Question: Given the joint probability density function $f(x, y) = 8xy\mathbf{1}_D(x, y)$, where $D = \{(x, y) \in \mathbb{R}^2 \mid 0 < x < y < 1\}$, calculate the probability density function of $Z = \max\{|X|, |Y|\}$. Express your answer in terms of $z$ for $z \in (0, 1)$. Provide the final expression for $f_Z(z)$.

Answer: 4z^3


Question: In a quadrilateral with known angles, where edges BC and CD are of equal length, determine the ratio in which the diagonals divide the angle \( \alpha \) into \( \alpha_1 \) and \( \alpha_2 \).

Answer: \dfrac{\sin \beta}{\sin \delta}


Question: Let $f : \mathbb{R} \to \mathbb{R}$ be a measurable function such that $F(x) = \int_0^x f(y)\ \mathrm{d}y$ exists and is bounded, i.e., $\|F\|_\infty = M$. Must $F$ be continuous? Provide a justification for your answer.

Answer: Yes


Question: Find the maximum value of the function \( f(x) = 2(a-x)(x+\sqrt{x^2+b^2}) \), where \( a, b, \) and \( x \) are real numbers.

Answer: a^2 + b^2


Question: Find the sum of the absolute values of all real numbers $x$ such that both fractions $\frac{x^2+4x−17}{x^2−6x−5}$ and $\frac{1−x}{1+x}$ are integers. Express this sum as $\frac{a}{b}$, where $a$ and $b$ are coprime positive integers. What is $a+b$?

Answer: 7


Question:
Determine the last digit of the following expression for any natural numbers \(n\) and \(m\):
\[
({}_{4n+1} C_0 )^{4m+1} + ({}_{4n+1} C_1 )^{4m+1} +({}_{4n+1} C_2 )^{4m+1} + \cdots + ({}_{4n+1} C_{4n+1} )^{4m+1}.
\]

Answer: 2


Question: Given the differential equation \( y'' = y' + y \) with initial conditions \( y(0) = 4 \) and \( y'(0) = 7 \), find the value of \( y^{(5)}(0) \), the fifth derivative of \( y \) evaluated at zero.

Answer: 47


Question: Consider a holomorphic function \( f: \mathbb{C}^n \to \mathbb{C}^n \). If \( f \) vanishes on a set \( E \subset \mathbb{R}^n \subset \mathbb{C}^n \) such that \( E \) has a limit point in \( \mathbb{R}^n \), is \( f \) identically zero?

Answer: Yes


Question:
Evaluate the double integral \( \int \int_D (1-y) \, dx \, dy \) over the region \( D \) defined by the inequalities:

1. \( x^2 + y^2 \leq 2y \)
2. \( y \leq x^2 \)
3. \( x \geq 0 \)

Determine the appropriate domain for \( x \) and \( y \) and compute the integral.

Answer: \dfrac{1}{15}


Question: Determine the number of complex numbers \( z \) such that \( |z| < \frac{1}{3} \) and \( \sum_{r=1}^{n} a_r z^r = 1 \), where \( |a_r| < 2 \).

Answer: 0


Question: Given that \(a + b + c = 1\), \(ab + bc + ca = 2\), and \(abc = 3\), find the value of \(a^4 + b^4 + c^4\).

Answer: 13


Question: Evaluate the limit \( \lim_{x \to 0} x \ln(\sin(x)) \) without using L'Hôpital's Rule.

Answer: 0


Question: Calculate \(5^{5^{33}} \mod 100\).

Answer: 25


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

Answer: 4\pi i


Question: Does there exist a complex function $f: \mathbb{C} \to \mathbb{C}$ such that $f$ is differentiable at exactly one point $a \in \mathbb{C}$ and is nowhere else continuous?

Answer: Yes


Question: Determine the number of solutions for the complex equation \(z^3 + \overline{z} = 0\).

Answer: 5


Question: Let $X$, $Y$, and $Z$ be independent random variables uniformly distributed on the interval $(0,1)$. Calculate the probability that $X + Y > Z$. Express your answer as a single probability value.

Answer: \dfrac{5}{6}


Question: Find a polynomial \( f \in \mathbb{Z}[x] \) that is irreducible over \( \mathbb{Q} \) but whose reduction modulo \( p \) is reducible for the primes \( p = 2, 3, \) and \( 5 \).

Answer: x^4 + 1


Question: Determine whether there always exists a subfield of \(\mathbb{C}\) that serves as a splitting field for a polynomial \(f(x) \in \mathbb{Q}[X]\).

Answer: Yes


Question: Given a matrix $A \in M_3(\mathbb{R})$ such that $AA^T = A^TA = I$ and $\det(A) = 1$, determine if $p_A(1) = 0$, where $p_A(x)$ is the characteristic polynomial of $A$. Provide a justification for your answer.

Answer: Yes


Question: Two equally strong tennis players play against each other until one player wins three games in a row. Each game is independent, and each player has a probability of \(\frac{1}{2}\) of winning any given game. What is the expected number of games they will play?

Answer: 7


Question: Consider a continuous function $f$ defined on the interval $[a,b]$ such that $|f(x)| < 1$ for all $x \in [a,b]$. Does the sequence of functions $[f(t)]^n$ converge uniformly to $0$ on $[a,b]$? Justify your answer.

Answer: Yes


Question: Determine the value of \( \liminf f_n \) for the sequence of functions \( f_n = \chi_{[n, n+1]} \).

Answer: 0


Question: Let $L$ be the line of intersection of the planes $ax+by+cz=1$ and $a'x+b'y+c'z=1$. Given that $$(a-a')^2+(b-b')^2+(c-c')^2=16$$ and $$(ab'-a'b)^2+(bc'-b'c)^2+(ac'-a'c)^2=9,$$ find the perpendicular distance of $L$ from the origin.

Answer: \dfrac{4}{3}


Question: Let $G$ be a finite abelian group with $|G| = n$. Determine the modulus of the determinant of the character table of $G$. Express your answer in terms of $n$.

Answer: n^{n/2}


Question: Let \( f : \mathbb{R} \to \mathbb{R} \) be a function that satisfies the intermediate value property. For every integer \( n \ge 2 \), does the function \( f_n : \mathbb{R} \to \mathbb{R} \) defined by \( f_n(x) = (f(x))^n \) for all \( x \in \mathbb{R} \) also satisfy the intermediate value property?

Answer: Yes


Question: Evaluate the integral \( \oint_C \frac{dz}{z-3} \) where \( C \) is the circle \( |z-2|=5 \). Determine if the result contradicts Cauchy's theorem.

Answer: 2\pi i


Question: Let $F$ be a one-to-one function such that $F = f \circ g$. Given that $g(1) = 2$, $g'(1) = 3$, $f(2) = 3$, and $f'(2) = 4$, find $(F^{-1})'(3)$. Provide your answer as a single numerical value.

Answer: \dfrac{1}{12}


Question: Let $G$ be a finite group and $x, y$ be distinct elements of $G$ with the same order. Is it possible to embed $G$ into a finite group $G_1$ such that $x$ and $y$ become conjugate in $G_1$?

Answer: Yes


Question: At a party, 32 people are invited, and each person shakes hands with every other person multiple times. Each pair of people shakes hands at least twice and up to X times, with every pair exchanging a different number of handshakes than any other pair. What is the minimum possible value of X?

Answer: 497


Question: Given a function $u$ defined on a connected open set $\Omega$ of $\mathbb{R}^n$ with $u \in C^2(\Omega)$ and $u(0)=0$, consider the hypersurface $X=\{(x,u(x))~|~x\in\Omega\}$ and the ball $B_r(0)=\{ x \in \mathbb{R}^{n+1}~|~|x|<r \}$. Determine if there exists a small $r>0$ such that $X \cap B_r(0)$ is connected.

Answer: Yes


Question:
Find the function \( f: \mathbb{R}^+ \to \mathbb{R}^+ \) that satisfies the following conditions:

1. \( f(xf(y)) = yf(x) \) for all \( x, y \in \mathbb{R}^+ \).
2. \( \lim_{x \to \infty} f(x) = 0 \).

Determine \( f(x) \).

Answer: \dfrac{1}{x}


Question: Let $A = \overline{\bigcup A_n}$ and $B = \overline{\bigcup B_n}$ be AF algebras with the same Bratelli diagram. Consider an isomorphism $\phi : A \rightarrow B$. If $x \in \bigcup A_n$, is it always true that $\phi(x) \in \bigcup B_n$?

Answer: Yes


Question:
Determine if the following matrix is invertible over the ring of integers modulo a certain integer:
\[
\begin{pmatrix}
54401 & 65432 & 45530 & 45678\\
34567 & 12121 & 11111 & 12345\\
12345 & 76543 & 98760 & 65456\\
43211 & 45678 & 88888 & 98765
\end{pmatrix}
\]

Answer: Yes


Question: Evaluate \( \lim_{x \to \infty} e^{-x^2} \int_x^{x + \ln(x)/x} e^{-t^2} \, dt \).

Answer: 0


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

Answer: -1


Question: Calculate the value of \( \frac{113A}{61} \) where \( A = \frac{(4\cdot2^4 + 1)(4\cdot4^4 + 1)(4\cdot6^4 + 1)}{(4\cdot1^4 + 1)(4\cdot3^4 + 1)(4\cdot7^4 + 1)} \).

Answer: 41


Question: Let $A$ and $B$ be square real symmetric matrices of the same degree $n \geq 3$, without common isotropic vectors (i.e., there is no nonzero vector $x \in \mathbb{R}^n$ such that $x^T A x = x^T B x = 0$). Are all the roots of the equation $\det(A - \lambda B) = 0$ with respect to $\lambda$ real?

Answer: Yes


Question: Let $G_1$ and $G_2$ be limit groups, and let $C_1$ and $C_2$ be cyclic subgroups of $G_1$ and $G_2$, respectively. If $G$ is the amalgamated product of $G_1$ and $G_2$ with amalgamated subgroups $C_1$ and $C_2$, is it true that $d(G) = d(G/G^{'})$? Here, $d(G)$ denotes the minimum number of generators for the group $G$.

Answer: Yes


Question: Determine whether the following statement is true or false: Let $A, B, C \in M_n(\mathbb{R})$ be matrices such that $A$ commutes with $B$, $B$ commutes with $C$, and $B$ is not a scalar matrix. Then $A$ commutes with $C$. Provide a justification for your answer.

Answer: False


Question: Evaluate the surface integral \(\iint_S (x, y, 3z) \, dS\) over the region \(S\), where \(S\) is the finite portion of the surface \(z = 4x^2 + y^2\) cut off by the plane \(z = 8x + 4y - 4\), and the normal to \(S\) points upwards.

Answer: 20\pi


Question: Suppose $f\colon [0,\infty)\to\mathbb{R}$ is continuously differentiable and both $f$ and $f'$ are Lebesgue integrable on $[0,\infty)$. Prove that $\lim_{x\to\infty}f(x)=0$. Provide a detailed explanation of your reasoning.

Answer: 0


Question:
Given the heat equation with von Neumann boundary conditions:
\[ u_t - \alpha u_{xx} = 0, \quad 0 < x < L, \ t > 0 \]
\[ u_x(0,t) = u_x(L,t) = 0, \quad t > 0 \]
and the initial condition:
\[ u(x,0) = \begin{cases} 0, & 0 < x < L/2 \\ 1, & L/2 < x < L \end{cases} \]

Using the ansatz:
\[ u(x,t) = \sum_{n=1}^\infty C_n \cos \frac{n\pi}{L} x e^{-\alpha \frac{n^2\pi^2}{L^2} t} \]

Find the correct value of the Fourier coefficient $C_0$ such that the solution satisfies the initial condition. Explain why $C_0$ should be $\frac{1}{2}$.

Answer: \frac{1}{2}


Question:
Solve for $x$ in the equation:
\[ \left(\sqrt{49+20\sqrt6}\right)^{\sqrt {a \sqrt{a\sqrt{a\cdots \infty}}}}+(5-2\sqrt{6})^{x^2+x-3-\sqrt{x\sqrt{x\sqrt{x}\cdots \infty}}}=10 \]
where $a=x^2-3$ and $a \neq 0$. Determine the value of $x$.

Answer: 2


Question: Given the sequence defined by $u_1 = -2$, $u_2 = -1$, and $u_{n+1} = \sqrt[3]{n(u_n^2 + 1) + 2u_{n-1}}$, find the value of $u_{2021}$. Provide your answer as a single number.

Answer: 2018


Question: Let $x, y, z, t$ be positive real numbers such that $x + y + z + t = 4$. Determine whether the inequality \[ x\sqrt{y} + y\sqrt{z} + z\sqrt{t} + t\sqrt{x} \leq 4 \] holds for all such $x, y, z, t$. Provide a proof or a counterexample.

Answer: 4


Question: Determine the degree of the differential equation for the family of circles that touch the lines $y^2 - x^2 = 0$ and lie in the first and second quadrants, with centers on the $y$-axis.

Answer: 2


Question: Evaluate the boolean sum \( \sum_{x_1, x_2, x_3, x_4, x_6, x_7} \neg(x_1 \oplus x_4 \oplus x_3 \oplus x_6) \land \neg(x_4 \oplus x_3 \oplus x_2 \oplus x_7) \). Determine the number of assignments to the variables \( x_1, x_2, x_3, x_4, x_6, x_7 \) that make the expression true.

Answer: 16


Question: Let $p, q, r$ be distinct primes and $a, b$ be natural numbers greater than 1. Must there exist some $a, b$ such that $p \mid (q^a - r^b)$?

Answer: Yes


Question: Find the minimum value of \(a+b+c\) given that the quadratic equations \(x^2+3x+5=0\) and \(ax^2+bx+c=0\) have a common root, where \(a, b, c \in \mathbb{N}\). Use the condition for a common root: \((3c-5b)(b-3a)=(c-5a)^2\).

Answer: 9


Question: Find a basis for the vector space of functions from the integers to the reals, subject to the condition that for any integer $n$, the function satisfies $f(n) = \frac{f(n + 1) + f(n - 1)}{2}$. What is a basis for this space of functions?

Answer: \{1, n\}


Question: Determine whether there exists a natural number \( k \) such that for all primes \( p \), the expression \((p+1)k-1\) is composite.

Answer: No


Question: Calculate the degree of the field extension \(\mathbb{Q}(a,b)\mid\mathbb{Q}(b)\), where \(a=\sqrt{5}+\sqrt{-5}\) and \(b=5^{\frac{1}{4}}\).

Answer: 2


Question: Calculate the area between the parametric curve \( p(t) = \begin{pmatrix} a(t-\sin(t)) \\ a(1-\cos(t)) \end{pmatrix} \) for \( t \in (0, 2\pi) \) and the x-axis.

Answer: 3\pi a^2


Question: A particle moves along a straight line with a resistance force proportional to its velocity. Initially, the particle has a velocity of 1,000 ft/s. When the particle has traveled 1,200 ft, its velocity is 900 ft/s. Calculate, to the nearest hundredth of a second, the time it took to travel this distance.

Answer: 1.26


Question: Find the total differentiation of the expression $x^T A$ with respect to the vector $x$, where $x$ is a vector and $A$ is a matrix.

Answer: A^\top


Question: Determine whether there is exactly one symmetric positive definite matrix $X$ such that $F(X) = X^n - \sum_{i=0}^n C_i \circ X^i = 0$, where $C_i$ are symmetric positive definite matrices, $\circ$ denotes the Schur (component-wise) product, and exponentiation is with respect to that same product.

Answer: Yes


Question: Is there a compact Hausdorff space $X$ and a continuous injection $\iota: X \hookrightarrow X$ such that $\iota(X)$ has empty interior relative to $X$?

Answer: Yes


Question: Calculate the arc length of the curve given by the polar equation \( r = 1 + \cos\theta \) from \( \theta = 0 \) to \( \theta = \pi \). Use the formula for arc length in polar coordinates: \[ \int_{0}^{\pi} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta \] and then multiply the result by 2 to account for the symmetry of the curve.

Answer: 8


Question: Find the value of the expression \( \left(\frac{i+\sqrt{3}}{2}\right)^{200} + \left(\frac{i-\sqrt{3}}{2}\right)^{200} \), given that \( \left(\frac{i+\sqrt{3}}{2}\right)^{100}\left(\frac{i-\sqrt{3}}{2}\right)^{100} = 1 \).

Answer: -1


Question: Calculate the volume of the solid formed by rotating the region enclosed by the curves \(xy=1\), \(x = y^{1/2}\), and \(y = 2\) about the \(y\)-axis.

Answer: \pi


Question: Triangle $ABC$ has sides of length $5$, $6$, and $7$, with vertices $A$, $B$, and $C$ located on the positive $x$-axis, $y$-axis, and $z$-axis, respectively. Let $O$ be the origin. Calculate the volume of tetrahedron $OABC$. Express your answer in simplest form.

Answer: \sqrt{95}


Question: Let $G$ be a finite group and $p$ a prime that divides $|G|$. Let $n$ denote the number of elements in $G$ that have order $p$. If the Sylow $p$-subgroup of $G$ is normal, determine whether $p$ divides $n+1$. Justify your answer.

Answer: Yes


Question: Given a proper field extension $L/K$, is it possible for $L$ to be isomorphic to $K$ via a field isomorphism?

Answer: Yes


Question: Determine the value of \(\aleph_1^{\aleph_0}\) in set theory.

Answer: 2^{\aleph_0}


Question: Determine the dimension of the vector space \(Pol(\mathbb{Z}_3)\), where \(Pol(\mathbb{Z}_3)\) consists of polynomial functions of one variable over the field \(\mathbb{Z}_3\).

Answer: 3


Question: Evaluate the integral \( \oint_C \frac{z^2-1}{z^2+2} \, dz \), where \( C \) is a circle of radius 2 centered at \( z=0 \), using the Cauchy Integral Formula.

Answer: 0


Question: Given a circular list of coins, all initially showing Tails, determine the minimum number of moves required to make all coins show Heads. In each move, flipping the coin at position $i$ also flips the coins at positions $i-1$ and $i+1$. What is the minimum number of moves needed to achieve this for a list of $N$ coins?

Answer: \dfrac{N}{\gcd(N, 3)}


Question: In a bipartite graph, consider a matching \( M \) and an alternating path \( P \) with respect to \( M \) that includes a \((1,1)\) edge, where both endpoints of this edge prefer to be matched together rather than with their partners in \( M \). Will the operation \( M \oplus P = M' \) always result in a matching \( M' \) that is more popular than \( M \)?

Answer: No


Question: A fair die has two faces each of brown, black, and white colors. Let Y be the random variable representing the number of tosses needed for all three colors to appear at least once, given that two of the three colors appear in the first k-1 tosses and all colors appear in the first k tosses. Calculate the expected value of Y, E[Y].

Answer: \dfrac{11}{2}


Question: Define polynomials \(f_n(x)\) for \(n \geq 0\) by \(f_0(x) = 1\), \(f_n(0) = 0\) for \(n \geq 1\), and \(\frac{d}{dx}(f_{n+1}(x)) = (n + 1)f_n(x + 1)\) for \(n \geq 0\). Find, with proof, the explicit factorization of \(f_{100}(1)\) into powers of distinct primes.

Answer: 101^{99}


Question: Determine whether it is possible to uniquely identify the symmetric matrix $B$ given that $C = BAB$, where $A$ is a known diagonal matrix, $C$ is a known symmetric matrix, and all matrices are positive semi-definite and of full rank.

Answer: Yes


Question:
Evaluate the limit of the series as \( n \to \infty \):
\[ \lim_{n \to \infty} \frac{1^3 + 4^3 + 7^3 + \ldots + (3n-2)^3}{\left(1 + 4 + 7 + \ldots + (3n-2)\right)^2} \]

Answer: 3


Question: Given the equation \(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{23} = \frac{a}{23!}\), find the remainder when \(a\) is divided by 13.

Answer: 7


Question: Let \(\omega\) be the 2-form in \(\mathbb{R}^2\) given by \(\omega = dx \wedge dy\). Consider the map \(\Phi: (0, \infty) \times (0, 2\pi) \to \mathbb{R}^2\) defined by \(\Phi(r, \theta) = (r\cos \theta, r\sin \theta)\). Compute the pullback \(\Phi^*\omega\). Then, evaluate \((\Phi^*\omega)_p(v, w)\) at the point \(p = \left(1, \frac{\pi}{2}\right)\) with vectors \(v = (1, 0)\) and \(w = (3, 1)\).

Answer: 1


Question: Consider 8 distinct boxes placed in a line, each containing between 1 and 8 red balls. The difference in the number of balls between any two adjacent boxes must be at most 1. Determine the number of distinct ways to distribute the balls among the boxes.

Answer: 11814


Question: Let $A$ be a finite abelian group and $p$ a prime number. Define $A^p = \{pa : a \in A\}$ and $A_p = \{a \in A : pa = 0\}$. Prove that $A/A^p \cong A_p$ by finding a surjective homomorphism $f: A \to A_p$ with kernel $A^p$.

Answer: A/A^p \cong A_p


Question: Find an element $a \in \mathbb{F} = \mathbb{Z}_2[x]/(x^3+x+1)$ such that the order of $a$ is 7.

Answer: x


Question:
Given the set of formulas \( \Gamma = \{ S \rightarrow R, \neg R \rightarrow P, S \lor \neg P \} \), deduce the formula \( R \) using the following rules of deduction:

1. You may define any tautology, e.g., write \((S \lor \neg P) \rightarrow (P \rightarrow S)\).
2. You may write any formula either previously deduced or in \( \Gamma \).
3. You may use modus ponens, e.g., if you know \( \beta_1 \rightarrow \beta_2 \) and \( \beta_1 \), then you also know \( \beta_2 \).

Answer: R


Question:
Determine the value of \( \alpha \in \mathbb{R} \) such that the wave equation
\[
\begin{align*}
 & u_{tt}-c^2u_{xx}=0 & \text{in } Q \\
 & u(t,0)=0 & t \ge 0 \\
 & u_t(t,ct)=\cos t - \alpha & t \ge 0
\end{align*}
\]
has a solution in \( C^2(Q) \cap C^1(\overline{Q}) \), where \( c > 0 \) and \( Q = \{(t,x) \in \mathbb{R}^+ \times \mathbb{R}^+ : 0 < x < ct\} \). Also, provide the explicit form of the solution. The general solution is of the form
\[
u(t,x) = u_1(x-ct) + u_2(x+ct), \text{ where } u_1,u_2 \in C^2.\]

Answer: 1


Question: Airplanes enter a rectangular land with a sector length of $50 \text{ nm}$. The spacing between airplanes as they enter is $20 \text{ nm}$ plus an exponentially distributed random variable with a mean of $1 \text{ nm}$. Each airplane travels at $300 \text{ nm}$ per hour. Calculate the average number of airplanes in the land.

Answer: \dfrac{50}{21}


Question: Is there an algorithm to determine the rank of a group generated by side-pairing isometries of a compact convex polyhedron in \( \mathbb{H}^3 \)?

Answer: Yes


Question: Find the integer \(a\) such that the repeating decimal \(0.aaa\ldots\) equals \(\frac{1}{aaa\ldots a}\) for some integer \(k\).

Answer: 3


Question: Suppose $X$ and $Y$ are correlated random variables, each with mean zero, such that $E(XY) \neq 0$. Is it true that $E(X|Y) \neq 0$? Justify your answer.

Answer: Yes


Question: A player starts a game with $i$ tokens, and there are $n$ tokens in total. The game lasts for a maximum of $t$ rounds. In each round, the player wins one token with probability $\frac{1}{3}$ or loses one token with probability $\frac{2}{3}$. The game ends when one of the following conditions is met: the player runs out of tokens, the player collects all $n$ tokens, or $t$ rounds are completed. At the end of the game, the player's prize is calculated as $prize = 2^{result}$, where $result$ is the number of tokens the player has at the end. Calculate the expected value of the prize, $E[prize]$. Provide your answer as a single numerical value.

Answer: 2^i


Question: Evaluate the line integral \( \int_C \vec{F}(x,y) \cdot \,d\vec{r} \) where \( \vec{F}(x,y) = \begin{bmatrix} 5y \\ 10x \end{bmatrix} \) and \( C \) is the quarter-circle arc centered at \((0,2)\) going from the origin to \((2,2)\). Use polar coordinates to solve the integral.

Answer: 20 + 5\pi


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left( \frac{7x+10}{1+7x} \right)^{x/3} \]

Answer: e^{\frac{3}{7}}


Question: Consider the following statement about connected sets in a metric space: Let $S_1, S_2, \ldots, S_n$ be connected sets such that $S_i \cap S_{i+1} \neq \emptyset$ for all $1 \le i \le n-1$. Then $S = \bigcup_{i=1}^n S_i$ is connected. Is this statement true? Additionally, does the same reasoning hold for topological spaces?

Answer: True


Question: If \(\sum_{n=1}^\infty\tan^{-1}\left(\frac{4}{n^2+n+16}\right)=\tan^{-1}\left(\frac{\alpha}{10}\right)\), find the value of \(\alpha\).

Answer: 40


Question: Evaluate the limit: \[ \lim_{x \to \pi} \left[ \tan\left(\frac{x}{4}\right) \right]^{\frac{1}{2(x-\pi)}} \] without using L'Hôpital's rule.

Answer: e^{\frac{1}{4}}


Question: Find the remainder when \((2014^{16} + 2016^{32})^9\) is divided by 23.

Answer: 15


Question: Find the coefficient of \(x^{41}\) in the expansion of \((x^5 + x^6 + x^7 + x^8 + x^9)^5\).

Answer: 70


Question: Given two 20-sided dice, X and Y, calculate the expected value of X given that X is greater than Y, denoted as E(X | X > Y).

Answer: 14


Question: What is the lowest dimension of a faithful ordinary representation of the projective unitary group $\rm{PU}(d)$?

Answer: d^2 - 1


Question: A right triangle has integer side lengths and the ratio between its perimeter and area is 2:3. Find the maximum possible perimeter of the triangle.

Answer: 56


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^{\infty}\frac{\sin (n)+11^n+6^n}{(2n)!}x^{2n} \).

Answer: \infty


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

Answer: e^3


Question: Determine the points where the function \( f(x+iy) = x^4y^5 + ixy^3 \) is complex differentiable. Find the derivative of the function at these points.

Answer: 0


Question: Calculate the residue of the function \( \frac{(z^6-1)^2}{z^5(2z^4-5z^2+2)} \) at the pole \( z=0 \).

Answer: \dfrac{21}{8}


Question: Given a positive real number $l$, determine whether there exists a closed hyperbolic surface $X$ such that the injectivity radius of $X$ is not less than $l$. Provide a justification for your answer.

Answer: Yes


Question: Determine the possible cardinalities of the equivalence classes formed by the relation $x \sim y \iff d(x,y) \in \mathbb{Q}$ on the set of real numbers $\mathbb{R}$. Provide a justification for your answer.

Answer: \aleph_0


Question: Determine whether the equality-free theory of the class of strict linear orders is the same as the equality-free theory of the class of strict partial orders in first-order logic without equality.

Answer: No


Question: Given that \(x = 2 + i\) and \(\gcd(a, b, c) = 1\), where \(a, b, c\) are integers, solve the equation \(ax^4 + bx^3 + cx^2 + bx + a = 0\) to find \(|c|\).

Answer: 42


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{dx}{x^2 + 1} \) using the Residue Theorem.

Answer: \pi


Question: Expand \( \frac{1}{x} \) in a power series centered at \( c = 2 \), integrate the series, and use the result to compute \( \log 3 \) to 4 significant figures.

Answer: 1.099


Question: Does there exist a prime number $p$ that does not divide $2^n - 1$ for any natural number $n$?

Answer: 2


Question: Let $G$ be an abelian group generated by elements $x$ and $y$, where the order of $x$ is 16, the order of $y$ is 24, and $x^2 = y^3$. Determine the order of the group $G$. \( \boxed{} \)

Answer: 48


Question: Let \( f(x) = \int_2^x \frac{dt}{\sqrt{1+t^4}} \) and let \( g \) be the inverse of \( f \). Find the value of \( g'(0) \).

Answer: \sqrt{17}


Question: Find the integral of \( \frac{1}{x} \) using first principles, without using the Fundamental Theorem of Calculus.

Answer: \ln|x| + C


Question: Determine the number of inequivalent two-dimensional complex representations of the group $Z_4$. Provide your answer as a single integer.

Answer: 10


Question: Is there a Tychonov space $(X, \mathcal{T})$ with points $a, b \in X$ such that $a$ has a countable neighborhood basis while $b$ does not have any countable neighborhood basis?

Answer: Yes


Question: Is a finite dimensional CW complex with the same homotopy groups as $S^n$ homotopy equivalent to $S^n$?

Answer: Yes


Question: Find the length of the loop of the curve given by the parametric equations: \( x = 3t - t^3 \) and \( y = 3t^2 \). Use the arc length formula \( \int \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \) to determine the correct length.

Answer: 12\sqrt{3}


Question: Let $K \subset \mathbb{C}$ be a closed subset of the complex plane, and let $U$ be the interior of $K$. Consider a continuous bounded function $f: K \to \mathbb{C}$, which is holomorphic on $U$. Additionally, assume that for every closed curve $\gamma \subset K$, the integral $\int_\gamma f(z) \, dz$ vanishes. Can $f$ be uniformly approximated by entire functions?

Answer: No


Question: Does there exist a nowhere continuous monotonic function on a closed interval?

Answer: No


Question:
Let $A=(a_{ij})$ be a $2018 \times 2018$ matrix defined by:
\[
a_{ij} = \begin{cases}
2 & \text{if } i+1=j,\\
\frac{1}{3} & \text{if } i =j+1,\\
0 & \text{otherwise}
\end{cases}
\]
Let $B$ be the leading principal minor of $A$ of order $1009$. What is the rank of $B$?

Answer: 1008


Question: Let $X$ be a compact Hausdorff topological space, and $C(X)$ denote the ring of complex-valued, continuous functions on $X$. Let $A$ be a matrix with entries from $C(X)$ of size $m \times n$ and $b \in \mathbb{C}^{m \times 1}$. Suppose that for each $x \in X$, the equation $A(x) v = b$ has a solution $v \in \mathbb{C}^n$. Does there exist a $V \in C(X)^{n \times 1}$ such that $AV = b$?

Answer: Yes


Question: Given the function \( f(x) = \frac{1+10x}{10-100x} \), and the composition \( f^n = \underbrace{f\circ f\circ f \circ\cdots \circ f}_{n} \), find the sum \( f(1/2) + f^2(1/2) + f^3(1/2) + \cdots + f^{6000}(1/2) \).

Answer: 595


Question:
Let \(n\in\mathbb{N}\) and \(X\) be a complete metric space. Assume there exists \(\epsilon>0\) such that \(\dim B_\epsilon(x)\le n\) for any \(x\in X\). Is it true that \(\dim X\le n\)?

*Note: \(\dim\) refers to the topological dimension, and \(X\) is not assumed to be separable.*

Answer: Yes


Question: Let $u \in H^1(U)$ with $Du = 0$ and $Tu = 0$. Given that $u$ is constant on each connected component of $U$ and $u \in H^1(U) \cap C(\overline{U})$, can you conclude that $u = 0$?

Answer: 0


Question: Given two independent exponential random variables $X_1$ and $X_2$ with mean 1, find the probability that the larger of the two, $X_{(2)}$, is greater than three times the smaller, $X_{(1)}$. That is, calculate $P(X_{(2)} > 3X_{(1)})$. Assume $X_1$ and $X_2$ are identically distributed with rate parameter $\lambda = 1$. Provide your answer as a probability value.

Answer: \dfrac{1}{2}


Question: Find the minimum value of $x$ that minimizes the expression $$x+[c+p(x)wy]\left[1-\frac{p(x)y}{y^{\max}} \right]$$ given that $w=0$. Assume all other variables are constants.

Answer: 0


Question:
Calculate the value of the expression \( \eta_{\mu\nu}\eta^{\mu\nu} \), where \( \eta \) is the Minkowski metric tensor given by:
\[ \eta = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \]

Answer: 4


Question:
Determine whether the following statement is correct:

Given a continuous function \(f(x,y): \mathbb{R}^+ \times \mathbb{R}^+ \rightarrow \mathbb{R}^+_0\) and a function \(g(x)\) such that \(f(x,g(x)) \geq f(x,z)\) for all \(z \geq 0\), is \(f(x,g(x))\) continuous?

Answer: Yes


Question: Let $X$ and $Y$ be independent and identically distributed random variables, each following a uniform distribution on the interval $(0,1)$. Calculate the expected value of $X$ given that $X > Y$, i.e., find $E(X|X>Y)$. Provide your answer as a single value.

Answer: \dfrac{2}{3}


Question: Let \( A = (a_{ij}) \in \mathbb{R}^{n \times n} \) be a symmetric matrix satisfying the following conditions: \( a_{1i} \neq 0 \) for all \( i \), the sum of each row equals zero, and each diagonal element is the sum of the absolute values of the other entries in the row. Determine the dimension of the eigenspace corresponding to the smallest eigenvalue of \( A \).

Answer: 1


Question: Calculate the triple integral \( \iiint_S z^2 \, dx \, dy \, dz \) where \( S \) is the region bounded by \( z = 0 \), \( x^2 + z = 1 \), and \( y^2 + z = 1 \).

Answer: \dfrac{1}{3}


Question: Calculate the flux of the vector field \( \vec{F}(x,y,z) = \langle x, y, 3 \rangle \) out of the region \( T \) bounded by \( z = x^2 + y^2 \) and \( z = 4 \).

Answer: 16\pi


Question: Calculate the limit of the sequence \( a_n = n^{\frac{2}{3}} \cdot ( \sqrt{n-1} + \sqrt{n+1} - 2\sqrt{n} ) \) as \( n \to \infty \).

Answer: 0


Question: Let $\mathsf{C}$ be an abelian category in which every object has finite length, and let $S$ be a simple object in $\mathsf{C}$. For any object $X \in \mathsf{C}$, denote by $[X\colon S]$ the number of times $S$ is a composition factor of $X$. Consider two morphisms $\phi,\psi \colon X \rightarrow Y$ in $\mathsf{C}$. Is the inequality \([\operatorname{im}(\phi +\psi)\colon S] \leq [\operatorname{im}(\phi)\colon S]+[\operatorname{im}(\psi)\colon S]\) true?

Answer: Yes


Question: Given a sequence of sets $E_n \subset [0,1]$ such that $\sum m^*E_n = \infty$, determine whether $m^*(\limsup E_n) > 0$, where $\limsup E_n = \cap_{k=1}^{\infty}\cup_{n=k}^{\infty}E_n$. Justify your answer.

Answer: 0


Question: Compute the fundamental group of the space obtained by attaching a 2-cell to the torus \( \mathbb{S}^{1} \times \mathbb{S}^{1} \) via the attaching map \( f: \mathbb{S}^{1} \to ( \mathbb{S}^{1} \times \mathbb{S}^{1} ) \) defined by \( f(\zeta) = (\zeta^{2}, \zeta^{3}) \).

Answer: \mathbb{Z}


Question:
Let \( X \) be a random variable with the probability density function (p.d.f.) given by:
\[
f_X(x) = \begin{cases}
3x^2, & \text{if } 0<x<1 \\
0, & \text{otherwise}
\end{cases}
\]
If \( W = -\frac{1}{2}\log(X) \), find the probability density function \( f_W(w) \) for \( W \).

Answer: 6e^{-6w}


Question: Given that \( \frac{x^2}{by+cz} = \frac{y^2}{cz+ax} = \frac{z^2}{ax+by} = 2 \), find the value of \( \frac{c}{2c+z} + \frac{b}{2b+y} + \frac{a}{2a+x} \).

Answer: \dfrac{1}{2}


Question:
Given a string \( w \), perform the following operations alternately each day:

1. Remove the last \( m \) characters from \( w \) and prepend them to \( w \). \( m \) is less than the length of \( w \).
2. Remove the last \( n \) characters from \( w \) and prepend them to \( w \). \( n \) is less than the length of \( w \).

Determine the number of days required for the string \( w \) to return to its original form. For example, if \( w \) is "abcde", \( m \) is 2, and \( n \) is 3, the string returns to its original form after 2 days.

Answer: 2


Question:
Consider the differential equation \( \frac{dy}{dx} = y^2 - 1 + \cos x \) with the initial condition \( y(0) = 0 \). Determine which of the following statements is correct about the solution \( y(x) \) on the open interval \((-c, c)\), where \(c\) is a positive real number:

(A) \( y(x) \) has a local maximum at the origin.
(B) \( y(x) \) has a local minimum at the origin.
(C) \( y(x) \) is strictly increasing on the open interval \((-\delta, \delta)\) for some positive real number \(\delta\).
(D) \( y(x) \) is strictly decreasing on the open interval \((-\delta, \delta)\) for some positive real number \(\delta\).

Answer: D


Question: If two graphs are isomorphic, do their adjacency matrices remain identical up to a permutation of both rows and columns? Specifically, if we write the adjacency matrix of the second graph by arranging the rows and columns in a way that corresponds to the arrangement of vertices in the first graph, should the matrices be equivalent up to congruence, i.e., is it true that \( A \rho B \iff A = P^tBP \) for some non-singular matrix \( P \)?

Answer: Yes


Question:
Let \(\mathbb{R}^3\) be equipped with the inner product \(\langle \cdot, \cdot \rangle\) defined by \(\langle \mathbf{u}, \mathbf{v} \rangle = 2u_1v_1 + 5u_2v_2 + 3u_3v_3\) for any vectors \(\mathbf{u} = (u_1, u_2, u_3)\) and \(\mathbf{v} = (v_1, v_2, v_3)\). Determine which of the following subsets of \(\mathbb{R}^3\) are bases of the orthogonal complement of the line \(L = \text{span}\{(1, -1, 1)\}\) in \(\mathbb{R}^3\) relative to \(\langle \cdot, \cdot \rangle\):

(i) \(\{(1, 1, 0), (0, 1, 1)\}\)
(ii) \(\{(-2, 4, 8), (3, -6, -12)\}\)
(iii) \(\{(3, 6, -12), (1, 2, -4)\}\)
(iv) \(\{(-2, 4, 8), (1, 1, 1)\}\)
(v) \(\{(-2, 4, 8), (0, 0, 0)\}\)

(A) Just i.
(B) Just iii.
(C) Just iv.
(D) ii and iv.
(E) ii, iv and v.

Answer: C


Question: Determine the rank of the set of functions from \(\omega\) to \(\omega\).

Answer: \omega + 2


Question: Determine the number of non-negative integer solutions to the equation \(2x + 7y + 11z = 42\).

Answer: 9


Question: A regular octahedron [ABCDEF] is positioned such that its vertices lie on the coordinate axes. The face [ABC] is numbered with the number 1. Number the remaining seven faces with the numbers 2 to 8, using a different number for each face. How many different ways can the remaining faces be numbered such that at least three of the faces concurrent at vertex A have odd numbers?

Answer: 1872


Question: Find the value of \( R_u \) that maximizes the expression \( P = \left(\frac{OCV}{R_i + R_u}\right)^2 R_u \), where \( OCV \) is a constant.

Answer: R_i


Question: Determine the digit in the 2019th decimal place of the number \((5+\sqrt{26})^{2019}\).

Answer: 0


Question: Given a region \( W \) in \( \mathbb{R}^3 \) with volume \( \sqrt{\pi} \), compute the volume of \( f(W) \), where \( f: \mathbb{R}^3 \to \mathbb{R}^3 \) is defined by \( f(x_{1},x_{2},x_{3}) = (x_{1}-x_{2},3x_{1}-x_{3},2x_{2}-x_{3}) \).

Answer: \sqrt{\pi}


Question: Use Taylor series to show that the expression \( P(Dc|-) = \frac{0.98-0.98p}{0.98-0.93p} \) can be approximated by \( 1 - 0.056p \) in the interval \([0, 0.1]\).

Answer: 1 - 0.056p


Question: Evaluate the integral \( \int^{1}_{0}\left(\frac{1}{1-x}+\frac{1}{\ln x}\right)dx \).

Answer: \gamma


Question:
Evaluate the infinite product \( P = \prod_{n=2}^\infty \frac{n^3-1}{n^3+1} \). Choose the correct answer from the options below:

(A) \( 1 \)
(B) \( \frac{2}{3} \)
(C) \( \frac{7}{3} \)
(D) None of the above

Answer: B


Question: Find the coefficient of \(x^{18}\) in the expansion of \((1+x^{3}+x^{5}+x^{7})^{100}\).

Answer: 1254792000


Question: Determine the radius of convergence of the series \( \sum (n^2 \sin(2/n^2))^{n^2+2} x^{n^2} \).

Answer: \dfrac{1}{2}


Question:
Let $f \in L^2([0,1])$ be a square-integrable function. Define the sequence of step functions $f_n$ by
\[ f_n(x) = \sum_{i=1}^n \mathbf{1}_{((i-1)/n,i/n]}(x) \left(n \int_{(i-1)/n}^{i/n} f(t) \, dt\right) \]
for $x \in [0,1]$. Does the sequence $f_n$ converge to $f$ in the $L^2$ norm as $n \to \infty$?

Answer: Yes


Question: Determine the number of real solutions to the equation \(2^{\sin x} + 2^{\cos x} = 2^{3x^2 - 2x^3 - 1}\).

Answer: 1


Question: Determine whether the operator \( T: \mathcal{l}^2 \to \mathcal{l}^2 \), defined by \((Tu)_n = \left(\frac{1}{2} + \frac{1}{n^2}\right)u_n\), is continuous.

Answer: Yes


Question: Calculate the Lebesgue integral of \( \frac{1}{\sqrt{x}} \) over the interval \((0,1]\).

Answer: 2


Question: Determine the order of the zero at \( z = 0 \) for the function \( f(z) = (e^z - 1)^{12} \sin z \).

Answer: 13


Question: Given that $a^2 + b^2 + c^2 = 4$ and $a^3 + b^3 + c^3 = 8$, find the value of $a^4 + b^4 + c^4$. Express your answer as a single number.

Answer: 16


Question: Determine whether there exists a real number \(\alpha\) such that for all \(x > -1\), the inequality \(\ln(1+x) \leq x - \frac{x^2}{2} + \alpha x^3\) holds. If such an \(\alpha\) exists, find its value.

Answer: \dfrac{1}{3}


Question: Calculate the flow of fluid through the closed surface S, which is the boundary of the region bounded by the half-spheres $z = -\sqrt{4-x^{2}-y^{2}}$ and $z = \sqrt{9-x^{2}-y^{2}}$, and the plane $z = 0$. The velocity field of the fluid is given by $\vec{v}(x,y,z)= (y^{3}e^{-z^{2}} + x)\vec{i} + (ze^{x} + y^{2})\vec{j} + (\cos(x^{2}+ y^{2}) +2z)\vec{k}$. Use the divergence theorem to find the flow of fluid through the surface.

Answer: 70\pi


Question: Find the value of \( a \) such that the gradient of the function \( f(x,y) = \ln(2x^4 + ax^2y^2 + 2y^4) \) is orthogonal to the circle centered at the origin with radius \( r = \sqrt{x^2 + y^2} \) at each point \((x, y) \neq (0, 0)\).

Answer: 4


Question: Suppose that \( f: \mathbb{R} \rightarrow \mathbb{R}^n \) is a differentiable function and that \( \lim_{t \rightarrow \infty} f(t) = 0 \). Does it follow that \( \lim_{t \rightarrow \infty} f'(t) = 0 \)? Provide a justification for your answer.

Answer: No


Question: Given a random variable \( X_n \) with \( n \geq 2 \), where \( P\left(X_n = \frac{1}{n}\right) = 1 - \left(\frac{1}{n^2}\right) \) and \( P(X_n = n) = \frac{1}{n^2} \), find \( \lim_{n \to \infty} \operatorname{Var}(X_n) \).

Answer: 1


Question: Find \( \lim_{x \to 0^+} f(x) \) for the function \( f(x) = \begin{cases} \left| \sin\frac{\pi}{2x} \right|^x, & x > 0 \text{ and } x \neq \frac{1}{2n}, \; n \in \mathbb{N} \\ 1, & x = \frac{1}{2n}, \; n \in \mathbb{N} \;. \end{cases} \)

Answer: 1


Question: Let \( \alpha, \beta, \gamma \) be the roots of the polynomial \( x^3 - 3x + 1 = 0 \). Construct a monic polynomial of degree 3 whose roots are \( 1 - \alpha^{-1}, 1 - \beta^{-1}, 1 - \gamma^{-1} \).

Answer: x^3 - 3x + 1


Question: Evaluate the integral \( \int_{|z|=1} \frac{d z}{\sin z} \).

Answer: 2\pi i


Question: Evaluate the line integral \( \oint_{C}\mathbf{F}\cdot\mathrm{d}\mathbf{r} \), where \( \mathbf{F}(x,y) = (2y^2+e^{3x})\mathbf{i} + 3xy\mathbf{j} \) and \( C \) is the curve tracing the x-axis and the circumferences of the circles \( x^2+y^2=4 \) and \( x^2+y^2=16 \) in the upper half-plane, with an anticlockwise orientation.

Answer: -\dfrac{112}{3}


Question: Compute \( \int_C \frac{f'(z)}{f(z)} \, dz \), where \( C \) is the positively oriented disk \( \{ z: |z|=4\} \) and \( f(z)=\frac{(z-8)^2z^3}{(z-5)^4(z+2)^2(z-1)^5} \).

Answer: -8\pi i


Question: Consider a nonlinear operator $N: E \rightarrow E$ on a Banach space $E$ that is cone preserving, meaning it maps the cone $K \subseteq E$ into itself, i.e., $N(K) \subseteq K$. Determine whether $N(0) = 0$ for such an operator.

Answer: No


Question: Identify subfields of \(\mathbb{C}\) other than \(\mathbb{R}\) that are connected as topological spaces with the induced topology from \(\mathbb{C}\).

Answer: \mathbb{C}


Question:
Given Laplace's equation in polar coordinates:
\[ \frac {1}{r} \frac {\partial} {\partial r}  \left(r \frac {\partial U} {\partial r}\right) +  \frac {1} {r^2}  \frac {\partial^2 U} {\partial \theta^2} = 0 \]
with the boundary conditions:
\[ U(a,\theta)=\begin{cases}
2\theta & 0 < \theta <\pi\\
0 & \pi< \theta < 2\pi\\
\end{cases} \]
Calculate \( U(0, \theta) \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the surface integral \( \iint_S x^2 z \,d\rho \), where \( S \) is the part of the cylinder \( x^2 + z^2 = 1 \) that is above the \( xy \)-plane and between the planes \( y = 0 \) and \( y = 2 \).

Answer: \dfrac{4}{3}


Question: Let $A$ be a domain with field of fractions $K$, and let $I$ be a non-zero fractional $A$-ideal. Define $I^{-1}$ as the fractional ideal $\{a \in K \mid aI \subseteq A\}$. Assume $II^{-1} \subseteq \mathfrak{p}$ for some non-zero prime ideal $\mathfrak{p}$ of $A$. If $a(I A_\mathfrak{p}) \subset A_\mathfrak{p}$ for some $a \in K$, does it follow that $a \in I^{-1}A_\mathfrak{p}$?

Answer: Yes


Question: Calculate \(\lim_{n \rightarrow \infty}n^x (a_1 a_2\dots a_n)^{\frac{1}{n}}\) given that \(\{a_n\}\) is a sequence such that \(\lim_{n \rightarrow\infty} {n^x}a_n=a\) for some real \(x\).

Answer: a e^x


Question:
Consider a four-dimensional random vector \(\boldsymbol{X}\) with a covariance matrix given by:
\[
\begin{bmatrix}1&\rho&\rho&\rho\\
\rho&1&\rho&\rho\\
\rho&\rho&1&\rho\\
\rho&\rho&\rho&1\\
\end{bmatrix}
\]
where \(\rho < 0\). Determine the range of the variance \(v\) of the first principal component. Choose the correct statement:

- \(v\) cannot exceed \(5/4\).
- \(v\) can exceed \(5/4\), but cannot exceed \(4/3\).
- \(v\) can exceed \(4/3\), but cannot exceed \(3/2\).
- \(v\) can exceed \(3/2\).

Answer: B


Question: Assume that $\lim_{n\rightarrow\infty}\frac{1}{n}a_n=0$ for a positive increasing sequence $a_n$. Determine whether $\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^n \frac{1}{a_k}$ is always greater than zero, or provide a counterexample where this limit equals zero.

Answer: 0


Question: Evaluate the contour integral \( \int_{|z|=2} \frac{1}{z^3+z^2+z+1} \, dz \), where the contour \(|z|=2\) is traversed counterclockwise.

Answer: 0


Question: Determine the order of the singularity at \( z_0 \) for the function \( \frac{f(z) + g(z)}{r(z) + h(z)} \), where \( f(z) \), \( g(z) \), \( r(z) \), and \( h(z) \) are analytic near \( z_0 \). At \( z_0 \), \( f(z) \) has a pole of order 2, \( g(z) \) has a zero of order 3, \( r(z) \) has a zero of order 2, and \( h(z) \) has a zero of order 1.

Answer: 3


Question:
Consider a connected bipartite graph $G = (V_1 \cup V_2, E)$ with the following properties:
1. The number of vertices in $V_1$ is equal to the number of vertices in $V_2$, i.e., $|V_1| = |V_2| = N$ for some $N \in \mathbb{N}$.
2. For every vertex $v \in V_1$, the degree of $v$ satisfies $2 \leq \mathrm{deg}(v) \leq N$; the same holds for every vertex in $V_2$.
3. For $2 \leq i \leq N$, the number of vertices of degree $i$ in $V_1$ is equal to the number of vertices of degree $i$ in $V_2$.

Does there always exist a perfect matching between $V_1$ and $V_2$ for any graph $G$ that satisfies these conditions?

Answer: Yes


Question:
Consider a Markov chain where each day there is a 0.2 probability of rain in the morning. The probability of rain in the afternoon given it rained in the morning is 0.6, and the probability of rain in the afternoon given it did not rain in the morning is 0.3. John goes to the office in the morning and leaves in the afternoon, and he has a total of 3 umbrellas. Let \( X_n \) be the number of umbrellas in his office at the \( n \)-th night. Given the transition probability matrix:
\[
P_{transition} = \begin{pmatrix}0.92 & 0.08 & 0 & 0 \\ 0.24 & 0.68 & 0.08 & 0 \\ 0 & 0.24 & 0.68 & 0.08 \\ 0 & 0 & 0.24 & 0.76 \end{pmatrix}
\]
What is the proportion of time John has 0 umbrellas in his office at night?

Answer: \dfrac{27}{40}


Question: Given the functions \( \varphi_1(\vec{x}) \) and \( \varphi_2(\vec{x}) \), define the vector field \( \vec{w} = \vec{\nabla}\varphi_1 \times \vec{\nabla}\varphi_2 \). Determine \( \vec{\nabla} \cdot \vec{w} \).

Answer: 0


Question: Determine whether there exist integer solutions \((x, y)\) to the equation \(2004x^2 + 2005y = 1\).

Answer: Yes


Question: Find the maximum value of the function \( A = \sqrt{|x-y|} + \sqrt{|y-z|} + \sqrt{|z-x|} \) given that \( 0 \le x, y, z \le 2 \).

Answer: 2 + \sqrt{2}


Question: Calculate the surface integral of the vector field \( F = \frac{x - (0,0,-1)}{||x - (0,0,-1)||^3} \) over the unit sphere, excluding the point \((0,0,-1)\).

Answer: 2\pi


Question: Solve the congruence $3^x \equiv 2 \pmod{29}$ using Shank's Baby-Step Giant-Step method. Use $k=6$ for your calculations.

Answer: 17


Question: What is the minimum size of a partial order that is universal for all partial orders of size 5, i.e., any partial order of size 5 embeds into it, preserving the order? Compute the value of $f(5)$. It is known that $f(5) = 11$.

Answer: 11


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty}\frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)} \]

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^\infty x^3 \delta(x^2-2)dx \).

Answer: 0


Question: Let $X$ be an infinite-dimensional vector space and $P$ be a subspace of $X$. If $P$ is isomorphic to $X$, must $P$ be equal to $X$? Answer with 'True' or 'False' and justify your answer.

Answer: False


Question: Given the joint probability density function \( f(x,y) = \begin{cases} 8xy & 0<x<1 \text{ and } 0<y<x \\ 0 & \text{elsewhere} \end{cases} \), find the value of \( P(Y < \frac{1}{8} | X < \frac{1}{2} ) \).

Answer: \dfrac{31}{256}


Question: Find the ratio \( \frac{R}{r} \) where \( R \) is the radius of the circumscribed circle of a trapezoid and \( r \) is the radius of the inscribed circle of the same trapezoid.

Answer: \sqrt{2}


Question: Let $U$ be an open and connected subspace of the Euclidean plane $\mathbb{R}^2$, and let $A \subseteq U$ be a subspace homeomorphic to the closed unit interval. Is the set $U \setminus A$ necessarily connected?

Answer: Yes


Question:
Consider the infinite system of equations given by:
\[
p(k+2)q(k+2) - 3p(k+1)q(k+1) + 2p(k)q(k) = 0
\]
for every natural number \(k\), where \(p(k)\) and \(q(k)\) are natural numbers greater than 2. Does there exist a pair of integer sequences \(p(k)\) and \(q(k)\) that satisfy this system for all \(k\)?

Answer: Yes


Question: Let \( f(x) = |x| + |x - 1| + |x - 3| + |x - 6| + \cdots + |x - (1 + 2 + \cdots + 101)| \) for all \( x \in \mathbb{R} \). Define \( m \) as the number of points where \( f(x) \) is non-differentiable, and \( n \) as the number of integral points where \( f(x) \) is minimum. Calculate the value of \( \frac{m+n-10}{18} \).

Answer: 8


Question: Find the derivative of the function \( F(x) = \int_{-\infty}^{-x} f(-u) \, du \) with respect to \( x \).

Answer: -f(x)


Question: Evaluate the infinite series: \[ \sum_{n=1}^{\infty} \frac{2^{\langle n\rangle}+2^{-\langle n\rangle}}{2^n}, \] where \( \langle n \rangle \) denotes the integer nearest to \( \sqrt{n} \).

Answer: 3


Question: Determine whether the following statement is true or false: There cannot exist topologies \(\tau, \tau^{'}\) on an infinite set \(X\) such that the product topology for \((X,\tau)\) and \((X,\tau^{'})\) coincides with the cofinite topology on \(X\times X\).

Answer: A


Question: Evaluate the limit: $$\lim_{x \to 0}\frac{x+2-\sqrt{2x+4}}{3x-1+\sqrt{x+1}}$$

Answer: \dfrac{1}{7}


Question: Let $X$ be a normally distributed random variable representing the IQ of a person, with mean $\mu = 100$ and standard deviation $\sigma = 15$. Compute $\operatorname{Var}(X^2)$. Use the properties of variance and binomial expansion in your solution.

Answer: 9101250


Question: Provide an example of a group that is not abelian but has all of its subgroups as normal subgroups.

Answer: Q_8


Question: Determine the number of independent parameters required to specify all possible conditional probabilities for a set of $n$ events $X_1, X_2, ..., X_n$. Consider all probabilities such as $P(X_i)$, $P(X_i, X_j)$ for $i \neq j$, $P(X_i | X_j)$ for $i \neq j$, $P(X_i, X_j, X_k)$ for $i \neq j \neq k$, $P(X_i | X_j, X_k)$ for $i \neq j \neq k$, and $P(X_i, X_j | X_k)$ for $i \neq j \neq k$. How many of these probabilities must be specified to determine all others using the standard laws of probability?

Answer: 2^n - 1


Question: Determine whether the sequence \(z_n = n\left\{1 - \cos\left(\frac{\theta}{n}\right) - i\sin\left(\frac{\theta}{n}\right)\right\}\), where \(\theta\) is a fixed constant, converges or diverges. If it converges, find its limit.

Answer: -i\theta


Question: Let $V$ be a subspace of $\Bbb R^{13}$ with dimension 6, and $W$ be a subspace of $\Bbb R^{31}$ with dimension 29. Determine the dimension of the space of all linear maps from $\Bbb R^{13}$ to $\Bbb R^{31}$ whose kernel contains $V$ and whose image is contained in $W$. Provide your answer as a single number.

Answer: 203


Question: Evaluate the double integral \( \int_0^1 \int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \, dx \, dy \).

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


Question: Let \( u: \mathbb{R}^N \to \mathbb{R} \) be a smooth function. Determine whether the gradient \( \nabla u \) has a null tangential component to \( \mathrm{supp}\, u \). Specifically, does \( \frac{\partial u}{\partial \nu} = 0 \) imply \( |\nabla u| = 0 \) on \( \mathrm{supp} \ u \), where \( \nu \) denotes the outer normal to \( \mathrm{supp} \ u \)?

Answer: Yes


Question: Given that \( a, b, c, d \) are the roots of the biquadratic equation \( x^4 + px^3 + qx^2 + rx + s = 0 \), find the value of \( \Sigma a^2b^2 \).

Answer: q^2 - 2pr + 2s


Question: Determine the range of the function \( f(x) = 2 \arctan x + \arcsin \frac{2x}{1+x^2} \).

Answer: [-\pi, \pi]


Question: Let $S$ be a subspace of a topological space $X$, and let $E$ be a subset of $S$. Prove that the relative closure of $E$ in $S$ is $E^* \cap S$, where $E^*$ is the closure of $E$ in $X$.

Answer: E^* \cap S


Question: Given the functional equation \( P(P(x)-1) = 1 + x^{36} \), find the value of \( P(2) \).

Answer: 65


Question: Given a multivariate normal random vector $x = [X_1, X_2, X_3, X_4]^T$ with mean vector \( \mu = \begin{bmatrix} 0 \\ -1 \\ 2 \\ 0 \end{bmatrix} \) and covariance matrix \( \Sigma = \begin{bmatrix} 4 & 1 & 0 & 2 \\ 1 & 4 & 3 & 0 \\ 0 & 3 & 9 & 3 \\ 2 & 0 & 3 & 9 \end{bmatrix} \), define \( Y_1 = X_1 + X_3 \) and \( Y_2 = X_2 - X_4 \). If \( Y \sim N(A\mu, A \Sigma A^T) \) with \( A\mu = \begin{bmatrix} 2 \\ -1 \end{bmatrix} \) and \( A \Sigma A^T = \begin{bmatrix} 13 & -1 \\ -1 & 13 \end{bmatrix} \), find the correlation between \( Y_1 \) and \( Y_2 \).

Answer: -\dfrac{1}{13}


Question: Given the parametrization of the curve of intersection of the surfaces $x^2 = 3y$ and $2xy = 9z$ as $x = t$, $y = \frac{t^2}{3}$, $z = \frac{2}{27}t^3$, determine the limits of integration for finding the length of the curve from the point $(0,0,0)$ to $(3,3,2)$. Confirm whether the integration should be from $t = 0$ to $t = 3$.

Answer: [0, 3]


Question: Calculate the probability of reaching state 5 before reaching state 0 when starting from state 3 in a Markov chain, where the process \( \{X_n : n = 0, 1, 2, \ldots\} \) is defined as follows: If \(|X_n|\) is even, then \(P(X_{n+1} = i + 1 | X_n = i) = 3/4\) and \(P(X_{n+1} = i - 1 | X_n = i) = 1/4\). If \(|X_n|\) is odd, then \(P(X_{n+1} = i + 1 | X_n = i) = 1/4\) and \(P(X_{n+1} = i - 1 | X_n = i) = 3/4\). Use one-step analysis to find this probability.

Answer: \dfrac{5}{9}


Question: Find the value(s) of $a$ such that \( \lim_{x\to 0}\frac{a^{\sin x}-a^{\tan x}}{\tan x -\sin x} = 1. \)

Answer: \dfrac{1}{e}


Question: Let $K = \mathbb{Z}_2[x]/\langle x^4 + x^2 + x \rangle$. Find an element $g \in K$ such that $g^2 = x^3 + x + 1$. Provide your answer as a polynomial in $K$.

Answer: x^3 + x + 1


Question: Let $\phi$ be a bilinear form on a vector space with basis $\{x_i\} \cup \{y_j\}$. Suppose $x \in \operatorname{Span}\{x_i\}$ and $y \in \operatorname{Span}\{y_i\}$ satisfy $\phi(x, \operatorname{Span}\{x_i\}) + \phi(y, \operatorname{Span}\{y_i\}) = 0$. Given that $\phi$ is non-degenerate on $\operatorname{Span}\{x_i\}$, can you conclude that $x = 0 = y$?

Answer: No


Question: Calculate the area of the domain D bounded by the loop of the curve \( (x+y)^5 = x^2y^2 \).

Answer: \dfrac{1}{1260}


Question: If $x$ is a real number and $x \in \mathbb{T}$, is $x^n \in \mathbb{T}$ for every $n \in \mathbb{N}$? Here, $\mathbb{T}$ denotes the set of real transcendental numbers.

Answer: Yes


Question: Determine the radius of convergence for the series \( \sum\limits_{n=0}^{\infty} ni^n z^n \).

Answer: 1


Question: Evaluate the integral \( \displaystyle \int_C \frac{1}{e^z-1} \, dz \) where \( C \) is a circle with center \( 2i \) and radius \( 4 \).

Answer: 2\pi i


Question: Let \( A = \{ x \mid 20\{x\} = x + [x] + [x + 0.5] \} \), where \([x]\) is the integer part of \(x\) and \(\{x\} = x - [x]\). If \( S \) is the sum of elements of \( A \), find \([S]\).

Answer: 21


Question: If $y = \sqrt{5+\sqrt{5-\sqrt{5+ \cdots}}}$, find the value of $y^2 - y$. \( \)

Answer: 4


Question: Determine the cardinality of the set $S$, where $S = \{T: \mathbb{R}^3 \to \mathbb{R}^3 \mid T \text{ is a linear transformation over } \mathbb{R}, T(1,0,1)=(1,2,3), \text{ and } T(1,2,3)=(1,0,1)\}$. Find $|S|$.

Answer: 2^{\aleph_0}


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

Answer: e^{-8}


Question: Determine the rank of the elliptic curve given by the equation \( Y^2 = X^3 + p^2X \) where \( p \equiv 5 \pmod{8} \).

Answer: 0


Question: Calculate the limit: \[ \lim_{n\to\infty} \frac{\gamma(n,n)}{\Gamma(n)} \] where \( \gamma(s,x) = \int_0^x t^{s-1} e^{-t} \, dt \).

Answer: \dfrac{1}{2}


Question: Find a set $X$ such that $X \cap P(X) \neq \emptyset$. Provide an example of such a set and justify your choice.

Answer: \{\emptyset\}


Question: A line passes through the point (1, 2). The abscissa of the intersection of the tangent to this line with the x-axis is half the abscissa of the intersection of the tangent with the line itself. Determine the abscissa of the intersection point of the tangent with the x-axis.

Answer: \dfrac{1}{2}


Question: Find the constant \( c \) such that \( \text{P}(\limsup\limits_{n\to \infty} X_n/\sqrt{\log n}=c)= 1 \), where \( X_n \) are a sequence of independent random variables with \( X_k\sim\mathcal{N}(0,1) \). Use the fact that \( \sqrt{2\pi}\xi\exp\left({\xi^2\over 2}\right)\text{P}(X\geq \xi)\to 1 \) as \( \xi \to \infty \).

Answer: \sqrt{2}


Question: Consider 150 identical persons voting for one of three politicians (a, b, c). Determine the number of ways to distribute the votes such that none of the politicians receives an absolute majority (76 votes or more). Show a bijection between these distributions and the number of ways to pick two distinct integers between 0 and 76.

Answer: 2926


Question: Determine the values of \( \alpha \in \mathbb{R} \) for which the integral \( \int_{\mathbb{R}^2} \frac{\sin(x^2+y^2)}{(x^2+y^2+1)^\alpha} \, dx\,dy \) converges.

Answer: \alpha > 0


Question: Find the probability density function for the random variable \( U = \frac{X}{X+Y} \), where \( X \) and \( Y \) are independent random variables with the density function \( f(x) = e^{-x} \) for \( x > 0 \) and \( f(y) = e^{-y} \) for \( y > 0 \).

Answer: 1


Question:
Verify whether the following integral calculation is correct for a smooth function $u: \mathbb{R} \to [0,1]$:
\[ \int\limits_0^1 \big[\chi_{\{u(x)<t\}}(x)\chi_{\{u(y)\geq t\}}(y)+\chi_{\{u(x)\geq t\}}(x)\chi_{\{u(y)<t\}}(y)\big]\mathrm{d}t =|u(x)-u(y)|. \]

Answer: |u(x) - u(y)|


Question: A spider starts at vertex A of a cube and moves randomly along the edges of the cube. Vertex B is the vertex opposite to A. Calculate the expected number of moves the spider will make to reach vertex B for the first time.

Answer: 10


Question: Does the definition of a convex function, which satisfies \(f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)\) for all \(0\leq t\leq 1\), imply that for any non-negative weights \(t_1, t_2, \ldots, t_k\) such that \(\sum_i t_i=1\), the inequality \[f(t_1x_1+t_2x_2+\cdots+ t_kx_k)\leq t_1f(x_1)+t_2f(x_2)+\cdots +t_kf(x_k)\] holds?

Answer: Yes


Question: Given two independent standard normal random variables, X and Y, calculate the probability that \( X^2 + Y^2 \leq 1 \).

Answer: 1 - e^{-1/2}


Question: If sets $A$ and $B$ in Euclidean space are closed, have the same boundary, and their interiors intersect non-trivially, must $A$ equal $B$?

Answer: A = B


Question: Calculate the mass of an object with density function \( \rho(x,y,z) = z \) over the volume \( V = \{ (x,y,z) \mid x^2 + y^2 + z^2 \leq 2, z \geq 0, x^2 + y^2 \geq 1 \} \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit: $$\lim_{x \to 1} (x-1) \zeta (x)$$

Answer: 1


Question: Let \( K = \mathbb{Q}_p \) where \( p \) is a prime number such that \( p-1 \) is divisible by 4. Let \( F/K \) be the unramified extension of degree 4. Determine whether \( F/K \) always has a generator of the form \( \sqrt[4]{a} \) for some integer \( a \).

Answer: Yes


Question: Determine the number of non-isomorphic groups of order 21.

Answer: 2


Question: Find the number of permutations \((a_1, a_2, a_3, a_4, a_5, a_6)\) of \((1,2,3,4,5,6)\) such that \[\frac{a_1 + 1}{2} \cdot \frac{a_2 + 2}{2} \cdot \frac{a_3 + 3}{2} \cdot \frac{a_4 + 4}{2} \cdot \frac{a_5 + 5}{2} \cdot \frac{a_6 + 6}{2} > 6!.\]

Answer: 719


Question: Determine the number of real roots of the quintic polynomial \(f(x) = x^5 + x^3 - 2x + 1\).

Answer: 1


Question: How many distinct ways are there to arrange 48 staves in a circular barrel, where each stave can be oriented either upside up or upside down?

Answer: 5864062367252


Question: In an Erdős–Rényi random graph $G(n, m)$, set $n = 4$ and $m = 5$. What is the probability that the chromatic number of $G(4, 5)$ is exactly 2? In other words, calculate $Pr(\chi(G(4, 5)) = 2)$. The chromatic number of a graph $G$ is the smallest number $k$ for which $G$ is $k$-colorable.

Answer: 0


Question: Evaluate the limit \( \lim_{n \to \infty}{\int_{0}^{1}{\frac{x^n}{1+x}dx}} \).

Answer: 0


Question:
Determine the smallest integer n such that there exists a Pythagorean n-tuple \((X_1, X_2, \ldots, X_n)\) where each \(X_i\) is a prime number and satisfies the equation:
\[ X_1^2 + X_2^2 + \cdots + X_{n-1}^2 = X_n^2 \]

Answer: 6


Question: Determine the radius of convergence of the series \( \sum_{n=0}^\infty \left(\frac{z^2-1}{z^2+1}\right)^n \) using the ratio test.

Answer: 1


Question: Calculate \( Q''\left(\frac{3}{2}\right) \) for the function \( Q(x) = 5 - 3(x-2) + \sum_{n=2}^{\infty}\frac{(x-2)^n}{n(n-1)} \).

Answer: \dfrac{2}{3}


Question: Compute the triple integral \( \int_D f(x, y, z) \, dV \) where \( D \) is the region in the first octant \((0, \infty)^3\) between the cone \( z^2 = (x - 1)^2 + y^2 \) and the sphere \( x^2 + y^2 + z^2 = 1 \). The function \( f : D \to \mathbb{R} \) is given by \( f(x, y, z) = z \sqrt{x^2 + y^2} \).

Answer: \dfrac{2}{75}


Question: Calculate the operator norms \( ||S||, ||T||, ||ST||, \) and \( ||TS|| \) for the operators \( S, T: C[0,1] \rightarrow C[0,1] \) defined by \( S(x)(z) = z \int_{0}^{1} x(t) \, dt \) and \( T(x)(z) = z \cdot x(z) \). It is known that \( ||T|| = ||S|| = ||TS|| = 1 \). Determine \( ||ST|| \).

Answer: \dfrac{1}{2}


Question:
Consider a collection \( S \) of pairwise disjoint open sets in \( \mathbb{R}^2 \). Determine which of the following statements is true:

1. \( S \) cannot be finite.
2. \( S \) cannot be countably infinite.
3. \( S \) cannot be uncountably infinite.
4. \( S \) is empty.

Provide a justification for your answer.

Answer: 3


Question: Define a function $f(a, b)$ that represents $a$ raised to the power of itself $b$ times. For example, calculate $f(4, 3)$. What is the value of $f(4, 3)$?

Answer: 4^{256}


Question:
Let \( \Omega \) be an open subset of \( \mathbb{R}^{d} \). Consider the Sobolev space \( H^{1}(\Omega) := \{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), j=1,\ldots,n \} \) with the norm \( \|u\|^{2}_{H^{1}(\Omega)} := \|u\|^{2}_{L^{2}(\Omega)} + \sum_{j=1}^{n}\|D_{j}u\|_{L^{2}(\Omega)}^{2} \), where \( D_{j}u = \partial u/ \partial x_{j} \) is the distributional derivative. Define the subspaces:

- \( X := \text{closure of }\{ u|{}_{\Omega} \in H^{1}(\Omega): u \in C_{c}(\bar{\Omega})\} \text{ in } H^{1}(\Omega) \)
- \( Y := \text{closure of }\{ u|{}_{\Omega} \in H^{1}(\Omega): u \in C(\bar{\Omega})\} \text{ in } H^{1}(\Omega) \)

where \( C_{c}(\bar{\Omega}) \) denotes the space of all continuous real-valued functions on \( \bar{\Omega} \) with compact support, and \( C(\bar{\Omega}) \) denotes the space of all continuous real-valued functions on \( \bar{\Omega} \). Prove or disprove that \( X = Y \) for an arbitrary open subset \( \Omega \).

Answer: X = Y


Question: Calculate the order of 3 modulo 257.

Answer: 256


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

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to0}{{\left(\frac{\tan{(1+\tan x)}}{\tan{(1+\sin x)}}\right)}^{1/x^3}} \]

Answer: e^{\frac{1}{\sin 2}}


Question: Find the orthogonal trajectories of the family of curves given by the equation \( y^2 = kx^3 \).

Answer: 2x^2 + 3y^2 = C


Question: Find the value of \(x\) such that \[ x! = \frac{(10!)!}{10!} \]

Answer: 3628799


Question: Suppose \( \mu \) and \( \nu \) are measures such that \( \nu(\Omega) = 2 \). Let \( f \) be the Radon-Nikodym derivative of \( \mu \) with respect to \( \mu + \nu \). Find \( \nu(\{x: f(x) < 1\}) \).

Answer: 2


Question: In a $C^*$-algebra, if an element $a$ satisfies $b^*a^*ab = 0$ for all elements $b$ in the algebra, does it necessarily follow that $a = 0$?

Answer: 0


Question: Determine whether the sum \( \sum_{n \neq 0}\left(\frac{1+|a|}{1+|a-n|}\right)^{100}e^{-n^2} \) is bounded independently of \(a\), where \(a\) is a fixed real number.

Answer: Yes


Question: Compute \( \int _0^\infty\frac{x \sin x}{1+x^2}dx \) using the residue theorem.

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


Question: In \(\triangle ABC\) with \(AB=AC\) and \(\angle BAC=20^\circ\), point \(D\) is on \(AC\) such that \(BC=AD\). Find \(\angle DBC\).

Answer: 70^\circ


Question: Let \( p \) be a prime number and \( F \) be a field with \( p^{23} \) elements. Define \( \phi: F \to F \) as the field automorphism that sends \( a \) to \( a^{p} \). Let \( K = \{ a \in F \mid \phi(a) = a \} \). Determine the value of \([K: \mathbb{F}_{p}]\).

Answer: 1


Question:
Compute the following limit related to a series:
\[ \lim_{N\rightarrow\infty}N^2\sum^{N-1}_{k=1}\left(\frac{k}{N}\right)^{N\ln N} \]

Answer: \infty


Question: True or False: If $E_1, E_2, \ldots$ are finite sets, then the set $E = E_1 \times E_2 \times \ldots = \{(x_1, x_2, \ldots) : x_j \in E_j \text{ for all } j \in \mathbb{N}\}$ is countable.

Answer: B


Question: Among the $3 \times 3$ invertible matrices with entries from the field $\mathbb{Z}/3\mathbb{Z}$, how many matrices are similar to the matrix \( \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)?

Answer: 117


Question: Does there exist a valuation ring \((R, \mathfrak{m})\) with a principal maximal ideal and infinite global dimension?

Answer: No


Question: Maximize \(x^2 + y^2 + z^2\) subject to the constraint \(x^2 + y^2 + 4z^2 = 1\).

Answer: 1


Question: Does there exist an irrational number \( \zeta \) such that the sequence \( [a(\zeta)]_{1}^{\infty} \), constructed by taking consecutive digits from \( \zeta \) without repetition, does not cover all natural numbers?

Answer: Yes


Question: Can a Riemann-integrable function have uncountably many discontinuities in its domain of integration?

Answer: Yes


Question: Compute the limit \( \lim_{x \to 0^+} \frac{\tan(x) - x}{x^3} \) without using L'Hôpital's rule.

Answer: \dfrac{1}{3}


Question: Let $X$ be an irreducible variety, and let $U \subset X$ be an open set. Consider the sheaf of regular functions $O_X(U)$ on $U$. If $U' \subset U$ is an open subset, is the restriction map $O_X(U) \to O_X(U')$ injective? Justify your answer.

Answer: Yes


Question: Let \( \{x_n\}_{n \ge 1} \) be a sequence of positive real numbers. If the sequence \( \{x_n^2\}_{n \ge 1} \) is convergent, does it necessarily follow that the sequence \( \{x_n\}_{n \ge 1} \) is also convergent? Provide a justification for your answer.

Answer: Yes


Question: Given the extension \(\dfrac{\mathbb{Z}_7[x]}{\langle 3+x^3\rangle}:\mathbb{Z}_7\) and \(\alpha =x^2+\langle 3+x^3\rangle \in \dfrac{\mathbb{Z}_7[x]}{\langle 3+x^3\rangle}\), find \(\alpha^{-1}\).

Answer: 2x


Question: Find the derivative of the integral \( \frac{d}{dx}\int_{-x}^{x} f(t) \, dt \).

Answer: f(x) + f(-x)


Question: Let \( p \) be a matrix in \( M_n(\mathbb{C}) \) such that \( p \) is almost self-adjoint (i.e., \( ||p-p^*||<\epsilon \)) and almost idempotent (i.e., \( ||p-p^2||<\epsilon \)). Is there a constant \( k \in \mathbb{N} \) and a projection \( q \) (i.e., \( q=q^*=q^2 \in M_n(\mathbb{C}) \)) such that \( ||q-p||<k\epsilon \), where \( k \) does not depend on \( \epsilon \)?

Answer: Yes


Question: Find the residue of the function $g(z) = f(z^2)$ at $z_0 = 0$, where $f(z)$ is analytic in the region $0 < |z| < R$.

Answer: 0


Question: Determine if the recursive formula \( P_m = P_{m-1}G_{m-1,m}A_m + A_0\sum_{i=0}^{m-2}P_i G_{i,m} \) for \( m \geq 2 \) can be expressed solely in terms of \( P_0 \), without involving other \( P_j \) terms.

Answer: Yes


Question: Consider a 3-manifold $M$ that is a solid torus $T = S^1 \times D^2$. Perform a surgery by removing the interior $A = S^1 \times D^2$ and replacing it with $B = D^2 \times S^1$. What is the resulting manifold $M'$?

Answer: S^3


Question: Use Stokes' Theorem to evaluate the line integral \( \int_C{ \vec{F} \cdot \vec{dr}} \), where \( \vec{F} =  \langle x+y^2, y+z^2, z+x^2 \rangle \) and \( C \) is the triangle with vertices (3, 0, 0), (0, 3, 0), and (0, 0, 3), oriented counterclockwise as viewed from above.

Answer: -27


Question: Let $f$ be a differentiable function on $(a,b)$ with $c \in (a,b)$. Suppose $f$ and $f'$ are non-zero on a deleted neighborhood of $c$, and \( \lim_{x \to c} f(x) = 0 \). Determine \( \lim_{x \to c} |f(x)|^{|f(x)|} \).

Answer: 1


Question:
Given that \( f: \mathbb{R} \to \mathbb{R} \) is a differentiable function satisfying the functional equation \( f(x+2y) = f(x) + f(2y) + 4xy \), determine the correct relationship between the derivatives of \( f \) at 0 and 1. Choose the correct option:

1) \( f'(0) = f'(1) + 2 \)
2) \( f'(0) = f'(1) - 2 \)
3) \( f'(0) = f'(1) + 1 \)
4) \( f'(0) = f'(1) - 1 \)

Answer: 2


Question: Given that \(x + y + z + w = 1\) and \(x, y, z, w \geq 0\), is it always true that \(|xy - zw| \leq \frac{1}{4}\)?

Answer: Yes


Question: Given that \( \int_0^4 e^{(x-2)^4}\,dx = k \), find the value of \( \int_0^4 xe^{(x-2)^4}\,dx \).

Answer: 2k


Question:
In the category $\mathsf{C}_{\alpha, \beta}$, what does a morphism correspond to? Consider the following options:

1. A morphism $\sigma: Z_1 \to Z_2$ such that $f_1=f_2\sigma$ and $g_1=g_2\sigma$.
2. A morphism $\sigma: Z_1 \to Z_2$ such that $\alpha f_1=\alpha f_2\sigma$ and $\beta g_1=\beta g_2 \sigma$.
3. A morphism $\sigma: Z_1 \to Z_2$ such that $\alpha f_1=\beta g_2\sigma$ and $\beta g_1=\alpha f_2 \sigma$.

Determine which option correctly describes a morphism in this category.

Answer: 1


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{\lambda x \ln(1+x)+(1-\lambda) x \ln x}{x- \lambda \ln(1+x)+(1-\lambda)\ln x} \] where \( \lambda \in (0,1) \).

Answer: 0


Question: Given the joint density function \( f(y_1, y_2) = 3y_1 \) for \( 0 \leq y_2 \leq y_1 \leq 1 \) and zero elsewhere, find the probability that more than 1/2 of the items are sold given that 3/4 of the items are stocked. Express your answer as a fraction.

Answer: \dfrac{1}{3}


Question: Find the minimum path length between the points \(P_1(\sqrt{2},0,-1)\) and \(P_2(0,\sqrt{2},1)\) on the hyperbolic surface defined by \(S = \{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 - z^2 = 1\}\).

Answer: 2\sqrt{2}


Question: Let \((x_{n})\) be a sequence of non-negative real numbers diverging to infinity. Determine whether there exists a real number \(r\in \mathbb{R}\) such that \(x_{n}-x_{m}\neq r\) for all \(n,m\in \mathbb{N}\).

Answer: Yes


Question: Determine whether the function $e(x,y) = d(f(x),f(y))$ defines a metric on the space $(X,d)$ for any function $f: X \rightarrow X$. Justify your answer by checking the properties of a metric.

Answer: No


Question: Given a circle with the equation \(x^2 + y^2 = 1\), fix a point \(X = (-1,0)\). Draw a random angle \(\theta\) from a uniform distribution \(U(0,\pi)\) to create a second point \(Y = (\cos \theta, \sin \theta)\) on the circle. Choose a third point \(Z\) on the circumference such that the area of the triangle \(XYZ\) is maximized. What is the expected value of the area of this triangle?

Answer: \dfrac{3}{\pi}


Question: Determine whether the limit \( \lim_{(x,y) \to (0,0)} \frac{x^3 + y^3}{x^2 + y^2} \) exists.

Answer: 0


Question: Evaluate the integral \( \int_{0}^{\infty} \left( \int_{0}^{\infty} 2x\sqrt{y}e^{-x^2\sqrt{y}-y} \, dy \right) dx \).

Answer: 1


Question: Determine the smallest integer $n$ such that the Simpson's Rule approximation for the integral \( \int_0^1 9e^{x^2} \, dx \) is accurate to within 0.0001.

Answer: 18


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

Answer: 1


Question: Find the limit of \( \frac{\tan(x)-x}{x^3} \) as \( x \) approaches 0 without using L'Hospital's Rule or Taylor series.

Answer: \dfrac{1}{3}


Question: A teller has a servicing time that follows an exponential distribution with a mean of 2 minutes. If a customer, who started being serviced at noon, is still being serviced at 12:05, what is the expected total time the customer will spend in the system? Express your answer as \( E[T_A|T_A \geq 5] \).

Answer: 7


Question: Evaluate the improper integral: $$\int_{0}^{\infty}\frac{\cos(x)-\cos(2x)}{x}dx.$$

Answer: \ln 2


Question: Determine the number of invariant subspaces of a transformation $T$ that maps $v \mapsto J_{\lambda,n}v$, where $J_{\lambda,n}$ is a Jordan block. Provide your answer as a single integer.

Answer: n+1


Question: Find the coefficient of \(x^7y^6\) in the expansion of \((xy+x+3y+3)^8\).

Answer: 672


Question: Consider a non-constant morphism $f: X \to Y$ between integral curves, where $Y$ is non-singular. Determine if it is possible for $X$ to be singular.

Answer: Yes


Question: Find the coefficient of \(x^{19}\) in the expansion of the generating function \(f(x) = \frac{(1-x^{5})^5}{(1-x)^{10}}\).

Answer: 3300000


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

Answer: 2e


Question:
Determine which of the following power series expressions for \( \sin^2(x) \cos(x) \) is correct:

1. \( \sin^2(x) \cos(x) = \sum x^2\frac{(-4)^n (9^{n+1}-1)x^{2n}}{(2n+2)!4^{n+1}} \)

2. \( \sin^2(x) \cos(x) = \frac{1\sum{(-1)^n x^{2n}(1-9)^n}}{4\cdot(2n)!} \)

Provide the correct expression.

Answer: 1


Question: Given that $\sin \angle BOC = \frac{4}{5}$ and $AC=24$, find the area of $\triangle BDE$. Let $B$ be a point on the circle centered at $O$ with diameter $AC$. Let $D$ and $E$ be the circumcenters of the triangles $OAB$ and $OBC$, respectively. Calculate the area of $\triangle BDE$.

Answer: 45


Question: Determine the Galois group of the polynomial \( f(x) = x^3 + 2x + 2 \) over \( \mathbb{Q} \).

Answer: S_3


Question: Find the real part of the expression \(\sum_{k=1}^{9} (-1)^{k+1} \beta^k\), where \(\beta = \cos\frac{9\pi}{19} + i\sin\frac{9\pi}{19}\).

Answer: \dfrac{1}{2}


Question: Find the kernel of the homomorphism \( \phi: \mathbb{C}[x,y] \to \mathbb{C} \) defined by \( \phi(z)=z \) for all \( z \in \mathbb{C} \), \( \phi(x)=1 \), and \( \phi(y)=0 \).

Answer: (x - 1, y)


Question: Determine whether there exists a linear algebraic group $G$ over an algebraically closed field $k$ such that all semisimple elements with infinite order lie in the identity component $G^0$. Consider the case where $G$ is not connected, and $1_G \neq s \in G$ is semisimple. The identity component $G^0$ is normal and of finite index $n = [G : G^0]$ in $G$, so $s^n \in G^0$ is semisimple. Is it possible for $s^n = 1_G$ for all semisimple $s \in G - G^0$?

Answer: Yes


Question: In how many ways can you distribute 6 red balls, 7 green balls, and 8 blue balls into 3 different boxes such that each box contains at least one ball of each color?

Answer: 3150


Question: Given two graphs, $G_1=(V_1,E_1)$ with a chromatic number of 7, and $G_2=(V_2,E_2)$ with a chromatic number of 5, where $|V_1 \cap V_2| = 2$, determine the chromatic number of the graph $G_3 = (V_3, E_3)$ where $V_3 = V_1 \cup V_2$. Can the chromatic number of $G_3$ be determined?

Answer: 7


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^\infty \left(\int_0^n\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt\right)x^n \).

Answer: 1


Question: Consider a Markov chain defined on a tree $T$ with countably many nodes, where each node has $n$ neighbors. The chain starts at a random vertex and moves to any of the $n$ neighboring nodes with equal probability $1/n$ at each step. Determine the values of $n$ for which this Markov chain is transient.

Answer: n \geq 3


Question: Simplify the expression: \[ B = \frac{\sqrt{2} + 2(\cos 20^\circ + \cos 25^\circ)}{\sin \left(90 - \frac{45}{2}\right) \sin 55^\circ \sin 57.5^\circ} \]

Answer: 8


Question: Given that the function $f(x,y)$ is differentiable at $(1,1)$ and $f(1,1)=\frac{\partial f}{\partial x}(1,1)=\frac{\partial f}{\partial y}(1,1)$, find the value of $\frac{d}{dx}f(x,f(x,x))$ when $x=1$. Provide your answer as a single number.

Answer: 3


Question: In a gambling game with 100 rounds, a random number is drawn from an unknown continuous distribution at the start. In each subsequent round, a new random number is drawn from the same distribution. Gambler A bets that the next random number will be greater than the previous one. What is the expected number of rounds in which gambler A is correct?

Answer: 49.5


Question: If \( \int_1^\infty f(x) x^n \, dx = 0 \) for all integers \( n > 1 \), is it true that \( f(x) = 0 \) for a continuous function \( f \)?

Answer: 0


Question: Let \( f(x) = \log_3(\sqrt{4-x} + 81) \) and \( g(x) = 3^{x-2} + 1 \). Given that \( g(a) = f^{-1}(b) \) and \( f(b) = g^{-1}(c) \), find the value of \( a + b + c \).

Answer: 17


Question: Given the equation \( h = g^a \mod p \), where \( p = 541 \), \( g = 2 \), and \( h = 419 \), find the integer \( a \).

Answer: 66


Question: Find the point on the surface defined by \( x^2 + y^2 + z^2 = 8 \) that is nearest to the point \( P = (0, 1, 1) \) using Lagrange multipliers.

Answer: (0, 2, 2)


Question: Suppose for a fixed integer $m$, the inequality $a_{n,m} < b_{n,m}$ holds for all $n \geq 1-m$. Given that \( \limsup_{n \to \infty} \frac{1}{n} \ln b_{n,m} = c \), determine whether \( \limsup_{n \to \infty} \frac{1}{n} \ln a_{n,m} < c \) or \( \limsup_{n \to \infty} \frac{1}{n} \ln a_{n,m} \leq c \).

Answer: \limsup_{n \to \infty} \frac{1}{n} \ln a_{n,m} \leq c


Question: Suppose we have a probability space \((\Omega, \mathcal{F}, \mu)\), where \(\Omega\) is a set, \(\mathcal{F}\) is a \(\sigma\)-algebra, and \(\mu\) is a probability measure. Let \(f: \Omega \rightarrow \mathbb{R}\) be a non-negative bounded function. Consider a sequence of sets \(A_i\), \(i=1, 2, \dots\), in \(\mathcal{F}\) such that \(\int_{A_i} f\ d\mu \leq a \mu(A_i)\) for each \(i\) and \(\mu\left(\bigcup_{i=1}^{\infty} A_i\right) = 1\). Is \(\int_{\Omega} f\ d\mu\) upper bounded by a scalar multiple of \(a\)?

Answer: a


Question:
Find the sum of all integral values of \( k \) such that the curve \( x^4 + 3kx^3 + 6x^2 + 5 \) is not situated below any of its tangent lines. Choose the correct answer from the following options:

(A) 5
(B) 2
(C) 0
(D) -2

Answer: C


Question: Determine whether the inequality \( p_{n+2} \leq 4(1+2+\ldots+n)+4 \) holds for all natural numbers \( n \), where \( p_{n+2} \) is the \((n+2)\)-th prime number.

Answer: Yes


Question:
Let \( n \) be a positive integer with exactly six divisors, \( 1 = d_1 < d_2 < \dots < d_6 = n \). Find all such \( n \) that satisfy the equation:
\[ \frac{1}{d_1} + \frac{1}{d_2} + \dots + \frac{1}{d_6} = 2. \]

Answer: 28


Question: Find all eigenvalues \( \lambda \) of the linear transformation \( f: \mathbb{R}^N \to \mathbb{R}^N \) defined by \( f((a_n)_{n \ge 1}) = (a_{n+1} - a_n)_{n \ge 1} \).

Answer: -1


Question: Verify that the function $f \star \phi$, where $\phi = \frac{1}{2} e^{-|x|}$, is a solution to the differential equation $u - u'' = f$. Provide your reasoning.

Answer: f \star \phi


Question:
Given a matrix $A$ of size $(m, n)$, a column vector $b$ of size $m$, and a column vector $x$ of size $n$, consider the least-squares solution $\begin{bmatrix} x \\ a \end{bmatrix}$ to the linear matrix equation:
\[
\begin{bmatrix}
A & \begin{matrix}
-1 \\
-1 \\
... \\
-1
\end{matrix}
\end{bmatrix}
\begin{bmatrix}
x \\
a
\end{bmatrix} = b
\]
Does this imply that the arithmetic mean of $Ax - b$ equals $a$?

Answer: Yes


Question: Does there exist a category \( C \) such that there is no functor \( F: C \rightarrow D \) with \( D \not\cong C \) that has a left or right adjoint?

Answer: Yes


Question: Evaluate the integral \( \frac{1}{2\pi} \int_0^{2\pi} \left(z_0 + re^{i\theta}\right)^j d\theta \) where \( U \subseteq \mathbb{C} \) is an open set, \( z_0 \in U \), \( r > 0 \), \( \{z:|z-z_0|\le r\} \subseteq U \), and \( j \in \mathbb{Z}^+ \).

Answer: z_0^j


Question: Calculate the value of $y_{500}(2)$, where the sequence of functions $\{y_i\}$ is defined recursively by $y_1(x) = \dfrac{x + 1}{x - 1}$ and $y_{i+1} = (y_1 \circ y_{i})(x)$ for $i \ge 1$. It is known that $y_2(x) = x$, and the sequence repeats on a 2-cycle.

Answer: 2


Question: Let $A$ and $B$ be complex square matrices such that $A^*A = B^*B - BB^*$, where $A^*$ denotes the conjugate transpose of $A$. Prove that $A = 0$.

Answer: A = 0


Question: Is the set of compactly supported functions $C^\infty_c((0,1])$ dense in the space $\{u \in H^1(0,1) \mid u(0)=0\}$?

Answer: Yes


Question: Given a sample of random variables \((X_1, \ldots, X_n)\), uniformly distributed in the interval \([0, 2\vartheta]\) with \(\vartheta > 0\) unknown, determine the value of the parameter \(\gamma\) such that the estimator \(\theta_3 = \gamma\left(\max_{i=1,\ldots,n}X_i + \min_{j=1,\ldots,n}X_j\right)\) is unbiased.

Answer: \dfrac{1}{2}


Question: Evaluate the line integral \( \int_C \left( \frac{1}{x+y}\mathbf{i} + \frac{1}{x+y}\mathbf{j} \right) \cdot \text{d}r \) where \( C \) is the segment of the unit circle going counterclockwise from \((1,0)\) to \((0,1)\). Use parameterization to solve the integral.

Answer: 0


Question: Determine the values of \( \alpha \) for which the integral \( \iint_{\mathbb{R}^2} \frac{dx\,dy}{(x^2 - xy + y^2 + 1)^\alpha} \) converges.

Answer: \alpha > 1


Question: If $p$ is a prime number, determine the number of elements of order $p$ in the group $\mathbb{Z}_{p^2} \oplus \mathbb{Z}_{p^2}$. Provide your answer as a single integer.

Answer: p^2 - 1


Question:
Determine the number of terms needed in the Maclaurin series for \( \arctan(x) \) to approximate \( \pi \) with an error less than \( 10^{-3} \), given that \( \tan(\pi/4) = 1 \). Use the series:
\[ \arctan(x) = \sum_{i=1}^\infty (-1)^{i+1} \frac{x^{2i-1}}{2i-1} \]
for \(-1 < x \leq 1\).

Answer: 2000


Question: Let $a, b, c, d, e$ be natural numbers in an arithmetic progression such that $a + b + c + d + e$ is the cube of an integer and $b + c + d$ is the square of an integer. What is the least possible value of the number of digits of $c$?

Answer: 3


Question: In a restaurant, clients arrive following a homogeneous Poisson process with an average rate of 10 clients per hour. Each client independently has a 20% probability of eating at the restaurant and an 80% probability of taking away. On average, how many clients should be expected before one eats at the restaurant?

Answer: 4


Question: Calculate the value of the integral \( \int_{0}^{20} \{x\}^{2} + \left \lfloor{x}\right \rfloor^{2} \, dx \).

Answer: \dfrac{7430}{3}


Question: Let \( \triangle ABC \) be an acute triangle with altitudes \( BB_1 \) and \( CC_1 \). Let \( B_2 \) and \( C_2 \) be the midpoints of \( AC \) and \( AB \), respectively. If \( B_1C_2 \) and \( B_2C_1 \) are perpendicular, find all possible values of \( \angle BAC \).

Answer: 30^\circ


Question: Find the maximum value of \(m+n\) for natural numbers \(m\) and \(n\) such that \(1 + 2^m = 3^n\).

Answer: 5


Question:
Find the derivative of the function \( f(x) \) at \( x = 0 \), where:
\[
f(x) = \begin{cases}
  e^{-\frac{1}{x^2}}\sin\frac{1}{x}, & \text{if } x \neq 0 \\
  0, & \text{if } x = 0
\end{cases}
\]

Answer: 0


Question: Let \( f \) be a holomorphic function on the unit disk \( \mathbb{D} \) such that \( |f(z)| < 1 \) for all \( |z| < 1 \). Given that \( f(0) = \frac{1}{2} \), find a bound for \( |f'(0)| \).

Answer: \dfrac{3}{4}


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

Answer: \dfrac{\pi}{e}


Question: Let $A$ be an $n \times n$ matrix where each entry is either $1$ or $-1$. Prove that $\det(A)$ is divisible by $2^{n-1}$.

Answer: 2^{n-1}


Question:
Find the coefficient $a_{20}$ in the Laurent series expansion of the function $f(z) = \frac{1}{\sin(z)}$ around $z = 0$ using contour integration. Use the formula for the coefficient of the $k$-th term in a Laurent series:
\[ a_k = \frac{1}{2\pi i}\int_{\Gamma} \frac{f(z)}{z^{k+1}}dz \]
Evaluate this integral for $k = 20$.

Answer: 0


Question: Find the limit of the function \((-0.1)^x\) as \(x\) approaches infinity: \[ \lim_{x \to \infty} (-0.1)^x \]

Answer: 0


Question: Given the polynomial \( f(x) = ax^2 - bx + c \) where \( a, b, \) and \( c \) are positive integers, and \( f(p) = f(q) = 0 \) for \( 0 < p < q < 1 \), find the minimum possible value of \( a \). The vertex of the polynomial is at \( \frac{b}{2a} \) and lies between 0 and 1. Additionally, \( f(0) > 0 \), \( f(1) > 0 \), and \( b^2 - 4ac > 0 \). Determine the smallest integer value of \( a \) that satisfies these conditions.

Answer: 5


Question: Determine the number of ordered triples \((a, b, c)\) of positive integers such that \(a \leq b \leq c\) and \(a \cdot b \cdot c \leq 1000\).

Answer: 5708


Question: Calculate the curvature at each point of the Lobachevsky plane \(\mathbb{R}^2_+ = \{(x,y) \in \mathbb{R}^2; y > 0\}\) with the metric given by \(g_{11} = g_{22} = \frac{1}{y^2}, g_{12} = 0\). Use the formula \(K = \frac{R_{1212}}{g_{11}g_{22}-g^2_{12}}\), where \(R_{klij} = \frac{1}{2}(\frac{d^2}{dx^jdx^k}g_{il} + \frac{d^2}{dx^idx^l}g_{jk} - \frac{d^2}{dx^jdx^l}g_{ik} - \frac{d^2}{dx^idx^l}g_{jl} + g_{rs}\Gamma^r_{jk}\Gamma^s_{il} - g_{rs}\Gamma^r_{jl}\Gamma^s_{ik} )\).

Answer: -1


Question: Find the function $g(x)$ that satisfies the equation: $$g(x) = a\sin(x) + \int_0^x \sin(x-u)g(u) \, du$$ using the Laplace Transform and the Convolution Theorem.

Answer: a x


Question: Determine whether the following statement is true or false: Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function. The function $f$ is strictly increasing on $\mathbb{R}$ if and only if $f'(x) \geq 0$ on $\mathbb{R}$ and the set $X = \{x \in \mathbb{R} \,| \, f'(x) = 0\}$ is countable.

Answer: B


Question: Compute \( \iint_S \mathrm{curl} \ F \cdot n \, dS \) where \( S \) is the portion of the surface \( z = x^2 + y^2 \) below \( z = 16 \), with \( n \) as the downward-pointing normal vector, and \( F = (z, x, y^2) \).

Answer: -16\pi


Question: Consider a polynomial $p(x)$ in $n$ variables with complex coefficients, of even degree $2m$, satisfying the strong elliptic condition: \( \text{Re} \, p(x) \ge C|x|^{2m} \) for any \( x \in \mathbb{R}^n \). For any non-zero \( y \in \mathbb{R}^{n-1} \), does the number of roots (counting multiplicities) of \( p(y,z)=0 \) in the upper half of the complex plane equal the number in the lower half?

Answer: Yes


Question: Find the distance from the point \((2, 3, 4)\) to the line given by \(\frac{x+3}{3} = \frac{y-2}{6} = \frac{z}{2}\), measured parallel to the plane \(3x + 2y + 2z + 5 = 0\).

Answer: \sqrt{33}


Question: Find a non-trivial integer solution to the equation $5x^2 + 7y^2 = 3z^2$.

Answer: (1, 1, 2)


Question: Find the limit of the expression \( x^{\left(\frac{1}{x}\right)^{\frac{1}{x}}} - x \) as \( x \to \infty \).

Answer: -\infty


Question:
Let $X$ be a measure space. Consider an increasing sequence of $\sigma$-algebras $S_j$, $j \in \mathbb{N}$, on $X$ such that $S := \bigcup_{j \geq 0} S_j$ is a $\sigma$-algebra. For each $j$, let $\mu_j$ be a probability measure on $S_j$. Let $f_{ij}$ $(i, j \in \mathbb{N})$ be a double-indexed sequence of functions such that for every $j$, $f_{ij}$ converges $\mu_j$-a.e. to an $S_j$-measurable function $f_j$. Suppose there exists a probability measure $\mu$ on $S$ such that $f_j$ converges $\mu$-a.e. to a function $f$. Additionally, assume:

- The restriction of $\mu$ to $S_j$ is absolutely continuous with respect to $\mu_j$ for every $j$.
- $f_{ij}, f_j, f$ are $\mu$-integrable, and $f_{ij}$ is $\mu_j$-integrable for every $i, j$.
- $\int f_{ij} \, d \mu_j \to \int f_j \, d \mu_j$ for every $j$.
- $\int f_j \, d \mu \to \int f \, d \mu$.

Is it true that there exists an increasing function $b: \mathbb{N} \to \mathbb{N}$ such that $\int f_{n, b(n)} \, d \mu$ converges to $\int f \, d \mu$?

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\pi} \exp\left(\cos\left(t\right)\right)\cos\left(\sin\left(t\right)\right)\,\mathrm{d}t \).

Answer: \pi


Question: Evaluate the derivative of the function \( y = \left(1 + \frac{1}{x}\right)^x + x^{1+\frac{1}{x}} \) at \( x = 1 \). Find \( y'(1) \).

Answer: 1 + 2 \ln 2


Question:
Determine the points at which the function \( f : \mathbb{R} \rightarrow \mathbb{R} \) defined by
\[
f(x) =
\begin{cases}
x^2,  & x\in \mathbb{Q} \\
-x^2, & x\in \mathbb{R} \setminus \mathbb{Q}\
\end{cases}
\]
is differentiable.

Answer: 0


Question: Determine the number of ordered triplets \((a, b, c)\) such that the function \(f(x) = a|\sin x| + b \cdot e^{-|x|} + c \cdot |x|^{2015}\) is differentiable at \(x = 0\), where \(0 \leq a, b, c \leq 10\) and \(a, b, c \in \mathbb{Z}\).

Answer: 121


Question:
Given the joint distribution of two random variables $X$ and $Y$:
\[ f(x, y) = \frac{3}{2}x + \frac{y}{2} \] for $x,y \in [0,1]$ and $0$ otherwise, calculate the probability:
\[ P\left(XY < \frac{1}{2}\right) \]

Answer: \dfrac{3}{4}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^{1} nx^n e^{x^2} \, dx. \]

Answer: e


Question: Calculate \( \lim_{n\to \infty} \int_0^1 \frac{nx\sin(x)}{n^3x^2 +1}\,dx \).

Answer: 0


Question: Is it true that if $k$ is a prime knot, then the double branched cover $M(k)$ of $S^3$ along the knot $k$ is a prime 3-manifold?

Answer: Yes


Question: Evaluate the definite integral: \[ \int _{-1}^{3} \left[\arctan \left(\dfrac{x}{x^2+1}\right) + \arctan \left(\dfrac{x^2 + 1}{x}\right)\right]dx. \]

Answer: \pi


Question: Find the greatest common divisor (gcd) of $3 + \sqrt{13}$ and $2 + 5\sqrt{13}$ in the ring $\mathbb{Z}[(1+\sqrt{13})/2]$. Express your answer in the form $a + b\sqrt{13}$, where $a$ and $b$ are integers.

Answer: 1


Question: Let $V$ and $W$ be linear operators on a finite-dimensional vector space over a field of characteristic $0$. Prove that $VW - WV \neq I$, where $I$ is the identity operator.

Answer: VW - WV \neq I


Question: Determine the splitting field $L$ of the polynomial $x^4 - 7x^2 + 10$ over $\mathbb{Q}$ and find the degree $[L: \mathbb{Q}]$.

Answer: 4


Question: Does the general linear group $GL_n(\mathbb{R})$ have any subgroups of finite index?

Answer: Yes


Question: In triangle \( \Delta ABC \), let \( AB = 2017 \text{ units} \). Point \( O \) is the midpoint of \( AB \). An interior point \( D \) is chosen such that \( \angle DAP = \angle DBQ \). Find the value of \( OP - OQ \).

Answer: 0


Question: Evaluate the limit using L'Hospital's rule: \[ \lim_{x\to 0} \left( \frac{1-x}{\sqrt{1-\cos x}} - \frac{1+x}{\arctan x} \right) \]

Answer: \infty


Question: Let \( \mathcal{F} \) be the family of holomorphic functions that map the open unit disc to itself and vanish at the origin along with their first three derivatives. Find \( \sup_{f \in \mathcal{F}} |f(1/2)| \).

Answer: \dfrac{1}{16}


Question: Consider a locally finite, vertex-transitive graph \( \Gamma = (V\Gamma, E\Gamma) \) where each vertex has degree \( d \). Is it possible to color the edges of \( \Gamma \) using \( d+1 \) colors? Assume \( V\Gamma \) is countably infinite.

Answer: Yes


Question: Find the product of all natural numbers \( n \) that satisfy the equation \( P(n) = n^2 - 42n + 440 \), where \( P(n) \) is the product of the digits of \( n \).

Answer: 8640


Question: Evaluate the limit: $$\lim\limits_{x\to 0}\frac{(1+x)^{\frac{\ln x}{x}}-x}{x^2\ln x}$$ You may use Taylor expansion.

Answer: -\dfrac{1}{2}


Question: Evaluate the line integral \( \int_{C} xydx + (x+y)dy \) along the curve \( y = x^2 \) from \((-2, 4)\) to \((1, 1)\).

Answer: -\dfrac{21}{4}


Question: Solve for \(x\) in the equation: \[ 4^{x-1} = 9 \cdot x^{3-x} + 7 \]

Answer: 3


Question: Solve the exponential equation \(2^{a+1} + 2^{b+1} = 9 \times 2^{\frac{a+b-1}{2}}\) and find the value of \(|a-b|\).

Answer: 3


Question: Determine whether the condition $a^m \in \mathbb{R}$ and $a^n \in \mathbb{R}$ is sufficient for $(a^m)^n = a^{mn}$ to hold for all $a, m, n \in \mathbb{R}$, considering the usual rules of exponentiation in real numbers.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n,m\to\infty}\sin^{-1}\left(\frac{(-1)^{m-4n}(1-n^2)}{3mn^2-5n^2+9nm-15n}\right) \] where \(n, m \in \mathbb{N}\).

Answer: 0


Question: Let $S = \bigoplus_{n \geq 0} S_n$ be a graded ring. Suppose $S$ is generated by $S_1$ as an $S_0$-algebra. Is it true that the subring $R = \bigoplus_{n \geq 0} S_{dn}$, for a fixed $d > 0$, is generated by $S_d$ as an $S_0$-algebra?

Answer: Yes


Question: Determine the number of triples \((a, b, p)\) of natural numbers with \(a > b > 0\) such that the equation \[ \left(\sum_{k=1}^n k^a\right)^p = \sum_{k=1}^n k^b \] holds for all natural numbers \(n\). Identify these triples.

Answer: 0


Question: Calculate the surface area of the portion of the cylinder defined by $x^2 + y^2 = 8y$ that is located inside the sphere $x^2 + y^2 + z^2 = 64$. Provide your answer in terms of the appropriate units.

Answer: 256


Question: Calculate the surface integral \( \int_{D} xyz \, dS \), where the surface \( D \) is the part of the sphere \( x^2+y^2+z^2=4 \) located above the region defined by \( y \leq x, \ y \leq 0, \ 0 \leq x^2+y^2 \leq 4 \).

Answer: -2


Question: Determine the degree of accuracy for an $n$-point Gauss-Legendre quadrature when applied to the multidimensional integral \( \int_{a}^{b}{\int_{p}^{q}{f(x,y) \,dx \,dy}} \).

Answer: 2n - 1


Question: Determine the value of \( p \) such that \( \lim_{x \to \infty} \left( x^p \left( \sqrt[3]{x+1} + \sqrt[3]{x-1} - 2\sqrt[3]{x} \right) \right) \) is a finite non-zero number.

Answer: \dfrac{5}{3}


Question: Consider a continuous function from the closed unit disk to itself that is the identity map on the boundary of the disk. Must this function be surjective?

Answer: Yes


Question: Find the integer value of $x$ that satisfies the equation $9 = 2^x \pmod{11}$. This is relevant in the context of the Diffie–Hellman key exchange algorithm.

Answer: 6


Question: Find the integer \(a\) such that \(0 \leq a < p^2\) and \((2p-1)! \equiv a \mod{p^2}\), where \(p\) is an odd prime.

Answer: p


Question: Let $X_1$ and $X_2$ be independent random variables, each following a normal distribution with mean 0 and variance 4, i.e., $X_1, X_2 \sim \mathcal{N}(0,4)$. Calculate the probability $P(X_1^2 < 36.84 - X_2^2)$. Provide your answer as a decimal rounded to four significant figures.

Answer: 0.9900


Question: Let \( g(\theta) = \tan \theta \) be a strictly increasing continuous function mapping \([0, b]\) onto \([0, \tan b]\), where \( b \in (0, \pi/2) \). Verify the equation \( \int_0^b \tan \theta \, d\theta + \int_0^{\tan b} \tan^{-1} y \, dy = b \tan b \).

Answer: b \tan b


Question: Prove or disprove the claim: \( \lim_{n\to\infty} \frac{\gamma(n+1,n)}{\Gamma(n+1)} = \frac{1}{2} \), where \( \gamma(s,x) = \int_{0}^{x}t^{s-1}e^{-t}\,dt \) is the incomplete gamma function and \( \Gamma(s) = \int_{0}^{\infty}t^{s-1}e^{-t}\,dt \) is the gamma function.

Answer: \dfrac{1}{2}


Question: Find the prime number \( p \) such that \( p > 5 \) and \( p \mid (3^{p+1} + 5^{p-1} + 1) \).

Answer: 11


Question: Find the value of $x$ such that $\ln(x) = \frac{1}{x}$.

Answer: 1.763


Question: Let $\alpha$ be a root of the polynomial $x^5 + 6x^3 + 8x + 10$. Determine the number of $\mathbb{Q}$-embeddings of the field $\mathbb{Q}[\alpha, \sqrt{7}]$ into $\mathbb{C}$. Provide your answer as a single integer.

Answer: 10


Question: Let $\ell^\infty$ be the Banach space of real bounded sequences with its usual norm, and let $S \subset \ell^\infty$ be the space of convergent sequences. Define $f: S \rightarrow \mathbb{R}$ by \( f(x) = \lim_{n \rightarrow \infty} x_n \), where \( x = (x_1, x_2, \ldots) \). Extend $f$ to a bounded linear functional $F: \ell^\infty \rightarrow \mathbb{R}$ using the Hahn-Banach Theorem. Determine whether $F$ belongs to $\ell^1$. Provide a justification for your answer.

Answer: No


Question: Let \( \{x_n\} \) be a sequence that converges to zero. If \( \lim_{n \to \infty} \frac{x_{n+1}}{x_n} = x \) exists, prove that \( x \in [-1, 1] \).

Answer: [-1, 1]


Question:
Evaluate the double integral \( \iint_{R} f \cdot g \, dx \, dy \) over the circular disk \( R \) with boundary \( x^2 + y^2 = 1 \), where the vector fields \( f \) and \( g \) are defined as follows:

\[ f(x,y) = v(x,y) \mathbf{i} + u(x,y) \mathbf{j} \]
\[ g(x,y) = \left(\frac{\partial u}{\partial x} - \frac{\partial u}{\partial y}\right) \mathbf{i} + \left(\frac{\partial v}{\partial x} - \frac{\partial v}{\partial y}\right) \mathbf{j} \]

Given that on the boundary of \( R \), \( u(x,y) = 1 \) and \( v(x,y) = y \), find the value of the integral.

Answer: -\pi


Question: The surface given by the equation \( z = x^3 + xy^2 \) intersects the plane given by \( 2x - 2y = 1 \) in a curve. Determine the slope of this curve at the point where \( x = 1 \) and \( y = -\frac{1}{2} \).

Answer: \dfrac{17}{4}


Question: Let \( P(x) \) be a polynomial in \( \mathbb{R}[x] \) with a leading coefficient of 1. Determine the leading coefficient of the polynomial \( P(2x+1) - 2P(x) \).

Answer: 2^n - 2


Question: Calculate the limit: \[ \lim_{t\to \infty}\frac{1}{t^2}\int_0^t \ln(e^x+x^2)\,dx \]

Answer: \dfrac{1}{2}


Question: Determine the capacity of the channel defined by the convolution operator \( C: \ell^1(G) \to \ell^1(G) \), where \( f \mapsto \mu * f \) and \( \mu \) is a probability measure on a finite group \( G \).

Answer: \log |G| - H(\mu)


Question: Determine the number of conjugacy classes of order 3 matrices in the group \( PGL(3, \mathbb{Z}) \).

Answer: 2


Question: Does there exist a field $K$ and a finite-dimensional $K$-division algebra $D$ that has two maximal separable subfields of different dimensions?

Answer: No


Question: Find all values of \( a > 0 \) such that the functions \( f'(x) \) and \( 1 - f''(x) \) have the same range over the domain \((a, \infty)\), where \( f(x) = \ln(xe^x) \).

Answer: 1


Question: Let \(x_1 = a > 0\) and \(x_{n+1} = x_n + \frac{1}{x_n}\) for \(n \in \mathbb{N}\). Determine whether the sequence \((x_n)\) converges or diverges.

Answer: \infty


Question: Find the sum of the series \( \sum_{n=1}^{\infty} u_n \), where \( u_n = \frac{a+n-1}{\prod_{j=1}^n (a+j)} \) and \( a > 0 \).

Answer: 1


Question: If ZFC has a model, does it have a countable model whose Turing degree is bounded by $\emptyset^{(n)}$ for some finite $n$?

Answer: Yes


Question: Is the complex projective plane a compact manifold without boundary?

Answer: Yes


Question: Let \( G \) be a cyclic group of order 7, generated by \( a \) such that \( a^7 = e \). Consider the mapping \( \phi: a^i \to a^{2i} \). Prove that \( \phi \) is an automorphism of \( G \) and determine its order.

Answer: 3


Question: Calculate the expected number of collisions when hashing 80,000 unique entries using a 36-bit hash function.

Answer: 0.0466


Question: Determine the set of all real numbers \(x\) for which the series \(\sum_{n=1}^{\infty} x^{n^2} n!\) is convergent.

Answer: (-1, 1)


Question: Determine the number of prime numbers that do not have any multiple consisting solely of the digit '1'.

Answer: 2


Question: In an acute-angled triangle $ABC$, let $H$ be the orthocenter. From points $A$ and $C$, tangents $AK$ and $CT$ are drawn to the circle with diameter $BH$. Given that the lengths of these tangents are $15$ and $19$ respectively, find the square of the smallest possible length of side $AC$. Express your answer as a single integer.

Answer: 586


Question: Let \( \xi \in T_2(V) \) where \( V = M_n(\mathbb{C}) \) and \( \xi (A,B) = n \cdot \text{tr}(AB) - \text{tr}(A)\text{tr}(B) \). Find the dimension of the orthogonal complement \( V_{\bot\xi} = \{ A \in M_n(\mathbb{C}) \mid \xi(A,B) = 0, \forall B \in M_n(\mathbb{C}) \} \).

Answer: 1


Question: Does there exist a set with non-empty interior whose Lebesgue measure is 0?

Answer: No


Question: Given a real-valued function $f(x)$ defined for all real numbers $x$ such that \(|f(x) - f(y)| \le \frac{1}{2}|x-y|\) for all $x,y$, determine the number of points where the graph of $y = f(x)$ intersects the line $y = x$. Provide your answer as a single integer.

Answer: 1


Question: Evaluate the integral \( \int_0^{2\pi} e^{2i\theta} f(e^{i\theta})\,d\theta \) using Cauchy's Theorem, given that \( f(z) \) is analytic on the disk \( |z| \leq 2 \).

Answer: 0


Question: Let $f(z)$ be a holomorphic function with a zero of order $n \geq 1$ at $z = 0$. Define $h(z) = \frac{f(z)}{f'(z)}$. Compute $h'(0)$ in terms of $n$. (Conjecture: $h'(0) = \frac{1}{n}$.)

Answer: \dfrac{1}{n}


Question: Find the maximum value of \( |xy(x^2+y^2)| \) subject to the constraint \( x^2 + y^2 + xy = 1 \).

Answer: 2


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

Answer: \pi


Question: Determine whether the following statement is true in general: If two singular matrices $A$ and $B$ exist such that the determinant of $EA - B$ is identically zero for all real matrices $E$, then either $A = YB$ or $B = ZA$, where $Y$ and $Z$ are undetermined matrices.

Answer: True