Question: Determine the number of edges in a $k$-dimensional cube $Q_k$ using the recurrence relation: $e(Q_1)=1$; $e(Q_n)=2e(Q_{n-1})+2^{n-1}$ for $n \ge 2$. Use the generating function technique to solve this recurrence relation and find the number of edges in $Q_k$.

Answer: k \cdot 2^{k-1}


Question: Can the diagonal sequence fail to converge for pointwise limits? Specifically, is it possible that \( f_n(x_n) \nrightarrow f(x) \) as \( x_n \to x \)?

Answer: Yes


Question: Let $B(t)$, $t \ge 0$ be the standard Brownian motion. Define the Brownian bridge as $X(t) = B(t) - tB(1)$ for $0 \le t \le 1$. Given $0 < t_1 < t_2 < \dots < t_n < 1$, is the vector $(X(t_1), X(t_2), \dots, X(t_n))$ independent of $B(1)$?

Answer: Yes


Question: The first $2n$ natural numbers are divided into two groups of $n$ numbers each. The first group is arranged in increasing order as $a_1 < a_2 < \ldots < a_n$, and the second group is arranged in decreasing order as $b_1 > b_2 > \ldots > b_n$. Find the sum \( \sum_{i=1}^n |a_i - b_i| \).

Answer: n^2


Question: Find the maximum error bound for the Taylor polynomial approximation of the function \( f(x) = x\sin(3x) \) at \( a = 0 \) with \( n = 4 \) over the interval \(-0.7 \le x \le 0.7\). The fourth-degree Taylor polynomial is given by \( T_4(x) = 3x^2 - \frac{9x^4}{2} \). Use the fifth derivative of \( f(x) \), which is \( 81(3x\cos(3x) + 5\sin(3x)) \), to find the error bound \( |R_4(x)| \le \) some error value.

Answer: 0.587


Question: Let $A \subseteq \mathbb{R}$ and $a \in A$. Define $A^+ := \{x \in A : x > a\}$ and let $f: A \to \mathbb{R}$. Assume that the right-hand derivative \( f'(a^+) = \lim_{x \to a^+} \frac{f(x) - f(a)}{x - a} \neq \pm \infty \) exists. Can we conclude that \( f'(a^+) = c \in \mathbb{R} \) always holds?

Answer: Yes


Question: Evaluate the integral \( \int \limits_{\Gamma} \frac{\sin(\pi z^2)+\cos(\pi z^2)}{(z-1)(z-2)}dz \) using Cauchy's integral formula, where the contour \( \Gamma \) is parameterized by \( \gamma(\theta) = 3e^{i\theta} + 1 \) for \( \theta \in [-\pi, \pi] \). Note that the function has a singularity at \( z=2 \), which lies inside the contour.

Answer: 4\pi i


Question: Let \( F = \langle a, b \rangle \) be a free group. If \( N \) is the normal subgroup of \( F \) generated by the set \( \{ ab, ab^{-1} \} \), find the index of \( N \) in \( F \).

Answer: 2


Question: Let $f$ be a holomorphic function in the disk $D(0, p)$, and consider the contour $\gamma(t) = re^{it}$ for $0 \leq t \leq 2\pi$, where $0 < r < p$. If $b$ is not in $D(0,p)$, show that $\int_{\gamma}\dfrac{f(z)}{z-b} \, dz = 0$. Justify your answer.

Answer: 0


Question: Express the integral \(\int_{0}^{1/2} \frac{1}{1+x^6} \, dx\) as a series. Determine the minimum number of terms needed in the series to ensure that the error is less than \(10^{-5}\).

Answer: 2


Question: Let \(f(x):[0,2] \to \mathbb{R}\) be a continuous function satisfying the equation \[ \int_{0}^{2} f(x)(x-f(x)) \,dx = \frac{2}{3}. \] Find the value of \(2f(1)\).

Answer: 1


Question: Let \( \{a_n\}_{n=1}^{\infty} \) be a sequence defined by \( a_1 = 1, a_2 = 1 \), and \( a_{n+2} = 2a_{n+1} + a_n \) for all \( n \ge 1 \). Calculate the value of \( 47 \sum_{n=1}^{\infty} \frac{a_n}{2^{3n}} \).

Answer: 7


Question: Consider a measurable space $(\Omega,\Sigma)$ with two probability measures $\nu_1$ and $\nu_2$. For $i=1,2$, define the linear map $L_{\nu_i}:L^{1}(\nu_i)\rightarrow\mathbb{R}$ by $L_{\nu_i}(g)=\int_\Omega g\,d\nu_i$. For the equivalence class $[1_E]_{\nu_i}$ in $L^1(\nu_i)$ of the characteristic function $1_E$, does there always exist a linear map $M:L^1(\nu_1)\rightarrow L^1(\nu_2)$ such that $$L_{\nu_2}\circ M([1_E]_{\nu_1})=L_{\nu_1}([1_E]_{\nu_1})$$ for all $E\in\Sigma$? Assume $\nu_1$ and $\nu_2$ are singular measures.

Answer: Yes


Question:
Find the asymptotic expansion of the expression involving modified Bessel functions:
\[
\frac{I_1\left(\frac{2\sqrt{MN}}{N-1}\right)K_{0}\left(\frac{2\sqrt{M}}{N-1}\right)+I_0\left(\frac{2\sqrt{M}}{N-1}\right)K_{1}\left(\frac{2\sqrt{MN}}{N-1}\right)}{I_1\left(\frac{2\sqrt{MN}}{N-1}\right)K_{1}\left(\frac{2\sqrt{M}}{N-1}\right)-I_1\left(\frac{2\sqrt{M}}{N-1}\right)K_{1}\left(\frac{2\sqrt{MN}}{N-1}\right)}
\]
as \( N \) approaches 1, and show that it becomes \( \coth{\sqrt{M}} \).

Answer: \coth{\sqrt{M}}


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function satisfying the inequality \(|f(x) - f(y)| \ge \frac{1}{2} |x - y|\) for all real numbers $x$ and $y$. Is the function $f$ onto?

Answer: Yes


Question: Calculate \( \mathbb{E}(N(1,4] \mid N(3,10] = 7) \) where \((N_t, t \geq 0)\) is a Poisson counting process with rate \(\lambda = 1\).

Answer: 3


Question: Let $G$ be a simple, undirected graph such that for any two distinct vertices, there is a path of length at most 2 connecting them. Determine if there exists a set $X$ and a full collection of subsets ${\cal C}$ on $X$ such that the intersection graph $G_{\cal C}$ is isomorphic to $G$. Provide a justification for your answer.

Answer: Yes


Question:
Given that \( \int_{1}^{2} e^{x^2} \mathrm{d}x = a \), find the value of \( \int_{e}^{e^4} \sqrt{\ln x} \mathrm{d}x \). Choose the correct option:

(A) \( e^4-a \)
(B) \( 2e^4 - a \)
(C) \( e^4 - e - 4a \)
(D) \( 2e^4-e-a \)

Answer: D


Question: Evaluate the double integral: $$\int^{\infty}_{-\infty}{\int^{\infty}_{0}{xe^{-x^2e^y}}}dydx$$

Answer: 0


Question: Does there exist a continuous function \( F: [a, b] \rightarrow \mathbb{R} \) on a set \( [a, b] \subseteq \mathbb{R} \cup \{-\infty, \infty\} \) such that the derivative \( F'(x) := \lim_{h\rightarrow 0}\frac{F(x+h)-F(x)}{h} \) exists in \( \mathbb{R} \cup \{-\infty, \infty\} \) for every \( x \in (a, b) \) and the set \( \{F(x): x \in (a, b) \land |F'(x)|= \infty\} \) has positive Lebesgue outer measure?

Answer: No


Question: Determine if the sum \( \sum_{i=1}^{\lg\lg n} n^{1-\frac{1}{2^i}} \) can be bounded by \( O(n) \).

Answer: O(n)


Question: Determine the number of distinct terms in the expansion of the expression \( (1+x)^{2012} + (1+x^2)^{2011} + (1+x^3)^{2010} \).

Answer: 4023


Question:
Given a deterministic, increasing function \( f: \mathbb{R} \to \mathbb{R} \) and independent, identically distributed noise variables \( \epsilon_1 \sim g \) and \( \epsilon_2 \sim g \), with known constants \( a, b \in \mathbb{R} \) such that \( a > b \), it is known that:
\[ P_{g}(f(a+\epsilon_1)>f(b+\epsilon_2)) \geq P_{g}(f(a+\epsilon_1)<f(b+\epsilon_2)). \]
If it is also known that:
\[ E_{g}[f(a+\epsilon_1)] > E_{g}[f(b+\epsilon_2)], \]
can you conclude that:
\[ P_{g}(f(a+\epsilon_1)>f(b+\epsilon_2)) > P_{g}(f(a+\epsilon_1)<f(b+\epsilon_2))? \]

Answer: Yes


Question: Assess the probability that there exists a number $n_0$ such that for every $n > n_0$, the event $A_n = \{|\frac{S_n}{n} - p| \le \epsilon\}$ occurs, where $S_n$ is the number of successes in a Bernoulli scheme with success probability $p$. Use the result $P(|\frac{S_n}{n} - p| \ge \epsilon) = e^{-\frac{n\epsilon^2}{4}}$ to aid in your calculation.

Answer: 1


Question: Determine the number of field homomorphisms from a finite field with 9 elements to a finite field with 4 elements.

Answer: 0


Question: Determine the values of the parameter \( p \) for which the series \( \sum_{n=1}^{\infty} \sin \frac{1}{n^p} \) is convergent.

Answer: p > 1


Question: A Martian writes down the equation \(x^2 - 19x + 76 = 0\). When asked to write down the difference between the larger and smaller root, the Martian writes 9. How many fingers do Martians have?

Answer: 12


Question: If \( L = \lim_{m \to \infty} \sum_{p=1}^m \frac{p}{2p+m+m^2} \), find \( \lfloor L \rfloor \).

Answer: 0


Question: Calculate the covariance \( \text{Cov}(U,V) \) where \( U = \frac{1}{2}(X+Y) \) and \( V = \frac{1}{2}(X-Y) \), given that \( X \) and \( Y \) are independent random variables with moment generating functions \( M_{X}(t) = (1-t)^{-1}, \ t<1 \) and \( M_{Y}(t) = (1-2t)^{-1}, \ t< 0.5 \).

Answer: -\dfrac{3}{4}


Question: Does there exist a deterministic push-down automaton (DPDA) for the language \( L = ba^nb^n \cup bba^nb^{2n} \)?

Answer: Yes


Question: Let X be a Banach space and let its dual space be denoted by $X^* = L(X, \mathbb{R})$. Suppose $x, y \in X$ such that $T(x) = T(y)$ for all $T \in X^*$. Determine whether $x = y$ or $x \neq y$.

Answer: x = y


Question: Find all pairs of prime numbers \(p, q\) such that \(p^3 - q^5 = (p+q)^2\).

Answer: (7, 3)


Question: Reverse the order of integration for the integral \( \int_{-1}^4\int_{y-4}^{4y-y^2} (y+1)\,dx\,dy \) and evaluate the resulting integral.

Answer: \dfrac{625}{12}


Question: Let $X$ be a Lusin space. If we adjoin an isolated point $\Delta$ to $X$, is $X_{\Delta}:=X\cup\{\Delta\}$ a Lusin space?

Answer: Yes


Question: Evaluate the limit: $$\lim_{(x,y,z)\to(0,0,0)}\frac{x^3yz+xy^3z+xyz^3}{x^4+y^4+z^4}.$$

Answer: 0


Question: Evaluate the contour integral \( \oint_{|z|=4} \frac{\sin^2(z)}{(z-\frac{\pi}{6})^2(z+\frac{\pi}{6})} \, dz \).

Answer: 3\sqrt{3}i


Question: Let $F = \langle a, b \rangle$ be a free group. If $N$ is the normal subgroup of $F$ generated by the set $\{a^3, b^2, aba^{-1}b^{-1}\}$, find the index of $N$ in $F$. Express your answer as an integer.

Answer: 6


Question: Find all positive integer solutions $(x, y)$ to the Diophantine equation $x^2 + 2 = y^3$.

Answer: (5, 3)


Question: Let \( f \colon \mathbb{R} \to \mathbb{R} \) be a real analytic function, and let \( A \subseteq \mathbb{R} \) be such that the set \( A' \) of all accumulation points of \( A \) is not empty. If \( f(a) = 0 \) for all \( a \in A \), is it necessary that \( f(t) = 0 \) for all \( t \in \mathbb{R} \)?

Answer: Yes


Question: Suppose \( f \in \mathbb{C}[X_1, \ldots, X_n] \) is a complex polynomial in \( n \) variables such that \( f(P) = 0 \) for all \( P \in \mathbb{R}^n \). Must \( f \) necessarily be the zero polynomial?

Answer: Yes


Question: Let $V$ be a vector space over $\mathbb{R}$ with basis $B = \{v_1, v_2, \ldots, v_n\}$. Prove that the intersection of all subspaces of $V$ with dimension $n-1$ is $\{0\}$. Use the fact that if you exchange a non-zero vector $\overline{v} \in V$ with a $v_i \in B$, you obtain another basis of $V$. Show that if there exists a non-zero vector common to all $(n-1)$-dimensional subspaces, it leads to a contradiction.

Answer: \{0\}


Question: Calculate the limit: \[ \lim_{x\to0}\left(\frac{1}{x^5}\int_0^x e^{-t^2}\,dt-\frac{1}{x^4}+\frac{1}{3x^2}\right) \]

Answer: \dfrac{1}{10}


Question: Find the coefficient of \(x^2\) in the polynomial \((1-x)(1+2x)(1-3x)\cdots(1+14x)(1-15x)\).

Answer: -588


Question:
Evaluate the limit using Taylor series:
\[ \lim_{x\to 0 }\frac{2\exp(\sin(x))-2-x-x^2-\arctan (x) }{x^3} \]

Answer: \dfrac{1}{3}


Question:
Determine whether the following limit is equal to zero:
\[ \lim_{R \to \infty} \int_S \frac{jRe^{j\theta}}{1+R^4e^{j4\theta}}d\theta \]
where \( S \) is a semicircle. Does the presence of \( j \) in the differential make any difference to the limit?

Answer: 0


Question: Let $x$ and $y$ be positive integers such that $21x^2 + 16xy + 3y^2 = 324,000$. What is the probability that both $x$ and $y$ are positive even integers?

Answer: 1


Question: Evaluate the double integral: \[ \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \exp\left[{-\frac{2}{3}}(y^2-yz+z^2)\right] \,dy\,dz \]

Answer: \pi \sqrt{3}


Question: Consider a complete graph $G$ with $2n \ge 6$ vertices. All edges in the graph have weights of $2$, except for the edges in the cycles $v_1, v_2, \ldots, v_n, v_1$ and $v_{n+1}, v_{n+2}, \ldots, v_{2n}, v_{n+1}$, which have weights of $1$. Determine the total number of minimum spanning trees in this graph.

Answer: n^4


Question: Solve the equation \( \arcsin{(2x^2-1)} + 2\arcsin{x} = -\frac{\pi}{2} \) for \( x \) in the domain \([0, 1]\).

Answer: 0


Question: Determine the range of values for \( p \geq 1 \) such that the function \( f(x) = \frac{1}{x^{1/3} (1 + x^{5/3})} \) is an element of the \( L^p \) space on the interval \([0, \infty)\).

Answer: [1, 3)


Question: Let $X$ be a Riemann Surface with genus $g$, and let $S^g(X)$ be the symmetric power of $X$, identified with the set of effective divisors of degree $g$. Consider the Abel-Jacobi map $A: S^g(X) \to Jac(X)$. For any $D \in S^g(X)$, $A^{-1}A(D)$ is given by the linear system $|D|$. Given that $A$ has constant rank along $A^{-1}A(D)$, does this imply that $A^{-1}A(D)$ is a submanifold of $S^g(X)$?

Answer: Yes


Question: Find \( \frac{m}{n} \) given that \( \lim_{x\to 0}\frac{e^{\cos(x^n)}-e}{x^m}=-\frac{e}{2} \).

Answer: 2


Question: Show that \(\lim_{x\to 0^+}\frac{x}{a}\cdot\left\lfloor\frac{b}{x}\right\rfloor=\frac{b}{a}\).

Answer: \dfrac{b}{a}


Question: Given two sheaves of modules $\mathcal{F}$ and $\mathcal{G}$ over a scheme $X$, consider their injective resolutions $I^*_\mathcal{F}$ and $I^*_\mathcal{G}$. Is the direct sum $I^*_\mathcal{F} \oplus I^*_\mathcal{G}$ an injective resolution of the sheaf $\mathcal{F} \oplus \mathcal{G}$?

Answer: Yes


Question: Find the sum of the series \( \sum_{n=1}^{\infty}(-1)^{n+1}\left(\frac{1}{n}\right) = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots \) using Abel's Theorem.

Answer: \ln 2


Question: Find the sum of the last two digits of the number $3^{1234}$. What is the value of $q + r$ if $3^{1234}$ can be expressed as $abcdef...qr$?

Answer: 15


Question: Let $\Omega$ be a convex subset of $\mathbb{R}^N$ with a smooth boundary. Consider the space $M = \{f : \Omega \to \mathbb{R} \mid \lvert|f\rvert|_{W^{1,N+1}} \in \mathbb{R}^+\} \setminus I$, where $I$ is the set of all functions that have at least one isolated discontinuity. Consider any sequence $\{f_n\}$ where $f_n \in M$. Is it true that if $f_n \to f$ in the norm $\lvert|\cdot\rvert|_{W^{1,N+1}}$, then $f \in M$?

Answer: Yes


Question: Find the volume of the solid in 3-space bounded by the surfaces \( z = x^2 - 1 \), \( z = 1 - x^2 \), \( x + y + z = 2 \), and \( x - 2y - 2z = 10 \).

Answer: \dfrac{56}{3}


Question:
If the roots of the equation \(z^2 + \alpha z + \beta = 0\) lie on the unit circle \(|z| = 1\), which of the following is true?\
\(\text{(A) } 2|\text{Im}(\alpha)| = 1 - |\beta|^2\)\
\(\text{(B) } 2|\text{Im}(\alpha)| = |\beta|^2 - 1\)\
\(\text{(C) } \text{Im}(\alpha) = 0\)\
\(\text{(D) None of these}\)

Answer: D


Question: Let \((a_k)_{k\geq 1}\) be a positive sequence such that the series \(\sum_{k\geq 1} a_k\) converges to \(L\). Define \(f(n):=\sum_{k=1}^n \frac{1}{a_k}\). Prove or disprove that there exists an \(n_0\) such that for all \(n \geq n_0\), \(f(n) > n^2\ln(n)\).

Answer: True


Question: Find the orthogonal projection of the function $g(x) = x^2$ onto the subspace $V = \text{span}(f)$, where $f(x) = x$, in the space $L^2[0,1]$. Verify that the projection $Pg$ is correct if $(g - Pg)(x) = \frac{3x}{4}$ for $x \in [0,1]$. Provide the expression for $Pg$. \(V\) is a closed subspace of \(L^2[0,1]\).

Answer: \dfrac{3}{4}x


Question: Let $\mathcal{A}$ be an abelian category and $\mathbb{K}(\mathcal{A})$ the homotopy category of complexes of $\mathcal{A}$. Consider a functor $F: \mathbb{K}(\mathcal{A}) \to \mathbb{E}$, where $\mathbb{E}$ is a triangulated category, such that $F$ takes quasi-isomorphisms to isomorphisms. By the universal property of the derived category $\mathbb{D}(\mathcal{A})$, there exists a functor $F': \mathbb{D}(\mathcal{A}) \to \mathbb{E}$ induced by $F$. Is $F'$ a triangulated functor, meaning it commutes with the translation functor and takes distinguished triangles to distinguished triangles?

Answer: No


Question: Consider the process defined by \( Y_{t} = X_{t} + w_{t} \) where \( X_{t} = X_{t-1} + e_{t} \), and both \( e_t \) and \( w_t \) are independent white noise processes. Is the process \( \Delta Y = Y_t - Y_{t-1} \) an MA(1) process? Justify your answer.

Answer: Yes


Question:
Given the Gaussian measure \( \mu: \mathbb{R} \rightarrow \mathbb{R} \) defined by
\[ \mu(B) = \int_{B} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^{2}}{2}} \, dx \]
does there exist a measure-preserving, non-trivial map \( T: \mathbb{R} \rightarrow \mathbb{R} \) such that \( T \) is ergodic? Specifically, for every Borel set \( B \) such that \( T^{-1}(B) = B \), is it true that \( \mu(B) \in \{0,1\} \)?

Answer: Yes


Question: Let \( f(x) = (x^2 + 6x + 9)^{50} - 4x + 3 \), and let \( r_1, r_2, \ldots, r_{100} \) be the roots of \( f(x) \). Compute the value of \( (r_1 + 3)^{100} + (r_2 + 3)^{100} + \cdots + (r_{100} + 3)^{100} \).

Answer: -1500


Question: Given that \( f(x) \) is a twice differentiable function on \([0, \pi]\), with \( f(\pi) = 2 \) and \( \int_0^\pi \left[f(x) + f''(x)\right]\sin(x)\,dx = 5 \), find the value of \( f(0) \).

Answer: 3


Question: Let $S = \sum_{n \in \mathbb{Z}} S_n$ be a commutative graded ring, and let $I$ be a homogeneous ideal of $S$. If $J$ is the radical of $I$, defined as $J = \{x \in S \mid x^n \in I \text{ for some } n > 0\}$, is $J$ a homogeneous ideal?

Answer: Yes


Question: Find the smallest positive integer \(x\) such that \(x^x \equiv 0 \pmod{y}\), where \(y\) is a given constant. For example, if \(y = 420\), determine the smallest \(x\) that satisfies this condition.

Answer: 210


Question: Determine the eigenvalues of the operator $A: L^2(0,2\pi) \rightarrow L^2(0,2\pi)$ defined by $$(Au)(x)=\sin x \int_{0}^{2\pi} u(y)\cos y \, dy$$ for $u\in L^2(0,2\pi)$ and $x\in [0,2\pi]$. What are the eigenvalues of $A$?

Answer: 0


Question: Let \( R = \{a_1 + a_2x^2 + a_3x^3 + \ldots + a_nx^n; a_i \in \mathbb{R}, n > 2\} \) be a polynomial ring. Consider the ideal \( I \) generated by \( \langle x^2 + 1, x^3 + 1 \rangle \). Determine whether \( I = R \).

Answer: I = R


Question: Given a strictly decreasing sequence \( \{q_{k}>0;k\geq 1\} \) converging to zero, and a set of \( M \) fixed distinct real values \( \{t_{1}<t_{2}<\cdots<t_{M}\} \) greater than 1, determine if it is always possible to select \( M \) indices \( k_{1},\cdots,k_{M} \) such that the matrix \( \begin{bmatrix} q_{k_{1}}^{t_{1}} & q_{k_{2}}^{t_{1}} & \cdots & q_{k_{M}}^{t_{1}} \\ q_{k_{1}}^{t_{2}} & q_{k_{2}}^{t_{2}} & \cdots & q_{k_{M}}^{t_{2}} \\ \vdots & \vdots &\cdots & \vdots\\ q_{k_{1}}^{t_{M}} & q_{k_{2}}^{t_{M}} & \cdots & q_{k_{M}}^{t_{M}} \end{bmatrix} \) is full-ranked.

Answer: Yes


Question: Determine whether there exists a non-zero function \( f(x) \in L^2(0,1) \) such that \[ \int_{1}^{\infty}\frac{f(\frac{1}{t})}{t}dt - \sum_{k=1}^{\infty}k\int_{k}^{k+1}\frac{f(\frac{1}{t})}{t^2}dt = 0. \]

Answer: Yes


Question: Does there exist an \(\mathcal{O}(n^3)\) algorithm for deciding whether there exists a permutation matrix \(P\), a lower unit triangular matrix \(L\), an upper unit triangular matrix \(U\), and a block-diagonal matrix \(D\) with \(1\times1\) and \(2\times2\) blocks such that for a given real square matrix \(A\), the equation \(P A P^T = LDU\) holds?

Answer: Yes


Question: What is the result of the sum of three Cantor sets, denoted as \( C + C + C \), where \( C \) is the usual Cantor set obtained from the interval [0,1]? Express your answer as a set.

Answer: [0, 3]


Question: Evaluate the limit: \[ \lim_{n\to \infty} \left(\frac{\sin \frac{e}{n}}{\sin \frac{1}{n}} + \frac{\ln{\pi n}}{\ln{n}}\right) \] without using l'Hospital's rule.

Answer: e + 1


Question: Let $f \in C^0(\mathbb{R}^2, \mathbb{R})$ be a continuous function such that there exists an $M > 0$ with $f(x,y) \geq M$ for all $(x,y) \in \mathbb{R}^2$. For any $g \in C^0(\mathbb{R}, \mathbb{R})$, does there exist a unique $h \in C^0(\mathbb{R}, \mathbb{R})$ such that \( \int_0^{h(a)} f(a,x)\,\text{d}x = g(a) \) for all $a \in \mathbb{R}$?

Answer: Yes


Question: Find the value of \( g^{-1}(1) \) if \( g(x) = e^{-x} \sin x - \cos x \).

Answer: \pi


Question: Consider a function $L:(0,\infty)\rightarrow (0,\infty)$ that is slowly varying at infinity. It is known that for any $\delta>0$, if $x$ is large enough, then $L(x)\leq x^\delta$. Does this imply that there exists a positive constant $C$ (which may depend on $\delta$) such that $L(x)\leq C\, x^\delta$ for all $x\geq 1$?

Answer: Yes


Question: Determine the number of roots of the polynomial \( f(z) = z^{10} + 10z + 9 \) inside the unit disk \( D(0,1) \) using Rouche's theorem.

Answer: 0


Question:
Suppose $A, B \in \mathcal{M}_{n \times n}(\mathbb{R})$ satisfy the following conditions:

1. $ABA = A$
2. $BAB = B$
3. $A^2 = A$

Is it necessarily true that $B^2 = B$? Justify your answer.

Answer: No


Question:
Let \( f(x) = [n + p \sin x] \), where \( x \in (0, \pi) \), \( n \in \mathbb{Z} \), \( p \) is a prime number, and \([x]\) represents the greatest integer less than or equal to \(x\). Find the number of points at which \( f(x) \) is not differentiable. Choose the correct answer from the following options:

(A) \( p \); (B) \( p-1 \); (C) \( 2p+1 \); (D) \( 2p-1 \).

Answer: D


Question: Is a maximal square covering of a shape in 2-dimensional space unique? A square covering of a shape X is defined as a set of axis-aligned squares whose union exactly equals X. A maximal square covering is one where each square is necessary, meaning for every square, there is a point in X covered only by that square, and no two squares can be replaced by a larger square contained in X. Can there be two different maximal square coverings of the same shape?

Answer: Yes


Question:
Calculate the following limit:
\[ \lim_{||x||\to\infty} \sup_{0 \leq t \leq 2\pi} \frac{||f(t,x)||}{||x||^7} \]
where \(x = (x_1,x_2,x_3)\) and \(f(t,x)= ((x_2^2+x_3^2+1)\cos(t),(x_1^2+x_3^2)\sin(t),(x_1^2+x_2^2)\cos(t))\).

Answer: 0


Question: Consider a complex connected reductive Lie group $G$ with an involution $\theta$, and let $K = G^\theta$ be the fixed-point subgroup. Given a $\theta$-stable Borel subgroup $B$ and maximal torus $T$, the map $\varphi: K\backslash G/B \to B\backslash G/B \cong W := N(T)/T$ is defined. If $v$ is a closed orbit in $K \backslash G/B$, does the restriction of $\varphi$ to the order ideal $\{u \in K \backslash G/B : u \geq v \}$ become injective?

Answer: Yes


Question: Let $f : (0,1) \to \mathbb{R}$ be the derivative of some function $F$. Suppose $f$ is square integrable on $(0,1)$ and \( \int_0^1 f(x)^2 \, \mathrm{d}x = 0 \). Can we conclude that $f(x) = 0$ for all $x \in (0,1)$?

Answer: Yes


Question:
Let $F$ be a vector bundle over a smooth curve $X$, and consider the extension:
\[0\rightarrow F\rightarrow V \rightarrow \mathcal{O}_X\rightarrow 0\]
associated with an element $e \in H^1(F)$. Consider a quotient $Q$ of $F$, i.e., $p: F \rightarrow Q \rightarrow 0$. The map $p$ induces a map $\tilde{p}: H^1(F) \rightarrow H^1(Q)$. Let $f = \tilde{p}(e)$ and let $W$ be the extension associated with $f$, i.e.,
\[0\rightarrow Q\rightarrow W\rightarrow \mathcal{O}_X\rightarrow 0\]
Is $W$ a quotient of $V$?

Answer: Yes


Question: Determine the radius of convergence for the power series \( \sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n. \)

Answer: 2


Question: Find the limit \( \lim _{n \to \infty}(n!)^{{\frac{1}{n^2}}} \).

Answer: 1


Question: Determine the points on the helix parameterized by $x=\cos(t)$, $y=\sin(t)$, $z=t$ where the function $f(x,y,z)$, with partial derivatives $f_x=\cos(t)$, $f_y=\sin(t)$, and $f_z=t^2$, can take an extreme value.

Answer: (1, 0, 0)


Question: Determine the minimum length of a sequence of numbers from the set \(\{0, 1, 2\}\) that is guaranteed to open an electronic lock. The lock opens when any three consecutive numbers in the sequence match the correct password, which is a sequence of three numbers from \(\{0, 1, 2\}\).

Answer: 29


Question: A large cube has an edge length of \(2x^2 + 1\). A smaller cube has a volume of \(x^2 - 1\). When the large cube is filled with smaller cubes of the same size, the remaining space has a volume that is 4 cubic units less than the volume of the smaller cube. What is the edge length of the larger cube?

Answer: 65


Question: Determine whether a simply connected complete Riemannian manifold with a positive upper curvature bound has a positive injectivity radius.

Answer: Yes


Question: Find the smallest ordinal \( \beta \) such that \( \omega + \beta = \beta \).

Answer: \omega^2


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

Answer: e^{-\gamma}


Question: Evaluate the double integral \( \iint |x+y| \, dx \, dy \) over the region where \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\).

Answer: \dfrac{8}{3}


Question: Compute the norm of the operator \(T: \ell_2 \to \ell_2\) defined by \(T(x_1, x_2, x_3, \dots) = \left(x_1, \frac{x_2}{2}, \frac{x_3}{3}, \dots\right)\).

Answer: 1


Question: Let $R$ be a commutative ring with a maximal ideal $m$. Consider the quotient ring $R/m^k$ for some integer $k$. Is $R/m^k$ a local ring, meaning that $m/m^k$ is the only maximal ideal in $R/m^k$?

Answer: Yes


Question: Compute the sum of all 4-digit numbers that can be formed using the digits from the multiset \(\{1^2, 2^2, 3^2, 4^2\}\), where each digit can appear at most twice.

Answer: 566610


Question: Is the orthogonal group $O(n)$ normal in the general linear group $GL(n)$? Provide a justification for your answer.

Answer: No


Question: Let \(x, y, z \in \mathbb{R^+}\) such that \(x \geq y \geq z\) and \(x^2+y^2+z^2 \geq 2xy+2xz+2yz\). Find the minimum value of \(\frac{x}{z}\).

Answer: 4


Question: Find the remainder when \( 13^{2019} \) is divided by 95.

Answer: 12


Question: Calculate the volume of the solid that is above the $xy$-plane and bounded by the paraboloid $z = x^2 + y^2$ and the cylinder $x^2 + y^2 = 2y$. Express your answer in terms of $oxed{}$.

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


Question: Find the limit: $$ \lim_{n \to \infty} \max \left \{ \cos \left( \alpha - 2\pi \frac{i-1}{n} \right) \right \}_{i=1}^{n} $$

Answer: 1


Question: Evaluate the surface integral \( \iint_{S}\left(\left(\frac{2x}{\pi}+\sin(y^2)\right)x + \left(e^z-\frac{y}{\pi}\right)y + \left(\frac{2z}{\pi}+\sin^2y\right)z\right)\,d\sigma \) over the unit sphere \( S = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\} \).

Answer: 4


Question: Find the smallest integer \( N_0 \) such that for all \( N > N_0 \), the inequality \( |N!/e - D_N| < 2^{-k} \) holds, where \( D_N \) is the number of derangements of \( N \) objects and \( k \) is a given positive integer.

Answer: 2^k - 1


Question: Consider the sequence defined by the recurrence relation: \( b_1 = \sqrt{2}, \) \( b_2 = \sqrt{2+\sqrt{2}}, \) and in general, \( b_{n+1} = \sqrt{2+b_n}. \) Determine the behavior or limit of the sequence \( b_n \) as \( n \to \infty. \)

Answer: 2


Question: Compute \( \lim\limits_{k\to\infty} \int_0^k \left(1 + \frac{x}{k}\right)^k e^{-2x} \, dx \) using the Dominated Convergence Theorem.

Answer: 1


Question: Determine the maximum value of the expectation of the conditional variance \( E[\operatorname{Var}(X+Z_1 \mid X+Z_2)] \), where \( X, Z_1, Z_2 \) are three mutually independent random variables. \( Z_1 \) and \( Z_2 \) are normally distributed with mean 0 and variance 1, i.e., \( Z_1, Z_2 \sim N(0,1) \). The random variable \( X \) has a unit second moment, \( E[X^2] = 1 \), but can have any distribution. What is the maximum value of this expectation?

Answer: \dfrac{3}{2}


Question: Let $R$ be a commutative ring with unity that has property P, meaning there exists a unique field among all proper subrings of $R$ (with the same unity as that of $R$). Assume $R$ is not a field. Does the polynomial ring $R[X]$ also have property P?

Answer: Yes


Question: Determine the values of \(m\) for which the polynomial \(P(x) = x^4 + (m-3)x + m^2\) has four distinct real roots.

Answer: \emptyset


Question: Let $R = \mathbb{C}[t]$ be the ring of polynomials in $t$. Consider the module $M = R[x]/(x-t)$ as an $R$-module. Prove or disprove that $M$ is isomorphic to $R$ as an $R$-module.

Answer: M \cong R


Question: Prove that \( \lim_{n\to\infty}\frac{a_1 b_1 + \ldots + a_n b_n}{b_n}=0 \) given that \( \sum a_n \) converges, \( b_n>0 \) is monotone increasing, and \( \lim b_n=\infty \).

Answer: 0


Question: Evaluate the integral \( \int\limits_6^{16}\left(\frac{1}{\sqrt{x^3+7x^2+8x-16}}\right)\,\mathrm{d}x \) and express it in the form \( \frac{\pi}{k} \). Find the value of the constant \( k \).

Answer: 6\sqrt{5}


Question: Evaluate the definite integral: \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\sin^2 x}{2^x + 1} \, dx. \]

Answer: \dfrac{\pi}{4}


Question: Calculate \( E(B_{1}^2B_{2}B_{3}) \), where \( B_{i} \) represents Brownian motion.

Answer: 4


Question: Evaluate the limit: \( \lim_{x \to 0} x \left[ \frac{1}{x} \right] = 1 \)

Answer: 1


Question: Compute the line integral of the vector field \( F = \langle 6y, -6x \rangle \) over the circle \( x^2 + y^2 = 4 \), oriented clockwise.

Answer: 48\pi


Question:
Verify whether the following property holds for every real polynomial $P(x)$ of degree $n$:
\[ \sum_{i=0}^{n+1}(-1)^i\binom{n+1}{i}P(i)=0 \]

Answer: 0


Question: Let \( f:(0,\infty) \to \mathbb{R} \) be a differentiable function that is increasing and bounded above. Determine whether \( \lim_{x \to \infty} f'(x) = 0 \).

Answer: No


Question: Calculate the measure of the set \(A = ([-1,2] \times [0,3] \times [-2,4]) \cup ([0,2] \times [1,4] \times [-1,4]) \setminus [-1,1]^3\).

Answer: 60


Question: Consider a continuous function $f$ such that $D^+ f(x) \geq a > 0$ for all $x \in \mathbb{R}$, where $a$ is a constant and $D^+ f(x)$ denotes the Dini's derivative defined as $D^+ f(x) = \lim\sup_{h \rightarrow 0^+} \frac{f(x+h) - f(x)}{h}$. Is the function $f$ necessarily monotone?

Answer: Yes


Question:
Let $X_1, X_2, \ldots$ be i.i.d. random variables with $E[X_i] = 0$ and $E[X_i^2] = \sigma^2 > 0$. Define $S_n = X_1 + \cdots + X_n$. Let $N_n$ be a sequence of nonnegative integer-valued random variables and $a_n$ a sequence of integers such that $a_n \to \infty$ and $N_n/a_n \to 1$ in probability. Prove that:

$$ \frac{S_{N_n}}{\sigma\sqrt{a_n}} \xrightarrow{d} \chi $$

where $\chi$ is a standard normal random variable. Use Kolmogorov's inequality to show that:

$$ \frac{S_{N_n}}{\sigma \sqrt{a_n}} - \frac{S_{a_n}}{\sigma\sqrt{a_n}} \to 0 $$

in probability.

Answer: \chi


Question: Evaluate the divergent series $2 + 2 + 2 + 2 + \ldots$ using zeta function regularization. What is the value of this series?

Answer: -1


Question: A monkey starts at position 0 on the real line. In each period $t \geq 0$, it moves 1 unit to the right with probability $p$ and 1 unit to the left with probability $1-p$, where $p \in [1/2, 1]$. What is the probability that the monkey will reach a positive integer $N$ at some period $t > 0$?

Answer: 1


Question: Determine whether there exist disjoint sets $A$ and $B$ in the power set of the real numbers, $\mathcal{P}(\mathbb{R})$, such that the outer measure $\lambda^*$ satisfies the inequality $\lambda^*(A \cup B) < \lambda^*(A) + \lambda^*(B)$. Provide a justification for your answer.

Answer: Yes


Question: Can there be a simple and closed geodesic on the Poincaré disk model of the Hyperbolic plane, given the metric \(ds^{2}=\frac{4dx^{2}+4dy^{2}}{(1-x^{2}-y^{2})^2}\)? Justify your answer using the Gauss-Bonnet Theorem.

Answer: No


Question: Determine the values of $x$ for which the series \( \sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n + 1/n} \) is convergent.

Answer: (-1, 1]


Question: Find the remainder when the number consisting of 105 consecutive ones is divided by 107.

Answer: 32


Question: Evaluate the integral \( \int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2} \, dx \).

Answer: \pi \ln 2


Question: Determine whether it is true that for any constant $0 < c < 1/2$, there exists a constant $d < 2$ such that for all sufficiently large integers $n$, the inequality \( \binom{n}{cn} < d^n \) holds, assuming $cn$ is an integer.

Answer: True


Question: Given a scheme $S$, does there exist an open atlas for $S$ consisting only of morphisms representable by an affine?

Answer: Yes


Question: Evaluate the infinite series: \[ \sum_{r=2}^{\infty} \frac{2-r}{r(r+1)(r+2)} \]

Answer: -\dfrac{1}{6}


Question: Find the smallest positive integer $k$ such that the $k$th derivative of the function $F(x) = x \sin(x^N)$ evaluated at $x=0$ is non-zero, where $N$ is a natural number.

Answer: N + 1


Question: Determine the angle between the left and right tangents to the graph of the function \( g(x) = \frac{1}{\sqrt{3}}\arcsin \frac{2x}{1+x^{2}} \) at the point \( \left( 1, \frac{\sqrt{3}\pi }{6} \right) \).

Answer: \dfrac{\pi}{3}


Question: Calculate the limit: \( \lim_{x \to 0} \frac{1 - \cos(x^6)}{x^{12}} \).

Answer: \dfrac{1}{2}


Question: Calculate the limit of the sequence \((a_n)_{n=1}^\infty\) where \[a_n=\frac{1^2}{n^3}+\frac{3^2}{n^3}+\frac{5^2}{n^3}+\ldots+\frac{(2n-1)^2}{n^3}\] as \(n\) approaches infinity.

Answer: \dfrac{4}{3}


Question: Find the volume of the solid enclosed by the surface \( \rho = 1 + \cos \phi \) in spherical coordinates.

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


Question: Let $N$ be the set of natural numbers. Suppose there is a bijection between $N$ and set $X$, and there is also a bijection between $N$ and set $Y$. Is there a bijection between $N$ and $X \cup Y$?

Answer: Yes


Question: Determine whether the following statement is true or false: If there are no nilpotent elements in $R/I$, where $R$ is a commutative ring and $I$ is an ideal of $R$, then $R/I$ is an integral domain.

Answer: B


Question:
Find the volume of the region $V \subset \mathbb{R}^3$ bounded by the paraboloid $x^2 + y^2 \le z$ and the plane $z \le x + 2$. Choose the correct volume from the options below:

(A) $\frac{75}{8}\pi$
(B) $\frac{81}{32}\pi$
(C) $\frac{9}{4}\pi$
(D) $\frac{45}{8}\pi$

Answer: B


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

Answer: \sqrt{2}


Question: In the random graph model $G(n, K/n)$, where $K > 1$ is a constant, consider a vertex to be "small" if it is part of a connected component of size at least 2 and at most $Cn$, with $C = e^{-1000}$. Determine if there exists a constant $B$ such that the probability of having at least $Bn$ small vertices in $G(n, K/n)$ approaches 1 as $n$ approaches infinity.

Answer: Yes


Question: Calculate the shortest distance between the curves \( y = x^2 \) and \( y = 2x - 6 \).

Answer: \sqrt{5}


Question: A particle is placed on a frictionless parabolic slide described by the equation \( y = x^2 \) at the point \((1,1)\). The particle is subjected to a constant gravitational acceleration of \( g = 9.8 \) units/s². Determine the time it takes for the particle to reach the bottom of the slide.

Answer: 0.66


Question: Find the coefficient of the $x^3$ term in the expansion of $(x^2 + x - 5)^7$. \( \text{(Use the multinomial theorem if necessary.)} \)

Answer: -109375


Question: A line with direction ratios $2:2:1$ intersects the lines $\frac{x-7}{3} = \frac{y-5}{2} = \frac{z-3}{2}$ and $\frac{x-1}{2} = \frac{y+1}{4} = \frac{z+1}{3}$ at points A and B, respectively. Find the distance AB.

Answer: 5


Question: Given \(\lim_{n\to\infty}\frac{1^a+2^a+\ldots+n^a}{(n+1)^{a-1}\cdot((na+1)+(na+2)+\ldots+(na+n))}=\frac{1}{60}\), find the value of \(a\).

Answer: 7


Question: Determine whether the following statement is true or false: Let \( f_n(x) \to f(x) \) uniformly on the domain \( D = (0,1) \), and assume that \( f(x) \) is continuous and bounded. Then there exists an \( N \) such that for all \( n > N \), \( f_n(x) \) is a continuous function.

Answer: B


Question: Determine the number of integer solutions to the equation \(a + b + c + d = 30\) where \(0 \leq a, b, c, d \leq 9\).

Answer: 84


Question: Calculate the limit: \[ \lim_{n\to\infty}  \int_{0}^{\pi}\sqrt[n]{\sin(x)}\,dx \]

Answer: \pi


Question: Evaluate the surface integral \( \iint_S(x\,dy\,dz+y\,dx\,dz+z\,dx\,dy) \) where \( S \) is the outer surface of the ellipsoid \( \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 \) that lies above the \( xy \)-plane.

Answer: 2\pi abc


Question: Let $Y$ and $Z$ be random variables, and let $X$ be a set of random variables including $Z$ but not $Y$. Assume $Y$ is conditionally independent of $X$ without $Z$ given $Z$. Can we conclude that $Y$ is conditionally independent of $X$ given $Z$? Justify your answer.

Answer: Yes


Question: Express the function $f = u + iv$, where $u = x^3 - 3xy^2 + 2xy$ and $v = -y^3 + 3x^2y - x^2 + y^2$, in terms of the complex variable $z = x + yi$. What is $f(z)$?

Answer: z^3 - iz^2


Question: Evaluate the limit: $$\lim_{n \to \infty}\left(\frac{n}{n^3+1}+\frac{2n}{n^3+2}+ \dots + \frac{n^2}{n^3+n}\right)$$

Answer: \dfrac{1}{2}


Question: Find the minimum value of the expression \( A = \frac{a(x+a)^2}{\sqrt{x^2-a^2}} \), where \( a \) is a constant.

Answer: 3\sqrt{3}a^2


Question: Given a group $G$ and a $2$-cocycle $a: G \times G \to U(1)$, is it always possible to find a cohomologous $2$-cocycle $a'$ such that $a'(s, s^{-1}) = 1$ for all $s \in G$?

Answer: Yes


Question: Let $(X,A,\nu)$ be a probability space and $T\colon X\to X$ a measure-preserving transformation with respect to $\nu$. Consider a measurable partition $P=\{P_0,\dots,P_{k-1}\}$. Define $I$ as the set of all possible itineraries, $I=\{(i_1,\dots,i_n,\dots)\in k^N; \text{ there exists } x\in X \text{ such that } T^n(x)\in P_{i_n} \text{ for all } n\in\Bbb N\}$. Suppose that $I$ is countably infinite. Is it true that the entropy of $T$ with respect to $P$ is $0$, i.e., $h(T,P)=0$?

Answer: 0


Question: Let $\omega_n$ and $\omega$ be measurable subsets of $[0,1]$. Suppose the indicator function $\chi_{\omega_n}$ converges to $\chi_{\omega}$ in the weak*-topology on $L^\infty(0,1)$, meaning that \(\int_0^1 f(x)(\chi_{\omega_n}(x)-\chi_{\omega}(x))\,dx \to 0\) for all $f \in L^1(0,1)$. Determine whether this implies that \(\int_0^1 |\chi_{\omega_n}(x)-\chi_{\omega}(x)|^2\,dx \to 0.\)

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}\sin(2\pi\cdot n!\cdot e) \]

Answer: 0


Question: Let $X \sim \mathcal{N}(0,1)$ and $Y \sim \text{Unif}(0,1)$. Assuming $X$ and $Y$ are independent, calculate $\text{Var}(XY)$. Express your answer in terms of integrals if necessary.

Answer: \dfrac{1}{3}


Question: Consider a sequence of nonzero representations \(\phi_n: A \rightarrow B(H_n)\), where \(A\) is a nonunital \(C^*\)-algebra, \(H_n\) is a Hilbert space, and \(P_n\) is a sequence of projections on \(H_n\). Determine whether there exists an element \(a_0 \in A\) such that the sequence \(\{P_n\phi_n(a_0)\}\) is norm convergent.

Answer: Yes


Question: Determine the values of $a$ for which the series \( \sum_{n=1}^\infty \frac{(\ln n)^{2014}}{n^a} \) converges.

Answer: a > 1


Question: Let $f:X\to Y$ be a finite etale Galois morphism of varieties over $\mathbb{C}$. If $C$ is a smooth quasi-projective connected curve in $X$, is $f(C)$ a smooth curve?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to +\infty}\left[\sqrt[4]{x^4+x^3+x^2+x+1}-\sqrt[3]{x^3+x^2+x+1}\cdot \frac{\ln(x+e^x)}{x}\right]. \]

Answer: -\dfrac{1}{12}


Question: Find all analytic functions $f$ on the unit disk such that $|f(z^2)| \geq |f(z)|$ for all $z$ in the unit disk.

Answer: f(z) = c


Question: Determine whether the following statement is true or false: Let $f$ be a non-negative function continuous on $[0, \infty)$ and differentiable on $(0, \infty)$ such that $f(0)=0$ and $f'(x) \le f(x)$ for all $x \in (0, \infty)$. Then $f \equiv 0$. Provide a justification for your answer.

Answer: True


Question: Evaluate the limit: \[ \lim_{x \to 0}\frac{1}{x} \int_0^{x}(1+u)^{\frac{1}{u}}du \] without using L'Hôpital's rule.

Answer: e


Question: Is the differential $d_{p}F$ of a smooth function between manifolds invertible if and only if its matrix representation $D\hat{F}(\psi(p))$ is invertible, regardless of the choice of charts?

Answer: Yes


Question: Determine the type of undirected, 2-connected n-vertex graph $G$ for which the sum of the shortest paths between all pairs of vertices, $S(G) = \sum_{x,y \in V[G]} d_G(x,y)$, is maximized.

Answer: C_n


Question: Evaluate the limit: \[ \lim_{z \to 0} \frac{-\pi^2}{\sin^2(\pi z)} \]

Answer: -\infty


Question:
Evaluate the following logarithmic expression and choose the correct answer from the options provided:

$$\frac{\log _{10}\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...\right)}{\log _{10}\left(2+\frac{2}{3}+\frac{2}{9}+...\right)} \cdot \left(\log _2\left(3\right)+\log _3\left(4\right)+\log _{16}\left(3\right)+...+\log _{2^{2n}}\left(3\right)+...\right)$$

Options:
A) 2
B) -1
C) -2
D) \(\frac{1}{2}\)

Answer: A


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(xy) \geq f(x+y) \) for all real numbers \( x \) and \( y \).

Answer: f(x) = c


Question: Derive a recursive formula for the integral \( I(n) = \int_0^1 x^m \ln^n(x) \, dx \) and solve the integral for \( m = 0 \).

Answer: (-1)^n n!


Question: Find the inverse Laplace transform of the function \( F(s) = \frac{2(3s^2+1)}{(s^2-1)^3} \).

Answer: t^2 \sinh(t)


Question: A business executive has $100,000 to invest in two plans. Plan A guarantees a return of $0.70 for each dollar invested after one year, and investments can be made annually. Plan B guarantees a return of $2 for each dollar invested after two years, and investments can only be made for periods that are multiples of two years. How should the executive allocate the $100,000 between these two plans to maximize the total earnings at the end of 3 years?

Answer: 410000


Question: Determine whether every prime of the form $4k+1$ divides a number of the form $4^n+1$. Provide a proof or counterexample.

Answer: 73


Question: Let $W = \langle (1,1,1,1), (1,1,2,2) \rangle$ be a subspace of the Euclidean inner product space $\mathbb{R}^{4}$. Consider the projection map $T_{P}: \mathbb{R}^{4} \to \mathbb{R}^{4}$ onto $W$. If $P$ is the matrix representation of $T_{P}$, find the value of $\det(P^{2} + 2P + 3I)$, where $I$ is the $4 \times 4$ identity matrix.

Answer: 324


Question: Let \( \mathbb{B} \) be a Boolean algebra and \( F \) a proper filter in \( \mathbb{B} \) such that \( 0 \notin F \). Let \( I \) be the dual ideal of \( F \). Suppose there exists an element \( a \in \mathbb{B} \) such that \( a \notin F \cup I \). Does there exist a Boolean algebra \( \mathbb{B}' \) and a homomorphism \( h: \mathbb{B} \to \mathbb{B}' \) such that \( h(a) \in h(F) \)?

Answer: Yes


Question: Evaluate the limit: \[ \lim _{x\to 0}\left(\dfrac{\sin\alpha x-\sin\beta x}{e^{\alpha x}-e^{\beta x}}\right) \]

Answer: 1


Question: Evaluate the limit: $$\lim_{(x,y)\rightarrow(0,0)}\frac{e^{xy}-1}{y}$$

Answer: 0


Question: Consider the sequence of functions defined by \( f_n(x) = n^2 x (1 - nx) \) for \( 0 \leq x \leq \frac{1}{n} \) and \( f_n(x) = 0 \) for \( \frac{1}{n} < x \leq 1 \). Determine the pointwise limit of \( f_n(x) \) as \( n \to \infty \) and whether the convergence is uniform on the interval \([0, 1]\).

Answer: 0


Question: Use a triple integral to find the volume of the solid enclosed by the paraboloids \(y = x^2 + z^2\) and \(y = 8 - x^2 - z^2\).

Answer: 16\pi


Question: Let $R$ be a commutative semi-local ring such that $R/P$ is finite for every prime ideal $P$ of $R$. Is it true that $R$ is an Artinian ring?

Answer: No


Question: Calculate \( 4064569^{4513230} \mod n \), where \( n = 1039 \times 4283 \). Use Fermat's Little Theorem and the Chinese Remainder Theorem to find the result.

Answer: 3293631


Question: Determine whether the statement is true or false: Two atlases \(\mathcal{A}_1\) and \(\mathcal{A}_2\) are compatible if and only if their associated maximal atlases \(\mathcal{A}_1^+\) and \(\mathcal{A}_2^+\) are equal.

Answer: True


Question: Determine whether the following statement is true or false: If $z_n \to z$ strongly in $C_0([0,T]; W^{1,\infty}(\Omega))$, then for any continuous function $f \in C_0(\mathbb{R})$, $f(z_n) \to f(z)$ strongly in $L^\infty(\Omega \times (0,T))$. Here, $\Omega$ is a bounded smooth domain of $\mathbb{R}^N$. Provide a justification for your answer.

Answer: True


Question: Let \( \mu_1, \mu_2 \in \mathcal{M}(\Omega) \) be Radon measures such that \( \mu_2 \ge \mu_1 \ge 0. \) Given that \( \|\mu_2\| = \| \mu_1\| \) and \( 0 \le \mu_1 \le \mu_2 \), is it true that \( \mu_1 = \mu_2 \)? Justify your answer.

Answer: \mu_1 = \mu_2


Question: Determine whether the polynomial \(x^3 - 9\) is irreducible over the integers modulo 31.

Answer: irreducible


Question: Three lattice points are chosen at random with replacement in the interior of the square defined by \(-99 \le x \le 100\) and \(-99 \le y \le 100\). The probability that the area of the triangle formed by these points is an integer can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\).

Answer: 13


Question: Let \(f: [0, 1] \to \mathbb{R}\) be a continuous function such that \(\int_{0}^{1} f(x) \, dx = \frac{1}{2}\). Prove that there exists a \(c \in [0, 1]\) such that \(f(c) = c\).

Answer: c


Question: Suppose $f:V \to V$ is a linear transformation such that the matrix representation of $f$ with respect to every basis is skew-symmetric. Does this imply that $f$ must be the zero linear transformation?

Answer: 0


Question: Calculate the integral \(I = \int_0^\infty x^3 e^{-x} \, dx\) by introducing an additional parameter \(\lambda\) and rewriting the exponential function as \(e^{-x} = e^{-\lambda x}\) with \(\lambda = 1\). Use the property \(\frac{d}{d\lambda} e^{-\lambda x} = -x e^{-\lambda x}\) to simplify and calculate the integral.

Answer: 6


Question: Consider the set $X$ of all $2 \times 2$ real matrices with trace zero and bounded entries. Let $Y = \{ \det(A) : A \in X \} \subseteq \mathbb{R}$. Does there exist $\alpha < 0$ and $\beta > 0$ such that $Y = [\alpha, \beta]$?

Answer: Yes


Question: Find the maximum value of the expression \((1 - x)(2 - y)^2 (x + y)\) given the constraints \(x < 1\), \(y < 2\), and \(x + y > 0\).

Answer: \dfrac{81}{64}


Question: Find the value of \(\alpha\) such that the roots of the polynomial \(x^5 + 5\lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8\lambda + 3)x + \lambda\alpha - 2 = 0\) are independent of \(\lambda\).

Answer: -3


Question: For a fixed odd integer \( k > 1 \), determine if there exists a positive integer \( n \) such that the sum of the binary digits of \( nk \) is less than the sum of the binary digits of \( n \).

Answer: Yes


Question: Given symmetric positive semidefinite matrices $A$ and $B$ of equal size, is the matrix $2AB$ also symmetric positive semidefinite?

Answer: No


Question: Show that the space $X+_fY$, where $X=[0,1]$, $Y=[2,3]$, $A=\{0,1\}$, and $f\colon A\to Y$ is defined by $f(0)=2$, $f(1)=3$, is homeomorphic to the circle $\mathbb{S}^1$. Use the known result that if $X$ is any space, $A$ is a subset of $X$, and $p\notin X$, the space $X+_f\{p\}$ resulting from the function $f$ which takes $A$ to $\{p\}$ is homeomorphic to the quotient space of $X$ obtained by identifying $A$ to a single point.

Answer: \mathbb{S}^1


Question: Determine whether the functional equation \( f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+} \) with the conditions \( f(x^2) = xf(x) \) and \( f(x+1) = f(x) + 1 \) can be solved.

Answer: f(x) = x


Question: Calculate the area enclosed by the curve \(a^2x^2 = y^3(2a-y)\) using a double integral. The curve is symmetric about the \(y\)-axis and intersects the \(y\)-axis at \(y=0\) and \(y=2a\). Set up and evaluate the double integral to find the total area.

Answer: \pi a^2


Question: Calculate the value of the infinite series \( \sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)} \), where \( H_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \) is the \( n \)-th harmonic number.

Answer: 2


Question: For which values of \( \alpha > 0 \) is the integral \( \int_{D_{\alpha}} f \, d\lambda < + \infty \) finite, where \( D_{\alpha} = \{(x,y) \in \mathbb{R}^2 \mid 0 < y, 0 < x < y^{\alpha} < 1 \} \) and \( f(x,y) = \frac{1}{(x+y)^2} \)?

Answer: \alpha > 1


Question: Let \( k \) be a number field and \( v \) a discrete non-archimedean valuation on \( k \). Let \( k_v \) be the completion of \( k \) with respect to \( v \), and let \( \mathcal{O}_v \) be the valuation ring of \( k_v \) with \( \mathcal{M}_v \) as its unique maximal ideal. Determine whether the factor ring \( \mathcal{O}_v/\mathcal{M}_v^n \) is finite for each positive integer \( n \).

Answer: Yes


Question:
Is every subset of the natural numbers a semi-linear set? A subset $X$ of $\mathbb{N}^n$ is linear if it can be expressed as:
\[ u_0 + \langle u_1,...,u_m \rangle = \{ u_0 + t_1 u_1 + ... + t_m u_m \mid t_1,...,t_m \in \mathbb{N}\} \]
for some $u_0,...,u_m \in \mathbb{N}^n$. A set $X$ is semi-linear if it is the union of finitely many linear subsets. Determine if any subset of the natural numbers can be expressed as a finite union of linear sets.

Answer: No


Question: Does there exist a non-trivial topological space $X$ with $|X| > 1$ such that every real-valued continuous function on $X$ is constant?

Answer: Yes


Question: Find the limit of \( \left(\frac{nx}{1+nx}\right)^{n} \) as \( n \to \infty \) for \( x > 0 \).

Answer: e^{-1/x}


Question: Evaluate the integral \( \int_0 ^{2\pi} \frac{1}{4\cos^2t+9\sin^2t} \, dt \).

Answer: \dfrac{\pi}{3}


Question: Let $X$ and $Y$ be independent exponential random variables with means 1 and 2, respectively. Define $Z = 2X + Y$. Calculate the expected value $E(X|Z)$.

Answer: \dfrac{Z}{4}


Question: Evaluate the integral \( I = \int_{C(i,2)} \frac{1}{(z^2+4)^2} \, dz \), where \( C(i,2) \) is the circle centered at \( i \) with radius 2.

Answer: \dfrac{\pi}{16}


Question: Let $f(x)$ be a real function not identically zero, such that for all $x, y \in \mathbb{R}$, the following holds: \[ f(x+y^{2n+1}) = f(x) + (f(y))^{2n+1} \] where $n \in \mathbb{N}$. If $f'(0) \ge 0$, find $f'(6)$.

Answer: 1


Question: Consider the graph $G = (V, E)$ where $V$ is the set of all functions $f: \mathbb{N} \to \mathbb{N}$. Two distinct functions $f, g: \mathbb{N} \to \mathbb{N}$ form an edge if and only if they differ at exactly one input $n \in \mathbb{N}$. Determine if the chromatic number $\chi(G)$ is $\aleph_0$.

Answer: \aleph_0


Question: Let $f$ be a function defined on the positive integers such that for any positive integers $a$, $b$, and $n$ with $a + b = 2^n$, it holds that $f(a) + f(b) = n^2$. Determine the value of $f(2021)$. \( \)

Answer: \dfrac{203}{2}


Question: Let $A$ and $B$ be positive definite Hermitian matrices. Define the sequences $\{A_n\}$ and $\{G_n\}$ by $A_0 = A$, $G_0 = B$, $A_{n+1} = \frac{A_n + G_n}{2}$, and $G_{n+1} = A_n \natural G_n$, where $A_n \natural G_n$ denotes the geometric mean as defined in the referenced paper. Do the sequences $\{A_n\}$ and $\{G_n\}$ converge to the same matrix?

Answer: Yes


Question: Determine the cardinality of the set of recursive subsets of the natural numbers.

Answer: \aleph_0


Question: Find the value of \( A \) that satisfies the equation: \[ \sqrt[3]{A-15\sqrt{3}} + \sqrt[3]{A+15\sqrt{3}} = 4 \]

Answer: 26


Question: Calculate the value of the 1000th derivative of the function \( g(x) = \exp(x^{999}) \) at \( x = 0 \).

Answer: 0


Question: Can a non-constant holomorphic function $f$ on the complex plane $\mathbb{C}$ have no zeroes? Provide a justification for your answer.

Answer: Yes


Question: Determine the structure of the quotient module $\mathbb{Z}^3/K$, where $K$ is generated by $(2,1,-3)$ and $(1,-1,2)$. Express the structure as a direct sum of simpler modules.

Answer: \mathbb{Z}


Question: Determine the number of nonisomorphic normal subgroups of the symmetric group $S_n$ for $n \geq 4$. Provide your answer as a single integer.

Answer: 3


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{\sqrt{1 + \tan(x)} - \sqrt{1 + \sin(x)}}{x^3} \]

Answer: \dfrac{1}{4}


Question: Given that \( g(x) = \int_0^x f(y) (y - x) \, dy \) is exactly 3 times continuously differentiable, find the greatest integer \( n \) for which \( f \) must be \( n \) times continuously differentiable.

Answer: 1


Question:
Evaluate the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) using Stokes' Theorem, where \( C \) is the curve defined by \((x-1)^2 + (y-2)^2 = 4\) and \( z=4 \), oriented counterclockwise when viewed from above the z-axis. The vector field is given by:
\[ \mathbf{F}(x,y,z) = (z^2 + y^2 + \sin x^2)\mathbf{i} + (2xy + xz)\mathbf{j} + (xz + 2yz)\mathbf{k} \]

Answer: 16\pi


Question: Evaluate the expression: \[ \lg_{2} \left( \prod\limits_{a=1}^{2015} \prod\limits_{b=1}^{2015} (1 + e^{\frac{2\pi iab}{2015}}) \right) \]

Answer: 13725


Question: Determine the number of whole number terms in the expansion of \((\sqrt[11]{5} + \sqrt[5]{11})^{2015}\).

Answer: 37


Question: Evaluate the integral \( \int_{S} x^{4} \sin (x^{3}z^{5})\,dx\,dy\,dz \) where \( S \) is the subset of the sphere defined by \( x^{2} + y^{2} + z^{2} = 1 \) and \( z > 0 \).

Answer: 0


Question: Determine whether $R^n$ is projective as a module over $M_n(R)$. Provide a justification for your answer.

Answer: Yes


Question: Given that $P(A)=\frac{1}{3}$, $P(A\cup C)=\frac{7}{9}$, $P(B^c|A^c\cap C^c)=\frac{3}{4}$, and $P(B|A)=P(B|C)=P(B|A\cap C)=\frac{1}{2}$, with $A$ and $C$ being independent events, calculate $P(A|B)$. Provide your answer as a fraction.

Answer: \dfrac{3}{8}


Question: Evaluate the double integral \( \iint_R \sqrt{\frac{x+y}{x-2y}} \, dA \) using a change of variables, where \( R \) is the region bounded by the lines \( y - \frac{x}{2} = 0 \), \( y = 0 \), and \( x+y = 1 \).

Answer: \dfrac{1}{3}


Question: Let \( P: H \to H \) be a bounded linear operator such that \( P^2 = P \) and \( P^\star = P \). Show that the operator norm \( \|P\| \le 1 \).

Answer: \|P\| \leq 1


Question: If $A$ and $B$ are two simultaneously diagonalizable normal matrices, can the common unitary matrix $U$, formed by their common eigenvectors, be found by diagonalizing their product $AB$?

Answer: No


Question: Let $A$ be an $m \times n$ matrix of rank $k$ over a field $F$. Can $A$ be expressed as $A = BC$, where $B$ is an $m \times k$ matrix of rank $k$, and $C$ is a $k \times n$ matrix of rank $k$? Justify your answer.

Answer: Yes


Question: Find the last three digits of \(3^{999}\).

Answer: 667


Question: Let $A$ and $B$ be subsets of a topological space $X$. Prove that if $A$ is connected, $B$ is both open and closed, and $A \cap B \neq \emptyset$, then $A \subset B$. Provide a proof by contradiction.

Answer: A \subset B


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{kn} \right) \] where \( k > 1 \) is an integer.

Answer: \ln k


Question: Determine whether the mapping $u \mapsto u(0)$ from $H^1(0, 1)$ into $\mathbb{R}$ is a continuous linear functional on $H^1(0, 1)$. Provide a justification for your answer.

Answer: Yes


Question: Find all prime numbers \( p \) such that \( p^4 + p^3 + p^2 + p + 1 \) is a perfect square.

Answer: 3


Question:
Find the maximum value of \( S = \sum_{k=1}^{100} x_k^2 \) subject to the following constraints:

1. \( x_1 \ge x_2 \ge x_3 \ge \dots \ge x_{100} \ge 0 \)
2. \( x_1 + x_2 \le 100 \)
3. \( x_3 + x_4 + \dots + x_{100} \le 100 \)

where \( x_k \in \mathbb{R} \) for \( 1 \le k \le 100 \).

Answer: 10000


Question: Let $M$ and $M'$ be simply connected manifolds of dimension $m > 0$. Is it true that if $M$ is homotopic to $M'$, then for $k \geq m$, the spaces $Imm(M, \mathbb{R}^{m+k})$ and $Imm(M', \mathbb{R}^{m+k})$ are homotopic? In other words, if $k \geq m$, does $M \simeq M'$ imply $Imm(M, \mathbb{R}^{m+k}) \simeq Imm(M', \mathbb{R}^{m+k})$?

Answer: Yes


Question:
Given a matrix where the value at each index is determined by the XOR operation between the column and row indices, and a fixed amount is subtracted from each element (ensuring non-negative results), calculate the sum of all elements in the matrix. For example, consider a matrix with 5 rows and 8 columns:

\[
\begin{bmatrix}
0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
1 & 0 & 3 & 2 & 5 & 4 & 7 & 6 \\
2 & 3 & 0 & 1 & 6 & 7 & 4 & 5 \\
3 & 2 & 1 & 0 & 7 & 6 & 5 & 4 \\
4 & 5 & 6 & 7 & 0 & 1 & 2 & 3 \\
\end{bmatrix}
\]

After subtracting a fixed amount from each element (ensuring non-negative results), the matrix becomes:

\[
\begin{bmatrix}
0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
0 & 0 & 2 & 1 & 4 & 3 & 6 & 5 \\
1 & 2 & 0 & 0 & 5 & 6 & 3 & 4 \\
2 & 1 & 0 & 0 & 6 & 5 & 4 & 3 \\
3 & 4 & 5 & 6 & 0 & 0 & 1 & 2 \\
\end{bmatrix}
\]

Find the sum of all elements in the modified matrix.

Answer: 105


Question: Determine whether the limit \( \lim_{(x,y) \to (0,0)} y\sin(1/x) \) exists.

Answer: 0


Question: Is the Birch and Swinnerton-Dyer conjecture known in positive characteristic?

Answer: Yes


Question: Is it true that between any non-prime square-free number and its double, there is another non-prime square-free number?

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} \int_{0}^{\infty} \exp((\cos^n x) - x) \, dx \).

Answer: 1


Question: Determine the maximum number of sides for a regular n-polygon that can be constructed using standard LEGO bricks, ensuring the polygon remains sturdy when pushed around. The construction should not use bent or angled bricks, and should not rely on non-obvious LEGO limitations. All side lengths must be natural or rational numbers, and all angles must be right angles or determined by some construction.

Answer: 4


Question: Find the directional derivative of the function implicitly defined by \( z + x\ln(z) + xe^{xy} - 1 = 0 \) at the point \( P = (0, 1) \) in the direction of the vector \( \mathbf{v} = \langle 4 \sqrt{3}, 3 \sqrt{3} \rangle \).

Answer: -\dfrac{4}{5}


Question: If \(f:[0,1] \rightarrow \mathbb{R}\) is a strictly increasing discontinuous function, can \(f([0,1])\) be a subset of \(\mathbb{Q}\)?

Answer: No


Question: Evaluate the variance of the expression \((X_1 + X_2)X_3\), where \(X_1, X_2, X_3\) are independent random variables with \(X_i \sim \exp(1)\).

Answer: 8


Question: Is the identity matrix the only matrix $A \in \mathbb{R}^{n \times n}$ with real positive eigenvalues that is equal to its inverse?

Answer: Yes


Question: Find the probability density function (PDF) of the random variable \( Z = \frac{X^2 + Y^2}{2} \), where \( X \sim N(0,1) \) and \( Y \sim N(0,1) \) are independent standard normal random variables.

Answer: e^{-z}


Question: Determine all weak limit points of the sequence of measures \( \{\mu_n\} \) on \((\mathbb{R}, B)\), where \( \mu_n = U_{\{\frac{1}{n}, \frac{2}{n}, \dots, \frac{n-1}{n}, 1\}} \) and \( U \) denotes the uniform distribution.

Answer: U_{[0,1]}


Question: Evaluate the limit: $$\lim_{n\to\infty}{\frac{n!\cdot (2n)!}{(3n)!}}$$

Answer: 0


Question: Evaluate the integral \( \int_C \frac{8z^{11}-3z^6+1}{z^7-1} \, dz \), where \( C \) is the positively oriented circle \( |z| = 4 \).

Answer: -6\pi i


Question: Is it possible for a transcendental number raised to the power of another transcendental number to be an algebraic number? Provide a justification for your answer.

Answer: Yes


Question:
Determine the values of \(a\) for which the matrix \(A\) is diagonalizable, where
\[ A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 0 & 2 & a & 5 \\ 0 & 0 & 2 & 6 \\ 0 & 0 & 0 & 7 \end{pmatrix} \in \mathbb{R}^{4 \times 4}. \]

Answer: 0


Question: Let \( f \in L^1[0,1] \). Determine whether the following limit holds: \[ \lim_{t\to\infty}t\cdot|\{x:|f(x)|>t\}|=0. \]

Answer: 0


Question: What is the universal covering space of a closed surface of genus 2?

Answer: \mathbb{H}^2


Question: Evaluate the limit: $$\lim_{x \to 0^+}e^{-ax\sqrt{2^{b+c/x}-1}},$$ where $a$, $b$, and $c$ are positive constants.

Answer: 0


Question:
Is it valid to change the limits of integration from \([-P/2, P/2]\) to \([0, P]\) for the Fourier series coefficient \(a_n\) of a function \(g(x)\) with period \(P\), given by:
\[
a_n = \frac{2}{P}\int_{-P/2}^{P/2} g(x) \cos\left(\frac{2\pi n}{P} x\right) \, dx
\]
to:
\[
a_n = \frac{2}{P}\int_{0}^{P} g(x) \cos\left(\frac{2\pi n}{P} x\right) \, dx
\]?

Answer: Yes


Question: Let $K$ be a field with an infinite number of elements, and let $f(x), g(x) \in K[x]$ be polynomials such that $f(\alpha) = g(\alpha)$ for all $\alpha \in K$. Prove that $f(x) = g(x)$. Provide a proof by contradiction.

Answer: f(x) = g(x)


Question: Determine the number of pairs of positive integers \((x, y)\) such that both \(x\) and \(y\) are prime numbers and satisfy the equation \(x^2 - 2y^2 = 1\).

Answer: 1


Question: Determine the signature of the quadratic form \( f(x, y, z) = xy + yz + xz \).

Answer: (1, 2)


Question: Determine the number of distinct real roots of the equation: \[ (x^2 + x - 2)^3 + (8 - 2x^2)^3 = (x^2 + 3x + 2)^3. \]

Answer: 2


Question: Determine the values of \(a > 0\) for which the series \(\sum_{n=1}^\infty \frac{a^{na}}{n^{2a}}\) converges.

Answer: (0, 1]


Question: Find the value of \( k \) such that the inequality \[ \frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{c}} \ge \sqrt{a}+\sqrt{b}+\sqrt{c}+k \] holds for all positive \( a, b, c \) with \( abc = 1 \).

Answer: 3


Question: Given that $x_1, x_2, \dots, x_n$ are the roots of the equation $x^n - 3x^{n-1} + 2x + 1 = 0$, find the value of \( \sum_{k=1}^n \frac{x_k}{x_k-1} \).

Answer: 3n - 5


Question: Let $G$ be a finite group and $A$ a subset of $G$ such that $A$ is symmetric (i.e., $A = A^{-1}$). Suppose there exists a subset $B \subseteq G$ such that $G = AB$ and $|AB| = |A||B|$. Is $A$ necessarily a subgroup of $G$?

Answer: No


Question:
Given the differential equation \(x' = 0.02310x - 0.01d\) with the initial condition \(x(0) = 1\), and the solution formula:
\[x = \left(1 - \frac{0.01d}{0.02310}\right)e^{0.02310t} + \frac{0.01d}{0.02310}\]
Determine the critical dosage \(d\) in milligrams that stabilizes the infection, knowing that it lies between 1.5g and 3.0g.

Answer: 2310


Question: Let $A$ be an $n \times n$ matrix such that for any $n \times n$ matrix $B$, the equation $\det(A+B) = \det(A) + \det(B)$ holds. Determine whether this implies that $A = 0$ or $\det(A) = 0$. Provide a justification for your answer.

Answer: A = 0


Question: Evaluate the integral by changing the order of integration: $$\int_{0}^{9}\int_{0}^{3}\int_{2y}^{6} \frac{2\cos x^2}{3\sqrt{z}}\,dx\,dy\,dz$$

Answer: \sin 36


Question:
Determine whether the following inequality holds for any real numbers \(u, v, w\) such that \(u < 0 < v < w\):
\[
(v-u)^2+(w-u)^2+(w-v)^2 \\
+\left(\sqrt{\frac{1+u^2}{1+v^2}}+\sqrt{\frac{1+v^2}{1+u^2}}\right) (w-u)(w-v) \\
-\left(\sqrt{\frac{1+u^2}{1+w^2}}+\sqrt{\frac{1+w^2}{1+u^2}}\right) (w-v)(v-u) \\
-\left(\sqrt{\frac{1+w^2}{1+v^2}}+\sqrt{\frac{1+v^2}{1+w^2}}\right) (v-u) (w-u)>0
\]

Answer: True


Question: Calculate the limit \( \lim_{n \to \infty } \frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{\ln n} \) using the Stolz-Cesàro theorem.

Answer: 1


Question: Compute the integral \( \int_{|z|=2} \frac{dz}{z^2-1} \) along the positively oriented circle of radius 2 centered at the origin, without using knowledge beyond the Cauchy Theorem and Cauchy Integration Formula.

Answer: 0


Question: Determine the value of \(a\) such that the equation \(f(x) = a\) has a solution for \(x \in (0,1]\), where \(f: (0,\infty) \to \mathbb{R}\) is defined by \(f(x) = x^n - n\ln x + 1\) and \(n \neq 0\), \(n \in \mathbb{N}\).

Answer: 2


Question: Find the function \( y(x) \) that satisfies the differential equation \( y'' + yy' = 0 \) with the initial conditions \( y(0) = 5 \) and \( y'(0) = 0 \).

Answer: 5


Question: Evaluate the limit \( \lim_{n\to\infty}\int_0^n\frac{\mathrm{d}x}{n+n^2\sin\frac{x}{n^2}} \).

Answer: \ln 2


Question: Determine whether there exists an integer \( x \) such that for all positive integers \( n \), the expression \( 2^n \) divides \( 3^n(x+1) \).

Answer: -1


Question: Calculate the probability that the total weight of apples, each with a weight following an exponential distribution with parameter \( \lambda = \frac{1}{3} \), exceeds 7 kilograms, given that you continue adding apples until the total weight exceeds 5 kilograms.

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


Question: Let $F$ and $E$ be fields with $F < E$. Suppose $V$ is a vector space over $F$ with $\mathrm{dim}_F(V) = n$. Consider $V^E = V \otimes_F E$, viewing $V$ as a vector space over $E$. What is $\mathrm{dim}_E(V^E)$?

Answer: n


Question: Evaluate the limit: \[ \lim_{t \to 0} \left(\int_{0}^{1}(1+x)^{t}dx\right)^{1/t} \]

Answer: \dfrac{4}{e}


Question:
For the given discrete-time system:
\[
\begin{bmatrix}
x_1(k+1)\\
x_2(k+1)
\end{bmatrix} = \begin{bmatrix}
0 & 1\\
-2 & 0
\end{bmatrix}\begin{bmatrix}
x_1(k)\\
x_2(k)
\end{bmatrix} + \begin{bmatrix}
0\\
1
\end{bmatrix}u(k)
\]
\[
y_1 = \begin{bmatrix}
1 & 1
\end{bmatrix}\begin{bmatrix}
x_1\\
x_2
\end{bmatrix}
\]
Design a state feedback control in the form:
\[u(k) = v(k) + \begin{bmatrix}k_1 & k_2\end{bmatrix}\begin{bmatrix}x_1(k)\\x_2(k)\end{bmatrix}\]
so that the system has a double pole at \( \beta \). The feedback is active for one time step and then inactive for two time steps before re-activating. Determine the value of \( \beta \) that maintains the stability of the system.

Answer: \dfrac{1}{2}


Question: Find \( \limsup_{n\to\infty} \sqrt[n]{\frac{(n!)^3}{(3n)!}} \).

Answer: \dfrac{1}{27}


Question: Determine the values of \(r\) and \(s\) for which the integral \( \int_{0}^{2} \frac{(1+x)^r-(1+x)^s}{x^2} \, dx \) converges.

Answer: r = s


Question: Determine the radius of convergence for the power series \( \sum\limits_{n=0}^{\infty} \frac{\sin(n!)}{n!} x^n \).

Answer: \infty


Question: Determine the values of \( p \in \mathbb{R} \) for which the integral \( \int_1^\infty x^p \cos(\ln x) \, dx \) converges absolutely and conditionally.

Answer: p < -1


Question: Let \( P(z) \) be a monic polynomial with complex coefficients of degree \( n \) such that \( |P(z)| \le 1 \) for all \( |z| \le 1 \). Is it true that \( P(z) = z^n \)?

Answer: P(z) = z^n


Question: Find all distributions $f \in D'(\mathbb{R})$ such that $e^x f = 0$.

Answer: 0


Question: In how many ways can 15 different books be distributed to 15 children such that each child receives exactly one book, and then the books are collected and redistributed so that no child receives the same book they initially received?

Answer: 15! \times !15


Question: Evaluate the limit of the integral: \[ \lim_{n \to \infty} \int_{-1}^{1} f(t) \cos^{2}(nt) \, dt \] where \( f: [-1,1] \to \mathbb{R} \) is a continuous function such that \( \int_{-1}^{1} f(t) \, dt = 1 \).

Answer: \dfrac{1}{2}


Question: Let \( f(x) = x^6 + a_5x^5 + \ldots + a_0 \) be a polynomial such that \( f(0) = f(1) \) and each root of \( f(x) = 0 \) is real and lies between 0 and 1. Find the minimum value of the reciprocal of the product of the roots of \( f(x) = 0 \).

Answer: 64


Question: Evaluate the contour integral \( \oint_{c(2,1)} \frac{z^{7}+1}{z^{2}(z^{4}+1)}\,\mathrm{d}z \), where the contour \( c(2,1) \) is the circle centered at 2 with radius 1, parametrized by \( t \rightarrow 2 + e^{it} \).

Answer: 0


Question: Let \( G = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mid a, b, c, d \in \mathbb{Z}, ad - bc = 1 \right\} \) be the group of 2x2 integer matrices with determinant 1. Let \( H = \left\{ \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix} \mid n \in \mathbb{Z} \right\} \) be a subgroup of \( G \). Determine the index \([G:H]\).

Answer: \infty


Question:
Determine the smallest degree $n$ of a Taylor polynomial centered at $0$ for the function $f(x) = (1-x)^{-1}$, such that the error in the interval $[0, 0.5]$ is no greater than $10^{-8}$. Use the remainder formula:
\[ \frac{|f^{(n+1)}(c)|}{(n+1)!} \cdot |(x-\bar{x})^{(n+1)}| \le 10^{-8} \]

Answer: 27


Question:
Given a random sample \(Y_1, Y_2, \ldots, Y_n\) from a population with the probability density function:
\[
f(y : \alpha, \beta) =
\begin{cases}
\alpha\beta^\alpha y^{-(\alpha + 1)},  & \text{if } y > \beta \\
0, & \text{otherwise}
\end{cases}
\]
where \(\alpha > 0\) and \(\beta\) is known, determine the method of moments estimator for \(\alpha\):

(a) \(\widehat \alpha = \frac{M_1}{(M_1 - \beta)}\), where \(M_1 = \frac {1}{n} \sum_{i=1}^n Y_i \)

(b) \(\widehat \alpha = \frac{M_1}{(\beta - M_1)}\), where \(M_1 = \frac {1}{n} \sum_{i=1}^n Y_i \)

(c) \(\widehat \alpha = \frac{2M_2}{(M_2 + \beta^2)}\), where \(M_2 = \frac {1}{n} \sum_{i=1}^n Y_i^2 \)

(d) None of the above

Select the correct option.

Answer: a


Question: What is the coefficient of \(x^n\) in the power series expansion of the function \(\frac{x}{1-x-x^2}\) about the origin?

Answer: F_n


Question: Evaluate the limit: \[ \lim_{n\to \infty} \int_{0}^{\pi/3} \frac{\sin^n x}{\sin^n x + \cos^n x} \, dx \]

Answer: \dfrac{\pi}{12}


Question: Determine whether the expression \(2^{2n+2} + 2^{m+2} + 1\) is a perfect square if and only if \(m = n\), given that \(m\) and \(n\) are natural numbers satisfying \(m \leq 2n\).

Answer: m = n


Question: Determine whether the following limit is equal to zero: \[ \lim_{x\to0}\left(\frac{\sin(x)}{x}\right)^{\frac{1}{x^2}} = 0 \]

Answer: e^{-\frac{1}{6}}


Question: Suppose $X \times Y$ and $X \times Z$ are homeomorphic product spaces, where $X$, $Y$, and $Z$ are topological spaces. Determine whether $Y$ and $Z$ must be homeomorphic.

Answer: No


Question: Let $f: U \rightarrow \mathbb{C}$ be an entire function such that $f(1/\sqrt{n})=1/n$ for all positive integers $n$. Determine the value of $f(-i)$. Assume the context of the Identity Theorem in complex analysis.

Answer: -1


Question: Evaluate the limit of the series: \[ \lim_{k \to \infty}\sum_{n=1}^{\infty}\frac{k^3+n^3}{k^32^n+1} \]

Answer: 1


Question: If $A$ is a non-separable $C^*$-algebra, can we deduce that all the representations of $A$ are infinite dimensional?

Answer: No


Question: Given the differential equation \((2xy - y^2 - y) \, dx = (2xy + x - x^2) \, dy\) with the initial condition \(y(1) = 1\), find the value of \(12|y(-1)|\).

Answer: 12


Question: What is the value of \(1^i\)?

Answer: 1


Question: Find a positive integer whose first digit is 1 such that if this digit is moved to the end of the number, the resulting number is three times the original number.

Answer: 142857


Question: Find the limit \( \lim\limits_{x \to \infty} \int_0^{x} \cos\left(\dfrac{\pi t^2}{2}\right) \, dt \).

Answer: \dfrac{1}{2}


Question: Determine the number of points at which the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) defined by \[ f(x) = \begin{cases} x^6 - 1 & x \in \mathbb{Q} \\ 1 - x^6 & x \notin \mathbb{Q} \end{cases} \] is continuous.

Answer: 2


Question: Let the series \(\sum_{k=0}^\infty a_k x^k\) have a radius of convergence 1. Assume further that \(\sum_{k=0}^\infty a_k = \infty\). Determine whether \(\lim_{x\to 1^-} \sum_{k=0}^\infty a_k x^k = \infty\).

Answer: \infty


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{\sum_{r=0}^n\binom{2n}{2r}3^r}{\sum_{r=0}^{n-1}\binom{2n}{2r+1}3^r} \]

Answer: \sqrt{3}


Question: Determine if there exists a positive integer other than 34 such that each of the numbers, the number minus one, and the number plus one, is the product of two distinct prime numbers.

Answer: 86


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

Answer: \dfrac{5}{6}


Question: Find the surface area of the portion of the plane \(x + y + z = 1\) that lies inside the cylinder \(x^2 + y^2 = 4\).

Answer: 4\sqrt{3}\pi


Question: For which complex numbers \( w \) does the equation \( \exp(z) = w \) have solutions?

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


Question: Determine whether a nonconstant entire function $f: \mathbb{C} \to \mathbb{C}$ can be extended to a holomorphic function $\bar{f}: S^2 \to S^2$ such that $\bar{f}|_\mathbb{C} = f$. Is this statement true or false?

Answer: B


Question: Determine the largest number in the infinite sequence \(1, \sqrt{2}, \sqrt[3]{3}, \sqrt[4]{4}, \ldots, \sqrt[n]{n}\).

Answer: \sqrt[3]{3}


Question: Find the coefficient \( c_{-1} \) in the Laurent series expansion of \( \frac{\cos z}{z^5} \) around \( z = 0 \).

Answer: \dfrac{1}{24}


Question: Compute the total curvature of the unit sphere $S^2$ in $\mathbb{R}^3$, given by the integral \( \int_{S^2} K(p) \, dA \), where \( K(p) \) is the Gaussian curvature. Use the stereographic coordinates with the spherical metric \( g_{uu} = g_{vv} = \frac{4}{(1+u^2 + v^2)^2} \) and \( g_{uv}=0 \). Show that the total curvature is \( 4\pi \).

Answer: 4\pi


Question: Evaluate the surface integral \( \iint_S (x^2 + y^2) \, dS \) over the surface \( S: x^2 + y^2 + z^2 = 2z \) using spherical coordinates. Express your answer in terms of \( \pi \).

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


Question: Consider two urns, A and B. Initially, urn A contains 2 red marbles, and urn B contains 2 white marbles. In each trial, one marble is drawn from each urn and switched. Let \( X_n \) represent the number of red marbles in urn A after \( n \) trials. Given that urn A contains at least one red marble after a long time, calculate the probability that it contained exactly one red marble before the previous trial, i.e., find \( P(X_{n-1} = 1 \mid X_n \geq 1) \) as \( n \to \infty \). The stationary distribution is \([1/6, 2/3, 1/6]\) and the transition matrix is \( \begin{bmatrix} 0 & 1 & 0 \\ 1/4 & 1/2 & 1/4 \\ 0 & 1 & 0 \end{bmatrix} \).

Answer: \dfrac{3}{5}


Question: Let $\sigma$ and $\rho$ be two probability measures on the complex unit circle $\mathbb{T}$ such that $\sigma \ll \rho$. Consider a sequence of complex functions $(f_n)$ in $L_2(\rho)$ such that $f_n \rightarrow 1$ in measure with respect to $\rho$. Is it true that $f_n \rightarrow 1$ in measure with respect to $\sigma$ as well, given that $|f_n(x)| = 1$ for every $x \in \mathbb{T}$ and $n \in \mathbb{N}$?

Answer: Yes


Question: Find the number of rational numbers \( \frac{m}{n} \) such that \( m \) and \( n \) are relatively prime positive integers, \( m < n \), and \( mn = 25! \).

Answer: 256


Question: Consider a sequence \( x \in \ell^2(\mathbb{N}) \) and define the partial sums \( s_n = \sum_{i=1}^n x_i \). Determine whether the sequence \( \frac{1}{\sqrt{n}} s_n \) converges.

Answer: 0


Question: Determine the largest exponent \(\alpha\) such that the ratio \(\frac{n^{\alpha}}{\frac{\sqrt{n}}{\log{\sqrt{n}}}}\) remains asymptotically bounded as \(n\) approaches infinity, where \(n\) is a positive integer.

Answer: \dfrac{1}{2}


Question: Determine the least degree of the splitting field for the polynomial $x^n - 1$ over a finite field $F$ of order $q$, given that $(q, n) = 1$. Express your answer as the smallest integer $f$ such that $q^f \equiv 1 \pmod{n}$.

Answer: f


Question: Determine the domain of the function \( f(x) = \int_{1}^{x} \frac{e^t}{t} \, dt \).

Answer: (0, \infty)


Question: Can the generalized Laguerre function \( L^{\alpha}_n(x) \) be defined for \( \alpha = -1 \) for all \( n \geq 0 \) and \( x > 0 \)?

Answer: Yes


Question: Let $X_{(1)}, X_{(2)}, X_{(3)}$ be the order statistics of three independent random variables $X_1, X_2, X_3$ uniformly distributed over $[0,1]$. Find the correlation coefficient between $X_{(1)}$ and $X_{(3)}$. Use the fact that $X_{(k)} \sim \text{Beta}(k, 4-k)$ and the formula for correlation: \[ \text{Corr}(X_{(1)}, X_{(3)}) = \frac{\text{Cov}(X_{(1)}, X_{(3)})}{\sqrt{\text{Var}(X_{(1)})} \sqrt{\text{Var}(X_{(3)})}}. \]

Answer: \dfrac{1}{3}


Question: Let \( X \) and \( Y \) be continuous random variables with the joint density function \( f(x,y) = 24xy \) on the domain \( x, y > 0, 0 < x + y < 1 \) and \( 0 \) otherwise. Calculate the conditional probability \( P(X > 1/2 \mid X+Y < 3/4) \).

Answer: \dfrac{1}{9}


Question: Given that $a, b \in \mathbb{R}$ and $a \neq 0$, when the polynomial $x^2 - 2ax + a^4$ is divided by $x + b$, the remainder is 1. Additionally, the polynomial $bx^2 + x + 1$ has $ax - 1$ as a factor. Determine the value of $b$. \( \text{(Express your answer as a real number.)} \)

Answer: -2


Question:
Find the sum of the rearranged alternating harmonic series given by:
\[ 1 + \frac{1}{3} + \frac{1}{5} - \frac{1}{2} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} - \frac{1}{4} + \frac{1}{13} + \frac{1}{15} + \frac{1}{17} - \frac{1}{6} + \cdots \]
Consider partial sums up to $4n$ terms to find the sum of the series.

Answer: \ln(2\sqrt{3})


Question: Does the short exact sequence \(0\rightarrow \mathbb{Z}_3 \rightarrow G \rightarrow \mathbb{Z}_2\oplus \mathbb{Z} \rightarrow 0\) always split?

Answer: Yes


Question: Determine whether there exists a $T_0$-hypergraph $H=(V,E)$ such that $2^{|E|} < |V|$. Provide a justification for your answer.

Answer: No


Question: Does the set of continuous functions with compact support, $C_c(X)$, separate points in a separable, infinite-dimensional Banach space $X$? Provide a justification for your answer.

Answer: No


Question: Let $I$ be an uncountable directed set, and let $T = \{E_{\alpha}: \alpha \in I\}$ be a collection of countably infinite sets such that $E_{\alpha} \subset E_{\beta}$ whenever $\alpha \le \beta$. The collection $T$ is totally ordered with respect to set inclusion. Determine the cardinality of $\bigcup_{\alpha \in I} E_{\alpha}$.

Answer: \aleph_1


Question: Calculate the following limit, given that the function \( f(x,y) \) is continuous at \((0,0)\) and integrable in the vicinity of \((0,0)\): $$\lim\limits_{r\to 0^+}\dfrac{1}{\pi r^2}\iint\limits_{|(x,y)|\le r} f(x,y) \, dS.$$

Answer: f(0,0)


Question: A criminal must pass 10 lie detector tests successfully to escape from jail. The probabilities of each lie detector detecting a lie are as follows: 0, \( \frac{1}{2} \), \( \frac{2}{3} \), \( \frac{3}{4} \), \( \frac{4}{5} \), \( \frac{5}{6} \), \( \frac{6}{7} \), \( \frac{7}{8} \), \( \frac{8}{9} \), \( \frac{9}{10} \). If a detector indicates a lie, the criminal must start over from the first detector. Each test takes 1 second. Calculate the expected time in seconds for the criminal to pass all tests and escape.

Answer: 9864100


Question: Compute the limit: \[ \lim_{n\rightarrow\infty}\sum_{i=1}^n\sin\left(\frac{i}{n^2}\right) \]

Answer: \dfrac{1}{2}


Question: Given a random variable $X$ such that $E(X) = 1$ and $E(X^2) = 1$, find the expected value $E(X^{100})$. Assume $X$ is a real-valued random variable.

Answer: 1


Question:
Find an expression for the number of binary strings $b_k \cdots b_0$ such that the number of 0s, $m$, and the number of 1s, $n$, satisfy $m + 2n \leq L$. Provide a simplified or alternative expression to the following sum:
$$\sum_{n=0}^{\lfloor L/2 \rfloor} \sum_{m=0}^{L-2n} {n+m \choose n}$$

Answer: F_{L+3} - 1


Question: Is the least supercompact cardinal number necessarily greater than the least measurable cardinal number?

Answer: Yes


Question: Is there a fixed integer $k$ and a $<_k$-increasing sequence of functions $\{ f_n : n \in \mathbb{N} \}$ from $\mathbb{N}$ to itself, such that there exists a function $g$ with $g >_k f_n$ for all $n$?

Answer: Yes


Question: A set contains 60 items, with 24 items of type A, 8 items of type B, and 4 items of type C. If 7 items are randomly selected from the set, what is the probability that the selection includes at least one item of each type?

Answer: \dfrac{11248768}{48275865}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( \sum_{r=1}^{n} \left(\frac{2r+5}{r^2+r}\right) \left(\frac{3}{5}\right)^{r+1} \right) \]

Answer: \dfrac{9}{5}


Question: Find the minimum value of \(a + b + \frac{1}{ab}\) given that \(a^2 + b^2 = 1\) and \(a, b > 0\).

Answer: 2 + \sqrt{2}


Question: Evaluate the triple integral \( \iiint z \, dz \, dy \, dx \) over the region defined by \( \sqrt{x} + \sqrt{y} + \sqrt{z} \leq 1 \).

Answer: \dfrac{1}{840}


Question: Evaluate the integral \( \int_{|z|=1} \frac{30z^2 - 23z + 5}{(2z-1)^2(3z-1)} \, dz \).

Answer: 5\pi i


Question: Determine the result of multiplying two multi-valued quantities: If $\sqrt{-1} = \{i, -i\}$, what is $\sqrt{-1} \times \sqrt{-1}$? Is it $\{1, -1\}$ or $\{-1\}$?

Answer: \{1, -1\}


Question: A random sample of size 7 is drawn from a distribution with probability density function \( f_{\theta}(x) = \frac{1+x^2}{3\theta(1+\theta^2)} \) for \( -2\theta \le x \le \theta \) and \( x > 0 \), and zero otherwise. The observations are 12, -54, 26, -2, 24, 17, -39. What is the maximum likelihood estimate of \( \theta \)?

Answer: 27


Question: Find the value of \( k \) such that \[ \lim_{n \to \infty} \left(\frac{n^{k-1}}{n^k - (n-1)^k}\right) = \frac{1}{2005}. \]

Answer: 2005


Question: Let $f$ be a non-degenerate bilinear form on a finite-dimensional vector space $V$ over $\mathbb{Z}_2$. Determine whether $f(x,x) = 0$ for every $x \in V$. Justify your answer.

Answer: No


Question:
For the series \( e^{\tan x} \), estimate the series by finding the best integer value \( k \) such that:
\[ e^{\tan x} = 1 + x + \frac{x^2}{2!} + o(x^k) \]
as \( x \to 0 \). Determine the largest integer \( k \) for which this statement holds true.

Answer: 2


Question: If \( \omega \) is a cube root of 1 and \( \alpha, \beta, \gamma \) are cube roots of \( p \), then for any real numbers \( x, y, z \), find the value of \( \frac{x\alpha + y\beta + z\gamma}{x\beta + y\gamma + z\alpha} \).

Answer: \omega^2


Question: Let \( \langle a_{n} \rangle \) be an increasing sequence of positive real numbers such that the series \( \sum_{k=1}^{\infty}{a_{k}} \) is divergent. Define \( s_{n} = \sum_{k=1}^{n} a_{k} \) for \( n = 1, 2, 3, \ldots \) and \( t_{n} = \sum_{k=2}^{n} \frac{a_{k}}{s_{k-1}s_{k}} \) for \( n = 2, 3, 4, \ldots \). Find \( \lim_{n \to \infty} t_{n} \).

Answer: \dfrac{1}{a_1}


Question: Suppose \(A\) is an open set in \(\mathbb{R}^n\) under the standard topology and is homeomorphic to a set \(B \subset \mathbb{R}^n\). Must \(B\) necessarily be open? Provide a justification for your answer.

Answer: Yes


Question:
Find all continuous functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the equation:
\[ f(xy) + f(x+y) = f(xy+x) + f(y) \quad \forall x, y \in \mathbb{R}. \]

Answer: f(x) = ax + b


Question: Evaluate the expression \(\left(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}\right)\cdot \left(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c}\right)\) given that \(a + b + c = 0\).

Answer: 9


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

Answer: 0


Question:
Consider a nonzero sequence of nonnegative real numbers \((u_n)_{n\in\mathbb{Z}}\) that satisfies a linear recurrence relation with constant coefficients:
\[
u_{n+r} = \sum_{j=0}^{r-1} a_j u_{n+j} \quad (n \in \mathbb{Z})\]
where \(a_0, a_1, \ldots, a_{r-1}\) are fixed real constants. If all the roots of the characteristic polynomial \(X^r - \sum_{j=0}^{r-1} a_j X^j\) have modulus greater than 1, does it follow that \(u_n \to +\infty\) as \(n \to +\infty\)?

Answer: Yes


Question: Solve the partial differential equation \( xu_x + u_y = x \sinh(y) + u \) with the initial condition \( u(0, y) = 0 \) using the method of characteristics. Verify if the solution is \( u = 0 \).

Answer: u = x \cosh(y)


Question: Does there exist a topologically transitive map $f: \mathbb{R}^n \to \mathbb{R}^n$ for $n \geq 2$?

Answer: Yes


Question: Given that the alternating subgroup \(\mathbb{A}_n\) has index 2 in the symmetric group \(\mathbb{S}_n\), does it follow that every element \(\sigma \in \mathbb{S}_n\) can be expressed as \(\sigma = \widetilde{\sigma}\alpha_{\sigma}\), where \(\widetilde{\sigma}\) is a fixed element in \(\mathbb{S}_n\) and \(\alpha_{\sigma} \in \mathbb{A}_n\)?

Answer: No


Question: Given an \( n \times n \) matrix \( A \), it is known that \( A \cdot \begin{bmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix} = \begin{bmatrix} p \\ p \\ \vdots \\ p \end{bmatrix} \) and \( A^* \cdot \begin{bmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix} = \begin{bmatrix} q \\ q \\ \vdots \\ q \end{bmatrix} \), where \( A^* \) is the adjugate matrix of \( A \), and \( A^{-1} \) exists. Find the value of \( \det(A) \).

Answer: pq


Question: Given the random variables $x$ and $y$ defined as follows: $x$ is sampled from a uniform distribution over the interval $[-1, 1]$, and $s$ is a random variable that takes the value $1$ with probability $\frac{1}{2}$ and $-1$ otherwise. The variable $y$ is defined as $y = sx$. Show that the covariance $\text{Cov}(x, y) = 0$. Derive this result using the expression $E[(x-E[x])(y-E[y])] = E[(x-E[x])(sx-E[sx])]$. Provide a detailed derivation.

Answer: 0


Question: Find the value of \( k \in \mathbb{N} \) such that the polynomial \( x^{2k} + 1 + (x+1)^{2k} \) is not divisible by \( x^2 + x + 1 \).

Answer: 3


Question: Find the smallest positive distance $d$ such that for any line in the plane not parallel to the $x$- or $y$-axis, there exists a lattice point whose distance to the line does not exceed $d$. Express your answer in simplest form.

Answer: \dfrac{\sqrt{2}}{4}


Question: Let \(a, b, c\) be real numbers such that \(a < 3\) and all the zeros of the polynomial \(x^3 + ax^2 + bx + c\) are negative real numbers. Determine the possible values of \(b + c\).

Answer: (0, 4)


Question:
Consider two Riemannian manifolds $(M,g)$ and $(N,g)$, and their Riemannian product manifold $M \times N$. Let $Ric_1(X_1,Y_1)$ denote the Ricci curvature on $M$ for vector fields $X_1$ and $Y_1$, and $Ric_2(X_2,Y_2)$ denote the Ricci curvature on $N$ for vector fields $X_2$ and $Y_2$. Determine which of the following expressions correctly represents the Ricci curvature $Ric$ on the product manifold $M \times N$:

1. $Ric(X_1+X_2,Y_1+Y_2)=Ric_1(X_1,Y_1)+Ric_2(X_2,Y_2)$
2. $Ric(X_1+X_2,Y_1+Y_2)=Ric_1(X_1,Y_1) \oplus Ric_2(X_2,Y_2)$

Choose the correct expression.

Answer: 1


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \frac{d\theta}{1+\sin^2(\theta)} \) using residue theory.

Answer: \pi \sqrt{2}


Question: Determine whether the following statement is true for $1 \leq p<\infty$: For $f \in L^p(\mathbb{R})$ and $q$ such that $1/p + 1/q = 1$, is it true that \( \|f\|_p = \sup \left\{ \int_{\mathbb{R}} fg \, d\mu : g \in L^q(\mathbb{R}), \|g\|_q \leq 1 \right\} \)? The integral is with respect to the Lebesgue measure.

Answer: True


Question: Find the minimum value of the expression \(x_1x_2 + x_2x_3 + \cdots + x_{2014}x_{2015} + x_{2015}x_1\) given that each \(x_i\) is in the interval \([-1, 1]\) for \(i = 1, 2, \ldots, 2015\).

Answer: -2013


Question: Determine the metric completion \( \hat{M} \) of the universal covering of the punctured plane \( \mathbb{C} \setminus \{0\} \).

Answer: \mathbb{C}


Question: Calculate the expected number of draws from a standard 52-card deck (with replacement) until you have exactly two Kings and two Jacks. Note that Kings and Jacks are not replaced once drawn, and the deck is shuffled after each draw. If you draw a third King or Jack, it is put back into the deck.

Answer: \dfrac{845}{21}


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

Answer: -1


Question: Find all positive integer pairs \((a, b)\) such that \(2^a + 17 = b^4\).

Answer: (6, 3)


Question: Let $G$ be a group with a normal subgroup $N$ such that the quotient group $G/N$ is cyclic of order 18. Prove that $G$ has a normal subgroup $H$ such that the index $|G:H| = 9$.

Answer: 9


Question: Calculate the value of the contour integral \( \oint_{|z|=2} \tan(z)\,dz \) using the residue theorem.

Answer: -4\pi i


Question: Find a complex number \( z \) such that \( z^3 = -8i \) and \( z^4 = -8 - 8i\sqrt{3} \).

Answer: \sqrt{3} - i


Question: Evaluate the integral \( \int_\pi^{3\pi} \frac{dx}{5\cos x+13} \) using the Residue Theorem.

Answer: \dfrac{\pi}{6}


Question: Evaluate the Riemann-Stieltjes integral: \( \int_{-1.2}^{3.9} x d[x] \).

Answer: 5


Question: If $G$ is a simple non-abelian group and $H$ is a subgroup with $[G:H]=7$, what is the number of conjugates of $H$ in $G$?

Answer: 7


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

Answer: 1


Question: Consider a solvable group $G$ of order $p^4 (p+1)(p^2+1)(p^4+1)$, where $p=2^n-1$ is a Mersenne prime. Determine whether $G$ can have an irreducible complex character of degree $p^4$.

Answer: No


Question: Determine the number of rotational symmetries of a dodecahedron. Consider the symmetries that include the identity rotation, rotations about axes through opposite edges, opposite faces, and opposite vertices. Verify that the total number of rotational symmetries is 60.

Answer: 60


Question: Let $f$ be an analytic function on the annulus $\{z : 1/2 < |z|<2\}$ except for a simple pole at $z=1$ with residue 1. The Laurent expansions of $f$ on the annuli $\{ z: 1/2<|z|<1\}$ and $\{z: 1<|z|<2\}$ are given by $\sum a_n z^n$ and $\sum b_n z^n$, respectively. Compute $b_n - a_n$ for every integer $n$. Provide your answer as a single expression or value.

Answer: 1


Question: In the hyperbolic plane, is it possible for a geodesic to intersect orthogonally two distinct geodesics $L$ and $L'$ that do not intersect each other at infinity?

Answer: Yes


Question: Evaluate the integral \( I = \frac{1}{2\pi i} \int_{\gamma} z^7 \cos \frac{1}{z^2} \, dz \), where \( \gamma = \{z \in \mathbb{C} : |z| = 2\} \) is oriented counterclockwise. Determine if the Cauchy Integral Theorem can be applied to conclude that the integral is zero.

Answer: \dfrac{1}{24}


Question: Let \( A \) be an \( n \times n \) matrix defined by \([a_{ij}]_{n\times n}=\frac{((-1)^i)(2i^2+1)}{4j^4+1}\). Compute the value of \(1+ \lim_{n\to \infty}\left(\text{tr}(A^n)^{1/n}\right)\).

Answer: \dfrac{1}{2}


Question: Assume that \(X\) and \(Y\) are independent random variables, each following a geometric distribution with parameter \(p\). Compute the conditional expectation \(E(X|X+Y=k)\) for all \(k = 2,3,4,\ldots\).

Answer: \dfrac{k}{2}


Question: Determine the number of positive integer pairs \((x, y)\) that satisfy the equation \(y^2 - x^2 = 2y + 7x + 4\).

Answer: 1


Question: Calculate the degree of the field extension \( \mathbb{Q}(\sqrt[80]{43},\sqrt[84]{43})/\mathbb{Q} \).

Answer: 1680


Question: Evaluate the integral: \[ \int_0^1 \left( \prod_{i=1}^n (x+i) \right) \left( \sum_{j=1}^n \frac{1}{x+j} \right) \, dx \] without using Gamma notation or other complicated integral methods.

Answer: n \cdot n!


Question: Let $q$ be a prime such that $q \equiv 1 \pmod{12}$. Find the sum of all solutions $\lambda_1, \lambda_2, \ldots, \lambda_k$ to the congruence $x^3 \equiv 8 \pmod{q}$, expressed modulo $q$. Provide your answer as a single integer modulo $q$. \( \text{Note: } q \equiv 1 \pmod{12} \text{ ensures that the equation has solutions.} \)

Answer: 0


Question:
Determine whether the barycenter of a convex plane curve is Lipschitz with respect to the Hausdorff distance. Specifically, for a convex curve $C$, define its barycenter as
$$b(C) = \frac{1}{\mathcal H^1(C)} \int\limits_C x \, d \mathcal H^1(x).$$
Is there a constant $L$ such that for any convex curves $C_1$ and $C_2$, the inequality $|b(C_1) - b(C_2)| \leq L \, d_H(C_1,C_2)$ holds?

Answer: Yes


Question: Estimate the number of ways to choose integers \( x_1, x_2, \ldots, x_k \) such that \( 1 \leq x_j \leq p \) for each \( j \), and there are at most \( n-1 \) distinct integers among them. Show that this number is at most \( n^k p^{n-1} \).

Answer: n^k p^{n-1}


Question: Determine whether the following statement is true: If $f \in L^{1}((0,1))$, then $\lim\limits_{n\to+\infty}\int_{0}^{\frac{1}{n}}f(x)\mathrm{d}x=0$.

Answer: True


Question: Let $X = \{(x,y) : x^2 + y^2 < 5\}$ and $K = \{(x,y) : 1 \leq x^2 + y^2 \leq 2 \text{ or } 3 \leq x^2 + y^2 \leq 4\}$. Determine the number of relatively compact connected components of $X \setminus K$ in $X$. Provide your answer as a single integer.

Answer: 2


Question: Given $P = \tan(3^{n+1}\theta) - \tan\theta$ and $Q = \sum_{r=0}^n \frac{\sin(3^r\theta)}{\cos(3^{r+1}\theta)}$, find the relationship between $P$ and $Q$.

Answer: P = 2Q


Question: Let \( F(x) = \int_0^x e^{e^t} \, dt \). Express \( \int_1^2 \frac{e^x}{x} \, dx \) in terms of \( F \).

Answer: F(\ln 2)


Question: Determine whether the matrices with minimal norm among the set of special linear matrices (matrices with determinant 1) are the special orthogonal matrices (matrices where \( A^TA = I_n \)).

Answer: Yes


Question: Determine the number of natural number solutions for the equation \( p(n) = n + 1 \), where \( p(n) \) follows the Fibonacci sequence.

Answer: 0


Question: For a random variable \( X \sim \mathcal{N}(-2, 2) \), find the expected value \( E(X^3) \).

Answer: -20


Question: The parabola \( y = 12 - 3x^2 \) intersects the x-axis at points \( A \) and \( B \). A horizontal line \( y = k \) (where \( 0 < k < 12 \)) intersects the parabola at points \( C \) and \( D \). Determine the value of \( k \) such that the area of the trapezoid \( ABCD \) is maximized.

Answer: \dfrac{32}{3}


Question: If a hypergraph $H=(V,E)$ is chromatically rigid, does there always exist a subset $E_0 \subseteq E$ such that $(V,E_0)$ is chromatically rigid and $E_0$ is minimal with respect to this property?

Answer: Yes


Question: Let $X$ be a separable Banach space and $\{x_n\}$ be a bounded, relatively weakly compact sequence in $X$. Define $y_n = \frac{1}{n}\sum_{i=1}^{n}{x_i}$. Suppose every weak limit point of $\{y_n\}$ is equal to some element $y \in X$. Does it follow that $\{y_n\}$ converges weakly to $y$?

Answer: Yes


Question: Let \( f(z) = \frac{z+1-i}{z-1+i} \) be a map. Determine the image of \( f(S) \), where \( S = \{ z \in \mathbb{C} \mid \text{Im}(z) > \text{Re}(z) \} \).

Answer: \{ w \in \mathbb{C} \mid |w| < 1 \}


Question: Use a surface integral to find the surface area of the surface defined by \( S = \{(r\cos\theta, r\sin\theta, 3-r) : 0 \leq r \leq 3, 0 \leq \theta \leq 2\pi \} \).

Answer: 9\sqrt{2}\pi


Question: Is the expression \( \langle X,Y\rangle = \int_0^1 \int_0^1 X^1(x,y)Y^1(x,y) + X^2(x,y) Y^2(x,y) \, dx \, dy \) the standard definition of the inner product for the space \( L^2([0,1]^2, \mathbb{R}^2) \), where \( X = (X^1(x,y), X^2(x,y)) \) and \( Y = (Y^1(x,y), Y^2(x,y)) \)?

Answer: Yes


Question: Given the sequences \( a_m = \sum_{k=0}^{m} (\sqrt{k} + \sqrt{k+1})^{-1} \) and \( b_m = \sum_{k=0}^{m} (\sqrt{k} + \sqrt{k+2})^{-1} \), define \( c_n = \prod_{m=0}^{n} \frac{a_m}{a_m - 2(b_m - b_{m+1})} \). Find the smallest natural number \( n \) for which \( c_n < 0.01 \).

Answer: 139


Question: Determine the value of \( \Gamma(0^+) \). Is it \(-\infty\) or \(+\infty\)?

Answer: +\infty


Question: Determine the asymptotic behavior of the recurrence relation \( T(n) = T(n-1) + (n-1)! \cdot n^k \) in terms of \( \Theta \) or \( O \) notation, where \( k \) is usually 0, 1, or 2.

Answer: \Theta(n! \cdot n^{k-1})


Question: Evaluate the summation \( \sum^{2020}_{r=0} (-1)^{r} (r + 1) {2020 \choose r} \).

Answer: 0


Question: Find the complex power series expansion of \( \frac{e^z}{1+z} \) centered at \( z=0 \) and determine its radius of convergence.

Answer: 1


Question: Assume the Generalized Continuum Hypothesis (GCH). Calculate \( \aleph_{\omega_1}^{\aleph_0} \).

Answer: \aleph_{\omega_1}


Question: In a Poisson process with parameter \( \lambda \), let \( S_1, S_2, \ldots \) be the arrival times. Given the time of the \( n \)-th arrival, find the expected value \( E(S_1|S_n) \).

Answer: \dfrac{S_n}{n}


Question: Find the minimum value of the expression \(2a^8 + 2b^6 + a^4 - b^3 - 2a^2 - 2\), where \(a\) and \(b\) are real numbers.

Answer: -\dfrac{11}{4}


Question: Suppose $X$ and $Y$ are sets, and $L^0(\sigma(X)) \subset L^0(\sigma(Y))$, where $L^0(\sigma(X))$ and $L^0(\sigma(Y))$ are the sets of measurable functions with respect to the $\sigma$-algebras generated by $X$ and $Y$, respectively. Can you conclude that $X \subseteq Y$?

Answer: No


Question: Let $R$ be a polynomial ring $R=k[X_1,X_2, \ldots ,X_n]$. Let $I$ be an ideal of $R$ such that any two elements of $I$ have a non-constant gcd. Does it follow that there is a non-constant polynomial $D$ dividing all the elements of $I$?

Answer: Yes


Question: Find the number of non-negative integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 = 12\) given that \(x_1 = 2x_2 + 2\) and \(x_3 \le x_4\).

Answer: 14


Question: Evaluate the expression: \( \sqrt{2 + \sqrt{3} - \sqrt{4 + \sqrt{5} + \sqrt{17 - 4\sqrt{15}}}} \)

Answer: 1


Question: Determine the minimum capacity required for each of 10 boxes such that in 90% of the games, no box overflows. In each of the 50 trials, 4 boxes are selected at random, and one ball is placed into each selected box. There are 200 balls in total. What is the minimum capacity of each box?

Answer: 28


Question: Provide an example of a bipartite connected graph with an even number of vertices that has an Eulerian circuit but does not have a perfect matching.

Answer: K_{2,4}


Question: Find the smallest positive integral solution (x, y, z) to the equation \( \frac{1}{x^2} + \frac{1}{y^2} = \frac{1}{z^2} \).

Answer: (15, 20, 12)


Question: Let $M$ and $N$ be closed, connected, smooth manifolds that are homotopy equivalent. Must $M$ and $N$ have the same dimension?

Answer: Yes


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

Answer: e


Question: Show that if $f$ is differentiable and $f''(0)$ exists, then \[ \lim _{h \rightarrow 0} \frac{f(h)-2f(0)+f(-h)}{h^2}=f''(0). \] Do not use L'Hospital's rule in your proof.

Answer: f''(0)


Question: Is the Sobolev inequality valid for a cube in $\mathbb{R}^n$? Specifically, determine if there exists a constant $C > 0$ such that for all $u \in W^{1,p}(Q)$, the inequality \( ||u||_{L^{p^\ast}(Q)} \leq C ||u||_{W^{1,p}(Q)} \) holds, where $Q$ is a cube in the maximum norm.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k=0}^n \frac{\binom{n}{k}}{k2^n+n} \]

Answer: 0


Question: Let \( K = \mathbb{R}(X) \) be the rational function field, and \( F = \mathbb{R}(X^4 - \frac{1}{X^4}) \). Let \( L \) be the Galois closure of the extension \( K/F \). Determine the extension degree \([L:F]\).

Answer: 16


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

Answer: \sqrt{2} - 1


Question: Given a quadratic function \( f(x) \), define \( F(x) = \int_0^x f(t) \, dt \). Assume \( a \) is a positive number and \( F(x) \) has extreme values at \( x = -2a \) and \( x = 2a \). Find the local maximum value of the function \( \frac{F(x)}{F'(0)} \).

Answer: \dfrac{4a}{3}


Question: Let $B$ be a Banach space with a continuous projection onto a closed subspace $X \subseteq B$. Does there exist a closed subspace $Y \subseteq B$ such that $X \cap Y = \{0\}$ and $X + Y = B$?

Answer: Yes


Question: Solve for \(n\) in the equation: \[ 2^{2n-3} = 32(n-1) \]

Answer: 5