}{n+1} \)
(c) \( \frac{n}{n+1} \)
(d) \( 1+\frac{1}{2}+\cdots +\frac{1}{n} \)

Answer: C


Question: Given a symmetrically and continuously distributed random variable $X$ with mean $E(X) = 0.01$ and variance $Var(X) = 0.04$, calculate the probability $P(X < -0.03)$. The correct result is $0.125$. Verify this result.

Answer: 0.125


Question: Determine the radius of convergence for the series \( \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{3\cdot6\cdot9\cdots(3n)}x^n \).

Answer: \dfrac{3}{2}


Question: Determine the number of integer pairs \((x, n)\) such that \(x^2 + 615 = 2^n\), where \(x \in \mathbb{Z}\) and \(n \in \mathbb{N}\).

Answer: 2


Question: Determine whether there exists a function \( f: \mathbb{Q}_p^\star \to \mathbb{R} \) that satisfies the functional equation \( f(xy) = f(x) + f(y) \) for all \( x, y \in \mathbb{Q}_p^\star \), and is not similar to the known solutions \( f(x) \equiv 0 \) and \( f(x) = \log |x|_p \). Here, \( p \) is a prime and \( \mathbb{Q}_p \) is the field of all \( p \)-adic numbers.

Answer: Yes


Question: What is the cardinal number associated with the ordinal number $\omega + 1$?

Answer: \aleph_0


Question: Determine whether the following statement is true: \( \lim_{n \to \infty} n^{\phi(n)} = 1 \) given that \( \phi(n) \to 0 \) as \( n \to \infty \).

Answer: False


Question:
Given the functions:
\[ F(x) = \int_1^x f(t) \, \mathrm{d} t \]
\[ f(t) = \int_x^{t^2} \frac{\sqrt{7+u^4}}{u} \, \mathrm{d} u \]
Find \( F''(1) \).

Answer: 0


Question: Find the sum of the series \( \sum_{k=1}^{\infty} \frac{k^4}{3^k} \).

Answer: 15


Question: Determine the values of \( x > 0 \) for which the series \( \sum_{n=1}^\infty \frac{\ln(1+nx)}{nx^n} \) converges.

Answer: (1, \infty)


Question: Two particles are placed on opposite vertices of the same face of a cube. If they are on the same edge at time \( t \), they remain in the same position at time \( t+1 \). If not, one of the particles is chosen at random and moves to one of its three neighboring vertices with equal probability at time \( t+1 \). What is the expected value of the minimum time until the particles are on the same edge?

Answer: 2


Question: Let $f:\mathbb{R}^n\longrightarrow\mathbb{R}^n$ be a continuous function with $n\geq 2$. Suppose that for every line of the form $x+tv$, where $x\in \mathbb{R}^n$, $t\in \mathbb{R}$, and $v\in B(0,1)$, the image under $f$ is also a line. Is $f$ necessarily an affine map?

Answer: Yes


Question: Evaluate the limit: \[ \lim _{N\rightarrow \infty } \left( \,{\frac {{N}^{3}}{8\sum _{k=0}^{N-1}({k}^{2}+\left\lfloor k/2\right\rfloor)}} \right) \]

Answer: \dfrac{3}{8}


Question: Find the maximum possible value of the integral \( \int_0^1  x f(x) - \int_0^x f(y) \, dy \, dx \), where \( f : [0,1] \to [0,1] \) is a weakly increasing function.

Answer: \dfrac{1}{4}


Question: Compute the complex integral \( \oint_C \frac{dz}{z(2z+1)} \) over the unit circle \( C \).

Answer: 0


Question: Compute the sum \( \sum_{k=1}^{10}\frac{1-\overline{\alpha_k}}{1+\alpha_k} \), where \( \alpha_k \) are the roots of unity of order 11, excluding 1.

Answer: 11


Question: Let $A$ be a complex unital $C^*$-algebra, and let $G(A)$ denote the set of invertible elements in $A$. Given a non-zero self-adjoint element $a \in A$, consider the element $x = \frac{1}{1+ia}$. Determine if $x$ belongs to the connected component of $G(A)$ containing the identity element $1$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the exact value of the norm \( \|\delta\| \) for the functional \( \delta = 2\delta_{-1} - 3\delta_0 + \delta_1 \), where \( \delta_c \) is the Dirac functional at \( c = -1, 0, 1 \) in the space \( \mathcal{C}[-1,1]' \).

Answer: 6


Question: Determine whether the following statement is true: For every homogeneous polynomial $P$ with real coefficients in at least three variables, and for every $\epsilon > 0$, there exists a $\delta > 0$ such that for every $x$ with $|P(x)| < \delta$, we have $d(x, Z) < \epsilon$, where $Z = P^{-1}(\{0\})$ is the set of roots of $P$, and $d$ is the standard distance.

Answer: True


Question: The numbers from 1 to 8 are divided into two sets, A and B. The product of the numbers in set A is denoted as \(a\), and the product of the numbers in set B is denoted as \(b\). Determine the smallest possible value of the larger of the two products, \(a\) and \(b\).

Answer: 210


Question: A sphere of radius $R$ has a charge $Q$ distributed uniformly over its surface. Determine the radius of a sphere that contains 90% of the electrostatic energy stored in the field of this charge distribution.

Answer: 10R


Question:
Consider sequences \(a_n\) and \(b_n\) such that \(\sum a_n = 1\) and \(b_n\) tends to infinity. Determine if there exists a sequence \(c_n\) such that:
1. \(c_n \to \infty\)
2. \(c_n \le b_n\)
3. \(c_n \cdot \max\{ a_k : b_n \le k \le 2b_n \} \to 0\)
Does such a sequence \(c_n\) exist?

Answer: Yes


Question: Suppose that the price of a company's stock increases if the latest news is positive. News articles arrive following a Poisson distribution with an intensity of 2 per day, and good news arrives with a probability of 2/3. Determine the Q matrix and the long-term proportion of time that the stock price is increasing.

Answer: \dfrac{2}{3}


Question: Given \((x, y) \in \mathbb{R}\), if \(x^2 + y^2 + xy = 4\) and \(x^4 + y^4 + (xy)^2 = 8\), calculate the value of \(x^6 + (xy)^3 + y^6\).

Answer: 19


Question: 322 mathematicians walk into a bar, each picking someone other than themselves at random and writing down that person's number on a piece of paper. The barman calls each mathematician in order, and if the person they wrote down has not yet been ordered a beer, they order a beer for them. How many mathematicians will be left without beer in expectation? Provide your answer as a whole number.

Answer: 118


Question: Let $(n_k)$ be an increasing sequence of positive integers. Define $E$ as the set of all $x$ such that the sequence $\sin(n_k x)$ converges. Prove that the Lebesgue measure of $E$, $\mu(E)$, is zero.

Answer: 0


Question: Calculate the residue of the function \( \frac{4-3z}{z(z-1)} \) at \( z = 0 \).

Answer: -4


Question: Let $P$ be an integer-valued polynomial with degree greater than 1. Is it true that the values $P(n)$ have arbitrarily large prime factors?

Answer: Yes


Question: Let $T^2$ be a compact smooth surface and let $p \in T^2$. Suppose that $T^2$ admits a symmetric $(0,2)$-tensor which is a flat Riemannian metric restricted to $T^2 - \{p\}$. Determine if it is true that the Euler characteristic $\chi(T^2) = 0$. Provide a justification for your answer.

Answer: 0


Question: Find the limit: \[ \lim_{n\rightarrow\infty} \prod_{k=1}^{n} \left(1+\frac{k}{n^2}\right) \]

Answer: \sqrt{e}


Question:
Find the radius of convergence for the series \( \sum_{n=0}^{\infty} (n+2^n)z^n \) using the formula:
\[ R = \lim_{n \rightarrow \infty} \left| \frac{n+2^n}{(n+1)+2^{n+1}} \right| \]

Answer: \dfrac{1}{2}


Question: A fair coin is flipped 30 times. Let \( X \) denote the number of heads among the first 20 coin flips, and \( Y \) denote the number of heads among the last 20 coin flips. Compute the correlation coefficient of \( X \) and \( Y \).

Answer: \dfrac{1}{2}


Question: Given the joint probability density function \(f(x,y) = y^{-1}e^{-x/y}e^{-y}\) for \(x,y > 0\) and \(0\) elsewhere, find \(\mathbb{P}(\{X>1\} \mid \{Y = y\})\) for \(y > 0\).

Answer: e^{-1/y}


Question: Suppose we have two linear maps $S$ and $T$ from a vector space $V$ to itself such that $\text{Ker}(S) = \text{Ker}(T)$ and $\text{Range}(S) = \text{Range}(T)$. Additionally, $V$ is the direct sum of $\text{Ker}(S)$ and $\text{Range}(S)$. Does this imply that $S = T$?

Answer: No


Question: Determine the number of integer values of $a$ for which the quadratic equation \( x^2 + 7x - 14(a^2 + 1) = 0 \) has at least one integer root.

Answer: 0


Question: Let $a \in \mathbb{Z}$ with $a \ge 2$, and let $A$ be an $n \times n$ matrix where each entry is either $\pm a$ or $0$. Prove that $\det(A)$ is divisible by $a^n$.

Answer: a^n


Question:
Evaluate the limit \( \lim_{x\to0}\bigl(\mathrm{Ci}(x) - \mathrm{Ci(3x)}\bigr) \), where \( \mathrm{Ci}(x) \) is the Cosine Integral defined as:
\[ \mathrm{Ci}(x) = -\int_x^{\infty} \frac{\cos(t)}{t}dt \]

Answer: -\ln 3


Question:
Let $A$ and $B$ be matrices in $M_{m \times n}(K)$, where $m < n$ and $K$ is a field. Determine which of the following statements is not always true:

1. $\det(A^tB) = 0$
2. $\det(AB^t) = 0$

Answer: 2


Question: Determine the number of solutions to the equation \( \alpha_1x_1 + \alpha_2x_2 + \ldots + \alpha_nx_n = 0 \) in the vector space \( \mathbb{F}_p^n \), where \( \mathbb{F}_p \) is a Galois field with \( p \) elements and not all \( \alpha_i \in \mathbb{F}_p \) are zero.

Answer: p^{n-1}


Question:
Let $f: \mathbb{R}^+ \to \mathbb{R}$ be a differentiable function such that $\lim_{x \to \infty} (f(x) + f'(x)) = 0$. Define $g(x) = e^x \cdot f(x)$. Evaluate the limit:
\[ \lim_{x, y \to \infty} \frac{g(x) - g(y)}{e^x - e^y} \]

Answer: 0


Question: Find the constant \( c \) such that \( \sum_{k=1}^{\infty}\frac{c \sin (x+k\pi)}{2^k} = c\sin x \).

Answer: 0


Question: Let $S = \{P_1, P_2, P_3\}$ be a set of three consecutive prime numbers. Consider the expression $X = xP_1 + yP_2 + zP_3$, where $x, y, z \in \mathbb{Z}^+$. Determine if there exist values for $x, y, z$ such that $X$ is divisible by a prime number greater than $P_1, P_2, \text{and } P_3$. Provide a proof or counterexample.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left(1+\frac{x}{5x^3+x^2+8}\right)^ {\dfrac{x^3+8}{x}} \]

Answer: e^{\frac{1}{5}}


Question: Consider the following greedy algorithm: Let \( p(n) \) be the number of unrestricted partitions of \( n \), with \( p(0) = 1 \). Start with two empty sets, Set 1 and Set 2. Place \( p(n) \) into Set 1. For each successive step \( k = 1, 2, 3, \ldots, n \), place \( p(n-k) \) into the set with the smaller sum of elements, or into Set 1 if the sums are equal. After the algorithm completes, will the sum of elements in Set 1 always differ from the sum of elements in Set 2 by at most 1?

Answer: Yes


Question: Let $(X, \mathscr{B})$ be a measurable space and $Y \subseteq X$ with $Y \in \mathscr{B}$. Define the relative $\sigma$-algebra on $Y$ as $\mathscr{B}_Y = \{Y \cap B \mid B \in \mathscr{B}\}$. Suppose $\mathscr{G} \subseteq \mathscr{B}$ generates the $\sigma$-algebra $\mathscr{B}$. Define $\mathscr{G}_Y = \{Y \cap G \mid G \in \mathscr{G}\}$. Does $\mathscr{G}_Y$ generate the relative $\sigma$-algebra $\mathscr{B}_Y$ on $Y$? Answer true or false.

Answer: True


Question: Let $X$, $Y$, and $Z$ be three independent uniform random variables on $[0, 1]$. Compute the probability $P(XY < Z^2)$. Provide your answer as a single probability value.

Answer: \dfrac{5}{9}


Question: Evaluate \( \lim_{n\rightarrow \infty}\frac{I_{n+1}}{I_{n}} \) where \( I_{n}=\int^{1}_{0}x^2(1-x^2)^n\,dx \).

Answer: 1


Question: Calculate the expected value of the minimum of three independent exponential random variables, where \( X_1 \sim \text{exp}(1) \), \( X_2 \sim \text{exp}(2) \), and \( X_3 \sim \text{exp}(3) \).

Answer: \dfrac{1}{6}


Question: Assume the Generalized Continuum Hypothesis (GCH) holds and \(\kappa\) is a regular cardinal. Determine the number of non-isomorphic partial orders of size less than \(\kappa\).

Answer: \kappa


Question: Find the smallest value of \( k \) such that the function \( f(x,y) = \frac{|x|^k |y|^k}{x^2+y^2} \) is continuous at the point \((0,0)\), given that \( f(0,0) = 0 \).

Answer: 2


Question: Given a continuous function $f$ on $[0,1]$ such that \[\int_0^{1} f(x) \, dx = \frac{1}{3} + \int_0^{1} f^2(x^2) \, dx,\] find the value of $f\left(\frac{1}{4}\right)$. Assume $f(x) = \frac{-1}{6} + x$ satisfies the condition.

Answer: \dfrac{1}{12}


Question:
Consider the 1-dimensional stochastic differential equation (SDE):
\[ dX_t = a(X_t,t) dt + \sigma(X_t,t) dW_t \]
with initial condition \( X_0 = x_0 \in \mathbb{R} \), where \( a, \sigma: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) are smooth functions and \( \sigma > 0 \). Let \( W_t \) be the standard Brownian motion. Fix \( y \in \mathbb{R} \) and \( t > 0 \). Show that:
\[ \mathbb{P}(\{ \omega \in \Omega : X_t = y \}) = 0 \]
where \( (\Omega, \mathcal{F}, \mathbb{P}) \) is the probability space being considered.

Answer: 0


Question: Calculate \( \lim_{n\to\infty}\int_{[0,\infty)}\frac{n\sin\frac{x}{n}}{x(1+x^2)}dx \).

Answer: \dfrac{\pi}{2}


Question: Is 1024 the last power of 2 that can be doubled without carrying in base 10? If not, how many more such powers of 2 exist?

Answer: 0


Question: Determine the cardinality of the set of subgroups of the free group on two generators, $F_2 = \mathbb{Z} * \mathbb{Z}$.

Answer: 2^{\aleph_0}


Question: In a Dedekind domain $R$, is it true that there are at most finitely many prime ideals with a given finite norm, where the norm is defined as the index of the ideal in $R$?

Answer: Yes


Question: A fair coin is tossed 30 times. What is the expected value of the last digit of the number of heads obtained?

Answer: 4.72


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

Answer: 4\pi


Question: Compute the series \( \sum_{j=1}^k \cos^n\left(\frac{j\pi}{k}\right)\sin\left(\frac{nj\pi}{k}\right) \).

Answer: 0


Question: In a rhombus $ABCD$, consider two circles: circle $(C_1)$ centered at $B$ passing through $C$, and circle $(C_2)$ centered at $C$ passing through $B$. Let $E$ be one of the intersection points of $(C_1)$ and $(C_2)$. The line $ED$ intersects $(C_1)$ again at point $F$. Determine the measure of angle $\angle AFB$. \( \text{Express your answer in degrees.} \)

Answer: 60


Question: Determine whether there exists a strictly increasing function \( f : \mathbb{R} \to \mathbb{R} \) such that its image is algebraically independent over \( \mathbb{Q} \).

Answer: Yes


Question:
Calculate the derivative of the integral with respect to its upper limit:
\[ \frac{d}{dx} \int_{0}^{x} e^{-\lambda t} \, dt \] Justify all operations used in the calculation.

Answer: e^{-\lambda x}


Question: Let \( J \) be a \( 101 \times 101 \) matrix with all entries equal to 1, and let \( I \) denote the identity matrix of order 101. Find the determinant of the matrix \( J - I \).

Answer: 100


Question: Let $V$ be a vector space with two norms $\|\,\|_1$ and $\|\,\|_2$. Denote by $V_1$ and $V_2$ the completions of $V$ with respect to these norms, respectively. Given that $\|v\|_1 \leq \|v\|_2$ for all $v \in V$, consider the map $V_2 \to V_1$ induced by the identity on $V$. Is it possible for this map to be non-surjective?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to \infty} \left( x(16x^4 + x^2+1)^{1/4} - 2x^2 \right) \]

Answer: \dfrac{1}{32}


Question: In a ternary tree of height 3, where each node except the root has three children, how many pairs of vertices (u, v) are there such that the distance between u and v is 4? Note that (u, v) and (v, u) are considered the same pair.

Answer: 162


Question:
Consider the space of all $4 \times 3$ matrices with entries in the finite field of three elements. Determine the number of matrices in this space that have rank 3. Choose the correct answer from the following options:

A. $(3^4-3)(3^4-3^2)(3^4-3^3)$

B. $(3^4-1)(3^4-2)(3^4-3)$

C. $(3^4-1)(3^4-3)(3^4-3^2)$

D. $(3^4)(3^4-1)(3^4-2)$

What is the correct answer?

Answer: C


Question: Determine all values of \( a \) for which the equation \( a\sin{x}+(a+1)\sin^2{\frac{x}{2}} + (a-1)\cos^2{\frac{x}{2}} =1 \) has at least one solution for \( x \).

Answer: [0, \infty)


Question: Find the minimum value of the expression $xy + yz + zx$ where $x, y, z \in [-1, 1]$. Provide a rigorous justification for your answer.

Answer: -1


Question: Let $X$ be a random variable uniformly distributed on $[0, \pi]$. Calculate the conditional expectation $\mathbb{E}[X \mid \sin(X)]$.

Answer: \dfrac{\pi}{2}


Question: Solve the equation \(3^{(x-3)^{(x+3)}} = (3x+2)^{(x-3)}\) for \(x\).

Answer: 3


Question: Is $R[x]/\langle p(x) \rangle$ always a finite integral extension of $R$ given a commutative ring $R$ and a monic polynomial $p(x) \in R[x]$?

Answer: Yes


Question: Provide an example of a group whose group of automorphisms is non-abelian.

Answer: V_4


Question: Is it possible to construct a holomorphic function \( f : \mathbb{C} \rightarrow \mathbb{C} \) such that \( f(n) \neq 0 \) for a given integer \( n \), but \( f(m) = 0 \) for any integer \( m \neq n \)?

Answer: Yes


Question: Find the largest possible area of a triangle in the first quadrant that is bounded by the x-axis, y-axis, and a tangent line to the curve \( y = e^{-5x} \).

Answer: \dfrac{2}{5e}


Question: Solve Burgers' equation \(u_t + uu_x = 0\) with the initial condition \(u(x,-1) = x^2\). Find the value of \(u(1,2)\).

Answer: \dfrac{7 - \sqrt{13}}{18}


Question: Are continuous functions measurable with respect to the sigma-algebra generated by analytic sets, denoted as ${\cal B}^1$?

Answer: Yes


Question:
For the function \( g(x) = x^{1/3} \), centered at \( a = 1 \), and using a Taylor polynomial of degree 3, the polynomial is given by:
\[ p_3(x) = 1 + \frac{x-1}{3} - \frac{(x-1)^2}{9} + \frac{5(x-1)^3}{81} \]

Use the error formula for the Taylor polynomial of degree 3 to find an upper bound on:
\[ |g(x) - p_3(x)|, \quad 1 \le x \le 1.5 \]

The fourth derivative of \( g(x) \) is \( g''''(x) = -\frac{80}{81}x^{-11/3} \). Determine the maximum value of the error term over the interval \( 1 \le x \le 1.5 \) to find the smallest possible upper bound.

Answer: \dfrac{5}{1944}


Question: Find the smallest integer \(n > 1\) such that \(2^n > n^{1000}\).

Answer: 13747


Question: Given a sequence of identical and independent events \(x_n\) with \(P(x_n) > r > 0\), prove that \(P(x_n \text{ happens for infinitely many } n) = 1\).

Answer: 1


Question: Determine the values of real $x$ for which the series \(\sum_{n=1}^{+\infty} \left (\frac{x^2n^2-2|x|^3n}{1+2xn^2} \right)^{7n}\) is convergent.

Answer: (-2, 2)


Question: Determine the isomorphism class of the quotient group \( \mathbb{Z}^3 / M \), where \( M \) is the subgroup of \( \mathbb{Z}^3 \) generated by the vectors \((13,9,2), (29,21,5), (2,2,2)\).

Answer: \mathbb{Z}_{16}


Question:
Evaluate the expression \(A + B + C + AB + BC + AC + ABC\) where:
\[ A = \frac{2\sqrt{3}}{3}i \cdot \cos\left(\frac{\pi}{6} + \frac{i}{3}\text{arcsinh}\left(\frac{3\sqrt{3}}{2}\right)\right) \]
\[ B = \frac{2\sqrt{3}}{3}i \cdot \cos\left(\frac{5\pi}{6} + \frac{i}{3}\text{arcsinh}\left(\frac{3\sqrt{3}}{2}\right)\right) \]
\[ C = \frac{2\sqrt{3}}{3}i \cdot \cos\left(\frac{3\pi}{2} + \frac{i}{3}\text{arcsinh}\left(\frac{3\sqrt{3}}{2}\right)\right) \]

Answer: 0


Question:
Determine the values of $t$ for which the function \( f(x) \) is differentiable at \( x = 0 \), where
\[
f(x) =
\begin{cases}
x^t \sin \frac{1}{x^2} & \text{if } x \ne 0 \\
0 & \text{if } x = 0
\end{cases}.
\]

Answer: t > 1


Question: Find the total differential \( \frac{dw}{dp} \) of the function \( w = \sqrt{u^2 + v^2} \), where \( u = \cos\left(\ln\left(\tan\left(p + \frac{1}{2}\pi \right)\right)\right) \) and \( v = \sin\left(\ln\left(\tan\left(p + \frac{1}{2}\pi \right)\right)\right) \).

Answer: 0


Question:
Calculate the minimum value of the expression:
\[ \frac{(2x^2 + y)(4x + y^2)}{(2x + y - 2)^2} + \frac{(2y^2 + x)(4y + x^2)}{(2y + x - 2)^2} - 3(x + y) \]
where \(x\) and \(y\) are positive numbers such that \(2x + y \neq 2\) and \(x + 2y \neq 2\).

Answer: -1


Question: Find the minimum value of the expression \[ \frac{1}{|x_1 - x_2||x_1 - x_3|} + \frac{1}{|x_2 - x_1||x_2 - x_3|} + \frac{1}{|x_3 - x_1||x_3 - x_2|} \] where \(x_1, x_2, x_3 \in [-1, 1]\).

Answer: 2


Question:
Determine the number of integer solutions \((a, b)\) with \(a > b > 0\) that satisfy the equation:
\[ 7 \times 13 \times 19 = a^2 - ab + b^2. \]

Answer: 8


Question: Let \( f(x) \) be a polynomial with integer coefficients. Determine whether there exists a point \( z \) on the unit circle \( T \) in the complex plane such that \( |f(z)| \geq 1 \).

Answer: Yes


Question:
In a bipartite graph $G = (A \cup B, E)$, where $A$ represents persons and $B$ represents clubs, determine the smallest integer $K$ such that there exists a $B$-covering matching $M$ satisfying the following conditions:

1. Each person in $A$ can be a member of at most 50 clubs.
2. Each club in $B$ must have a president who is a member of the club.
3. Each person in $A$ can be the president of at most 5 clubs.
4. Each club in $B$ must have at least $K$ members.

Find the minimum value of $K$ that ensures such a matching exists.

Answer: 10


Question: Let \(T\) be a constant stopping time. Find the sigma-algebra \(\mathcal{F}_T\).

Answer: \mathcal{F}_c


Question: Determine the Krull dimension of the ring \( \mathbb{R}[x,y,z] /(x^2+y^2+z^2+1) \).

Answer: 2


Question: Let $G=\{z\in\mathbb{C}: |z-2|<1\}$ and assume $f$ is a holomorphic function on the closed disk $\overline{G}$ except for a simple pole $z_0\in G$. If $|f(z)|=1$ for every $z\in \partial G$, show that for every complex number $a$ with $|a|>1$, the pre-image $f^{-1}(a)$ contains exactly one point.

Answer: 1


Question: Let $B_t$ be a Brownian motion. Calculate the probability $P(3B_2 > 4)$. Assume $B_2 \sim N(0, 2)$. Use the property that if $X \sim N(\mu, \sigma^2)$, then $kX \sim N(k\mu, k^2\sigma^2)$ for a constant $k$. Determine the correct variance for $3B_2$ and find $P(3B_2 > 4)$. Provide the final probability value.

Answer: 0.173


Question: Determine whether the following statement is true: \( \lim_{n\rightarrow\infty}(1+1/a_n)^n=\lim e^{n/a_n} \), where \( a_n \) is a positive sequence and \( \lim_{n\rightarrow\infty}{a_n}=\infty \).

Answer: True


Question: Consider a simple symmetric random walk \(S_n = X_1 + \ldots + X_n\) with \(S_0 = 0\), where \(\mathbb{P}(X_i = \pm 1) = \frac{1}{2}\). Define \(T = \inf\{n : S_n = 1\}\). Compute \(\mathbb{E}(T)\).

Answer: \infty


Question:
Find an explicit formula for the function \( f(x) \) given by the integral equation:
\[ f(x) = \int_{0}^{x} e^{f(k)} \, dk \]

Answer: -\ln(1 - x)


Question: Is the open subset \( U = E \setminus A \) affine, where \( A \) is a finite set of closed points of an elliptic curve \( E \) over an algebraically closed field?

Answer: Yes


Question: How many homomorphisms are there from the cyclic group \( \mathbb{Z}/3\mathbb{Z} \) into the symmetric group \( S_3 \)?

Answer: 3


Question: Find the expected value of \( \max(x, y) \) where \((x, y)\) is a uniformly random point in a square of unit length centered at \((0,0)\) with edges parallel to the axes.

Answer: \dfrac{1}{6}


Question: Determine whether the polynomial \(y^3 + ny + 1\) is reducible over \(\mathbb{Z}\) for infinitely many integer values of \(n\). Answer Yes or No.

Answer: No


Question:
Calculate the matrix norm \( \Vert A \Vert \) for the matrix \( A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \) using the definition:

\[ \left\Vert A \right\Vert = \max_{\mathbf{x} \ne 0} \frac{\left\Vert A\mathbf{x} \right\Vert}{\left\Vert \mathbf{x} \right\Vert} \]

Use the hint that \( \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \begin{bmatrix} 1 & 2 \end{bmatrix} \) to assist in your calculation.

Answer: 5


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

Answer: \dfrac{4}{e}


Question: Determine the values of \( n > 0 \) for which the series \( \sum_{k=1}^{\infty}\frac{1}{k}\sin\left(\frac{k\pi}{n}\right) \) converges.

Answer: n > 0


Question: Find the minimum value of \( \frac{a^3}{b} + 2\frac{b^2}{c^2} + \frac{c}{2a^3} \) where \( a, b, \) and \( c \) are positive real numbers.

Answer: \dfrac{5}{2}


Question: Determine whether the limit \( \lim_{x \to \infty} \frac{\pi(x)}{x^\delta} \) exists for all real \( \delta \) in the interval \((0, 1)\), where \( \pi(x) \) is the prime counting function.

Answer: \infty


Question: Evaluate the infinite product: \[ \frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{2+\sqrt{2}}} \cdot \frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}} \cdots \]

Answer: \dfrac{\pi}{2}


Question: What is the maximum number $m$ such that there exists a simplex with $n$ vertices in $n-1$ dimensions whose projection onto a 2D plane forms the boundary of a regular $m$-gon?

Answer: n


Question: Given that \( \sum_{k=-\infty}^{\infty}c_ke^{ikx} \) is a Fourier Series for \( f(x) \), determine the function to which \( \sum_{k=-\infty}^{\infty}c_ke^{i4kx} \) converges.

Answer: f(4x)


Question: Find the product of the 5th roots of \((1+\sqrt{2}i)^3.\)

Answer: -5 + \sqrt{2}i


Question: Evaluate the limit: \[ \lim_{t \to 0^+}\frac{(p-2)t^{p-2} + (q-2)t^{q-2}}{t^{p-2} + t^{q-2}} \] where \( 1 < q < p < \infty \).

Answer: q - 2


Question: What is the maximum number of prime implicants for a conditional statement in a Karnaugh map with \( n \) variables?

Answer: 2^{n-1}


Question: Determine the second homology group $H_2(\mathbb{Q},\mathbb{Z})$ where the action is trivial.

Answer: 0


Question: Evaluate the limit \( \lim_{x \to 0} \sum_{n=1}^{\infty} \frac{\sin(nx)}{n^2} \).

Answer: 0


Question: Given that $a + b + c = 2001$ with $a > b > c > 0$, determine the number of possible ordered triples $(a, b, c)$. Provide your answer as an integer.

Answer: 332667


Question: Determine the number of real roots of the polynomial \(x^p - p^p x + p\) for a prime \(p \geq 5\).

Answer: 3


Question: What are the last two digits of \(3^{3^{100}}\)?

Answer: 03


Question:
Find the sum of the series \(\sum_{n=1}^{\infty} a_n\) where:
\[a_n=\Bigl(2\pi n+\frac{1}{n}\Bigr)\Bigl\lvert \sin\Bigl(\frac{1}{n}\Bigr)\Bigr\rvert\]

Answer: \infty


Question: Determine the number of solutions to the quadratic congruence $x^2 \equiv 182 \pmod{727}$. Note that $727$ is a prime number.

Answer: 2


Question: Find the number of 4-tuples \((a, b, c, d)\) where each element is an integer between 1 and 10, inclusive, such that the sum \(a + b + c + d = 21\).

Answer: 660


Question: If \(\arctan(4) = 4 \arctan(x)\), find the value of the expression \(x^5 - 7x^3 + 5x^2 + 2x + 9870\).

Answer: 9871


Question: Consider a sequence of compact metrizable groups $(G_n, \mu_n)$, where each $\mu_n$ is a probability (right-invariant) Haar measure on $G_n$. Is the product measure $\otimes_n \mu_n$ on $\prod_n G_n$ the unique probability (right-invariant) Haar measure on the product group?

Answer: Yes


Question: Evaluate the integral \( \int_C e^{-1/z}\sin(1/z)\,dz \) over the unit circle \(|z|=1\).

Answer: 2\pi i


Question: Let \( \alpha = \sqrt{2} + \sqrt{3} \) and consider a field \( K \) such that \( \mathbb{Q} \subset K \subset \mathbb{Q}[\alpha] \). Given that the degree of the minimum polynomial of \( \alpha \) over \( \mathbb{Q} \) is 4, determine the degree of the minimum polynomial of \( \alpha \) over \( K \).

Answer: 2


Question: Determine the nilpotency class of the dihedral group $D_{16}$ using the upper central series.

Answer: 3


Question: What is the remainder when $40!$ is divided by $1763$?

Answer: 1311


Question: Is the Berge-Fulkerson conjecture still open for bridgeless cubic planar graphs?

Answer: No


Question: Evaluate the limit \( \lim_{n \to \infty} \dfrac{1}{n^2} \sum_{k=1}^{n-1} k \cdot \left \lfloor x + \dfrac{n-k-1}{n} \right \rfloor \) to three decimal places, where \( x = \sqrt{2015} \).

Answer: 22.395


Question: Find the number of distinct quadruples \((x_1, x_2, x_3, x_4, x_5)\) of non-negative integers such that \(x_1 \geq 3\), \(x_2 \geq 3\), \(x_4 \geq 8\), \(x_5 \leq 3\), and \(x_1 + x_2 + x_3 + x_4 + x_5 \leq 23\).

Answer: 1750


Question: Evaluate the integral \( \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(5x^2-6xy+5y^2)} \, dx \, dy \).

Answer: \dfrac{\pi}{4}


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

Answer: 90


Question: In triangle \( \triangle ABC \), the side lengths are \( AB = 17 \), \( BC = 28 \), and \( AC = 25 \). The altitude from \( A \) to \( BC \) and the angle bisector of angle \( B \) intersect at point \( P \). If the length of \( BP \) can be expressed as \( \frac{a\sqrt{b}}{c} \) for positive integers \( a, b, c \) where \( \gcd(a, c) = 1 \) and \( b \) is not divisible by the square of any prime, find \( a + b + c \).

Answer: 47


Question: Evaluate the triple integral: \[ \int\int\int_\text{whole space}3(\vec{r}+2\vec{a})^2\delta^3(\vec{r}-\vec{a})d\tau, \] where \(\vec{a}\) is a fixed vector and \(\delta\) is the Dirac delta function.

Answer: 27|\vec{a}|^2


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

Answer: 1


Question: Find the product of all values of \((1+i\sqrt{3})^{\frac{3}{4}}\).

Answer: 8


Question: Are there any infinite dimensional irreducible complex representations of the symmetric group $S_n$?

Answer: No


Question: Calculate the area of the spherical triangle with vertices at the points \((0,0,1)\), \(\left(\frac{1}{\sqrt2},0, \frac{1}{\sqrt2}\right)\), and \((0,1,0)\).

Answer: \dfrac{\pi}{4}


Question: For a unit octahedron defined by \(O^2 = \{(x,y,z): |x|+|y|+|z|=1\}\), consider the set \(L(r) = \{(x,y,z) \in O^2: P(x,y,z)=r\}\), where \(P(x,y,z)\) is the sum of the positive coordinates of \((x,y,z)\). Given \(r \in (0,1)\), determine the length of \(L(r)\). Are these lengths the same for all \(r\) in this range, and how can they be calculated?

Answer: 3\sqrt{2}


Question: Find the number of 3x3 matrices \( A \) with integer entries such that the trace of \( A^t A \) is 6, where \( A^t \) denotes the transpose of matrix \( A \).

Answer: 7392


Question: Find the limit of the sequence of polynomials \( \{f_n\} \) defined recursively by \( f_0(x) = 0 \) and \( f_{n+1}(x) = f_n(x) + \frac{x^2 - f_n^2(x)}{2} \) for \( n = 0, 1, 2, \ldots \), in the supremum norm on the interval \([-1, 1]\).

Answer: |x|


Question: Let $f$ and $g$ be entire functions such that $f(0) = g(0) \neq 0$ and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$. Prove that $f = g$.

Answer: f = g


Question: Find the remainder when \(6^{7^n}\) is divided by 43.

Answer: 6


Question: Evaluate the integral \( \int \limits_{B(0, \pi / 4)} \frac{1}{z \tan(z)} \, \mbox{d}z \) using the residue theorem. Determine the value of the integral.

Answer: 0


Question:
Let $U$ be an open disc centered at $0$ with radius $R > 0$ in the complex plane, and let $f: U \setminus \{0\} \rightarrow \mathbb{C}$ be a holomorphic function. Suppose that for some $a \in \mathbb{C}$, the radial limits of $f$ at $0$ exist and are equal to $a$ for any $\theta \in [0, 2\pi)$:
\[
\lim_{r \rightarrow 0} f(r e^{i \theta}) = a,
\]
where $r$ is a positive real number. Determine whether the limit of $f$ at $0$ exists.

Answer: a


Question: Evaluate the surface integral \( \int\int_S z^2 \, dS \) over the triangle with vertices \((1,0,0)\), \((0,2,0)\), and \((0,0,3)\) in \(\mathbb{R}^3\).

Answer: \dfrac{21}{4}


Question:
Consider the function $u : \mathbb{R}^2 \to \mathbb{R}$ with the following properties:

1. Every level curve is a function from $\mathbb{R}$ to $\mathbb{R}$. That is, a level curve $U(x,y)= c$ can be written as $y = f(x)$ for some $f : \mathbb{R} \to \mathbb{R}$.
2. For all $p \in \mathbb{R}^2$, the sets $L(p)$ and $U(p)$ are closed, where $L(p) := \{(x,y) \in \mathbb{R}^2 : u(x,y) \leq u(p)\}$ and $U(p) := \{(x,y) \in \mathbb{R}^2 : u(x,y) \geq u(p)\}$.

Can $u$ have a level curve $y = f(x)$ such that $f$ is discontinuous?

Answer: No


Question: The plane \(4x-3y+8z=5\) intersects the cone \(z^2=x^2+y^2\) in an ellipse. Let \(P\) and \(Q\) be the highest and lowest points on the ellipse. If \(O\) denotes the origin, what is the angle between the vectors \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\)?

Answer: \dfrac{\pi}{2}


Question:
Let $A$ be a full triangulated subcategory of $B$, and let $u: A \rightarrow B$ be the corresponding embedding. Consider a triangulated functor $f: B \rightarrow A$ satisfying the following conditions:

1. $f \circ u = \text{id}$
2. For any $b \in B$, if $f(b) = 0$, then $b = 0$.

Determine whether $K_0(A) = K_0(B)$. Provide a proof or counterexample.

Answer: K_0(A) = K_0(B)


Question: Is there a field of characteristic two that is not isomorphic to \( \mathbb{Z}/2\mathbb{Z} \)?

Answer: \mathbb{F}_4


Question:
Consider the continuous-time state-space representation:
\[ \frac{d}{dt}x(t)=Ax(t)+Bu(t), \quad y(t)=Cx(t), \quad t\in \mathbb{R}^{+} \]
with the matrices given by:
\[ A=\begin{bmatrix}0&-1&0&0\\-2&0&0&0\\1&0&-1&0\\0&0&0&-2 \end{bmatrix}, \quad B=\begin{bmatrix}0\\0\\-1\\1 \end{bmatrix}, \quad C=\begin{bmatrix}1&0&0&-1 \end{bmatrix} \]
The controllability and observability matrices are:
\[ \mathcal{C}=\begin{bmatrix}0&0&0&0\\0&0&0&0\\-1&1&-1&1\\1&-2&4&-8 \end{bmatrix}, \quad \mathcal{O}=\begin{bmatrix}1&0&0&-1\\0&-1&0&2\\2&0&0&-4\\0&-2&0&8 \end{bmatrix} \]
Their reduced row-echelon forms are:
\[ \text{rref}(\mathcal{C})=\begin{bmatrix}1&0&-2&6\\0&1&-3&7\\0&0&0&0\\0&0&0&0 \end{bmatrix}, \quad \text{rref}(\mathcal{O})=\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&0&0 \end{bmatrix} \]
A Kalman decomposition of this state space realization can be obtained using a similarity transformation with the transformation matrix $M$ given by one of the following options:

A) \( M= \begin{bmatrix}0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0 \end{bmatrix} \)

B) \( M= \begin{bmatrix}0&0&1&1\\0&1&0&0\\-1&0&0&0\\1&1&1&1 \end{bmatrix} \)

C) \( M= \begin{bmatrix}0&-1&1&2\\-2&0&1&-2\\1&0&1&2\\0&0&1&-2 \end{bmatrix} \)

D) \( M= \begin{bmatrix}1&0&2&1\\0&0&-2&1\\2&1&2&1\\1&1&-2&1 \end{bmatrix} \)

E) \( \text{None of the above} \)

Determine the correct transformation matrix $M$. \( \boxed{?} \)

Answer: E


Question: Let \( \omega \) and \( z \) be complex numbers such that \( |\omega| = 1 \) and \( |z| = 10 \). Define \( \theta = \arg\left(\frac{\omega - z}{z}\right) \). Determine the maximum possible value of \( \tan^2 \theta \).

Answer: \dfrac{1}{99}


Question:
A machine consists of two components with lifetimes described by the joint density function:
\[
f(x,y)=
\begin{cases}
1/50, & \text{for }x>0,y>0,x+y<10 \\
0, & \text{otherwise}
\end{cases}
\]
The machine operates until both components fail. Calculate the expected operational time of the machine.

Answer: 5


Question: Given a positive bounded sequence where the average converges to zero, does the average of the square of the sequence also converge to zero? Specifically, if \( \frac{\sum\limits_{i=1}^n |x_i|}{n}\rightarrow 0 \) and \( |x_i|\leq C \), does \( \frac{\sum\limits_{i=1}^n |x_i|^2}{n}\rightarrow 0 \) also hold true?

Answer: Yes


Question:
Let $X$ be a complete separable non-compact metric space, and let $C_b(X)$ denote the space of all continuous bounded real-valued functions on $X$. Consider a bounded linear operator $A$ from $C_b(X)$ to itself. Let $X^* = X \cup \{\infty\}$ be the Alexandroff compactification of $X$. Define the space $C^*_b(X^*)$ as follows:
\[
C^*_b(X^*) = \{f:X^*\to\mathbb{R} \text{ such that } f|_X \in C_b(X)\}
\]
This is the space of all extensions of continuous bounded functions from $X$. Extend the operator $A$ to $A^*$ on $C^*_b(X^*)$ by:
\[
(A^*f)(x) = \begin{cases}
(A(f|_X))(x) & \text{if } x \neq \infty, \\
f(\infty) & \text{if } x = \infty.
\end{cases}
\]
Does the operator $A^*$ map $C_b(X^*)$ into itself?

Answer: No


Question: Calculate the marginal density of $x$ given the conditional density $f(x|\theta) = \frac{1}{\theta}$ for $0 \leq x \leq \theta$ and the prior distribution $\pi(\theta) = \theta e^{-\theta}$ for $\theta > 0$. Express the marginal density $f(x)$ in terms of $x$.

Answer: e^{-x}


Question: Find the cumulative distribution function (CDF) of the sum of K independent uniform random variables, each distributed uniformly over the interval [0,1]. Express the CDF of the sum \( S = X_1 + X_2 + \cdots + X_k \) for \( z \leq 1 \).

Answer: \dfrac{z^K}{K!}


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

Answer: \dfrac{1}{e}


Question: Determine the cardinality of the set \( S = \mathbb{N} \times A \), where \( A \) is a set with infinitely many elements and \( \mathbb{N} \) is the set of natural numbers.

Answer: |A|


Question: Determine the radius of convergence for the series: $$\sum_{n=1}^{\infty} \frac{z^n}{[1,2,...,n]}$$ where $[a_1,a_2,a_3,...,a_n]$ denotes the least common multiple of the numbers $a_1,a_2,...,a_n$. Provide your answer as a single numerical value.

Answer: e


Question: Evaluate the double integral \( \iint_D \frac{xy}{\sqrt{1-y^2}} \, dx \, dy \), where \( D \) is the region in the first quadrant bounded by the unit circle \( x^2 + y^2 = 1 \).

Answer: \dfrac{1}{6}


Question: Evaluate the infinite series: \[ \sum_{i=0}^\infty \frac{i^2}{4^i} \]

Answer: \dfrac{20}{27}


Question: Let $X_n$ be a sequence of independent and identically distributed random variables. Assume $X_1$ is symmetric, meaning $X_1$ and $-X_1$ have the same distribution. Define $Y_n = X_n 1_{[|X_n| \le 1]}$. Prove that $\frac{\sum_{i=1}^n Y_i}{n} \to 0$ almost surely.

Answer: 0


Question: Determine the value of \( c \) such that the expression \( c(12X_1^2 + 6X_2^2 + 4X_3^2) \) has a \( \chi_3^2 \) distribution, where \( X_1, X_2, \) and \( X_3 \) are independent random variables with \( X_k \sim N(0,k) \).

Answer: \dfrac{1}{12}


Question: A square $ABCD$ has all its vertices on the curve $x^2y^2=1$. The midpoints of its sides also lie on the same curve. Determine the area of square $ABCD$. \( \text{(Express your answer as a single number.)} \)

Answer: 4\sqrt{5}


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

Answer: \pi


Question: Let \( G \) be an abelian group of order 80. Given that there are more than 33 elements of order 20 in \( G \) and that \( G \) has no element of order 40, find the number of elements of order 20 in \( G \).

Answer: 48


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

Answer: 1


Question: Determine the values of \( p \) for which the series \( \sum_{k=1}^{\infty} \frac{k}{\sqrt{4+k^p}} \) converges.

Answer: p > 4


Question: Evaluate the expression \( \frac{\sum_{k=1}^{n} \binom{n}{2k-1}(n-(2k-1))^3}{n^2(n+3)2^n} \).

Answer: \dfrac{1}{16}


Question: In a circle with a diameter of 8 meters, points A and B are located on the diameter, each 1 meter from the center. Through point B, draw any chord PC. Calculate the sum of the squares of the medians of triangle APC.

Answer: \dfrac{147}{2}


Question: Find the last two digits of the perfect number given by the expression $2^{126}(2^{127}-1)$. Express your answer as a number modulo 100.

Answer: 28


Question: Solve the recurrence relation: \[ t(n) = n + \sum_{j=1}^n t(n-j) \]

Answer: 2^n - 1


Question: Evaluate the limit \( \lim_{|z|\to 0}\left|\frac{1-e^{2iz}}{z}\right| \).

Answer: 2


Question: Use the Taylor series formula to find the quadratic approximation of the function \( g(x,y) = e^x \ln(2x+y+1) \) at the origin, and use this approximation to estimate \( g\left(\frac{1}{10}, \frac{1}{5}\right) \).

Answer: \dfrac{9}{25}


Question: Find the value of \( a \) such that \[ \lim_{x \to \infty} \left(1 + \frac{a}{x} - \frac{4}{x^2}\right)^{2x} = e^3. \]

Answer: \dfrac{3}{2}


Question: Find the limit \( \lim_{x \to 2} \frac{\cos \frac{\pi}{x}}{x-2} \) using the substitution \( t = \frac{\pi}{2} - \frac{\pi}{x} \) and trigonometric identities.

Answer: \dfrac{\pi}{4}


Question: Simplify the expression: \[ \frac{\sqrt{\sqrt[4]{27}+\sqrt{\sqrt{3}-1}}-\sqrt{\sqrt[4]{27}-\sqrt{\sqrt{3}-1}}}{\sqrt{\sqrt[4]{27}-\sqrt{2\sqrt{3}+1}}} \]

Answer: \sqrt{2}


Question:
Find the limit of the sequence defined recursively by:
\[ x_n = x_{n-1} \cdot \left(2-\frac{\tan^{-1}(n) \cdot n^2 - 3}{3 + n^2}\right) \]
with initial condition \( x_1 = 2-\frac{\tan^{-1}(1) \cdot 1^2 - 3}{3 + 1^2} \).

Answer: 0


Question: Let $R$ be a commutative ring with unity, which is a subring of a commutative ring $A$ with unity, and $A$ is a subring of $R[x,y]$. Is there an ideal $J$ of $A$ such that $A/J$ is isomorphic to $R$?

Answer: Yes


Question: Let $\{a_n\}$ be a sequence such that \(\lim_{n\to \infty}\left|a_n+3\left(\frac{n-2}{n}\right)^n\right|^\frac{1}{n}=\frac{3}{5}\). Calculate \(\lim_{n\to \infty}a_n\).

Answer: -\dfrac{3}{e^2}


Question: Determine the number of abelian groups of order \(10^5\), up to isomorphism.

Answer: 49


Question: Determine whether there exists an integer $k > 1$ and a connected $k$-regular graph $G = (V, E)$ such that there are distinct vertices $v \neq w \in V$ with the same neighborhood set $N(v) = N(w)$. Note that $v$ and $w$ cannot be adjacent.

Answer: Yes


Question: A particle is located at a vertex of an infinite hexagonal grid, where each edge has a length of 1 and all angles are 120°. Each second, the particle randomly moves to one of its neighboring vertices. After n seconds, what is the expected value of the square of the distance from the starting position?

Answer: n


Question: An urn contains 4 blue and 4 red marbles. Initially, a marble is drawn (without looking) and removed from the urn. Then, a marble is drawn from the urn, its color recorded, and put back in the urn. This process is repeated 1000 times. Let event D be that between 450 and 550 blue marbles are drawn, and let event E be that between 379 and 479 blue marbles are drawn. Which event is more likely, D or E?

Answer: E


Question: Find the natural number \( a \) such that \( \sum_{k=1}^{n} f(a+k) = 16(2^n - 1) \), where the function \( f: \mathbb{N} \to \mathbb{N} \) satisfies \( f(x+y) = f(x)f(y) \) for all \( x, y \in \mathbb{N} \) and \( f(1) = 2 \).

Answer: 3


Question: Let $P(X)$ be a polynomial with positive integer coefficients that splits over $\mathbb{Q}$ and satisfies $P(0) = 1$. Determine if there exist positive integers $k_1, k_2, \ldots, k_d$ such that $P(X) = \prod_{i=1}^{d} (k_iX + 1)$. Provide a justification for your answer.

Answer: Yes


Question:
Which of the following sets cannot be expressed as the intersection of countably many open sets in \( \mathbb{R} \)?

- \( \mathbb{Z} \)
- \( \mathbb{Z}' \)
- \( \mathbb{Q} \)
- \( \mathbb{Q}' \)
- \( [0,1) \)

Answer: C


Question:
Given that $f(0)=0$ and $f'(x) \in (0,1]$ for all $x \in [0,1]$, determine which of the following values the expression \( \frac{\left(\int_0^1 f'(x) \, dx\right)^2}{\int_0^1 (f(x))^3 \, dx} \) cannot take.

A) \( \frac{1}{2} \)

B) \( 2 \)

C) \( \frac{1}{4} \)

D) \( 1 \)

Answer: B


Question:
Let $y = f(x)$ be a differentiable function that satisfies the equation \[ f(x) = x + \int_0^1 x^2 z f(z) \, dz + \int_0^1 x z^2 f(z) \, dz. \] Determine the value of \( f\left(-\frac{9}{4}\right) \).

A) $-\frac{4}{9}$
B) $\frac{4}{9}$
C) $-1$
D) $0$

Answer: D


Question: Determine the radius of convergence of the power series: \[ \sum_{n=2}^\infty (\log n)^n(z+1)^{n^2} \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{n\to \infty}\int^n_{-n}e^{-x^2}dx \]

Answer: \sqrt{\pi}


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

Answer: 2


Question: Calculate the limit \( \lim_{t\to\infty}\left(\sum_{m=1}^{t}\left(\frac{(\pi \xi)^{2m-1}}{(2m-1)!}\;\frac{1}{1-2^{\beta-2t+2m-2}}\right)\right) \) where \( \beta \simeq 0.7 \) and \( \xi \in \mathbb{C} \).

Answer: \sinh(\pi \xi)


Question: Evaluate the limit: $$\lim_{t \rightarrow 0} \frac{e^{t^3}-1-t^3}{\sin(t^2)-t^2}.$$

Answer: -3


Question: Determine the number of terms containing powers of $x$ in the expansion of \((1+x)^{100}(1+x^2-x)^{101}\). Choose the correct answer from the following options: A: 202, B: 302, C: 301, D: 101.

Answer: B


Question: Evaluate the limit: \[ \lim_{n\to \infty} \left( \sum_{r=0}^n \frac {2^r}{5^{2^r}+1}\right) \]

Answer: \dfrac{1}{4}


Question:
Given the transition matrix of a Markov chain:
\[
\begin{bmatrix}
\frac{1}{5} & \frac{4}{5} & 0 & 0 & 0 \\
\frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\
\frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} \\
0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 \\
0 & 0 & 0 & 0 & 1
\end{bmatrix}
\]
find the expected long-run proportion of time the chain spends at state $a$, given that it starts at state $b$. Use the stationary distribution to determine this proportion.

Answer: \dfrac{5}{13}


Question: Determine whether the function \( H_{\lfloor x \rfloor} - \log(x) \), where \( H_n \) is the n-th harmonic number, converges as \( x \to \infty \).

Answer: \gamma


Question: A die is thrown repeatedly. If the result is 1, 2, or 3, you score 1 point. If the result is 4, you score 2 points. If the result is 5 or 6, you score 3 points. Let \( N \) be the number of throws needed for the product of scores to exceed 100,000. Calculate \( P[N > 25] \). (Hint: It may be useful to use the normal distribution in this case.)

Answer: 0.416


Question: If $I$ is an integral ideal of $O_K$ for $K=\mathbb{Q}(\theta)$, and $a \in O_K$, does $N(I)|N(\langle a \rangle)$ imply $I \mid a$?

Answer: No


Question: Given the function \( f(x) = (x+4)(x+1)(6-x) \) defined on the interval \((-10, 6]\), determine the value of \( mn \) if \( m-n < 0 \) and the integral \( \int_{n}^{m} f(x) \, dx \) is maximized.

Answer: 4


Question: Let $M$ be a closed (compact, without boundary) topological manifold. Determine whether there exists a subset $A$ of $M$ such that $M$ deformation retracts onto $A$. Provide a justification for your answer.

Answer: Yes


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

Answer: -\infty


Question: Compute the matrix exponential $e^{At}$ for the matrix $A = I + N$, where $N^2 = 0$, in terms of $N$.

Answer: e^{t}(I + tN)


Question: Let $X$ be a topological space and let $\mathcal{F}$ be a sheaf of abelian groups on $X$. Suppose $\mathcal{G}$ is a subsheaf of abelian groups of $\mathcal{F}$. Construct the quotient presheaf $\mathcal{F}/\mathcal{G}$ by defining $$(\mathcal{F}/\mathcal{G})(U) := \mathcal{F}(U)/\mathcal{G}(U)$$ for an open set $U \subseteq X$. Is this presheaf always separated? In other words, is the group homomorphism $$(\mathcal{F}/\mathcal{G})(U) \to \prod_{x \in U} (\mathcal{F}/\mathcal{G})_x$$ always injective?

Answer: Yes


Question: Determine the number of digits in the number \(2^{2^{22}}\) using number theory techniques, without employing logarithms.

Answer: 1262612


Question: Compute \( \operatorname{Ext}_{k[x] / \langle x^2 \rangle}(k,k) \) where \( k \) is a field.

Answer: k


Question: Evaluate the limit of $i^{1/n}$ as $n$ approaches infinity. Determine whether the expression remains an imaginary number or converges to a real number.

Answer: 1


Question: Find the coefficient of \(x^{46}\) in the generating function \(\frac{1}{1 - x^3 - x^4 - x^{20}}\).

Answer: 3224


Question: Evaluate the limit: \[ \lim_{x \to 3} \frac{\sqrt{19-x} - 2\sqrt[4]{13+x}}{\sqrt[3]{11-x} - x + 1} \]

Answer: \dfrac{9}{52}


Question: Provide an example of a group \( G \) that contains an element \( a \) of order 3 and an element \( b \) of order 4, such that the order of the product \( ab \) is less than 12.

Answer: S_4


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

Answer: \dfrac{\pi}{4}


Question: Let \( T \) be an invertible linear operator on \( \mathbb{R}^2 \). Determine if it is true that if \( T \) has determinant \( \pm 1 \), then \( T \) and \( T^{-1} \) have the same norm (the usual operator norm).

Answer: True


Question: Find the extrema of the implicit function defined by the equation \( x^2 + y^2 - z^2 = 0 \).

Answer: (0, 0, 0)


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{\sqrt{2\pi} x - x\int_{-\infty}^x e^{-\frac{t^2}{2}} dt}{\exp\left(-\frac{x^2}{2}\right)} = 1. \]

Answer: 1


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a continuously differentiable periodic function with period $1$. Prove that \( n\int_{0}^{1} f(x) \sin(2\pi nx)\, \mathrm{d}x \to 0 \) as \( n \to \infty \).

Answer: 0


Question: Let $F$ be a field and $f(x) \in F[x]$ be a polynomial. Fix an algebraic closure $\overline{F}$ of $F$. Are any two splitting fields $E, E' \subseteq \overline{F}$ for $f$ equal as sets, in addition to being isomorphic?

Answer: Yes


Question: Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) \neq x$ for all $x \in \mathbb{R}$ and for every $a \in \mathbb{R}$ there exists a sequence $\{x_n\}$ such that \(\lim_{n \to \infty} x_n = f(a)\) and \(\lim_{n \to \infty} f(x_n) = a\)?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int_0^\infty\frac{n\cos^2(x/n)}{n+x^4}dx. \]

Answer: \infty


Question: Evaluate the integral \( \int_{0.1}^{1} |(\pi)( x^{-2}) \sin(\pi \cdot x^{-1})| \, dx \) without using a calculator.

Answer: 18


Question: Determine the degree of the field extension \(\mathbb{Q}(\sqrt{2}, \sqrt[4]{2}, \sqrt[8]{2})\) over \(\mathbb{Q}\). Choose the correct option: 4, 8, 14, 32.

Answer: 8


Question: Determine whether the integral \( \int_{0}^{\infty} \frac{\sin^2 x }{x^2}dx \) is equal to the integral \( \int_{0}^{\infty} \frac{\sin x }{x}dx \).

Answer: Yes


Question: Let \( \phi \) be a programming system satisfying the UTM Theorem, meaning \( \phi \) is a 2-ary partial recursive function such that the list \( \phi_0, \phi_1, \ldots \) includes all 1-ary partial recursive functions, where \( \phi_i = \lambda x.\phi(i,x) \) for all \( i \in \mathbb{N} \). Suppose that for all recursive functions \( f \) with finite range, there exists an \( n \in \mathbb{N} \) such that \( \phi_n = \phi_{f(n)} \). Determine if, for any recursive function \( g \) (with finite or infinite range), there exists an \( n \in \mathbb{N} \) such that \( \phi_n = \phi_{g(n)} \).

Answer: Yes


Question: Determine whether the sequence \( \{f_n\} \) has a convergent subsequence in the space \( V = C_c(\mathbb{R}) \), where \( f_n(x) = f(x-n) \) and \( f \) is a continuous function vanishing outside the interval \([0,1]\). The space \( V \) is endowed with the metric \( d(f,g) = \left(\int_{\mathbb{R}} |f(t) - g(t)|^2 dt\right)^{1/2} \). Answer True or False.

Answer: B


Question:
Find the sup-norm \( \|f\|_{\sup} \) of the function \( f(x) \) defined by:
\[
f(x) = \begin{cases}
0, & x \in \mathbb{Q} \\
-x^2, & x \not\in \mathbb{Q}
\end{cases}
\]

Answer: \infty


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

Answer: 1


Question: Determine the Krull dimension of the ring \( R = \mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4] \).

Answer: 3


Question: If $(x_n)$ and $(y_n)$ are sequences of non-zero real numbers such that $\lim (x_n-y_n)=0$, is it true that $\lim \left(\frac{x_n}{y_n}\right)=1$? Justify your answer.

Answer: No


Question:
Solve the following system of equations for \((x, y)\):
\[\begin{cases}
3y^3+3x\sqrt{1-x}=5\sqrt{1-x}-2y\\
x^2-y^2\sqrt{1-x}=\sqrt{2y+5}-\sqrt{1-x}
\end{cases}\]

Answer: (-3, 2)


Question: Calculate the limit of the sequence defined by $e_0=0$ and for any $n\geq 1$, \[e_n=\frac{1}{2^n}\left(\sum_{i=0}^{n}C_n^i (e_i +1)\right)\] as \(n\) approaches infinity: \[\lim_{n\rightarrow\infty}\frac{e_n}{\log_2 n}\]

Answer: 1


Question: Find the residue of the function \( \frac{1}{z^2 \sin(z)} \) at the point \( z = 0 \).

Answer: \dfrac{1}{6}


Question: Given vectors \( \mathbf{a} \) and \( \mathbf{b} \), the vector \( \mathbf{c} = \mathbf{a} + \mathbf{b} \) makes an angle of \( \frac{\pi}{3} \) with \( 2\mathbf{b} \). The magnitude of \( \mathbf{c} \) is 6, and the area of the parallelogram determined by \( \mathbf{a} \) and \( \mathbf{b} \) is \( 2\sqrt{3} \). Find the magnitude of \( \mathbf{b} \).

Answer: \dfrac{2}{3}


Question: Determine the number of elements of order 6 in the automorphism group \( \text{Aut}(\mathbb{Z}_{720}) \).

Answer: 30


Question: Given the transfer function \( G(s) = \frac{1}{(s+7)(s+12)} \) and the input \( x(t) = \left(1 + \frac{1}{t}\right)^t \), calculate the limit of the output \( y(t) \) as \( t \to \infty \).

Answer: \dfrac{e}{84}


Question: Find all pairs of positive integers \((m, n)\) such that \(2^{m+1} + 3^{n+1}\) is a perfect square.

Answer: (3, 1)


Question: Evaluate the limit: \[ \lim_{(x,y) \to (0,0)} \frac{xy-\sin(x)\sin(y)}{x^2+y^2} \]

Answer: 0


Question: Determine the cardinality of the set \( A = \left \{(x_n)_{n \geq 1}\ |\ x_n \in \{0,1 \}\ \text {for all}\ n \geq 1\  \text {and}\ x_n = 0\ \text {for all but finitely many}\ n \in \Bbb N \right \} \).

Answer: \aleph_0


Question: Compute \( \lim_{|r|\to\infty}\int_{\Bbb R}|f(x+r)+f(x)|^p\ dx \) for a function \( f \in L^p(\Bbb{R}) \) where \( 1\leq p<\infty \).

Answer: 2 \|f\|_p^p


Question:
Let \(\mathcal{F}\) be the set of functions analytic on a neighborhood of the closed unit disk in \(\mathbb{C}\). Determine the value of:
\[\sup\{|f(0)| \mid f(1/2)=0=f(1/3) \text{ and } |f(z)|\leq 1 \text{ for } |z|=1\}\]
Is the supremum attained? Consider the function \(\frac{z-1/2}{1-z/2}\frac{z-1/3}{1-z/3}\).

Answer: \dfrac{1}{6}


Question: Calculate the limit: $$\lim_{(x,y)\to(0,0)}\frac{-|x+y|}{e^{x^2+2xy+y^2}}$$

Answer: 0


Question:
Determine the radius of convergence for the formal power series:
\[ f(x) = x\sqrt{1} + x^2\sqrt{2} + x^3\sqrt{3} + \cdots + x^n\sqrt{n} + \cdots \]

Answer: 1


Question: Find the normal closure of the field extension \( \mathbb{Q} \subset \mathbb{Q}(t) \), where \( t \) is a root of the polynomial \( x^3 - 3x^2 + 3 \).

Answer: \mathbb{Q}(t)


Question: Given the joint density function \( f(x,y) = xe^{-x(y+1)} \) for \( 0 < x, 0 < y \), compute the density function of \( Z = XY \).

Answer: e^{-z}


Question: Find the area of the surface that is part of the cylinder \(x^2+z^2=a^2\) and lies inside the cylinder \(x^2+y^2=a^2\).

Answer: 8a^2


Question: Solve the differential equation \( x^2y'' + xy' + 2xy = 0 \) at \( x = 0 \) using the Frobenius method. Determine the value of \( r \) from the indicial equation.

Answer: 0


Question: Given a sample $X_1, X_2, \ldots, X_{15}$ from a distribution with probability density $\exp(-(x-\theta))$ for $x \ge \theta$, test the hypothesis $H_0: \theta=0$ against $H_1: \theta>0$. The test statistic is $T = \min\{x_1, x_2, \ldots, x_{15}\}$. If the observed value of $T$ is $t=0.1$, what is the p-value of this test? (Hint: If $X_1, X_2,\ldots,X_n$ is a sample from an $\operatorname{Exp}(\lambda)$ distribution, then $\min\{X_1, X_2,\ldots,X_n\}$ has an $\operatorname{Exp}(n\lambda)$ distribution.)

Answer: e^{-1.5}


Question: Find the sum of the series \( \sum_{n=1}^{\infty} \frac{n^2+1}{5^n} \).

Answer: \dfrac{23}{32}


Question: Determine whether the statement $|S_{n}| = O( \frac{1}{\sqrt n} )$ is true, where $|S_{n}|$ represents the surface area of the n-dimensional sphere.

Answer: True


Question: Let $A$ be an $n \times n$ matrix such that $A^r = I$ and $A$ has exactly one eigenvalue. Prove that $A = \lambda I$ for some scalar $\lambda$. Provide a detailed explanation of your reasoning.

Answer: A = \lambda I


Question: Determine the maximum number of real zeros that a function of the form \( \lambda_1e^{x} + \lambda_2e^{2x} + \cdots + \lambda_ne^{nx} \), where \( \lambda_1, \lambda_2, \ldots, \lambda_n \in \mathbb{R} \) and \( x \in [0, \infty) \), can have.

Answer: n-1


Question: Given the equation \(z = (x+i)^2\) where \(x > 0\) is a real number, and \(\arg(z) = \frac{\pi}{3}\), solve for \(x\).

Answer: \sqrt{3}


Question: Calculate the limit \( \lim_{(h,k) \to (0,0)} \frac{f(h+k) - f(k) - f(h) + f(0)}{hk} \) given that \( f \) is twice differentiable at 0.

Answer: f''(0)


Question: Determine the number of zeros in the right half-plane for the polynomial \(z^4 + 2z^3 - 2z + 10\).

Answer: 2


Question: Find the volume of the solid bounded by the cylinder \(x^2 + y^2 = 1\) and the planes \(x + y + z = 1\) and \(4x + 2y + z = 7\).

Answer: 6\pi


Question: Find the largest possible integer value of \( n \) such that \( x^2 = n^2 + 2016n \) for some integer \( x \) and \( n > 0 \).

Answer: 253009


Question: Is each diagonal entry of a positive semidefinite doubly stochastic matrix the largest element in its respective row?

Answer: Yes


Question: Given that \( A \) is an invertible \( 3 \times 3 \) matrix and \( B \) is another \( 3 \times 3 \) matrix, determine whether there exists an integer \( n \) such that the matrix \( A + nB \) is invertible.

Answer: Yes


Question: Does there exist a compact metric space $X$ that contains countably infinitely many clopen subsets?

Answer: Yes


Question: Find the smallest integer $B$ such that $4B$ is not in the image of Euler's totient function $\phi(n)$. What is the value of $B$?

Answer: 17


Question: Find the value of \( \lim _{x\to \infty }\left(1+\frac{4}{\ln x}\right)^{\ln\left(x^2+x\right)} \).

Answer: e^8


Question: What is the expected number of non-empty columns (or sampled dishes) $\Lambda_t$ in an Indian Buffet Process with $T$ customers and concentration parameter $\alpha$?

Answer: \alpha \sum_{t=1}^T \frac{1}{t}


Question: Verify that \( \int \int_{S} (x^2 + y^2) \, d\sigma = \frac{9\pi}{4} \) for the surface \( S = \{(x,y,z) : x>0, y>0, 3>z>0, z^2 =3(x^2+y^2)\} \). Use the formula \( \int \int_{S} f(x,y,z) \, d\sigma = \int \int_{D} f(x, y, g(x,y)) \sqrt{\left(\frac{\partial z }{\partial x}\right)^2 + \left(\frac{\partial z }{\partial y}\right)^2+1} \, dA \) to set up and evaluate the integral.

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


Question: Determine the number of solutions to the equation \( \cos^7x + \cos^7(x+2\pi/3) + \cos^7(x + 4\pi/3) = 0 \) within the interval \([0, 2\pi]\).

Answer: 6


Question: Let $X$ and $Y$ be independent random variables, each with the probability density function $f(x) = e^{-x}x^{-rac{1}{2}}$. Find the probability density function of the random variable $Z = X + Y$. Express your answer in terms of $z$.

Answer: e^{-z}


Question:
Let \(a, b, c, d \in \mathbb{R}\) and consider the following system of equations:
\[ a \sec(200^\circ) - c \tan(200^\circ) = d \]
\[ b \sec(200^\circ) + d \tan(200^\circ) = c \]
If \(\frac{a^2 + b^2 + c^2 + d^2}{ac - bd} = \lambda \csc(200^\circ)\), find the value of \(\lambda\).

Answer: 2


Question: Evaluate the complex integral \( \int_{\gamma} \frac{1}{z^{2}-1} \, dz \), where the contour \( \gamma:[-\pi,\pi]\to\mathbb{C} \) is defined by \( \gamma(t)=2e^{it} \).

Answer: 0


Question:
Consider a random variable $X \le 0$ with a density function $f_X$ having a bounded support. If $g(t) = \infty$ for $t \in (-\infty,0]$, evaluate the integral:
\[ \int_{-\infty}^0 g(t) \; f_X(t) \; dt. \] Determine if this integral equals infinity.

Answer: \infty


Question: Evaluate the integral: \[ \int_{-\infty}^\infty x^2 \frac{1}{\sqrt{2\pi}}e^{-x^2/2} \, dx \]

Answer: 1


Question: Find the limit of the sequence \((-1)^n \frac{3^{2n+1}n^{2n}}{(4n-1)^{2n}}\) as \(n\) approaches infinity.

Answer: 0


Question: Evaluate the double integral \( \int\limits_1^2\int\limits_{1/y}^y\sqrt{\frac{y}{x}}e^{\sqrt{xy}}\,dx\,dy \).

Answer: 2e^2 - 4e


Question: Calculate the residue of the function \( f(z) = \frac{1}{(z+i)^7} - \frac{3}{z-i} \) at the pole \( z = -i \).

Answer: 0


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

Answer: 0


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

Answer: \sqrt{2}


Question: Let $Z$ be a Poisson distributed random variable with parameter $\Lambda$. In turn, $\Lambda$ is a Poisson distributed random variable with parameter $\mu=2$. Compute $E[Z]$. (Hint: $Z|\Lambda=\lambda$ is Poisson distributed with parameter $\lambda$.)

Answer: 2


Question:
Find a mapping \( f: \mathbb{R}^{n} \rightarrow \mathbb{R} \) such that for any points \( a, b, c \in \mathbb{R}^{n} \) with \( ||a-b|| \le ||a-c|| \le ||b-c|| \), the following inequality holds:
\[ |f(a)-f(b)| \le |f(a)-f(c)| \le |f(b)-f(c)| \]

Answer: f(x) = 0


Question: Find the residue of the function \( z \cos\left(\frac{1}{z}\right) \) at \( z=0 \).

Answer: -\dfrac{1}{2}


Question: Find the volume of the region enclosed by the circle \(x^2+y^2=2x\), below the plane \(3x+4y+z=12\), and above the \(xy\)-plane. Use polar coordinates to set up the integral and determine the correct limits for \(\theta\).

Answer: 9\pi


Question: Determine the quotient space of the torus $S^1 \times S^1 = \{(e^{2\pi ix}, e^{2\pi iy}) \mid x, y \in \mathbb{R}\}$ under the equivalence relation $(w, z) \sim (iw, iz)$. Express your answer in terms of a known topological space or provide a description of the resulting space.

Answer: S^1 \times S^1


Question: Given that the quadratic equation \((a+b+c)x^2 - 2(ab+bc+ca)x + ab^2 + bc^2 + ca^2 = 0\) has at least one real solution, find the value of \(\frac{a+5b}{c}\), where \(a\), \(b\), and \(c\) are real and positive numbers.

Answer: 6


Question: Let $f$ be a differentiable function of one variable and $u=\frac{x}{y}f(\frac{y}{x})$. If $u$ satisfies the partial differential equation \( \frac{x}{y}u_{x} + u_{y}= G \), find the function $G$. \( \)

Answer: 0


Question: Assume $K$ is an imaginary quadratic extension of $\mathbb{Q}$, and $E$ is an elliptic curve defined over $\mathbb{Q}$. Let $p \neq l$ be primes in $\mathbb{Q}$ where $E$ has good reduction. Assume $p$ splits in $K/\mathbb{Q}$ and $l$ stays inert. Denote by $K_{l}$ the localization of $K$ at the place $l$. Is there always an isomorphism \(H^1(K_{l},E)_{p^n}\rightarrow E(K_l)/p^nE(K_l)\), at least for large enough $n$?

Answer: Yes


Question: Let $f$ be a continuous function on $(-1,1)$. Suppose that for every sequence $\{x_n\}$ of rational numbers converging to $0$, the limit \( \lim_{n \rightarrow \infty }\frac{f(x_n)-f(0)}{x_n}=1 \) holds. Determine whether $f$ is differentiable at $0$.

Answer: Yes


Question: Let $X\subset\mathbb{R}^n$ be a nonempty convex set with a nonempty interior, and let $f:X\rightarrow \mathbb{R}$ be a continuous function. If $f$ is convex on the interior of $X$, is $f$ convex on the entire set $X$?

Answer: Yes


Question:
Find the square matrix $A$ of minimal order such that the vector function \( \vec{X}_0(t) = \vec{W}(t) + \vec{V}e^{2t} \) satisfies the differential equation \( \vec{X}_0'(t) = A \cdot \vec{X}_0(t) \), where:

\[ \vec{W}(t) = \begin{pmatrix} \sin(t) + \cos(t) \\ \cos(t) - \sin(t) \\ 0 \\ \vdots \\ 0 \end{pmatrix} \quad \text{and} \quad \vec{V} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \\ \vdots \\ v_n \end{pmatrix} \]

Determine the smallest possible value of \( n \) for which such a matrix \( A \) exists.

Answer: 3


Question: Let $A$ be a Dedekind domain, and $B$ a finite torsion-free generically separable $A[[t]]$-algebra. If the preimage of $\text{Spec } B$ over $t = 0$ is disconnected, must $\text{Spec } B$ also be disconnected? Answer yes or no.

Answer: Yes


Question: Let $F$ be a field with $5^{12}$ elements. Determine the total number of proper subfields of $F$. Provide your answer as a single integer.

Answer: 5


Question: An ordered vertical stack of \( n \) books is on a desk. Each day, one book is picked uniformly at random from the stack and placed on top. What is the expected number of days for the books to return to their original order?

Answer: n!


Question: Jack needs to assign 25 job students to 25 jobs. Each student is suited for at least 4 jobs, and each job can be performed by at most 4 students. Can Jack assign each student to a job they are suited for? Justify your answer using graph theory principles.

Answer: Yes


Question: Determine the number of non-isomorphic semidirect products between the groups \( \mathbb{Z}_2 \) and \( SL(2,3) \).

Answer: 3


Question:
Let \( n \) be a positive integer and \( a_1, a_2, \ldots, a_n \) be non-negative numbers such that \( a_1 + a_2 + \cdots + a_n = n \). Find the maximum value of the expression:
\[
(n^{n-1}-1)\sqrt[n]{a_1 a_2 \cdots a_n} + \sqrt[n]{\frac{a_1^n + a_2^n + \cdots + a_n^n}{n}}
\]

Answer: n^{n-1}


Question:
Given the function \( f : \mathbb{R} \to \mathbb{R} \) with the properties:

- \( f(1) = 1 \)
- \( f(x+y) = f(x) + f(y) \) for all \( x, y \in \mathbb{R} \)
- \( f\left(\frac{1}{x}\right) = \frac{f(x)}{x^2} \) for all \( x \in \mathbb{R} \setminus \{0\} \)

Find \( f(2021) \).

Answer: 2021


Question: In triangle ABC, points D, E, and F are on sides BC, CA, and AB, respectively. The cevians AD, BE, and CF intersect at point P. The ratios of division are given as \( \frac{BD}{DC} = \frac{1}{2} \), \( \frac{CE}{AE} = \frac{3}{2} \), and \( \frac{AF}{FB} = \frac{4}{3} \). If the area of triangle ABC is 45, find the area of quadrilateral CEPD.

Answer: 22


Question: Let \( G = (\mathbb{Z} \bigoplus \mathbb{Z}) \star (\mathbb{Z} \bigoplus \mathbb{Z}) \), where \( \star \) denotes the free product, and let \( F \) be the commutator subgroup of \( G \). By Kurosh's theorem, \( F \) is a free group. Find a proper normal subgroup of \( F \) (other than the trivial one) that has infinite index.

Answer: [F, F]


Question: Determine whether a rational function of the form \( f(x)=\frac{\sum_{i=0}^n a_i x^i}{\sum_{j=0}^m b_j x^j} \), with the constraints \( \sum_{j=0}^m b_j x^j \not\equiv 1 \) and \( \sum_{i=0}^n a_i x^i \not\equiv 0 \), can have a horizontal inflection point.

Answer: Yes


Question: Find the limit of the sequence \(a_{n+1} = \int_{0}^{a_n} \sin(t^2) \, dt\) with \(a_1 = 1\) as \(n\) approaches infinity.

Answer: 0


Question: Consider the stochastic process $X := (X_T)_{T \in [0,\infty)}$ where $X_T(\omega) := \int_0^T f(t, \omega)\ dt$ for a $(\mathcal{B}_{[0,\infty)}\otimes\mathcal{A})/\mathcal{B}$-measurable function $f:[0,\infty)\times\Omega\rightarrow\mathbb{R}$. Is the function $X: [0,\infty)\times \Omega \to \mathbb{R}$ $(\mathcal{B}_{[0,\infty)}\otimes \mathcal{A})/\mathcal{B}$-measurable?

Answer: Yes


Question: Find the remainder when \( \binom{2013}{101} \) is divided by 101.

Answer: 19


Question: Use the squeeze theorem to find the limit \( \lim_{n\to\infty} \frac{1}{n^2} \sum_{k=n}^{5n} k \).

Answer: 12


Question: Evaluate the contour integral \( \oint_{|z-1|=1} \sec(z) \, dz \).

Answer: -2\pi i


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

Answer: \dfrac{1}{2}


Question:
Let $f$ and $g$ be meromorphic functions on $\mathbb{C}$. If $f$ has a zero of order $k$ at $a$ and $g$ has a pole of order $m$ at $z=0$, determine the nature of $g(f(z))$ at $z=a$. Choose the correct option:

- A zero of order $km$ at $z=a$
- A pole of order $km$ at $z=a$
- A zero of order $|k-m|$ at $z=a$
- A pole of order $|k-m|$ at $z=a$

Answer: B


Question: Determine the largest prime number \( p \) such that \( J_0(p) \) completely splits into elliptic curves over \( \mathbb{Q} \).

Answer: 37


Question: Find the degree of the field extension \([\mathbb{Q}(\sqrt{5},\sqrt[3]{2}): \mathbb{Q}(\sqrt{5})]\).

Answer: 3


Question: Evaluate the limit: \[ \lim_{n \to +\infty}\int _{-1}^1 f(t)\cos^2(nt) \, \mathrm{d}t \] where \( f:[0,1]\rightarrow \mathbb{R} \) is continuous and \( \int_{-1}^{1}f(t)\,\mathrm{d}t=1 \).

Answer: \dfrac{1}{2}


Question:
Find the last three digits of the sum:
\[1! + 3! + 5! + 7! + \ldots + 2013! + 2015! + 2016!\]

Answer: 647


Question: Determine the number of solutions \((\alpha, \beta)\) in the finite field \(F_{q^2}\) such that \(\alpha + \alpha^q + \beta \beta^q = 0\).

Answer: q^3


Question: Let $f(z)$ be an analytic function in the unit disc $\{z: |z| \leq 1\}$ such that $|f(z) - z| < |z|$ on the boundary $\{z: |z| = 1\}$. Determine the number of zeroes of $f(z)$ inside the unit disc $\{z: |z| < 1\}$.

Answer: 1


Question: Evaluate the limit: \[ \lim_{x\to 0}\ \frac{\dfrac{\sin x}{x} - \cos x}{2x \left(\dfrac{e^{2x} - 1}{2x} - 1 \right)} \]

Answer: \dfrac{1}{6}


Question: Consider the martingale defined by \(X_t = \exp(B_t - \frac{t}{2})\), where \((B_t)\) is a Brownian motion. Determine whether \(\lim_{t \rightarrow \infty} X_t\) exists, and if so, find its value.

Answer: 0


Question: Evaluate the limit of the sequence defined by \( y_1 = 1 \) and \( y_{n+1} = y_n + \frac{1}{n y_n} \) as \( n \to \infty \).

Answer: \infty


Question: An urn contains 30 balls, with 10 balls of color X and 8 balls of color Y. Let the random variables $X$ and $Y$ denote the count of balls of color X and Y, respectively, when 12 balls are selected without replacement. Find the expected value $E[XY]$ by conditioning on $Y$. Specifically, first find $E[X|Y = y]$ and use it to compute $E[XY]$. Assume a hypergeometric distribution for the selection process.

Answer: \dfrac{352}{29}


Question: Evaluate the integral \( \int_{0}^{1}{(\{2x\}-1)(\{3x\}-1)}\,dx \), where \( \{x\} \) denotes the fractional part of \( x \).

Answer: \dfrac{19}{72}


Question:
Evaluate the limit of the following expression as \( n \to \infty \):
\[ \frac{ \sqrt{\frac{3 \pi}{2n}} - \int_0^{\sqrt 6}( 1-\frac{x^2}{6} +\frac{x^4}{120})^n dx}{\frac{3}{20}\frac{1}{n} \sqrt{\frac{3 \pi}{2n}}} \] and show that it equals 1.

Answer: 1


Question: In a group of order 16 generated by elements $a$ and $b$ with $o(a)=8$, $o(b)=2$, and $bab^{-1}=a^{-1}$, how many elements of order 2 are there?

Answer: 9


Question: Let $H$ be a Hilbert space and $T \in B(H)$ a bounded normal operator. Let $\mathscr{A}$ be the von Neumann algebra generated by $T$. Determine whether $\mathscr{A}$ contains every orthogonal projection that commutes with $T$. Provide a justification for your answer.

Answer: No


Question: Evaluate \( A^2 \) if \( A = \lim_{n\to\infty} \left(\prod_{r=0}^n \binom{n}{r}\right)^\frac{1}{n(n+1)} \).

Answer: e


Question: Given a rectangle PQRS in the first quadrant with adjacent sides PQ and QR having slopes 1 and -1, respectively, and the function \( u(x,t) \) satisfying the wave equation \( \frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}=0 \), where \( u(P)=1 \), \( u(Q)=1/2 \), and \( u(R)=1/2 \), find \( u(S) \).

Answer: 1


Question: Evaluate the surface integral \( \iint_S \vec{A} \cdot \hat{n} \, dS \), where \( \vec{A} = 18z\hat{i} - 12\hat{j} + 3y\hat{k} \) and \( S \) is the part of the plane \( 2x + 3y + 6z = 12 \) located in the first octant.

Answer: 24


Question: In the projective plane $\mathbb{P}_2(\mathbb{K})$ with $\mathbb{K} \ne \mathbb{Z}_2$, consider the points $A(1,0,0)$, $D(1,-1,0)$, $B(0,1,0)$, and $C(1,1,0)$ on a line. Verify that $D$ is the harmonic conjugate of $C$ with respect to $A$ and $B$ by showing that the cross-ratio $(A,B;C,D) = -1$. Compute the cross-ratio to confirm this property.

Answer: -1


Question: Let $C \in (0, 1)$ and $g(x) = |x|^{-C}$ for $x \in \mathbb{R}$. If $f$ is nonnegative, real-valued, and integrable on $\mathbb{R}$, does it follow that the function $h(x) := \int f(x -t)g(t)\,dt$ is finite almost everywhere?

Answer: Yes


Question:
Calculate the determinant of the following $5 \times 5$ trigonometric matrix (unit in radians):
\[ \mathbf{A} = \begin{pmatrix} \sin(1) & \sin(2) & \sin(3) & \sin(4) & \sin(5) \\ \sin(2) & \sin(3) & \sin(4) & \sin(5) & \sin(6) \\ \sin(3) & \sin(4) & \sin(5) & \sin(6) & \sin(7) \\ \sin(4) & \sin(5) & \sin(6) & \sin(7) & \sin(8) \\ \sin(5) & \sin(6) & \sin(7) & \sin(8) & \sin(9) \end{pmatrix} \]

Answer: 0


Question:
Given vectors \(a\), \(b\), and \(c\) such that \(|a| = |b| = 1\) and \(c = a \times b\), find the maximum value of the expression:
\[
|(a+2b+3c) \cdot ((2a+3b+c) \times (3a+b+2c))|
\]

Answer: 18


Question: Evaluate the line integral \( \int\limits_C e^{x^2-y^2}(\cos(2xy)dx+\sin(2xy)dy) \), where \( C \) is the unit circle with positive orientation.

Answer: 0


Question: Let $(X, Y)$ be a 2D random variable with $E(X) = E(Y) = 3$, variance of both $X$ and $Y$ is $1$, and covariance between $X$ and $Y$ is $1/2$. Calculate $P(|X-Y| > 6)$. Use Chebyshev's inequality for multidimensional random variables if applicable.

Answer: \dfrac{1}{36}


Question: Compute the value of the Legendre symbol \( \left(\frac{59}{619}\right) \).

Answer: -1


Question: Find the supremum of the expression \( \frac{ |x + 2y| + 2|y|}{|x| + |y|} \) for \(x, y \in \mathbb{R}\).

Answer: 4


Question: Determine whether the limit \( \lim_{(x,y) \to (0,0)} \frac{2xy^2\sin^2(y)}{(x^2+y^2)^2} \) exists.

Answer: 0


Question: Let $(e_n)$ be an orthonormal basis of $l^2$, and fix arbitrary complex numbers $(\lambda_n)$. Define the operator $T: l^2 \to l^2$ by \[ T\left(\sum x_n e_n\right) = \sum \lambda_n x_n e_n, \] with domain \[ D(T) = \left\{ \sum x_n e_n : \sum |\lambda_n x_n|^2 < \infty \right\}. \] Determine the adjoint operator $T^*$ and its domain. Is it true that $D(T^*) = D(T)$?

Answer: D(T^*) = D(T)


Question: Suppose $A$ is a self-adjoint operator and $B$ is a compact operator. Is the operator $A+B$ necessarily self-adjoint?

Answer: No


Question: Let \( f \) be a homomorphism from the real numbers under addition to the nonzero complex numbers under multiplication. Determine the image of \( f \).

Answer: \{ e^{zt} \mid t \in \mathbb{R} \}


Question: Determine the degree of the splitting field of the polynomial \((x^2+x+1)(x^3+x+1)\) over \(\mathbb{Q}\).

Answer: 12


Question: Determine the largest interval around \(x=0\) where the function \(f(x) = x^5 - 5x + 3\) is invertible.

Answer: (-1, 1)


Question: Given that \( A \) is a positive definite matrix and \( I \) is the identity matrix, determine if there exists a scalar \( a > 0 \) such that the matrix \( A - aI \) is positive semi-definite.

Answer: Yes


Question: Let $X$ be a locally compact Hausdorff space and $m$ a positive regular Borel probability measure such that $m(Y)$ is 0 or 1 for any Borel set $Y$ of $X$. Does it necessarily follow that $m$ is a Dirac delta measure?

Answer: Yes


Question: Let $A$ be an $n \times n$ complex matrix whose characteristic polynomial has no repeated roots. How many $n \times n$ matrices over $\mathbb{C}$ are there that are both similar to and commute with $A$?

Answer: n!


Question: Let $M$ be an $n \times n$ idempotent real matrix with rank $p$. Given that the trace of $M$ is $p$, determine the trace of $M^TM$ in terms of $p$. Note that $M$ is not necessarily an orthogonal projection, so $M \neq M^T$ in general.

Answer: p


Question: Calculate the line integral \( \int_C Pdx + Qdy \) for the vector field \( F = \left( \frac{x+y}{x^2+y^2}, \frac{-x+y}{x^2+y^2} \right) \), where \( C \) is the line \( y = \frac{x+25}{7} \) that begins at the point \((3,4)\) and ends at \((-4,3)\).

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


Question: Consider a sequence of i.i.d. random variables $Y_i$, $i \in \mathbb{N}$, with $E[Y_1]=0$ and $Var[Y_1]=1$. For $0<s<t$, does the sum \( \frac{1}{\sqrt{n}}\sum_{[ns]+1}^{[nt]}Y_i \) converge in distribution to a normally distributed random variable? If so, determine the variance of this normal distribution.

Answer: t - s


Question: Determine whether there exists an open subset of \( \mathbb{R} \) with Lebesgue measure 0.5 whose closure has Lebesgue measure 1. Provide a justification for your answer.

Answer: Yes


Question:
Let \( f, g \in \mathcal{L}(E) \) where \( E \) is a complex vector space of dimension \( n \), satisfying the relation:
\[ f \circ g - g \circ f = f \]
Determine the value of \( \det(f) \).

Answer: 0


Question: Does there exist a normed vector space \((E, ||\cdot||)\) over \(\mathbb{R}\) such that for every other normed vector space \((F, ||\cdot||_F)\), there is a linear map \(T_F: F \to E\) that is an isometry onto its image, i.e., \(||T_F(x)|| = ||x||_F\) for all \(x \in F\)?

Answer: No


Question: Determine the structure of the quotient ring \( \frac{\mathbb{C}[u]}{u\mathbb{C}[u]} \), where \( \mathbb{C}[u] \) is the algebra of polynomials with the indeterminate \( u \).

Answer: \mathbb{C}


Question: Find a formula that gives the minimum number of moves required to invert a triangular array of coins, where the triangle is arranged such that the first row has 1 coin, the second row has 2 coins, the third row has 3 coins, and so on. For example, for a triangle with 4 rows, the minimum number of moves to invert it so that it points downwards is 3. Express the formula in terms of the number of rows or the total number of coins.

Answer: n - 1


Question: Let $X$ and $Y$ be two independent random variables, each following an exponential distribution with parameter 1, i.e., $X, Y \sim \text{Exp}(1)$. Determine the probability density function (pdf) of the random variable $U = \frac{X}{X+Y}$. Provide your answer in terms of $u$, where $0 < u < 1$.

Answer: 1


Question: Given a $4\times4$ grid, determine the number of unique ways to arrange the numbers $1, 2, 3,$ and $4$. Repetition of numbers is allowed, and not all numbers need to appear. A grid filled entirely with $1$'s is permissible. Rotations of a grid do not count as unique, but mirrored images do. For example, a grid with fifteen $3$'s and a single $2$ in the $(1,1)$ position is considered the same as having the $2$ in the $(1,4)$, $(4,4)$, or $(4,1)$ positions. However, a grid with fifteen $3$'s and a single $2$ in the $(2,1)$ position is distinct from having the $2$ in the $(2,4)$ position. Find the total number of such unique arrangements.

Answer: 1073758336


Question:
Given the sequence \( \{a_n\}_{n=1}^\infty \) defined by:
\[ a_1 = 2 \]
\[ a_{n+1} = 4 - \frac{3}{a_n} \text{ for all } n \in \mathbb{N} \]
Find \( \lim_{n \to \infty} a_n \).

Answer: 3


Question: Find the function \( f(x) \) that satisfies the following conditions: \( f(ax) = \frac{1}{a}f(x) \) and \( \int_{a}^{ab} f(x) \, dx = \int_{1}^{b} f(x) \, dx \).

Answer: \dfrac{C}{x}


Question: Using Lagrange multipliers, find the maximum value of the function \( f(x_1, x_2, \ldots, x_n) = \sqrt[n]{x_1 x_2 \cdots x_n} \) subject to the constraint \( x_1 + x_2 + \cdots + x_n = c \), where \( x_1, x_2, \ldots, x_n \) are positive numbers and \( c \) is a constant.

Answer: \dfrac{c}{n}


Question: Calculate \(5^{3^{1000}} \mod 101\). Use Fermat's Little Theorem to simplify the computation.

Answer: 5