Question: In the normed linear space \(\mathbb{R}^2\) with the norm \(\|(x,y)\|= |x|+|y|\), consider the subspace \(M = \{(x,y)\in \mathbb{R}^2 : x = y\}\). Let \(f\) be the linear functional on \(M\) given by \(f(x,y) = 3x\). If \(g(x,y) = ax+by\), where \(a, b \in \mathbb{R}\), is a Hahn-Banach extension of \(f\) to \(\mathbb{R}^2\), find the value of \(a-b\).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{x^2}. \]

Answer: 3


Question: Determine the possible values of the real number $k$ such that the inequality \( r^k \le C(1+r)^k \) holds for all \( r > 0 \), where \( C \) is a constant independent of \( r \).

Answer: [0, \infty)


Question: Consider an irreducible plane real algebraic curve $R$ without isolated points, such that for any point $(x,y)$ on the curve, the point $(x,-y)$ is also on the curve. Is it possible to find a polynomial $f(x,y)$ defining the zero set $R$ such that $f(x,y) = f(x,-y)$ for all $(x,y) \in \mathbb{R}^2$?

Answer: Yes


Question: In a 3x3x3 tic-tac-toe game with three players A, B, and C taking turns in order, how many unique board states are possible after 6 moves? Assume that each player places their mark in an empty cell during their turn.

Answer: 26640900


Question: Does there exist a binary operation on the set of natural numbers such that it forms a non-commutative group?

Answer: Yes


Question: Calculate the moment of inertia about the x-axis using double integration for the area bounded by the curves \(x = 2y - y^2\) and \(x = y^2\), given that the density at any point \((x, y)\) is \(y + 1\).

Answer: \dfrac{1}{6}


Question: Determine if the derivative of the matrix multiplication map \( \psi: \text{GL}(n,\Bbb R) \times \text{GL}(n,\Bbb R) \rightarrow \text{GL}(n,\Bbb R) \), defined by \( \psi(A,B) = AB \), at the identity is given by \( d_{id,id}\psi(A,B) = A + B \).

Answer: A + B


Question: For a smooth complex projective variety $X$ and an ample divisor $L$, consider holomorphic maps $u: \Sigma \to X$ with degree $\deg(u^*L)$. Given a positive integer $M$, determine if there exists a positive integer $N$ such that for a generic smooth hypersurface $D$ in the linear system $|NL|$, there are no nontrivial holomorphic maps of both degree and genus less than $M$ into $D$.

Answer: Yes


Question: Given that \( g(x) = \frac{P(x)}{x^2-4} \), \( \lim_{x\rightarrow\infty}g(x)=5 \), and \( \lim_{x\rightarrow2}g(x)=\frac{9}{4} \), find the polynomial \( P(x) \).

Answer: 5x^2 - 11x + 2


Question: For the equation \( e^{-0.0042x}(1+0.0042x) = 0.032 \), the general solution is given by \( x = -\frac{5000}{21}(W_{n}(-\frac{4}{125e})+1) \), where \( n \) is an integer. If \( n = 0 \) yields the solution \( x = -235.259 \), determine the integer value of \( n \) that results in the solution \( x = 1256.97 \).

Answer: -1


Question: Let $(X \times Y, \mathcal{F}, P)$ be a probability space where $P$ is a product measure on $X \times Y$. Consider the event $E = \cup_{x \in X} E_x$, where for each $x \in X$, $E_x$ is of the form $\{x\} \times B_x$ with $B_x \subseteq Y$ and the marginal probability $P_Y(B_x) \geq \frac{1}{2}$. Prove or disprove that $P(E) \geq \frac{1}{2}$. Assume the spaces are not necessarily discrete.

Answer: P(E) \geq \frac{1}{2}


Question: Determine the cardinality of the set \( A = \{ f \in C'[0,1] : f(0)=0, f(1)=1, |f'(t)| \leq 1 \text{ for all } t \in [0,1] \} \).

Answer: 1


Question:
Let \( f \in \mathcal{C}^1(\mathbb{R}^3,\mathbb{R}) \) be a function such that:
\[ f(0,1,1)=0, \quad \frac{\partial f}{\partial x}(0,1,1)=1, \quad \frac{\partial f}{\partial y}(0,1,1)=2, \quad \frac{\partial f}{\partial z}(0,1,1)=3. \]
Determine the limit:
\[ \lim_{t \to 0} \frac{f(t^2,\cosh t,e^t)}{f(t,\cos t,\cosh t)}. \]

Answer: 3


Question: Determine if the function $f(x) = x^2$ is Riemann-Stieltjes integrable with respect to the function $g(x)$ defined as $g(x) = 3x$ for $x \in [0,1)$ and $g(1) = 4$. If it is integrable, find the value of the integral $\int_0^1 f(x) \, d(g(x))$. Assume $h(x) = g(x) - 3x$ and use the partition $P = \{0, 1-\epsilon, 1\}$ to evaluate the integral.

Answer: 2


Question: Let $K$ be a field, and consider the category $K \text{-vect}$ of $K$-vector spaces. There is an adjunction $[-, K]_{K \text{-vect}} : K \text{-vect}^{op} \leftrightarrow K \text{-vect} : [-, K]_{K \text{-vect}}$, where $[-, K]_{K \text{-vect}}$ sends a vector space $V$ to $[V, K ]_{K \text{-vect}}$, the $K$ vector space of linear maps from $V$ to $K$. For each $a \in V$, define a map $\hat{a} : V^* \rightarrow K$ sending $\phi$ to $\phi(a)$. Give $V^*$ the weakest topology such that $\hat{a}$ is continuous for each $a \in V$. Form $[V^*, K]_{K \text{-topvect}}$, the vector space of continuous linear maps of topological vector spaces from $V^*$ to $K$. Is this vector space isomorphic to $V$?

Answer: V


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

Answer: \dfrac{\pi}{4}


Question: Show that the process \( M_t = \exp(\alpha B_t - \alpha^2 t/2) \) is a martingale and does not converge to zero in the \( L^1 \) norm. Specifically, verify that for \( t > s \), \( E[M_t|\mathcal{F}_s] = M_s \), and conclude that \( \lim_{t \to \infty} ||M_t||_1 = 1 \).

Answer: 1


Question: Find the sum of all possible values of \( b \) such that the equations \( f(x) = x^3 - 3x + b = 0 \) and \( g(x) = x^2 + bx - 3 = 0 \) have a common root.

Answer: 0


Question: Determine whether there exist complex number series $a_n$ and $b_n$ such that the series $\sum_{n=1}^\infty a_n z^n$ and $\sum_{n=1}^\infty b_n z^n$ both have a radius of convergence equal to 1, while the series $\sum_{n=1}^\infty (a_n + b_n) z^n$ has a radius of convergence equal to 3. Provide a justification for your answer.

Answer: Yes


Question: Find the derivative of the inverse function \((f^{-1})'(y)\) at \(y = \frac{1}{2} \sqrt{2}\) for \(f(x) = \cos x\) in the interval \([0, \pi]\).

Answer: -\sqrt{2}


Question: Approximate the integral \( \int_1^\frac{3}{2}\frac{\ln(x+1)-\ln(2)}{x-1}dx \) to an accuracy of 0.001.

Answer: 0.236


Question: Consider a function $f : \mathbb{R} \rightarrow [0,1]$ with $f'(x) \geq 0$. Does the weak monotonicity and the bounds on the function imply that $f'(x) \leq 1$?

Answer: No


Question: Determine the smallest integer $n$ such that the error $|f(x) - Q_n(x)|$ in the Lagrange interpolation of the function $f(x) = e^x$ over the interval $[-4, 0]$ is less than 0.1, where $Q_n(x)$ is the Lagrange interpolating polynomial with equally spaced nodes.

Answer: 4


Question: Find the limit \( \lim_{x \to 0^+} x\log\left(\frac{1}{x}\right) \exp((\log x)^{\alpha}) \) for \( \alpha \in (0,1) \).

Answer: 0


Question:
Identify the incorrect statement about K-theory as a generalized cohomology theory:

1. A generalized cohomology theory (on well-behaved topological spaces) is determined by its values on a point.
2. Reduced complex $K$-theory $\tilde K$ and reduced real $K$-theory $\widetilde{KO}$ are generalized cohomology theories (on well-behaved topological spaces).
3. $\tilde K(*)= \widetilde{KO} (*)=0$.

Note: It is known that $\tilde K \neq \widetilde{KO}$. Which statement is incorrect?

Answer: 1


Question: Find the residue of \( \frac{\sin(z^2)}{z^7} \) at \( z = 0 \).

Answer: -\dfrac{1}{6}


Question: In triangle $ABC$, point $M$ is on segment $BC$ such that $CM:MB = 3:5$. Point $P$ divides segment $AM$ in the ratio $AP:PM = 2:3$. Line $BP$ intersects segment $AC$ at point $Q$. Using vectors, find the ratio $AQ:QC$. Express your answer as a simplified ratio.

Answer: 5:12


Question: Evaluate the limit: \(\lim_{x\to +\infty}\frac {\zeta(1+\frac 1x)}{\Gamma(x)}\)

Answer: 0


Question: Compute the integral \( \int_{-\infty}^{+\infty} x^2 e^{-x^2/2} \, dx \).

Answer: \sqrt{2\pi}


Question: Determine the number of terms required in the series expansion to estimate the integral \( \int_{0}^{0.4}\sqrt{1+x^4} \, dx \) such that the absolute error is less than or equal to \( \frac{0.4^9}{72} \).

Answer: 2


Question: Evaluate the definite integral \( \int^{4}_0 (16-x^2)^{\frac{3}{2}}\,dx \) using an appropriate method. The final answer is \( 48\pi \).

Answer: 48\pi


Question: If a rope has $n$ sides and is rotated $k$ times, determine the number of colors needed to color the rope such that each side is uniquely colored. Express your answer in terms of $n$ and $k$. \[ \text{Number of colors} = \gcd(n, k) \]

Answer: \gcd(n, k)


Question: Find a function \( f(x) \) such that \( 10 - f(x) = 2\int_0^x f(t) \, dt \).

Answer: 10e^{-2x}


Question: If the normalizer of a subgroup in a group is equal to the subgroup itself, is the subgroup necessarily abelian? Specifically, if $H$ is a proper subgroup of $G$ such that $H = N_G(H)$, is it true that $H$ must be abelian?

Answer: No


Question: Given the recurrence relation for the sequence $a_n = 3a_{n-1} - 3a_{n-2} + a_{n-3}$ for $n \geq 3$, with initial conditions $a_0 = 2$, $a_1 = 2$, and $a_2 = 4$, find a non-recursive formula for $a_n$ using the generating function approach.

Answer: n^2 - n + 2


Question: Determine the initial conditions for which the solution to the differential equation \( y' = y(y-4)(y+4)(y-6) \) satisfies \( \lim_{t \rightarrow \infty} y(t) = 4 \).

Answer: (0, 6)


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

Answer: 0


Question: Consider a hypergraph $H=(V,E)$ where $|V| \geq \omega$, $|E| = |V|$, and $|e| = |V|$ for all $e \in E$. Is such a hypergraph $2$-colorable?

Answer: Yes


Question: Evaluate the limit \( \lim_{(x,y)\to(0,0)} g \left(\frac{x^4 + y^4}{x^2 + y^2}\right) \) given that \( \lim_{z\to 0}\frac{g(z)}{z}=2. \)

Answer: 0


Question: Let $G$ be a finite group and $p$ a prime divisor of $|G|$. Let $\alpha$ be the greatest power of $p$ dividing $|G|$, and let $n_k$ denote the number of subgroups of $G$ of order $p^k$ for $k=1,\dots,\alpha$. Is it true that $n_k \equiv 1 \pmod{p}$ for every $k=1,\dots,\alpha$?

Answer: Yes


Question: Evaluate \( \int_{\partial \mathbf{D}} f(z) \, dz \) where \( f \) is a meromorphic function in \( \mathbb{C} \) satisfying \( |f(z) z^2| \leq 1 \) for \( |z| \geq 1 \). Here, \( \mathbf{D} \) represents the unit disk.

Answer: 0


Question: Determine if the derivative of the integral \( I = \int_1^n (t-[t])(st^{s-2} - (1-s)t^{-s-1}) \mathrm{d}t \), where \( s \in \mathbb{C} \), with respect to \( s \) can be zero. Specifically, is it possible that \( \frac{dI}{ds} = 0 \)?

Answer: Yes


Question: Given three points $A$, $B$, and $C$ on a geometric surface, where the geodesic distances $AB$ and $AC$ are known, determine if the geodesic distance $BC$ satisfies the triangle inequality $AC \leq AB + BC$. Provide a justification for your answer.

Answer: AC \leq AB + BC


Question: Is the unit circle the only infinite compact subgroup of the multiplicative group of non-zero complex numbers? Specifically, if $G$ is an infinite subgroup of $(\mathbb{C} \setminus \{0\}, \cdot)$ that is compact as a subset of $\mathbb{C}$, is it true that $G = S^1$?

Answer: S^1


Question: Each of two teams has seven players numbered 1 to 7. In the first game, the two players numbered 1 play each other. The loser of each game is eliminated and replaced by the next player of the same team, until all players from one team have been eliminated. Let \( N \) be the number of possible sequences of games. Find the remainder when \( N \) is divided by 1000.

Answer: 432


Question:
Let $V = \{(x, y, z) \in \mathbb{R}^3 : y - z = 0 \}$, and let $f : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be an endomorphism such that $V = \text{ker}(f)$ and $-1$ is an eigenvalue of $f$. Which of the following is true?

(a) $\text{dim}(\text{Im}(f)) = 2$

(b) The characteristic polynomial of $f$ can be $-t(t + 1)^2$

(c) $f$ has three distinct eigenvalues

(d) $f$ is simple

Select the correct answer.

Answer: D


Question: Determine the number of roots of the polynomial \( f(x) = x^{12} + x^8 + x^4 + 1 \) in the finite field \( \mathbb{F}_{121} \).

Answer: 4


Question:
Given two random variables $X$ and $Y$ with the joint probability density function:
\[f_{X,Y}(x,y) = \begin{cases}3y, & -1\leq x \leq 1, 0\leq y \leq |x|\\0, & \text{otherwise}\end{cases}.\]
Calculate the probability $P(X \geq Y + 0.5)$. Use the marginal distributions:
\[f_X(x) = \begin{cases}\frac{3x^2}{2}, & x \in [-1, 1]\\0, & \text{otherwise}\end{cases}\]
and
\[f_Y(y) = \begin{cases}6y(1-y), & y \in [0, 1]\\0, & \text{otherwise}\end{cases}.\]

Answer: \dfrac{1}{16}


Question: Determine the number of ordered pairs of real numbers \((a, b)\) that satisfy the following limit equation: $$ \lim_{x\to 0} \frac{\sin^{2}x}{e^{ax}-2bx-1}=\frac{1}{2}. $$

Answer: 2


Question: Let $p_1, p_2 \in \mathbb{C}$ be linearly independent over $\mathbb{R}$. Consider an entire function $f: \mathbb{C} \to \mathbb{C}$ such that $f(0) = f'(0) = 1$, and $f(z + p_1) = a f(z)$ and $f(z + p_2) = b f(z)$ for all $z \in \mathbb{C}$, where $a, b \in \mathbb{C}$ are constants. Prove or disprove that $f(z) = e^z$ for all $z \in \mathbb{C}$, i.e., $f'(z) = f(z)$ for all $z \in \mathbb{C}$ and $f(z + w) = f(z)f(w)$ for all $z, w \in \mathbb{C}$. \[ \text{Hint: Consider the implications of the conditions on $f$.} \]

Answer: f(z) = e^z


Question: Determine whether there exists a continuous function \( f : \mathbb{R}^2 \to \mathbb{R} \) such that \( f \equiv 1 \) on the set \( \{(x, y) \in \mathbb{R}^2 : x^2+y^2 = 3/2 \} \) and \( f \equiv 0 \) on the set \( B \cup \{(x, y) \in \mathbb{R}^2 : x^2+y^2 \geq 2\} \), where \( B \) is the closed unit disk. Answer True or False.

Answer: True


Question:
Determine the order of convergence of the series \( S_N(x) = \sum_{n=1}^{N}\frac{\cos(x n)}{n} \) to its limit \( -\log\left(2\sin\left(\frac{x}{2}\right)\right) \) as \( N \to \infty \). Find a function \( g(N) \) such that:
\[ S_N(x) = -\log\left(2\sin\left(\frac{x}{2}\right)\right) + \mathcal{O}(g(N)), \]
or equivalently,
\[ \left|S_N(x) + \log\left(2\sin\left(\frac{x}{2}\right)\right)\right| \leq C g(N), \]
for some constant \( C > 0 \).

Answer: \dfrac{1}{N}


Question: Calculate the work done by the force field \( \mathbf{F}(x,y,z) = z^2~\mathbf{i}+ 2xy~\mathbf{j}+ 4y^2~\mathbf{k} \) on a particle moving along the path defined by the line segments connecting the points \((0,0,0)\), \((1,0,0)\), \((1,2,1)\), \((0,2,1)\), and back to \((0,0,0)\). Use Stokes' theorem to find the work done.

Answer: 3


Question:
Solve the equation involving the floor function:
\[ \left\lfloor \frac{x}{1!} \right\rfloor + \left\lfloor \frac{x}{2!} \right\rfloor + \left\lfloor \frac{x}{3!} \right\rfloor + \cdots + \left\lfloor \frac{x}{10!} \right\rfloor = 1001, \]
where \( \lfloor r \rfloor \) denotes the integer part of a real number \( r \). Find the integer value of \( x \).

Answer: 584


Question: Consider a circle with radius $R$. Two points, $A$ and $B$, are chosen at random within this circle. Determine the probability that the circle with center $A$ and radius equal to the distance $AB$ is entirely contained within the original circle. The circles may be tangent.

Answer: \dfrac{1}{6}


Question: Find all natural number solutions \((m, n, p)\) to the Diophantine equation \(5^m + n^2 = 3^p\), where \(m, n, p \in \mathbb{N} - \{0\}\).

Answer: (1, 2, 2)


Question: In a bipartite graph with two sides $R$ and $L$, where each node on side $R$ is connected to $k$ nodes on side $L$, $|R| < k < |L|$, and $k$ is much larger than $|R|$, determine the minimal number of edges in the maximal biclique of the graph.

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \to a} \frac{\ln\frac{x}{a}}{x-a} \] without using L'Hôpital's rule.

Answer: \dfrac{1}{a}


Question: Given the second-order linear differential equation \( \frac{d^2x}{dt^2} + p(t)\frac{dx}{dt} + q(t)x = 0 \), find the function \( p(t) \) if the Wronskian of two solutions \( x_1(t) \) and \( x_2(t) \) is a non-zero constant. Use Abel's theorem to determine \( p(t) \).

Answer: 0


Question: Find all functions \( f: \mathbb{R}^{+} \to \mathbb{R} \) such that for any \( x, y \in \mathbb{R}^{+} \), the following equation holds: \[ f(x+y) = f(x^2 + y^2). \]

Answer: f(x) = c


Question:
Let $T$ be the linear transformation represented by the matrix
$$ \left( \begin{array}{cccc}
1 & 1 & 0 & 3 \\
1 & 1 & 1 & 5 \\
2 & 2 & 1 & 8 \end{array} \right) $$
from $\mathbb{R}^4$ to $\mathbb{R}^3$. Consider the space $V$ of linear maps $B$ from $\mathbb{R}^2$ to $\mathbb{R}^4$ satisfying $AB=0$. Determine the dimension of the space $V$.

Answer: 4


Question: Evaluate the limit using a power series approach: $$\lim_{x \to 0} \left[\frac{2+x}{2x*(e^x-1)}-\frac{1}{x^2}\right]$$

Answer: -\dfrac{1}{6}


Question:
Given the system of equations for real numbers \(x, y, z\):
\[ x + y + z = 3 \]
\[ x^2 + y^2 + z^2 = 9 \]
\[ xyz = -2 \]
Find the value of \(x^4 + y^4 + z^4\).

Answer: 57


Question: Determine whether the following statement is true or false: If $M$ and $N$ are two smooth surfaces and $f: M \to N$ is a homeomorphism such that $\det(J_f)$, the determinant of the Jacobian of $f$, has a constant sign, then $f$ is a diffeomorphism.

Answer: B


Question: Consider a non-complete inner product space $V$. Let $x_n$ be a weakly Cauchy sequence in $V$, meaning that for every $y \in V$, the sequence $\langle x_n, y \rangle$ converges. Is it true that $x_n$ is necessarily bounded?

Answer: No


Question: Evaluate the complex integral \( \int_{\gamma}\frac{dz}{1+z^2} \) where \( \gamma \) is a rectifiable curve from 0 to 1 that does not pass through \( \pm i \).

Answer: \dfrac{\pi}{4}


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

Answer: 1


Question: Determine whether there exists a group with elements $a$ and $b$ such that the cyclic subgroups generated by $ab$ and $ba$ are equal, i.e., $\langle ab \rangle = \langle ba \rangle$, yet the cyclic subgroups generated by $a$, $b$, and $ab$ are distinct. Provide a justification for your answer.

Answer: Yes


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

Answer: \dfrac{\pi}{4}


Question: Find the smallest positive value of \( a \) such that \( \left(1+\frac{1}{x}\right)^{x+a} > e \) for all \( x \geq 1 \).

Answer: \dfrac{1}{2}


Question: Consider a Markov chain with state space $S=\{0, 1, 2, \ldots \}$ and transition probabilities $p(0,0)=3/4$, $p(0,1)=1/4$, $p(x, x-1)=3/4$, and $p(x,x+1)=1/4$ for $x \geq 1$. Let $e(x)$ denote the expected number of steps to reach the origin starting from state $x$. Find $e(1)$. (Hint: First consider the expected return time starting at the origin and express $e(1)$ in terms of this.)

Answer: 2


Question: Find the minimum value of the expression \( \frac{\int_{0}^{1}{[f'(x)]^2 \, dx}}{\int_{0}^{1}{[f(x)]^2 \, dx}} \), where \( f(x) \) is a twice differentiable function on \([0,1]\) with \( f(0) = f(1) = 0 \) and \( f(x) \) is not identically zero.

Answer: \pi^2


Question: Does the map \( \langle f, g \rangle = \int_0^1 \left( \left( f(x) - \frac{d}{dx}f(x) \right) \left( g(x) - \frac{d}{dx}g(x) \right) \right) dx \) define an inner product on the set of all polynomial functions of order less than or equal to \( n \)?

Answer: Yes


Question:
Consider the function \(f(x) = x^3 + 3x^2 - x + 1\) and the point \(c = 2\). In the context of the \(\epsilon-\delta\) definition of limit, find the least upper bound on \(\delta\) such that \(f(x)\) is within \(\epsilon\) of \(L\) for all sufficiently small \(\epsilon > 0\). Choose the correct answer from the following options:

(A) \(\frac{\epsilon}{3}\)
(B) \(\frac{\epsilon}{23}\)
(C) \(\frac{\epsilon^3}{4}\)
(D) \(\frac{\epsilon^2}{16}\)
(E) \(\frac{\epsilon}{19}\)

Answer: B


Question: Find the maximum integer $k$ such that \((7!)!\) is divisible by \((7!)^{k!} \cdot (6!)!\).

Answer: 6


Question: Let $A$ be a symmetric matrix of order $n$ with eigenvalues $\lambda_1, \ldots, \lambda_n$. If the main diagonal of $A$ is $\lambda_1, \ldots, \lambda_n$, is $A$ necessarily a diagonal matrix? Provide a justification for your answer.

Answer: Yes


Question: Find the maximum value of \( \boldsymbol{x}^T \boldsymbol{A} \boldsymbol{x} \) subject to the constraints \( \boldsymbol{q}^T \boldsymbol{x} = 1 \) and \( x_i \geq 0 \) for all \( i \), where \( \boldsymbol{x} \in \mathbb{R}^n \), \( \boldsymbol{A} \) is a positive definite matrix, and \( \boldsymbol{q} \) is a given vector. For example, if \( \boldsymbol{A} = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix} \) and \( \boldsymbol{q} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \), determine the solution that maximizes the objective function.

Answer: 2


Question: Determine the radius of convergence of the Maclaurin series for the function \( f(z) = \frac{2}{1 + \cosh(z)} \).

Answer: \pi


Question: Given the parametric equation \( r(t) = \cos(-7t)\mathbf{i} + \sin(-7t)\mathbf{j} + 6t\mathbf{k} \), compute the normal component of the acceleration vector.

Answer: 49


Question: If the abelianization of a group $G$ exists, is it unique up to isomorphism? Specifically, if $A$ and $A'$ are two abelianizations of $G$ with homomorphisms $\varphi: G \to A$ and $\varphi': G \to A'$ respectively, is there an isomorphism $\rho: A' \to A$ such that $\rho \varphi' = \varphi$?

Answer: Yes


Question: Find the value of \( \lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}} \).

Answer: 1


Question: Let $k$ be an algebraically closed field and $R = k[x]$. For a given $\lambda_0 \in k$, determine the localization of the $R$-module \( M = \bigoplus_{\lambda \in k} k[x]/(x-\lambda) \) at the maximal ideal $(x-\lambda_0)$. Provide your answer in terms of the structure of the localized module.

Answer: k


Question: Does there exist an increasing sequence \(a_1, a_2, \dots\) of positive integers such that each positive integer appears exactly once among the differences \(a_2-a_1, a_3-a_2, \dots\) and for some \(n\), each positive integer at least \(n\) appears exactly once among the differences \(a_3-a_1, a_4-a_2, a_5-a_3, \dots\)?

Answer: No


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{C} \) be a differentiable function. Is it true that for all \( a, b \in \mathbb{R} \), there exists some \( c \in (a, b) \) such that \[ \left| \frac{f(b) - f(a)}{b - a} \right| \leq |f'(c)|? \]

Answer: Yes


Question:
Consider the set $U$ of all $n \times n$ matrices $A$ with complex entries such that $A$ is unitary (i.e., $A^* A = I_n$). Determine which of the following statements is true about $U$ as a topological subspace of $\mathbb{C}^{n^2}$:
1. Compact, but not connected
2. Connected, but not compact
3. Connected and compact
4. Neither connected nor compact

Answer: C


Question: Solve the equation for exact solutions: \( \arccos x + \arccos 2x = \arccos \frac{1}{2} \).

Answer: \dfrac{1}{2}


Question: Find the limit of the sequence \(a_n = \frac{2^{3n} - n3^n}{n^{1729} + 8^n}\) as \(n\) approaches infinity.

Answer: 1


Question: Determine the number of digits in the number $10000!$ when expressed in base 10.

Answer: 35660


Question: Evaluate the integral \( \int_{-1}^{1} x^2 \sin^{2019}(x) e^{-x^4} \, dx \).

Answer: 0


Question: Find the equation of the plane tangent to the surface given by \(x^3+y^3+z^3-3xyz =0\) at the point \((1,1,1)\).

Answer: x + y + z = 3


Question: Let $H$ be a Hilbert space and $X$ be a subset of $H$. If $X$ is connected and compact, is $X$ necessarily path-connected?

Answer: No


Question: Find the coordinates of the symmetric point $P^{\prime}$ of the point $P(4,3,10)$ with respect to the line given by \( \frac{x-1}{2}=\frac{y-2}{4}=\frac{z-3}{5} \).

Answer: (2, 9, 6)


Question: Find the area under one arch of the cycloid given by the parametric equations \(x=6(t-\sin(t))\) and \(y=6(1-\cos(t))\).

Answer: 108\pi


Question: Three particles A, B, and C are moving in the plane along parallel lines at constant speeds. Initially, the area of triangle ABC is 2. Five seconds later, the area of triangle ABC is 3. Determine the sum of all possible values of the area of triangle ABC after an additional five seconds.

Answer: 12


Question: Let $M$ be a compact manifold and $f : M \rightarrow M$ be a homeomorphism isotopic to the identity. Can $f$ be fragmented, meaning for any open cover $\mathcal{U}$ of $M$, there exist homeomorphisms $f_1,\ldots,f_n$ isotopic to the identity and open sets $U_1,\ldots,U_n \in \mathcal{U}$ such that the support of $f_i$ is in $U_i$ and $f = f_1 \cdots f_n$?

Answer: Yes


Question:
Let $(f_n)_n$ be a sequence of real-valued continuous functions on a compact, Hausdorff space $K$ with pairwise disjoint (closed) supports, satisfying
$$0<\inf_n \|f_n\|\leq \sup_n\|f_n\|<\infty.$$
Given a sequence $(a_n)\in c_0$, where $\lim_{n\to \infty}a_n=0$, does the series
$$\sum_{n=1}^\infty a_n f_n$$
define an element of $C(K)$, with convergence in the norm of $C(K)$?

Answer: Yes


Question: Determine the radius of convergence for the series \( \sum_{n=0}^\infty \frac{n!\,z^{2n}}{(1+n^2)^n} \).

Answer: \infty


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

Answer: \dfrac{1}{e}


Question: Find the point on the ellipse formed by the intersection of the cylinder \(x^2 + y^2 = 1\) and the plane \(x + z = 1\) that is furthest from the origin.

Answer: (-1, 0, 2)


Question: Evaluate the limit: \( \lim_{{x, y \to 0}}(x^2+y^2)^{x^2y^2} \)

Answer: 1


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right ) \]

Answer: -\infty


Question: Determine the number of distinct prime ideals in the ring \( \mathbb{Q}[x]/(x^5 - 1) \).

Answer: 2


Question: Determine whether every open subset in an arbitrary metric space necessarily contains one of its accumulation points. Justify your answer.

Answer: No


Question: Determine the total number of sets of 3 distinct factors of the number \( X = 26 \times 34 \times 52 \) such that the greatest common divisor (gcd) of any two factors in each set is 1. Choose the correct answer from the following options: 104, 360, 380, or 236.

Answer: 236


Question: Determine the range of values for the parameter \( \alpha \) such that the curve \( \gamma_{\alpha}(t) = (t \cdot \cos(\alpha t), t \cdot \sin(\alpha t), \alpha t) \) is a geodesic on the parameterized surface \( X(u,v) = (u \cdot \cos v, u \cdot \sin v, v) \).

Answer: 0


Question: Does a finitely generated group always have a proper subgroup with finite index?

Answer: No


Question: Determine whether every $\mathbb{R}$-algebra homomorphism $h: \mathbb{R}[V] \to \mathbb{R}$ is of the form $ev_x$ for some point $x \in V$, where $ev_x(f) = f(x)$ for $f \in \mathbb{R}[V]$. Provide a justification for your answer.

Answer: Yes


Question: Given a function $f \in L^2(K)$, where $K \subseteq \mathbb{R}^n$ is compact, is it possible to construct a sequence of continuous functions $f'_n$ that converges to $f$ pointwise almost everywhere, with the property that $f'_n(x) \geq f'_m(x)$ whenever $n \geq m$?

Answer: Yes


Question: Evaluate the limit: \( \lim_{x\to \pi/2}\cos(x)^{2x-\pi} \)

Answer: 1


Question: Determine the values of $x$ for which the series \( \sum_{n=0}^\infty \frac{x^n}{n\#} \) converges, where \( n\# \) is the product of all primes less than or equal to \( n \).

Answer: (-e, e)


Question: Let \( \{f_n : n \in \mathbb{N}\} \subset C^1([0, 1]) \) be a sequence of functions such that \( f_n(0) = 0 \) and \( \sup_{n \in \mathbb{N}} \left\| \frac{d}{dx}f_n\right\|_2 < \infty. \) Determine whether \( \{f_n : n \in \mathbb{N}\} \) has a convergent subsequence in \( L^2((0, 1)) \).

Answer: Yes


Question: Consider a surface $S$ that is the graph of a continuous function $f: U \rightarrow \mathbb{R}$ on an open set $U \subset \mathbb{R}^2$. Let $p = (x, y, f(x, y))$ be a point on $S$. Suppose there exists a plane $P$ through $p$ such that for every smooth curve $\gamma: I \rightarrow S$ passing through $p$, the tangent line to $\gamma(I)$ at $p$ lies in the plane $P$. Is this property sufficient to conclude that $f$ is differentiable at $(x, y)$?

Answer: Yes


Question:
Find the coefficient of $x^{28}$ in the expansion of $(1+x+x^{2}+x^{3}+
obreak	ext{...}+x^{27})(1+x+x^{2}+x^{3}+
obreak	ext{...}+x^{14})^{2}$.

Answer: 224


Question: Evaluate the integral \( \int_{|z+2|=1}\frac{e^{z^2}}{z^{1995}}dz \), where \( z \in \mathbb{C} \).

Answer: 0


Question: Find the variance of the distribution of the number of Bernoulli trials needed to achieve 100 successes, given that the probability of success in each trial is 0.01.

Answer: 990000


Question: Let \( \phi: \mathbb{Z}[x] \rightarrow \mathbb{R} \) be the homomorphism defined by \( \phi(f(x)) = f\left(\frac{1}{2} + \sqrt{2}\right) \). Determine if the kernel of \( \phi \) is a principal ideal. If it is, find a generator for this ideal.

Answer: 4x^2 - 4x - 7


Question: Evaluate the integral \( \int_C \frac{2z-1}{z(z-1)} \,dz \) where \( C \) is the circle \(|z|=2\).

Answer: 4\pi i


Question: Determine the angle of rotation for the conic section given by the equation: \( x^2 + 4xy + y^2 + 8x + 4y - 2 = 0 \). Use the formula for rotation angle \( 2\theta = \arctan\left(\frac{b}{a-c}\right) \), where \( a, b, \) and \( c \) are the coefficients of \( x^2, xy, \) and \( y^2 \) respectively.

Answer: \dfrac{\pi}{4}


Question:
Determine which of the following implications are true for a function $f: \mathbb{R} \to \mathbb{R}$:

1. $A \Rightarrow B$: If $f$ is continuous almost everywhere, then there exists a continuous function $g: \mathbb{R} \to \mathbb{R}$ such that $f = g$ almost everywhere.
2. $B \Rightarrow A$: If there exists a continuous function $g: \mathbb{R} \to \mathbb{R}$ such that $f = g$ almost everywhere, then $f$ is continuous almost everywhere.

Answer: B \Rightarrow A


Question: Let \( \{f_n\} \) be a sequence of continuous, strictly positive functions on \( \mathbb{R} \) which converges uniformly to the function \( f \). Suppose that all the functions \( \{f_n\}, f \) are integrable. Determine whether \( \lim_{n \rightarrow \infty} \int f_n(x) \, dx = \int f(x) \, dx \).

Answer: No


Question: Find the limit of the function \( f(x) = \frac{\cos(\sin x) - \cos x}{x^4} \) as \( x \to 0 \).

Answer: \dfrac{1}{6}


Question: Determine whether a homeomorphism maps $F_\sigma$ sets to $F_\sigma$ sets in general topological spaces.

Answer: Yes


Question: Find the value of \( \limsup\limits_{n\rightarrow\infty}\left\{\frac{p_{m+1}}{p_m}\middle|m\in \mathbb{N},m\geq n\right\} \), where \( p_i \) is the \( i \)-th prime number.

Answer: 1


Question: Calculate the contour integral of the vector field \( \mathbf{F}(x, y, z) = z^2 \mathbf{e}_1 + x^2 \mathbf{e}_2 + y^2 \mathbf{e}_3 \) along the circle of radius \( a \) centered at \( (1, 1, 0) \) in the plane \( z = 0 \). Use the parameterization \( \mathbf{x}(t) = (1 + a\cos(t), 1 + a\sin(t), 0) \) for \( t \) from \( 0 \) to \( 2\pi \).

Answer: 2\pi a^2


Question:
Suppose \( f: S_3 \rightarrow \mathbb{R}^* \) is a homomorphism. Determine the size of the kernel of \( f \). Choose the correct option:

1. Always at most 2 elements
2. Always at most 3 elements
3. Always at least 3 elements
4. Always exactly 6 elements

Answer: C


Question:
Consider the boundary value problem \(x^2y''-2xy'+2y=0\) with the boundary conditions \(y(1)+\alpha y'(1)=1\) and \(y(2)+\beta y'(2)=2\). Determine the values of \(\alpha\) and \(\beta\) for which the problem has a unique solution. Choose the correct option:

1. \(\alpha=-1,\beta=2\)
2. \(\alpha=-2,\beta=2\)
3. \(\alpha=-1,\beta=-2\)
4. \(\alpha=-3,\beta=2/3\)

Answer: 1


Question: Determine whether the following statement is true or false: If a sequence of continuous functions $f_n$ converges pointwise to a function $f$, then the supremum norms $\|f_n\|_{\infty}$ converge to $\|f\|_{\infty}$. Provide a justification for your answer.

Answer: False


Question: Find the z-coordinate of the center of mass of a solid with constant density \(\rho\), bounded below by the disk \(R: x^2 + y^2 \leq 4\) in the plane \(z=0\) and above by the paraboloid \(z = 4 - x^2 - y^2\). (Hint: By symmetry, \(\overline{x} = \overline{y} = 0\), so you only need to compute \(\overline{z}\).)

Answer: \dfrac{4}{3}


Question:
Find the $m \times n$ binary matrix $X$ that minimizes the sum $\sum_{i=1}^m v_i$, where $v = b - Xa$, subject to the following constraints:

1. The sum of each column in $X$ must be either $0$ or $1$.
2. $v_i \geq 0$ for all $i$.

Here, $a$ is an $n \times 1$ positive vector, and $b$ is an $m \times 1$ positive vector.

Answer: X


Question: Let \(X_1, X_2, X_3\) be independent random variables, each uniformly distributed on \((0, 1)\). Let \(X_{(1)}, X_{(2)}, X_{(3)}\) be the order statistics of these variables. Compute the probability \(P(X_{(1)} + X_{(3)} \le 1)\).

Answer: \dfrac{1}{2}


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

Answer: \dfrac{\pi}{16}


Question: Compute \( 29^{2013^{2014}} - 3^{2013^{2014}} \pmod{22} \).

Answer: 4


Question: Compute the limit: \( \lim_{h\rightarrow0}\frac{2^{2+h}-2^2}{(2+h)-2} \)

Answer: 4 \ln 2


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

Answer: 1


Question: Determine the largest radius of a disk centered at the origin such that the function \( f(z) = z^2 + z \) is injective within that disk.

Answer: \dfrac{1}{2}


Question:
For which value of $n$ does the finite field $\mathbb{F}_{5^n}$ contain a non-trivial 93rd root of unity?

(A) 92

(B) 30

(C) 15

(D) 6

Answer: D


Question: Given the points \((k,1), (5k,2), (10k, 3)\) where \(k\) is an unknown constant, determine if it is possible to find a linear regression equation that passes through the origin.

Answer: Yes


Question: For the function \( f(z) = 2x^3 + 3iy^2 \), find \( f'(x + ix^2) \).

Answer: 6x^2


Question: Evaluate the limit: \[ \lim_{x \to \pi} \frac{1}{x-\pi} \int_{0}^{\sin(x)} e^{t} \, dt \]

Answer: -1


Question: Calculate the outward flux of the vector field \( F(x,y,z) = (xy+2xz, x^2+y^2, xy-z) \) through the surface of the cylinder \( x^2+y^2=4 \) bounded by the planes \( z=y-2 \) and \( z=0 \), without using the Divergence Theorem.

Answer: -40\pi


Question: In the triangle shown, \( CZ \) is perpendicular to \( XY \), and the ratio of \( AZ \) to \( ZB \) is \( 1:2 \). The angle \( ACX \) is \( \alpha \) and the angle \( BCY \) is \( \beta \). Find an expression for the angle \( AZC \) in terms of \( \alpha \) and \( \beta \).

Answer: \alpha + \beta


Question:
Suppose a function is expressed by:
  \[f(x)=f(x+1) + f(x-1)\]
  and
  \[f(16)=20 , f(20)=16.\]
  What is \(f(20162016)\)?

Answer: -36


Question: In an imaginary quadratic field $K$, is there an element $\alpha \in K$ that is not a unit in the ring of integers of $K$ and has a finite multiplicative order? Specifically, does there exist $n \in \mathbb{N}$ such that $\alpha^n = 1$?

Answer: No


Question: Find the function \( f : \mathbb{R} \rightarrow \mathbb{R} \) that satisfies the conditions \( f(-1)=0 \), \( f(0)=1 \), \( f(1)=0 \), and \( f(x) = 0 \) if \(|x|>1\), and minimizes the integral \( \int_{\mathbb{R}} |f'(x)| \, dx \). What is the minimum possible value of this integral?

Answer: 2


Question: What is the maximal number of spanning trees that can be constructed in the complete graph \( K_9 \)?

Answer: 4


Question: Let $G$ be a finite non-nilpotent group. Determine whether there exists a non-normal maximal subgroup $M$ such that the index $|G:M|$ is a prime power.

Answer: No


Question: Find the minimum value of the expression \( P = \frac{x}{2x+3y} + \frac{y}{y+z} + \frac{z}{z+x} \) given that \( x, y, z \in [1, 4] \) and \( x \geq y \) and \( x \geq z \).

Answer: \dfrac{34}{33}


Question: Find the position on the helix \( r(t) = (12\sin(t)) \mathbf{i} - (12\cos(t)) \mathbf{j} + 5t \mathbf{k} \) that is located at an arc-length distance of \(-13\pi\) units along the helical curve from the initial position \( r_0 = (0, -12, 0) \).

Answer: (0, 12, -5\pi)


Question: Compute the residue of the function \( \frac{1+2z+3z^2}{1+z+z^2-3z^3} \) at \( z = 1 \).

Answer: -1


Question: Suppose \( f: \mathbb{R}^n \rightarrow \mathbb{R}^n \) is a continuously differentiable function such that the Jacobian determinant of \( f \) is zero everywhere in \( \mathbb{R}^n \). For \( n \ge 2 \), is it true that \( f \) cannot be one-to-one? Justify your answer.

Answer: Yes


Question: Let $M$ be a compact orientable manifold with boundary $\partial M$. Determine whether $H_n(M;\mathbb{R})$ is always zero.

Answer: 0


Question: Find all solutions near the identity matrix $I$ of the equation $P^2 = I$ using the exponential function.

Answer: I


Question: Let $A$ be a commutative reduced ring, and let $S$ be the set of all non-zerodivisors of $A$. Determine the Krull dimension of the total ring of fractions $S^{-1}A$. Provide your answer as a single integer.

Answer: 0


Question: Let $\mathcal{P}$ be a directed family of seminorms on a topological vector space $X$, and let $\tau_{\mathcal{P}}$ be the locally convex topology on $X$ generated by $\mathcal{P}$. If $q$ is a $\tau_{\mathcal{P}}$-continuous seminorm on $X$ with $q \not\in \mathcal{P}$, does there exist a $p \in \mathcal{P}$ and a constant $c > 0$ such that $q(x) \leq c p(x)$ for all $x \in X$?

Answer: Yes


Question: In triangle $\triangle ABC$, where $AC = BC$ and $\angle C = 20^{\circ}$, point $M$ is located on side $AC$ and point $N$ is located on side $BC$. Given that $\angle BAN = 50^{\circ}$ and $\angle ABM = 60^{\circ}$, determine the measure of $\angle NMB$ in degrees.

Answer: 30


Question: Determine the radius of convergence for the series \( \sum_{n=1}^\infty \left (1+{1 \over 2}+\ldots+{1 \over n} \right )x^n \).

Answer: 1


Question: Assume \( f(x) = \frac{2^{x+1}}{2^x+1} \), and \( \{a_n\} \) is an arithmetic sequence with a common difference of 1. Given that \( f(a_1) + f(a_2) + \cdots + f(a_{2017}) = 2017 \), find the value of \( f^2(a_{1009}) - a_{1008}a_{1010} \).

Answer: 2


Question: Find the limit: \[ \lim_{x\to0^+}\frac{1}{\sin^{2}x}\int_{\frac{x}{2}}^x \sin^{-1}t \, dt \]

Answer: \dfrac{3}{8}


Question:
Let \( K \subset \mathbb{R}^n \) be a compact set. For each \( r > 0 \), construct a partition of unity \( \sum_{i=1}^N \xi_i = 1 \) on the set \( K \) with the following properties:

1. The support of \( \xi_i \) is contained in a ball with radius \( r \).
2. \( |D\xi_i| \le C r^{-1} \), where \( C \) is a constant depending only on \( n \).
3. For each \( x \in K \), the number of functions \( (\xi_i)_i \) such that \( \xi_i(x) \neq 0 \) is bounded by \( C \).

Determine whether there exists an upper bound for the number of overlapping balls, \( M(x, r, n) \), that is independent of \( x \) and \( r \).

Answer: Yes


Question: Evaluate the integral \( \int _0^{\pi/4}\:\frac{dx}{\cos^4x-\cos^2x\sin^2x+\sin^4x} \). Determine if the solution method affects the result.

Answer: \dfrac{\pi}{2}


Question: Let $X_1, \dots, X_n$ be an i.i.d. sequence of discrete random variables with entropy $H(X)$. Define the set $C_n(t) = \{ x^n \in X^n : p(x^n) \ge 2^{-nt} \}$. Determine the values of $t$ for which $P[X^n \in C_n(t)] \rightarrow 1$ as $n \rightarrow \infty$. Provide a justification for your answer.

Answer: t \geq H(X)


Question: Compute the line integral \( \int \limits_{C} F \cdot dr \) for the vector field \( F(x,y) = (y,x) \) where \( C \) is the curve given by \( r = 1 + \theta \) for \( \theta \in [0, 2\pi] \).

Answer: 0


Question: In a school with 100 students forming 450 clubs, any two clubs have at least 3 students in common, and any five clubs have no more than 1 student in common. Must there exist four clubs that have exactly 1 student in common?

Answer: Yes


Question: Given a graph $G$ with a maximum degree greater than 0, suppose $G$ contains a perfect matching $P$ and that the graph $G-P$ is bipartite. Determine the chromatic index (edge chromatic number) of $G$ in terms of its maximum degree.

Answer: \Delta


Question:
If $a_1, a_2, a_3, \\ldots, a_n$ are $n$ distinct odd natural numbers not divisible by any prime number greater than 5, then what is the maximum value that $\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$ can be less than?

(a) $\frac{15}{8}$
(b) $\frac{8}{15}$
(c) $\frac{7}{4}$
(d) $\frac{4}{7}$

Answer: A


Question: In a minimal uniquely ergodic system \((X,T)\) with measure \(\mu\), does there exist a point \(p \in X\) such that \(\mu(\partial B(p,t)) = 0\) for all \(t > 0\) with respect to some metric \(d\) that induces the same topology?

Answer: Yes


Question:
Let \( f_n:[a,b] \to \mathbb{R} \), \( n \in \mathbb{N} \), be a sequence of \( L^p \) functions for some \( p \in (1,\infty) \). For every fixed \( m \in \mathbb{N}^* \), suppose that the sequence of functions \( \{f_{n}\psi_m(f_n)\}_{n \in \mathbb{N}} \) has a strongly convergent subsequence in \( L^p([a,b]) \). Here, \( \psi_m \) is a smooth function such that
\[
\psi_m(f) =
\begin{cases}
1 & \text{if } |f| \ge 1/m \\
0 & \text{if } |f| \le 1/(2m)
\end{cases}
\]
and \( 0 \le \psi_m \le 1 \).

Is it true that \( \{f_n\}_{n \in \mathbb{N}} \) also has a strongly convergent subsequence in \( L^p([a,b]) \)?

Answer: Yes


Question: Find all bijective functions \( f : [0,1] \to [0,1] \) such that \( x = \frac{1}{2} \big(f(x) + f^{-1}(x)\big) \) for all \( x \in [0,1] \).

Answer: f(x) = x


Question: Evaluate the limit: \[ \lim_{t \to 0} \int_0^t \frac{x^d}{e^x-1}\,dx. \]

Answer: 0


Question: Find all functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) that satisfy the equation \( f(m-n+f(n))=f(m)+f(n) \) for all \( m,n \in \mathbb{N} \).

Answer: f(n) = 2n


Question: Determine if 24 is the largest integer such that for every integer \( a \) coprime to 24, \( a^2 \equiv 1 \pmod{24} \).

Answer: 24


Question: Evaluate the double integral \( \iint_D e^{-x^2-4y^2} \, dx \, dy \), where \( D = \{(x,y) : 0 \leq x \leq 2y\} \).

Answer: \dfrac{\pi}{16}


Question: Let $X_1, X_2, X_3, X_4, X_5, X_6$ be independent samples from a normal distribution $N(0, 1)$. Define $Y = (X_1 + X_2 + X_3)^2 + (X_4 + X_5 + X_6)^2$. Find the constant $C$ such that $CY \sim \chi^2$. Express your answer as a fraction.

Answer: \dfrac{1}{3}


Question: Find a nontrivial solution to the initial-value problem \( y' = ty^a \) with \( y(0) = 0 \) and \( a > 1 \).

Answer: 0


Question: If $z$ lies on the circle $|z-2i|=2\sqrt{2}$, find the value of $\arg\left(\frac{z-2}{z+2}\right)$. \(\text{Express your answer in degrees.}\)

Answer: 45


Question: Given two independent random variables $x_i$ and $x_j$ sampled from the same distribution $X$ with finite moments, find the expected value of their product, $\mathbb{E}[x_ix_j]$. Assume $x_i$ and $x_j$ are independent.

Answer: (\mathbb{E}[X])^2


Question: Let \( X(t) \) be a Poisson process with rate \( \lambda \). Define \( T_n = \min\{t \geq 0: X(t) = n\} \) as the waiting time for the \( n \)-th event. Calculate \( \mathbb{E}[T_1|X(1)=2] \).

Answer: \dfrac{1}{3}


Question: Consider a finite simple graph \( G \). Let \( \mathcal{O} \) be the set of graphs without isolated vertices, up to isomorphism, whose line graph is isomorphic to \( G \). Can \( \mathcal{O} \) contain more than one element?

Answer: Yes


Question:
Given the joint probability density function of random variables $X$ and $Y$:
\[ f(x,y) = \begin{cases} 2 & 0<x<y<1 \\ 0 & \text{elsewhere} \end{cases} \]
find the conditional distribution of $X$ given $Y$. Use this distribution to calculate $P(X > 0.5 \mid Y = 0.7)$. Provide your answer as a probability.

Answer: \dfrac{2}{7}


Question: Determine whether the converse of the Cauchy Integral Formula is true. Specifically, if a function satisfies the integral equation \( f(z)=\frac{1}{2\pi i}\int_{\partial B_r(z_0)}\frac{f(w)}{w-z}dw \) for every \( r>0 \) with \( \overline{B_r(z_0)} \subset D \), is the function necessarily analytic? Provide a justification for your answer.

Answer: Yes


Question: Calculate the total number of ways to distribute 10 different cards to 2 players, considering that the order of cards matters in each player's hand.

Answer: 39916800


Question: Let \( f : \mathbb{R} \to \mathbb{R} \) be a function that is continuous at 0 with \( f(0) = 1 \). The function satisfies the relation \( f(x) - f\left(\frac{x}{2}\right) = \frac{3x^2}{4} + x \) for all \( x \). Find \( f(3) \).

Answer: 16


Question: Given the function \( f(x) = \frac{\sqrt{2-e^{2x}}\sqrt[4]{2-e^{4x}}\cdot\ldots\cdot\sqrt[50]{2-e^{50x}}}{(2-e^x)\sqrt[3]{2-e^{3x}}\cdot\ldots\cdot\sqrt[99]{2-e^{99x}}} \), find the derivative \( f'(0) \).

Answer: 25


Question: Evaluate the derivative \( y' = \frac{2xy^3-6x(x^2+y^2-1)^2}{6y(x^2+y^2-1)^2-3x^2y^2} \) at the point \((1,0)\) for the implicit function \((x^2+y^2-1)^3=x^2y^3\). Given that the slope of the tangent line at this point is 2, verify this result.

Answer: 2


Question: Let \( D = \{(x,y) : x^2 + y^2 \leq 1\} \) and \( T \) be a transformation of class \( C' \) on an open set containing \( D \), defined by \( T:\begin{cases} u = f(x,y) \\ v = g(x,y) \end{cases} \), where the Jacobian of \( T \) is never zero in \( D \). Given that \(|T(p) - p| \leq \frac{1}{3}\) for all \( p \in D \), prove that there exists a point \( p_0 \in D \) such that \( T(p_0) = (0,0) \).

Answer: (0,0)


Question: Consider a stationary ergodic process $(X_1, X_2, \ldots)$ where each $X_n$ is a real random variable taking values in $[-1, +1]$ and $\mathbb{E}[X_n] = 0$. Define $S_n = \sum_{k=1}^n X_k$. Is the process $(S_1, S_2, \ldots)$ necessarily recurrent, meaning that there exists some $M$ such that almost surely $|S_n| \leq M$ infinitely often?

Answer: Yes


Question: Given that \( f(0) = 0 \), \( f(1) = 2 \), and \( \int_0^1 f(x) \, dx = 3 \), find the value of \( \int_0^1 x f(x) \, dx \).

Answer: \dfrac{5}{3}


Question:
Given the joint probability density function of the lifetimes of two circuits $C1$ and $C2$ in an electronic device:
\[
f(x,y) = \begin{cases}
\frac{24}{(x+y)^4},  & \text{x,y > 1} \\
0, & \text{otherwise}
\end{cases}
\]
where $X$ and $Y$ are continuous random variables representing the lifetimes of $C1$ and $C2$, respectively. Calculate the expected value of the time until the device is completely corrupted.

Answer: 4


Question: Given a sample of 7 elements with an inter-quartile range (IQR) of 4, find the minimum possible value of the variance of the sample.

Answer: \dfrac{8}{3}


Question:
Consider the sequence \( a_n = \frac{n}{10^{\lceil \log_{10} n \rceil}} \) for each natural number \( n \ge 1 \). Determine which of the following statements are true:

(a) \( \liminf a_n = 0 \)
(b) \( \liminf a_n \) does not exist
(c) \( \liminf a_n = 0.15 \)
(d) \( \limsup a_n = 1 \)

Answer: d


Question: Let \( A \in GL(n, \mathbb{C}) \) be such that \( 0 \notin \{x^*Ax : x^*x = 1\} \). Is it true that \( 0 \notin \{x^*A^{-1}x : x^*x = 1\} \)?

Answer: Yes


Question: Let $F$ be an algebraically closed field. Prove that the quotient ring $F[x]/(x-\alpha)$ is isomorphic to $F$ for any $\alpha \in F$. Justify your answer by considering the map $f(x) \to f(\alpha)$. What does this imply about the quotient rings of $F[x]$?

Answer: F


Question: Evaluate the limit using L'Hopital's rule: \[ \lim_{x \to 0} \log (\cot x)^{\tan x} \]

Answer: 0


Question: Find the value of \(x_0\) that minimizes the area of the pedal curve of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) with respect to the point \((x_0, 0)\). The area of the pedal curve is given by the expression \(\frac{1}{2}\pi(a^2+b^2+x_0^2)\).

Answer: 0


Question:
Consider a family of parabolas defined by the equation \( p(x) = ax^2 + bx + c \) for \( x \in [0,1] \), where \( a, b, c \) are real numbers. The parabolas satisfy the following conditions:

1. The area under the parabola from 0 to 1 is 1: \( \int_0^1 p(x)\, dx = 1 \).
2. The parabola has a minimum point: \( a > 0 \).
3. The parabola does not intersect the x-axis: \( b^2 - 4ac < 0 \).

Determine the upper bound of the maximum values of these parabolas on the interval \([0,1]\).

Answer: 4


Question: How many number pairs \((n - 2, n)\) exist, where \(n - 2\) is a prime number and \(n\) is a composite number?

Answer: \infty


Question: Suppose that \(u_1, u_2 \in C^2(\mathbb{R}^{d+1})\) are real-valued functions satisfying the wave equation \(\partial_t^2 u = \Delta u\), and that \(|u_1(t, x)| = |u_2(t, x)|\) for all \((t, x) \in \mathbb{R}^{d+1}\). Does it follow that either \(u_1 = u_2\) or \(u_1 = -u_2\)?

Answer: Yes


Question: Given the differential equation \( \frac{d^2y}{dx^2}+\frac{1}{y}\left (\frac{dy}{dx}\right)^2-\frac{dy}{dx}=0 \) with boundary conditions \( y(0)=0 \) and \( y(1)=1 \), find the value of \( y(2) \).

Answer: \sqrt{e + 1}


Question: Let $X_i$, $i \geq 0$ be independent and identically distributed random variables with probability mass function $p(j) = P\{X_i=j\}$ for $j=1,...,m$, where $\sum^{m}_{j=1}P(j)=1$. Find $E[N]$, where $N=\min\{n>0:X_n=X_0\}$.

Answer: m


Question: Given \( F(x) = \int_{1}^{x}{f(t)}\,dt \), where \( f(t) = \int_{1}^{t^2}{\frac{\sqrt{9+u^4}}{u}}\,du \), find \( F''(2) \).

Answer: \sqrt{265}


Question: Evaluate the limit of the Riemann sum: \[ \lim_{n\to \infty} \frac{1^p+2^p+\ldots+n^p}{n^{p+1}}. \]

Answer: \dfrac{1}{p+1}


Question: Find the minimum polynomial of $\sqrt[3]{2}$ over the field $\mathbb{Q}(i)$. Provide your answer as a polynomial in terms of $X$.

Answer: X^3 - 2


Question: Evaluate the integral \( \int_{0}^{\infty}\frac{\ln x \, dx}{x^2+e^2} \) and determine the greatest integer value of the result, denoted as \([k]\).

Answer: 0


Question: Given two functions $f(t)$ and $g(t)$, let $h(t) = f(t) * g(t)$ denote their convolution. Express $f(t-1) * g(t+1)$ in terms of $h(t)$. Provide your answer as a function of $h(t)$.

Answer: h(t)


Question: Evaluate the definite integral: $$\int _0 ^{2\pi} y e^{-a\cos\left( y\right)}\, \mathrm{d}y$$ where $a$ is a constant.

Answer: 2\pi^2 I_0(a)


Question: Evaluate the limit without using L'Hospital's rule: $$ \lim_{x\to 1}\frac{\sqrt[m]{x} - 1}{\sqrt[n]{x} - 1}. $$

Answer: \dfrac{n}{m}


Question: Determine whether there exists a smooth, closed, non-orientable 6-manifold $M$ such that $H_4(M;\mathbb{Z})=\mathbb{Z}/2$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0}\left(\frac{1}{(\sin^{-1}(x))^2} - \frac{1}{x^2}\right) \] without using L'Hôpital's rule.

Answer: -\dfrac{1}{3}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=0}^n\binom{n}{k}\frac{3k}{2^n(n+3k)} \]

Answer: \dfrac{3}{5}


Question: Let $A$, $B$, and $C$ be groups where $C$ is finite and $A$ and $B$ are infinite. If $A \oplus C \cong B \oplus C$, is it true that $A \cong B$? Justify your answer.

Answer: A \cong B


Question: What is the coproduct in the category of commutative rings?

Answer: A \otimes_{\mathbb{Z}} B


Question: Find \( \lim_{x\to 0^+}(\sin x)^\frac{1}{\ln x} \).

Answer: e


Question: Evaluate the infinite sum \( \sum_{k=1}^{\infty} \left( \sqrt{k} - 2\sqrt{k+1} + \sqrt{k+2} \right) \).

Answer: 1 - \sqrt{2}


Question:
Given the function \( v = \frac{x-y}{x^2+y^2} \), find the analytic function \( f(z) = u(x, y) + iv(x, y) \). Choose the correct option:

(a) \( z+c \)
(b) \( z^{-1}+c \)
(c) \( \frac{1-i}{z}+c \)
(d) \( \frac{1+i}{z}+c \)

Answer: D


Question: Let $A$ and $B$ be non-negative matrices such that $0 \leq B \leq 1$. Determine whether the inequality $BAB \leq A$ holds in the quadratic form sense. Note that $A$ and $B$ do not need to commute.

Answer: No


Question:
Find the limit of the sequence defined by the infinite nested radical expression:
\[ \sqrt{2^{\sqrt{2^{\sqrt{2^{\sqrt{\ldots}}}}}}} \]

Answer: 2


Question: Evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \) where \( \mathbf{F}(x,y,z) = \langle 2x, 2y, z \rangle \) and \( S \) is the portion of the plane \( x+y+z=1 \) in the first octant, oriented upward.

Answer: \dfrac{5}{6}


Question: Given a random variable \( X \) that is exponentially distributed with rate parameter \( \lambda \), find the variance of \( \frac{1}{X} \), i.e., \( \operatorname{Var}\left(\frac{1}{X}\right) \).

Answer: \infty


Question:
Given the sequence of real numbers \(a_1, a_2, \ldots, a_n\) such that:
\[ \sqrt{a_1} + \sqrt{a_2 - 1} + \sqrt{a_3 - 2} + \cdots + \sqrt{a_n - (n-1)} = \frac{1}{2}(a_1 + a_2 + \cdots + a_n) = \frac{n(n-3)}{4} \]
Compute the value of \(\sum_{i=1}^{100} a_i\).

Answer: 4850


Question: Show that the limit \(\lim_{(h,k)\to (0,0)}\dfrac{|hk|^{\alpha} \log(h^2+k^2)}{\sqrt{h^2+k^2}} = 0\) exists for \(\alpha > \frac{1}{2}\).

Answer: 0


Question: Let \( f(x) \) be a function in \( C^\infty(\mathbb{R}) \) such that \( f(a) = 0 \). Is it true that \( f(x) = (x-a)g(x) \) for some \( g(x) \in C^\infty(\mathbb{R}) \) with \( g(a) = f'(a) \)?

Answer: Yes


Question: Given $a$ apples and $b$ mangoes, where $a$ and $b$ are non-negative integers, you can convert one apple to a mango or one mango to an apple in one move, each with equal probability. What is the expected number of moves required until only one type of fruit remains, i.e., either all apples or all mangoes?

Answer: ab


Question:
Find the solution of the partial differential equation \( \frac{\partial u}{\partial x}=4\frac{\partial u}{\partial y} \) given the initial condition \( u(0, y)=8e^{-3y} \). Choose the correct option:

(a) \( u(x, y)=8e^{-12x-3y} \)

(b) \( u(x, y)=8e^{3x-4y} \)

(c) \( u(x, y)=8e^{-3y-4x} \)

(d) \( u(x, y)=8e^{-12x+3y} \)

Answer: a


Question: Evaluate the determinant of the matrix $A$, where $A$ is an $n \times n$ matrix defined by $a_{ij} = \min\{i, j\}$ for all $i, j \in \{1, \ldots, n\}$. For example, for $n=2$, $A_2 = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}$, and for $n=3$, $A_3 = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{pmatrix}$. Determine $\det(A)$ for a general $n$. \[ \]

Answer: 1


Question: Given the function \(f(x) = x^2 - 3x + 2\), find the sum of all the roots of the equation \(f^{[5]}(x) = 0\), where \(f^{[n]}(x)\) denotes the \(n\)-th iterate of \(f\).

Answer: 48


Question: Determine the number of ordered pairs \((a, b, c)\) of non-negative integers such that \(2^a + 2^b = c!\).

Answer: 5


Question: Let \( a \) be an element of a group with infinite order. Determine the number of generators of the cyclic group \( \langle a \rangle \).

Answer: 2


Question: Determine whether the statement \( \sum_{n\geq1}\frac{1}{n^2}=\sum_{n\leq x}\frac1{n^2}+\mathcal O(1/x) \) is true as \( x\to\infty \).

Answer: True


Question: The acceleration of a body moving in a straight line is given by \(a = 4v^2\). Given that the velocity \(v = e\) when \(x = 1\), find the velocity of the body when \(x = 2\).

Answer: e^5


Question: In a cube-shaped room with mirrors on all six faces, if you stand at the center of the room, how many reflections of yourself can you see in one mirror?

Answer: \infty


Question: Evaluate the limit \(\lim_{x\to 0}\frac{\sqrt[5]{1+\sin x}-1}{\ln(1+\tan x)}\) without using L'Hôpital's rule, relying only on algebraic manipulation and known limits.

Answer: \dfrac{1}{5}


Question: Evaluate the sum \( \sum_{k=1}^{10}\left(\sin\frac{2k\pi}{11}+i\cos\frac{2k\pi}{11}\right) \).

Answer: -i


Question: Evaluate the integral \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dfrac{x^{2}\cos(x)}{1+\exp(x^{2}\sin(x))}\,dx \) and express it in the form \(\frac{\pi^a - b}{c}\), where \(a, b, c \in \mathbb{Z}\). Find the value of \(a + b + c\).

Answer: 14


Question: Let $X$ and $Y$ be two compact metric spaces. Suppose there is a sequence of bi-Lipschitz homeomorphisms $f_n: X \to Y$, with a constant $c \in (0,1]$, such that for any $x_1, x_2 \in X$, the inequality \( c \cdot d(x_1, x_2) \le d(f_n(x_1), f_n(x_2)) \le \frac{1}{c} \cdot d(x_1, x_2) \) holds. Is it true that there exists a subsequence $f_{n_i}$ that uniformly converges to a bi-Lipschitz homeomorphism $f$ with the same constant $c$?

Answer: Yes


Question: Let $k$ be a field and $\alpha$ be algebraic over $k$. Let $K$ be the Galois closure of $k(\alpha)$, obtained by adding all conjugates of $\alpha$. If $f(x) \in k[x]$ is irreducible over $k[\alpha]$, is $f(x)$ also irreducible over $K$? Answer yes or no.

Answer: No


Question: Evaluate the line integral \( \int_C \vec{F} \cdot d\vec{s} \) where \( \vec{F}(x,y,z) = \left\langle\frac{z^2}{x},\frac{z^2}{y},2z\ln(xy)\right\rangle \) and \( C \) is the path consisting of straight line segments from \( P = (1, 2, 1) \) to \( Q = (4, 1, 7) \) to \( R = (5, 11, 7) \), and then back to \( P \).

Answer: 0


Question: Let $X$ be a complete metric space, and let $A$ and $B$ be disjoint closed subsets of $X$ such that neither $A$ nor $B$ has a limit point in $X$, and both $A$ and $B$ are infinite. Does there exist a uniformly continuous function $f: X \to \mathbb{R}$ such that $d(f(A), f(B)) := \inf \{|f(a) - f(b)| : a \in A, b \in B\} > 0$?

Answer: No


Question: Consider a sample $X_1, \ldots, X_n$ from a Uniform Distribution $(0, \theta)$ where $\theta > 0$ is an unknown parameter. Two estimators for $\theta$ are given: \( \hat{\theta}_1 = 2\bar{X} \) based on the sample mean, and \( \hat{\theta}_2 = \frac{n+1}{n}X_{(n)} \) based on the nth order statistic. Determine which estimator is better in terms of Mean Square Error (MSE).

Answer: \hat{\theta}_2


Question: Let $R$ be a regular local ring and $I$ a non-zero radical ideal that is height unmixed. Suppose $J$ is a radical ideal contained in $I$ with the same height as $I$. Is there always a non-minimal prime $P \in \operatorname{Ass}(R/J)$?

Answer: No


Question: Evaluate the limit of the sequence \( a_n = \frac{2^n + n! + n}{5^n + 3n^2 - 1} \) as \( n \to \infty \).

Answer: \infty


Question: Find the derivative of the function \( g(t) = \int_t^{t^2} \cos(tx) \, dx \) and evaluate it at \( t = 0 \).

Answer: -1


Question: Let $K$ be a local field of characteristic zero, $k$ its residue field, $R$ its ring of integers, and $p$ the characteristic of the residue field $k$. Let $G$ be the Galois group of $K$, $I \subset G$ the inertia group, and $P$ the maximal pro-$p$ subgroup of $I$. Let $I_t := I/P$. Let $A_0$ be an abelian scheme over $R$ with generic fibre $A$. Then $A[p]$ is an $I$-module. Let $V$ be a Jordan-Hölder quotient of the $I$-module $A[p]$. Is it true that $P$ acts trivially on $V$?

Answer: Yes


Question: What is the expected number of tosses needed to roll three consecutive sixes with a fair six-sided die?

Answer: 258


Question: Find the degree of the splitting field of the polynomial $x^6 - 1$ over $\mathbb{Q}$.

Answer: 2


Question: Evaluate the integral \(\int_\Gamma \frac{2z+j}{z^3(z^2+1)}\,\mathrm{d}z\) where \(\Gamma\) is the positively oriented circle \(|z-1-i| = 2\). Use residue theory to find the value of the integral.

Answer: -\pi


Question: Let $M^n$ be a smooth closed embedded hypersurface in $\mathbb{R}^{n+1}$. Denote by $D$ the bounded connected component of $\mathbb{R}^{n+1} \setminus M$. Assume that $\mathbb{R}^{n+1} \setminus D$ is simply connected. Let $B = B_R^n$ be the ball centered at the origin with radius $R \gg 1$, and let $S$ denote its boundary. Determine whether there exists a smooth map \( F: M \times [0,1] \rightarrow \bar{B} \setminus D \) such that: \( F(x,0) = x \) and \( F(x,1) \in S \) for all \( x \in M \), and for any fixed \( x \in M \), \( F(x,\cdot): [0,1] \rightarrow \bar{B} \setminus D \) is injective.

Answer: Yes


Question: A flame detector has a lifespan that is exponentially distributed with parameter \( \lambda = 0.1/\text{year} \). The number of events that activate the flame detector in an interval of length \( t \) is Poisson distributed with parameter \( \mu t \), where \( \mu = 0.1/\text{year} \). What is the probability that the flame detector never makes an alarm during its entire lifespan?

Answer: \dfrac{1}{2}


Question: Use the divergence theorem to evaluate the surface integral \( \iint_S (2x+2y+z^2) \, dS \), where \( S \) is the sphere defined by \( x^2+y^2+z^2 = 1 \).

Answer: \dfrac{4}{3}\pi


Question: Find the volume of the region in the first octant (where $x \ge 0$, $y \ge 0$, $z \ge 0$) bounded by the cylinder $x^2 + y^2 = 4$ and the planes $z = 2$ and $y + z = 4$. Evaluate the integral \( \int_0^2 \int_0^{4-z} \int_0^{\sqrt{4-y^2}} dx \, dy \, dz \) to find the volume.

Answer: 2\pi


Question: Determine the minimum number of weighings necessary to identify two groups of 4 boxes each, such that the total weight of the boxes in each group is equal, using a balance scale with two pans.

Answer: 4


Question: Calculate \((14^{2014^{2014}}) \pmod{60}\).

Answer: 16


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

Answer: 8\pi


Question: Compute the line integral \( \int_{C}(x+y){\mathbf{i}}+(x-y){\mathbf{j}}\ d\alpha \) around the ellipse \( b^2x^2+a^2y^2=a^2b^2 \) counterclockwise.

Answer: 0


Question: Solve the recurrence relation \(a_n = 2a_{n-1} + 2^n\) with the initial condition \(a_0 = 1\) using generating functions.

Answer: (n + 1)2^n


Question: Find \( \lim\limits_{x\to 0} \frac{a^x-1}{x} \) without using L'Hopital's rule or series expansion.

Answer: \ln a


Question: Find the limit as \( n \to \infty \) of the series \( \sum_{k=1}^n \frac{n^{1/k}}{k} \).

Answer: \infty


Question: Find the value of \( \arctan(\cosh u) \) as \( u \to -\infty \).

Answer: \dfrac{\pi}{2}


Question: Evaluate \( \lim_{a \to \infty} a \cdot P(|X| > a) \) for a random variable \( X \) in \( L^1(P) \).

Answer: 0


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function such that \( f'(x) = 0 \) for all \( x \in \mathbb{R} \setminus \mathbb{Q} \). Does it follow that \( f \) is a constant function? Justify your answer.

Answer: Yes


Question: What is the largest possible order of the finite field needed to contain all the roots of an irreducible polynomial \( f(x) \) of degree 11 over \( \mathbb{F}_5 \)?

Answer: 5^{11}


Question: Let $f \in C^0([0, 1]) \cap C^1(0, 1)$ be a function such that $f(0)=0$. Let $g \in H^1_{loc}(0, 1)$ be another function such that $fg \in C^0([0, 1])$. Determine if $(fg)(0) = 0$.

Answer: 0


Question: Evaluate the improper integral \( \int_{0}^{\infty} \log(x) \, dx \).

Answer: \infty


Question: For which values of \( p \in [1, +\infty) \) does the sequence of functions \( f_h(x) = x^{\frac{1}{h}} \log(x) \) converge in \( L^p \) for \( 0 < x < 1 \)? The pointwise limit of \( f_h \) is \( f(x) = \log(x) \). Analyze the convergence of \( ||x^{\frac{1}{h}} \log(x) - \log(x)||_p \).

Answer: [1, +\infty)


Question: Calculate the limit: \[ \lim_{n\rightarrow\infty}\sqrt[n]{\sum_{k=1}^n \left(k^{999} + \frac{1}{\sqrt{k}}\right)} \]

Answer: 1


Question:
Find a closed form for the expression:
\[ \sum\limits_{k=1}^n \frac{{n \choose k} \cdot {(-1)^{k}}}{k+1} \]

Answer: -\dfrac{n}{n + 1}


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

Answer: 1


Question: Determine if the series \( X_{k+1} = 1 + x + x^{2} + \cdots + x^{2^{k+2}-3} \) can be expressed as \( (1 - x)^{-1} \) as \( k \to \infty \), given that \( \|x\| < 1 \).

Answer: (1 - x)^{-1}


Question: Determine the number of pairs of positive integers \((p, q)\) that satisfy the equation \((p+1)! + (q+1)! = p^2 q^2\).

Answer: 2


Question: Find the natural number \( n \) such that the ring \( (\mathbb{Z}_n, +, \cdot) \) has exactly 4 invertible elements and 5 zero-divisors.

Answer: 10


Question: Let \( A \in M_5(\mathbb{R}) \) be a matrix such that \( A^2 - 4A - I = 0 \). If \( a_1, a_2, \ldots, a_5 \) are the eigenvalues of \( A \), find the value of \( (a_1 - \frac{1}{a_1}) + (a_2 - \frac{1}{a_2}) + \ldots + (a_5 - \frac{1}{a_5}) \).

Answer: 20


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

Answer: (-1, 1)


Question: Consider the abelian group \( G \) given by the real line with addition, and let \( H \) be an infinite torsion abelian group. If the order of the elements of \( H \) is not uniformly bounded by a constant, is it true that there can only be the trivial group homomorphism from \( G \) to \( H \), which sends every element of \( G \) to the identity of \( H \)?

Answer: Yes


Question: Let \( X \) be an exponential random variable with mean 2. Define the random variable \( Y = X - 2 \) given that \( X > 2 \). Find the expected value of \( Y \), \( E(Y) \).

Answer: 2


Question: Let $a < b$ be two real numbers and $f(x,y)$ be a bivariate polynomial. Suppose that for $x < 0$, $f(x,.)$ has no real roots in the interval $[a,b]$, and for $x > 0$, $f(x,.)$ has at least one real root in the interval $[a,b]$. Does it follow that $f(0,.)$ either vanishes at one of the endpoints (i.e., $f(0,a)=0$ or $f(0,b)=0$), or $f(0,.)$ has a double root in the open interval $(a,b)$? Provide a justification for your answer.

Answer: Yes


Question: Let $X$ be an exponential random variable with rate parameter $\lambda = 5$, and let $Y$ be a uniformly distributed random variable on the interval $(-3, X)$. Calculate the expected value $\mathbb{E}(Y)$. Use the law of total expectation and conditional expectations to find the solution.

Answer: -\dfrac{7}{5}


Question: Let $G$ be a finite group acting on a finite set $X$, with the action of $g \in G$ on $x \in X$ denoted by $gx$. For each $x \in X$, the stabilizer of $x$ is the subgroup $G_x = \{g \in G : gx = x\}$. Given $x, y \in X$ and $y = gx$ for some $g \in G$, express the stabilizer $G_y$ in terms of $G_x$.

Answer: G_y = g G_x g^{-1}


Question: Evaluate the contour integral \( \int_\gamma \frac{z}{e^z-1}dz \) where \( \gamma \) is a closed contour enclosing two poles and one removable singularity.

Answer: 0


Question: Given the joint moment generating function \(M_{X,Y}(t,u) = \exp[2t+3u+t^2+atu+2u^2]\), determine the value of \(a\) that makes the random variables \(X + 2Y\) and \(2X - Y\) independent.

Answer: \dfrac{4}{3}


Question: Consider the sequence of measures on $[-1, 1]$ given by $\mu_n = n \phi(nx) dx$, where $\phi \in C_c^{\infty}([-1,1])$ is a smooth, compactly supported function with integral $2$, and $dx$ is the Lebesgue measure. Determine the limit measure of $\mu_n$ as $n \to \infty$ in the weak-* topology.

Answer: 2\delta_0


Question: Determine whether the statement $X-A \subset \overline{X-\operatorname{Int}(A)}$ is true or false, where $A$ is a proper subset of the topological space $X$. Provide a proof or a counterexample to support your answer.

Answer: True


Question:
Calculate the value of the following double sum without using the binomial distribution:
\[ \sum_{s=0}^\infty\sum_{t=0}^s e^{-3}st\frac{1}{t!}\frac{2^{s-t}}{(s-t)!} \]

Answer: 4


Question: Determine the number of groups of order 88 that have at least one element of order 8. Consider the groups \( \mathbb{Z}_{88} \) and \( \mathbb{Z}_{11} \rtimes_{\phi}\mathbb{Z}_8 \), where \( \phi:\mathbb{Z}_{8}\rightarrow \text{Aut}(\mathbb{Z}_{11}) \) is defined by \( y\mapsto \phi(y)(a)=a^{10} \) for \( a\in \mathbb{Z}_{11} \). Are there any other groups of order 88 with this property?

Answer: 2


Question: Find the limit of the sequence \( \prod_{k=n+1}^{k=2n} \left(1-\frac{1}{k^2}\right) \) as \( n \to \infty \).

Answer: 1


Question: Does there exist a Hilbert space \( \mathcal{H} \) such that a sequence \( \{ x_n \} \subset \mathcal{H} \) is weakly convergent but not convergent in the Hilbert space norm?

Answer: Yes


Question: Solve the equation \(x! = n\) for a given value of \(n\). For example, find \(x\) when \(n = 6\).

Answer: 3


Question: Let $X_1, X_2, X_3, X_4, X_5$ be a random sample from a standard normal distribution $N(0,1)$. Define $T = \sum_{i=1}^5 (X_i - \bar{X})^2$, where $\bar{X}$ is the sample mean. Calculate $E(T^2 \bar{X}^2)$. Provide your answer as a single value.

Answer: \dfrac{24}{5}


Question: Determine the number of non-zero functions \( f: \mathbb{Z} \to \mathbb{C} \) that are periodic with period \( a \) and satisfy the functional equation \( f(x+y) = f(x)f(y) \).

Answer: a


Question:
Evaluate the limit:
\[ \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{D(\epsilon)} \frac{1 + \sin(x) + \sin(y)}{\sqrt{x^2 + y^2}} \, d(x,y) \]
where \( D(\epsilon) = \{(x,y) \in \mathbb{R}^2 \mid 0 < x^2 + y^2 \le \epsilon^2\} \).

Answer: 2\pi


Question:
Consider the two circles given by the equations \(x^2+y^2+\lambda_1(x-y)+c=0\) and \(x^2+y^2+\lambda_2(x-y)+c=0\), where \(\lambda_1, \lambda_2 \in \mathbb{R}\). If one of these circles lies within the other, determine the correct condition for \(c\):

\(\text{A) } c < 0\)

\(\text{B) } c = 0\)

\(\text{C) } c > 0\)

\(\text{D) None of these}\)

Answer: C


Question: Evaluate whether the expression \( \epsilon^{\rho\sigma\mu\nu}\epsilon_{\rho\sigma\mu\nu} \) is zero, given that \( \epsilon^{\rho\sigma\mu\nu}\epsilon_{\mu\nu\rho'\sigma'} = -2(\delta^{\rho}_{\rho'}\delta^{\sigma}_{\sigma'}-\delta^\rho_{\sigma'}\delta^{\sigma}_{\rho'}) \).

Answer: -24


Question: Determine whether the following statement is true or false: If a square matrix $B$ is obtained from a permutation matrix $A$ by swapping its rows, then there exists an invertible matrix $T$ such that $T^{-1}AT = B$. Provide a justification for your answer.

Answer: False


Question: Let $G$ be the group of invertible matrices of the form \( \begin{bmatrix}a & b & c\\0 & d & e\\0 & 0 & f\end{bmatrix} \) where \( a,b,c,d,e,f \in \mathbb{Z}_3 \). Determine the composition length of $G$ and identify its composition factors in terms of known groups, specifying which groups occur as composition factors and how many times each occurs in the composition series.

Answer: 6


Question: For positive numbers $a$, $b$, and $c$ such that $\frac{a}{2} + b + 2c = 3$, find the maximum value of $\min\left\{ \frac{1}{2}ab, ac, 2bc \right\}$.

Answer: 1


Question: For what value(s) of \( a \), if any, is the function \( g(x, y) = \begin{cases} \frac{\sin(2x^2+2y^2)}{x^2+y^2},& (x, y) \neq (0,0) \\ a,& (x, y) = (0,0) \end{cases} \) continuous at \((0, 0)\)?

Answer: 2


Question: Let \( V \) be a finite-dimensional vector space, and let \( \mathcal{L}(V) \) denote the space of all endomorphisms of \( V \). For any \( \phi \in \mathcal{L}(\mathcal{L}(V)) \), there is a unique bilinear map \( \mathcal{L}(V) \otimes \mathcal{L}(V) \to \mathcal{L}(V) \) (denoted by \( \Phi(\phi) \)) satisfying \( \Phi(\phi) (a \otimes b) = \phi(ab) \) for any \( a, b \in \mathcal{L}(V) \). Determine the rank of the linear map \( \Phi \).

Answer: n^4


Question: Evaluate the infinite sum \( \sum_{n=1}^{\infty}\frac{(n-1)!}{(x+1)(x+2)(x+3)\cdots(x+n)} \) for a positive real number \( x \).

Answer: \dfrac{1}{x}


Question: Given that $(x_1, y_1)$ and $(x_2, y_2)$ are independent and identically distributed (i.i.d.) pairs, and that the conditional expectations are $E(y_1|x_1) = \alpha + \beta x_1$ and $E(y_2|x_2) = \alpha + \beta x_2$, determine if there is enough information to compute $E(y_1y_2|x_1,x_2)$ in terms of $x_1, x_2, \alpha, \beta$. Answer YES or NO.

Answer: YES


Question: Let $X$ be a set of points in $\mathbb{R}^2$, and define $g(X) = \arg \min_{y \in \mathbb{R}^2} \sum_{x_i \in X} \| x_i - y \|_2$ as the geometric median of $X$. If $X$ and $X'$ are neighboring sets, differing by only one element (denoted $X \sim X'$), compute $\sup_{X':X \sim X'} \| g(X) - g(X') \|_2$. Determine the supremum of the Euclidean distance between the geometric medians of these neighboring sets.

Answer: \infty


Question: Calculate the double integral \( \int\int_D \frac{x}{4x^2 + y^2} \, dA \), where \( D \) is the region in the first quadrant bounded by the ellipses \( 4x^2 + y^2 = 16 \) and \( 4x^2 + y^2 = 1 \).

Answer: \dfrac{3}{4}


Question: Calculate the limit \( \lim_{x\to 0} \frac{\sin(\sin x) - \sin x}{x^3} \) using the Maclaurin expansion of \( \sin(\sin x) \).

Answer: -\dfrac{1}{6}


Question:
Determine which of the following numbers cannot be the order of the quotient group $G/Z(G)$, where $Z(G)$ is the center of the group $G$:

(a) 15,
(b) 25,
(c) 6,
(d) 4.

Answer: A


Question: Show that the image of the unit disk $B_1(0)$ under the function $f(z) = 6z^6 - 2z^2 + 1$ contains the ball $B_4(1)$ using Rouché's theorem.

Answer: B_4(1)


Question: Let $p_1, \dots, p_r$ be $r$ distinct odd primes. Define the group \( G = \prod_{i=1}^r \mathrm{SL}_3(\mathbb{F}_{p_i}) \). Determine the maximum possible rank of a subgroup of \( G \), denoted as \( \max \{d(H) : H \leq G\} \), where \( d(K) \) is the smallest cardinality of a generating set of a group \( K \).

Answer: 2r


Question: In the floating point number system \( F(10,10,-98,100) \), calculate the number of significant decimal digits lost when subtracting \( \sqrt{1001} \) from \( \sqrt{1002} \).

Answer: 3


Question: Let \( f(x) = e^{x^2} \) and \( F(x) \) be an antiderivative of \( f \). Calculate the limit: \[ \lim_{x \rightarrow \infty} \frac{xF(x)}{f(x)}. \]

Answer: \dfrac{1}{2}


Question: Classify the group \((\mathbb{Z}_4 \times \mathbb{Z}_2)/({0} \times \mathbb{Z}_2)\). Determine whether it is isomorphic to the cyclic group \(\mathbb{Z}_4\) or the Klein-4 group.

Answer: \mathbb{Z}_4


Question: Determine the Galois group of the polynomial \(x^4 - x^2 - 3\).

Answer: D_4


Question: Let \((S,\tau)\) be a topological space and \(X \subseteq S\). Define a sequence of sets as follows: \(a_0 = X\) and \(a_{n+1} = \partial a_n\), where \(\partial\) denotes the boundary operator. Must there exist a natural number \(k\) such that \(a_k = a_{k+1}\)?

Answer: Yes


Question: Evaluate the integral: \[ \int_{-1}^{1} \frac{x^4}{a^x+1} \, dx \]

Answer: \dfrac{1}{5}


Question: Evaluate the definite integral \( \int_0^\infty e^{-ax} x^4 \, dx \) for \( a > 0 \).

Answer: \dfrac{24}{a^5}


Question: Evaluate the expression \(1^{2017} + 2^{2017} + \cdots + 1000^{2017} \pmod{2016}\) without using a calculator.

Answer: 784


Question: If $f$ is holomorphic on a domain $D$ and $f$ is bounded on $D$, does this imply that $f'$ is also bounded on $D$? Justify your answer.

Answer: No


Question: Let $A \in \mathbb{R}^{m \times n}$ be a rank-$k$ matrix with real coefficients. Suppose that \( A = L_1R_1^T \) and \( A = L_2 R_2^T \), where $L_1, L_2$ are $m \times k$ real matrices with orthonormal columns (i.e., $L_1^T L_1 = L_2^T L_2 = I_k$), and $R_1, R_2$ are $k \times n$ real matrices. Determine if there exists a $k \times k$ orthogonal matrix $U$ such that $L_2 = L_1 U$. If so, provide the reasoning.

Answer: Yes


Question: Assuming the axiom of choice, determine whether every quasi-connected graph has a spanning tree, defined as a subgraph where every pair of vertices is connected by a unique finite or infinite path.

Answer: Yes


Question: Consider two real matrices $A$ and $B$ of dimension $n \times n$, where $A$ is non-Hurwitz and $B$ is Hurwitz. Is it true that there exists a $\phi > 0$ such that $A + \theta B$ is Hurwitz for all $\theta \geq \phi$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \displaystyle\int\limits_\gamma \frac{2z+1}{e^{\pi z}-1}dz \) along the specified path \( \gamma \).

Answer: 2i


Question: Determine the value of $b_{32}$ such that the polynomial product \( \prod_{n=1}^{32}(1-z^n)^{b_n} \equiv 1-2z \pmod{z^{33}} \), where each \( b_n \) is a positive integer. Express your answer in exponential notation.

Answer: 2^{27} - 2^{11}


Question: Find the greatest value of the function \( f(x) = \text{Mid}\{ x-1, (x-3)^2, 3-\frac{(x-3)^2}{2} \} \) for \( x \in [1, 4] \), where \( \text{Mid}\{ g(x), h(x), p(x) \} \) denotes the median of the three values \( g(x), h(x), \) and \( p(x) \) at a particular \( x \).

Answer: 1 + \sqrt{3}


Question: Find the volume of the region bounded by the cylinder $x^2 + y^2 = 2y$ and the planes $z = 0$ and $x + y + z = 7$. Express your answer as a single numerical value.

Answer: 6\pi


Question: Calculate the volume of the solid formed by rotating the curve \( y = \frac{1}{1+x^2} \) for \( x \ge 0 \) about the y-axis.

Answer: \infty


Question: For which values of \( \alpha \in \mathbb{R} \) is the matrix \( A \) diagonalizable? \[ A=  \begin{pmatrix} 1+\alpha & -\alpha & \alpha \\ 2+\alpha & -\alpha & \alpha-1 \\ 2 & -1 & 0 \end{pmatrix} \]

Answer: 0


Question:
Determine the set of discontinuities \( D_f \) for the function \( f: \mathbb{R} \to \mathbb{R} \) defined by:
\[
f(x) = \begin{cases} 0 & x \notin \mathbb{Q} \\ \frac{1}{n} & x=r_n \in \mathbb{Q} \end{cases}
\] where \( \mathbb{Q} = \{r_1, r_2, \ldots\} \) is the set of rational numbers.

Answer: \mathbb{Q}


Question: Determine if there exists a reflection $R$ such that $T^{-1}ST=R$, where $T: \mathbb{R}^2 \to \mathbb{R}^2$ is the counterclockwise rotation by $\frac{\pi}{2}$ and $S: \mathbb{R}^2 \to \mathbb{R}^2$ is the reflection with respect to the line $x+3y=0$. If such a reflection $R$ exists, find the line with respect to which the reflection $R$ occurs.

Answer: 3x - y = 0


Question: Evaluate the limit: \( \lim_{x \rightarrow e} \frac{e-x}{\ln x -1} \) without using L'Hopital's rule.

Answer: -e


Question: A monkey is sitting at a keyboard with only the keys "a", "b", and "c". The monkey presses the keys at random. Let \( X \) be the number of keys pressed until the monkey has pressed all the different keys at least once. What is the expected value of \( X \)?

Answer: \dfrac{11}{2}


Question: Find the probability density function of the random variable $U = XY$, where $X$ and $Y$ are independent random variables uniformly distributed over the interval $[0,1]$. Provide the expression for the p.d.f. of $U$.

Answer: -\ln u


Question:
Consider \( \mathbb{R}^2 \) with the \( \|\cdot\|_1 \)-norm and the subspace \( M = \{(x,0) \mid x \in \mathbb{R}\} \). Define the function \( g: M \to \mathbb{R} \) by \( g(x,0) = x \). Which of the following is a Hahn-Banach extension \( f \) of \( g \) to \( \mathbb{R}^2 \)?

(a) \( f(x,y) = 2x \)
(b) \( f(x,y) = x+y \)
(c) \( f(x,y) = x-2y \)
(d) \( f(x,y) = x+2y \)

Answer: B


Question: Provide an example of a non-abelian group that contains an element which is its own inverse and is not the identity element.

Answer: S_3


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

Answer: \dfrac{4}{e}


Question:
Consider a short exact sequence of groups:
\[ 1 \to F \to G \to \mathbb{Z}^2 \to 1, \]
where $F$ is a locally finite group. Determine whether $G$ necessarily contains a subgroup isomorphic to $\mathbb{Z}^2$. Provide a justification for your answer.

Answer: Yes


Question: Find the supremum of the set \( \{ \operatorname{Re}(iz^3+1) : |z|<2 \} \).

Answer: 9


Question: Does there exist a Galois extension $L/\mathbb{Q}$ with Galois group $A_4$ such that all the decomposition groups are cyclic?

Answer: Yes


Question: Consider a power series \( \sum_{n \ge 0} a_n z^n \) where all coefficients \( a_i \) are non-zero complex numbers, and the radius of convergence is finite and non-zero, specifically \( R = 2 \). Is it necessary that the limit \( \lim_{n \rightarrow \infty} \frac{|a_{n+1}|}{|a_n|} \) exists?

Answer: No


Question: Let \( A, B, \) and \( X \) be \( 7 \times 7 \) matrices such that \( \det A = 1 \), \( \det B = 3 \), and \( A^{-1} X B^t = -I_7 \), where \( I_7 \) is the \( 7 \times 7 \) identity matrix. Calculate \( \det X \).

Answer: -\dfrac{1}{3}


Question: Determine the minimum number of coin flips $n$ required to ensure that the estimated probability $\hat{p} = X/n$ of a biased coin is within $5\%$ of the true probability $p$, given that $0.4 \leq p \leq 0.6$. Ensure that the probability of this confidence interval, $\mathbb{P}[0.95p \leq \hat{p} \leq 1.05p]$, is at least 0.95.

Answer: 2305


Question:
Given a continuous function $u(x,y)$ on $B^+ \cup B^0$ that is harmonic in $B^+$ and satisfies $\frac{\partial u}{\partial y}(x,0) = 0$ for all $-1 < x < 1$, determine if $u$ can be extended harmonically to the whole unit ball $B$ using the even extension:
\[ w(x,y) := \begin{cases} u(x,y), & y \geq 0 \\ u(x,-y), & y < 0 \end{cases}. \]

Answer: Yes


Question: Determine the number of ways to partition the set \( S = \{1, 2, \ldots, 2020\} \) into two disjoint subsets \( A \) and \( B \) such that for any \( a \in A \) and \( b \in B \), the sum \( a + b \) is never a multiple of 20. Note that the order of \( A \) and \( B \) does not matter, and either set can be empty.

Answer: 1024


Question: Determine the cardinality of the set of all sequences in \(\mathbb{R}\) that converge to a real number \(a\).

Answer: \mathfrak{c}


Question: Find the value of \( a \) such that the line integral \( I = \int_\Gamma x(a+y)(ydx+xdy) \) over the path \( \Gamma \) equals 36. The path \( \Gamma \) consists of three segments: from \( O = (0,0) \) to \( A = (3,0) \), from \( A = (3,0) \) to \( B = (0,3) \), and from \( B = (0,3) \) back to \( O = (0,0) \). The segment \( AB \) is characterized by the quarter-circle \( \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 = 9, x \ge 0, y \ge 0\} \).

Answer: 4


Question: Determine whether the sequence \( \frac{f(n)}{2^n} \), where \( f(n) \) is the \( n^{th} \) Fibonacci number, converges as \( n \to \infty \).

Answer: 0


Question: Find the sum of the coefficients of $x^{20}$ and $x^{21}$ in the power series expansion of $\frac{1}{(1-x^3)^4}$.

Answer: 120


Question: Evaluate the limit using Taylor series: \[ \lim_{x\to0} \frac{\sqrt{1+ x\sin(x)} - \sqrt{\cos(2x)}}{ \tan^2(x/2)} \]

Answer: 6


Question: Let $L/K$ be a Galois extension of number fields with $G = \text{Gal}(L/K)$ being abelian. If $\mathcal{O}_L$ is free as an $\mathcal{O}_K[G]$-module, determine if it necessarily has rank 1.

Answer: Yes


Question: Construct a finite field of order $2^6$. Provide a polynomial that can be used to construct this field and explain why it is suitable.

Answer: x^6 + x + 1


Question: Calculate the integral \( \int_{0}^{441}{\frac{\pi\sin(\pi \sqrt{x})}{\sqrt{x}}}{dx} \) using the method of differentiating under the integral sign (Feynman's Trick).

Answer: 4


Question: A random glass breaks at an exponentially distributed time, with a mean of 24 months. Calculate the probability that, out of 500 glasses, at most 210 break within a year.

Answer: 0.896


Question: Determine whether there exist non-negative integers $a$ and $b$, with neither $a$ nor $b$ equal to zero, such that $a^{2n+1} + b^{2n+1}$ is a perfect square for all non-negative integers $n$. Provide a justification for your answer.

Answer: Yes


Question:
Let \( y(t) \) be a continuous function on \([0, \infty)\) whose Laplace transform exists. If \( y(t) \) satisfies the integral equation
\[
\int_{0}^{t} \left[1 - \cos(t - \tau)\right] y(\tau) \,\mathrm{d}\tau = t^{4},
\]
find \( y(1) \).

Answer: 28


Question: If $\chi_1$ and $\chi_2$ are primitive Dirichlet characters with moduli $q_1$ and $q_2$ respectively, and $\chi_1(n) = \chi_2(n)$ for all integers $n$, must $q_1 = q_2$? Justify your answer.

Answer: Yes


Question: Find the limit of the function \( p_n = \frac{n}{(2n+1)^2} \frac{1}{\sum_{d=1}^n \dfrac{1}{(2n+d)^2} } \) as \( n \to \infty \).

Answer: \dfrac{3}{2}


Question: If \( \frac{n^2}{n^2+c}(an+b) \) is the general term of an arithmetic series, can we deduce that \( c = 0 \)? Assume \( a, b, c \in \mathbb{R} \).

Answer: Yes


Question:
If \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 - 3x^2 + 3x + 7 = 0 \), and \( \omega \) is a cube root of unity, find the value of the expression:
\[
\frac{\alpha - 1}{\beta - 1} + \frac{\beta - 1}{\gamma - 1} + \frac{\gamma - 1}{\alpha - 1}
\]
Choose the correct answer:
(a) \( \frac{3}{\omega} \)
(b) \( \omega^2 \)
(c) \( 2\omega^2 \)
(d) \( 3\omega^2 \)

Answer: D


Question: Evaluate the triple integral \( \iiint_{x^2+y^2+z^2<z} \sqrt{x^{2}+y^{2}+z^{2}}\, dV \) using spherical coordinates.

Answer: \dfrac{\pi}{10}


Question: Find the remainder when the product of the first 92 positive integers is divided by 97.

Answer: 4


Question: Consider a measure arising from Riesz's theorem on the representation of positive linear functionals on a locally compact space, as stated in Rudin's "Real and Complex Analysis." If this measure is restricted to Borel sets, do the integrals of Borel functions with respect to this restricted measure remain equal to their integrals with respect to the original measure?

Answer: Yes


Question: Find the smallest positive integer $x$ such that $x \equiv 55! \pmod{61}$. Use Wilson's theorem to assist in your calculation.

Answer: 30


Question: Determine whether the inequality \( p_k + p_l + 1 \leq p_{k+l+1} \) holds for all natural numbers \( k \) and \( l \), where \( p_n \) denotes the \( n \)-th prime number.

Answer: Yes


Question: Determine the degree of the minimal polynomial of $e^{\frac{2\pi i}{3}}\sqrt[3]{2}$ over the field $\mathbb{Q}(\sqrt[3]{2})$. Provide your answer as an integer.

Answer: 2


Question: Let \( R \) be a unique factorization domain. If \( P \) is a prime ideal minimal over a principal ideal, is it true that the height of \( P \) is at most 1? Justify your answer.

Answer: Yes


Question: Evaluate the limit \( \lim_{t\to0}\frac{f(4t, e^t)}{f(\sin2t, \cos2t)} \) where \( f:\mathbb{R^2}\to \mathbb{R} \) is a function with continuous partial derivatives, \( f(0,1)=0 \), \( f_x(0,1)=1 \), and \( f_y(0,1)=2 \).

Answer: 3


Question: Find the limit of the sequence defined by \(a_1 = 2\) and \(a_{n+1} = \frac{1}{2} (a_n + \frac{1}{a_n})\) if it exists.

Answer: 1


Question: Let $p(x)$ be a polynomial with integer coefficients. Given that $p(a) = p(b) = p(c) = -1$ for three distinct integers $a$, $b$, and $c$, determine the number of integer roots of $p(x)$. Provide your answer as an integer.

Answer: 0


Question: Let $X$ and $Y$ be topological spaces with $n$ and $m$ path components, respectively. Assume $n$ and $m$ are finite. Determine the number of path components in the product space $X \times Y$. Is it true that the number of path components in $X \times Y$ is $nm$?

Answer: nm


Question: Evaluate the limit: $$\lim\limits_{n\to\infty}\sum\limits_{r=0}^\infty \left(\frac{r}{n}\right)^n$$

Answer: \infty


Question: Given a positive integer $n$, let $P(n)$ be the product of the non-zero digits of $n$ (if $n$ is a one-digit number, $P(n)$ is $n$ itself). Let $S = P(1) + P(2) + P(3) + \ldots + P(999)$. Determine the largest prime factor of $S$. \( \boxed{?} \)

Answer: 103


Question: Consider the sequence of the first 101 integers. Prove that in any permutation of these integers, there exists either an increasing or a decreasing subsequence of at least 11 integers.

Answer: 11


Question: Does there exist a metric \( d \), equivalent to the standard Euclidean metric on the real numbers, such that \((\mathbb{Q}, d)\) is a complete metric space?

Answer: No


Question: Determine whether the following statement is true: If $X_i$ are i.i.d. random variables, is it true that \[ E\left[\frac{1}{\sum_{i=1}^{n}X_i}\right] = \frac{1}{\sum_{i=1}^{n} E\left[X_i \right]} \] where $E[X]$ denotes the expected value of a random variable $X$?

Answer: False


Question:
Let $A$ be the set of all functions $f:\mathbb{R}\to\mathbb{R}$ that satisfy the following two properties:
1. $f$ has derivatives of all orders.
2. For all $x,y \in \mathbb{R}$, $f(x+y)-f(y-x)=2xf'(y)$.

Which of the following statements is true?
(a) Any $f \in A$ is a polynomial of degree less than or equal to 1.
(b) Any $f \in A$ is a polynomial of degree less than or equal to 2.
(c) There exists $f \in A$ which is not a polynomial.
(d) There exists $f \in A$ which is a polynomial of degree 4.

Answer: B


Question: Calculate the surface integral of the vector field \( \mathbf{F}(x,y,z) = (z,x,y) \) over the surface \( S \), which is delimited by the elliptical cylinder \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) and the plane \( z + y = 9 \).

Answer: 0


Question:
Using Stokes' theorem, evaluate the closed loop integral of the vector field \( \vec{F}(x,y,z) = \langle yx-yz , 2x^2+z^2, y^3 \cos(xz)\rangle \) over the boundary of the square defined by:
\[
\begin{cases}
0 \leq x \leq 2\\
0 \leq y \leq 2\\
z=5
\end{cases}
\]

Answer: 32


Question: Given a field extension $K \subseteq F$ and an extension $K(\alpha)/K$ of degree 4, where $\alpha^2$ is not a root of the minimal polynomial $m_{(\alpha,K)}(x)$, calculate the degree of the extension $[K(\alpha^2):K]$. Use the tower formula: $$[K(\alpha^2):K]=[K(\alpha^2):K(\alpha)][K(\alpha):K]$$ and the fact that $[K(\alpha):K]=4$. Determine $[K(\alpha^2):K(\alpha)]$ to find $[K(\alpha^2):K]$. Provide your answer as a single number.

Answer: 2


Question: Determine which function grows faster as $n$ becomes very large: $f(n) = 2^{2^{2^n}}$ or $g(n) = 100^{100^n}$.

Answer: f(n)


Question: Does there exist a non-negative continuous function \(f: \mathbb{R}_+ \rightarrow \mathbb{R}_+\) such that the integral \( \int_0^\infty f \, dx < \infty \) converges, \( f(x) < 1 \) for all \( x \in \mathbb{R}_+ \), and there exists a strictly increasing sequence \( \{q_n\}_{n=0}^\infty \subset \mathbb{R}_+ \) for which \( \sum_{n=0}^\infty f(q_n)=\infty \)?

Answer: Yes


Question: Let \( \mathbb{D} \) be the open unit disc in \( \mathbb{C} \). Let \( w_1, w_2 \in \overline{\mathbb{D}} \) with \( w_1 \neq w_2 \). Define \( \tilde{c}(w_1, w_2) = \sup_f\left\{\left|\frac{f(w_1)-f(w_2)}{1-\overline{f(w_2)}f(w_1)}\right| : f\in \tilde{A}(\mathbb{D})\right\} \), where \( \tilde{A}(\mathbb{D}) = \{ f \in \text{Disc algebra} : f(\overline{\mathbb{D}}) \subset \mathbb{D} \} \). Is it true that \( \tilde{c}(w_1, w_2) \) is always strictly less than 1?

Answer: No


Question:
Given a circle with the equation \(x^2 + y^2 = 25\), let \(M(x_0, y_0)\) be a point on the circle such that \(MN\) is a chord perpendicular to the x-axis. Let \(P(x, y)\) be the intersection of lines \(BM\) and \(NA\), where \(B\) and \(A\) are points on the x-axis. The equations of the lines are given by:
\[ MB: y = \frac{y_0}{x_0+5}x + \frac{5y_0}{x_0+5} \]
\[ NA: y = \frac{y_0}{5-x_0}x - \frac{5y_0}{5-x_0} \]
Find the locus of the points \(P\).

Answer: x^2 - y^2 = 25


Question: Let \( \{z_n\}_{n \geq 1} \) be a sequence of complex numbers and \( z \in \mathbb{C} \). Suppose that for a given \( \varepsilon > 0 \), there exists \( k \in \mathbb{N} \) such that for all \( n \geq k \), we have \( |z_n - z| < \varepsilon \sqrt{1 + |z_n|^2} \). Can it be concluded that \( z_n \to z \) as \( n \to \infty \)? Justify your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim _{n \rightarrow \infty}\left(\frac{\sin \left\{\frac{2}{n}\right\}}{\left[2 n \tan \frac{1}{n}\right]\left(\tan \frac{1}{n}\right)}+\frac{1}{n^{2}+\cos n}\right)^{n^{2}} \] where \(\{\}\) and \([\]\) denote the fractional part function and the greatest integer function, respectively.

Answer: 1


Question: Compute the limit: \[ \lim_{n \to \infty} \left( 1-\frac{2t}{n^2} \right)^{-n/2} \]

Answer: 1


Question: Find the function $f(x)$ such that \( \sum_{n=0}^{\infty} \frac{n+1}{n!}x^n \) is the Taylor series of $f(x)$. Provide your answer in terms of elementary functions.

Answer: (x + 1)e^x


Question: Evaluate the integral \( \frac{2}{\pi} \int_{-\pi}^{\pi} \frac{\sin\left(\frac{9x}{2}\right)}{\sin\left(\frac{x}{2}\right)} \, dx \).

Answer: 4


Question: Evaluate the limit: \[ \lim_{x\to\infty} \frac{(x!)^{\frac{1}{x}}}{x} \]

Answer: \dfrac{1}{e}


Question: Evaluate the contour integral \( \int_C \frac{z+1}{z^2-2z} \, dz \) using Cauchy's residue theorem, where \( C \) is the circle \( |z|=3 \).

Answer: 2\pi i


Question: Can exotic \(\mathbb{R}^4\) admit a metric such that its Riemann tensor is zero everywhere?

Answer: No


Question: Evaluate the arc length of the curve given by \( \alpha(t) = (\alpha \cos^3 t, \alpha \sin^3 t) \) for \( \alpha > 0 \) and \( t \in [0, 2\pi] \).

Answer: 6\alpha


Question: Find the minimum value of \( \frac{\sqrt[3]{x-1}+\sqrt[3]{y-1}}{\sqrt[3]{x+y-2}} \) given that \( y > 1 \) and \( x > 1 \).

Answer: 1


Question: Determine the values of \(\delta\) for which the series \(\sum_{n=1}^{\infty}\frac{x}{n(1+n^{\delta}x^{2})}\) is uniformly convergent on \(\mathbb{R}\).

Answer: \delta > 0


Question: Determine whether a finitely abelian group of the form \(\mathbb{Z} \times G\), where \(G\) is a finite abelian group, has only finitely many retracts \(H\). Consider retracts as subgroups \(H \subset G\), not "up to isomorphism."

Answer: Yes


Question: Determine the limit of the function \( f(x,y) = \frac{y}{x} \) as \((x,y)\) approaches \((0,0)\) with the restriction that \(|y| < |x^5|\).

Answer: 0


Question: Let $h: \mathbb{R}^3 \rightarrow \mathbb{R}$ be defined by $h(x,y,z)=g(x^2-y^2,y^2-z^2)$, where $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ is a differentiable function with $\nabla g(0,0)=(1,2)$. Determine the directional derivative of $h$ at the point $(1,1,1)$ in the direction of the vector $\vec{v}=(2,0,1)$. Use the Jacobian matrix in your approach.

Answer: 0


Question: Calculate the integral of \( f(z) = \frac{z^2}{e^{\frac{1}{(z+i)^2}}} \) over the contour \(|z|=2\).

Answer: -4\pi


Question: Find the smallest integer $n > 2$ such that the sum of its non-divisors is a multiple of $n$. Provide the value of $n$.

Answer: 24


Question: Let \( f \) be an alternating tensor in \( A^2(\mathbb{R}^3) \). Given vectors \( v_1 = (2,3,1) \), \( v_2 = (2,-5,4) \), \( v_3 = (6,1,6) \), and \( v_4 = (4,6,2) \), and that \( f(v_1, v_2) = 3 \), determine the value of \( f(v_3, v_4) \).

Answer: -6


Question: Find the norm of the operator \(L: \ell_2 \to \mathbb{R}\) defined by \(L((x_n)) = \sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n(n+1)}}\), where \((x_n) \subset \ell_2\).

Answer: 1


Question: Is it always true that the Schur Multiplier $H_2(H,\mathbb{Z})$ of a finitely generated normal subgroup $H$ of a finitely presented group $G$ is a direct product of finitely generated abelian groups?

Answer: No


Question: Solve the congruence equation: \(2^{53x} \equiv 3 \pmod{23}\). Given that \(2^{11} \equiv 1 \pmod{23}\), find the value of \(53x \equiv 8 \pmod{11}\).

Answer: 7


Question: Find the joint probability density function of $X_1$ and $X_2$, where $X_1$ and $X_2$ are independent and uniformly distributed on the interval $[0, 1]$, conditioned on the event $X_1 \leq X_2$. Express your answer in terms of $x_1$ and $x_2$.

Answer: 2


Question: Determine the order of the group \( G = \langle a, b : a^3 = b^2 = (ab)^3 = e \rangle \).

Answer: 12


Question: Let $K$ be a field and $x$ be transcendental over $K$. Compute the degree of the field extension $[K(x):K(\frac{x^5}{1+x})]$. Show that this degree is 5.

Answer: 5


Question: Solve the convolution equation \((y * \sin)(t) = t^2\) using Laplace transforms and verify that the solution is \(y(t) = t^2 + 1\).

Answer: y(t) = t^2 + 2


Question: Evaluate the limit: $$\lim_{h\to0}\frac {2y}{h^3}e^{-\frac{y^2}{h^2}}$$

Answer: 0


Question:
Let $R = \mathbb{C}[x_1, \ldots, x_n]$ and $S = \mathbb{C}[y_1, \ldots, y_m]$ be polynomial rings, and consider a ring homomorphism $f: R \to S$. Suppose that:

1. $S/(f(x_1), \ldots, f(x_n))$ is a finite-dimensional $\mathbb{C}$-vector space.
2. $S/(f(x_1), \ldots, f(x_n))$ is a complete intersection ring, meaning $(f(x_1), \ldots, f(x_n))$ is a regular ideal, or equivalently, $(f(x_1), \ldots, f(x_n)) = (u_1, \ldots, u_m)$ for some regular sequence $u_1, \ldots, u_m$.

Is it true that the ideal $\text{ker}(f)$ of $R$ can be generated by $n-m$ elements?

Answer: Yes


Question: Compute the integral \( \int_{D} (1-x^{2})\:dx\wedge dy \), where \( D = \{(x,y) \in \mathbb{R}^{2} \mid x^{2}+y^{2} \leq 1\} \).

Answer: \dfrac{3\pi}{4}


Question:
Calculate the determinant of the following matrix for \( 2 \leq n \) and \( 0 \leq k \leq n-2 \):
\[
\begin{vmatrix}
1^k & 2^k & 3^k & \cdots & n^k \\
2^k & 3^k & 4^k & \cdots & (n+1)^k\\
3^k & 4^k & 5^k & \cdots & (n+2)^k\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
n^k & (n+1)^k & (n+2)^k & \cdots & (2n-1)^k
\end{vmatrix}
\]

Answer: 0


Question: Determine the poles and calculate the residue of the function \( f(z) = \frac{\sin(z)}{z^4} \) at \( z = 0 \).

Answer: -\dfrac{1}{6}


Question: Determine the radius of convergence of the series \( f(z) = 10 + \sum_{n=1}^\infty \frac{p(n)}{n!}z^{n!} \), where \( p \) is a polynomial with positive integer coefficients.

Answer: 1


Question: Determine all values of \(x\) for which the power series \(\sum_{n=0}^\infty \frac{(x-2)^n}{4^n\sqrt{n}}\) is convergent.

Answer: [-2, 6)


Question: Let $T > 0$ and consider a sequence of continuous functions $f_n : [0,T] \rightarrow \mathbb{R}$. Suppose that \( \sum_{k = 0}^{\infty} \sup_{t \in [0,T]} \lVert f_{k+1}(t) - f_{k}(t) \rVert < \infty. \) Does this imply that there exists a continuous function $f : [0,T] \rightarrow \mathbb{R}$ such that $(f_n)$ converges uniformly to $f$ on $[0,T]$?

Answer: Yes


Question: Calculate the integral \( \int_C P\,dx + Q\,dy \) where \( C = (x-1)^2 + (y-1)^2 = 1 \), \( Q(x,y) = x \cdot y^2 + y \ln(x + \sqrt{x^2 + y^2}) \), and \( P(x,y) = \sqrt{x^2 + y^2} \). Use Green's theorem to evaluate the integral.

Answer: \dfrac{5\pi}{4}


Question:
Determine the behavior of the error function \( \text{erf}(z) \) as \( z \to -\infty \). The error function is defined as:
\[ \text{erf}(z) = \frac{2}{\sqrt{\pi}} \int_0^{z} e^{-s^2} \, \mathrm{d}s \]

Answer: -1


Question: Is it possible to construct a sequence \(\{M_k\}\) of commuting factors of type \(\mathrm{I}_2\) within a uniformly hyperfinite factor \(M\) of type \(\mathrm{II}_1\) such that their infinite tensor product \(\bigotimes_{k=1}^{\infty}M_k\) coincides with \(M\)? In other words, can \(M\) be regarded as the infinite tensor power of \(\mathrm{Mat}(2,\mathbb{C})\)?

Answer: Yes


Question: Evaluate the complex line integral \( \oint_C \frac{dz}{z-2} \) where \( C \) is the circle \( |z-1|=5 \) and the square with vertices \( 3+3i, 3-3i, -3+3i, -3-3i \). Determine the value of the integral for both paths.

Answer: 2\pi i


Question: Find a formula involving the integers \(n\) and \(k\) for the number of connected components in the graph \(G\), where \(1 \le k < n\). The graph \(G\) has vertices labeled \(0, 1, 2, \ldots, n-1\), and an edge exists between vertices \(a\) and \(b\) if \(a - b \equiv \pm k \pmod{n}\). For example, if \(n = 20\) and \(k = 6\), vertex \(2\) is adjacent to vertices \(8\) and \(16\). Is the formula \(n \bmod k\) correct for determining the number of connected components of \(G\)?

Answer: \gcd(n, k)


Question: Calculate the integral \( I = \int_C \frac{z^5}{1-z^3} \, dz \), where \( C \) is the circle \( |z| = 2 \).

Answer: -2\pi i


Question: Determine the fundamental group of the space \(X = \left\{\left(\sqrt{x^2+y^2}-2\right)^2 + z^2 = 1\right\}\cup \left\{(x,y,0)\; :\; x^2 + y^2 \leq 9\right\}\subset\mathbb R^3\).

Answer: \mathbb{Z}


Question: Find the value of \( \lfloor{100N}\rfloor \) where \( N = \lim\limits_{a\,\rightarrow\,\infty}\sum\limits_{x=-a}^{a}\frac{\sin x}{x} \).

Answer: 314