Question:
Find the limit in $L^2$ as $n \rightarrow \infty$ of the expression:
\[ A_n = \sum_{i=1}^n \left(W^a\left(\frac{it}{n}\right) - W^a\left(\frac{(i-1)t}{n}\right)\right)\left(W^b\left(\frac{it}{n}\right) - W^b\left(\frac{(i-1)t}{n}\right)\right) \] where $W^a$ and $W^b$ are Wiener processes.

Answer: 0


Question: Determine the asymptotic complexity of the recurrence relation \( T(n) = 2T(n/2) + T(n/4) + n^2 \). Is the solution \( \Theta(n^2) \)?

Answer: \Theta(n^2)


Question: Evaluate the limit: \[ \lim_{x\to 0^+} \frac{\sqrt{1+2x} - e^x}{x\arctan{x}} \]

Answer: -1


Question: Find the number of elements in the group \((\mathbb{Z}\times\mathbb{Z})/M\), where \(M\) is the subgroup generated by the elements \((2,4)\) and \((4,2)\).

Answer: 12


Question: Find \( \lim_{n \to \infty} \frac{n!}{2^n+1} \).

Answer: \infty


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

Answer: 2


Question: Find the area enclosed by the curve defined by the equation \( x^2 + (y + \sqrt[3]{|x|})^2 = 1 \).

Answer: \pi


Question: Calculate the probability that for randomly chosen pairs \((x, y)\) with \(x, y \in [0, 1]\), the angle \(\theta = \cos^{-1}\left(\frac{-y^2 + y - x^2}{\sqrt{x^2 + (y-1)^2} \sqrt{x^2 + y^2}}\right)\) is less than \(90^\circ\).

Answer: \dfrac{\pi}{8}


Question: Determine whether the number of conjugacy classes of the group \(\operatorname{SL}_2 \mathbb{F}_p\) is \(p+4\) for an odd prime \(p\).

Answer: p + 4


Question: Evaluate \( \lim_{x \to 0} \frac{P_n}{x} \) where \( P_n = a^{P_{n-1}} - 1 \) for all \( n = 2, 3, 4, \ldots \) and \( P_1 = a^x - 1 \), with \( a \in \mathbb{R}^+ \).

Answer: (\ln a)^n


Question: Let $a$ and $b$ be two coprime natural numbers greater than 1. Does there exist a prime $p$ such that $\frac{\operatorname{ord}_{p}(a)}{\operatorname{ord}_{p}(b)} > 1$?

Answer: Yes


Question: Calculate the number of ways to obtain exactly one set of three of a kind when rolling 9 dice.

Answer: 4082400


Question: Let $p(x)$ be a polynomial of degree 2019 with all roots real. Determine the number of real roots of the polynomial $p(x) + p'(x)$. Provide your answer as a single integer.

Answer: 2019


Question: Consider a linear optimization problem where the right-hand side of the constraints depends on a parameter $a$, given by $Ax = b + ae_j$, where $e_j$ is a unit vector in the direction of $x_j$. Suppose that for some $a_0 \in [\alpha, \beta]$, the problem has been solved using the tableau version of the simplex algorithm, and the value of the objective function does not change for all $a \in [\alpha, \beta]$. Assume the initial basis is given by the matrix $B$, and $B^{-1}$ and $c_B$ can be found in the final tableau. Now consider the dual problem. For $a = a_0$, an optimal solution is $y^T = c_B^T B^{-1}$. Does this solution remain optimal for all $a \in [\alpha, \beta]$?

Answer: Yes


Question: Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the equation \( f(xy) = f\left(\frac{x^2 + y^2}{2}\right) + (x-y)^2 \) for all real numbers \( x \) and \( y \).

Answer: f(x) = -2x + C


Question: Let \( \{X_n\}_{n\ge1} \) be a sequence of independent and identically distributed random variables with \( \mathbb{P}(X_n=1)=\mathbb{P}(X_n=-1)=\frac{1}{2} \). Determine the probability \( \mathbb{P}\left(\sum\limits_{n\ge1}\frac{X_n}{n} \text{ converges}\right) \).

Answer: 1


Question: Find the coefficient of \(x^{25}\) in the expansion of \((1 + x^2 + x^4 + x^7)^{10}\).

Answer: 34750


Question: Find the volume of one of the regions bounded by the planes $x=0$, $y=0$, $z=0$, $x=y$, $y=z$, $x=z$, $x=1$, $y=1$, and $z=1$.

Answer: \dfrac{1}{6}


Question:
Consider the initial value problem given by the partial differential equation \( u_{x} + u_{y} = 1 \) with the initial condition \( u(s, s) = 1 \) for \( 0 \leq s \leq 1 \). Determine the number of solutions this problem has:

(a) Two solutions
(b) A unique solution
(c) No solution
(d) Infinitely many solutions

Select the correct option.

Answer: C


Question: Is it possible to cover the entire line segment [0, 1] with a set of segments whose total length is less than 1?

Answer: No


Question: Given two random variables $X$ and $Y$ that are both distributed as $N(0,1)$ with a correlation of $0.5$, find the probability $P(0 < Y < X)$. Express your answer as a probability value.

Answer: \dfrac{1}{6}


Question: Evaluate the line integral \( \int_C y(x^2+y^2)dx-x(x^2+y^2)dy+xydz \) where \( C \) is parametrized by \( r(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t\mathbf{k} \) for \( -\pi\leq t\leq \pi \).

Answer: -2\pi


Question: Consider two Sobolev spaces $H^s(\mathbb{R}^n)$ and $H^t(\mathbb{R}^n)$ where $s > t$. Let $V$ be a linear subspace of $H^s(\mathbb{R}^n)$ such that there exists a constant $C$ for which $\|f\|_{H^s} \leq C\|f\|_{H^t}$ for any $f \in V$. Is $V$ necessarily finite-dimensional?

Answer: No


Question: In an election with 15 voters and 4 candidates, each voter must cast exactly one vote, and candidates can vote for themselves. Determine the number of ways the votes can be cast such that at least one candidate receives no votes. Use the principle of inclusion-exclusion to find the solution.

Answer: 57199024


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

Answer: \dfrac{\pi}{e}


Question:
Find all positive real solutions to the system of equations:
\[\begin{cases}
x_1 + x_2 + \cdots + x_{1994} = 1994 \\
x_1^4 + x_2^4 + \cdots + x_{1994}^4 = x_1^3 + x_2^3 + \cdots + x_{1994}^3
\end{cases}\]

Answer: 1


Question: Find the coefficient of $x^{10}$ in the generating function $$\frac{1+x^3}{1-2x+x^3-x^4}.$$

Answer: 631


Question: Evaluate the integral: \[ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx \]

Answer: \dfrac{\pi}{4}


Question: Find the vector \(a \in \mathbb{R}^p\) that maximizes the expression \(\frac{a'Ca}{a'Ba}\), where \(C\) and \(B\) are \(p \times p\) symmetric matrices.

Answer: a


Question:
Identify which of the following subspaces is not dense in the Hilbert space $l^2$ of square summable sequences, where $e_k$ denotes the coordinate vector with 1 in the $k$th position and 0 elsewhere:

1. $\operatorname{span}\{e_{1}-e_{2},e_{2}-e_{3},e_{3}-e_{4}, \ldots\}$
2. $\operatorname{span}\{2e_{1}-e_{2},2e_{2}-e_{3},2e_{3}-e_{4},\ldots\}$
3. $\operatorname{span}\{e_{1}-2e_{2},e_{2}-2e_{3},e_{3}-2e_{4},\ldots\}$
4. $\operatorname{span}\{e_{2},e_{3},\ldots\}$

Provide a justification for your answer.

Answer: 3


Question:
Let \( f:[0,10] \rightarrow [10,30] \) be a continuous and twice differentiable function such that \( f(0)=10 \) and \( f(10) =20 \). Suppose \( |f'(x)| \leq 1 \) for all \( x \in [0,10] \). Determine the value of \( f''(5) \).

(a) 0
(b) \( \frac{1}{2} \)
(c) 1
(d) Can't be determined

Answer: A


Question: At the corners of an equilateral triangle, three mice are positioned. At time $t = 0$, each mouse begins crawling directly towards the next mouse: Mouse 1 towards Mouse 2, Mouse 2 towards Mouse 3, and Mouse 3 towards Mouse 1. The speed of each mouse is proportional to the distance to the mouse it is chasing. Describe the trajectories of the mice by setting up and solving a system of differential equations. What happens to the positions of the mice as $t \to +\infty$? Use complex numbers to represent the positions of the mice, and choose an appropriate origin for the coordinate system.

Answer: 0


Question:
Suppose we have a periodic function $f$ of period $2\pi$ with Fourier coefficients $a_n$ and $b_n$ defined as follows:
\[a_n:=\frac{1}{\pi}\int_0^{2\pi}f(x)\cos(nx)\,\mathrm{d}x,\quad b_n:=\frac{1}{\pi}\int_0^{2\pi}f(x)\sin(nx)\,\mathrm{d}x.\]
The function $f(x)$ can be expressed as:
\[f(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty}\left[a_n\cos(nx)+b_n\sin(nx)\right].\]
It is known that if $f\in C^\infty(\mathbb{R})$, then:
\[\lim_{n\to\infty}n^{2k}(a_n^2+b_n^2)=0,\:\forall k>0.\]
Is the converse also true?

Answer: Yes


Question: Let $\mathbb{\hat{Z}}$ be an algebraic extension of $\mathbb{Z}$. Suppose $H$ is a finite index subgroup of $SL_n(\mathbb{\hat{Z}})$. Determine whether $H \cap SL_n(\mathbb{Z})$ has finite index in $SL_n(\mathbb{Z})$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the limit: \[ \lim_{n \to \infty} \frac{1}{n^n} \int_0^n x^n \arctan(x^{1/n}) \, dx. \]

Answer: \dfrac{\pi}{4}


Question: An open can of oil is accidentally dropped into a lake, causing the oil to spread over the surface as a circular disc of uniform thickness. The radius of the disc increases at a steady rate of 10 cm/sec. At the moment when the radius is 1 m, the thickness of the oil slick is decreasing at a rate of 4 mm/sec. Given that the volume of oil remains constant, how fast is the thickness of the oil slick decreasing when the radius is 2 m?

Answer: 0.5


Question: Find all entire functions \( f \) such that \( f(f(z)) = f'(z) \) for all complex numbers \( z \).

Answer: 0


Question: Determine the region in the complex plane where the series \( \sum_{n=0}^{\infty} \frac{z^2}{(1+z^2)^n} \) is uniformly convergent.

Answer: \{ z \in \mathbb{C} \mid |1 + z^2| > 1 \}


Question: Evaluate the limit \( \lim_{n\rightarrow\infty}\int_0^2 f_n(x) \, dx \), where the sequence of functions \( f_n: [0,2] \rightarrow \mathbb{R} \) is defined by \( f_n(0) = 0 \) and \( f_n(x) = \frac{\sin(x^n)}{x^n} \) for all other \( x \).

Answer: 1


Question:
Determine the range of real roots for the equation \(e^{x} - ax - b = 0\). Choose the correct option based on the conditions for \(a\) and \(b\):

(A) One real root if \(a \le 0\).
(B) One real root if \(b > 0, a \le 0\).
(C) Two real roots if \(a > 0, a \log_e a \ge a - b\).
(D) No real root if \(a > 0, a \log_e a < a - b\).

Answer: D


Question: Given segments $a$ and $b$, construct a segment $x$ such that $x = \sqrt{a^2 + ab + b^2}$. Provide a geometric construction for this segment.

Answer: x


Question: Determine whether there exist matrices $A, B \in \mathcal{M}_n (\mathbb{K})$ such that $\rho(A) + \rho(B) < \rho(A+B)$, where $\rho(A)$ and $\rho(B)$ are the absolute values of the largest eigenvalues of $A$ and $B$, respectively.

Answer: Yes


Question: Prove that a 20-degree angle is not constructible using a straightedge and compass.

Answer: 20^\circ


Question: Let $S$ be the sum of the digits of $2014^{2014}$. Let $T$ be the sum of the digits of $S$. Let $U$ be the sum of the digits of $T$. Find the value of $U$. \( \boxed{} \)

Answer: 7


Question: Consider the statement: Given a function $f: A \rightarrow \mathbb{R}$ such that \( \forall \delta>0\ (\exists \epsilon>0\ (|x-y|<\delta \implies |f(x)-f(y)|<\epsilon) ) \). Does this statement hold for uniformly continuous functions defined on $A$? Provide a justification for your answer.

Answer: Yes


Question: Determine the number of terminal zeros in the result of the subtraction \(100^{100} - 100!\).

Answer: 24


Question: Find the function \( f(x, y) \) that solves the partial differential equation \( x\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}=0 \), given that \( f = e^{-2y} \) when \( x = 1 \). Use the substitutions \( u = xe^{y} \) and \( v = xe^{-y} \).

Answer: x^2 e^{-2y}


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

Answer: 2


Question: Calculate the remainder when \(12^{34^{56^{78}}}\) is divided by 90.

Answer: 36


Question:
Two subgroups H and K of a group G have orders 12 and 30, respectively. Which of the following could NOT be the order of the subgroup of G generated by H and K?

A. 30
B. 60
C. 120
D. 360
E. Countable infinity

Answer: A


Question: Determine the minimum number of weighings required to order 5 coins of different weights using a balance scale.

Answer: 7


Question: How many permutations of the word "arrangement" have exactly two of the following consecutive letter pairs: aa, ee, rr, nn?

Answer: 332640


Question: Compute the integral \( \int_{0}^{2\pi}(1+2\cos(t))^n\cos(nt)\,\mathrm{d}t \) using the substitution \( z=e^{it} \), where \( n \) is a positive integer.

Answer: 2\pi


Question: Locate the poles and find the residues of the function \( \frac{e^{5z}}{(z+i\pi)^3} \).

Answer: -\dfrac{25}{2}


Question: Compute the limit: $$ \lim _{n \to \infty} \frac{1}{n}\sum_{k=0}^{n-1}\left[\frac{k+1}{n}-\frac{1}{2}\right]\log(2n-2k-1) .$$

Answer: -\dfrac{1}{4}


Question: Find the value of $5^{2003} \pmod{1001}$ given the following congruences: \[ 5^{2003} \equiv 3 \pmod{7} \] \[ 5^{2003} \equiv 4 \pmod{11} \] \[ 5^{2003} \equiv 8 \pmod{13} \] Use the Chinese Remainder Theorem to solve the problem.

Answer: 983


Question: Calculate the limit: \[ \lim_{n \to \infty} \frac{\ln(2^{\frac{1}{n}}) - \ln(n^2)}{1 + \frac{1}{2} + \cdots + \frac{1}{n}}. \]

Answer: -2


Question:
Let $f$ be a function satisfying $f(x+y) = f(x) f(y)$ with $f(0) = 1$, and let $g$ be a function that satisfies $f(x) + g(x) = x^2$. Determine the value of the integral $\int_0^1 f(x) g(x) \, dx$. Choose the correct answer from the following options:

(a) $e - \frac{e^2}{2} - \frac{5}{2}$

(b) $e + \frac{e^2}{2} - \frac{3}{2}$

(c) $e - \frac{e^2}{2} - \frac{3}{2}$

(d) $e + \frac{e^2}{2} + \frac{5}{2}$

Answer: C


Question: Let $V$ be an infinite-dimensional subspace of $\ell^2$ such that there does not exist a sequence $v \in V$ with only finitely many nonzero elements. Let $W = \overline{V}$ be the closure of $V$. Can the space $W$ contain a sequence with only finitely many nonzero elements?

Answer: Yes


Question:
Calculate the limit:
\[ \lim_{n\to\infty} \left(\sum_{k=0}^n \frac{{(1+k)}^{k}-{k}^{k}}{k!}\right)^{1/n} \]

Answer: e


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

Answer: 1 - e^{-t}


Question: Find the mass of the region $E$ bounded by the spheres $x^2+y^2+z^2 = 4z$ and $x^2+y^2+z^2 = 2z$, given that the density of the region is $\rho(x,y,z) = (x^2+y^2+z^2)^{3/2}$.

Answer: 192\pi


Question: Find the equation of the hyperplane in \(\mathbb{R}^4\) that passes through the points \(P_1 = (1, 2, 1, 4)\), \(P_2 = (0, 1, 2, 1)\), \(P_3 = (-2, 1, 1, 0)\), and \(P_4 = (1, 1, 1, 0)\).

Answer: 4y + z - w = 5


Question: Determine whether there are any solutions to the functional equation \( f(f(x)-x) = f(f(x)) \) other than the constant solution \( f(x) = k \), where \( k \in \mathbb{R} \).

Answer: f(x) = k


Question: Two players take turns drawing one number randomly from the set \( \{1, 2, 3, 4\} \), with each number having a probability of \( \frac{1}{4} \) of being drawn. The winner is the player who draws a number that makes the sum of all numbers drawn by both players divisible by 3. What is the probability that the first player will be the winner?

Answer: \dfrac{13}{23}


Question: Let \( f \) be a function that has derivatives of all orders for all real numbers, and let \( P_3(x) \) be the third-degree Taylor polynomial for \( f \) about \( x=0 \). The Taylor series for \( f \) about \( x=0 \) converges at \( x=1 \), and \( |f^{(n)}(x)|\leq\frac{n}{n+1} \), for \( 1\leq n\leq4 \) and all values of \( x \). What is the smallest value of \( k \) for which the Lagrange error bound guarantees that \( |f(1)-P_3(1)|\leq k \)?

Answer: \dfrac{1}{30}


Question: Evaluate the limit: $$\lim_{x\rightarrow0}\frac{x^{672}-2\cos\left(x^{1007}\right)-\sin(x^{672})+2}{x^{2014}}$$

Answer: 1


Question: Let $F$ be a self-adjoint linear operator on an inner product space $V$. If $F$ satisfies the condition \( P F P^T = F \) for any orthogonal linear transformation \( P: V \to V \), prove or disprove that \( F = cI \) for some constant \( c \in \mathbb{R} \).

Answer: F = cI


Question: Find the smallest natural number \( m \) that is the product of three distinct prime numbers and for which the equation \( x^2 + 1 \equiv 0 \pmod{m} \) has exactly 8 solutions modulo \( m \).

Answer: 1105


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

Answer: e^{-4}


Question: Arrange the numbers 21, 31, 41, 51, 61, 71, and 81 such that the sum of every four consecutive numbers is divisible by 3. How many such arrangements are possible?

Answer: 144


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

Answer: 6


Question:
Given an orthonormal positively oriented basis \(B=(\mathbf{i}, \mathbf{j}, \mathbf{k})\) and a real number \(h\), let
\[\mathbf{v} = \mathbf{i}-h\mathbf{k}\]
\[\mathbf{w} = 2\mathbf{i}-\mathbf{j}+\mathbf{k}\]
\[\mathbf{z}= h\mathbf{i}+\mathbf{j}-\mathbf{k}\]
Determine the values of \(h\) for which \((\mathbf{v},\mathbf{w},\mathbf{z})\) is a positively oriented basis.

Answer: (-2, 0)


Question: Given a Cauchy sequence of matrices $C_i \in \mathbb{R}^{p \times q}$ such that $\lim_{n\rightarrow \infty} \| C_{n+1} - C_{n} \| = 0$ for any norm, and knowing that $\|C\|_{1,1} \leq t$ for a fixed $t$, can you conclude that the sequence $C_i$ converges?

Answer: Yes


Question: If \( \{a_1, \dots, a_{\phi(n)}\} \) is a reduced residue system modulo \( n \), what is \( a_1 \cdots a_{\phi(n)} \) congruent to modulo \( n \)? Prove that \( a_1 \cdots a_{\phi(n)} \equiv (-1)^{N/2} \pmod{n} \), where \( N \) is the number of solutions to \( x^2 \equiv 1 \pmod{n} \).

Answer: (-1)^{N/2}


Question: Given the differential equation \( y'(x) + y(x)g'(x) = g(x)g'(x) \) with the initial condition \( y(0) = 0 \), where \( g(x) \) is a non-constant differentiable function on \( \mathbb{R} \) with \( g(0) = g(2) = 0 \), find the value of \( y(2) \).

Answer: 0


Question: Let $(X,Y)$ be a random variable with the joint probability density function (PDF): \[ f(x,y) = \begin{cases} e^{-x-y} & \text{if } x,y \ge 0 \\ 0 & \text{otherwise} \end{cases} \] Calculate the conditional expectation $E(X+Y|X<Y)$. Express your answer in terms of integrals if necessary.

Answer: 2


Question: Find \( \lim\limits_{n\to\infty}\left(\frac{a_1}{a_0S_1}+\frac{a_2}{S_1S_2}+...+\frac{a_n}{S_{n-1}S_n}\right) \) where \( a_n=2015^n \) and \( S_n=\sum\limits_{k=0}^{n}a_k \). Express \( S_n \) as the geometric sum \( S_n=\frac{2015^{n+1}-1}{2014} \).

Answer: 1


Question:
Consider the function \( \lim_{x\to c^-}\{\ln x\} \) and \( \lim_{x\to c^+}\{\ln x\} \), where \( \{\cdot\} \) denotes the fractional part function. If both limits exist finitely but are not equal, determine the nature of the values that \( c \) can take. Choose the correct option:

(a) \( c \) can take only rational values
(b) \( c \) can take only irrational values
(c) \( c \) can take infinite values in which only one is irrational
(d) \( c \) can take infinite values in which only one is rational

Answer: D


Question: Find the mean value of the discrete signal \(X(n) = A\cos(\omega_0 n + \Phi)\), where the probability density functions for \(A\) and \(\Phi\) are given by \( \rho_A(a) = \frac{1}{4} \) for \(0 \le a \le 4\) and \( \rho_\Phi(\phi) = \frac{1}{2\pi} \) for \(0 \le \phi \le 2\pi\).

Answer: 0


Question:
Given the sequence $(u_n)$ defined recursively by:
\[\begin{cases}
u_1=\frac{1}{1+x}\\ \frac{1}{u_{n+1}}=\frac{1}{u^2_{n}}-\frac{1}{u_n}+1, &\forall n\geq 1\end{cases}\]
where $x\in\mathbb{R}, x\neq 1$, let $S_n = u_1 + u_2 + \ldots + u_n$ and $P_n = u_1 u_2 \ldots u_n$. Determine the value of the expression $xS_n + P_n$. \(x \in \mathbb{R}, x \neq 1\).

Answer: 1


Question:
Evaluate the integral \( \int_{0}^{\pi/2} \frac{\sin(nx)}{\sin(x)}\,dx \). For every odd \( n \geq 1 \), the answer is \( \frac{\pi}{2} \). For every even \( n \geq 2 \), choose the correct answer from the following options:

A) \( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots + (-1)^{n/2+1}\frac{1}{n-1} \)

B) \( 3\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots + (-1)^{n/2+1}\frac{1}{n-1} \right)\)

C) \( 2\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots + (-1)^{n/2+1}\frac{1}{n-1} \right)\)

D) \( \frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots + (-1)^{n+1}\frac{1}{n-1} \right)\)

E) \( 1-\frac{1}{3}-\frac{1}{5}-\frac{1}{7}-\cdots - \frac{1}{n-1} \)

Does the recurrence \( (n-1)(I_{n}-I_{n-2})=2\sin(n-1)x \) help in solving the problem?

Answer: C


Question: A lane runs perpendicular to a road that is 64 feet wide. If a pole 125 feet long can be carried horizontally from the road into the lane, what is the minimum width of the lane?

Answer: 27


Question: Determine the cardinality of the set of functions from the natural numbers \( \mathbb{N} \) to the real numbers \( \mathbb{R} \), assuming the generalized continuum hypothesis (GCH).

Answer: \aleph_1


Question: Determine whether a harmonic function on an open connected set is analytic. Provide a justification for your answer.

Answer: Yes


Question: Determine the order of the automorphism group of \( \mathbb{Z}_5 \times \mathbb{Z}_5 \).

Answer: 480


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

Answer: -\infty


Question: Compute the covariance: \[ \text{cov}[W(1), \int_0^1 rW(r) \, dr] \] where \( W(t) \) is a standard Brownian motion.

Answer: \dfrac{1}{3}


Question: Determine the condition on the matrix $C$ such that the linear map $\varphi_C : V \to V$, defined by $A \mapsto CA$, is selfadjoint with respect to the scalar product $\langle A, B \rangle = \operatorname{tr}(A^t B)$, where $V = M_{n\times n}(\mathbb{R})$. Provide your answer in terms of properties of $C$.

Answer: C = C^\top


Question: Determine if the kernel function \( k(x,y) = \prod_{i=1}^n (1 + x_i y_i) \) can be expressed as a dot product \( \langle \phi(x), \phi(y) \rangle \), where \( \phi(x) \) is a feature mapping that depends only on \( x \).

Answer: Yes


Question: Evaluate the line integral of the vector field \( \mathbf{F}(x,y) = x^2y^2 \mathbf{\hat{i}} + x^3y \mathbf{\hat{j}} \) over the square with vertices (0,0), (1,0), (1,1), (0,1) in the counterclockwise direction. Calculate the integral by hand.

Answer: \dfrac{1}{6}


Question: Does there exist an infinite magma such that for any positive integer $n \geq 2$, and for all distinct elements $x_1, x_2, \ldots, x_n$, all products formed by using each of $x_1, x_2, \ldots, x_n$ exactly once are distinct?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty} \sum_{r=0}^{\infty} \frac{^n\text{C}_{r}}{n^r(r+4)} \]

Answer: 6 - 2e


Question: Evaluate the limit: \[ \lim_{x\to0}\left(\frac{1+\tan x}{1+\sin x}\right)^{\frac{1}{\sin x}} \] using the rule \( \lim\limits_{x\to \infty}\left(1+\frac{1}{x}\right)^x= e \).

Answer: 1


Question: Determine if there exists a lower bound for \( \sigma^2 = \sigma^2(p) \) such that for all \( t \in \mathbb{R} \), the inequality \( pe^t + qe^{-t} \leq \exp\left(\frac{\sigma^2}{2}t^2\right) \) holds, given that \( 0 < p, q < 1 \) and \( p + q = 1 \).

Answer: 1


Question:
Given the function \( f(x) = x^3 - 6x^2 + 9x - 5 \), evaluate the following limit:
\[
\lim_{h \to 0} \frac{[f'(1+2h) + f'(3-3h)]}{2h}
\]

Answer: -15


Question:
Evaluate the limit of the integral:
\[
I = \lim_{\epsilon\to 0} \frac{1}{2\pi\epsilon}\int_{-\infty}^\infty\int_{-\infty}^\infty f(x,y)\frac{y}{\epsilon}\exp{\frac{-(x^2+y^2)}{2\epsilon}}dxdy
\]
where \(f\) and \(\nabla f\) are continuous and bounded.

Answer: \dfrac{\partial f}{\partial y}(0,0)


Question: Show that the function \( f(x) = x^{3}\cos{\left(\frac{1}{x^{2}}\right)} \) for \( x \neq 0 \) and \( f(0) = 0 \) is differentiable at \( x = 0 \) using the definition of the derivative. Find \( f'(0) \).

Answer: 0


Question:
Let \( A = [a_{ij}] \in \text{Mat}_n(\mathbb{R}) \) be a matrix defined by:
\[
a_{ij} = \begin{cases}
  n+1, & \text{if } i = j \\
  1, & \text{if } i \neq j
\end{cases}
\]
Calculate \( \det(A) \).

Answer: 2n^n


Question: Let \( R = \mathbb{Z}_7[x] \) and \( I = \langle x^2 - 6 \rangle \) be a maximal ideal. Determine if \( x^2 - 1 + I \) is invertible in \( \frac{R}{I} \) to \( 4 + I \).

Answer: No


Question: Evaluate the limit: \[ \lim_{z\to \infty} \ln \left( \frac{z+\sqrt{z^2+p^2}}{-z+\sqrt{z^2+p^2}} \right) \]

Answer: \infty


Question: Find the intersection \( \bigcap_{k \in \mathbb{Z}^+} B_k \) where \( B_k = \left[ \frac{3}{k}, \frac{5k+2}{k} \right) \cup \{ 10+k \} \).

Answer: [3,5]


Question: Find the inverse of the polynomial \(x^2 + x + 1\) in the field \(\mathbb{Z}_2[x]/\langle x^3 + x + 1 \rangle\).

Answer: x^2


Question: Find the smallest positive integer \( n \) such that \( x^n \equiv 1 \pmod{101} \) for every integer \( x \) between 2 and 40.

Answer: 100


Question:
Given a random vector $(X,Y,Z)$ with the density function:
\[ f_{X,Y,Z}(x,y,z)=\frac{1}{\sqrt{56\pi^3}} e^{-\frac{1}{2}\left[\frac{(x+1)^2}{7}+y^2+5(z-2)^2+4y(z-2)\right]} \]
compute the variance of $X \cdot Y$. The covariance matrix is given by:
\[ \begin{bmatrix} 7 & 0 & 0 \\ 0 & 5 & -2 \\ 0 & -2 & 1 \end{bmatrix} \]

Answer: 40


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

Answer: 0


Question: Determine whether the following statement is true or false: If $K$ is a perfect field of characteristic $p>0$, then $K$ is algebraic over the prime field $\mathbb{F}_p$. Provide a justification for your answer.

Answer: False


Question:
Consider a continuous function \( \chi \) defined on \( GL(\mathbb{R}^n) \) by:
\[
\chi(\alpha) := \left(1+\max_{1\leq i,j\leq n}|\alpha_{ij}|\right)^\ell\left(1+\max_{1\leq i,j\leq n}|\alpha^{-1}_{ij}|\right)^m|\det\;\alpha|^{1/p},
\]
where \( \alpha \in GL(\mathbb{R}^n) \) with coefficients \( \alpha=[\alpha_{ij}]_{i,j=1,\ldots,n} \), and \( \alpha^{-1} \) has coefficients \( ( \alpha^{-1}_{ij}) \). Here, \( m, \ell \) are positive integers and \( 1\leq p\leq \infty \).

If \( \alpha \) is an element of a compact subset \( \mathcal{G} \) of \( GL(\mathbb{R}^n) \), defined as:
\[
\mathcal{G} := \left\{ b : |\det\;b|\geq\varepsilon>0,\;|b_{ij}|\leq\frac{1}{t_o} \right\},
\]
is it true that \( \chi \) is bounded for all \( \alpha \in \mathcal{G} \)?

Answer: Yes


Question: Evaluate the contour integral \( \oint_{\gamma} \frac{d z}{z^{2}+4} \), where \( \gamma = |z-2i|=1 \) is a closed loop counterclockwise, using the Cauchy integral formula.

Answer: \dfrac{\pi}{2}


Question: Given a real vector space $V$ and vectors $v_1, \dots, v_{16} \in V$, assume that the equation $\sum\limits_{i=1}^8 a_{2i-1} v_{2i-1} = 0$ has infinitely many solutions, and the equation $\sum\limits_{i=1}^8 a_{2i} v_{2i} = 0$ has a unique solution. Determine the maximum possible dimension of $W = \text{Span}\{v_1, \dots, v_{16}\}$.

Answer: 15


Question: Evaluate the line integral \( \oint_{C} xy^2 \, dx + 2x^2y \, dy \) where \( C \) is the triangle with vertices \((0,0)\), \((2,2)\), and \((2,4)\).

Answer: 12


Question:
Consider the nonautonomous ordinary differential equation (ODE):
\[ \dot{x}(t) = A(t)x(t) \]
where \( A(t) \) is an \( n \times n \) matrix that depends continuously on \( t \) and is positive definite for \( |t| \geq m \) for some \( m \geq 0 \). The matrix norm of \( A(t) \) is bounded by a constant. Assume that \( A(t) \to A^{\pm} \) as \( t \to \pm\infty \) for some positive definite matrices \( A^{\pm} \). Prove or disprove: \( x = 0 \) is the only bounded solution to this ODE.

Answer: x = 0


Question:
Find all positive integers \( n \) such that there exist consecutive positive integers \( a, b, c \) satisfying the equation:
\[ 2018^n = a^4 + b^4 + (b^2 + c^2)^2 \]

Answer: 1


Question: Let $V$ be a finite-dimensional vector space, and let $T \in \mathcal{L}(V)$ be a diagonalizable linear operator with all eigenvalues having absolute value less than one. Given $\phi \in V'$ (the dual space of $V$) and $v \in V$, find $\lim_{n \rightarrow \infty} \phi(T^n v)$. \( \text{Provide your answer in the form of a limit.} \)

Answer: 0


Question: Let \( f \in L^1(\mathbb{R}^3) \) be a function such that \( \oint_{B} \int_{\mathbb{R}^3} f(x+y) \, dx \, dy = 0 \) for any bounded set \( B \) of \( \mathbb{R}^3 \). Prove or disprove that \( \int_{\mathbb{R}^3} f(x) \, dx = 0 \).

Answer: 0


Question: Let $K_1$ and $K_2$ be disjoint convex sets in $\mathbb{C}$. Let $z_1 \in \partial K_1$ and $z_2 \in \partial K_2$ be the points that minimize the distance $\text{dist}(K_1, K_2) = \inf{|x-y|}$ where $x \in K_1$ and $y \in K_2$. Can the separating hyperplane be chosen to be orthogonal to the line connecting $z_1$ and $z_2$?

Answer: Yes


Question: Find the maximum value of \( x_1^2 + x_2^2 + \cdots + x_{2014}^2 \) given that \( x_i \in [-11, 5] \) for \( i = 1, 2, \ldots, 2014 \) and \( x_1 + x_2 + \cdots + x_{2014} = 0 \).

Answer: 110710


Question: Determine whether the set \( \{(x,y) \in [0,1]^2 : 0 \leq y \leq f(x) \} \), where \( f \) is the Cantor Lebesgue function, is a Borel set in \( \mathbb{R}^2 \).

Answer: Yes


Question:
Let $x_0$ be a transcendental number, and consider the sequence defined by $x_{n+1} = \frac{3-x_n}{x_n^2 + 3x_n - 2}$. Determine the limit of the sequence $x_n$. You may find the following expressions helpful:

- $f(x) = \frac{3-x}{x^2 + 3x - 2}$
- $f'(x) = \frac{(x-7)(x+1)}{(x^2 + 3x - 2)^2}$
- $f(x) - x = \frac{-(x-1)(x+1)(x+3)}{x^2 + 3x - 2}$
- $f(x) + 1 = \frac{(x+1)^2}{x^2 + 3x - 2}$

Answer: -1


Question:
If \(\theta_{1},\theta_{2},\theta_{3}\) are three distinct values in the interval \([0,3\pi)\) for which \(\tan \theta =\frac{7}{10}\), find the greatest integer value of the expression:
\[
\tan\frac{\theta_{1}}{3}\tan \frac{\theta_{2}}{3}+\tan\frac{\theta_{2}}{3}\tan \frac{\theta_{3}}{3}+\tan\frac{\theta_{3}}{3}\tan \frac{\theta_{1}}{3}
\]

Answer: -3


Question: Given that $x^2 y + y^2 z + z^2 x = 2186$ and $xy^2 + yz^2 + zx^2 = 2188$, where $x, y, z$ are integers, find the value of $x^2 + y^2 + z^2$.

Answer: 245


Question:
Let $f$ be the function defined on the real line by
\[ f(x) = \begin{cases} 2x^2, & \text{if $x \in \mathbb{Q}$} \\ -3x^2, & \text{if $x \notin \mathbb{Q}$} \end{cases} \]
Which of the following is true?
A. $f$ is not continuous and not differentiable everywhere
B. $f$ is continuous only at $x=0$ and not differentiable everywhere
C. $f$ is continuous and differentiable only at $x=0$
D. $f$ is continuous and not differentiable everywhere
E. $f$ is continuously differentiable everywhere

Answer: C


Question: Evaluate the path integral \( \int_{C(0,1)} \sqrt{z+2} \, dz \), where \( C(0,1) \) is the circle centered at the origin with radius 1.

Answer: 0


Question: Calculate the value of \( \lim_{m \to \infty} \lim_{n \to \infty}(1+\cos^{2m}(n!\pi x)) \) when \( x \in \mathbb{Q} \).

Answer: 2


Question: Find the prime decomposition of 2 in the cyclotomic field \( \mathbb{Q} ( \xi_5 ) \).

Answer: (2)


Question: Find the sum of the series \( \sum_{x=5}^{\infty} \frac{C}{(x+1)(x+2)} \), where \( C \) is a constant.

Answer: \dfrac{C}{6}


Question: Evaluate the limit: \[ \lim\limits_{x \to 0} \int_{0}^{x}\frac{\cos t^3}{t+x}dt \]

Answer: \ln 2


Question: Prove that the smallest integer greater than \((3+\sqrt{5})^n\) is divisible by \(2^n\).

Answer: 2^n


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

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


Question: Given the modular exponentiation equations \(m^5 \equiv 7 \pmod{7769}\) and \(m^7 \equiv 252 \pmod{7769}\), find the value of \(m\).

Answer: 6


Question: Find a natural number $0 < n < 30,000$ such that $\sqrt[3]{5n} + \sqrt{10n}$ is a rational number.

Answer: 25000


Question: Provide an example of a continuous function \( f: \mathbb{R} \to [0, \infty) \) that is unbounded, but such that \( f^2 \) is uniformly continuous on \( \mathbb{R} \).

Answer: f(x) = \sqrt{|x|}


Question: Calculate the expected stopping time for a symmetric random walk to reach either -3 or 5. That is, find \( E[S] \) for \( S = \min\{ t : S_t = -3 \text{ or } S_t = 5 \} \).

Answer: 15


Question:
Find the limit of the expected average gain of citizens as the number of citizens, $n$, goes to infinity, given that the gain from a policy is uniformly distributed on $[-1, 1]$ and the policy is implemented if more than half of the citizens vote for it. The expected gain is given by the formula:
\[ \sum_{k=\lceil \frac{n}{2} \rceil }^n \frac{C_n^k (2k-n)}{n 2^{n+1}} \]

Answer: 0


Question: Evaluate the double integral: \[ \int_0^2\int_0^{2-x}(x+y)^2 e^{\frac{2y}{x+y}}\,dy\,dx. \]

Answer: 2(e^2 - 1)


Question: Given the symmetry results of positive solutions of second order elliptic equations as presented in the paper by Gidas, Ni, and Nirenberg, does Theorem 1, which asserts that the solution must be radial and decreasing in $r$, imply that the solution $u=u(r)$ cannot take an infinite value at $r=0$? In other words, must the solution $u=u(r)$ be bounded for $0 \leq r \leq R$?

Answer: Yes


Question: Let $G$ be a group of prime order $p$. Prove that if $G$ can be embedded into the symmetric group $S_m$, then $m \ge p$. In other words, a group of prime order cannot be embedded into symmetric groups of degree smaller than its order.

Answer: m \ge p


Question: A normal to the hyperbola \( \frac{x^2}{4} - \frac{y^2}{1} = 1 \) has equal intercepts on the positive x and y axes. If this normal also touches the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), find the value of \( a^2 + b^2 \).

Answer: \dfrac{25}{3}


Question: Find the volume of the solid bounded above by the plane \( z = y \) and below by the paraboloid \( z = x^2 + y^2 \) using cylindrical coordinates.

Answer: \dfrac{\pi}{32}


Question: Let $U_1, U_2, W$ be subspaces of a vector space $V$ such that $V = U_1 \oplus W = U_2 \oplus W$. Prove that $U_1 = U_2$. Provide a detailed proof to support your conclusion.

Answer: U_1 = U_2


Question: For a two-by-two matrix $A$, determine the flux of the linear vector field ${\bf F}(x,y)=A\begin{bmatrix}x\\ y\end{bmatrix}$ through a unit circle centered at the origin in $\mathbb{R}^2$, expressed in terms of the entries of $A$. Assume the unit circle is parameterized by $x = \cos t$ and $y = \sin t$ for $0 \leq t < 2\pi$. Provide your answer in terms of the matrix entries.

Answer: \pi(a + d)


Question: Determine the value of \( \gamma(0) \) if \( \gamma(t) = e^{2\pi it \sin(1/t)} \) for \( t \in (0, 2\pi] \).

Answer: 1


Question: Let $S\subset \mathbb{R}^d$ and define $v(S)$ as the volume of the set $(S+\mathbb{Z}^d)\cap [0,1]^d$, where $[0,1]^d$ is the unit cube. Consider a linear transformation $T: \mathbb{R}^d \rightarrow \mathbb{R}^d$ with determinant one and all eigenvalues equal to one. Does $v(S)$ equal $v(T(S))$?

Answer: No


Question: Let \( z \) be a complex number satisfying \( \frac{1}{2} \leq |z| \leq 4 \). Determine the sum of the maximum and minimum values of \( \left| z + \frac{1}{z} \right| \).

Answer: \dfrac{17}{4}


Question: Determine the degree of the splitting field of the polynomial \((x^{15}-1)(x^{12}-1)\) over the finite field \(\mathbb{F}_7\).

Answer: 4


Question: Evaluate the integral \( \int_\gamma \frac{e^{iz}}{z^2} \, dz \), where \( \gamma: [0, 2\pi] \to \mathbb{C} \) is defined by \( \gamma(t) = e^{it} \).

Answer: -2\pi


Question: Find the area of a wall where the base is part of the circle of radius 1 centered at the origin, lying in the first quadrant. The height of the wall is given by the function \( f(x, y) = y + 4x \). Use line integrals to determine the area.

Answer: 5


Question: Determine the $2$-adic valuation of $x^2 - 3$ in the field extension $\mathbb{Q}[x]/(x^3 - 3)$, given that $x$ has $2$-adic valuation $0$.

Answer: 1


Question: Solve the equation for $x$: $$(x^3-4)^3=(\sqrt[3]{(x^2+4)^2}+4)^2$$

Answer: 2


Question: Determine the value of \(\alpha\) for which the series \(\sum \frac{\sqrt{a_{n}}}{n^{\alpha}}\) converges, given that \(\sum a_{n}\) converges and \(a_{n}\) are positive.

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


Question: Determine the cardinality of the Mandelbrot set. Is it countable or does it have the cardinality of the continuum, $2^{\aleph_0}$?

Answer: 2^{\aleph_0}


Question: Let $f_1,f_2\in\mathbb{Z}[X]$ be two different irreducible monic polynomials. Determine if it is true that for almost all primes $p$ (that is, for all but a finite number of primes), the polynomials $\bar{f}_1$ and $\bar{f}_2$ have no common roots in $\mathbb{F}_p$. Here, $\bar{f}$ denotes the reduction modulo $p$. Answer with "True" or "False".

Answer: A


Question:
Consider the sequence of prime numbers \((P_k)_{k\geq1} = 2, 3, 5, 7, 11, 13, \ldots\). Determine whether the following limits exist:
\[ \lim_{n\to\infty}\frac{P_n}{P_{n+1}} \quad \text{and} \quad \lim_{n\to\infty}\frac{P_{n+1}}{P_n}. \]

Answer: 1


Question: Let $f$ be an analytic function on $\mathbb{C}$ such that $f(e^{it})=e^{it}$ for all $t \in \mathbb{R}$. Determine $f(z)$ for all $z \in \mathbb{C}$. \( \text{Provide your answer in the form of a function.} \)

Answer: z


Question: Determine the values of $x > 0$ for which the series \( \sum_{n=1}^{\infty} x^{1+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}+...+\frac{1}{\sqrt n}} \) is convergent.

Answer: (0, 1)


Question: Evaluate the integral \( I = \int_0^{+\infty} u^3 e^{-u(1+i)} \, du \), where \( i^2 = -1 \).

Answer: -\dfrac{3}{2}


Question: Let $f(z)$ be an entire function such that $f(z) = f(z+1)$ for all $z$. If there exists a constant $c \in (0,1)$ such that $|f(z)| \le e^{c|z|}$ for all $z$, prove that $f(z) = 0$ for all $z$.

Answer: 0


Question:
Given that $a, b, c, d$ are solutions to the equation $x^4 - bx - 3 = 0$, determine the form of the equation whose solutions are $\frac{a+b+c}{d^2}, \frac{a+b+d}{c^2}, \frac{a+c+d}{b^2}, \frac{b+c+d}{a^2}$. Choose the correct form from the following options:

A) $x^4 - 3bx + 1 = 0$
B) $3x^4 - bx^3 - 1 = 0$
C) $3x^4 + bx^3 - 1 = 0$
D) $3x^4 - bx + 1 = 0$
E) $3x^4 + bx + 1 = 0$
F) There's no right answer.

Answer: B


Question:
Evaluate the limit as \( n \to \infty \) of the expression:
\[
\left[\left(\frac{2\cdot 1+1}{2\cdot 1+3}\right)^{1^2}\cdot\left(\frac{2\cdot 2+1}{2\cdot 2+3}\right)^{2^2}\cdot \ldots \cdot\left(\frac{2\cdot n+1}{2\cdot n+3}\right)^{n^2}\right]^{1/n}
\]

Answer: 0


Question:
Consider the polynomial equation \(x^4 + ax^3 + bx^2 + cx + 1 = 0\), where \(a, b, c\) are real numbers. Given that the equation has no real roots and at least one root has a modulus of one, determine the correct relationship among the coefficients:

(a) \(b = c\)
(b) \(a = c\)
(c) \(a = b\)
(d) None of the above

Select the correct option.

Answer: B


Question: Determine whether the following limit is true: \( \lim_{a\to -\infty}\int_{-\infty}^a f(x)\, dx=0 \).

Answer: 0


Question: Let \( X \) and \( Y \) be two continuous random variables with the joint probability density function (PDF) given by \( f(x,y) = c e^{-x} \) for \( x > 0 \) and \( |y| < x \). Find the value of the constant \( c \) and determine the joint cumulative distribution function (CDF) of \( X \) and \( Y \).

Answer: \dfrac{1}{2}


Question: Let $a, b, c \in \mathbb{R}$ such that $\cos(a-b) + \cos(b-c) + \cos(c-a) = -\frac{3}{2}$. Find the value of $\sin(a+b) + \sin(b+c) + \sin(c+a)$. \(\text{Express your answer in terms of known constants.}\)

Answer: 0


Question:
Find the seven-digit phone number \( \overline{a_0a_1a_2a_3a_4a_5a_6} \) such that moving the last four digits to the front results in a number that is one more than twice the original number. Formally, solve for the phone number satisfying:
\[ 2 \cdot \overline{a_0a_1a_2a_3a_4a_5a_6} + 1 = \overline{a_3a_4a_5a_6a_0a_1a_2} \] where each digit \( a_i \) is between 0 and 9.

Answer: 4358717


Question: Calculate \( \lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=1}^{n}k\sin\left(\frac{a}{k}\right) \).

Answer: a


Question: Find the annual yield on a 4-year zero-coupon bond given the following information: You have two 4-year annual-coupon bonds, each with a face value and redemption value of 8000. The first bond has a coupon rate of 7% and a price of 7908.57, while the second bond has a coupon rate of 8% and a price of 8101.55.

Answer: 5.10\%


Question: Find the residue of the function \( f(z) = e^{-\frac{3}{z^2}} \) at \( z = 0 \).

Answer: 0


Question: Evaluate the integral \( \int_\Gamma \frac{3z-2}{z^2-z} \, dz \), where \( \Gamma \) is a simple, closed, positively oriented contour enclosing the points 0 and 1.

Answer: 6\pi i


Question: Find the Cauchy principal value of the integral \( \int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx \).

Answer: \pi


Question:
Determine the radius of convergence of the power series \(\sum_{n=0}^\infty a_nz^{n^2}\) where \(a_0 = 1\) and \(a_n=3^{-n}a_{n-1}\) for \(n\in \mathbb{N}\). Choose the correct answer from the following options:

(A) \(0\), (B) \(\sqrt{3}\), (C) \(3\), (D) \(\infty\).

Answer: B


Question: Determine whether the following limit is greater than zero: \[ \lim_{n\to\infty} \log \log n \prod_{i=1}^{\lfloor \log n \rfloor} \frac{p_i-1}{p_i} > 0. \]

Answer: Yes


Question: Find the limit of \( \frac{e^{xy} - 1}{y} \) as \((x, y) \to (0, 0)\) using \(\epsilon-\delta\) arguments.

Answer: 0


Question: Evaluate the line integral \( \int _C \vec F \cdot d\vec r \) using Stokes' Theorem, where \( \vec F = (x+y^2, y+z^2, z+x^2) \) and \( S \) is the triangle with vertices \((1,0,0), (0,1,0), (0,0,1)\).

Answer: -1


Question: Determine the number of solutions of the equation \( e^z = 2z + 1 \) within the open unit disc \( \{ z \in \mathbb{C} : |z| < 1 \} \).

Answer: 1


Question: For any k-dimensional subspace $L$ of the tangent space $T_p M$ at a point $p$ on an $n$-dimensional manifold $M$, can a k-dimensional submanifold $R$ of $M$ be found such that $p \in R$ and $T_p R = L$?

Answer: Yes


Question: Let $V$ be a 4-dimensional vector space over $\mathbb{R}$. Consider the affine planes $L_1$: $x_0=0, x_2=0$ and $L_2$: $x_1=0, x_3=0$ in $V$. Given the vectors $v_1 = (1, 0, 1, 1)$ and $v_2 = (1, 1, 0, 1)$, find the intersection of the affine planes $L_1 + v_1$ and $L_2 + v_2$. Determine the coordinates of the intersection point.

Answer: (1, 1, 1, 1)


Question: Consider a game where you flip a coin. If it lands on heads, you win $1 from the house, but if it lands on tails, you lose $1 to the house. Starting with $n, where the house has infinite money, determine the expected number of coin flips before you lose all your money.

Answer: \infty


Question: Compute the norm \( \|[x]\| \) in the quotient space \( l_\infty/c_0 \) for \( x = (1,1,1, \ldots) \).

Answer: 1


Question: Given a nonnegative random variable $X$ such that $Z = \log(X)$, where $E(Z) = 0$ and $V(Z) = \sigma^2 > 0$, determine whether $E(X)$ is greater than, equal to, or less than 1.

Answer: E(X) > 1


Question: Evaluate the limit of the infinite product: \[ \lim_{a \to \infty} \prod^{\infty}_{i=a} i \] Determine whether the limit is infinity, undefined, or something else.

Answer: \infty


Question: Find the largest integer value of $X$ such that \( \frac{4X-1}{5} \), \( \frac{4X-2}{6} \), and \( \frac{4X-3}{7} \) are all integers, and \( 4X + 2 \) is a four-digit number.

Answer: 2414


Question: Determine the rank of the function $f: \mathbb{F}^{2020 \times 2020} \to \mathbb{F}^{2020 \times 2020}$ defined by $f(X) = X - X^T$, where $\mathbb{F}$ is a field.

Answer: 2039190


Question: Let \( \{X_n\} \) be a sequence of i.i.d. random variables uniformly distributed on \((0,1)\). Define \( \tau = \inf\{n \mid X_1 + X_2 + \cdots + X_n \ge 1\} \). Find the expected value \( E[\tau] \).

Answer: e


Question: Determine the number of positive integral solutions for the equation \( \frac{xy}{x+y} = 2^4 \cdot 3^5 \cdot 5^4 \).

Answer: 891


Question: Determine the Hausdorff dimension of the graph of the function \( y = \log(1+x)\sin\frac{1}{x} \) for \( x > 0 \).

Answer: 1


Question:
Given a function $f(x)$, if for any $k \in (0,1)$, the following conditions hold:
\[ \lim_{x\to \infty} \left(f(x) - k\ln(x)\right) \leq 0 \]
and
\[ \lim_{x\to \infty} \left(f(x) - \ln(x)\right) \geq 0 \]
are these conditions sufficient to conclude that:
\[ \lim_{x\to \infty} \left(f(x) - \ln(x)\right) = 0 \]?
Evaluate the validity of the proof provided for this conclusion.

Answer: 0


Question: Given a set of $2m$ elements, they are randomly grouped into $m$ pairs. Then, the elements are randomly paired again to form another $m$ pairs. Is it always possible to select one element from each pair in both groupings such that the original set is reconstituted? Provide a proof or counterexample.

Answer: Yes


Question:
Given a Markov chain $(X_n)_{n\geq1}$ with state space $S=\{A,B,C,D,E \}$ and the transition matrix
$$ P = \left( \begin{matrix} 0.6 & 0.4 & 0 & 0 & 0 \\ 0.3 & 0.7 & 0 & 0 & 0 \\ 0.2 & 0 & 0.4 & 0 & 0.4 \\ 0.2 & 0.2 & 0.2 & 0.2 & 0.2 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} \right) $$
compute $\lim_{n \to \infty} P(X_n=A|X_0=C)$.

Answer: \dfrac{1}{7}


Question: Find the remainder when $35^{32} + 51^{24}$ is divided by 1785.

Answer: 1


Question:
Consider the matrices:
\[ H=\begin{pmatrix} A & 0 & \cdots & \cdots & 0 \\ I & A & 0 & \cdots & 0 \\ 0 & I & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & I & A \end{pmatrix}, \quad K=\begin{pmatrix} A & 0 & \cdots & \cdots & 0 \\ U & A & 0 & \cdots & 0 \\ 0 & U & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & U & A \end{pmatrix}. \]
Let \( T \) be an \( nk \times nk \) nonsingular matrix with integer entries for which the minimal polynomial is \( p(x)^n \), where \( p(x) \) is irreducible of degree \( k > 1 \). In the matrices \( H \) and \( K \), \( A \) denotes the companion matrix of \( p(x) \), \( U \) is a matrix with every entry being \( 0 \) except for the entry \( 1 \) in the upper right corner, and \( I \) is the identity matrix. Given that \( K \) is similar to \( T \), and \( p(x)^n \) is the minimal polynomial of \( K \), determine if \( p(x)^n \) is also the minimal polynomial of \( H \).

Answer: Yes


Question:
Evaluate the limit \( L \) as \( h \to 0 \):
\[ L = \lim_{h\to 0} \frac{1}{h}\frac{a^4(a+h)-a(a+h)^4}{(a+h)^3+(-a)^3} \] where \( a \in \mathbb{R} \).

Answer: -2a


Question: Evaluate the limit: \[ \lim_{x\to0^+} (x^{x^x-1}) \]

Answer: 1


Question: Given the sequence defined by $a_{1}=1$ and $a_{n+1}=a_{n} + \frac{1}{a_{n}}$, find $\lim \limits_{n\to\infty}\frac{a_{n}}{n}$.

Answer: 0


Question: Is it correct to say that for every algorithm solving the Tower of Hanoi problem, the recursive relation \( T_n = 2T_{n-1} + c \) holds, where \( c \) is the number of moves for moving the last disk from A to C? Justify your answer.

Answer: No


Question: Find the value to which the sum \( \frac{1}{n^3}\sum_{i=1}^n (2i^2 - i) \) converges as \( n \to \infty \).

Answer: \dfrac{2}{3}


Question: Professor Liyung wants to form a math club with his 40 students. Each student is enemies with two other students, and no student will join the club if any of their enemies is already a member. Let \( M \) be the maximum number of members the club can have. Find the sum of all possible values of \( M \).

Answer: 119


Question: Find all finite sets \(A\) such that for every \(x \in A\), the expression \(x^3 - 3|x| + 4\) is also in \(A\).

Answer: \emptyset


Question: Find the least value of the expression \(ab - cd\) given that \(a, b, c, d \in \mathbb{R}\) satisfy \(a^2 + b^2 = c^2 + d^2 = 4\) and \(ac + bd = 0\).

Answer: -4


Question: Let $X$ be a discrete random variable with $\\mathbb{P}(X=2)=\\mathbb{P}(X=4)=\\mathbb{P}(X=6)=1/3$, and let $Y$ be a continuous random variable uniformly distributed on the interval $[3,5]$. Compute $\\mathbb{E}(\\min(X,Y))$.

Answer: \dfrac{13}{4}


Question: Let \( \{a_n\}_{n\ge 1} \) be a sequence of non-zero real numbers. Does there exist a subsequence \( \{a_{k_n}\}_{n\ge 1} \) such that either \( \left\{\frac {a_{k_{n+1}}}{a_{k_n}}\right\}_{n\ge 1} \) converges to \(1\) or \(0\), or \( \left\{\frac {a_{k_{n}}}{a_{k_{n+1}}}\right\}_{n\ge 1} \) converges to \(0\)?

Answer: Yes


Question: Find the value of the 100th derivative of the function \((\sinh(x)\cosh(x))^2\) at the point \(x=0\).

Answer: 2^{197}


Question: Calculate the value of \( \sum_{k = 1}^{p-1} k^n \mod p \) given that \( n \not\equiv 0 \mod (p-1) \), where \( n \geq 1 \) and \( p \) is an odd prime.

Answer: 0


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

Answer: -\dfrac{2}{27}


Question: Determine whether there is a closed-form expression for the series \( \sum_{k=0}^{\infty} \frac{1}{(k!)^2} \).

Answer: I_0(2)


Question: Consider a disk with an even number of points $n$ randomly distributed uniformly within it. These points are paired to form $n/2$ line segments, creating a perfect matching. Construct the intersection graph $G$ where each node represents a segment, and an edge exists between two nodes if their corresponding segments intersect properly. Determine the diameter of the graph $G$ as $n$ approaches infinity.

Answer: 2


Question: Let $X$ be a Banach space and $X^{*}$ its dual space. Consider a subspace $M$ of $X^{*}$ and a weak*-continuous linear functional $L$ defined on $M$. If $M$ is weak*-closed, there exists an element $x \in X/{M}_\perp$ such that $L(m) = m(x)$ for all $m \in M$, where $M_\perp$ is the pre-annihilator of $M$ in $X$. Is this conclusion still valid if $M$ is not weak*-closed?

Answer: Yes


Question: For every prime number \( p \), is there always a larger prime \( q \) such that \( q = p + 4n \) for some integer \( n \geq 2 \)?

Answer: No


Question: Find the sum of the maximum and minimum curvature of the ellipse given by the equation \(9(x-1)^2 + y^2 = 9\). Use the parametrization \(x(t) = 1 + \cos(t)\).

Answer: \dfrac{28}{9}


Question: Calculate the limit: \[ \lim_{x \to 0}\frac{x \tan2x-2x\tan x}{(1-\cos2x)^2} \]

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{n \to +\infty} \sum_{k=0}^n \left(\sqrt[n]{k} - \sqrt[n]{n}\right). \]

Answer: -2


Question: Find the function \( \phi(xy) \) such that \( \phi(xy) \) is harmonic. Assume \( \phi \) is an unknown function and use the Laplacian equation to determine the form of \( \phi(xy) \).

Answer: \phi(xy) = A xy + B


Question: Determine the order of the element \( \frac{3}{4} + \mathbb{Z} \) in the group \( \mathbb{Q}/\mathbb{Z} \).

Answer: 4


Question: Find the derivative of the function \( f(x) = \dfrac{3(\sqrt[3]{5x-2})^2}{(2x-3)^5(x-1)^4} \) at \( x = 2 \).

Answer: -163


Question: In the vector space \(\mathbb{R}^{2n}\), is the canonical basis a symplectic basis?

Answer: Yes


Question: Determine the cardinality of the set \( M = \{\mu : \mathcal{B}(\mathbb{R}) \to \mathbb{R} : \mu \text{ is a measure}\} \). What is \(|M|\)?

Answer: 2^{\aleph_0}


Question:
Find the value of \( \alpha \) that satisfies the equation:
\[ 2\left(\frac{1}{\alpha}\right) \ln\left(\frac{1}{1-\alpha}\right) = \frac{1}{1-\alpha} \]

Answer: 0.715


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

Answer: a > 1


Question: Find the smallest positive integer \( n \) such that \( n/1 \) generates the ideal \( \langle 4/17^8 \rangle \) in the ring \( \mathbb{Z}[1/17] \). Explain why the generator of the ideal can be expressed as \( n/1 \).

Answer: 4


Question: Suppose \(A \subset \mathbb{R}\) is the domain of a function \(f(x)\) on which it is uniformly continuous. The function \(f(x)\) satisfies the property that for each \(x_0 \in \overline{A} - A\), \(\lim_{x \to x_0^-} f(x) = \lim_{x \to x_0^+} f(x)\). Extend the domain of \(f(x)\) to \(\overline{A}\) by defining \(f(x_0) = \lim_{x \to x_0} f(x)\) for each \(x_0 \in \overline{A} - A\), ensuring that \(f(x)\) is continuous on \(\overline{A}\). Is \(f(x)\) uniformly continuous on \(\overline{A}\) when the cardinality of \(\overline{A} - A\) is infinite?

Answer: Yes


Question: Provide an example of a set with cardinality \(\aleph_2\).

Answer: \omega_2


Question: Consider a sequence of independent random variables $X_1, X_2, \ldots$ where $X_n \sim \text{Unif}\{1, \ldots, n\}$. Calculate the probability that the set $\{X_1, X_2, \ldots\}$ equals the set of natural numbers $\mathbb{N}$.

Answer: 1


Question: Given a matchstick of length 1 and an infinite grid of 1x1 squares, what is the probability that the matchstick lands overlapping at least one of the matchsticks in the grid?

Answer: \dfrac{3}{\pi}


Question: Let \( M \) be a 4-manifold homeomorphic to \( S^4 \). Does \( M \) always admit a Riemannian metric with positive scalar curvature, regardless of the differentiable structure given to \( M \)?

Answer: Yes


Question: Evaluate the integral \( \int_C \frac{1}{z^2+1} \, dz \) where \( C \) is the square contour centered at the origin with side length 2, oriented positively.

Answer: 0


Question: A stick of length 2 meters is made of uniformly dense material. A point is chosen randomly on the stick, and the stick is broken at that point. The left portion of the stick is discarded. This process is repeated indefinitely on the remaining portion of the stick. What is the probability that one of the discarded left parts has a length greater than 1 meter?

Answer: \ln 2


Question: Find the radius of convergence of the complex power series \( \sum_{n=1}^{\infty} 3^{n} z^{n^{2}} \) using the Cauchy-Hadamard formula.

Answer: 1


Question: Let $x, y, z$ be three integers in a geometric progression such that $y - x$ is a perfect cube. Given that \( \log_{36} x^2 + \log_{6} \sqrt{y} + 3 \log_{216} z \sqrt{y} = 6 \), find the value of $x + y + z$.

Answer: 189


Question: In a metric space $(S,d)$, consider points $a, b, c \in S$ such that for all $x \in S$, $d(a,x) \leq d(a,b)$, and for all $y \in S$, $d(b,y) \leq d(b,c)$. Does it follow that for all $x, y \in S$, $d(x,y) \leq d(b,c)$? In other words, is $d(b,c)$ the maximum possible distance between points in $S$?

Answer: No


Question: Determine the best approximation of \(|f^{(4)}(0)|\) if \(f\) is analytic in the whole plane and satisfies the inequality \(\left|\frac{f^{(4)}(0)}{4!}\right| \leq \frac{e^{p^2}}{p^4}\) for \(p > 0\).

Answer: 6e^2


Question: Find the minimum value of the length of a segment of a tangent to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) intercepted by the coordinate axes. The length of the segment is given by \(\sqrt{a^2\sec^2\theta+b^2\csc^2\theta}\).

Answer: a + b


Question: Determine whether the function \( f(x) = \begin{cases} \frac{(-1)^p}{\sqrt{q}} & \text{if } x = \frac{p}{q} \text{ with } \gcd(p,q)=1 \\ 0 & \text{if } x \text{ is irrational} \end{cases} \) is Riemann integrable on the interval \([0,1]\). If it is integrable, compute the integral \( \int_0^1 f(x) \, \text{d}x \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{1}{n^{\frac{3}{2}}}\sum_{k=0}^{n}k\left(\sqrt{k+a}-\sqrt{k}\right) \] for a fixed real number \(a > 0\).

Answer: \dfrac{a}{3}


Question: In the finite complement topology on the real numbers \( \mathbb{R} \), to which point or points does the sequence \( \left\{ \frac{1}{n} \right\} \) converge?

Answer: \mathbb{R}


Question: Evaluate the double integral \( \iint \limits_D 2x^2e^{x^2+y^2}-2y^2e^{x^2+y^2} \, dy \, dx \) where \( D \) is the region defined by \( x^2+y^2=4 \).

Answer: 0


Question:
Determine whether there exist real numbers \(a, b, c, d, e, f\) such that the following conditions are satisfied:
\[
\begin{cases}
ax^2+by^2+cxy+dx+ey+f > 0 \quad \forall 0 < x \le 1, 0 < y \le 1, \\
a+b+c+d+e+f \le 1, \\
a+d+f \le 0, \\
b+e+f \le 0, \\
f \le 0
\end{cases}
\]

Answer: Yes


Question: Let $k$ be either the field $\Bbb C$ of complex numbers or the field $\Bbb R$ of real numbers. Let $X$ be a quasi-projective and smooth algebraic variety over $k$. Consider the set of $k$-points $X(k)$ with the usual topology. Is $X(k)$ homotopically equivalent to a CW complex?

Answer: Yes


Question: Find the area of the surface obtained by rotating the circle \((x-a)^2 + (y-b)^2 = r^2\), where \(0 < r < a\), about the line \(x = -a\).

Answer: 8\pi^2 a r


Question: Given a point $A$ on a sheet of paper and two lines $L_1$ and $L_2$ that intersect at a point $B$ outside the sheet, describe how to draw the line $AB$ using only a ruler.

Answer: AB


Question: Let \( \{x_{n}\}_{n=0}^{\infty} \) be a decreasing sequence of non-negative real numbers. Suppose there exist constants \( a, s > 0 \) and \( b > 1 \) such that \( x_{n+1} \leq ab^{n}x_{n}^{1+s} \) and \( x_{0} \leq a^{-1/s}b^{-1/s^{2}} \). Determine whether \( \lim_{n \to \infty} x_{n} = 0 \).

Answer: 0


Question: Find the directional derivative of the function \( f(x,y,z) = x^2+y^2-z^2 \) at the point \((3,4,5)\) along the curve of intersection of the surfaces \( 2x^2+2y^2-z^2=25 \) and \( x^2+y^2=z^2 \).

Answer: 0


Question: Calculate \( \lim_{n \to \infty} \sqrt[n]{\frac{(2n)!}{(n!)^2}}. \)

Answer: 4


Question: Find the volume of the solid bounded by the sphere \(x^2 + y^2 + z^2 = 4\) and the cones \(z = \sqrt{x^2 + y^2}\) and \(z = -\sqrt{3x^2 + 3y^2}\) using spherical coordinates. Determine the bounds for \(\phi\) for each cone, assuming \(0 \leq \rho \leq 2\) and \(0 \leq \theta \leq 2\pi\).

Answer: \dfrac{8\pi}{3} \left( \sqrt{3} + \sqrt{2} \right)


Question: Evaluate the line integral \( \int_C y \, dx + z^2 \, dy + x \, dz \) over the curve \( C \), which is the intersection of the surfaces \( z = \sqrt{x^2 + y^2} \) and \( z = 6 - (x^2 + y^2) \).

Answer: -4\pi


Question:
Find the circumference of a melon if the boundary line of the melon is described by a double cycloid with the following parametric equations:

\[ x(t) = 5(t - \sin(t)), \quad y(t) = 5(1 - \cos(t)) \]
\[ x(t) = 5(t - \sin(t)), \quad y(t) = -5(1 - \cos(t)) \]

Calculate the circumference of the melon.

Answer: 80


Question: Evaluate the expression $\exp(\log(3+4i))$ using the principal logarithm, and express your answer in the form $x+iy$. What is the value of the expression?

Answer: 3 + 4i


Question: Solve the equation \( \sqrt{x + 3 - 2\sqrt{x + 2}} + \sqrt{x + 27 - 10\sqrt{x + 2}} = 4 \) and find the solution set \( S \).

Answer: [-1, 23]


Question: Find the minimum value of the expression \( \frac{\sec^4 \alpha}{\tan^2 \beta} + \frac{\sec^4 \beta}{\tan^2 \alpha} \) for \( \alpha, \beta \in \left(0,\frac{\pi}{2}\right) \).

Answer: 8


Question: Evaluate the integral \( \int_{0}^{2\pi} e^{-i\theta}e^{e^{i\theta}} \, d\theta \).

Answer: 2\pi


Question: Calculate the area between the function \( y = x + \sin x \) and its inverse function.

Answer: 4


Question: Determine the radius of convergence of the function \( \frac{z^3-1}{z^2+3z-4} \).

Answer: 4


Question: Does there exist a nowhere dense subset $A \subseteq \mathbb{R}$ such that the Lebesgue measure $m(A) = +\infty$?

Answer: Yes


Question: Evaluate the infinite series \( \sum_{n=1}^{\infty} \frac{1}{a_n} \) where \( a_1=2 \) and \( a_{n+1}=a_n^2-a_n+1 \) for \( n \geq 1 \).

Answer: 1


Question: Find the coefficient of \((z-\pi)^2\) in the Taylor series expansion of the function \(f(z) = \frac{\sin(z)}{z-\pi}\) around \(z = \pi\). Assume \(f(z) = -1\) if \(z = \pi\).

Answer: \dfrac{1}{6}


Question:
Prove that for a random graph $G \in \mathcal{G}(n,p)$, the expected number of edges is given by both the summation formula and the indicator random variable method, i.e., show that:
\[ \sum_{k=1}^m k \binom{m}{k} p^k (1-p)^{m-k} = mp \] for $0 < p < 1$, where $m = \binom{n}{2}$.

Answer: mp


Question: Evaluate the integral \( \int_{-1/2}^{1/\sqrt{2}} \frac{f(x)}{1-x^2} \, dx \) given that \( f:[0,1]\to [0,1] \) is a continuous function satisfying \( x^2 +(f(x))^2\le 1 \) for all \( x\in [0,1] \) and \( \int_0^1 f(x) \, dx=\frac{\pi}{4} \).

Answer: \dfrac{5\pi}{12}


Question:
Given the moment generating function of a random variable X:
\[ M_X(t) = \frac{1}{(1-2500t)^4} \]
calculate the standard deviation of the claim size for this class of accidents.

Answer: 5000


Question: Determine whether the sequence \( \{ f^{(n)}(0) \} \) is unbounded, where \( f(x) = \tan^{-1} x \) and \( x \in \mathbb{R} \). Answer "True" if the sequence is unbounded and "False" otherwise.

Answer: A


Question: Let $f$ be a function such that $f$ and $f''$ are continuous on $[0, \pi]$, $f(\pi) = 1$, and \( \int^{\pi}_0 (f(x) + f''(x)) \sin(x) \, dx = 2. \) Find $f(0)$. \( \)

Answer: 1


Question: Determine the number of 1-dimensional characters for a nonabelian group of order 57.

Answer: 3


Question:
Determine the minimum number of terms needed in the series that defines Euler's constant \( \gamma \) to approximate \( \gamma \) with an error less than or equal to \( 3 \times 10^{-3} \). Use the inequality for the error of a series:
\[
\left| \sum_{k=1}^{\infty}{f(k)} - \sum_{k=1}^{n}{f(k)} \right| \leq \int_{n}^{\infty}{f(t)dt}
\]
and the definition of \( \gamma \):
\[
\gamma = \lim_{n \rightarrow \infty}{\left( \sum_{k=1}^{n}{\dfrac{1}{k} - \log{n}} \right)}
\]

Answer: 167


Question: Does there exist a continuous function \( f \) on the interval \([0,1]\) such that \( f \) has an infinite number of roots in \([0,1]\) and is not constantly zero on any subinterval of \([0,1]\)?

Answer: Yes


Question: Evaluate the limit: \[ \lim _{x\to \infty} \left(\frac{1+2^{\left(\frac{1}{2}\right)}+3^{\left(\frac{1}{3}\right)}+\ldots +x^{\left(\frac{1}{x}\right)}}{x}\right) \]

Answer: 1


Question:
If $x_1, x_2, x_3, x_4$ are the roots of the polynomial $x^4 - x^3 + 2x^2 + 3x + 1$, find the value of the expression:
\[ \frac{\sum x_i^3 + \sum (x_i x_j x_k)^3}{\sum (x_i x_j)^3 - (x_1 x_2 x_3 x_4)^3} \]

Answer: -\dfrac{20}{49}


Question: Calculate the limit: \[ \lim_{N\to\infty} \frac{N!}{(N-k)!N^{k}} \] where \( k \) is a fixed integer such that \( 0 \leq k \leq N \).

Answer: 1


Question: Given a set of \( n \) non-zero real numbers \( a_1, a_2, \ldots, a_n \) and another real number \( b \), determine the maximum number of ways to assign a plus or minus sign to each \( a_i \) such that the equation \((+/-) a_1 (+/-) a_2 \ldots (+/-) a_n = b\) holds true. Assume \( b = 0 \) and all \( a_i \) are the same. What is the maximum number of ways to achieve this?

Answer: \dbinom{n}{n/2}


Question: Determine the number of (not necessarily abelian) extensions of $\mathbb{Z}$ by $\mathbb{Z}$ up to isomorphism.

Answer: 2


Question: In a circle with radius 2, a regular 11-gon $K_1 K_2 \dotsb K_{11}$ is inscribed. Let $L$ be a point such that the distance from $L$ to the circle's center is 3. Calculate the sum $LK_1^2 + LK_2^2 + \dots + LK_{11}^2$. Provide your answer.

Answer: 143


Question: Evaluate the integral \( I = \int_{-n}^{n} \frac{x + \tan{x}}{A + B(x + \tan{x})^{2n}} \, dx \) and show that it equals zero, where \( n \) is an integer with \( n \geq 1 \), and \( A \) and \( B \) are constants.

Answer: 0


Question: Is every order isomorphism between power sets induced by a bijection? Specifically, if $f$ is an order isomorphism from $\mathscr{P}(A)$ to $\mathscr{P}(B)$, where $A$ and $B$ are sets and the order is set inclusion $\subseteq$, does there always exist a bijection $F: A \rightarrow B$ such that $f(X) = F[X]$ for every $X \in \mathscr{P}(A)$?

Answer: Yes


Question:
Consider a random sample of size $r$ taken without replacement from a population of $n$ elements. The probability $u_r$ that $N$ specific elements are included in the sample is given by:
\[ u_r = \frac{{n - N \choose r-N}}{{n \choose r}}. \]
If $n \to \infty$ and $r \to \infty$ such that $r/n \to p$, show that $u_r \to p^N$. Prove this convergence.

Answer: p^N


Question: For a given lower triangular square matrix $T_1$, is there an orthogonal matrix $Q$ such that $QT_1$ is also lower triangular? Provide a justification for your answer.

Answer: Yes


Question: For a given \( n \times n \) array, the first row is \( x_1, x_2, x_3, \ldots, x_{n-1}, x_n \). Each successive row is obtained by a cyclic shift of \( r \) places from the previous row. Determine the values of \( r \) for which this construction results in a Latin square.

Answer: \gcd(r, n) = 1


Question: Let $A = \mathbb{R}[X,Y]$ be the ring of polynomials in two variables over the real numbers, and let $m \geq 1$. Consider the set \( \mathcal{S}_m = \{ X^m, X^{m-1}Y, X^{m-2}Y^2, \ldots, Y^m \} \) and let \( I \) be the ideal of \( A \) generated by \( \mathcal{S}_m \). Suppose \( S' \) is another finite set of polynomials in \( A \) such that \( I \) is also the ideal of \( A \) generated by \( S' \). Is it necessarily true that \( S' \) must have at least \( m+1 \) elements? Justify your answer.

Answer: Yes


Question:
Find an upper bound for the integral \( I_n = \int_0^1 x^n (1+x)^{0.5} \, dx \), where \( n = 0, 1, 2, \ldots \). Which of the following is an upper bound on \( I_n \)?

A. \( \frac{1}{n + 1} \);
B. \( \frac{1}{2(n + 1)} \);
C. \( \frac{1}{n + 1} + \frac{1}{n + 2} \);
D. \( n^n \).

Answer: C


Question: Let $f\colon \mathbb{R}\to\mathbb{R}$ be a function. Assume that for all sequences $(x_n)_{n\in\mathbb{N}}\subset\mathbb{R}$ with $x_n\to x$ in $\mathbb{R}$, there exists a subsequence $(x_{n_k})_k$ such that $f(x_{n_k})\to f(x)$ as $k\to \infty$. Does it follow that $f$ is sequentially continuous? Justify your answer.

Answer: Yes


Question: If $xf(x)$ approaches a limit as $x$ approaches infinity, does it necessarily follow that $f(x)$ approaches 0? Specifically, if $f: \mathbb{R} \to \mathbb{R}$ satisfies $\lim_{x \to \infty} xf(x) = \lambda$ for some $\lambda \in \mathbb{R}$, does it imply that $\lim_{x \to \infty} f(x) = 0$?

Answer: 0


Question:
Compute the integral \( \int_X d\omega \), where \( \omega \) is a 1-form on \( X \) and \( X \) is a 2-segment in \( \mathbb{R}^3 \) defined as follows:
\[ \omega=[e^{xz}+2y e^{x^2+y^2}] \, dx + [\ln (2+z)+3xe^{x^2+y^2}] \, dy+3x^2y \, dz \]
and
\( X:E \to \mathbb{R}^3 \) is such that \( X(r, \theta) = (r \cos\theta, r \sin\theta, 1-r^2) \) with \((r, \theta) \in [0,1] \times [0, 2\pi].\)

Use Stokes' theorem to evaluate the integral.

Answer: e \pi


Question: Find the maximum value of \(M = a^2 + b^2 + a - 3b\) given that \(a, b \in \mathbb{R}\) satisfy the inequality \(a^2 + b^2 \leq 40\).

Answer: 60


Question:
Find the sum of the roots of the equation:
\[(x − 1) + (x − 2)^2 + (x − 3)^3 + \ldots + (x − 10)^{10} = 0.\]

Answer: 99


Question: Determine the values of $z$ for which the function $g(z) = \cos(x)\cosh(y) + i\sin(x)\sinh(y)$ is differentiable.

Answer: z = n\pi


Question: Determine the \(\omega\)-limit set \(\mathcal{S}_\omega(\xi)\) for the solution \(\xi(t)\) of the dynamical system \(\frac{dx}{dt}(t) = \sin(x(t))\) with the initial condition \(\xi(0)=\pi/2\).

Answer: \{\pi\}


Question: Compute the integral \( \int\limits_{\mathbb{R}} \int\limits_{\mathbb{R}} \frac{1}{(1+(x-y)^2)^2} \, dx \, dy \).

Answer: \infty


Question: Determine whether there exists a compact manifold with Euler characteristic 0 that cannot be given a Lie group structure.

Answer: Yes


Question: Calculate the divergence of the vector field $v = \partial_\phi$ in polar coordinates on the manifold $M = \mathbb{R}^2 \setminus \{0\}$. Express your answer in terms of polar coordinates.

Answer: 0


Question: Determine whether the following statement is true or false: If \( \frac{a_{0}}{5}+\frac{a_{1}}{4}+\frac{a_{2}}{3}+\frac{a_{3}}{2}+a_{4}=0 \) with \( a_{4} \neq 0 \), then the equation \( a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{3}x+a_{4}=0 \) has at least one root in the interval \([0,1]\).

Answer: True


Question: Evaluate the integral \( \int_{\gamma} \frac{1}{\sin z} \, dz \) where \( \gamma \) is a simple, closed path with 0 in its interior and \( \{\pi n : n \in \mathbb{Z} \setminus \{0\}\} \subset \mathbb{C} \setminus |\gamma| \).

Answer: 2\pi i


Question: A rectangle is positioned such that its base lies on the $x$-axis and it is entirely contained within the region between the curve $y = \frac{x}{x^2+1}$ and the $x$-axis. By revolving this rectangle about the $x$-axis, a cylinder is formed. Determine the maximum possible volume of this cylinder.

Answer: \dfrac{\pi}{4}


Question: Let X, Y, and W be independent random variables where X follows a Gamma distribution with parameters 2 and 3, Y follows a Normal distribution with mean 1 and variance 4, and W follows a Binomial distribution with 10 trials and success probability 1/4. Define U = 2X - 3Y and V = Y - W. Calculate the covariance \( \text{cov}(U, V) \).

Answer: -12


Question: Determine the cardinality of the set of functions \( F: \mathbb{N} \longrightarrow \{0,1,2\} \).

Answer: 2^{\aleph_0}


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

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{x \to 0}\frac{1-\frac{1}{2} x^2 - \cos\left(\frac{x}{1-x^2}\right)}{x^4} \]

Answer: \dfrac{23}{24}


Question: Simplify the expression \(42^{25\sqrt{25^{25}}}\).

Answer: 42^{5^{27}}


Question: Evaluate the limit: $$\lim \limits_{(x, y) \to (0,0)} \frac{x^3 + \sin(x^2+y^2)}{y^4 + \sin(x^2+y^2)}$$

Answer: 1


Question: Let \( f(x) \) be a polynomial of degree 1007 such that \( f(k) = 2^k \) for \( 0 \le k \le 1007 \). Determine the value of \( f(2015) \).

Answer: 2^{2014}


Question: What is the OLS coefficient when regressing the residuals \( \hat{\epsilon} \) from an OLS regression of \( y \) on \( X \) back on \( X \)?

Answer: 0


Question: Find an upper bound for \( \log \operatorname{rad}(n!) \) that is better than \( n \log n \), where \( n > 1 \) is an integer and \( \operatorname{rad}(n!) \) denotes the radical of \( n! \). The radical of an integer \( m \) is the product of the distinct prime divisors of \( m \).

Answer: n \log 4


Question: Evaluate the integral \( \int \pi e^{\pi \overline{z}}\, dz \) over the square with vertices at \( z=0, z=1, z=1+i, z=i \).

Answer: 4(e^{\pi} - 1)


Question: Let $G$ be a group and $C$ a conjugacy class of $G$. Suppose $N$ is a normal subgroup of $G$ such that $N \cap C \neq \emptyset$. Prove that $C \subset N$.

Answer: C \subset N


Question: Find the remainder when \( P = 2008^{2007} - 2008 \) is divided by \( Q = 2008^2 + 2009 \).

Answer: 4032066


Question: Consider the graph $G$ obtained by deleting any edge from the complete bipartite graph $K_{7,8}$. Determine the number of spanning trees in the complement graph $\overline{G}$ of $G$. Provide your answer as a single integer.

Answer: 4405854208


Question: Find the area of the set \( M = \{(x,y): |x|^{\frac{2}{3}} + |y|^{\frac{2}{3}} \le 1\} \) using integration.

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


Question: Determine the order of the entire function \(f(z) = \cos\sqrt{z}\).

Answer: \dfrac{1}{2}


Question: For which values of \( n \) does the inequality \( 2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-1}{p_i}\right) \) hold, where \( p_i \) is the \( i \)-th prime number and \( A \) is defined by \( p_n = 6A + r \) with \( 0 \leq r < 6 \)?

Answer: n \geq 4


Question: Given the linear transformation \( T: \mathbb{R}^n \to \mathbb{R}_2[x] \) defined by \( T(1,0) = 1+x \) and \( T(1,1) = 1-x \), find \( T^{-1}(2x) \).

Answer: (0, -1)


Question: Determine the number of elements in the conjugacy class of the permutation \( \tau = (56789)(3456)(234)(12) \) in the symmetric group \( S_9 \).

Answer: 15120


Question: Find all prime numbers \( p \) such that \( p^p - 2 \) is a perfect square.

Answer: 3


Question: Determine if the function \( y = \ln(xy) \) is a solution to the differential equation \((xy-x)y''+xy'^2+yy'-2y'=0\).

Answer: Yes


Question:
Consider a function $u \in H^1(U)$ and a smooth, bounded, nondecreasing function $\phi$ such that $\phi'$ is bounded and $\phi(z) = z$ for $|z| \le 1$. Define $u^\epsilon(x) := \epsilon \phi(u/\epsilon)$. Determine whether the following integral converges to zero as $\epsilon \to 0$:
\[ \int_U Du^\epsilon \cdot Du\,dx = \int_U \phi'(u/\epsilon)|Du|^2\,dx \to 0. \]

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \to \infty} n \ln\frac{a(\sqrt{n}-1)}{a(\sqrt{n}-1)+b} \] where \(0 < a, b < 1\).

Answer: -\infty


Question:
Let $A$ be an $n \times n$ matrix over the real numbers such that $AB = BA$ for all $n \times n$ matrices $B$. Determine which of the following statements is true:

(a) $A$ must be the zero matrix.
(b) $A$ must be the identity matrix.
(c) $A$ must be a diagonal matrix.
(d) $A$ must be either the zero matrix or the identity matrix.

Answer: C


Question: Let \( \{\xi_t\} \) be a sequence of i.i.d. random variables, and \( \{z_t^{n}\} \) be a sequence of i.i.d. Bernoulli random variables such that for each \( t \), \( z_t^{n} = 0 \) with probability \( 1 - \frac{1}{\sqrt{n}} \) and \( z_t^{n} = 1 \) with probability \( \frac{1}{\sqrt{n}} \). The sequences \( \{z_t^n\} \) and \( \{\xi_t\} \) are independent. Determine whether the sequence \( \frac{1}{n} \sum_{t=1}^n z_t^{n}\xi_t \) converges in probability to 0 as \( n \to \infty \).

Answer: 0


Question: Let $x$ and $y$ be two non-zero natural numbers such that $7x^5=11y^{13}$. The lowest possible value for $x$ has a prime factorization of the form $a^cb^d$. What is the value of the sum $a+b+c+d$?

Answer: 31


Question: Evaluate the improper integral: $$\int\limits_0^{+\infty}\frac{\ln{x}}{x^2+x+1}\,\mathrm{d}x.$$

Answer: 0


Question: Determine the set of all possible values of \( \gcd(13x + 2, 5x - 1) \) as \( x \) ranges over all integers, using the Euclidean algorithm.

Answer: \{1, 23\}


Question: Determine whether every conformal diffeomorphism on a closed hyperbolic surface of genus $g \geq 2$ is an isometry. Justify your answer.

Answer: Yes


Question: Find all integers \(n > 2\) such that there exists an integer \(x\) satisfying \(n^{n-2} = x^n\).

Answer: 4


Question: Find the degree of the field extension \([ \mathbb{Q}(\sqrt[3]{3}, \eta) : \mathbb{Q} ]\), where \(\eta = (e^{\frac{\pi}{3}i})^2\).

Answer: 6


Question: Let \(X\) and \(Y\) be independent random variables, each following a standard normal distribution \(\mathcal{N}(0,1)\). Define \(D_r\) as the disk centered at the origin with radius \(r\). Determine the value of \(r\) such that the probability \(\mathbb{P}[(X,Y) \in D_r] = 0.3\).

Answer: \sqrt{-2 \ln 0.7}


Question:
Consider an arbitrary $k$-linear form $P$ on $\mathbb{R}^n$ given by:
\[ P(x_1, \ldots, x_k) = \sum_{i_1, \ldots, i_k=1}^n a_{i_1, \ldots, i_k} x_{1i_1} \cdots x_{ki_k}. \]
Define the quantity:
\[ I(P) := \sum_{i_1, \ldots, i_k=1}^n a_{i_1, \ldots, i_k}^2. \]
Is this quantity $I(P)$ invariant under the orthogonal group? Specifically, for $f \in O(n)$, if we define $P_f(x_1, \ldots, x_k) := P(f(x_1), \ldots, f(x_k))$, is it true that:
\[ I(P) = I(P_f)? \]
Provide a justification for your answer.

Answer: Yes


Question: Find the value of the limit \( \lim_{m,n\to\infty}\cos^{2m}(n! \cdot \pi \cdot x) \) where \( x \) is irrational.

Answer: 0


Question:
Let \( f(x) \) be a polynomial function such that \( f(x) + f'(x) + f''(x) = x^5 + 64 \). Evaluate the value of \( \lim_{x \to 1}\frac{f(x)}{x-1} \). Choose the correct answer from the options below:

\( \boxed{A) \; -15} \)

\( B) \; -60 \)

\( C) \; 60 \)

\( D) \; 15 \)

Answer: A


Question: Consider the representation $V(0)$ of $\mathfrak{sl}(2, \Bbb C)$. Is $V(0)$ a one-dimensional representation with basis $\{v_0\}$, where all elements $e$, $f$, and $h$ of the Lie algebra act as zero? In other words, does $V(0)$ have a single weight space $\mathfrak{g}_0 = \{x \in V : h(x) = 0\}$?

Answer: Yes


Question: Given the force of interest \( \delta_t = \frac{2t}{10+t^2} \) for \( t \geq 0 \), calculate the future value \( \ddot{s}_{\bar{2}|} \).

Answer: \dfrac{147}{55}


Question: Given a set \( A \subseteq \mathbb{N} \) with positive upper density, i.e., \( \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} > 0 \), determine if there exists an integer \( k \geq 2 \) such that the set \( M_A(k) = \{a \in A : ka \in A\} \) is infinite.

Answer: Yes


Question: Consider a connected, planar, directed graph G where each vertex has either degree 1 or 3. At each degree 3 vertex, either (1) exactly two of the three adjacent edges are oriented inwards and the third edge points outwards, or (2) two edges point outwards and one inwards. Assign a label of 1 or 2 to each degree 3 vertex based on which case applies. If such a labeling exists such that no two adjacent vertices have the same label, this provides a 2-coloring of G. Is it true that if such a graph G is 2-colorable, then there exists a 2-coloring that arises in this way?

Answer: Yes


Question: Suppose that \( A \in M_{n}(\mathbb{C}) \) is a nonsingular matrix and is similar to \( A^k \) for each \( k = 1, 2, 3, \ldots \). Can we conclude that 1 is the only eigenvalue of \( A \)?

Answer: Yes


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

Answer: 1


Question: In the figure, $AB = 200$ is a diameter of the circle. Points $A$ and $B$ are assigned the numbers $1$ and $1$. The two semi-circles are bisected at points $C$ and $D$, which are assigned the numbers $2$ and $2$. Each quarter circle is then bisected and assigned the number $3$. This process continues, with each arc being bisected and the new point being assigned the sum of the numbers at the endpoints of the arc. This process is repeated until the sum of all numbers on the circle is at least equal to the product of $4$ times the area of the circle and $rac{1}{3}$ of the radius. Find the least number of points on the circle. (Take $\pi$ to be $3.14$.)

Answer: 32768


Question: Determine all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \((x+y)(f(x)-f(y)) = f(x^2) - f(y^2)\) for all real numbers \(x\) and \(y\).

Answer: f(x) = ax + b


Question: Determine if there exist real numbers $x$, $y$, and $s$ with $s \in (0,1)$ such that the expression \[ \cos\left(\frac{x - y}{2}\right)\sin\left(\frac{x + y}{2}\right)\cos\left(sx + (1 - s)y\right) \] equals 1.

Answer: Yes


Question: Is it true that for distinct points $x_1, x_2 \in \mathbf{R}$ and for real sequences $(a_n)_{n \in \mathbf{N_0}}$ and $(b_n)_{n \in \mathbf{N_0}}$, there exists a smooth function $f: \mathbf{R} \rightarrow \mathbf{R}$ such that $f^{(n)}(x_1)=a_n$ and $f^{(n)}(x_2)=b_n$ for all $n \in \mathbf{N_0}$?

Answer: Yes


Question: Calculate the probability that the minimum of a Brownian motion \((W_t)_{t \in [0, T]}\) is zero, given that \(W_0 = 0\) almost surely. Express this probability as \(\mathbb{P} \left(\min_{t \in [0, T]} W_t = 0\right)\).

Answer: 0


Question: Consider a probability game where a target number \( x \) is chosen from the interval \((0, 1)\). A sequence of numbers \( c_1, c_2, \ldots \) is generated such that each \( c_i \) is picked uniformly at random from the interval \((0, c_{i-1})\), starting with \( c_1 \) from \((0, 1)\). The game stops when a number \( c_i \) is picked such that \( c_i < x \). What is the expected number of picks required to stop the game? Express your answer in terms of \( x \).

Answer: 1 - \ln x


Question: ABCD is a rectangle, and M is the midpoint of side CD. The inradii of triangles ADM and ABM are 3 and 4, respectively. Find the area of rectangle ABCD.

Answer: 216


Question: Find the eccentricity of the hyperbola that is the locus of the intersection of the lines \( \sqrt{3} x - y - 4\sqrt{3} t = 0 \) and \( \sqrt{3} tx + ty - 4\sqrt{3} = 0 \), where \( t \) is a parameter.

Answer: 2


Question: Two circles intersect orthogonally, and the length of their common chord is \( \frac{24}{5} \) units. If the radius of one of the circles is 3 units, find the length of the direct common tangent between the two circles.

Answer: 2\sqrt{6}


Question: Find the expression for $E[X_t \, dt]$ where $X_t$ is a drifted Brownian motion given by $X_t = \mu_x t + \sigma_x W_t^x$. Use stochastic calculus to determine the result.

Answer: \mu_x t \, dt


Question: Solve the nonlinear ordinary differential equation \( y''(x) + x^2 y'(x) + y(x)^3 = 0 \) with the initial conditions \( y(0) = 0 \) and \( y'(0) = 0 \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{t \to 0^+} \frac{e^{-1/t}}{t^2} \]

Answer: 0


Question: Find the exact value of the infinite series \( \sum_{x=1}^\infty \frac{x^2}{2^x} \).

Answer: 6


Question: Find the residue of the function \( f(z) = \frac{e^z}{\sin^2(z)} \) at each finite singularity.

Answer: e^{n\pi}


Question: Given the function equation \( f \left(\frac{\sqrt{4x} + 6}{3}\right) = \frac{x - 9}{36} \), find the value of \( f(4\sqrt{x}) \).

Answer: x - \sqrt{x}


Question: Suppose $H/P$ and $M/P$ are two subgroups of a group $G/P$ such that $(H/P) \cap (M/P) = \{P\}$. Is it true that $H \cap M = P$?

Answer: Yes


Question: Determine whether there exists a function \( f(x) \) on the real domain and real constants \( a \) and \( b \neq 0 \) such that the equation \( f(x) - f(x - \delta) + a + bx^2 = 0 \) holds for some real \( \delta \neq 0 \).

Answer: Yes


Question: In the accompanying figure, M and N are the respective hyperbolic midpoints of segments AB and AC in a hyperbolic triangle. Given the angle measures \(\phi\) and \(\theta\), determine which of the following is true: \(\phi = \theta\), \(\theta < \phi\), or \(\theta > \phi\). Provide a proof for your answer.

Answer: \theta > \phi


Question: If matrix $A$ is positive definite and matrix $B$ is positive definite, is the matrix $B^T A B$ positive definite?

Answer: Yes


Question: Evaluate the line integral \( \int _ {\gamma} (x-2y ^ {2}) \, dy \), where \( \gamma \) is the arc of the parabola \( y = x ^ {2} \) from \((-2,4)\) to \((1,1)\).

Answer: 48


Question: Let $G=(V,E)$ be a planar graph with a chosen planar representation such that $v-e+f=2$, where $v$, $e$, and $f$ are the number of vertices, edges, and faces, respectively. Does this imply that $G$ must be connected? Answer "yes" or "no."

Answer: yes


Question: Calculate the average distance between two randomly chosen points from the Cantor set.

Answer: \dfrac{2}{5}


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

Answer: 0


Question: Solve the equation: \( \sqrt[4]{3(x+5)} - \sqrt[4]{x+13} = \sqrt[4]{11-x} - \sqrt[4]{3(3-x)} \)

Answer: -1


Question: Determine whether the following conjecture is true or false: If two finite-dimensional matrix spaces \(\mathcal{A},\mathcal{B}\subseteq \mathrm{M}(n,\mathbb{F})\) have the same dimension, then there exist \(P,Q\in\operatorname{GL}(n,\mathbb{F})\) such that \(\mathcal{B}=Q^{-1}\mathcal{A}P\).

Answer: False


Question:
The angle between the pair of tangents drawn from a point P to the circle \(x^2 + y^2 + 4x - 6y + 9\sin^2\alpha + 13\cos^2\alpha = 0\) is \(2\alpha\). Determine the equation of the locus of the point P. Choose the correct option:

A) \(x^2 + y^2 + 4x - 6y + 4 = 0\)

B) \(x^2 + y^2 + 4x - 6y - 9 = 0\)

C) \(x^2 + y^2 + 4x - 6y - 4 = 0\)

D) \(x^2 + y^2 + 4x - 6y + 9 = 0\)

Answer: D


Question: Prove that \((5/p) = 1\) if \(p \equiv 1 \pmod{5}\) without using quadratic reciprocity.

Answer: 1


Question: Let $X$, $Y$, and $Z$ be independent Poisson random variables with parameter $\lambda = 1$. Calculate $E[X + Y \mid X + Y + Z = n]$. Provide your answer in terms of $n$.

Answer: \dfrac{2n}{3}


Question: Find all polynomials \( P(x) \) with integer coefficients such that the equation \( P(x)=0 \) has no positive integer solutions and for all positive integers \( n \), \( \phi(n) \mid \phi(P(n)) \). Assume \( P(n) > 0 \) for all positive integers \( n \).

Answer: P(x) = x^k


Question: Calculate the volume of the solid obtained by rotating the region bounded by the hyperbola \( y^2 - x^2 = 1 \) and the line \( y = 2 \) about the \( x \)-axis.

Answer: 4\pi\sqrt{3}


Question: Evaluate the definite integral: $$\int_{0}^{\infty}\frac{4x\ln (x)}{x^4+4x^2+1}dx$$

Answer: 0


Question: Find a measurable function \( g: \mathbb{R} \to \mathbb{R} \) such that if \( X \sim \mathcal{N}(0,1) \), then \( \mathbb{E}(g(X)) = 2 \).

Answer: 2


Question: Let $M \in M_n$ be a normal matrix with distinct singular values, and suppose $M^*M$ is a real matrix. Is $M$ necessarily symmetric? Justify your answer.

Answer: Yes


Question: Determine the asymptotic lower bound of the sum \( S = \frac{3}{2} + \sum_{k=3}^{n} \frac{n!}{k(n-k)!n^k} \).

Answer: \Omega(\log n)


Question: Given a linear operator $T$ on a complex vector space $V$ with characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^3(x-5)$, determine the nature of the operator induced by $T$ on the quotient space $V/\ker(T-5I)$. Choose the correct option: \(a\) The operator is nilpotent. \(b\) The operator is a scalar multiple of the identity operator.

Answer: a


Question: Determine the set of all points where the Taylor series of the function \( f(x)=\sum_{n=0}^{\infty} \frac {x^2}{(1+x^2)^n} \) around the point \( x=e \) converges to \( f(x) \).

Answer: \mathbb{R} \setminus \{0\}


Question:
Consider two CW-complexes $A \subset B$ where the inclusion is a retract. Let $H_{\ast}$ denote the ordinary homology with integral coefficients. Suppose there is a continuous map $\mu: B \times B \rightarrow B$ such that:

1. The restriction $\mu: A \times A \rightarrow A$ is well-defined.
2. There exists an element $b \in B$ such that $\mu(b, b') = \mu(b', b) = b'$ for any $b' \in B$.
3. There exists an element $a \in A$ such that $[a] = [b]$ in $H_{0}(B)$.

Does it follow that the map $\mu$ induces a map in homology $\bullet: H_{\ast}(A) \otimes H_{\ast}(A) \rightarrow H_{\ast}(A)$ such that $[a] \bullet [a_{n}] = [a_{n}] \bullet [a] = [a_{n}]$ for any $[a_{n}] \in H_{n}(A)$?

Answer: Yes


Question:
Given two sequences of positive numbers \((x_n)_{n\in\mathbb{N}}\) and \((y_n)_{n\in\mathbb{N}}\), suppose:

1. There exists a \(\gamma>1\) such that \(\gamma^n|x_n-y_n|\rightarrow0\) as \(n\rightarrow\infty\).
2. \(\prod_{n=1}^{\infty}x_n = 0\).

Can we conclude that \(\prod_{n=1}^{\infty}y_n = 0\)? Justify your answer.

Answer: Yes


Question: Let $f$ and $g$ be two polynomials in $k[X_1,X_2]$ such that \(\frac{\partial f}{\partial X_2} = \frac{\partial g}{\partial X_1}.\) Determine if there exists a polynomial $h$ in $k[X_1,X_2]$ such that \(\frac{\partial h}{\partial X_1} = f\) and \(\frac{\partial h}{\partial X_2} = g.\)

Answer: Yes


Question: If $H$ and $K$ are both normal subgroups of a group $G$, and every element of $G$ can be uniquely expressed as $hk$ with $h \in H$ and $k \in K$, does it follow that $H$ and $K$ commute, meaning $kh = hk$ for all $h \in H$ and $k \in K$?

Answer: Yes


Question: Using Rouché's theorem, determine the number of zeros of the polynomial \( z^7 - 5z + 1 \) within the annular region \( \{ z \mid 1 < |z| < 3 \} \).

Answer: 6


Question: Consider a sequence of Borel probability measures \( \mu_n \in \mathcal{M}(X) \) converging to \( \mu \in \mathcal{M}(X) \) in the weak\(^*\) topology, where \( X \) is a space and \( \mathcal{M}(X) \) denotes the set of Borel probability measures on \( X \). For each \( n \in \mathbb{N} \), there exists a finite set \( F_n \subseteq A \) such that \( \mu_n(F_n) = \mu_n(A) \). Is there a finite set \( F \subseteq A \) such that \( \mu(A) = \mu(F) \)?

Answer: No


Question: Find the derivative \( \frac{\partial h}{\partial U} \) where \( h(U) = \lambda \sum_{i,j} U_{i,j}^2 \). Express \( h(U) \) in terms of the trace and compute the derivative.

Answer: 2\lambda U


Question:
Consider the nested sets $A_n$ and $B_n$ for $n \in \mathbb{N}$, where $A_1 \supseteq A_2 \supseteq A_3 \supseteq \dots$ and $B_1 \subseteq B_2 \subseteq B_3 \subseteq \dots$. Let the cardinality of $A_n$ be $a_n$ and the cardinality of $B_n$ be $b_n$. Determine which of the following statements is correct:

1. The cardinality of $\cap_{n=1}^\infty A_n$ is $\lim\limits_{n\to \infty} a_n$.
2. The cardinality of $\cup_{n=1}^\infty B_n$ is $\lim\limits_{n\to \infty} b_n$.

Provide a justification for your answer.

Answer: 2


Question: Calculate the expectation \( E[\ln{x}] \) for the probability density function \( f(x;\theta) = \frac{1}{\theta^2}x^{\frac{1-\theta}{\theta}} \) where \( 0 < x < 1 \) and \( 0 < \theta < \infty \).

Answer: -1


Question: Find the extrema (maximum and minimum values) of the function \( f(x,y) = xy \ln(x^2+y^2) \) for \( x > 0 \) and \( y > 0 \).

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


Question: Evaluate the integral: \[ I = \int_{-2}^2 4(4-x^2)^{3/2} \, dx \]

Answer: 24\pi


Question: Find the remainder when \(4^{101}\) is divided by 101, without using Fermat's Little Theorem.

Answer: 4


Question: Determine the poles or removable singularities of the function \( f(z) = z \cot(z) \) and find the order of each pole.

Answer: 1


Question: Find the limit \( \lim\limits_{|z| \rightarrow \infty} e^{iaz^2} \) where \( a \) is a positive integer and \( 0 < \arg z \leq \pi /4 \).

Answer: 0


Question: Isaac is planning a nine-day holiday. Each day, he will either go surfing, go water skiing, or rest. He never does different water-sports on consecutive days. How many possible schedules are there for his holiday?

Answer: 3363


Question: In a finite abelian group \( G \), two subgroups \( H \) and \( K \) both have an index of 3 in \( G \). Determine the index of the intersection \( H \cap K \) in \( G \).

Answer: 9


Question: Find the expected value of the maximum of two independent exponentially distributed random variables X and Y, where X has parameter \( \lambda_1 = 1 \) and Y has parameter \( \lambda_2 = 2 \).

Answer: \dfrac{7}{6}


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

Answer: -\dfrac{2}{3}


Question:
Determine the number of real solutions to the equation:
\[ \sqrt{\frac{x-8}{1388}}+\sqrt{\frac{x-7}{1389}}+\sqrt{\frac{x-6}{1390}}=\sqrt{\frac{x-1388}{8}}+\sqrt{\frac{x-1389}{7}}+\sqrt{\frac{x-1390}{6}} \]

Answer: 1


Question: Find a bijection to demonstrate that the set \( B = \{ A \cup \mathbb{N}_{\text{even}} : A \subseteq \mathbb{N}_{\text{odd}} \} \) has cardinality \( \mathfrak{c} \).

Answer: \mathfrak{c}


Question:
Determine the number of real roots of the polynomial equation:
\[ x^9 + \frac{9}{8}x^6 + \frac{27}{64}x^3 - x + \frac{219}{512} = 0. \]

Answer: 3


Question: Evaluate the limit: \[ \lim_{x \rightarrow 0} (\tan(x) + \sec(x))^{\csc(x)} \]

Answer: e


Question: Evaluate the limit \( L = \lim\limits_{n \to \infty} \int_0^{n a} \exp\left(-\dfrac{t}{1+\frac{b t}{n}}\right) dt \), where \( a > 0 \) and \( b > 0 \).

Answer: 1


Question: Find the closest value to the integral \( \int_0^1 \sqrt{1+\frac{1}{3x}} \, dx \) from the following options: 1.6, 2, 1.2.

Answer: 1.6


Question: For the parabola $y = -x^2$, let $a < 0$ and $b > 0$. Define points $P(a, -a^2)$ and $Q(b, -b^2)$. Let $M$ be the midpoint of $PQ$, and $R$ be the point of intersection of the vertical line through $M$ with the parabola. Find the ratio of the area of the region bounded by the parabola and the line segment $PQ$ to the area of triangle $PQR$. Express your answer in terms of $a$ and $b$.

Answer: \dfrac{4}{3}


Question:
Find the limit of \( \left(S_n\right)^{\frac{1}{n}} \), where \( S_n=\frac{n^n}{(n+1)(n+2) \ldots(n+n)} \). Choose the correct option for the limit as \( n \to \infty \):

(1) \( \frac{e}{2} \)
(2) \( \frac{e}{4} \)
(3) \( \frac{e}{8} \)
(4) \( 0 \)

Answer: B


Question: Evaluate the integral \( \int_0^1 \frac{x^k-1}{\ln x} \, dx \) and verify that it equals \( \ln(k+1) \) using high school techniques.

Answer: \ln(k+1)


Question: Let $T$ be a self-adjoint trace-class operator on $L^2(\mathbb{R})$. Can $T$ be represented as an integral operator with the kernel given by \( k_T(x,y) = \sum_{i=1}^\infty \lambda_i \phi_i(x) \bar\phi_i(y) \), where \( \{\phi_i\} \) is an eigenbasis of $T$ and \( T=\sum_i \lambda_i |\phi_i\rangle\langle\phi_i| \)? Verify if this representation is correct.

Answer: Yes


Question: Consider a $(G,X)$ manifold $M$ with local charts $(U,\varphi_U)$, where $\varphi_U$ is a diffeomorphism onto an open subset of $X$ and the transition maps are locally-$G$. Let $\mathfrak{p}:\widetilde{M}\rightarrow M$ be the universal covering of $M$. The developing map theorem introduces a local diffeomorphism $dev:\widetilde{M}\rightarrow X$. Does the developing map locally commute with the (restricted) covering and local charts, i.e., for $\widetilde{U}$ (small enough), is $dev|_\widetilde{U}=\varphi_{U}\circ\mathfrak{p}|_\widetilde{U}$ for some $\varphi_U$? Provide a justification for your answer.

Answer: Yes


Question: Compute the double integral \( \iint_S yz \, dS \) over the surface of the half-cylinder defined by \( x^2 + z^2 = a^2 \), where \( z \le 0 \) and \( 0 \le y \le h \).

Answer: -a^2 h^2


Question: Let $A_n = K[[X_1, \ldots, X_n]]$ be a formal power series ring over a field $K$ of characteristic $p > 0$ in $n$ variables. A monomial $X_{i_1}^{e_{i_1}} \cdots X_{i_n}^{e_{i_n}}$ is called an $\epsilon$-monomial if it satisfies $e_{i_j}/p^{i_j} < \epsilon$ for all $j$. Let $\alpha$ and $\beta$ be two $\epsilon$-monomials. For $x, y \in (X_1, \ldots, X_n)$, the unique maximal ideal of $A_n$, define the product $P_{\alpha,\beta}(x,y) = (\alpha + x)(\beta + y)$. If $x$ and $y$ are composed of monomials different from $\alpha$ and $\beta$ respectively, does $P_{\alpha, \beta}(x, y)$ always contain a non-zero $2\epsilon$-monomial?

Answer: Yes


Question: Let $f_1,\ldots,f_n:\mathbb{R}^n\to \mathbb{R}$ be convex functions, and let $f=\sum_{i} f_i$. Assume each $f_i$ has a unique minimum $x^*_i$, and $f$ has a unique minimum $x^*$. Let $g_1,\ldots,g_n:\mathbb{R}^n\to \mathbb{R}$ be convex functions such that $f_i(x)=g_i(x)$ if $\|x-x^*_i\|\geq 1$, and $f_i(x)\leq g_i(x)$ if $\|x-x^*_i\|\leq 1$, where $\|\cdot\|$ is a norm. Let $g=\sum_{i} g_i$. Determine if there exists a minimum $x'^*$ of $g$ such that $\|x^*-x'^*\|\leq 2$. Provide a justification for your answer.

Answer: Yes


Question: A square pyramid has equilateral triangular faces. A cuboid with dimensions 25 (height) by 50 (width) by 50 (length) is placed inside the pyramid such that the bottom of the cuboid is 26 units above the base of the pyramid. Determine the side length of the square base of the pyramid required for the cuboid to be fully contained within the pyramid.

Answer: 50 + 51\sqrt{2}


Question: Does there exist a topological space $X$ that is not contractible, but whose singular chain complex $C_*(X)$ is contractible?

Answer: Yes


Question: Evaluate \( \lim_{n\rightarrow\infty} \int_{0}^{\pi} \frac {\sin x}{1+ \cos^2 (nx)} \, dx \).

Answer: \sqrt{2}


Question: Suppose $V_{n}$ is a decreasing sequence of bounded open sets in $\mathbb{R}^{m}$ with $m \geq 1$, such that the intersection of all $V_{n}$ is empty. Let $F$ be the intersection of the closures of $V_{n}$. Is there an integer $N$ such that every point $x$ in $F$ belongs to the boundary of $V_{n}$ for all $n \geq N$?

Answer: Yes


Question: Evaluate the double integral \( \int\int_R x^3 \, dA \), where the region \( R \) is bounded by the triangle with vertices at \((0, 0)\), \((7, 1)\), and \((1, 7)\).

Answer: 960


Question: Given that \(3\sin \alpha \sin \beta + 4\cos \beta + 5\cos \alpha \sin \beta = 5\sqrt{2}\) for all \(\alpha, \beta \in \mathbb{R}\), find the value of \(|adj (adjA)|\).

Answer: 0


Question:
Calculate the determinant of the following $2n \times 2n$ matrix $A$:
\[
A=\left(
\begin{array}{ccccccc}
a & 0 & 0 & \ldots & 0 & 0 & b \\
0 & a & 0 & \ldots & 0 & b & 0 \\
0 & 0 & a & \ldots & b & 0 & 0 \\
\ldots & \ldots & \ldots & \ldots \\
0 & 0 & b & \ldots & a & 0 & 0 \\
0 & b & 0 & \ldots & 0 & a & 0 \\
b & 0 & 0 & \ldots & 0 & 0 & a \\
\end{array}
\right)
\]

Answer: (a^2 - b^2)^n


Question: Determine the radius of convergence for the power series \( \sum_{n=0}^{\infty} 2^n z^{n^2} \).

Answer: 1


Question: Calculate the expected hitting time \( t_3 \) for a simple random walk on a 3-dimensional hypercube \( C_3 \), where \( t_3 \) is the expected time to hit a vertex \( B \) starting from a vertex \( A \) with Hamming distance 3. Assume the random walk has a \( 1/n \) chance of moving to each of the \( n \) adjacent vertices.

Answer: 10


Question:
Determine whether there exists a non-constant continuous function \( f: \mathbb{R} \rightarrow \mathbb{R} \) and matrices \( A = \begin{pmatrix} a_1 & 0 \\ 0 & a_2 \end{pmatrix} \) and \( B = \begin{pmatrix} b_1 & 0 \\ 0 & b_2 \end{pmatrix} \) such that there are no matrices \( C \in \mathrm{Mat}_{d\times 2} \), \( D \in \mathrm{Mat}_{2\times d} \) and vectors \( c \in \mathbb{R}^d \), \( e \in \mathbb{R}^2 \) with only non-zero entries, satisfying the equation:

\[
D f_d\left(C \begin{pmatrix} x \\ x \end{pmatrix} + c \right) + e = A f_2\left(B \begin{pmatrix} x \\ x \end{pmatrix} \right) \quad (\forall x \in \mathbb{R})
\]

where \( f_i(x) = (f(x_1), \dots, f(x_n)) \).

Answer: Yes


Question: Evaluate the integral of the function \( f(x, y) = x + 2y \) over the region \( I = [0, 2] \times [0, 1] \) using the Archimedes-Riemann theorem.

Answer: 4


Question: Consider a continuous map $f: X \to Y$ between topological spaces, where $Y$ is irreducible and $X$ has finitely many irreducible components. If $f$ is surjective, does there exist an irreducible component of $X$ such that the restriction of $f$ to this component is surjective onto $Y$?

Answer: No


Question: Consider a strictly increasing function \(f(x)\) defined on \(\mathbb{R}\). Determine whether the following statement is true or false: For any convergent strictly increasing sequence \(a_n\), the sequence \(f(a_n)\) is also convergent. Provide a justification for your answer.

Answer: True


Question: Consider a set of functions $p_k(t)$ that are linearly independent, with $k > 0$. Let $u(t)$ be any continuous and integrable function. Define $\alpha_k = \int_{0}^{T} p_k(T-\tau)u(\tau)\,d\tau$, where $T$ is a fixed number. Does the set of vectors of the form $(\alpha_1, \alpha_2, \ldots, \alpha_k)$, for different choices of $u(t)$, span $\mathbb{R}^k$?

Answer: Yes


Question: Evaluate \( \lim\limits_{x\to \frac{\pi}{2}} (\tan(x))^{\cos(x)} \).

Answer: 1


Question: Determine whether there exists a non-constant entire holomorphic function $f$ such that $|f(z)|$ is constant on an analytic simple closed curve $\gamma$ in $\mathbb{C}$ that surrounds the origin.

Answer: Yes


Question: Evaluate the double integral \( \int\int_D (1+xy) \, \mathrm{d}x \, \mathrm{d}y \) where \( D \) is the region defined by \( y \geq 0 \) and \( 1 \leq x^2 + y^2 \leq 2 \).

Answer: \dfrac{\pi}{2}


Question: Solve the equation \(\sqrt{x}^{\sqrt{x}^{\sqrt{x}}} = 2^{512}\).

Answer: 16


Question: Find the expected number of trials needed such that the sum of numbers picked from a uniform distribution over $[0, 1]$ is at least 1.

Answer: e


Question: Find the radius of convergence of the power series \(\sum_{n=1}^\infty a_{n}x^n\) given that \(\lim_{n\to \infty} |na_{n}|=1101.\)

Answer: 1


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0} \frac{(1+x)^{\frac{1}{x}}-e}{x} \]

Answer: -\dfrac{e}{2}


Question: Let $f: A \times B \rightarrow C$ be a continuous function such that for any closed subsets $R \subset A$ and $S \subset B$, the image $f(R \times S)$ is a closed subset of $C$. Given this, if $T$ is any closed subset of $A \times B$, is $f(T)$ a closed subset of $C$?

Answer: No


Question: Is there a connected topological space that is maximal compact, but not $T_2$? A space $(X,\tau)$ is said to be maximal compact if for any topology $\tau'$ on $X$ with $\tau'\supseteq \tau$ and $\tau'\neq \tau$, the space $(X,\tau')$ is not compact.

Answer: Yes