Question: Evaluate the limit: $$\lim_{x\to0} \frac{f(e^{5x} - x^2) - f(1)}{x}$$ given that $f'(1) = -2$. Use the information provided to find the limit.

Answer: -10


Question: Simplify the expression: \[ \tan(\theta) + 2\tan(2\theta) + 2^2\tan(2^2 \theta) + \cdots + 2^{14} \tan(2^{14}\theta) + 2^{15} \cot(2^{15} \theta) \]

Answer: \cot \theta


Question:
Suppose \( B=\{(A_1,A_2,A_3) \mid \forall i; 1\le i \le 3; A_i \subseteq \{1,\ldots,20\}\} \). If we have:
\[ T=\sum_{(A_1,A_2,A_3)\in B} |A_1\cup A_2 \cup A_3| \]
What is the value of \( T \)?

(a) \( 2^{59}-2^{56} \)
(b) \( 2^{60}-2^{57} \)
(c) \( 20(2^{59}-2^{56}) \)
(d) \( 20(2^{60}-2^{57}) \)

Answer: D


Question: How many bijective functions \( f: \mathcal{A} \rightarrow \mathcal{A} \) exist such that for each \( x \in \mathcal{A} \), \( f(x) \neq x \) and \( f(f(f(x))) = x \), where \( \mathcal{A} = \{1, 2, \ldots, 12\} \)?

Answer: 246400


Question: If $P$ and $Q$ are invertible matrices such that $PQ = -QP$, what can be said about the traces of $P$ and $Q$? Choose the correct option: (A) Both traces are 0, (B) Both traces are 1, (C) $\text{Tr}(Q) \neq \text{Tr}(P)$, (D) $\text{Tr}(Q) = -\text{Tr}(P)$. Provide a justification for your choice.

Answer: A


Question: Let $G$ be a group and $g_i \in G$ for $i = 1, \ldots, n$. For any permutation $\sigma \in S_n$, is it true that $g_1g_2\ldots g_ng^{-1}_{\sigma(1)}\ldots g^{-1}_{\sigma(n)} \in G'$?

Answer: Yes


Question: Determine the number of elements of order 9 in the group \(C_{135} \times C_6 \times C_9\).

Answer: 216


Question: If $q^k n^2$ is an odd perfect number with special prime $q$, is $\sigma(q^k)$ coprime to $\sigma(n^2)$? Assume $q \equiv k \equiv 1 \pmod{4}$ and $\gcd(q,n)=1$. The function $\sigma(x)$ denotes the sum of divisors of $x$. Determine if the conjecture $\gcd(\sigma(q^k),\sigma(n^2))>1$ holds true.

Answer: \gcd(\sigma(q^k), \sigma(n^2)) > 1


Question: In a game, the expression \( \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box \) is written on the blackboard. Two players, A and B, take turns. Player A goes first. On each turn, a player replaces a symbol \( \Box \) with a positive integer. After all symbols \( \Box \) are replaced, player A replaces each \( \pm \) with either + or -. Player A wins if the resulting expression is not divisible by any of the numbers 11, 12, ..., 18. Otherwise, player B wins. Determine which player has a winning strategy.

Answer: A


Question: Consider the recursive sequence \( \{a_n\} \) defined by \( a_1 = 0 \) and \( a_{n+1} = \left(\frac{1}{4}\right)^{a_n} \) for \( n \ge 1 \). Determine the limit \( l \) of the sequence as \( n \to \infty \).

Answer: \dfrac{1}{2}


Question: Given the function \( f(x) = (x-1)(x-2)(x-3)(x-4) \), prove that the derivative \( f'(x) \) has 3 distinct real roots.

Answer: 3


Question: Find the volume of the region in \( \mathbb{R}^3 \) bounded by the surfaces \( y = x^2 \), \( x = y^2 \), \( z = 0 \), and \( z = xy \).

Answer: \dfrac{1}{12}


Question: Is it possible to find a 1-dimensional nonsingular foliation on an orientable surface with one boundary component such that lines of the foliation are transverse to the boundary? Answer yes or no.

Answer: No


Question:
Given the matrix \( A = \begin{bmatrix} 4 & 1 & 0 \\ 2 & -1 & 1 \\ 0 & 2 & 1 \end{bmatrix} \) and the polynomial \( P(x) = x^3 - 3x + 4 \), calculate the determinant \( | P(A) + 2I | \). Choose the correct answer from the following options:

(a) 1648
(b) 1624
(c) 1528
(d) 1728
(e) 1696

Answer: A


Question:
Evaluate the limit of the expression as \( M \to \infty \):
\[
\lim_{M \rightarrow \infty} \frac{1}{\sum_{i=0}^{\infty} \frac{M!}{(M+i)!} x^i}
\]
where \( x \) is a positive real constant \( (x > 0) \).

Answer: 1


Question: Calculate the area enclosed by the curve \( \sqrt{|x|} + \sqrt{|y|} = \sqrt{a} \) for \( a = 1 \). Verify if the area is \( \frac{2}{3} \).

Answer: \dfrac{2}{3}


Question: Determine the values of $a > 0$ for which the series \( \sum_{n=1}^{\infty} \left(\frac{a n+2}{3n+1}\right)^{n} \) converges.

Answer: (0, 3)


Question: Express the ideal $(8)$ as a product of prime ideals in the ring $\mathbb{Z}[\zeta_{18}]$. Given that the minimal polynomial of $\zeta_{18}$ is $x^6 - x^3 + 1$, determine the factorization.

Answer: (2)^3


Question: For what values of \( \gamma \in \mathbb{C} \) do there exist non-singular matrices \( A, B \in \mathbb{C}^{n \times n} \) such that \( AB = \gamma BA \)?

Answer: \gamma^n = 1


Question: Solve the recurrence relation \( T(n) = n \cdot T(n/2)^2 \). Determine the asymptotic behavior of \( T(n) \).

Answer: \Theta\left(\frac{4^n}{n}\right)


Question: Estimate the sum \( \sum^{\infty}_{k=1} \frac{12(-1)^{k+1}}{k^2} \) using the alternating series test to within an error of \( \frac{2}{5} \). Which of the following is an estimate of the sum \( L \) within this error margin? Choices: 10, 9, 12, 11, \( \frac{31}{3} \).

Answer: 10


Question: Thirty people are seated at three round tables, each with ten seats, at an alumni dinner. After each time interval \( \Delta t \), everyone changes position simultaneously to sit next to different people on their left and right. What is the minimum number of seating configurations required for everyone to have sat next to every other person at least once? If it is not possible for everyone to sit next to each other just once, specify the minimum number of configurations needed for everyone to sit next to each other at least once.

Answer: 15


Question: Evaluate the integral \( I_n = \int_0^1 [\ln(x)]^n \, dx \) and verify that it equals \((-1)^n n!\).

Answer: (-1)^n n!


Question: Calculate the multivariable limit: \[ \lim_{(x,y) \rightarrow (0,0)} \frac{x^2-y^2}{\sqrt{x^2+y^2}} \]

Answer: 0


Question: Evaluate the limit: $$\lim\limits_{x\to \infty}\left ( \frac{1-3x}{1-2x} \right )^{\frac{-2x+1}{x}}$$

Answer: \dfrac{4}{9}


Question: Given the function \( h(x) = e^{2x} + x^3 \), find the value of \( (h^{-1})'(1) \).

Answer: \dfrac{1}{2}


Question: Find the area of the polygon with vertices defined by the roots of the equation \( \sqrt{7}+3i-x^{2n}=0 \) as \( n \to \infty \).

Answer: \pi


Question: Determine the radius of convergence of the series \( \sum_{n=1}^\infty a_n x^n \), where \((a_n)_n\) is a bounded sequence with \(L((a_n)_n) \subseteq \mathbb{R} \setminus \{0\}\).

Answer: 1


Question: Let $\mu$ be the Cantor measure on the Cantor set $C$ in $[0,1]$. Determine if the space of functions obtained by restricting smooth compactly supported functions to $C$ is dense in $L^p(\mu)$ for $1 \leq p < \infty$. Provide a justification for your answer.

Answer: Yes


Question:
Given the two-dimensional distribution of the random variables \(\xi\) and \(\eta\) specified by the distribution function:
\[
\mathbb{P}(\xi \leq x,\; \eta \leq y)  = F(x,y) = \begin{cases} 0 & \textrm{ if } \min(x,y) <0 \\ \min(x,y) & \textrm{ if } 0 \leq \min(x,y) \leq 1 \\ 1 & \textrm{ if } \min(x,y) > 1 \end{cases}.
\]
Find \(\mathbb{P}[ (\xi-\frac{1}{2})^2 + (\eta -\frac{1}{2})^2 \leq \frac{1}{4}].\) Assume \(\xi\) and \(\eta\) are independent and uniformly distributed over \([0,1].\)

Answer: \dfrac{\pi}{4}


Question: Consider a nonzero representation $(f,V)$ of the reductive group $GL_n(\mathbb{Q}_p)$, where $V$ is a complex vector space. Define the smooth subrepresentation $V' = \{v \in V \mid \text{Stab}(v) \text{ is open in } GL_n(\mathbb{Q}_p)\}$. Is it possible to endow $V$ with a topology such that $V'$ is dense in $V$?

Answer: Yes


Question:
Determine which of the following metric spaces $X$, with the metric induced by the usual Euclidean metric on $\Bbb R^n$, are complete:

A. $X=\Bbb Z \times \Bbb Z \subset \Bbb R \times \Bbb R$

B. $X=\Bbb Q \times \Bbb R \subset \Bbb R \times \Bbb R$

C. $X=(-\pi,\pi) \cap \Bbb Q \subset \Bbb R$

D. $X=[-\pi,\pi] \cap (\Bbb R - \Bbb Q) \subset \Bbb R$

Select the correct option(s).

Answer: A


Question: Compute $2^{65536} \mod 49$ using a method that leverages the known result of $2^{65536} \mod 7$. Provide the final result of the computation.

Answer: 23


Question: Evaluate the infinite product: \[ 3 \cdot 9^{\frac{1}{2}} \cdot 27^{\frac{1}{4}} \cdot 81^{\frac{1}{8}} \cdots \]

Answer: 81


Question: Let \( \mathcal{F} \) be a finite set of matrices in \( \mathcal{M}_n(\mathbb{R}) \) for some fixed \( n \). A set \( \mathcal{F} \) is called jointly nilpotent if there exists a finite product of elements in \( \mathcal{F} \) that equals zero, i.e., \( C_1C_2\ldots C_r=0 \) for some \( r \) and some \( C_i \in \mathcal{F} \). The smallest such \( r \) is called the nilpotency index of \( \mathcal{F} \). Define \( m(t) \) as the largest nilpotency index of a jointly nilpotent \( t \)-element set. Given that \( m(1)=n \), determine the values of \( m(t) \) for \( t \geq 2 \).

Answer: n


Question: Let \( U \) be a random variable uniformly distributed on \([0,1]\) and let \( V = U(1-U) \). Calculate the conditional expectation \( E(U | V) \).

Answer: \dfrac{1}{2}


Question: Let $X$ and $Y$ be Banach spaces, and let $U$ be an open set in $X$. For a fixed $y \in Y$, is the set $U \times \{y\}$ open in the product space $X \times Y$?

Answer: No


Question: Let $A \subseteq B$ be a ring extension where $A$ and $B$ are both finitely generated $\mathbb{C}$-domains of the same Krull dimension, and $A$ is regular. If $P$ is a prime ideal of $A$ with finitely many prime ideals of $B$ lying over it, does there exist an element $f \in A \setminus P$ such that every prime ideal of $A_f$ has finitely many prime ideals of $B$ lying over it?

Answer: Yes


Question: Suppose $(X, T_1)$ and $(X, T_2)$ are two topological spaces on the same set $X$. It is given that a sequence $\\{x_n\\}$ converges in $T_1$ if and only if it converges in $T_2$. Does this imply that $T_1 = T_2$? Provide a justification for your answer.

Answer: No


Question: Let $f$ be a continuous function on $[-1,1]$. Evaluate the limit: \[ \lim_{n\rightarrow \infty}n \int_{-\frac{1}{n}}^{\frac{1}{n}} f(x)(1-n|x|)dx \]

Answer: f(0)


Question: Given the matrix \( M = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix} \), compute the value of \( \frac{1}{e} \sum_{i=1}^{3} \sum_{j=1}^{3} b_{ij} \) where \( e^M = I + M + \frac{1}{2!}M^2 + \cdots \) and \( e^M = [b_{ij}] \).

Answer: \dfrac{11}{2}


Question: If $S \subset A^{\times}$, what is $S^{-1}A$ isomorphic to?

Answer: A


Question: Does there exist a family of regular dense graphs with girth at least 5, where every maximal independent set has a size of at least \( \Omega(n) \)? Assume the graphs have \( n \) nodes and \( m \ge n^{1+\epsilon} \) edges for some constant \( \epsilon > 0 \).

Answer: Yes


Question:
Evaluate the complex integral along the curve \( \gamma(t) = 2e^{it} \) for \( t \in [0, 2\pi] \):
\[ \int _\gamma \frac{2z}{z^2-2}dz \]

Answer: 4\pi i


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{1-\cosh(x^5)}{x^6\sin^2(x^2)} \]

Answer: -\dfrac{1}{2}


Question: Find a tight upper bound for the function \( f(n) \) defined by the recurrence relation \( f(n) = f(\sqrt{n}) + \frac{1}{n} \).

Answer: O(1)


Question: Calculate the expected number of draws needed to obtain a sum divisible by 3 when drawing numbers from a bag containing \(1, 3, 5, 7, 9\), with replacement after each draw.

Answer: 3


Question: Let \( \{f_n\} \) be a sequence of Lebesgue measurable functions on \([0,\infty)\) such that \( |f_n(x)| \le e^{-x} \) for all \( x \in [0,\infty) \). If \( f_n \rightarrow 0 \) almost everywhere, does \( f_n \rightarrow 0 \) almost uniformly?

Answer: Yes


Question: Determine the Hausdorff dimension of the set of rational numbers within a given interval on the real number line.

Answer: 0


Question: Evaluate the integral \( \int_C (x - y)(dx + dy) \) using Green's Theorem, where \( C \) is the semicircular part of \( x^2 + y^2 = 4 \) above the line \( y = x \), from \((-\sqrt{2}, -\sqrt{2})\) to \((\sqrt{2}, \sqrt{2})\).

Answer: -4\pi


Question: Prove that \( \operatorname{Hom}(\operatorname{Hom}(\mathbb{Q}/\mathbb{Z}, \mathbb{Q}), \mathbb{Z}) \cong \{0\} \) as \( \mathbb{Z} \)-modules.

Answer: \{0\}


Question: Evaluate the limit \( \lim_{n\to\infty}\frac{\sum_{k=1}^n s_k\mu(k)}{n} \), where \( s_k = 1 + \frac{2\pi k}{\log 2}i \) are the zeros of the Dirichlet eta function and \( \mu(k) \) is the Möbius function. Analyze the convergence and compute the limit.

Answer: 0


Question: Evaluate the limit: \[ \lim_{\alpha\to 0^+}\int_0^{\frac {1}{\alpha}}\sin(\alpha x) \,\mathrm{d}x \]

Answer: \infty


Question: Given the probability density function \( f(y) = e^y \) for \( y < 0 \) and \( f(y) = 0 \) elsewhere, find the expectation \( E[e^{\frac{3Y}{2}}] \).

Answer: \dfrac{2}{5}


Question: Evaluate the limit: \[ \lim_{x\rightarrow+\infty}(\sqrt[5]{x^5-x^4}-x) \]

Answer: -\dfrac{1}{5}


Question: Evaluate the surface integral \( \iint_\Sigma \mathbf{f} \cdot d \mathbf{a} \) where \( \mathbf{f}(x,y,z)=(x^2,xy,z) \) and \( \Sigma \) is the part of the plane \( 6x+3y+2z=6 \) with \( x,y,z\geq 0 \).

Answer: \dfrac{7}{4}


Question: Determine whether there exists a fixed positive integer \( n \) such that \( 2^{2017} \mid n^n - 2017 \).

Answer: Yes


Question: Let $X_1$ and $X_2$ be independent and identically distributed random variables. Find the probability $P(X_1 < X_2)$ using transformations.

Answer: \dfrac{1}{2}


Question: Find the determinant of the principal matrix solution \( \Pi(t,t_0) \) for the system of differential equations \( \dot{x} = A(t)x \) in \( \mathbb{R}^2 \), where the matrix \( A(t) \in GL(\mathbb{R}^2) \) has eigenvalues \( \alpha_1(t) = t \) and \( \alpha_2(t) = 0 \). Calculate \( \det(\Pi(1,0)) \).

Answer: \sqrt{e}


Question: Determine the value of \( b \) such that the equation \( x^3 + a(a+1)x^2 + ax - a(a+b) - 1 = 0 \) has a root that is independent of \( a \).

Answer: 2


Question: Evaluate the limit of the sequence \( b_n = \frac{n}{\sum_{i=1}^{n}\frac{1}{a_i}} \) where \( a_n = (2^n+3^n)^{1/n} \).

Answer: 3


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{n+n^{\frac{1}{2}}+n^{\frac{1}{3}}+\ldots+n^{\frac{1}{n}}}{n} \]

Answer: 2


Question: Let $f$ be a differentiable function defined on $[1, \infty)$ with a continuous derivative, satisfying the equation \( f'(x) = \frac{1}{x + 2010f(x)^2} \) for \( x \geq 1 \) and \( f(1) = 0 \). Determine \( \lim_{x \to \infty} \frac{f(x)}{x} \).

Answer: 0


Question: Evaluate the limit: $$ \lim_{(x,y) \to (1,0)} \frac{(x-1)^2\ln(x)}{(x-1)^2 + y^2} $$

Answer: 0


Question: Compute the value of the improper integral: $$\int_{0}^{\infty }\frac{(\ln x)^{40021}}{x}dx$$.

Answer: 0


Question: Find the value of $m + 2n$ for integer pairs $(m, n)$ that satisfy the equation \[ \frac{m^2 + mn + n^2}{m + 2n} = \frac{13}{3}. \]

Answer: 9


Question: Consider the sequence of random variables $X_n$ where $X_n \sim \mathrm{Normal}(1/n, 1/n)$. Determine if $X_n$ converges in distribution to a random variable $X$. If it does, specify the distribution of $X$.

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\ldots+\sqrt{n}-\frac{2}{3}n\sqrt{n}}{\sqrt{n}}. \]

Answer: \dfrac{1}{2}


Question: Calculate the flux of the vector field \( F = \frac{11x}{(x^2+y^2+z^2)^{3/2}} \vec{i} + \frac{11y}{(x^2+y^2+z^2)^{3/2}} \vec{j} + \frac{11z}{(x^2+y^2+z^2)^{3/2}} \vec{k} \) through the surface \( S: x^2+y^2+z^2 = 16 \).

Answer: 44\pi


Question: Determine whether the differential equation \( y'=(y-1)(y+1) \) with the initial condition \( y(0)=1 \) has a particular solution. The general solution is given by \( \frac{1-e^{2x+2C}}{1+e^{2x+2C}} \).

Answer: y = 1


Question: Let $s, u \in M_m(\mathbb{k})$ be a pair of commuting matrices where $s$ is a diagonal matrix and $u$ is a strictly triangular matrix. Define $a = s + u$. Determine whether there exists a polynomial $f(x) = c_1x + \cdots + c_dx^d \in \mathbb{k}[x]$, without a constant term, such that $s = f(a)$ holds as a matrix equality, where $f(a) := c_1a + \cdots + c_da^d$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of solutions to the congruence \(x^{p+1} \equiv 1 \mod p^{2017}\) in the set \(\{0, 1, \ldots, p^{2017} - 1\}\), where \(p\) is a prime number greater than 2.

Answer: 2


Question: Let $R = \mathbb{Z}$ and $S = \mathbb{Z}[i]$. Suppose $p \in R \setminus \{2\}$ and the ideal $Q \triangleleft S$ lies over $(p) \triangleleft R$. Show that the ramification index $e(Q|p) = 1$.

Answer: 1


Question: Find the continuous function \( f: \mathbb{R} \to \mathbb{R} \) that satisfies the functional equation \( f(2x) = \frac{f(x) + x}{2} \) for all \( x \in \mathbb{R} \).

Answer: \dfrac{x}{3}


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

Answer: 0


Question: Let F be a free group of rank n > 1. Prove that F and its abelianization F/F' have the same rank.

Answer: n


Question: Let $R$ be a ring and $M$ a semisimple left $R$-module such that $M = \bigoplus_{i \in I} M_i$, where each $M_i$ is a simple module. Is the length of $M$ equal to the cardinality of the index set $I$?

Answer: Yes


Question: Find the value of \( \lim\limits_{n\to\infty}\sum\limits_{k=2}^n \log_\frac 13 \left( 1-\frac{2}{k(k+1)}\right) \).

Answer: 1


Question: Is the space \((C[0,1], \lVert \cdot \rVert_1)\), where \(C[0,1]\) is the space of continuous functions on the interval \([0,1]\) and \(\lVert f \rVert_1 = \int_0^1 |f(x)| \, dx\) for \(f \in C[0,1]\), a countable union of nowhere dense sets?

Answer: Yes


Question: Calculate the path integral of the electric field \( E(x) = -kq\left(\frac{x}{(x^2+y^2)^{\frac{3}{2}}},\frac{y}{(x^2+y^2)^{\frac{3}{2}}}\right) \) along the straight line connecting the points \((1,0)\) and \((0,1)\).

Answer: 0


Question: Given the number \( n = 2^{14} \times 3^{9} \times 5^{8} \times 7^{10} \times 11^{3} \times 13^{5} \times 37^{10} \), find the number of positive divisors of \( n \) that are both perfect squares and divisible by \( 2^{2} \times 3^{4} \times 5^{2} \times 11^{2} \).

Answer: 9072


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \frac{1}{\sqrt{x}}\left(e^x + \frac{2\log(\cos(x))}{x^2}\right) \] without using l'Hopital's rule, Taylor theorem, or little/big o notation, if possible. You may use the following limits: \( \lim_{x \to 0^+} \frac{\log(x + 1)}{x} = 1 \), \( \lim_{x \to 0^+} \frac{e^x - 1}{x} = 1 \), and \( \lim_{x \to 0^+} \frac{1 - \cos(x)}{x^2} = \frac{1}{2} \).

Answer: 0


Question: Calculate \( \lim _{ n\to \infty } \int _{ |x|<n }{ \int _{ |y|<n }{ \sin(x^2+y^2)\,dx\,dy } } \).

Answer: \pi


Question:
Evaluate the inverse Fourier transform given by the integral:
\[ f(x) = \int_{-\infty}^\infty \frac{dk}{2\pi} e^{-ikx} \frac{\sin^2(k/2)}{(k/2)^2} \]
Determine the function \( f(x) \) for \( x \in [-1, 1] \).

Answer: 1 - |x|


Question: Let $K$ be the field $\mathbb{Q}(\sqrt{-15})$, and let $R = \mathcal{O}_K$ be the ring of integers of $K$. Define $\alpha = \frac{-1+\sqrt{-15}}{2}$. Consider the prime ideals $p = (2, \alpha)$ and $q = (17, \alpha + 6)$ of $R$. Construct an element of $R$ that admits two distinct factorizations into irreducibles using the prime ideals $p$ and $q$. Provide the element and its factorizations.

Answer: 34


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{10^x-2^x-5^x+1}{x\tan x} \]

Answer: (\ln 2)(\ln 5)


Question: Compute the value of the infinite product: \( \prod_{n=1}^\infty \mathrm{e}^{\mathrm{i}\pi/2^n} \).

Answer: -1


Question:
Let \((c_{n})_{n}\) and \((d_{n})_{n}\) be two sequences of complex numbers, and let \(N\) be a large natural number. Determine if the following inequality holds:
\[ \left|\sum_{n=1}^{N} c_{n}d_{n} \right| \leq \max_{n=1,\dots,N} |c_{n}| \left| \sum_{n=1}^{N} d_{n} \right|. \]

Answer: No


Question: Evaluate the series \(\sum_{n=1}^{\infty} \frac{n^2+2n+3}{2^n}\).

Answer: 13


Question: Calculate the expected value of the quantity \( Z_n := \sum_{i=1}^{n} (X_i - \bar{X})^2 \), where \( X_1, X_2, \ldots, X_n \) are i.i.d. random variables with mean \( \mu \) and variance \( \sigma^2 \), and \( \bar{X} \) is the sample mean. Express your answer in terms of \( n \), \( \mu \), and \( \sigma^2 \).

Answer: (n-1)\sigma^2


Question: Evaluate the limit: \[ l = \lim_{x\to 0}\frac{e^{\sin x}-e^{\tan x}}{e^{\sin 2x}-e^{\tan 2x}} \]

Answer: \dfrac{1}{8}


Question: Given the joint probability density function \( f(x, y) = 2 \) for \( 0 < x < y < 1 \) and \( f(x, y) = 0 \) otherwise, find the conditional probability \( P(0 < X < 1/2 \mid Y = 3/4) \).

Answer: \dfrac{2}{3}


Question: Evaluate the limit \( \lim_{x\rightarrow 0}\frac{\cos (\tan x)-\cos x}{x^4} \) without using L'Hôpital's Rule or series expansion.

Answer: -\dfrac{1}{3}


Question:
Find all polynomials \( f \in \mathbb{R}[x] \) such that:

1. \( xf(x-1) = (x-3)f(x) \)
2. \( x(x-1)f(x+1) = (x+2)(x+1)f(x-1) \)

Answer: 0


Question: Suppose $X_1, X_2, \ldots, X_n$ are non-constant real random variables such that $X_1 + X_2 + \cdots + X_n = 0$. Determine the largest possible independence depth of the set $\{X_1, X_2, \ldots, X_n\}$. Note that the independence depth is defined as the maximal number $k$ such that any $k$ distinct elements of the set are mutually independent.

Answer: 1


Question: Suppose $f \in L^\infty(\mathbb{R})$ and $f_h(x) = f(x + h)$ satisfies $$\lim_{h \to 0} \|f_h - f\|_\infty = 0.$$ Does there exist a uniformly continuous function $g$ on $\mathbb{R}$ such that $f = g$ almost everywhere?

Answer: Yes


Question: Solve the equation \( \frac{4}{3} \left( \frac{1}{x} - \lfloor \frac{1}{x} \rfloor \right) = x \) for a rational number \( x \) such that \( 0 < x < 1 \). Here, \( \lfloor \cdot \rfloor \) denotes the floor function.

Answer: \dfrac{2}{3}


Question: Given that \( \sum_{r=1}^n t_r = n(n+1)(n+2)(n+3) \), find the value of \( \lim_{n\to\infty} \sum_{r=1}^n \frac{1}{t_r} \).

Answer: \dfrac{1}{16}


Question: Compute the largest squarefree positive integer \( n \) such that \( \mathbb{Q}(\sqrt{-n})\cap \overline{\mathbb{Z}} \) is a unique factorization domain.

Answer: 163


Question: Determine whether there exists a natural number \( k \) such that all terms \( a_n \) of the sequence defined by \( a_1 = k \) and \( a_n = 2a_{n-1} + 1 \) for \( n \geq 2 \) are composite numbers.

Answer: Yes


Question: Find all integer solutions to the equation \( \cos(\sqrt{n^2-1}) = \frac{1}{n} \).

Answer: 1


Question: Determine the values of $p$ for which the integral \( \int_0^\infty \frac{\log(x)}{1+x^p} \, dx \) converges.

Answer: p > 1


Question: Let $S$ be a Boolean topos and $f : E \rightarrow S$ be a hyperconnected geometric morphism. For $s$ in $S$ and $x$ in $E$, let $\pi : f^* s \times x \rightarrow x$ be the projection in $E$. If $u \rightarrow f^* s \times x$ is a complemented subobject, is the image of $u$ along $\pi$ a complemented subobject of $x$?

Answer: Yes


Question: In how many ways can the numbers 1 to 12 be arranged on the sides of a square, with 5 places on each side (20 places total), leaving 8 places empty?

Answer: \dfrac{20!}{8!}


Question:
Consider the set \( U \) of all \( n \times n \) unitary matrices with complex entries. As a topological subspace of \( \mathbb{C}^{n^2} \), determine which of the following properties \( U \) possesses:

- Compact but not connected.
- Connected but not compact.
- Connected and compact.
- Neither connected nor compact.

Answer: C


Question: Consider a sequence \((M_n)_{n\in \mathbb{N}}\) of non-empty and compact subsets of \(\mathbb{C}\) such that \(M_1 \supset M_2 \supset M_3 \supset \ldots\). Define the diameter of a set \(M\) as \(d(M) := \max\{|z-w| : z,w \in M\}\). Suppose the sequence satisfies \(\lim_{n \to \infty} d(M_n) = 0\). Does this imply that \(\lim_{n \to \infty} |M_n| = 1\), where \(|M_n|\) denotes the cardinality of \(M_n\)?

Answer: No


Question: Solve the partial differential equation \( u_x + u_y = 1 \) with the boundary condition \( u(y, \frac{y}{2}) = y \).

Answer: y


Question: Evaluate the integral: \[ \int_0 ^{\pi} \left(\frac{\pi}{2} - x\right)\sin\left(\frac{3x}{2}\right)\csc\left(\frac{x}{2}\right) \, dx \]

Answer: 4


Question:
Given the equations:
\[ \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} = 9, \]
\[ \frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} = 32, \]
\[ \frac{a^3}{b+c} + \frac{b^3}{c+a} + \frac{c^3}{a+b} = 122, \]
find the value of \(abc\).

Answer: 13


Question:
Let \( f(x) = x^3 - x^2 - 3x - 1 \) and \( h(x) = \frac{f(x)}{g(x)} \) be a rational function such that:

(a) \( h(x) \) is continuous everywhere except at \( x = -1 \).
(b) \( \lim_{x \to \infty} h(x) = \infty \).
(c) \( \lim_{x \to -1} h(x) = \frac{1}{2} \).

Find \( \lim_{x \to 0} (3h(x) + f(x) - 2g(x)) \).

Answer: -\dfrac{39}{4}


Question: Evaluate the limit of the integral: \[ \lim_{k\rightarrow\infty}\int_\pi^{2\pi}{\sum_{n=1}^k}\left(\sin x\right)^n\ \text{dx.} \]

Answer: 2 - \pi


Question: Evaluate the limit: $$\lim_{n\rightarrow\infty} \frac{7^{\sqrt{n+1}-\sqrt{n}}\cdot(\frac{n+1}{2})!\cdot(\frac{n+1}{2})!}{(n+1)\cdot(\frac{n}{2})!\cdot(\frac{n}{2})!}$$

Answer: \dfrac{1}{2}


Question: For a complete graph $K_n$ with $n \ge 3$, consider the graph $G$ obtained by removing the edges of a cycle $C_n$. Determine the maximum integer $k$ such that $G$ is vertex $k$-connected.

Answer: n - 3


Question:
Calculate the following limit, where \( I_{n} = \int_{0}^{1} x^{n} \arctan(x) \, dx \):
\[ \lim_{n \rightarrow \infty} \left( n((n+1)I_{n}-\frac{\pi}{4}) \right) \]

Answer: -\dfrac{1}{2}


Question: Provide an example of an irreducible polynomial \(g\) over \(\mathbb{Q}\) such that the extension \((\mathbb{Q}[x]/(g))/\mathbb{Q}\) is not Galois.

Answer: x^3 - 2


Question: Determine the number of positive integral solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 = x_1 \cdot x_2 \cdot x_3 \cdot x_4 \cdot x_5\).

Answer: 40


Question: Find the limit of the sequence \(x_{n}=\frac{1}{n+2^{0}}+\frac{1}{n+2^{1}}+\ldots+\frac{1}{n+2^{n}}\) as \(n\) approaches infinity.

Answer: 0


Question: Let $X_1, X_2, \ldots, X_{10}$ be independent and identically distributed normal random variables with mean $0$ and variance $2$. Calculate the expected value of the expression \( \frac{X_1^2}{X_1^2 + X_2^2 + \cdots + X_{10}^2} \).

Answer: \dfrac{1}{10}


Question: Let \( f: (0,1] \to \mathbb{R} \) be a continuous function such that the improper integral \( \int_0^1 f(x) \, dx \) converges. Determine whether the following statement is true: \( \lim_{\eta \searrow 0} \int_0^\eta f(x) \, dx = 0. \) Justify your answer.

Answer: True


Question: An Abelian group $G$ is generated by elements $x$ and $y$, where $|x| = 16$, $|y| = 24$, and $x^2 = y^3$. Determine the order of the group $G$.

Answer: 48


Question: Evaluate the integral \( \int_{0}^{\pi}\frac{\log(1+a\cos x)}{\cos x}\,dx \) using the formula \( \frac{d}{dy}\int_{a}^{b}f(x,y)dx=\int_{a}^{b}\frac{\partial }{\partial y}f(x,y)dx \).

Answer: \pi \arcsin a


Question: Evaluate the line integral \( \int_C \vec{F} \cdot d\vec{r} \) where \( \vec{F} = \nabla f(x,y) = 6y \sin (xy) \vec{i} + 6x \sin (xy) \vec{j} \) and \( C \) is the segment of the parabola \( y = 5x^2 \) from the point \((2,20)\) to \((6,180)\).

Answer: 6\left( \cos(40) - \cos(1080) \right)


Question: Let \( f: \mathcal{l}^2 \to \mathcal{l}^2 \) be defined by \( f((x_n)) = \left( \frac{x_n}{n} \right) \). Determine the operator norm \( \|f\|_2 \).

Answer: 1


Question:
Given that \(ab + ac + bc = 2\), find the minimum value of the expression \(10a^2 + 10b^2 + c^2\). Choose the correct answer from the following options:

1) 3
2) 4
3) 8
4) 10

Answer: C


Question: Find the value of the limit \( S = \lim_{n \to \infty} \sum_{r=n^2}^{(n+2)^2} \frac{1}{\sqrt{r}} \).

Answer: 4


Question: Determine the values of \(a\) for which \(\int_{\gamma} \left(\frac{1}{z} + \frac{a}{z^3}\right)e^z \, dz = 0\) for every closed curve \(\gamma\) in \(\mathbb{C} \setminus \{0\}\) that is partially \(C^1\). For each of these values of \(a\), find a function \(F\) such that \(F'(z) = \left(\frac{1}{z} + \frac{a}{z^3}\right)e^z\) for \(z \neq 0\).

Answer: -2


Question:
Evaluate the contour integral \( \int \limits_\gamma \frac{1}{z-1} \, dz \) along the path defined by:
\[ \gamma(t) = \begin{cases}(1+i)t, & 0\leq t\leq 1 \\\ t+i(2-t), & 1\leq t \leq2\end{cases} \]

Answer: -i\pi


Question: Calculate the sum: \[ \cos^2\left(\frac{\pi}{101}\right) + \cos^2\left(\frac{2\pi}{101}\right) + \cos^2\left(\frac{3\pi}{101}\right) + \cdots + \cos^2\left(\frac{100\pi}{101}\right) \]

Answer: \dfrac{99}{2}


Question: Determine the number of functions \( f(x) \) that satisfy the differential equation \( f(x) = f'(x) \) with the conditions \( f(0) = 0 \) and \( f(1) = 0 \).

Answer: 1


Question:
Let \( f(x) = \ln|\sec x + \tan x| \) and \( g(x) = \sec x + \tan x. \) Find the fourth derivative of \( g(x) \) with respect to \( f(x). \)

A) \( f'(x) \)
B) \( f'(x)g(x) \)
C) \( g(x) \)
D) \( g'(x) \)
E) NOTA (None of the above)

Answer: C


Question: Find the limit: \( \lim_{n \to \infty} n!^{1/n} \).

Answer: \infty


Question: Let $X$ be a random variable with a continuous distribution function $F$. Find the distribution function for the random variable $F(X)$. Provide your answer in terms of $x$.

Answer: x


Question: Compute the limit \( \lim \limits_{x\to \infty} ((2x^3-2x^2+x)e^{1/x}-\sqrt{x^6+3}) \) using Taylor series.

Answer: \infty


Question: Let $f$ be an analytic function on $|z|<2$ such that $f(0)=f'(0)=0$, $f''(0) \neq 0$, $f(1/3)=i/12$, and $|f(z)|\le3$ for $|z|<2$. Find the value of $f(2i/3)$. \( \text{Provide your answer in the form } \boxed{\text{answer}}. \)

Answer: -\dfrac{i}{3}


Question: Evaluate the improper integral: $$\int_0^{+\infty}\frac{x\sin x}{x^2+1}\,dx$$

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


Question: Determine the spectrum of the $C^*$-algebra $C(K) \oplus C(K')$, where $K$ and $K'$ are compact Hausdorff spaces.

Answer: K \sqcup K'


Question: Find the number of different symmetric matrices $A \in \mathbb{R}^{3\times 3}$ such that $[1,2,2]^T$ and $[2,1,-2]^T$ are eigenvectors, $A$ has three distinct eigenvalues, and $A^n = A$ for some $n > 1$. How many different solutions are there?

Answer: 6


Question: Calculate the limit: \[ \lim_{n \to \infty} \int_{(0,1)^n} \frac{n}{x_1 + \cdots + x_n} \, dx_1 \cdots dx_n \]

Answer: 2


Question: Determine the cardinality of the set $A$ if $A = \{\{A\}\}$. Provide your answer as a single number.

Answer: 1


Question: Let $p_i$ $(1 \leq i \leq m)$ be primes such that $p_i < 2(3^{k-1})+1 $, where $k \in \mathbb{N}$. Let $a=p_1^{\alpha_1} \cdots p_m^{\alpha_m}$ for some $\alpha_i >0$. If $a > 2 (3^k)$, determine if there exists a $j$ such that $p_j^{\alpha_j-1}(p_{j}-1)>2(3^{k-1})$. Answer true or false.

Answer: False


Question: Find the supremum of the set \( \{ |\sin(nx) - \sin(mx)| : m,n \in \mathbb{N} \} \) for \( x \in [-1,1] \).

Answer: 2


Question: For which values of \( n \) is it possible to divide any given angle into \( n \) equal parts using only a compass and a straight edge?

Answer: 2^k


Question: Evaluate the limit: \[ \lim_{x \to 0} (\cos(x)+\tan^2(x))^{\csc(x)} \]

Answer: 1


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{(n+1)^n-(n+2)^n}{(n+2)^n-(n+3)^n}$$ without using L'Hôpital's rule.

Answer: \dfrac{1}{e}


Question: Does there exist a non-compact topology $E$ such that for every continuous function $f : E \to \mathbb{R}$, the maximum value of $f(E)$ exists?

Answer: Yes


Question: Let $f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x, y) = x - y$. If $K \subset \mathbb{R}$ is a closed set, is the set $f(K \times K)$ closed in $\mathbb{R}$?

Answer: No


Question:
Given the in-degree and out-degree sequences of a directed graph without loops or multiple edges, determine how many of the following sequences correspond to a valid graph:

(a) \(d_{in}=(0,1,2,3), d_{out}=(2,2,1,1)\)

(b) \(d_{in}=(2,2,1), d_{out}=(2,2,1)\)

(c) \(d_{in}=(1,1,2,3,3), d_{out}=(2,2,3,1,2)\)

Find the number of sequences that can form a valid directed graph.

Answer: 2


Question: Consider a scenario where there are \( n \) letters, each with a specific envelope. Each letter is randomly placed into an envelope. Determine the probability that at least one letter is placed in its correct envelope as \( n \to \infty \). Express your answer in terms of the mathematical constant \( e \).

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


Question:
Consider the sequence \( \langle x_n \rangle , \: n \geq 0 \) defined by the recurrence relation \( x_{n+1} = c \cdot x^2_n -2 \), where \( c > 0 \). Suppose there exists a non-empty, open interval \((a, b)\) such that for all \( x_0 \) satisfying \( a < x_0 < b \), the sequence converges to a limit. What is the limiting value of the sequence?

\( A. \frac{1+\sqrt{1+8c}}{2c} \)

\( B. \frac{1-\sqrt{1+8c}}{2c} \)

\( C. 2 \)

\( D. \frac{2}{2c-1} \)

Answer: B


Question:
Find all functions \( f(z) \) that are analytic everywhere in the complex plane and satisfy the following conditions:

1. \( f(2-i) = 4i \)
2. \( |f(z)| < e^2 \) for all \( z \) in the complex plane.

Answer: 4i


Question: Simplify the expression: \[ \frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}} \]

Answer: 1 + \sqrt[4]{5}


Question: Use Lagrange multipliers to find the maximum and minimum values of the function \( f(x,y) = e^{xy} \) subject to the constraint \( x^3 + y^3 = 16 \).

Answer: e^4


Question: Find \(\lim_{(x, y)\to(1, 1)}\frac{x^y-y(x-1)-1}{(x-1)^2 + (y-1)^2}\).

Answer: 0


Question: Evaluate the Fourier coefficient \( \hat{f}_k \) for the function \( f(t) = \frac{1-r^2}{1+r^2-2r\cos t} \) where \( t \in \mathbb{R} \) and \( |r| < 1/2 \). Verify that the correct result is \( \hat{f}_k = r^{|k|} \).

Answer: r^{|k|}


Question:
Determine whether the following statement is true:

Let $H$ be a real or complex Hilbert space and $R, S: H \to H$ be compact operators. For every $n \in \mathbb{N}$, does the inequality
\[ \sum_{j=1}^n s_j(RS) \leq \sum_{j=1}^n s_j(R)s_j(S) \]
hold true?

Here, $s_j(R)$ denotes the $j$-th singular value of the operator $R$, and the sequence of singular values decreases monotonically to zero.

Answer: True


Question: Let \( G \) be a simple group with a maximal subgroup of the form \( p:q \) (a semidirect product of \( \mathbb{Z}_p \) and \( \mathbb{Z}_q \)), where \( p \) and \( q \) are primes and \( q \mid (p-1) \). Suppose every Sylow \( p \)-subgroup of \( G \) is \( \mathbb{Z}_p \) and each is contained properly in exactly one maximal subgroup of \( G \). Prove that \( G \cong L_2(p) \).

Answer: L_2(p)


Question:
Let $X_1, X_2, \ldots$ be an infinite sequence of independent, identically distributed uniform random variables on $[0,1]$. For a given constant $x$, where $0 < x <1$, define
$$
M= \min\{n\ge1: X_1 + X_2 + \ldots + X_n > x\}.
$$
Find $P(M>k)$, where $k$ is a non-negative integer.

Answer: \dfrac{x^k}{k!}


Question: Suppose $R$ is a commutative ring, $A$ is an $R$-module, and $A'$ is an $R$-submodule of $A$. If for every $f \in \mathrm{Hom}_R(A,A')$ we have $f(A) = 0$, does this imply that $A' = 0$?

Answer: No


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

Answer: 0


Question: Given the system of equations: \[\begin{cases} x + y + z = 12 \\ x^2 + y^2 + z^2 = 12 \\ x^3 + y^3 + z^3 = 12 \end{cases}\] find the value of \(x^4 + y^4 + z^4\).

Answer: 1992


Question: Let \(a, b > 0\). Define the stopping time \(T_{a,b}\) for a Brownian motion with drift as \[ T_{a,b} := \inf\{t > 0 : B(t) = at - b\} \] where \(B(t)\) is a standard Brownian motion. Compute \(E[T_{a,b}]\).

Answer: \dfrac{b}{a}


Question: Given random variables $A$, $B$, and $X$ with $\text{corr}(A, X) = 0.03$, $\text{corr}(B, X) = 0.04$, and $\text{corr}(A, B) = 0$, determine the range of the correlation $\text{corr}(c_1A + c_2B, X)$ for arbitrary constants $c_1$ and $c_2$. Provide your answer as an interval.

Answer: [-0.05, 0.05]


Question: Given triangle \(\triangle ABC\) and point \(D\) on segment \(\overline{AB}\), construct point \(P\) on line \(\overleftrightarrow{CD}\) such that \(\angle APB = \angle BAC\).

Answer: P


Question: Calculate the Fourier coefficient $\hat{s}(-1)$ for the 1-periodic signal $s : \mathbb{R}/\mathbb{Z} \to \mathbb{C}$ defined by the equation $s(t) = (2\cos(\pi t))^{16}$.

Answer: 11440


Question: Calculate the Lebesgue integral \( \int_{[0,\pi]} \sin x \: \epsilon[\mathbb{R}\setminus\mathbb{Q}] \, d\mu,\) where \(\epsilon\) is the characteristic function of the irrationals. Provide your answer as a single number.

Answer: 2


Question: Determine whether the ideal \( I = (x, y) \) is a free module over the ring \( \mathbb{C}[x, y] \).

Answer: No


Question: Determine the largest integer $n$ such that $2^n$ divides the product $A = 63 \times 64 \times 65 \times \cdots \times 130$. What is the maximum value of $n$?

Answer: 71


Question: Consider the set $A = \bigcup_{n=1}^\infty ((2n+1) + V)$, where $V \subset [0,1]$ is the Vitali set. Given that $m^*(A) = \infty$, determine whether it is true that for any closed subset $F$ of $A$, the Lebesgue measure $m(F) = 0$. Provide a justification for your answer.

Answer: True


Question: Let \( \mu \) be a measure on \(((0,\infty),\mathbb{B}(0,1))\) such that \( M(t):=\int_{(0,\infty)}e^{tx}\mu(dx)<\infty \) for \( t\in (0,\epsilon] \). Prove that \( \lim_{t\to 0}M(t)=\mu(0,\infty) \) using the Dominated Convergence Theorem.

Answer: \mu(0,\infty)


Question:
For each nonnegative integer \( n \), there exists a polynomial \( p_n(x) \) such that
\[ \int x^n e^x \, dx = p_n(x) e^x + C \]
Let
\[ L = \lim_{n \to \infty} \frac{|p_n(2)|}{n!} \]
Find \( \lfloor 1000L \rfloor \).

Answer: 135


Question: If \( m^2\cos{\frac{2\pi}{15}}\cos{\frac{4\pi}{15}}\cos{\frac{8\pi}{15}}\cos{\frac{14\pi}{15}} = n^2 \), find the value of \( \frac{m^2 - n^2}{n^2} \).

Answer: 15


Question: Let $X$ be a possibly non-separable Banach space and $X^*$ its dual. Consider a sequence $(f_n)_{n=1}^\infty$ of linearly independent norm-one functionals in $X^*$. Determine whether there exists an element $x \in X$ such that $f_n(x) \in (0, \infty)$ for all $n$. Provide a justification for your answer.

Answer: Yes


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

Answer: (x + 1)e^x


Question: Calculate the sum of all natural numbers between 1 and \(3^{100000}\) that are not divisible by 3.

Answer: 3^{199999}


Question: Do Gromov-Witten invariants count the morphisms from a curve to a variety over \( \mathbb{C} \)?

Answer: Yes


Question: Is it true that a point $x^*$ is an accumulation point of the sequence $\{x_n\}$ if and only if there exists a subsequence $x_{a_n}$ such that $\lim_{n\rightarrow\infty} x_{a_n}=x^*$?

Answer: True


Question: Determine the number of quaternion solutions to the equation \( q^2 = 1 \), where \( q = x_0 + ix_1 + jx_2 + kx_3 \) and \( x_i \in \mathbb{R} \). The quaternions satisfy the relations \( i^2 = j^2 = k^2 = -1 \), \( ij = -ji = k \), \( jk = -kj = i \), and \( ki = -ik = j \).

Answer: 2


Question: Find the largest value of \( 0 < c < 4 \) such that \( 2^{n + \sqrt{n}} \in O((4-c)^n) \).

Answer: 2


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left(\frac{1}{n} + \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{6n}\right) \]

Answer: \ln 6


Question: Consider the series given by \[ f(x) = \sum_n f(n)\frac{\sin\pi(x-n)}{\pi(x-n)} \] where the Fourier transform of a function \( f \in L^2 \) has compact support with frequency \( \xi \) satisfying \( |\xi| \leq \pi \). Determine whether this series converges absolutely for all \( x \).

Answer: Yes


Question: Is it possible to construct a projection operator (which is not the identity operator) in \(\mathbb{R}^n\)? For example, consider the space \(\mathbb{R}^4\).

Answer: Yes


Question: Is a $CD(K,\infty)$ space a length space? Consider a complete and separable metric space $(X,d)$ with a nonnegative Borel measure $\mu$ that has support $X$ and satisfies $\mu(B(x,r))<\infty$ for every $x \in X$ and $r>0$. If $(X,d,\mu)$ is a $CD(K,\infty)$ space in the sense of Sturm, determine whether $(X,d)$ is a length space.

Answer: Yes


Question: Given a symmetric positive definite matrix $A$ with the spectral decomposition $A = PDP^T$, where $D$ is a diagonal matrix of eigenvalues, it is known that $A^{\frac{1}{2}} = Q D^{\frac{1}{2}}Q^T$. Is it also true that if $A = P D P^{-1}$, then $A^{\frac{1}{2}} = Q D^{\frac{1}{2}}Q^{-1}$? Justify your answer.

Answer: Yes


Question: Determine whether the sequence of functions \( g_n(\alpha) = \int_a^b \alpha(x) \sin(nx) \cos(nx) \, dx \) converges uniformly for all functions \( \alpha \) in the set \( X = \{ \alpha : [a,b] \to \mathbb{R} \} \), where \( \alpha'' \) exists and is continuous, and there exists a constant \( K > 0 \) such that \( |\alpha(x)|, |\alpha''(x)| \le K \) for all \( x \in [a,b] \).

Answer: 0


Question: Consider the sequence of random variables \( X_n = [A_n, B_n, C_n] \) in \( \mathbb{R}^3 \), where \( A_n, B_n, C_n > 0 \) almost surely, and \( X_n \overset{a.s.}{\to} x = [a, b, ab] \). Determine if the function \( g: \mathbb{R}^3 \to \mathbb{R} \) defined by \( g([p,q,r]) = \frac{pq}{r} \) is continuous at \( x \) such that the continuous mapping theorem can be applied, resulting in \( g(X_n) \overset{a.s.}{\to} 1 \).

Answer: 1


Question: A differentiable function \( f \) satisfies the relation \( f(x+y) = f(x) + f(y) + 2xy(x+y) - \frac{1}{3} \) for all real numbers \( x \) and \( y \). Additionally, \( \lim_{h \to 0} \frac{3f(h)-1}{6h} = \frac{2}{3} \). Determine the greatest integer value of \( f(2) \).

Answer: 8


Question: Find the remainder when \( 13^{16} - 2^{25} \cdot 15^{16} \) is divided by 37.

Answer: 26


Question:
Verify the correctness of the gradient calculation for the function \( \Phi_1 = \sum_{l=0}^{\infty} f(l) r^{l} P_l(\cos(\theta)) \), where \( P_n \) is the \( n \)-th Legendre polynomial. Determine which of the following expressions correctly represents \(-\nabla \Phi_1\):

1. \(-\nabla \Phi_1 = -\left(\begin{array}{c}\sum_{l=1}^{\infty} f(l) \cdot l \cdot r^{l-1} P_l(\cos(\theta)) \\
-\sum_{l=0}^{\infty} f(l) \cdot (l+1) \cdot r^{l-1} \frac{\cos(\theta) P_n(\cos(\theta)) - P_{n-1}(\cos(\theta))}{\sin(\theta)}\end{array}\right)\)

2. \(-\nabla \Phi_1 = -\left(\begin{array}{c}\sum_{l=1}^{\infty} f(l) \cdot l \cdot r^{l-1} P_l(\cos(\theta)) \\
\sum_{l=1}^{\infty} f(l) \cdot l \cdot r^{l-1} \frac{\cos(\theta) P_n(\cos(\theta)) - P_{n-1}(\cos(\theta))}{\sin(\theta)}\end{array}\right)\)

Note that the only difference is in the second component of the gradient expression.

Answer: 2


Question: Determine if there is a closed form for the integral \( \int_0^{\pi/2} \frac{1}{1+\tan^{s} x} \, dx \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit: \[ \lim_{x \to 0^{+}} (x^{x^{x}} - x^x) \]

Answer: -1


Question: Find the Maclaurin series of the function \( f(x) = \begin{cases} \frac{\sin(x)}{x},& x \neq 0 \\ 1,& x=0 \end{cases} \) and determine \( f^{(10)}(0) \).

Answer: -\dfrac{1}{11}


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty}{\left(\sum_{k=1}^{n}{\frac{1}{k}}\right)}z^n \).

Answer: 1


Question: Determine the number of primitive polynomials of degree 10 over the finite field \( F_2 \).

Answer: 60


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

Answer: -\dfrac{2}{e}


Question: Find a particular solution to the partial differential equation: \[ U_x + xU = x^3 + 3xy \]

Answer: x^2 + 3y - 2


Question: Compute the abelianization of the group given by the presentation \( G = \langle x,y \mid x^3 = y^5 \rangle \).

Answer: \mathbb{Z}


Question: Calculate the expectation of the product of Brownian motion terms: \( E[B(U)B(U+V)B(U+V+W)B(U+V+W+x)] \), given that \( U + V + W > U + V > U \) and \( x > 0 \).

Answer: 3U^2 + 3UV + UW


Question: Let \((X, S)\) be a measurable space. Suppose \(f\) is a function from \(X\) to \([-\infty, \infty]\) such that the inverse image of \((a, \infty)\) belongs to \(S\) for all \(a \in \mathbb{R}\). Is \(f\) an \(S\)-measurable function?

Answer: Yes


Question: Let \( f \in \mathbb{R}(x_1, \ldots, x_n) \) be a rational function that is continuous on \( \mathbb{R}^n \). Determine whether \( f \) must be Lipschitz on the unit ball.

Answer: Yes


Question: Evaluate the following limit of the integral: \[ \lim_{n \rightarrow \infty} \int_{0}^{\pi/4} \frac{\sin^2(\frac{nx}{2})}{n\,\sin^2(\frac{x}{2})} \,dx \]

Answer: \pi


Question: Compute the limit \( \lim_{n \to \infty} \frac{X(n)}{n} \) for a standard Brownian motion \( X(t), t \ge 0 \).

Answer: 0


Question: Evaluate the integral \( \int_0^{2\pi}\frac{d\theta}{5+4\sin\theta} \) using complex analysis methods.

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


Question: Determine if the series expansion \( \frac{1}{1-\frac{x^2}{n^2\pi^2}} = 1 + \left(\frac{x^2}{n^2\pi^2}\right) + \left(\frac{x^2}{n^2\pi^2}\right)^2 + \left(\frac{x^2}{n^2\pi^2}\right)^3 + \cdots \) is valid for \(|x| < \pi\). Justify your answer.

Answer: Yes


Question: Find the maximum area bounded by the curves \(y^2=4ax\), \(y=ax\), and \(y=\frac{x}{a}\) for \(a \in [1,2]\).

Answer: 84


Question: For which values of \( b > 0 \) is the set \( \{e_n(t) = e^{int} : n \in \mathbb{Z}\} \) a basis for \( L^2([0,b]) \), meaning that the orthogonal complement \( V^\perp = 0 \) where \( V = \text{span}\{e_n : n \in \mathbb{Z}\} \)?

Answer: 0 < b \leq 2\pi


Question: Evaluate the limit: \[ \lim_{x,y \to 0,0} \frac{x+y-\frac{1}{2}y^2}{\sin(y)+\log(1+x)} \]

Answer: 1


Question: Minimize the expression \( \sum\limits_{k=0}^{19} \sqrt{1+u_k^2} \) subject to the conditions \( x_0 = 0 \), \( x_{20} = 5 \), and \( x_{k+1} - x_k = u_k \) for \( k = 0, 1, 2, \ldots, 19 \).

Answer: 5\sqrt{17}


Question: Calculate the limit: \[ \lim_{n\to\infty}\left(2^n+n\cdot2^n\sin^2{\frac{n}{2}}\right)^{\left(\frac{1}{2n-n\cos{\frac{1}{n}}}\right)} \]

Answer: 2


Question: Evaluate the limit: \[ \lim_{n \to \infty} n^2 \int_{0}^{\infty} \frac{\sin(x)}{(1 + x)^n} \, dx \]

Answer: 1


Question: Determine the values of \( p \) for which the integral \( \int_{0}^{1}\frac{(x^p+x^{-p})\log(1+x)}{x}dx \) is convergent.

Answer: (-1, 1)


Question: Solve the partial differential equation \( \frac{\partial z}{\partial x} \cdot \frac{\partial z}{\partial y} = 1 \) with the initial condition \( z(0,0) = 0 \). Determine the value of \( z(0,1) \).

Answer: 0


Question: Compute the limit \( \lim_{x\rightarrow\infty}\frac{x}{\overline{F}_{X}(x)}\int_{x}^{\infty}\overline{F}_{X}(s)ds \), where \( X\sim\text{Norm}(0,1) \) is a standard normally distributed random variable and \( \overline{F}_{X}(x) = 1 - F_{X}(x) \) is the survival function of \( X \).

Answer: 1


Question:
Given a sequence of product probability measures $P_n=\times_{i=1}^n P$ and a sequence of measurable events $E_n$ such that $P_\infty(\liminf_{n\to \infty}E_n)=1$, determine whether for any other sequence of measurable events $S_n$, the following limit holds:
$$
\lim_{n \to \infty} \frac{P_n(E_n \cap S_n)}{P_n(S_n)}=1.
$$

Answer: No


Question: Calculate the work done by the force field \( F(x,y) = e^y\sin(x) \mathbf{i} - (e^y\cos(x) - \sqrt{1+y}) \mathbf{j} \) in moving a particle along the path \( y = x^2 \) from \((-\pi, \pi^2)\) to \((\pi, \pi^2)\).

Answer: 0


Question: Evaluate the complex integral \( \int_{|z|=2}\frac{1}{(z-1)^3}dz. \)

Answer: 0


Question: Find the area of the spherical triangle \( T = \{ x \in S^2 : d_S(x,u) \le \frac{\pi}{2}, d_S(x,v) \le \frac{\pi}{2}, d_S(x,w) \le \frac{\pi}{2} \} \), where \( u = \frac{1}{\sqrt{2}}(1,-1,0) \), \( v = \frac{1}{\sqrt{2}}(1,1,0) \), and \( w = \frac{1}{\sqrt{2}}(0,1,1) \).

Answer: \dfrac{\pi}{2}


Question: Verify the formula \( \sum\limits_{d|n} \mu (d) \lambda(d)= 2^{\omega (n)} \) for an integer \( n \geq 1 \).

Answer: 2^{\omega(n)}


Question: Find the smallest integer \( N \) such that the inequality \( \left|f(x)-T_{N} f(x, 0)\right| \leq 10^{-6} \) holds for all \( x \in \left[-\frac{1}{10}, \frac{1}{10}\right] \), where \( f(x) = \cos(x) e^x \). Also, specify the \( N \)-th Taylor polynomial of \( f(x) \) at \( x = 0 \). You may use the fact that \( \cos(x) \geq \frac{1}{2} \) for \( x \in \left[-\frac{1}{10}, \frac{1}{10}\right] \).

Answer: 4


Question:
Given the linear regression model with the design matrix \(X = [x_1 \mid x_2 \mid \cdots\mid x_p] \in\mathbb R^{n\times p}\), where the columns of \(X\) are centered (i.e., \(x^T_j 1_n = 0\) for \(j = 1, \ldots , p\)), and the model does not include an intercept, assume the following covariance matrix:
\[
\frac{1}{n}X^TX=\begin{bmatrix}I_{p-1} & z\\z & a^2\end{bmatrix}, z\in \mathbb{R}^{p-1}, a^2\in \mathbb{R}
\]
where \(a^2=10\|z\|^2\). Let \(R^2_j\) be the \(R^2\) obtained when regressing \(x_j\) onto the rest of the covariates \(\{x_1, \ldots , x_{j−1}, x_{j+1}, \ldots , x_p\}\). Find \(R^2_p\), the coefficient of determination resulting from regressing \(x_p\) onto \(\{x_1, \ldots , x_{p−1}\}\).

Answer: \dfrac{1}{10}


Question: Let $k$ be an integer greater than $1$. Initially, Tita the frog is sitting at the point $k$ on the number line. In one movement, if Tita is located on the point $n$, then she jumps to the point $f(n) + g(n)$, where $f(n)$ and $g(n)$ are the greatest and the least prime divisors (both positive) of $n$, respectively. Determine all the values of $k$ for which Tita will visit infinitely many distinct points on the number line.

Answer: \emptyset


Question: Given that \( \mathrm{P}(1) + \cdots + \mathrm{P}(n) = n^5 \) for a polynomial function \( \mathrm{P} \) defined on \( \mathbb{R} \to \mathbb{R} \), find the value of \( \int_{0}^{1} \mathrm{P}(x) \, \mathrm{d}x \).

Answer: \dfrac{1}{3}


Question: Determine the value of the step size \( h \) for which the iteration \( x_{n+1} = \phi(x_n) \) becomes unstable, given that \( \phi(x) = x + h \beta x (1 - \frac{x}{N}) \), \( \beta = 0.125 \), and \( N = 10000 \). The iteration is unstable when \( |\phi'(N)| \geq 1 \).

Answer: 16


Question: Let $E$ be a vector space over a field $F$. Define the algebraic dual $E^*$ as $\{f:E\rightarrow F : f \text{ is linear}\}$. Assume $E^* \neq \{0\}$, taking the axiom of choice for granted. Is it true that $E^*$ separates $E$? In other words, does the following hold: $$ \forall x\in E\backslash\{0\}, \; \exists f \in E^* \; \text{such that} \; f(x)\ne 0 \; ? $$

Answer: Yes


Question: Determine the Galois group of the polynomial \( f(t) = t^3 - 4t + 2 \) over \( \mathbb{Q} \).

Answer: S_3


Question: Consider a Banach space $X$ with a nondegenerate Gaussian probability measure $\gamma$ having mean 0. Determine whether \( \int_X{d\gamma(t)} = 0 \) or \( \int_X{d\gamma(t)} = 1 \).

Answer: 1


Question: Calculate the surface integral \( \int_\Sigma x \, dS \) over the surface \( \Sigma = \{(x,y,z):(x-2y)^2+(y-x)^2=1,\,0\le x+y+2z\le1\} \).

Answer: 0


Question:
Determine the value of $m$ such that the following series of binomial coefficients sums to zero:
\[
\left({m\choose0}+{m\choose1}-{m\choose2}-{m\choose3}\right) + \left({m\choose4}+{m\choose5}-{m\choose6}-{m\choose7}\right) + \cdots = 0
\]
Which of the following is true for some positive integer $k$?

(i) $m = 4k$.
(ii) $m = 4k + 1$.
(iii) $m = 4k - 1$.
(iv) $m = 4k + 2$.

Select the correct option.

Answer: iii


Question: In a street, all the houses are numbered continuously from $1$ to $1000$. Alice lives at number $6$, and the sum of the numbers from $1$ to $5$ is equal to that of the numbers from $7$ to $8$. Bob lives at number $35$, and the sum of the numbers from $1$ to $34$ is equal to that of the numbers from $36$ to $49$. Cedric lives at number $N$, different from Alice’s and Bob’s numbers. The sum of the numbers from $1$ to $(N - 1)$ is equal to that of the numbers from $(N + 1)$ to a certain number strictly greater than $(N + 1)$. What is the value of $N$?

Answer: 204


Question: Let \( k = \overline{k} \) and \( X \subset \mathbb{A}^2_k \) be the algebraic set \( V(y(y-x^2)) \), which represents the union of the \( x \)-axis and a parabola. Determine the radical ideal \( I(X) \subset k[x,y] \).

Answer: (y(y - x^2))


Question: Find the remainder when $2^{2021}$ is divided by $2700$. Provide your answer as a single integer.

Answer: 1652


Question: Is every odd order skew-symmetric matrix with complex entries singular? A square matrix $A$ is skew-symmetric if $A = -A^T$. A matrix is singular if its determinant is zero. Answer yes or no.

Answer: Yes


Question: Determine the number of spanning trees in the complete bipartite graph $K_{m,n}$, where $K_{m,n}$ is defined as a graph with $m + n$ vertices $a_1, \ldots, a_m$, $b_1, \ldots, b_n$ such that there is an edge between each $a_i$ and each $b_j$, but not between any two $a$'s or any two $b$'s.

Answer: m^{n-1} n^{m-1}


Question:
Given that $a$ and $n$ are coprime positive integers, and $k$ is a positive integer satisfying:
\[\begin{cases}
a^k \equiv 1\;(\mathrm{mod}\;n)\\
k\leqslant \varphi(n)
\end{cases}\]
Does it necessarily follow that $k$ divides $\varphi(n)$?

Answer: No


Question: Find the Maclaurin series for the function \( g(x) = \int_0^x \frac{1}{1+t^3} \, dt \). Determine the radius of convergence of this series.

Answer: 1


Question: Evaluate the integral \( \int_0^{2\pi}\sqrt{1+\sin x} \, dx \).

Answer: 4\sqrt{2}


Question: Given the function \( f(x) = \frac{1}{1-x-x^2} \), find the value of \( \frac{f^{(10)}(0)}{10!} \).

Answer: 89


Question: Evaluate the line integral \( \int_{T} (x-y)\,\text{d}x + (x+y)\,\text{d}y \), where \( T \) is the path counterclockwise around the triangle with vertices \((0,0), (1,0), (0,1)\).

Answer: 1


Question: Find the equilibrium points of the system of differential equations given by \( x' = y \) and \( y' = -x - y\ln(x^2 + 4y^2) \).

Answer: (0, 0)


Question: Given a batting average of 0.3085443, use continued fractions to estimate the number of at-bats the player had. What is the most likely number of at-bats?

Answer: 94


Question: Evaluate the contour integral \( \int_C \frac{z^2}{z^2 + 4} \, dz \) where \( C \) is the rectangle with vertices \(-2, 2, -2 + 4i, 2 + 4i\) traversed in the anticlockwise direction.

Answer: -2\pi


Question: Determine whether the following statement is true for all natural numbers \(m\) and \(n\): \[ \sum_{k=1}^{m+n} \binom {m+n}{k} k^m (-1)^k = 0. \] Note: \(0 \notin \mathbb{N}\).

Answer: True


Question: Compute the limit: \[ \lim_{x \to 0} \frac{e^{-1/x^2}}{x^2} \] without using L'Hopital's rule.

Answer: 0


Question: Evaluate the limit of the infinite summation: \[ \lim_{n\to\infty} \sum_{i=0}^n \frac{1}{i+n} \]

Answer: \ln 2


Question: Evaluate the contour integral \( \int_{\gamma(0;2)} \frac{e^{i\pi z/2}}{z^2-1} \, \mathrm{d}z \) using the Cauchy integral formula, where \( \gamma(0;2) \) is a contour enclosing the singularities at \( z=1 \) and \( z=-1 \).

Answer: -2\pi


Question: Identify the topological space that is homeomorphic to the partition of \(\mathbb{R}^2 - \{0\}\) into lines of the form \(y = mx\) for all real \(m\), including vertical lines.

Answer: S^1


Question: Let $U$ be an open subset of a manifold $X$. Prove that for any point $x \in U$, the tangent space $T_x(U)$ is equal to the tangent space $T_x(X)$. That is, show that $T_x(U) = T_x(X)$ for all $x \in U$.

Answer: T_x(U) = T_x(X)


Question: Find the residue of the function \( \frac{1}{1+e^z} \) at the pole \( z = i\pi \).

Answer: -1


Question: Evaluate the surface integral of the vector field \( A(x,y,z) = (xy, yz, x^2) \) over the sphere \( S \) defined by \( x^2 + y^2 + z^2 = R^2 \), where the normal vector is pointing outward. Use an appropriate parametrization to perform the integration.

Answer: 0


Question: Let $V = \mathbb{R}^n$ with a non-standard inner product defined by a metric matrix $G$. Let $L$ be the linear operator on $V$ represented by the matrix $A$. Find the matrix representation of the adjoint operator $L^A$ in terms of the matrix $A$ and the metric matrix $G$.

Answer: G^{-1} A^\top G


Question: Find the slope of the tangent line to the curve defined by \(x^y = y^x\) at the point \((c, c)\).

Answer: 1


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

Answer: 0


Question: Evaluate \( \lim_{x\to\frac{\pi}{2}} \left(\frac{\sin x}{1-\sin x}\right)^{\frac{1}{1-\sin x}} \).

Answer: \infty


Question: Suppose $X$ is a topological space, and $\{U_i \mid i \in I\}$ is a totally ordered family of open subsets of $X$. Denote by $U = \bigcup_{i \in I} U_i$. Is it possible to find a countable subset $J$ of $I$ such that $U = \bigcup_{j \in J} U_j$?

Answer: No


Question:
Consider a continuous function \( f : \mathbb{R} \rightarrow \mathbb{R} \). Let \( A \subset \mathbb{R} \) be defined as \( A = \{ y \in \mathbb{R} : y = \lim f(x_n), \text{ for some sequence } x_n \rightarrow \infty \} \). Determine which of the following statements about the set \( A \) is necessarily true:

A. \( A \) is a connected set.
B. \( A \) is a compact set.
C. \( A \) is a singleton set.
D. None of the above.

Answer: A


Question: Let \( G = ( \mathbb{Z} / 133 \mathbb{Z} )^{\times} \) be the group of units of the ring \( \mathbb{Z} / 133 \mathbb{Z} \). Determine the number of elements in \( G \) that have an order of 9.

Answer: 18


Question: Consider the set of numbers \( \{1, 2, 3, \ldots, 2n\} \). This set is randomly split into two subsets, each containing \( n \) elements. Each subset is then sorted, with one subset sorted in increasing order and the other in decreasing order. Let \( A = \{a_1, a_2, \ldots, a_n\} \) be the subset sorted in increasing order, and \( B = \{b_1, b_2, \ldots, b_n\} \) be the subset sorted in decreasing order. Define \( S = \sum_{i=1}^n |a_i - b_i| \). Find the expected value \( E(S) \).

Answer: n^2


Question: Is the HHH (three heights) a congruence criterion for triangles? In other words, can a triangle be uniquely determined by its three heights?

Answer: Yes


Question: Determine the number of different solutions for the equation \( \sqrt{3x^2 + 2x + 5} + \sqrt{x^2-4x+5} = \sqrt{2x^2 - 2x + 2} + \sqrt{2x^2+8} \).

Answer: 3


Question:
Consider a communication channel where a user is allowed to transmit only if the channel is detected to be vacant. If the channel is detected to be vacant and it is actually vacant, the user's transmission rate is $r = a$. If the channel is detected to be vacant but is actually occupied, the rate is $r = b$. Define $d_n=0$ as the event that the channel is detected to be vacant, and $d_n=1$ otherwise. Define $n=0$ as the event that the channel is actually vacant, and $n=1$ otherwise. Determine the correct expression for the expected value of the user's transmission rate from the following options:

1. $E[r] = Pr(n=0|d_n=0) \times a + Pr(n=1|d_n=0) \times b$
2. $E[r] = Pr(n=0,d_n=0) \times a + Pr(n=1,d_n=0) \times b + Pr(n=0,d_n=1) \times 0 + Pr(n=1,d_n=1) \times 0$
3. $E[r] = Pr(d_n=0|n=0) \times a + Pr(d_n=0|n=1) \times b$

Which expression correctly represents the expected value of the user's transmission rate?

Answer: 2


Question: Is there a constant \(c > 0\) such that for any integer \(n\), there exists a polynomial \(p(x) = a_nx^n + \cdots + a_0\) with coefficients \(-1, 0, 1\) where the root \(x=1\) has a multiplicity of at least \(c\sqrt{\frac{n}{\log n}}\)?

Answer: Yes


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

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


Question: Determine whether every group of order 56 has a proper non-trivial normal subgroup. Provide a justification for your answer.

Answer: Yes


Question: Use a line integral to find the area of the surface that extends upward from the semicircle \( y=\sqrt{4-x^2} \) in the \( xy \)-plane to the surface \( z=3x^4y \).

Answer: \dfrac{384}{5}


Question: Is the set of Gödel numbers of variable-free true sentences in the natural number structure \((N;+,\cdot,<,0,1)\) definable in Peano Arithmetic (PA)? Consider the set \(S\) of true first-order sentences that contain no variables. Let \(G(S)\) be the set of Gödel numbers of \(S\) for an arbitrary Gödel numbering \(G\). Determine if \(G(S)\) is definable in PA.

Answer: Yes


Question: Evaluate the line integral \( \oint x \, dy + y \, dx \) around the curve defined by \( x^4 + y^4 = 1 \).

Answer: 0


Question: How many unique permutations are possible for a 3x3x3 cube where each of the 27 smaller cubes has a distinct number, considering that simple rotations of the entire cube should not be counted as different permutations?

Answer: \dfrac{27!}{24}


Question: Determine whether the quantile function of an $L^1$ random variable is always in $L^2(]0,1[)$. Specifically, given a probability space $(\Omega, \mathcal A, P)$ and an $L^1(\Omega, \mathcal A, P)$ random variable $X$, with distribution function $F(x) = P(X \leq x)$ and quantile function $Q(\alpha) = \inf \{x \in \mathbb R : F(x) \geq \alpha \}$, is it true that $Q \in L^2(]0,1[)$, meaning $\int_0^1 Q(\alpha)^2 \, d\alpha < \infty$?

Answer: No


Question: Find the degree of the splitting field of the polynomial \(x^{16} - 1\) over the field \(\mathbb{F}_9\). Compute \([E: \mathbb{F}_9]\), where \(E/\mathbb{F}_9\) is the splitting field of \(x^{16} - 1\) over \(\mathbb{F}_9\).

Answer: 2


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\frac{\sin(t)}{t}f(t)dt \) given that the Fourier transform of \( f \) is \( \hat{f}(w)=\frac{w}{1+w^4} \).

Answer: 0


Question:
Determine whether the following limit is infinite:
\[ \lim_{x\to\infty}x\log(f(x)) \]
where
\[ f(x)=1+\frac{\log x}{x}+o\left(\frac{\log x}{x}\right), \quad x\to\infty. \]

Answer: \infty


Question: Let $X$ and $Y$ be independent random variables, each following a normal distribution $N(1,1)$. Calculate the probability $P(2-X < Y < X)$. Express your answer as a single probability value.

Answer: \dfrac{1}{4}


Question: In a non-standard model of Peano Arithmetic (PA), does there exist a non-zero number that is divisible by every standard natural number?

Answer: Yes


Question: Determine the minimum sample size \( n \) such that the probability that the sample mean \( \frac{Y_1 + Y_2 + \cdots + Y_n}{n} \) is within two standard deviations of the mean \( \mu \) is at least 98%, given that \( Y_1, Y_2, \ldots, Y_n \) are independent and identically distributed random variables.

Answer: 13


Question: Find the norm of the element \( t + M \) in the quotient space \( C[0,1]/M \), where \( X = C[0,1] \) and \( M = \{ f \in C[0,1] : f(0) = f(1) = 0 \} \). Specifically, determine the norm of the element in \( X/M \) that contains the function \( f(t) = t \).

Answer: 1


Question: Calculate the limit: $$\lim_{n \to \infty}\frac{1}{n}\left(\frac{n}{n+1}+\frac{n}{n+2}+...+\frac{n}{2n}\right).$$ Choose the correct answer from the following options: (A) ∞ (B) 0 (C) ln 2 (D) 1.

Answer: C


Question: Determine whether the function \( f(x,y) = \begin{cases} \frac{x^2y^{\frac{4}{3}}}{x^4+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} \) is continuous at the point \((0,0)\). Specifically, evaluate whether \( \lim_{(x,y)\to(0,0)} f(x,y) = 0 \).

Answer: 0


Question: Let \( f \) be a continuously differentiable function with \( f'(0) \neq 0 \). For \( x > 0 \), define \( \alpha = \alpha(x) \) such that \( \int_{0}^x f(u) \, du = f(\alpha) x \), where \( \alpha \) is a number in \([0,x]\). Find \( \lim_{x\to 0^+ } \frac{\alpha(x)}{x} \).

Answer: \dfrac{1}{2}


Question:
In the polynomial ring $\mathbb{R}[X,Y,Z]$, identify which of the following ideals is a prime ideal:

A) $\langle X^{2},Y+Z\rangle$
B) $\langle X^{2}YZ,Z\rangle$
C) $\langle XY,XZ+YZ\rangle$
D) $\langle XY,XZ\rangle$.

Answer: B


Question: Let $f$ be a holomorphic function in the right half-plane, with $|f(z)| < 1$ for all $z$ such that $\text{Re}(z) > 0$, and $f(1) = 0$. Determine the largest possible value of $|f(2)|$. \( \text{Re}(z) \) denotes the real part of $z$. \( \boxed{} \)

Answer: \dfrac{1}{3}


Question: Consider the fractional order algebraic equation \( x^{\pi} - 1 = 0 \). Let \( U \) be the set of unit magnitude complex numbers defined as \( U = \{ x \mid x \in \mathbb{C}, |x|=1, x^{\pi} \neq 1 \} \). What percentage of the unit circle does the set \( U \) constitute?

Answer: 100


Question: Calculate the value of \(321^{654} \mod 1013\).

Answer: 863


Question: Given an angle ∠ABC and a point D within the angle, construct a line segment using only a compass and straightedge such that one endpoint lies on AB, the other on BC, and D is the midpoint of the segment.

Answer: PQ


Question: Given the joint probability density function of two continuous random variables \( X \) and \( Y \), where \( f(x,y) = 8xy \) for \( 0 \leq y \leq x \leq 1 \), find the conditional probability \( P(X<3/4 | Y<1/2) \).

Answer: \dfrac{1}{2}


Question: Let $B = \{ v_i \}_{i=1}^n$ be a basis of a vector space $V$, and let $T: V \rightarrow V$ be a linear operator such that $T(v_1) = \sum_{i=1}^n v_i$. Find the sum of the elements of the matrix $[T^{-1}]_B$, where $[T^{-1}]_B$ is the matrix representation of the inverse of $T$ with respect to the basis $B$. Provide your answer as a single number.

Answer: 1


Question: Calculate the expected value of the following game: You roll a fair six-sided die up to 60 times and receive an amount in dollars equal to the last number rolled. You can choose to stop rolling at any time. What is the expected value of this game?

Answer: 6


Question: Determine whether the following limit exists: \[ \lim_{h \to -\frac{1}{2}} \int_0^{1+2h}\frac{\sin(x^2)}{(1+2h)^3}dx \]

Answer: \dfrac{1}{3}


Question: Let $K$ be an algebraically closed field of characteristic zero, and let $B$ be an integral domain that is finitely generated as a $K$-algebra. Suppose $D$ is a derivation on $B$. If there exists an element $x \in B$ and a natural number $k \geq 2$ such that $D^n(x^k) = 0$ for some $n$, is it true that there exists a natural number $m$ such that $D^m(x) = 0$?

Answer: Yes


Question: Find all entire functions $f$ such that $f(\frac{1}{n}) = f(\frac{1}{n^2})$ for all $n \in \mathbb{N}$. Provide a justification for your answer.

Answer: f(z) = c


Question: Evaluate the triple integral \( \iiint \frac{xz}{1+x^2+y^2} \, dz \, dy \, dx \) over the region where \( 1 \leq x^2 + y^2 \leq 3 \) and \( 0 \leq z \leq 3 \).

Answer: 0


Question: Let $G$ be a complex reductive group acting linearly on a complex affine variety $X$. Define the kernel of the action as $K:=\{g\in G:g\cdot x=x\text{ for all }x\in X\}$. Is the set $X_K:=\{x\in X:\mathrm{Stab}_G(x)=K\}$ Zariski-open in $X$?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\rightarrow \infty}\int_{a}^{b} \frac{\sin (nx)}{x} \, dx \) for \(0 < a < b\).

Answer: 0


Question: Let \( f : \mathbb{R}^+ \to \mathbb{R} \) satisfy the equation \( f(xy) = e^{xy - x - y} (e^y f(x) + e^x f(y)) \) for all \( x, y \in \mathbb{R}^+ \). Given that \( f'(1) = e \), find the function \( f(x) \).

Answer: e^x \ln x


Question: Let $X$ and $Y$ be normed linear spaces, and let $T: X \to Y$ be a linear map such that $T^{-1}(\{y\})$ is closed for every $y \in Y$ and $T$ has a closed graph. Is $T$ necessarily continuous?

Answer: No


Question: Consider the Cantor space $2^\omega$ with the topology generated by the basic open sets $N_s = \{ \alpha \in 2^\omega : s \text{ is an initial segment of } \alpha \}$ for every $s \in 2^{<\omega}$. Let $U$ be a principal ultrafilter on $\omega$, identified with $\{ A \subseteq \omega : n \in A \}$ for some $n \in \omega$. Is $U$ an open set in this topology?

Answer: Yes


Question: Determine the positive integers $x$ and $y$ such that $\sigma_x(n)$ divides $\sigma_y(n)$ for all positive integers $n$, where $\sigma_k(n) = \sum_{d \mid n} d^k$. Provide your answer in terms of $x$ and $y$.

Answer: x = y


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

Answer: \dfrac{\pi}{2}


Question: Given that the lines $x+y-2=0$, $2x-y+1=0$, and $px+qy-r=0$ are concurrent, find the slope of the line in the family $2px+3qy+4r=0$ that is farthest from the origin.

Answer: -\dfrac{3}{10}


Question: Evaluate the contour integral \( \frac{1}{2\pi i} \oint_y \frac{f(C)}{(C-z)^2}dC \), where \( f: \mathbb{C} \to \mathbb{C} \) is holomorphic and \( y \) is a closed circle centered at \( z \in \mathbb{C} \), traced anticlockwise. Determine if the value of this integral is \( f'(z) \).

Answer: f'(z)


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left( x^6 \left[ e^{-\frac{1}{2x^3}} - \cos\left(\frac{1}{x\sqrt{x}}\right) \right] \right) \]

Answer: \dfrac{1}{12}


Question: Given that \( x, y, z \) are three real numbers such that \( \frac{(x-y)}{(2+xy)} + \frac{(y-z)}{(2+yz)} + \frac{(z-x)}{(2+zx)} = 0 \), determine if it is true that at least two of the three numbers must be equal. Justify your answer.

Answer: Yes


Question:
Integrate the probability density function (PDF) of the Kumaraswamy distribution to find its cumulative distribution function (CDF). The PDF is given by:
\[
f(x) = abx^{a-1}(1-x^a)^{b-1}, \quad 0 < x < 1.
\]
Perform the integration and verify that the CDF is:
\[
1 - (1 - x^a)^b.
\]

Answer: 1 - (1 - x^a)^b


Question: Calculate the integer \( \lambda \) such that the group homomorphism \( \mathbb{Z} \cong \pi_{n+1}(S^{n+1}) \rightarrow \pi_n(SO_{n+1}) \rightarrow \pi_n(S^n) \cong \mathbb{Z} \) is multiplication by \( \lambda \). The map is induced by the matrix multiplication map \( \alpha: SO_{n+1} \rightarrow S^n \) and the connecting homomorphism of the fibration \( SO_{n+1} \rightarrow SO_{n+2} \rightarrow S^{n+1} \).

Answer: 2


Question: Find the sum of the infinite series: \[ \sum^{\infty}_{k=1} \frac{6^k}{(3^k-2^k)(3^{k+1}-2^{k+1})} \]

Answer: 2


Question: Evaluate the limit \( \lim_{n \to \infty} \int_0^1 \frac{n \sin x}{x (1+n^2 x^2)} \, dx \).

Answer: \dfrac{\pi}{2}


Question: Calculate the conditional expectation \( \mathbb{E}(X|X+Y=a) \) given that \( X \) and \( Y \) are independent and identically distributed non-negative random variables.

Answer: \dfrac{a}{2}


Question: Let $X$ be a random variable such that $X \sim Po(3)$. Calculate the variance of $2^X$, i.e., $Var[2^X]$. Provide your answer as a single numerical value.

Answer: e^9 - e^6


Question: Let $A = \{ z \in \mathbb{C} : |z-1| \leq |z-i| \text{ and } |z-(2+2i)| \leq 1 \}$. Find the modulus of $z \in A$ such that the argument of $z$ is minimized. Express your answer as a single value.

Answer: \sqrt{7}


Question: Let $R$ be an exponential random variable with parameter $\lambda > 0$, and let $X$ be a random variable uniformly distributed on $(0, 2\pi)$, independent of $R$. Calculate $\mathbb{E}[Re^{iX}]$. Express your answer in terms of $\lambda$.

Answer: 0


Question: Let $U \subset W \subset V$ be three linear spaces with dimensions 3, 6, and 10, respectively. Define $E \subset L(V,V)$ as the set of linear transformations $f: V \rightarrow V$ such that $f(U) \subset U$ and $f(W) \subset W$. Determine the dimension of $E$. \( \dim E \).

Answer: 67


Question: Evaluate the integral \( \int_{0}^{\infty}\frac{1-\cos x}{x^2}dx \) using contour integration techniques.

Answer: \dfrac{\pi}{2}


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty} a_n x^n \), where \( a_n \) is the nth prime number.

Answer: 1


Question:
Calculate the limit of the sequence as \( n \to \infty \):
\[ \lim_{n \to \infty} \frac{1\cdot3\cdot5\cdot\cdots\cdot(2n-1)}{2\cdot4\cdot6\cdot\cdots\cdot(2n)} \]

Answer: 0


Question: Evaluate the flux of the vector field \( \vec{F} = \langle x, y, 5 \rangle \) through the surface \( S \), which is the region enclosed by \( x^2 + z^2 = 1 \), \( y = 0 \), and \( x + y = 2 \).

Answer: 4\pi


Question: Given two complex numbers \(z_1\) and \(z_2\) such that \(|z_1+z_2|=1\) and \(|z_1^2+z_2^2|=25\), find the minimum value of \(|z_1^3+z_2^3|\).

Answer: 37


Question: Is it possible to have a $9\times9$ Sudoku puzzle with less than 8 different numbers as clues?

Answer: No


Question: Determine whether a cubic graph with $2n$ vertices can have a minimal vertex cover consisting of $n-1$ vertices.

Answer: No


Question: Can a two-variable harmonic function \( f(x, y) \) be zero on a curve with a cusp?

Answer: No


Question: Determine for which complex numbers \( z \) on the unit circle the series \( \sum_{n=1}^\infty \frac{z^n}{n} \) converges.

Answer: z \neq 1


Question: Determine whether the operator \( T: L^2(\mathbb{R}) \to \mathbb{R} \) defined by \( T(f) = \int_{\mathbb{R}} f(x) e^{-x^2} \, dx \) is a compact operator.

Answer: Yes


Question:
Calculate the Lebesgue integral of the function \( f \) over the interval \([0,1]\), where \( f \) is defined as follows:
\[
f(x)=\begin{cases}
x^{4} & \text{if } x \in [0,1] \setminus \mathbb{Q} \\
0 & \text{if } x \in [0,1] \cap \mathbb{Q}
\end{cases}
\]

Answer: \dfrac{1}{5}


Question: Compute the limit: $$ \lim_{n\to\infty}(n^8+2016^n)\left(1-\sqrt{1-\frac{1}{n!}}\right). $$

Answer: 0


Question: Find the coordinates of vertex \( p \) in a trapezoid with vertices \( A = (2,6,-3,9) \), \( B = (4,4,7,3) \), \( C = (8,2,7,-1) \), and \( p = (x, y, z, w) \). The lines \( BA \) and \( Cp \) are parallel. Use the parametric equations for these lines to determine the coordinates of \( p \).

Answer: (6, 4, -3, 5)


Question: Given the curve \( \vec{r}(t) = \left[\frac{4}{9}(1+t)^{3/2}, \frac{4}{9}(1-t)^{3/2}, \frac{1}{3}t\right] \) for \( t \in [-1, 1] \), find the value of \( t \) where the torsion \( \tau(t) \) is minimal.

Answer: 0


Question: For a positive integer $n$ less than $1000$, the decimal representation of $\frac{1}{n}$ is a repeating decimal with a period of 6, and the decimal representation of $\frac{1}{n+6}$ is a repeating decimal with a period of 4. Determine the interval in which $n$ lies: \(\textbf{(A)}\ [1,200] \qquad \textbf{(B)}\ [201,400] \qquad \textbf{(C)}\ [401,600] \qquad \textbf{(D)}\ [601,800] \qquad \textbf{(E)}\ [801,999] \)

Answer: B


Question: Find the value of \(a\) such that the linear transformation \(T: \mathbb{R}^3 \to \mathbb{R}^3\) is diagonalizable, given that \(T(1,1,1) = (1,1,1)\), \(T(1,0,1) = (-3,0,-3)\), and \(T(1,1,-1) = (a+1,a+1,a-1)\).

Answer: 0


Question: Given that 3 is a primitive root modulo 2011, find an element of order 67 modulo 2011.

Answer: 1116


Question:
Determine whether the function \( f: \mathbb{R}^2 \to \mathbb{R} \) defined by
\[
f(x,y) =
\begin{cases}
e^{\frac{-1}{x^2+y^2}}, & \text{if } (x,y) \neq (0,0) \\
0, & \text{if } (x,y) = (0,0)
\end{cases}
\]
is differentiable at the origin \((0,0)\).

Answer: Yes


Question: Evaluate the integral \( \int_0^{2\pi} \int_0^{\pi} \sin^3(\phi) \cos^2(\theta) \, d\phi \, d\theta \) to find the integral of \( G(x,y,z) = x^2 \) over the unit sphere \( x^2 + y^2 + z^2 = 1 \) using spherical coordinates.

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


Question:
Find the eigenvalues of \(\exp(tA)\), where \(\exp(tA)\) is defined by the convergent Taylor series:
\[ \exp(tA) = I + tA + \frac{t^2}{2!} A^2 + \cdots + \frac{t^n}{n!} A^n + \cdots = \sum_{n=0}^\infty \frac{t^n A^n}{n!} \]
Here, \(I\) is the identity matrix and \(t\) is a real parameter.

Answer: e^{t\lambda}


Question: Let $V$ be a vector space with dimension $N$. Define the multiplication operator $L_{\mathbf{b}}$ as $L_{\mathbf{b}}: \omega \to \mathbf{b} \wedge \omega$, where $\omega \in \wedge V$ (the entire exterior algebra) and $\mathbf{b} \in V$. Compute the trace of $L_{\mathbf{b}}$ using a coordinate-free method, given that $\dim(\wedge V) = 2^N$.

Answer: 0


Question: Is the norm, as a map between the tangent bundle of a Riemannian manifold and \( \mathbb{R} \), continuous?

Answer: Yes


Question:
Let $S$ be an infinite set and define the following:

- $W = \mathcal{P}(S)$
- $B = \mathcal{P}(W)$
- $F = \{p \in B : \exists s \in S \text{ such that } p = \{w \in W : s \in w\} \text{ or } p = \{w \in W : s \not\in w\}\}$
- $C = \{p \in B : \forall X \subseteq F \text{ such that } \bigcap X \subseteq p, \exists Y \subseteq X \text{ such that } \bigcap Y \subseteq p \text{ and } \forall Z \subset Y, \bigcap Z \not\subseteq p\}$

Is $C$ a Boolean subalgebra of $B$ under the natural set-theoretic operations?

Answer: Yes


Question: Let $A^*$ denote the complex conjugate transpose of a matrix $A$. In the Euclidean norm, if \( ||A^*A+AA^*||=||A^*A|| \), does this imply that \( ||A^*A+AA^*||=||A^*A-AA^*|| \)? Determine whether this implication holds.

Answer: Yes


Question: Determine the number of solutions to the congruence \(x^3 \equiv 1 \pmod{162}\).

Answer: 3


Question: Find the remainder when $40!$ is divided by $1763$.

Answer: 1311


Question: Compute the expected value of the surplus water when water flows in and out of a dam. The daily inflow is uniformly distributed on $[0, 2]$ liters, and the daily outflow is uniformly distributed on $[0, 1]$ liters, independently of the inflow. Let $X$ be the inflow and $Y$ be the outflow, with $X \sim \text{unif}[0,2]$ and $Y \sim \text{unif}[0,1]$. The surplus water is given by $Z = X - Y$. Determine the expected amount of surplus water in a given day.

Answer: \dfrac{1}{2}


Question: Consider the matrix $A = (a_{ij})$ where \( a_{ij} = \begin{cases} 1 & \text{if } i + j = n + 1 \\ 0 & \text{otherwise} \end{cases} \). Determine the minimal polynomial of the matrix $A$. Note that one eigenvalue can be found using the eigenvector \( \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{pmatrix} \).

Answer: x^2 - 1


Question: Determine the degree of the Fourier expansion of the function \( f: \{-1,1\}^n \rightarrow \{-1,1\} \), where \( f(x) = \operatorname{sgn}(x_1 + x_2 + \cdots + x_n) \) and \( n \) is odd. Is the degree equal to \( n \) or less than \( n \)?

Answer: n


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{1}{\sqrt{1+|x|}} \, \frac{1}{\big(\operatorname{dist}(x,S)\big)^{2}} \] where the sequence of pairs \(S=\left\{\left(n, \frac{-1}{\sqrt{n}}\right)\right\}_{n=1}^{\infty}\) is given in the \(xy\)-plane, and \(\operatorname{dist}(x,S)\) denotes the distance between the point \(x\) and the set \(S\), defined by \(\operatorname{dist}(x,S)=\inf\limits_{a_{n}\in S}\operatorname{dist} (x,a_{n})\). The distance between any two points \(P=(x_{1},y_{1})\) and \(Q=(x_{2},y_{2})\) is given by \(\operatorname{dist}(P,Q)=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}\).

Answer: 0


Question: Find the complex number \(a\) such that the linear transformation \(f_a: \mathbb{C}^4 \rightarrow \mathbb{C}^4\), defined by \(f_a(u_i) = au_{i+1}\) for \(1 \le i \le 3\) and \(f_a(u_4) = au_1\), is unitary, where \(\{u_1, u_2, u_3, u_4\}\) is an orthonormal basis of \(\mathbb{C}^4\).

Answer: |a| = 1


Question: Determine whether the limit \( \lim_{n \to \infty} \frac{n^3(\sqrt{2}+(-1)^n)^n}{3^n} \) exists. If it does, find its value.

Answer: 0


Question:
Construct an explicit deformation retraction of \(\mathbb{R}^n - \{0\}\) onto \(S^{n−1}\). Consider the homotopy given by:
\[ f_t(x) = (1-t)x + t \frac{x}{|x|}, \quad t \in [0,1], \]
where \(f_0(x) = x\) and \(f_1(x) = \frac{x}{|x|}\). Show that this defines a deformation retraction of \(\mathbb{R}^n - \{0\}\) onto \(S^{n−1}\).

Answer: f_t(x) = (1 - t)x + t \frac{x}{|x|}


Question: Determine the first fundamental group of the space $Z = X \bigsqcup Y / (x_1 \sim y_1, y_2 \sim x_2)$, where $X$ and $Y$ are copies of $S^2$, and $x_1, x_2 \in X$ and $y_1, y_2 \in Y$ are points identified as described.

Answer: 0


Question: Let $M$ be an embedded submanifold of dimension $k$ in $\Bbb R^n$. Given coordinates $u^1, \dots, u^k$ for a region of $M \subset \Bbb R^n$, the inclusion map $M \hookrightarrow \Bbb R^n$ determines $n$ smooth functions $x_1(u^1, \dots, u^k), \dots, x_n(u^1, \dots, u^k)$. Define $\vec{x} = (x_1, \dots, x_n)$ and $\frac{\partial \vec{x}}{\partial u^i} = (\frac{\partial x_1}{\partial u^i}, \dots, \frac{\partial x_n}{\partial u^i})$. Can coordinates $(u^1, \dots, u^k)$ be chosen such that the symmetric matrix $$(g_{ij}) = \left( \dfrac{\partial \vec{x}}{\partial u^i} \cdot \dfrac{\partial \vec{x}}{\partial u^j}\right)$$ becomes the identity matrix at a fixed point $q \in M$?

Answer: Yes


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

Answer: -\ln 2


Question: Is there an operator $T: X \rightarrow Y$ that factors through a Banach space $Z$ containing no complemented copy of $l_1$, but does not factor through any Banach space $W$ containing no copy of $l_1$?

Answer: Yes


Question: Given a sample of IID uniform random variables $X_1, X_2, \ldots, X_n$, find the probability density function (PDF) of the random variable $Y = \max(X_1, X_2, \ldots, X_n)$. Provide the expression for the PDF of $Y$.

Answer: n y^{n-1}


Question: Compute \(d\alpha(V, W)\) at the point \((0,1)\) on \(\mathbb{R}^{2}\), where \(V = x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}\), \(W = y \frac{\partial}{\partial y}\), and the differential 1-form \(\alpha\) satisfies \(\alpha(V) = 2x - y^{2}\) and \(\alpha(W) = 2y\).

Answer: 1


Question: Is the hypothesis that the characteristic of the field is not 2 necessary for the classification of affine conics over an algebraically closed field?

Answer: Yes


Question: Determine whether the function \( g(x) = x \sin\left(\frac{1}{x}\right) \) for \( x \neq 0 \) and \( g(0) = 0 \) is Riemann integrable on the interval \([-1, 1]\).

Answer: Yes


Question: Let $A$ be a commutative ring and $t$ an indeterminate over $A$. Consider the ring $A[t]/(t^m)$, where $m$ is a positive integer. Suppose $f$ is an automorphism of this ring such that $f(x) \equiv x \pmod{(t)}$ for each $x \in A[t]/(t^m)$. Can we conclude that $f(t) = t$?

Answer: No


Question: Find the mean of the random variable $Y = X_1 + X_2 + \cdots + X_N$, where $N$ follows a Poisson distribution with parameter $\lambda$, and each $X_i$ is an independent Bernoulli random variable with parameter $\rho$. Assume $N$ and $(X_i)$ are independent. Express your answer in terms of $\lambda$ and $\rho$. \( \text{Mean of } Y = \boxed{\rho \lambda} \)

Answer: \rho \lambda


Question: Find the limit of the sequence \( a_n = \frac{1}{\sqrt{n}} \left( 1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \right) \) as \( n \to \infty \).

Answer: 2


Question: Let $D$ be a divisor on an algebraic complex smooth projective surface $S$. Assume that the complete linear system $|D|$ is not empty and has no fixed component. Determine whether it is true that $D^2 \geq 0$, where $D^2$ is the self-intersection of $D$. Provide a justification for your answer.

Answer: D^2 \geq 0


Question: Given a unitary matrix \( U \), find the unitary matrix \( V \) that maximizes \( \Re \, \text{tr}(UV) \).

Answer: V = U^\dagger


Question:
Calculate the integral of the scalar field \( \phi(x,y,z) = xyz \) over the surface of the unit sphere. Express the integral in spherical coordinates and evaluate it:
\[ I = \int_{S} \phi \, dA \]
where \( S \) is the surface of the unit sphere.

Answer: 0


Question: Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the equation \( f\left(x^2 + \frac{1}{4}\right) = f(x) \) for all real numbers \( x \).

Answer: f(x) = c


Question: Calculate the fractional part of \( \left\{\frac{22^{56}}{17}\right\} \).

Answer: \dfrac{16}{17}


Question: Let $k$ be a field and consider the ring $R = k[x,y,z,t]/(xy-z^3,z^5,x^2t^3+y^2)$. Determine the Krull dimension of $R$. Provide your answer as a single integer.

Answer: 1


Question:
For which values of $p \in [1, \infty]$ does the function $u(x)$, defined on the square $\Omega = \{ (x_1, x_2) \in \mathbb{R}^2 \mid |x_1| < 1, |x_2| < 1 \}$ by:
\[
u(x) = \begin{cases}
1-x_1 & \text{if } |x_2| < x_1, \\
1+x_1 & \text{if } |x_2| < -x_1, \\
1-x_2 & \text{if } |x_1| < x_2, \\
1+x_2 & \text{if } |x_1| < -x_2,
\end{cases}\]
belong to the Sobolev space $W^{1,p}(\Omega)$?

Answer: [1, \infty]


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function such that \( \lim_{n \to \infty} \int_{0}^1 f \left(\frac{x}{n}\right) dx = 0 \). Determine the value of \( f(0) \).

Answer: 0


Question:
Calculate the order parameter \( S \) using the integral:
\[ S = 2\pi\int_0^\pi P_2(\cos\theta)f(\theta)\sin\theta \, d\theta \]
where \( P_2(\cos\theta) = \frac{3}{2}\cos^2\theta - \frac{1}{2} \) and \( f(\theta) = \frac{\delta(\cos\theta-1)}{2\pi} \). Verify that \( S = 1 \).

Answer: 1


Question: Let \( S_n = \frac{1}{n^6} \sum_{k=1}^{n} k^5 \) and \( T_n = \frac{1}{n^6} \sum_{k=1}^{n-1} k^5 \). Determine whether \( S_n + T_n > \frac{1}{3} \) or \( S_n + T_n < \frac{1}{3} \) as \( n \to \infty \).

Answer: S_n + T_n > \frac{1}{3}


Question: Let $f$ be a continuous function on $\mathbb{R}$ with compact support, and let $g \in \mathcal{S}(\mathbb{R})$ be a Schwartz class function. Is the convolution $f \ast g$ also in $\mathcal{S}(\mathbb{R})$?

Answer: Yes


Question: Calculate the dimension of the \( \mathbb{C} \)-vector space \( \mathfrak{m}/\mathfrak{m}^2 \), where \( A = \left(\frac{\mathbb{C}[x,y]}{(y^2-x^3-x^2)}\right)_{(x,y)} \) is the local ring and \( \mathfrak{m} = (x,y)A \) is the maximal ideal in \( A \).

Answer: 2


Question: What is the Grothendieck group of a commutative group? Is it the group itself?

Answer: G


Question:
Determine the set of points in the complex plane \(\mathbb{C}\) where the function \(f(z)\) is differentiable, given by:
\[
f(z) = \begin{cases}
\frac{z^4}{|z|^2}, & \text{if } z \neq 0 \\
0, & \text{if } z = 0
\end{cases}
\]

Answer: \{0\}


Question: Let \( X \sim N(50, 20.5) \) represent a population. If \( X_1, X_2, \ldots, X_n \) is a random sample from this population, determine the sample size \( n \) such that \( \mathbb{P}(X_1 + X_2 + \ldots + X_n > 2000) = 0.95. \)

Answer: 41


Question: Let $X$ and $Y$ be smooth complex varieties, and let $G$ be a smooth affine algebraic group acting transitively on both $X$ and $Y$, making them $G$-homogeneous spaces. Consider a smooth $G$-equivariant morphism $f: Y \to X$ and a sheaf $\mathcal{D}$ of $G$-homogeneous twisted differential operators on $X$. Define the pullback sheaf $f^{\#} \mathcal{D} := \text{Dif}_{f^{-1} \mathcal{D}}(f^* \mathcal{D}, f^* \mathcal{D})$ on $Y$. Determine whether $f^{\#} \mathcal{D}$ is a $G$-homogeneous twisted differential operator on $Y$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the line integral \( \int_{(0,1,0)}^{(1,0,2)} \frac{z}{y}\mathrm{d}x + (x^2+y^2+z^2)\mathrm{d}z \) along the curve in the first octant given by \(x^2+y^2=1, z = 2x\). Use an appropriate parametrization to find the correct value of the integral.

Answer: \dfrac{20}{3}


Question:
Find the continuous function \( f(x) \) that satisfies the equation:
\[
4(1-x)^{2} f \left(\frac{1-x}{2} \right)+16f \left(\frac{1+x}{2} \right)=16(1-x)-(1-x)^{4}
\]

Answer: 1 - x^2


Question: Let $I=[0,1]$ with Lebesgue measure and $1\leq p<\infty$. Consider a sequence $f_k$ in $L^p(I)$ such that $\|f_k\|_p\leq 1$ for all $k$. If $f(x)=\lim_{k\rightarrow\infty} f_k(x)$ exists almost everywhere, must $f$ belong to $L^p(I)$?

Answer: Yes


Question:
Find the time \( t^* \) and position \( x^* \) where the shock wave collides with the rarefaction wave for the equation:
\[ \partial_t u + \partial_x\left(\frac{u^2}{2}\right) = 0 \]
with the initial conditions:
\[
u(x,0) = g(x) = \begin{cases}
2, & \text{if} \ x<1 \\
1, & \text{if} \ 1<x<2 \\
2, & \text{if} \ x>2
\end{cases}\]
The characteristic equation is:
\[ x(t) = \xi + g(\xi)t = \begin{cases}
\xi + 2t, & \text{if} \ \xi \leq 1 \\
\xi + t, & \text{if} \ 1<\xi<2 \\
\xi + 2t, & \text{if} \ \xi>2
\end{cases}\]
The shock wave originates from \((1,0)\) and the rarefaction wave from \((2,0)\). The shock wave is described by:
\[ x = s(t) = \frac{3}{2}t + 1 \]
The rarefaction wave is between \(x(t) = 1+t\) and \(x(t) = 1+2t\) with:
\[ u(x,t) = \frac{x-2}{t} \]
Determine the intersection point \((x^*, t^*)\).

Answer: (4, 2)


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

Answer: \dfrac{2}{15}


Question: Find the value of \(x\) such that \(\sum_{n=1}^\infty\frac{n^x}{2^n n!} = \frac{1539}{64}e^{1/2}.\)

Answer: 6


Question:
Given integers \(n\) and \(k\), determine the minimum number of iterations \(r\) required such that the sequence defined by:
\[ n_0 = n \]
\[ n_1 = \lceil n_0^{1/k} \rceil \]
\[ n_2 = \lceil n_1^{1/k} \rceil \]
\[ \vdots \]
\[ n_r \leq 2 \]
Find the gap \( r - \frac{\log_2 \log_2 n}{\log_2 k} \).

Answer: 1


Question: Solve the recurrence relation: \[ T(n) = 2T(n - 1) + (\log n)T([\log n]) + 1 \]

Answer: T(n) = \Theta(2^n)


Question: Calculate the value of the limit \( \lim_{k \to \infty} I(k) \), where \( I(k) = \int_0^\infty \frac{e^{-k^2(1+x^2)}}{1+x^2} \, dx \).

Answer: 0


Question: Given a convergent sequence $(x_k)$, does the condition \(|x_{k+1}| \le C|x_k| + o(|x_k|)\) with \(|C| < 1\) imply that the rate of convergence of \((x_k)\) is less than \(C\)?

Answer: No


Question: Determine the domain of definition of the function \( \frac{S_1}{S_0} \) on the projective plane \( \mathbb{P}^2 \) over an algebraically closed field \( k \). Consider whether the domain is simply \( S_0 \neq 0 \) or if it extends further.

Answer: S_0 \neq 0


Question: Let \( A \) be a real, symmetric, positive definite \( n \times n \) matrix, and let \( B \) be a real, symmetric \( n \times n \) matrix. Determine whether there exists an \( \epsilon > 0 \) such that \( A + \epsilon B \) is positive definite.

Answer: Yes


Question: Is it possible to reconstruct an unweighted graph $G = (V, E)$ by making only polynomial (in $|V|$) many queries to its cut function $C:2^V \rightarrow \mathbb{Z}$, where $C(S) = |\{(u,v) \in E : u \in S \wedge v \not\in S\}|$?

Answer: Yes


Question: Prove that 7 cannot be expressed as the sum of the squares of two rational numbers.

Answer: 7


Question: Given that \((B_t)\;t>0\) is a Brownian motion, find the expected value \(E(B_2 B_3 B_5)\).

Answer: 0


Question: Is the Cantor cube \(C = \{0,1\}^I\) a closed subset of the Hilbert cube \(H = [0,1]^I\)? Provide a proof to support your answer.

Answer: Yes


Question: Find the value of the constant $K$ such that the function \( u(x,t) = \frac{2}{\sqrt{\pi}}\int_{0}^\frac{x}{\sqrt{t}} e^{-s^2}ds \) satisfies the partial differential equation \( \frac{\partial u}{\partial t} = K\frac{\partial^2u}{\partial x^2} \).

Answer: \dfrac{1}{4}


Question: Determine the number of group homomorphisms from $A_3$ to $S_3$. Provide your answer as a single integer.

Answer: 3


Question: Let \( f: \Bbb{N} \to \Bbb{N} \) be a strictly increasing function such that \( f(f(n)) = 3n \) for all \( n \in \Bbb{N} \). Find \( f(2001) \).

Answer: 3816


Question: Given the polynomial \( f(z) = z^4 - 4z^3 + 8z - 2 \), determine the number of zeros (including multiplicity) that the function has within the region \( \{ z \in \mathbb{C} : |z| < 3 \} \).

Answer: 3


Question: Determine whether it is true that a graph with more than six vertices of odd degree cannot be decomposed into three paths.

Answer: True


Question: In triangle $ABC$, the centroid is $G$. Points $P$ and $Q$ are located on segments $AB$ and $BC$, respectively, such that $BP = 4AP$ and $BQ = 8$. If points $P$, $G$, and $Q$ are collinear, find the length of segment $QC$. Express your answer as a fraction.

Answer: 6


Question: Consider a smooth manifold $M$ and a smooth function $f: M \rightarrow \mathbb{R}$. If $p \in M$ is a local extremum of $f$, must $p$ be a critical point? Determine whether Fermat's theorem about local extrema holds in this context.

Answer: Yes


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

Answer: e^2


Question: Is it possible to have a ring $A$, with identity, such that $A^m \cong A^n$ as left $A$-modules with $m \neq n$? Provide a proof or counterexample.

Answer: Yes


Question: Evaluate the limit of the infinite product: \[ \lim_{n \to \infty} \frac{1\cdot4\cdot7\cdot...\cdot(3n+1)}{2\cdot5\cdot8\cdot...\cdot(3n+2)} \]

Answer: 0


Question: Let $R$ be a regular local ring, not necessarily of dimension one. Consider a nodal curve $X \to \text{Spec } R$, where all geometric fibers are reduced and have at most nodal singularities. Assume $X$ admits a stable model $Y$ over $\text{Spec } R$. Is there necessarily a morphism $X \to Y$ over $\text{Spec } R$ that contracts unstable $\mathbb{P}^1$'s over each geometric point of $\text{Spec } R$?

Answer: Yes


Question: Let \( f \) be the real-valued function defined on the interval \((-1,1)\) such that \( e^{-x}f(x)=2+ \int_0^x\sqrt{t^4+1}dt \) for all \( x\in(-1,1) \). Let \( f^{-1} \) be the inverse function of \( f \). Find the value of \((f^{-1})'(2)\).

Answer: \dfrac{1}{3}


Question: Compute the integral \( \iint_D (3x^3 + 3x^2y + x + y) \, dA \) over the region \( D \) bounded by the curves \( y = x^3 - 4 \), \( y = x^3 + 4 \), \( x + y = 6 \), and \( x + y = 0 \), using the change of variables \( u = x^3 \) and \( v = x + y \).

Answer: 144


Question: Prove that \( \sum\limits_{b=0}^{p-1}\left(\frac{b}{p}\right) = 0 \), where \( p \) is a prime number and \( \left(\frac{b}{p}\right) \) is the Legendre symbol.

Answer: 0


Question: Compute the volume of the solid bounded by the surfaces: \( z = x^2 \), \( z = x + 2 \), \( y + z = 5 \), and \( y = -1 \).

Answer: \dfrac{99}{5}


Question: Determine the number of elements in the ring \( \mathbb{F}_2[x]/(x^5 + x^3 + 1) \) that are divisible by \( x^2 + 1 \).

Answer: 32


Question: Evaluate the integral \( \int _0 ^{2\pi} e^{\sin\theta} \sin(\cos \theta)\,d\theta \).

Answer: 0


Question: Prove by induction that the sequence \(x_n = 10^{(3n+2)} + 4(-1)^n\) is divisible by 52 for all natural numbers \(n\).

Answer: 52


Question: Evaluate \(\lim \limits_{n \to \infty} ((n+1)\ln(n+1))^c-(n\ln(n))^c\) for any \(c \in (0,1)\). Determine if this limit equals zero.

Answer: 0


Question: Given that \( a, b, c \in \mathbb{R}^+ \) satisfy the conditions \( a + \frac{b}{16} + \frac{c}{81} \leq 3 \), \( \frac{b}{16} + \frac{c}{81} \leq 2 \), and \( c \leq 81 \), find the maximum value of the function \( A = \sqrt[4]{a} + \sqrt[4]{b} + \sqrt[4]{c} \).

Answer: 6


Question: Determine whether there exists a homomorphism from the group $K = \langle u,v \mid u^2 = 1, v^4 =1, uv=vu\rangle$ to $\mathbb{Z}_4$. Provide a justification for your answer.

Answer: Yes


Question:
Calculate the limit using the Central Limit Theorem:
\[ \lim_{n \to \infty} \frac{8^n}{27^n}\sum_{k=0}^n\binom{3n}{k}\frac{1}{2^k} \]

Answer: \dfrac{1}{2}


Question: Given that \( f(x) = \int_{0}^{x} \sqrt{f(t)} \, dt \) and \( f(0) = 0 \), find the value of \( f(6) \). Assume \( f: [0, \infty) \rightarrow [0, \infty) \) is continuous on \([0, \infty)\) and differentiable on \((0, \infty)\).

Answer: 9


Question:
Evaluate the limit of the sequence of integrals:
\[ \lim_{n\rightarrow \infty}  \int _0^\infty \frac{(1-e^{-x})^n}{1+x^2}dx. \]
Use the Dominated Convergence Theorem to justify the interchange of the limit and the integral, and calculate:
\[ \int _0^\infty  \lim_{n\rightarrow \infty} \frac{(1-e^{-x})^n}{1+x^2}dx. \]

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to\infty}\lim_{x\to 0}\frac{1}{(1^{\cot^2x}+2^{\cot^2x}+3^{\cot^2x}+\ldots+n^{\cot^2x})^{\tan^2x}} \]

Answer: 0


Question:
Find the power of the signal \(x(t) = 2\operatorname{rect}(\sin(2\pi t))\), where the rectangular function is defined as:
\[ \operatorname{rect}(t) = \Pi(t) = \begin{cases} 0 &|t| >\frac{1}{2} \\ \frac{1}{2} & |t| = \frac{1}{2} \\ 1 & |t|< \frac{1}{2} \end{cases} \]
The power of a periodic signal \(x(t)\) is given by the formula:
\[P = \frac{1}{T_0}\int_{T_0}{|(x(t))|^2dt}\]
where \(T_0 = 1\) is the period. Calculate the power of the signal.

Answer: \dfrac{4}{3}


Question: Consider the operator \( T: L^2([0,1]) \to L^2([0,1]) \) defined by \( Tf(t) = \int_0^t f(s) \, ds \). Find \( \alpha < 1 \) such that \( ||T|| \leq \alpha \).

Answer: \dfrac{2}{\pi}


Question: Determine the cardinality of an infinite set $S$ of real positive numbers such that the sum of the elements of any non-empty finite subset of $S$ is less than or equal to 1. Assume the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC).

Answer: \aleph_0


Question: Given the function \(f(x) = \frac{ax+b}{cx+d}\) with \(abcd \neq 0\) and the condition \(f(f(x)) = x\) for all \(x\) in the domain of \(f\), find the value of \(a+d\).

Answer: 0


Question: Determine the number of polynomial functions \( f \) of degree at least 1 that satisfy the equation \( f(x^2) = (f(x))^2 = f(f(x)) \) for all real \( x \).

Answer: 1


Question:
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, and let $S$ and $\xi$ be independent real-valued $\mathbb{P}$-random variables. Consider the real-valued measurable functions $f$ and $g$ such that
\[f(S(\omega), \xi(\omega)) \leq g(S(\omega), \xi(\omega)), \quad \text{for almost all $\omega\in\Omega$}.\]
Assume that the distribution $\mathbb{P}_S$ of $S$ has a Lebesgue-density. Does it follow that for Lebesgue-almost all $s\in\text{supp}(\mathbb{P}_S)$, the inequality
\[f(s, \xi)\leq g(s, \xi)\quad \text{holds $\mathbb{P}$-almost surely}?\]

Answer: Yes


Question: Determine the radius of convergence for the power series \( \sum_{n=1}^{\infty} \frac{(-1)^{\lfloor \sqrt{n} \rfloor}}{n} x^n \), where \( \lfloor \sqrt{n} \rfloor \) denotes the floor function.

Answer: 1


Question: Evaluate the limit: $$\lim_{x,y\to 0}\frac{(x^3y+xy^3)\ln(1+x^2+y^4)}{x^4+6x^2y^2+y^4}$$

Answer: 0


Question: Find the coefficient of \(x^{30}\) in the expansion of \(\left(\frac{1-x^{11}}{1-x}\right)^3\left(\frac{1-x^{21}}{1-x}\right)\).

Answer: 1111


Question: In a quadrilateral $ABCD$ where $AB$ is not parallel to $CD$, use a compass and straightedge to construct a point $M$ such that triangles $AMD$ and $BMC$ are similar.

Answer: M


Question: Let \( S \) be an invertible \( n \times n \) matrix in \( M_{n \times n}(\mathbb{C}) \). Prove that if \( x \in \mathfrak{gl}_S(n, \mathbb{C}) \), then the trace of \( x \) is zero, where \( \mathfrak{gl}_S(n, \mathbb{C}) = \{ x \in M_{n \times n}(\mathbb{C}) \mid x^T S = -S x \} \).

Answer: 0


Question:
Let $X_1, X_2, \ldots$ be independent identically distributed (i.i.d.) random variables with $E(X_i) = 2$ and $\text{Var}(X_i) = 1$. Determine the almost sure limit of the sequence:
\[ Y_n = \frac{\sum_{i=1}^n X_i \sum_{i=1}^n (X_i - 2)^2}{n^2} \]

Answer: 2


Question: Evaluate the contour integral \( \int_{\gamma} \sin(z) \, dz \), where \( \gamma \) is a circle of radius 5 centered at the point \( i \) in the complex plane.

Answer: 0


Question: Consider a sequence \( \{x_n\} \) of real numbers. For any two subsequences \( \{u_n\} \) and \( \{v_n\} \) (with no repeated terms) of \( \{x_n\} \), it is given that \( |u_n - v_n| \to 0 \) as \( n \to \infty \). Can we conclude that the sequence \( \{x_n\} \) is convergent?

Answer: Yes