Question: Suppose you have a continuous probability density function $f_Y(y)$ and you take $n$ random observations from it. What is the probability that the last observation recorded is the smallest number in the entire sample?

Answer: \dfrac{1}{n}


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

Answer: 2\pi


Question:
Determine the number of solutions for the following system of partial differential equations. Suppose \( u \in C^2(\bar{B}) \), where \( B \) is the unit ball in \( \mathbb{R}^2 \), satisfies:
\[ \Delta u = f \quad \text{in } B \]
\[ \alpha u + \frac{\partial u}{\partial n} = g \quad \text{on } \partial B, \quad \alpha > 0 \]
where \( n \) is the unit outward normal to \( B \). If a solution exists, choose the correct statement:
1. It is unique.
2. There are exactly two solutions.
3. There are exactly three solutions.
4. There are infinitely many solutions.

Answer: 1


Question: Determine the smallest possible order of a finite group that contains elements of every order from 1 to 12.

Answer: 27720


Question: Compute the work done by the radial vector field \( F(x, y, z) = xI + yJ + zK \) on a particle moving along the curve \( C \) given by \( X(t) = (1 + 3\sin(t))I + (1 + 3\sin^2(t))J + (1 + 5\sin^3(t))K \) for \( 0 \leq t \leq \pi/2 \).

Answer: \dfrac{65}{2}


Question: In triangle $\triangle PQR$, it is given that $QR^2 + PR^2 = 5PQ^2$. Let $G$ be the intersection point of medians $PM$ and $QN$. Determine the measure of $\angle QGM$. Provide a solution using pure geometry.

Answer: 90^\circ


Question: Determine the order of the pole at $z=0$ for the function \( f(z)=\frac{1}{(2\cos(z)-2+z^2)^2} \).

Answer: 8


Question: Is convergence in distribution on a product space defined by the condition \( \mathbf{E}[f(X_n)g(Y_n)]\to\mathbf{E}[f(X)g(Y)] \) for all continuous, bounded functions \( f:S\to\mathbb{R} \) and \( g:T\to\mathbb{R} \)?

Answer: Yes


Question: Determine the range of $xy$ given the conditions: $x^2 - xy + y^2 = 9$ and $|x^2 - y^2| < 9$, where $x, y \in \mathbb{R}^+$. Find the range of $xy$. \( \text{(Express your answer as an interval.)} \)

Answer: (6, 9]


Question: Evaluate the binomial sum \( \sum_{k=0}^{n} k \binom{n}{k} (-1)^{k-1} 2^{n-k} \).

Answer: n


Question: Compute the dimension of the vector space \( \mathbb{Q}[s^{\pm 1},t^{\pm 1}]/(t^2-t+1,s^2t-st+s-1) \) over \( \mathbb{Q} \).

Answer: 4


Question: Given the summation \( S_n=\sum^{n-1}_{i=0}\sum^{i-1}_{j=0}\sum^{j-1}_{k=0} (i+j+k) \), there exist positive integers \( A \) and \( B \) such that \( \frac{1}{S_3}+\frac{1}{S_4}+\frac{1}{S_5}+\dots=A-\frac{2\pi^2}{B} \). Find the value of \( A+B \).

Answer: 10


Question: Consider the elliptic curve given by the equation \(X^3 + Y^3 = AZ^3\), where \(A\) is a non-zero element in the field \(K\) and \(O = (1, -1, 0)\) is a point on the curve. Show that the \(j\)-invariant of this elliptic curve is 0.

Answer: 0


Question: Given a finite set $X$ and a subset $S \subseteq X$, consider the equivalence relation $R$ on the power set $P(X)$ defined by: \((A, B) \in R \iff A \cup S = B \cup S\). Determine the size of the quotient set $P(X)/R$. Provide your answer as a single integer.

Answer: 2^{|X| - |S|}


Question: Determine the number of non-isomorphic groups of order 122.

Answer: 2


Question: Evaluate the integral \( \iint x^2 (1-x^2-y^2)\,dxdy \) over the interior of the circle of radius 1 centered at the origin.

Answer: \dfrac{\pi}{12}


Question: Given that \(ab \equiv 1 \pmod{m}\) and \(ord_m a = 10\), find \(ord_m b^2\).

Answer: 5


Question: A satellite of mass 'm' is revolving around Earth in a circular orbit of radius \( r_0 \) with speed \( v_0 \). At a certain point in its orbit, the direction of motion of the satellite is suddenly changed by an angle \( \theta = \cos^{-1}\left(\frac{3}{5}\right) \), while maintaining the same speed. As a result, the satellite enters an elliptical orbit around Earth. Calculate the ratio of the speed at perigee to the speed at apogee of the new elliptical orbit.

Answer: 9


Question: Given the function \( f(x) = 1 + \sum_{k=0}^{100}\frac{(-1)^{k+1}}{(k+1)!}x(x-1)(x-2)\cdots (x-k) \), find the value of \( S(f(x)) - S(f'(x)) \), where \( S \) denotes the sum of the real roots of the respective function.

Answer: 51


Question: Determine the values of \( p \in (1, +\infty) \) for which the Banach space \( l_p \) is a Fréchet differentiable space (FD-space).

Answer: (1, +\infty)


Question: Let $X$ be a continuous random variable with probability density function $f$ and cumulative distribution function $F$. Suppose that $f$ is continuous and $f(x) > 0$ for all $x \in \mathbb{R}$. Compute the probability density function of the random variable $F[X]$. Provide your answer in terms of $f$ and $F$.

Answer: 1


Question:
Given $u_n = \int_1^n e^{-t^2} \, dt$ for $n = 1, 2, 3, \ldots$, determine which of the following statements is true:

1) Both the sequence $u_n$ and the series $\sum u_n$ are convergent.
2) Both the sequence $u_n$ and the series $\sum u_n$ are divergent.
3) The sequence $u_n$ is convergent and $\sum u_n$ is divergent.
4) $\lim_{n \to \infty} u_n = \frac{2}{e}$.

Answer: 3


Question: Let \( f(x) \) be a real-valued function such that \( f(0) = 78 \), \( f(x+2) - f(x) \leq 3 \cdot 2^x \), and \( f(x+6) - f(x) \geq 63 \cdot 2^x \). Given that \( f(78) = k - 1 + 2^k \), find the value of \( k \).

Answer: 78


Question: Is the function \( f(x) = (x-a)^n g(x) \) \( n \) times differentiable at \( a \) if \( g \) is continuous at \( a \)? Justify your reasoning using the difference quotient operator \( \Delta^n_h f(x) = \frac{1}{h^n} \sum_{k=0}^n \binom{n}{k} f(x+(n-k)h)(-1)^k \).

Answer: Yes


Question: Compute the remainder when the power tower \(3^{3^{3^{\cdots}}}\) with a height of 2020 is divided by 46.

Answer: 41


Question: Consider two unknowns $y \in \mathcal{Y}$ and $s \in \mathcal{S}$. Let $f$ be a function of $(y,s)$ and $g$ be a function of $s$. Suppose the maximizers of the problem \( \max_{y \in \mathcal{Y}, s \in \mathcal{S}} f(y,s) - g(s) \) are unique and denoted by $(y^*, s^*)$. Now consider the problem \( \max_{y \in \mathcal{Y}} f(y, s^*) \) with a unique maximizer. Is it true that the maximizer of this problem, \( \arg\max_{y \in \mathcal{Y}} f(y, s^*) \), is equal to $y^*$?

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\sin^2 x}{x^2} \, dx \) using the principal value and complex analysis techniques.

Answer: \pi


Question: Evaluate the limit: \[ \lim_{n \to \infty}\left\{2n^2\cos\left(\frac{1}{n}\right)\right\}\cos\left(\frac{1}{n}\right) \] where \( \{x\} = x - \lfloor x \rfloor \) denotes the fractional part of \( x \).

Answer: 0


Question: Determine whether the parametric curve given by \( x(t) = 5t^3 \) and \( y(t) = 4t - \sin(4t) \) has a well-defined tangent line at the origin \((0, 0)\).

Answer: Yes


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

Answer: 1


Question: Given an invertible matrix $A$ such that all elements in $A$ and $A^{-1}$ are integers, find $|A^4|$.

Answer: 1


Question: Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a strictly convex continuous function that is differentiable at a point $x_0 \in \mathbb{R}$. Is $f$ differentiable on some neighborhood $U$ of $x_0$? Provide a justification for your answer.

Answer: No


Question: Determine whether 3 is the only natural number \( n \) such that \( 2^n - n - 2 \) is a prime number.

Answer: 3


Question: Let $\bar D=\{z\in \mathbb{C}: |z|\leq 1\}$. Consider a monic polynomial $p: \bar D \longrightarrow \mathbb{C}$ that has no roots in $\bar D$. Prove or disprove that the uniform norm of $p$, denoted by $\|p\|_\infty = \sup\{|p(z)| : z \in \bar D\}$, is greater than or equal to 2.

Answer: 2


Question:
Consider a matrix where each element $x_{ij}$, for $i=1,2,\ldots,n$ and $j=1,2,\ldots,m$, is an independent and identically distributed continuous random variable. Determine the probability that the following group of vectors is linearly independent:
\[
\left(\sum_{j=1}^m x_{1j}, \sum_{j=1}^m x_{2j}, \cdots, \sum_{j=1}^m x_{nj}\right), \\
\left(\sum_{j=1}^m x_{1j}^2, \sum_{j=1}^m x_{2j}^2, \cdots, \sum_{j=1}^m x_{nj}^2\right), \\
\vdots \\
\left(\sum_{j=1}^m x_{1j}^n, \sum_{j=1}^m x_{2j}^n, \cdots, \sum_{j=1}^m x_{nj}^n\right)
\]
Is this probability equal to 1?

Answer: 1


Question: Determine the radius of convergence of the series \( \sum\limits_{n=1}^\infty n!x^{n^2} \).

Answer: 1


Question: Determine whether there exist rational numbers $x, y, z$ with $xyz \neq 0$ such that $y^2 = x^3 + z^{4k}$ for some integer $k > 1$. Provide a justification for your answer.

Answer: Yes


Question: Does the unit interval [0, 1] have a finite partition such that no element of the partition contains an open interval? Provide a justification for your answer.

Answer: Yes


Question: Let \(a, b, c, d, e, f\) be real numbers such that the polynomial \( p(x) = x^8 - 4x^7 + 7x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f \) factorizes into eight linear factors \(x-x_i\), with \(x_i > 0\) for \(i = 1, 2, \ldots, 8\). Determine all possible values of \(f\).

Answer: \dfrac{1}{256}


Question: Evaluate the limit: \[ \lim_{x \rightarrow -\infty}\sum_{k=1}^{1000} \frac{x^k}{k!} \]

Answer: \infty


Question: True or False: No parametrized surface is closed in \(\mathbb{R}^3\). Determine whether it is true that no parametrized surface is topologically closed in \(\mathbb{R}^3\). Use the definition provided in Fitzpatrick's Advanced Calculus (2009).

Answer: B


Question: In Fisher's lady tasting tea experiment, consider a scenario where the lady can correctly guess the type of each cup with a probability of 90% (i.e., \( P(\text{guess } A|\text{true } A) = P(\text{guess } B|\text{true } B) = 0.9 \)). Given a total of \( N = 8 \) cups, with \( n = 4 \) cups of each kind, calculate the power of the test under this alternative hypothesis. Assume the lady knows there are \( n \) cups of one kind. What is the distribution of \( k \), the number of correctly guessed cups, under this alternative hypothesis?

Answer: 0.831


Question: Find the limit of the sequence \( A_n = \frac{n^3 + n!}{2^n + 3^n} \) as \( n \to \infty \).

Answer: \infty


Question: Find a closed form for the series \( S_a = \sum_{n=1}^\infty \frac{\ln(n)}{n^a} \) for \( a > 1 \).

Answer: -\zeta'(a)


Question:
Evaluate the limit:
\[ \lim_{n\to \infty}\lim_{m\to \infty} \frac{1+\sqrt[n]{1^n+2^n}+\sqrt[n]{2^n+3^n}+\sqrt[n]{3^n+4^n}+\cdots+\sqrt[n]{(m-1)^n+m^n}}{m^2} \]

Answer: \dfrac{1}{2}


Question: Find the constant \( c \) such that \( \omega \wedge \nu \wedge \gamma = c\, dx \wedge dy \wedge dz \), where \( \omega = dx + 5dy - dz \), \( \nu = 2dx - dy + dz \), and \( \gamma = -dx + dy + 2dz \).

Answer: -29


Question: Determine whether there exists a homeomorphism between the topological spaces $(X,T^1)$ and $(X,T^2)$, where $T^1$ and $T^2$ are topologies on the set $X$ such that $T^1$ is a proper subset of $T^2$. Provide a justification for your answer.

Answer: No


Question: Let $\{e_i\}$ be a basis of a $\mathbb{C}$-linear space $E$ and $h$ a Hermitian metric on $E$. Denote by $H=\{h_{ij}\}$ the matrix of $h$ for $\{e_i\}$, where $h_{ij}=h(e_i,e_j)$. Let $g\in GL(E)$ be a complex automorphism and consider the new basis $\{e'_i\}=\{ge_i\}$. Denote by $H'=\{h'_{ij}\}$ the corresponding matrix. Find the relation between $H$ and $H'$.

Answer: H' = G^* H G


Question: Determine the exact value of the Hales-Jewett number $HJ(2,r)$, where $HJ(k,r)$ is the smallest $n$ such that any coloring of the elements of $[k]^n$ with $r$ colors contains a monochromatic combinatorial line.

Answer: r


Question: Differentiate the function \( \log(\Gamma(x + r)) \) with respect to \( r \). Express your answer in terms of the digamma function.

Answer: \psi(x + r)


Question: Find the closed form of the summation \( \sum\limits _{i=0}^{\lfloor n/2 \rfloor} {n-i \choose i} \).

Answer: F_{n+1}


Question: Prove that \( \lim_{n \rightarrow \infty}\frac{1^k+2^k+\ldots+n^k}{n^{k+1}}=\frac{1}{k+1} \) using the concept of Riemann sums and the integral of \( x^k \).

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


Question: Is it true that every finite group of odd order is isomorphic to a subgroup of $A_n$? Justify your answer.

Answer: Yes


Question: Let $F: \mathcal{B} \to \mathbb{R}$ be a functional on a Banach space $\mathcal{B}$. For a fixed $x \in \mathcal{B}$ and $c \in \mathbb{R}$, define the map $\phi_x: \mathbb{R} \to \mathbb{R}$ by $\phi_x(c) = F(cx)$. Find an expression for the derivative $\frac{d}{dc}\phi_x(c)$. Assume $F$ is differentiable and provide the expression in terms of $F'(cx)$ and $x$. What is $\frac{d}{dc}\phi_x(c)$?

Answer: F'(cx)(x)


Question: In a metric space \((\mathbb{R}^n, d_p)\), where \(d_p\) is the metric induced by the \(p\)-norm for \(p \in (1, \infty]\), does the Hausdorff dimension of a subset \(A \subseteq \mathbb{R}^n\) depend on the choice of \(p\)?

Answer: No


Question:
Calculate the determinant of the matrix \( A \) given by:
\[
A = \begin{bmatrix}
a_1b_1+1 & a_1b_2 + 1 & \cdots & a_1b_n + 1 \\
a_2b_1+1 & a_2b_2 + 1 & \cdots & a_2b_n + 1 \\
\vdots & \vdots & \ddots & \vdots \\
a_nb_1+1 & a_nb_2 + 1 & \cdots & a_nb_n + 1
\end{bmatrix}
\]
Find the determinant of this matrix.

Answer: 0


Question: Find the sequence $a_n$ defined by the recurrence relation \[ a_n = \sqrt{\frac{a_{n-1}^2 + a_{n+1}^2}{2}} \] with the initial condition $a_1 = 10$, where $a_n \in \mathbb{N}$ for $n = 2, 3, 4, \ldots$. Determine the values of $a_n$.

Answer: 10


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

Answer: \dfrac{\pi}{4}


Question: Calculate \( \lim_{n\to \infty} \frac{\frac{2}{1}+\frac{3^2}{2}+\frac{4^3}{3^2}+...+\frac{(n+1)^n}{n^{n-1}}}{n^2} \).

Answer: \dfrac{e}{2}


Question:
Consider the group \( G = \mathbb{Q}/\mathbb{Z} \). Let \( n \) be a positive integer. Is there a cyclic subgroup of order \( n \) in \( G \)? Choose the correct answer:

- Not necessarily.
- Yes, a unique one.
- Yes, but not necessarily a unique one.
- Never.

Answer: B


Question: Let \(f : \mathbb{R}^6 \rightarrow \mathbb{R}\) be a function such that \(f(0) = 0\) and for any point \(c \in \mathbb{R}^6\) and any increment \(h\), \(\Vert Df(c)(h)\Vert \le \Vert h\Vert\). Prove that \(f(B_1(0)) \subseteq (-1, 1)\), where \(B_1(0)\) is the unit ball centered at the origin in \(\mathbb{R}^6\).

Answer: (-1, 1)


Question: Find the limit inferior of the sequence of functions \( f_n = \mathbb{1}_{[n,n+1]} \), where \( \mathbb{1}_{[n,n+1]} \) is the indicator function that takes the value 1 on the interval \([n, n+1]\) and 0 elsewhere. Determine \( \liminf_{n\to\infty} f_n \).

Answer: 0


Question: Determine the values of \( s \) for which the integral \( \int_{\mathbb{R}^n}\left(\frac{1}{1+|x|^2}\right)^{s}dx \) converges.

Answer: s > \dfrac{n}{2}


Question: A sample from a normal population has a variance of 4.0. Determine the sample size required so that the sample mean deviates from the population mean by no more than 2.0 with a probability of at least 0.95.

Answer: 4


Question: Given a bounded self-adjoint operator \( A = \begin{pmatrix}3 & 0 \\ 0 & 1\end{pmatrix} \) on \( \mathbb{R}^2 \) and a vector \( \psi = \begin{pmatrix}1 \\ 1\end{pmatrix} \), find the spectral measure \( \mu_\psi \) such that \( (\psi, f(A)\psi) = \int_{\sigma(A)} f(\lambda) \, d\mu_\psi(\lambda) \) for \( f(x) = x^2 \). Calculate the explicit expression for \( \mu_\psi \) and evaluate the integral over the spectrum \( \sigma(A) = \{3, 1\} \) to verify that it equals 10.

Answer: 10


Question: Consider a holomorphic vector bundle $E$ over the complex projective line $\mathbb{P}^1$. Let $c: \mathbb{P}^1 \rightarrow \mathbb{P}^1$ be the anti-holomorphic involution defined by $c(z) = \frac{-1}{\bar{z}}$. Suppose there exists an anti-holomorphic involution $\pi: E \rightarrow \bar{E}$ covering $c$. Is it true that the first Chern class $c_1(E)$ is even?

Answer: Yes


Question: Find the minimum value of the expression \( \frac{4}{4-x^2} + \frac{9}{9-y^2} \) given that \( x \) and \( y \) are in the interval \((-2, 2)\) and satisfy \( xy = -1 \).

Answer: \dfrac{12}{5}


Question: Suppose you select a coin from a pile where the probability of landing heads is distributed as \(p \sim \mathcal{N}(\frac{1}{2}, \frac{1}{100})\). The probability of landing heads three times in a row lies within \((\frac{1}{N+1}, \frac{1}{N})\), where \(N\) is a positive integer. Find the value of \(N\).

Answer: 7


Question: Given a colimit $X$ of a sequence of topological subspaces $X_1 \subset X_2 \subset \ldots$ and a map $f: X \to Y$, determine whether the following statement is true: $f$ is continuous if and only if $f_i = f_{|X^i}: X^i \to Y$ is continuous for all $i$.

Answer: True


Question: Compute the limit: \[ \lim_{(x,y)\to (0,0)} \frac{\sin(xy)-xy}{x^2y} \] using Taylor's theorem.

Answer: 0


Question: Find the value of \( k \) such that \( \int^{\pi}_0 x f (\sin x) \, dx = k \int^{\pi/2}_0 f (\sin x) \, dx \).

Answer: \pi


Question: Calculate the integral \( \int_{0}^{2\pi} \frac{\cos(3\theta)}{5-3\cos(\theta)} \, d\theta \) using the residue theorem.

Answer: \dfrac{\pi}{54}


Question: Find the area of the surface obtained by revolving the curve \( y = \sqrt{1-x^2} \) about the x-axis.

Answer: 4\pi


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{A^{2^{n+1}} + \frac{1}{2}}{\left(A^{2^{n}} + \frac{1}{2}\right)^2} \] where \( A \) is a constant greater than 1.

Answer: 1


Question: Given the joint probability density function \( f(x,y) = \frac{2y}{x^2}e^{-x} \) for \( 0 < y < x < +\infty \), find the expected value \( \mathbb{E}(X) \).

Answer: 1


Question: Given a figure where $\angle ERZ = 75^\circ$ and $EH = 6$, find the area of triangle $\triangle AEZ$. Assume all necessary geometric properties and relationships are provided in the figure.

Answer: 18


Question: Does there exist a non-recursive sequence of natural numbers that has no recursive subsequence?

Answer: Yes


Question: Given the series expansion of the rational polynomial \( \frac{p(x)}{q(x)} = c_0 + c_1 x + c_2 x^2 + \ldots \), is it possible to recover the original polynomials \( p(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_{n-1}x^{n-1} \) and \( q(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_{n}x^{n} \) using the coefficients \( c_n \) through the solution of a linear system?

Answer: Yes


Question: Consider a symmetric random walk starting at level 0, which ends when it reaches level 3 or level -1. Calculate the expected number of steps, denoted as $E[\tau]$, where $\tau$ is the stopping time.

Answer: 3


Question: Calculate the integral \( \int_0^1 \frac{dx}{\sqrt{-\ln(x)}} \) using Euler integrals, specifically the Gamma function \( \Gamma(x) \) and the Beta function \( B(x, y) \).

Answer: \sqrt{\pi}


Question:
Consider the Markov chain with the transition matrix:
\[
P=\begin{pmatrix}
0 & 1 & 0 & 0 & 0\\
\frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{4} & \frac{1}{4}\\
0 & \frac{1}{2} & 0 & \frac{1}{2} & 0 \\
0 & \frac{1}{2} & \frac{1}{2} & 0 & 0\\
0 & 1 & 0 & 0 & 0
\end{pmatrix}
\]
Calculate the expected number of steps to first reach state 2 from state 4.

Answer: 2


Question: Given a prime $p$ such that $3$ does not divide $p-1$, determine the order of the elliptic curve over $\mathbb{F}_p$ defined by $E(\mathbb{F}_p) = \{ (x,y) \in \mathbb{F}_p^2 \mid y^2 = x^3 + 7 \}$. Use Hasse's theorem to guide your calculation.

Answer: p + 1


Question: Let $1 \leq j < p-1$ with $p$ a prime number. Determine if the following statement is true: For any positive integer $n$ with $n \not\equiv j \pmod{p-1}$, the congruence \[ \sum_{r>1} \binom{n}{r(p-1)+j} \binom{r-1}{j} \equiv 0 \pmod{p} \] holds.

Answer: True


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

Answer: p > 1


Question: Prove that \( g(p, q) = (1 - p)\log_2 \left(\frac{1 - p}{1 - q}\right) + p\log_2\left(\frac{p}{q}\right) - \frac{2}{\ln(2)}(p - q)^2 \ge 0 \) for all \( p, q \in [0, 1] \).

Answer: g(p, q) \ge 0


Question: Given the function \( z(x,y) \) defined implicitly by the equation \( F(x,y,z) = 5x + 2y + 5z + 5\cos(5z) + 2 = 0 \), find the mixed partial derivative \( \frac{\partial^2 z}{\partial x \partial y} \) at the point \( \left(\frac{\pi}{5}, \frac{3}{2}, \frac{\pi}{5}\right) \).

Answer: -10


Question: Find \( F''(\pi) \) if \( F(x) = \int_{x}^{\sin(x)} \left(\int_0^{\sin(t)}\sqrt{1+u^4}\,\text{d}u\right)\text{d}t \).

Answer: 2


Question:
Let \( I_0 \) be a neighborhood of \( 0 \), and let \( g \in C^1(I_0) \) such that \( g(0) < 0 \). Define the function
\[
f_\beta(x) = \int_{\sin(x) - \beta x}^{x^2} g(y) \, dy \quad \forall \beta \in \mathbb{R}.
\]
Find the values of \( \beta \) such that \( x_0 = 0 \) is a local maximum for \( f_\beta(x) \).

Answer: 1


Question: Let $A$ be an $n \times n$ matrix with entries in $\mathbb{C}$. Suppose that the characteristic polynomial of $A$ is the same as the minimal polynomial of $A$, given by \(p_{A}(t) = m_{A}(t) = (t - \lambda)^{n}\). Determine the number of linearly independent eigenvectors for the eigenvalue $\lambda$. \(\boxed{\text{Your answer here}}\)

Answer: 1


Question: Determine the real values of $p$ for which the series \( \sum_{n=2}^{\infty}\frac{1}{(n^2\log n)\cdot(n^{\frac{1}{n}}-1)^p} \) is convergent.

Answer: (-\infty, 1]


Question: Calculate the limit: \[ \lim_{n\to\infty}\int_{0}^{\infty}\frac{e^{-x}\cos(x)}{\frac{1}{n}+nx^2}dx \]

Answer: \dfrac{\pi}{2}


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

Answer: 1


Question: Calculate the Lebesgue integral of the function \( f(x) = \sum_{k=0}^{120} \lfloor x \rfloor \cdot \chi_{(k-1,k)} (x) \) over the real line.

Answer: 7139


Question:
Find the marginal probability mass function of $x$ given the joint mass function:
\[ f_{x,y} (x,y) = \begin{cases} \frac{1}{x} & 0< y<x < 1 \\ 0 & \text{else} \end{cases} \]

Answer: 1


Question:
Suppose $a, b, c \in \mathbb{C}$ are three distinct complex numbers that do not lie on a straight line. Let $z, w \in \mathbb{C}$ be such that:
\[
|z-a| = |w-a|, \quad |z-b| = |w-b|, \quad |z-c| = |w-c|.
\]
Determine whether it is necessarily true that $z = w$. Provide a formal proof or counterexample.

Answer: z = w


Question: A function $f$, continuous on the positive real axis, satisfies the condition that for all $x > 0$ and $y > 0$, the integral \( \int_x^{xy} f(t) \, dt \) is independent of $x$. Given that $f(2) = 2$, compute the value of the integral \( A(x) = \int_1^x f(t) \, dt \) for all $x > 0$. Provide your answer as a function of $x$.

Answer: 4 \ln x


Question: Let $p$ be a prime and $F=\mathbb{Z}/p\mathbb{Z}$. Consider a nonzero multivariate polynomial $f \in F[x_1, \ldots, x_n]$ such that $\deg_i f < p$ for all $i=1, \ldots, n$. Can $f$ be identically zero as a polynomial function $F^n \to F$?

Answer: No


Question: Does there always exist an irreducible polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$, where $p$ is a prime and $d$ is a positive integer?

Answer: Yes


Question: Given that $Y_1 < Y_2 < \ldots < Y_8$ are the order statistics of $n$ independent observations from a continuous distribution with a 70th percentile $\pi_{0.7} = 27.3$, find the probability $P(Y_7 < 27.3)$. Assume no specific distribution is given.

Answer: 0.2553


Question: Determine whether the function \( f(x,y) = \frac{x^py^q + x^ry^s}{x^qy^p + x^sy^r} \) can be defined at \((0,0)\) so that it is continuous, where \(p, q, r, s > 0\).

Answer: 1


Question: Is every Borel set a countable union of intervals?

Answer: No


Question: Is there a function \( f(x) \) that is not continuous, but for which the function \( F(x) = \int_a^x f(t) \, dt \) is differentiable on \([a, b]\)?

Answer: Yes


Question: Determine the smallest integer \( n \) such that the limit \( \lim_{(x,y) \rightarrow (0,0)}\frac{x^r}{|x|^2+|y|^2} \) exists for all \( r \geq n \).

Answer: 3


Question: Determine whether the following statement is true or false: If \((u_n)_{n=1}^\infty\) is a sequence in \(\mathcal{L}^1\) that converges pointwise to 0, then \(\lim_{n \rightarrow \infty }\int u_n \, d\mu=0\). Justify your answer.

Answer: False


Question: Evaluate the error for the numerical integration method given by: $$\int_a^b \left( f'(x) \right)^2\ dx \approx h\cdot \sum_{i=0}^{n-1} \left(\frac{f(x_{i+1}) - f(x_i)}{h}\right)^2$$ where $h=\frac{b-a}{n}$ and $x_k = a+kh$. Assume that the derivatives are bounded. What is the error for this method?

Answer: O(h^2)


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

Answer: \dfrac{1}{2}


Question: Given the functions \( f(x) \) and \( g(x) \) such that \( f(g(x)) = x^2 \) and \( g(f(x)) = x^3 \) for all \( x \ge 1 \), and knowing that \( g(16) = 16 \), find the value of \( \log_2 g(4) \). Assume that \( f(x) \ge 1 \) and \( g(x) \ge 1 \) for all \( x \ge 1 \).

Answer: \dfrac{4}{3}


Question: Let $f:[0,+\infty)\rightarrow \mathbb{R}$ be a function such that for every sequence $\{x_n\}$ with $x_{n+1}=x_n+1$, it holds that $f(x_n)\rightarrow 0$. Does this imply that $\lim_{x\rightarrow +\infty}f(x)=0$? Justify your answer.

Answer: No


Question: Show that \(66! \equiv 68 \pmod{71}\).

Answer: 68


Question: Determine the values of \( \alpha \in \mathbb{R} \) for which the improper integral \( \int_{0}^{+\infty}\frac{1}{(\arctan x)^\alpha} \, dx \) converges.

Answer: \emptyset


Question:
Determine the minimal polynomial of the linear transformation \( \phi: \text{Mat}(n \times n, \mathbb{F}) \rightarrow \text{Mat}(n \times n, \mathbb{F}) \) defined by
\[
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n}\\
a_{21} & a_{22} & \cdots & a_{2n}\\
\vdots & \vdots & \quad & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn}
\end{bmatrix} \mapsto
\begin{bmatrix}
a_{21} & a_{22} & \cdots & a_{2n}\\
\vdots & \vdots & \quad & \vdots\\
a_{n1} & a_{n2} & \cdots & a_{nn}\\
a_{11} & a_{12} & \cdots & a_{1n}
\end{bmatrix}
\]

Answer: x^n - 1


Question: Is it possible to color each stick in a finite set of sticks placed on an infinitely long line either red or blue such that at every point on the line, the difference between the number of red and blue sticks is -1, 0, or 1?

Answer: Yes


Question: Consider the set $E = \{ \mathbb{R}^n \times \{(0,0,\ldots,0)\} \}$ in $\mathbb{R}^m$ with $n < m$, equipped with the subset topology $\tau_E$ induced from the standard topology $\tau_m$ on $\mathbb{R}^m$. Let $\tau_n$ be the standard topology on $\mathbb{R}^n$. Is the map $i: E \to \mathbb{R}^n$ given by $i((a_1,a_2,\ldots,a_n,0,0,\ldots,0)) = (a_1,a_2,\ldots,a_n)$ a homeomorphism with respect to these topologies?

Answer: Yes


Question: Determine whether the following statement is true: For a function \( f \in C^1[0,1] \), \[ \lim_{n\to\infty}\frac{\int_{\frac{i}{n}}^{\frac{i+1}{n}}f(x)dx-\frac{1}{2n}[f(\frac{i}{n})+f(\frac{i+1}{n})]}{\frac{1}{n^2}}=0 \] for \( 0\leq i\leq n-1, i\in \mathbf{N} \).

Answer: True


Question: Given that \( f(x) \) is a strictly convex function and \( \lim_{x\to \infty}\left(f(x) - x - ue^{x}\right) = w \) for some \( u \ge 0 \) and \( w \), determine whether the function \( g(x) = ve^{-x} + f(x) \) has exactly one minimum for \( x \ge 0 \), where \( v \) is a fixed real number.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{1}{n^2}\left(\frac{2}{1}+\frac{9}{2}+\frac{64}{9}+\cdots+\frac{(n+1)^{n}}{n^{n-1}}\right) \]

Answer: \dfrac{e}{2}


Question: Find the intersection multiplicity of the curves given by the equations \( f(x,y) = x^5 + x^4 + y^2 \) and \( g(x,y) = x^6 - x^5 + y^2 \) at the point \( P = (0,0) \).

Answer: 8


Question: Let $a_n$ and $b_n$ be two sequences with well-defined limits $a$ and $b$ in $\mathbb{R}$ as $n \to \infty$. Assume $a$ is positive. Determine whether the following statement is true: $$\lim_{n \to \infty} (a_n^{b_n}) = (\lim_{n \to \infty} a_n)^{\lim_{n \to \infty} b_n} = a^b.$$ If the statement is false, provide a counterexample.

Answer: True


Question: Determine the number of ways to select three subsets \(A\), \(B\), and \(C\) from the set \(\{1, 2, \ldots, n\}\) such that \(A \subseteq C\), \(B \subseteq C\), and \(A \cap B \neq \emptyset\).

Answer: 5^n - 4^n


Question:
Evaluate the integral \(\int_0^\infty f(x)\delta(x-1)dx\), where \(f(x)\) is defined as follows:
\[
f(x) = \begin{cases}
x^2, & 0 \le x < 1 \\
\sin 2, & x = 1 \\
x, & x > 1
\end{cases}
\]

Answer: \sin 2


Question: Let \( X \) be a compact manifold and \( \mathscr{D}^\prime(X \times X) \) be the space of tempered distributions on \( X \times X \). Given two test functions \( \varphi, \psi \in \mathscr{D}(X) \), an element \( T \in \mathscr{D}^\prime(X \times X) \) can be evaluated at the function \( \varphi \otimes \psi \) on \( X \times X \) defined by \( (\varphi \otimes \psi)(x, y) := \varphi(x) \psi(y) \). Suppose \( T_\lambda \in \mathscr{D}^\prime(X \times X) \), \( \lambda \in\mathbb{R} \), is a family of distributions such that \( \lambda \longmapsto T_\lambda[\varphi \otimes \psi] \) is a smooth function from \( \mathbb{R} \) to \( \mathbb{R} \) for any two \( \varphi, \psi \in \mathscr{D}(X) \). Does it follow that the function \( \lambda \longmapsto T_\lambda[\Phi] \) is smooth for every \( \Phi \in \mathscr{D}(X \times X) \)?

Answer: Yes


Question: Solve the equation \(\sqrt{x + 2} - \sqrt{3 - x} = x^2 - 6x + 9\).

Answer: 2


Question: Consider a discontinuous function $f$ defined on a compact set $K$. Assume that $\sup_{x \in K} f(x) < \infty$. Does $f$ have a maximum on $K$?

Answer: No


Question: Verify the calculation of the tangent vector as a linear map from a smooth function to the reals. Consider the function \( f(x,y)= e^{-((x+ 2.13)^2+(y+1.16)^{2})} \) and the curve \( \gamma(t)=(t, -0.16(t +2.5)^2 +1) \) on a manifold, with \( p = \gamma(0) = (0,0) \). Compute the directional derivative \( (f \circ \gamma)'(0) \) and determine if the result is \( -0.00670189 \).

Answer: -0.00670189


Question: Find the value of $a$ such that the polynomial $x^3 - 3x + a = 0$ has three distinct real roots.

Answer: (-2, 2)


Question: Determine whether the following statement is true or false: If $R$ is a ring, $x^3 \in Z(R)$ and $x^2 \in Z(R)$, then $x \in Z(R)$. Provide a justification for your answer.

Answer: False


Question: Let $f: [0,1] \to \mathbb{R}$ be a continuous function such that $f(0) = 0$ and \( \int_0^r f(x) \,dx \geq 0 \) for all \( r \in (0,1) \). Determine if there exists a $\delta > 0$ such that $f(x) \geq 0$ for all $0 \leq x < \delta$. Provide a justification for your answer.

Answer: Yes


Question: Consider the maximization problem: \( \max\limits_{a,b} \,\mathrm{Cov}(a\cdot X, b \cdot X) \) subject to the constraints \( \|a\|_2 = \|b\|_2 = 1 \), where \( X = (X_1,...,X_n) \) is a vector of \( n \) random variables and \( a \cdot X \) denotes the dot product between \( a \) and \( X \). Determine if there exists a solution such that \( a = b \).

Answer: Yes


Question: Determine whether the inequality \( \frac{(\alpha)^n (\beta)^n}{(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n}{(\delta+1)^n} \) holds for any integer \( n \), given that \( \alpha = (K-1)a \), \( \beta = K \), and \( \delta = Ka \) with \( K > a \ge 1 \) and \( \delta > \alpha > \beta \). In other words, is \( _2F_1(\alpha,\beta,\delta,x) > _2F_1(\alpha+1,\beta+1,\delta+1,x) \)?

Answer: No


Question: A circular helix in $xyz$-space is defined by the parametric equations $x(\theta) = 5\cos(\theta)$, $y(\theta) = 5\sin(\theta)$, and $z(\theta) = \theta$, where $\theta \in \mathbb{R}$. Let $L(\theta)$ be the arc length of the helix from the point $P(\theta) = (x(\theta), y(\theta), z(\theta))$ to the point $(5, 0, 0)$, and let $D(\theta)$ be the distance between $P(\theta)$ and the origin. If $L(\theta_0) = 26$, find $D(\theta_0)$. Express your answer in simplest form.

Answer: \sqrt{51}


Question:
Evaluate the limit of the sequence given by the general term:
\[ a_n = \left[\frac{1}{3} + \frac{n^{1000}}{2^n}\right]^n \]
Determine if the limit as \( n \to \infty \) is infinite.

Answer: 0


Question: In a metric space $(X,d)$, a set $S \subseteq X$ is defined as convex if for all $x,y \in S$, the metric segment $[x,y] = \{ z \in X : d(x,z) + d(z,y) = d(x,y) \}$ is contained in $S$. Consider the topology $\tau$ on $X$ induced by the metric $d$. Does there exist a family $\mathcal{B} \subseteq \tau$ of convex sets such that $\mathcal{B}$ is a basis for the topology $\tau$?

Answer: Yes


Question: Evaluate the integral: \[ \int _ {-\infty} ^ {\infty} \frac {e^{-x^2}}{\sqrt{\pi}} e^x\ dx. \]

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


Question:
Given that \(x_1, x_2, x_3\) are the roots of the polynomial \(x^3 + 3x + 5 = 0\), find the value of the expression:
\[ \left(x_1 + \frac{1}{x_1}\right) \left(x_2 + \frac{1}{x_2}\right) \left(x_3 + \frac{1}{x_3}\right) \]

Answer: -\dfrac{29}{5}


Question: Determine if there exists a natural number $n$ such that the equation $n = x^2 + y^2$ has more than three solutions in natural numbers $x$ and $y$. If such an $n$ exists, provide an example.

Answer: 65


Question: Compute the limit \( \lim_{k\to0}\frac{1}{2k^3}\int_0^k\frac{1-\cos 4x}{x^2+2}~\mathrm{d}x. \)

Answer: \dfrac{2}{3}


Question: Determine the cardinality of the set of all functions from \(\mathbb{Z} \to \mathbb{Z}\).

Answer: 2^{\aleph_0}


Question:
Suppose that \(X\) is a random variable such that, for some \(\alpha > 0\), the probability mass function is given by:
\[ \mathbb{P}(X=k)=e^{-2}\frac{2^{k-2}}{k!}(1+\alpha k), \quad k=0,1,2,\ldots \]
Determine the value of \(\alpha\) such that the total probability is 1.

Answer: \dfrac{3}{2}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{n! + 2^n \log(n)}{3n! + n^2} \]

Answer: \dfrac{1}{3}


Question: Determine the radius of convergence for the power series \( \sum_{k=0}^\infty 3^k \cdot z^{2^k} \).

Answer: 1


Question:
Find the value of $a$ that satisfies the equation for $x > 0$:
$$
6 + \int\limits_a^{x^2}\frac{f(t)}{t^2}dt = 2\sqrt{x}.
$$

Answer: 81


Question: Is the sum of the derivatives of a set of functions equal to the derivative of their sum?

Answer: Yes


Question: Compute the probability \( \mathbb{P}\{X_2 - X_1 < \frac{1}{4}, X_1 + X_2 < \frac{1}{2}\} \) where \( X_1 \) and \( X_2 \) are independent uniform random variables on \([0,1]\).

Answer: \dfrac{7}{64}


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{ \log ((k+x)^{b} - (k-x)^{b} )}{\log x} \] where \(k\) and \(b\) are constants in the interval (0,1).

Answer: 1


Question: Calculate the area of the region defined by the inequality \((x^{2}+y^{2})^{2} \leq a^{2}(x^{2}-y^{2})\).

Answer: a^2


Question: Calculate \( \lim_{n \to \infty} \cos(\sqrt{\lfloor(2n\pi)^2\rfloor}) \). Here, \( \lfloor . \rfloor \) denotes the floor function.

Answer: 1


Question: Given a linear modulo function of the form \( y = ax + b \pmod{n} \), where \( n \) is a known integer and \( \gcd(a, n) = 1 \), determine how many integer points \((x, y)\) are needed to uniquely determine the integers \( a \) and \( b \). Assume \( x \) and \( y \) are integers in the range from 0 to \( n-1 \).

Answer: 2


Question: Suppose that the joint distribution of $X$ and $Y$ is the uniform distribution on the circular disk defined by $x^2 + y^2 < 1$. Find the conditional expectation $E(X|Y)$.

Answer: 0


Question: Find the maximum value of \( f(z) = |z^3 + 3z + 2i| \) where \(|z| = 1\).

Answer: 3\sqrt{3}


Question: Let $X\subset\mathbb{P}^N_{\mathbb{C}}$ be a projective irreducible variety and $p\in X$. Define the set \( A = \{ x\in X : \langle x,p\rangle \subseteq X \} \), where \( \langle x,p\rangle \) denotes the line through $x$ and $p$. Is the set $A$ constructible?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \infty} e^{x^2} \int_{0}^{x} e^{-t^2} \mathrm{d}t \]

Answer: \infty


Question: Evaluate the limit: \[ \lim_{x \to 2, y\to 1}\frac{\sin^{-1}(xy-2)}{\tan^{-1}(3xy-6)} \]

Answer: \dfrac{1}{3}


Question: Find the differentiable function \( f(x) \) such that \( \frac{1}{2a}\int_{x-a}^{x+a}f(t)dt = f(x) \) for any \( a \neq 0 \), with the conditions \( f(0) = 1 \) and \( f(1) = 2 \).

Answer: x + 1


Question: Verify the divergence theorem for the vector field \( v(x,y,z) = (3x + z^2, 2y, R-z) \) over the cube with vertices \((0,0,0),(R,0,0),(0,R,0),(0,0,R),(R,R,0),(0,R,R),(R,0,R),(R,R,R)\). Calculate the surface integral \( \int\int_{S_R} v \cdot n \, dA \) and confirm that it equals the volume integral of the divergence of \( v \) over the cube.

Answer: 4R^3


Question: Determine the value of the sum \(\sum_{\textbf{t} \in \mathbb{Z}_{2}^{d}}(-1)^{\textbf{t}\cdot(\textbf{s} + \textbf{r})}\) when \(\textbf{s} \neq \textbf{r} \in \mathbb{Z}_{2}^{d}\).

Answer: 0


Question: Given a sequence \(a_1, a_2, a_3, \ldots\) such that \(\lim_{n\rightarrow \infty} a_n = a\), find the value of \(\lim_{n\rightarrow \infty}\frac{1}{\ln (n)}\sum^{n}_{k=1}\frac{a_{k}}{k}\).

Answer: a


Question: Let $X$ be an exponential random variable with rate $\lambda = 5$, and let $Y$ be a uniformly distributed random variable on the interval $(-3, X)$. Find the expected value $\mathbb{E}[Y]$. \(\mathbb{E}[Y] = ?\)

Answer: -\dfrac{7}{5}


Question: Let $A$ be an integrally closed domain, not necessarily Dedekind, with fraction field $K$. Consider a Galois extension $L/K$ and let $B$ be the integral closure of $A$ in $L$. Suppose $p$ is a prime ideal in $A$. We know that the Galois group $G$ acts transitively on prime ideals in the localization $B_{p}$ lying over the unique maximal ideal $m_p$ of $A_{p}$. Does $G$ also act transitively on the prime ideals in $B$ lying over $p$?

Answer: Yes


Question:
If \( x \) and \( y \) are of the same sign, find the value of the expression:
\[
\frac{x^3}{2} \csc^2 \left(\frac{1}{2}\tan^{-1} \frac{x}{y}\right) + \frac{y^3}{2} \sec^2 \left(\frac{1}{2}\tan^{-1} \frac{y}{x} \right)
\]
Choose the correct option:
(A) \((x-y)(x^2+y^2)\)
(B) \((x+y)(x^2-y^2)\)
(C) \((x+y)(x^2+y^2)\)

Answer: C


Question: Determine the dimension of the subspace of $M_n(\mathbb{C})$ consisting of matrices $B$ such that $AB = BA$, where $A$ is a fixed matrix in $M_n(\mathbb{C})$. Provide your answer as a single integer.

Answer: n


Question: Find the minimum value of the expression \(a^3+b^3+c^3-3abc\) given that \(a, b, c\) are three different positive integers satisfying \(ab+bc+ca \geq 107\).

Answer: 54


Question: In the group $G = \mathbb{Z}_p \times \mathbb{Z}_p$, where $p$ is a prime number, let $S$ and $H$ be subsets of $G$ such that $|S| = |H| = p$ and both contain the identity element $e$. If the product $HS$ is equal to the whole group $G$, does it follow that either $H$ or $S$ must be a subgroup of $G$?

Answer: Yes


Question: Consider two disjoint sets $A$ and $B$, each of size $n$. Some undirected edges connect nodes in $A$ with nodes in $B$. Suppose that each node (in $A$ or $B$) is adjacent to exactly $k$ of those edges, for some fixed $k \geq 1$. Determine whether there must be a perfect matching between $A$ and $B$. Provide a justification for your answer.

Answer: Yes


Question: If $f(x)$ is a differentiable function such that \( f(x) = \left(1 + \frac{x^3}{3}\right) + \int_0^x e^{-t} f(x-t) \, dt \), find the value of \( \int_0^1 f(x) \, dx \).

Answer: \dfrac{8}{5}


Question:
Consider a matrix \( A \in \mathbb{R}^{d \times m} \) with \( m \geq d \), and let its columns be denoted by \( a_i \). Assume that \( AA^T \) is invertible. Define the sum \( S(A) = \sum_{r=1}^m a_r a_r^T \), which is a \( d \times d \) matrix. Is it true that \( \text{rank}(S(A)) = d \)?

Example: For \( A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix} \), \( S(A) = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \) and \( S \) is full rank.

Answer: Yes


Question: Let $X$ be a Banach space. Determine whether there exists a nonzero compact operator $A: X \to X$ such that the spectrum of $A$ consists only of zero.

Answer: Yes


Question: Determine whether the following statement is true or false: In a finite Markov chain, if $i$ is a transient state, then there is at least one recurrent state $j$ such that $j$ is reachable from $i$.

Answer: True


Question: Does the ring $A = k(t)\langle {d\over{dt}}\rangle$, the $k$-algebra of differential operators with coefficients in $k(t)$, have any zero divisors?

Answer: No


Question:
Calculate the Riemann sum of \( \sin(x) \) given by:
\[ R = \frac{\pi}{n} \sum_{j=1}^n \sin\left(\frac{\pi}{n}\cdot j\right) \]
Determine the simplest method to evaluate this sum and find the limit as \( n \to \infty \) to obtain the value of the integral.

Answer: 2


Question: If the letters of the word 'mathematics' are arranged around a circle, what is the probability that no two identical letters are adjacent?

Answer: \dfrac{47}{90}


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

Answer: \dfrac{\pi}{2}


Question: Is every non-invertible element of a commutative von Neumann regular ring a zero-divisor?

Answer: Yes


Question: Let X be a random variable uniformly distributed between 0 and 1. Let Y be a random variable uniformly distributed between X and 1. Determine the probability density function (PDF) of Y.

Answer: -\ln(1 - y)


Question: Evaluate the integral: \[ \int_0^{2\pi} e^{\cos x} \cos(\sin x) \, dx \]

Answer: 2\pi


Question: Consider sequences $a_n > b_n > 0$ tending to infinity and a constant $\delta$. Let $x = a_n + \delta b_n$. Determine whether the integral \( \int_{a_n}^x o\left(\frac{t-a_n}{b_n^2}\right) \mathrm{d}t \) tends to 0 as $n \to \infty$. Justify your answer.

Answer: 0


Question: Determine the number of natural numbers \( n \) for which the expression \( n^6 + n^4 + 1 \) is a perfect square.

Answer: 1


Question: Consider a non-abelian group \(G\) of order \(p^3\), where \(p\) is a prime number. It is known that \(G\) has \(p^2 + p + 1\) subgroups of order \(p\). Prove or disprove the conjecture that \(G\) has \(p + 1\) subgroups of order \(p^2\).

Answer: p + 1


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

Answer: 0


Question: Determine whether the following statement is true or false: If $K$ is a compact set contained in the connected open subset $\Omega\subseteq \mathbb{R}^n$, then there exists an open cover of $K$ consisting of finitely many balls $(B_i)_{i = 0}^m$ strictly contained in $\Omega$, such that the center of the ball $B_{i}$ belongs to $B_{i+1}$.

Answer: True


Question: Determine whether the function \( f(n) = \sum_{k=1}^n \frac{\mu(k)}{k} \) changes sign infinitely many times as \( n \to +\infty \).

Answer: Yes


Question: Does there exist a metric \(d: (-1,1)\times (-1,1) \to \mathbb{R}\) that is complete and equivalent to the standard Euclidean metric on \((-1,1)\)?

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} \int_0^1 \sqrt[n]{x} e^x \, dx \).

Answer: e - 1


Question: How many closed spaces are formed when 5 general planes divide a 3-dimensional space?

Answer: 26


Question: Given that $X$ and $Y$ are independent random variables with cumulative distribution functions $F_X(x) = 2e^{-2x}$ for $x > 0$ and $F_Y(y) = 3e^{-3y}$ for $y > 0$, find $\mathbb{P}\{2X + 3Y > 2\}$.

Answer: 3e^{-2}


Question: Determine the value of \( n \) such that the series \( 1\cdot(0)!+3\cdot(1!)+7\cdot(2!)+13\cdot(3!)+21\cdot(4!)+\cdots \) up to \( n \) terms equals \( (4000)4000! \).

Answer: 4000


Question: Is the operator $(1-\Delta)^{-s}$ bounded on weak $L^p(\mathbb{R}^d)$ spaces for $p \in [1,\infty]$ when $s > 0$?

Answer: Yes


Question: Evaluate the series: \[ \sum_{k=0}^{\infty} (-1)^k \dfrac{\pi^{2k}}{(2k)!} \] and verify that it equals \(-1\).

Answer: -1


Question: Given the parametric equations \( x = 2\cos t - \cos 2t \) and \( y = 2\sin t - \sin 2t \), find the value of \( \frac{d^2y}{dx^2} \) when \( t = \frac{\pi}{2} \).

Answer: -\dfrac{3}{2}


Question: Find the element in the quotient ring $R/(x^3+x^2+x+1)$ whose image in $R/(x^2+1) \times R/(x+1)$ is $(2x+1+(x^2+1), 3+(x+1))$. Use the Chinese remainder theorem for polynomials to determine this element.

Answer: 2x^2 + 2x + 3


Question: Evaluate the limit \( \lim_{n\to\infty}\sum_{i=0}^n \frac{(-1)^i}{i!}f^{(i)}(1) \) where \( f(x)=\frac{1}{x^2+3x+2} \).

Answer: \dfrac{1}{2}


Question: Is a deformation (mod $h^{n+1}$) of a Hopf algebra $A$ as a bialgebra also automatically a Hopf algebra?

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\pi/2} \frac{dx}{1+\sqrt{\tan x}} \).

Answer: \dfrac{\pi}{4}


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

Answer: 1 - e


Question: Evaluate the convergence of the integral \( \int_{2}^{\infty} \frac{1}{x \ln^2 x} \, dx \) and determine if it equals \( \frac{1}{\ln 2} \).

Answer: \dfrac{1}{\ln 2}


Question: Find the Taylor series of the function \( f(x) = \frac{1}{x^2} \) at the point \( a = -1 \). Determine all \( x \in \mathbb{R} \) for which the series converges.

Answer: (-2, 0)


Question:
Let $X$ be a smooth projective variety with an exact sequence:
$$0\mapsto Aut^{o}(X)\rightarrow Aut(X)\rightarrow H\mapsto 0$$
where $Aut^{o}(X)$ is the connected component of the identity and $H$ is the group of connected components of $Aut(X)$. Assume there is a GIT quotient $Y:=X//Aut^{o}(X)$ which is also a smooth projective variety. Does there exist a morphism of groups $H\rightarrow Aut(Y)$?

Answer: Yes


Question: If the product space of two topological spaces $X$ and $Y$, denoted as $X \times Y$, is a manifold, must both $X$ and $Y$ individually be manifolds?

Answer: Yes


Question: Let $X$ be a complex inner product space and $T, S \in B(X)$. Prove that $T \equiv S$ if and only if $\langle Tx, x \rangle = \langle Sx, x \rangle$ for all $x \in X$. Specifically, show the implication: if $\langle Tx, x \rangle = \langle Sx, x \rangle$ for all $x \in X$, then $T = S$. (Note: This implication is not valid for real spaces.)

Answer: T = S


Question: Determine the values of \( \alpha > 0 \) for which the series \( \sum_{k=1}^{\infty} \frac{1}{(k+1)[\ln(k+1)]^{\alpha}} \) converges.

Answer: \alpha > 1


Question:
Evaluate the lambda calculus expression:
\[(\lambda x.\lambda y.x + ( \lambda x.x+1) (x+y)) ( \lambda z.z-4\ 5)\ 10\]

Answer: 13


Question: Solve the equation \(\frac{x^3+2x}{x^2-1}=\sqrt{x^2-\frac{1}{x}}\) for \(x\).

Answer: -\dfrac{1}{2}


Question: Find the residue of the function \( f(z) = \tan{z} \) at any of its poles.

Answer: -1


Question: Determine whether there exist matrices $A_1, \dots, A_d \in M_n(\mathbb{C})$ and $B_1, \dots, B_d \in M_n(\mathbb{C})$ such that for a given $\mathbb{C}$-linear map $f: M_n(\mathbb{C}) \to M_n(\mathbb{C})$, the equation \[ f(X) = \sum_{j = 1}^d A_jXB_j \] holds for all $X \in M_n(\mathbb{C})$. Provide a justification for your answer.

Answer: Yes


Question:
Consider the sequence \( \{x_n\}_{n \ge 0} \) defined by the iterative formula:
\[ x_{n+1} = x_n - m \frac{f(x_n)}{f'(x_n)} \]
where \( \xi \) is a root of multiplicity \( m \) of \( f(x) = 0 \). If \( x_0 \) is sufficiently close to \( \xi \), determine the order of convergence of the sequence. Assume that \( g(x) = x - m \frac{f(x)}{f'(x)} \) and analyze the conditions under which the order of convergence is at least 2.

Answer: 2


Question: Find the spectrum of the operator \( T \) defined on the space of double-sided sequences \( \ell_2(\mathbb{Z}) = \{ \{ a_n \}_{n = -\infty}^{\infty} ; \sum_{n= -\infty} ^{\infty} | a_n |^2 < \infty \} \) by \( Tx = y \) where \( y_n = -x_{n-1} + x_n - x_{n+1} \).

Answer: [-1, 3]


Question: Evaluate the surface integral \( \iint_S (x^2 y^2 + y^2 z^2 + x^2 z^2)\, dS \) using the divergence theorem, where \( S \) is the entire surface of the sphere of unit radius centered at the origin.

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


Question: Let $X$ and $Y$ be two random variables such that for all $1 \le m < k$, the joint probability is given by: \[ P(X=k, Y=m) = \frac{1}{16}\left(\frac{3}{4}\right)^{k-2} = \frac{1}{9}\left(\frac{3}{4}\right)^{k} \] Calculate the covariance $\text{Cov}(X,Y)$ using the formula: \[ \text{Cov}(X,Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y] \]

Answer: 12


Question: Determine the dimension of the vector space \( V = \{ \phi : \mathbb{R}^{11} \to \mathbb{R}^9 \text{ linear} : U \subseteq \ker \phi \} \), where \( U \subseteq \mathbb{R}^{11} \) is a subspace of dimension 4.

Answer: 63


Question: Find the residue of the function \( \frac{1}{\cos^2 z} \) at its poles, which occur at \( z = \frac{\pi}{2} + \pi k \), where \( k \) is an integer.

Answer: 0


Question: For large $n$, determine which is larger: $2^{p_n}$ or $p_{2^n}$, where $2 = p_1 < p_2 < \dots$ is the increasing sequence of all prime numbers.

Answer: 2^{p_n}


Question: Determine whether the sequence \( \frac{1}{n\sin(n)} \) converges to 0. If it does not converge to 0, find the upper limit of the sequence.

Answer: \infty


Question: Find a good asymptotic approximation for \((\log n)^{\log n}\).

Answer: n^{\log \log n}


Question: Calculate \( \frac{dF(x)}{dx} \bigg|_{x=0} \) for the function \( F(x) = \int_0^2 \sin(x+l)^2\ dl \).

Answer: \sin(4)


Question:
Determine the big-O bound for the following expression:
\[ \sum^{ N}_{k=0}{2^k \log_2\left(\frac{2^N}{2^k}\right)} \]
where \( N = \log M \).

Answer: O(M)


Question: Determine all values of \( \alpha \) for which some non-trivial solutions of the differential equation \( y'' + (2\alpha - 3)y' + \alpha(\alpha - 3)y = 0 \) remain bounded and some become unbounded as \( t \to \infty \).

Answer: [0, 3)


Question: Given that $|G|=30$ and $|Z(G)|=5$, determine the structure of the quotient group $G/Z(G)$. What is the isomorphism class of $G/Z(G)$?

Answer: S_3


Question: Calculate the limit: \[ \lim_{n \to \infty} \left( \sqrt[3]{n^3+3n^2} - \sqrt{n^2+2n} \right) \]

Answer: 0


Question: Given that $a, b, c$ are integers such that $a, b, c \geq 4$ and $4abc = (a+3)(b+3)(c+3)$, where $a, b, c$ are not all equal, find the value of $a + b + c$.

Answer: 16


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

Answer: 1


Question: Given the quartic equation \( x^4 - mx^2 - 4 = 0 \), find the value of \( m \) such that \( \sqrt[4]{3-2\sqrt{2}} + \sqrt[4]{3+2\sqrt{2}} \) is a root.

Answer: 4


Question: Determine whether the statement "A countable locally compact abelian (LCA) group can only be discrete" is true or false.

Answer: True


Question: Determine the cardinality of the set of all functions from the set \( \{0, 1\} \) to the set of natural numbers \( \mathbb{N} \).

Answer: \aleph_0


Question: Evaluate the limit \( \lim_{k\to\infty} \prod_{i=1}^{k}(1-\alpha_i+\alpha_i^2) \), where \( \alpha_k \in (0,1) \) for every \( k \in \mathbb{N} \) and \( \lim_{k\to\infty}\alpha_k=0 \).

Answer: 0


Question: Suppose that $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a differentiable function. The best linear approximation of $f$ near $(a,b)$ is the affine function $$T(x,y) = Df(a,b)(x-a, y-b) + f(a,b),$$ where $Df(a,b)$ is the $2\times2$ Jacobian matrix for $f$ at $(a,b)$. What should we expect the ratio of the area of $f(S)$ to the area of $S$ to be for small objects $S$ centered at $(a,b)$?

Answer: |\det Df(a,b)|


Question: Let $C \in \mathfrak{gl}(\mathbb{Z},n)$ be a symmetric full rank integer-valued matrix. Consider a full rank sublattice $\Lambda$ such that $C\mathbb{Z}^n \subseteq \Lambda \subseteq \mathbb{Z}^n$. Is there a symmetric matrix $C'$ such that $\Lambda = C'\mathbb{Z}^n$?

Answer: Yes


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables with the distribution: $P(X_1=0)=\frac{1}{18}$, $P(X_1=1)=\frac{8}{9}$, and $P(X_1=2)=\frac{1}{18}$. Define $T_n=100(X_1+\cdots+X_n)$. Determine the limit of $P(T_n>97n)$ as $n \to \infty$.

Answer: 1


Question:
Determine the time complexity of the following tasks using the binary representation of prime numbers from 3 to 2n, denoted as \( B_{p}(2n) \):

1. Counting the number of Goldbach partitions by counting the occurrences of '2' in \( B_{g}(2n) = B_{p}(2n) + \text{reverse}(B_{p}(2n)) \).
2. Finding the indices of '2's in \( B_{g}(2n) \).

Assume that \( B_{p}(2n) \) is already computed and given as input.

Answer: O(n)


Question: Evaluate the limit: \[ \lim_{n\to\infty} \int_0^\infty \frac{1}{1+\left(\frac{x}{a}\right)^n} \, dx \]

Answer: a


Question: Let $a$ be the greatest natural number such that $4^{2016} + 4^a + 4^{37}$ is the square of a natural number. Determine $a \pmod{37}$.

Answer: 35


Question: Find the smallest nonnegative value of the sum \( \sum_{k=1}^{1985} e_k k^5 \), where each \( e_k = \pm 1 \).

Answer: 1


Question: Evaluate the surface integral \( \iint\limits_S xy \sqrt{x^2+y^2+1}\,\mathrm d\sigma, \) where \( S \) is the surface cut from the paraboloid \( 2z=x^2+y^2 \) by the plane \( z=1 \).

Answer: 0


Question: Determine whether there exist coprime integers $a$, $b$, and $c$ such that $a^2 + b^2 = c^2 + 1$. Provide a proof or counterexample.

Answer: Yes


Question:
Suppose that $(\mathbb{Q},+)$ is the additive group of rational numbers and $H$ is a subgroup of it. Determine which of the following statements is not true:

(a) If $\mathbb{Q}/H \cong \mathbb{Q}$, then $H = 0$.
(b) If $H \neq 0$, then every proper subgroup of $\mathbb{Q}/H$ is of finite order.
(c) If $H \neq 0$, then every element of $\mathbb{Q}/H$ is of finite order.
(d) If $\mathbb{Q}/H$ is finite, then $H = \mathbb{Q}$.

Select the statement that is not true.

Answer: B


Question: Find the derivative of the integral \( L = \int f(g(x)) \, dx \) with respect to \( g(x) \). What is \( \frac{dL}{dg(x)} \)?

Answer: \dfrac{f(g(x))}{g'(x)}


Question: Find the supremum value of the function \( f(y) = \sup_{x>0} \left( \exp(|y|-|y-x|) \right) \).

Answer: \infty


Question: Let \( G(x, y) = ((x - 1)^2, y^4) \). Given the point \( t = (7, -3) \), find the largest real number \( L \) such that \( G \) is invertible in the neighborhood defined by \( \sqrt{(x - 7)^2 + (y + 3)^2} < L \).

Answer: 3


Question: Given a group $G$ with $|G| = 7!$, and a subgroup $H = \langle f, g \rangle$ where $|f| = 7$, $|g| = 3$, $|fg| = 3$, and $|gf| = 3$, determine the order of the subgroup $H$. Is the provided information sufficient to find $|H|$?

Answer: 21


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

Answer: \sqrt{2}


Question:
Consider a real, symmetric, $n \times n$ matrix $A$ with the following properties:
1) All diagonal elements $a_{ii}$ are strictly positive.
2) All off-diagonal elements $a_{ij}$ are non-positive.
3) The sum of the elements in each row (and therefore also in each column, since $A$ is symmetric) is nonnegative, with at least one row having a strictly positive sum.

Does it follow that $A$ has full rank?

Answer: No


Question:
Differentiate the risk-neutral price of a European call option, given by the Black-Scholes formula:
\[ C_t = S_tN(d_1) - e^{r\tau}KN(d_2) \]
where
\[ d_1 = \frac{\log\left(\frac{S_t}{K}\right) + \left(r + \frac{1}{2}\sigma^2\right)\tau}{\sigma\sqrt{\tau}} \]
and
\[ d_2 = d_1 - \sigma \sqrt{\tau} \]
with respect to \( S_t \), the stock price at time \( t \). Show that the derivative is:
\[ \frac{\partial C_t}{\partial S_t} = N(d_1) \]
where \( N(x) \) is the cumulative distribution function of the normal distribution. \( K \) is the strike price, \( r \) is the interest rate, and \( \tau = T - t \), where \( T \) is the end of the contract.

Answer: N(d_1)


Question: Determine whether there is an isomorphism \( \textrm{Hom}_{A\text{-mod}}(\textrm{Hom}_{A\text{-mod}}(A,A),A) \cong A \), where \( A \) is a commutative ring with identity and \( A \) is viewed as an \( A \)-module.

Answer: A


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

Answer: 1


Question: Assume $f(x) = \sum_{n=1}^{\infty} a_nx^n$ converges on $(−R, R)$. Show that $F(x) = \sum_{n=1}^{\infty} \frac{a_nx^{n+1}}{n+1}$ is defined on $(−R, R)$ and satisfies $F'(x) = f(x)$. Provide a proof for your answer.

Answer: F'(x) = f(x)


Question: Abraham and Blaise each have $10. They repeatedly flip a fair coin. If it comes up heads, Abraham gives Blaise $1. If it comes up tails, Blaise gives Abraham $1. What is the expected number of flips until one of them runs out of money?

Answer: 100


Question: Calculate the limit: \[ \lim_{x\to\infty} \int_{0}^{1} g(xz) \, dz \] where \( g(x) \) is a continuous function such that for each \( x \ge 0 \), \( \lim_{x\to\infty} g(x) = L \neq 0 \).

Answer: L


Question: Find all continuous functions \( f(x) \) for positive values of \( x \) that satisfy the equation \( \int\limits_x^{x^2} f(t)\, dt = \int\limits_1^x f(t)\, dt \).

Answer: \dfrac{C}{x}


Question: Calculate the flux of the vector field \( F = \langle x, -y, z \rangle \) through the surface of the cylinder defined by \( x^2 + y^2 = a^2 \) and \( 0 < z < b \).

Answer: 0


Question: Let \( X \) be a non-empty compact Hausdorff space with at least \( n \) points. Is the dimension of \( \mathcal{C}(X) \), the space of continuous real functions over \( X \), at least \( n \)?

Answer: Yes


Question: Determine the norm of the linear operator \( L(f) = f(x) - \int_{0}^{1}f(t)dt \) for \( f \in C^0([0,1],\mathbb{R}) \) with the supremum norm \( \|\cdot\|_{\infty} \).

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \to 0} \log_{\cos(3x)}\cos(2ix) \] where \( i = \sqrt{-1} \). Note: The use of L'Hospital's rule and expansion is prohibited.

Answer: -\dfrac{4}{9}


Question: Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x \in \mathbb{R} \), the equation \( x^2 \cdot f(x) + f(1-x) = 2x - x^4 \) holds.

Answer: 1 - x^2


Question: Let $p(x)$ be the minimal polynomial of $\alpha$ over a field $F$. If $f(x) \in F[x]$ and $f(\alpha) = 0$, prove that $p(x)$ divides $f(x)$. Show that $p(x) \mid f(x)$. \(p(x)\) is irreducible and minimal for $\alpha$, and $f(\alpha) = 0$ implies $p(x) \mid f(x)$.

Answer: p(x) \mid f(x)


Question:
Four points A, B, C, and D lie in that order on the parabola \(y = ax^2 + bx + c\). The coordinates of A, B, and D are \((-2, 3)\), \((-1, 1)\), and \((2, 7)\) respectively. Determine the coordinates of C for which the area of the quadrilateral ABCD is maximized. Choose from the following options:

(A) \(\left(\frac{1}{2}, \frac{7}{4}\right)\)
(B) \(\left(\frac{1}{2}, -\frac{7}{4}\right)\)
(C) \(\left(-\frac{1}{2}, \frac{7}{4}\right)\)
(D) \(\left(-\frac{1}{2}, -\frac{7}{4}\right)\)

Answer: A


Question: Evaluate the integral \( \int_{0}^{\pi / 2} \frac{\sin x \cos x}{1 + \sqrt{\tan x}} \, \text{d}x \).

Answer: \dfrac{1}{4}


Question:
Let $n \geq 4$ be a positive integer. A graph $G = (V, E)$ is called a $K^{(n)}_{n-1}$-graph if:

- There are $n$ subsets $S_1, \ldots, S_n$, each consisting of $n-1$ vertices, and each forms a clique.
- For positive integers $i < j \leq n$, the intersection $|S_i \cap S_j| = 1$.
- For every vertex $v \in V$, there exist positive integers $i < j \leq n$ such that $S_i \cap S_j = \{v\}$.

If $G_1$ and $G_2$ are both $K^{(n)}_{n-1}$-graphs, are they isomorphic? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sin^2(\pi \sqrt{n^2 + n}) \]

Answer: 1


Question: Let $p, q, r$ be distinct real numbers such that $ap^2 + bp + c = (\\sin(\\theta))p^2 + (\\cos(\\theta))p$, and similarly for $q$ and $r$. Here, $a, b, c \in \mathbb{R}$. Find the maximum value of the expression \( \frac{a^2 + b^2}{a^2 + 3ab + 5b^2} \).

Answer: 2


Question: Suppose $F$ is a field and $a, b \in F$ with relatively prime integers $m, n \geq 1$ such that $a^m = b^n$. Determine if there exists some $c \in F$ such that $c^{mn} = a^m = b^n$. Provide a justification for your answer.

Answer: Yes


Question: Let \( S = \{ (\pm1, \pm1, \pm1) \} \) be the set of 8 vertices of a cube in \( \mathbb{R}^3 \). Determine the number of matrices \( A \in \mathbb{R}^{3 \times 3} \) such that \( A \) is a bijective map that maps \( S \) to \( S \) and \( |\det(A)| = 1 \).

Answer: 48


Question: Evaluate the expression \( \frac{\sum_{k=0}^{6}\csc^2\left(x+\frac{k\pi}{7}\right)}{7\csc^2(7x)} \) when \( x=\frac{\pi}{8} \).

Answer: 7


Question: If $f \colon \mathbb{R} \to \mathbb{R}$ is a continuous function, which of the following statements implies that $f(0) = 0$?\n(A) $\int_0^1 f(x)^n \,dx \to 0$ as $n \to \infty$\n(B) $\int_0^1 f\left(\frac{x}{n}\right) \,dx \to 0$ as $n \to \infty$\n(C) $\int_0^1 f(nx) \,dx \to 0$ as $n \to \infty$\n(D) $\int_0^1 f(x+n) \,dx \to 0$ as $n \to \infty$\nChoose the correct option.

Answer: B


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

Answer: a


Question: Evaluate the integral \( \int_{2}^5 \frac{f'(x) \sin(f(x))}{1 + f^2(x)}\mathrm{d}x \), given that \( f \in C(\mathbb{R}) \), \( f(5) = 2 \), and \( f(2) = -2 \).

Answer: 0


Question: What is the Shannon entropy of a riskless random variable?

Answer: 0


Question: Find the convolution of the Dirac delta functions \( \delta(3-t) \) and \( \delta(t-2) \).

Answer: \delta(t - 5)


Question: Evaluate the integral: \[ I = \int_{0}^{\frac{\pi}{2}} \ln\frac{(1+\sin x)^{1+\cos x}}{1+\cos x} \, dx \]

Answer: 2 \ln 2 - 1


Question: Consider a sequence \( \{x_{n}\}_{n\in \mathbb{Z}^+} \) of complex numbers such that \( \sum\limits_{p\in \mathbb{Z}_{+}}\vert x_{p} \vert^2=1 \) and \( \sum\limits_{p\in \mathbb{Z}_{+} }x_{p} \overline{x_{p+r}}=0 \) for all \( r\geq 1 \). Determine whether this implies that at most finitely many \( x_{p} \) can be non-zero.

Answer: Yes


Question: Determine the minimum dimension n such that the system of differential equations \( x' = Ax \) is satisfied, where \( A \) is an \( n \times n \) matrix and the first coordinate of the solution \( x(t) \) is given by \( x_1(t) = 5t^2\cos(3t) + 2\sin(3t) - e^{2t}\sin(3t) + 4t \). What is the minimum value of n?

Answer: 10


Question: Compute the Ext functor \( \mathrm{Ext}^i_{\mathbb{Z}}(\mathbb{Q}, \mathbb{Z}/2\mathbb{Z}) \).

Answer: 0


Question: Given a skew quadrilateral $ABCD$, points $P$, $Q$, $R$, and $S$ lie on sides $AB$, $BC$, $CD$, and $DA$ respectively. Calculate the product \( \frac{AP}{PB} \cdot \frac{BQ}{QC} \cdot \frac{CR}{RD} \cdot \frac{DS}{SA} \).

Answer: 1


Question: Determine the convergence radius of the power series \( F(n,z) := \sum_{j=0}^n z^{3j^2} + 5z^{j^3} \).

Answer: 1


Question: Find the value of the constant $c$ such that the solution $y^2 + 2xy - ce^{2x} = 0$ satisfies the initial condition $y(0) = 0$.

Answer: 0


Question: Given that $abc \equiv x \pmod{p}$ and $x \not\equiv 0 \pmod{p}$, determine the number of solutions for $abc$ in terms of $x$ and $p$. Assume $p$ is a prime number.

Answer: (p-1)^2


Question: Let $k$ be an algebraically closed field, and let $X$ and $Y$ be schemes of finite type over $k$. Consider a morphism of schemes $f: X \to Y$ that is not surjective. Is there a closed point in the set $Y \setminus f(X)$?

Answer: Yes


Question: Find the smallest odd integer \( n > 1 \) such that the Mersenne number \( 2^n - 1 \) is divisible by twin primes \( p \) and \( q \) with \( p < q \).

Answer: 315


Question: Determine whether the following statement is true or false: If $u \in H^1(\Omega)$, then $\Delta u \in H^{-1}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain with a smooth boundary.

Answer: True


Question: Minimize \( \frac{1}{N}\left(\sum_{i=1}^{N} w_{i}\right)^{2} \) subject to the constraint \( \|\mathbf{w}\|_{2}^{2}=N \).

Answer: 0


Question: Find an integer $x \in \mathbb{Z}$ such that $|x^2 + 1|_5 \leq 5^{-4}$. If no such $x$ exists, provide a justification.

Answer: 182


Question:
Let $p \geq 23$ be a prime number such that the decimal expression of $\frac{1}{p}$ is periodic with period $p-1$. Let $(\frac{\Bbb{Z}}{p\Bbb{Z}})^*$ denote the multiplicative group of integers modulo $p$. Which of the following is correct about the order of $10$ in $(\frac{\Bbb{Z}}{p\Bbb{Z}})^*$?

1. The order of $10$ is a proper divisor of $(p-1)$.
2. The order of $10$ is $\frac{(p-1)}{2}$.
3. The order of $10$ is a generator of the group $(\frac{\Bbb{Z}}{p\Bbb{Z}})^*$.
4. The group $(\frac{\Bbb{Z}}{p\Bbb{Z}})^*$ is cyclic but not generated by the element $10$.

Select the correct statement.

Answer: C


Question: Let $G$ be a group and $X$ be a G-set. Determine whether the following statement is true or false: If $a \cdot x = b \cdot x$ for some $x \in X$, then $a = b$ in $G$. Justify your answer.

Answer: False


Question: Let $(X, \mu)$ be a measurable space and $f$ be a complex-valued measurable function on $X$. Suppose that for all $g \in L^2(X)$ with $\|g\| = 1$, the inequality $\left|\int_X f \bar{g} \, d\mu\right| \leq 1$ holds. Prove or disprove that $f$ is square integrable, i.e., $f \in L^2(X)$.

Answer: f \in L^2(X)


Question: Find a formula for the number of distinct cycle types in the symmetric group $S_n$. For example, the cycle types for $S_5$ are: \( (5), (41), (32), (311), (221), (2111), (11111) \).

Answer: p(n)


Question:
Given that \(x, y, z\) are the roots of the cubic polynomial \(t^3 - 2t^2 + \frac{1}{2}t - 4 = 0\), find the value of the cyclic sum:
\[
\sum_{\mathrm{cyc}} \frac{1}{xy + z - 1} = \frac{1}{xy + z - 1} + \frac{1}{yz + x - 1} + \frac{1}{zx + y - 1}.
\]

Answer: -\dfrac{2}{9}


Question: For which \( \alpha > 0 \) does the equality \( \int_0^1 \sum_{n=0}^\infty x^\alpha e^{-nx} \, dx = \sum_{n=0}^\infty \int_0^1 x^\alpha e^{-nx} \, dx \) hold?

Answer: \alpha > 0


Question: Compute the covariance \( \text{Cov}(X_t, X_s) \) for the martingale \( X_t = \int_0^t W_u^2 \, dW_u \). Use the fact that \( \text{Cov}\left(\int_0^T a(t) \, dW_t, \int_0^T b(t) \, dW_t\right) = \mathbb{E}\left(\int_0^T a(t)b(t) \, dt\right) \).

Answer: (\min(t, s))^3


Question: Let $X$ be a metric space and let $\{A_n\}_{n \in \mathbb{N}} \subset X$ be a sequence of non-empty nested subspaces of $X$, i.e., $A_{n+1} \subseteq A_n$. Define \( A = \bigcap_{n=1}^\infty \overline{A_n} \), where \( \overline{A_n} \) denotes the closure of \( A_n \). Is it true that for every \( a \in A \), there exists a sequence \( \{a_n\} \) with \( a_n \in A_n \) such that \( \lim_{n \to \infty} a_n = a \)?

Answer: Yes


Question: Determine the value of \( a \) such that \[ \lim _{x\to \infty }\left(a^{\frac{1}{x}}+\frac{1}{x}\right)^x=3. \]

Answer: \dfrac{3}{e}


Question: Let $A = \bigoplus_{i \in \mathbb{Z}} A_i$ be a strongly graded unital algebra over $\mathbb{C}$, with no zero divisors. Determine whether the map \( m: A_i \otimes_{A_0} A_j \to A_{i+j} \) is always an isomorphism.

Answer: Yes


Question: Evaluate the surface integral \( \iint_S \frac{1}{(1+4z)^2} \, \mathrm{d}S \) over the paraboloid surface \( S \) parametrized in polar coordinates by \( t(r,\phi)=(r\cos \phi,r \sin \phi,r^2) \), where \( r \geq 0 \) and \( 0 \leq \phi \leq 2\pi \).

Answer: \dfrac{\pi}{2}


Question: Is it true that if a function $f:[0,1]\to\mathbb{R}$ is in $L^1(0,1)$ and satisfies \( \int_0^t f(s) \, ds = 0 \) for all \( t \in [0,1] \), then \( f(t) = 0 \) almost everywhere?

Answer: Yes


Question: Given the equations \( a^3 - 3ab^2 = 10 \) and \( b^3 - 3ba^2 = 5 \), find the value of \( a^2 + b^2 \).

Answer: 5


Question: Compute \( \lim_{x \rightarrow +\infty} \frac{\left[\int^x_0 e^{y^{2}} dy\right]^2}{\int^x_0 e^{2y^{2}}dy} \).

Answer: 0


Question: Calculate the value of the integral expression \( \int_{[0,a]} f(x) \, dx + \int_{[0,f(a)]} f^{-1}(y) \, dy \) given that \( f \) is a strictly increasing real function with \( f(0) = 0 \).

Answer: a f(a)


Question: Determine whether the sequence \( \left\{\frac{p_{n+1}-p_n}{n}\right\} \) is bounded, where \( p_n \) denotes the \( n \)th prime number.

Answer: Yes


Question: Find the smallest integer \( n > 1 \) such that the average of the sum of the squares from 1 to \( n \) equals \( k^2 \), where \( k \) is an integer.

Answer: 337


Question: Evaluate the limit: \[ \lim_{n \to \infty}\left(8n-\frac{1}{n}\right)^\frac{(-1)^n}{n^2} \]

Answer: 1


Question: Determine whether the following statement is true or false: Given $n$ real matrices $A_1, A_2, \ldots, A_n$, if their $k$-th power sum is zero for all $k \in \mathbb{N}_{>0}$, then they are all nilpotent.

Answer: True


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{x^2}{(x^2 + a^2)^2} \, dx \).

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


Question:
Given the transition matrix of a Markov chain:
\[ P = \begin{pmatrix} 0.3 & 0.3 & 0.4 \\ 0.2 & 0.7 & 0.1 \\ 0.2 & 0.3 & 0.5 \end{pmatrix} \]
Determine the smallest integer \(n\) such that the probability of transitioning from state 1 to state 2 in \(n\) steps is 0.42.

Answer: 2


Question: Find the center of a circle that passes through the point \((x_p, y_p)\) and is tangent to the lines defined by the points \((x_1, y_1), (x_2, y_2)\) and \((x_3, y_3), (x_4, y_4)\).

Answer: (h, k)


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

Answer: \dfrac{1}{12}


Question: A die is rolled infinitely many times. What is the probability that the sequence (6,6) occurs before the sequence (1,2)?

Answer: \dfrac{6}{13}


Question: Let X be a random variable following a Binomial distribution with parameters n and p, where p is uniformly distributed over the interval (0,1). Show that the probabilities P(X = 0), P(X = 1), ..., P(X = n) are all equal. You are given that the integral of the binomial distribution over (0,1) is equal to \( \frac{i!(n-i)!}{(n+1)!} \).

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


Question: Determine the intersection number of the curve $y = x^3$ and its tangent at the inflection point. Given that the intersection number of this curve and its tangent at the origin is 3, calculate the intersection number at the inflection point.

Answer: 3


Question: Let $A$ be a bounded self-adjoint semi-positive linear operator on a Hilbert space $X$, such that $(Ax,x) \ge 0$ for all $x \in X$. It is known that $(I+A)^{-1}$ is well-defined with $\|(I+A)^{-1}\| \le 1$. Show that the operator $(I+A)^{-1}A$ satisfies $\|(I+A)^{-1}A\| \le 1$. Does this inequality hold in general, beyond the finite-dimensional case?

Answer: \| (I + A)^{-1} A \| \le 1


Question: A point \(a\) is chosen randomly from the interval \([0,5]\), and \(b\) is chosen randomly from the interval \([1,6]\). Find the probability that \(\int_{\sqrt{b}}^a x \, dx > \frac{3}{2}\).

Answer: \dfrac{37}{75}


Question: Evaluate the integral: \[ \int_{\pi/2}^{5\pi/2} \frac{e^{\tan^{-1}(\sin x)}}{e^{\tan^{-1}(\sin x)} + e^{\tan^{-1}(\cos x)}} \, dx \]

Answer: \pi


Question:
Let \( \triangle ABC \) be a triangle in \( \mathbb{R}^2 \). For a point \( D \) on side \( AB \), show that the following equation holds:
\[
\frac{|BC|^2}{|AB|\cdot|DB|}+\frac{|CA|^2}{|AB|\cdot|AD|}-\frac{|CD|^2}{|AD|\cdot|DB|}=1
\]

Answer: 1


Question:
Determine the geometric representation of the set of complex numbers \( z \) satisfying the equation:
\[(3+7i)z+(10-2i)\bar{z}+100=0\]
Choose the correct option:
A) A straight line
B) A pair of intersecting straight lines
C) A point
D) A pair of distinct parallel straight lines

Answer: C


Question:
Consider the system of stochastic differential equations given by:
\[
\begin{cases}
dX_t = a X_t \, dt + Y_t \, dW_t & ,X_0 = x\\
dY_t = a Y_t \, dt - X_t \, dW_t & ,Y_0 = y
\end{cases}
\]
where \(W=\{W_t:t \geq 0\}\) is the standard Brownian motion. Determine \(\mathbb{E}[X_t]\).

Answer: x e^{a t}


Question: Let $X_1, X_2, \ldots, X_n$ be i.i.d. integrable random variables. Find \( \lim_{n \rightarrow \infty} E(X_1 \mid X_1 + X_2 + \cdots + X_n) \).

Answer: E(X_1)


Question: Determine whether the set \( X = \{ f \in \mathbb{L}_1(\mathbb{R}^n) \mid m(f^{-1}((0,\infty))) = 0 \} \) is open, closed, or neither in \( \mathbb{L}_1(\mathbb{R}^n) \).

Answer: closed


Question: Evaluate the integral of the function \( \frac{z^3}{2z-i} \) around the unit circle using Cauchy's integral theorem or residue theorem.

Answer: \dfrac{\pi}{8}


Question:
Evaluate the complex line integral over the contour \( c = \{ z \mid \max\{|\text{Re}(z)|, |\text{Im}(z)|\} = 1 \} \), which is a square with positive orientation. Calculate:
\[ \int_c \frac{z \, dz}{\cos(z) - 1} \]

Answer: -4\pi i


Question: Let \(C(\mathbb{R})\) be the linear space of all continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\). Let \(S\) be the set of differentiable functions \(u(x)\) that satisfy the differential equation \(u' = 2xu + c\) for all real \(x\). For which value(s) of the real constant \(c\) is this set a linear subspace of \(C(\mathbb{R})\)?

Answer: 0


Question: Calculate the limit: $$\lim_{s\rightarrow\infty}\left(\frac{1}{(2-e^{t/\sqrt{2s}})e^{t/\sqrt{2s}}}\right)^{s}$$

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


Question: Evaluate the limit: \[ \lim_{n\to \infty} \frac{2^{\ln(\ln(n))}}{n\ln(n)} \]

Answer: 0


Question: Find the limit of the Laplace transform \( \lim_{\epsilon \to 0} \frac{1-e^{-p\epsilon}}{p\epsilon} \).

Answer: 1


Question: If a function \( f \) is of bounded variation on \([a, b]\) and has the intermediate value property, is it true that \( f \) must be continuous on \([a, b]\)?

Answer: Yes


Question: Let \( X \) be a random variable following a Poisson distribution with parameter 2. We roll \( X \) dice and let \( Y \) be the number of 1s obtained. Determine \( \mathbb{E}[Y \mid X] \).

Answer: \dfrac{X}{6}


Question: Let $\Gamma \subset F^*_p$ be a multiplicative subgroup of the integers modulo a prime number $p$. Suppose $A \subset F^*_p$ is a set such that $|A| < |\Gamma|$. Consider the set $A\Gamma = \{a\gamma: a \in A, \gamma \in \Gamma \}$. Is it true that $|A\Gamma| = |A|$?

Answer: No


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

Answer: 2 - \pi


Question: Let $X$ be the set of all upper triangular matrices in $GL_n(\mathbb{R})$. Does there exist a matrix $T \in GL_n(\mathbb{R})$ such that for every $x \in X$, the matrix $TxT^{-1}$ is a lower triangular matrix?

Answer: Yes


Question: Find the range of the function \( f(x) = \frac{(x-a)(x-b)}{(c-a)(c-b)} + \frac{(x-b)(x-c)}{(a-b)(a-c)} + \frac{(x-c)(x-a)}{(b-c)(b-a)} \), where \( a, b, c \) are distinct real numbers such that \( a \neq b \neq c \neq a \).

Answer: 1


Question: Determine the minimum possible order of a homogeneous linear differential equation with constant coefficients that has $x^2 \sin x$ as a solution.

Answer: 6


Question: Consider a line of \( n \) soldiers. They are to be divided into several non-empty units, and a commander is chosen for each unit. Determine the number of ways this can be done.

Answer: F_{2n}


Question: Evaluate the limit: \[ \lim_{n \to \infty }\underbrace{\sin \sin \dots\sin}_{\text{$n$ compositions}}\,n. \]

Answer: 0


Question: Find a particular solution to the differential equation: \[ y'' + 3y' + 2y = \exp(\mathrm{e}^x) \]

Answer: e^{\mathrm{e}^x - 2x}


Question: Let \( F_{p^n} \) be a finite field and consider a subfield of order \( p^d \) where \( d \mid n \). Suppose we have an irreducible polynomial of degree \( k \) over this subfield such that \( n = dk \). Is this polynomial a factor of \( X^q - X \) where \( q = p^n \)?

Answer: Yes


Question: Evaluate the series: $$\sum_{n=0}^{\infty} \frac{n^{2}-5n+2}{n!}$$

Answer: -e


Question: Compute the total area covered by a circle of radius \( r \) whose center moves along the path \( x = t, y = \sin(t) \) for non-negative \( t \). Consider any overlaps in the area covered by the circle during its motion.

Answer: \infty


Question: In the quarter circle defined by the equation $x^2 + y^2 = 16$, consider the quarter arc $AD$. Points $B$ and $C$ are arbitrary points on the arc $AD$, with $C$ located between $D$ and $B$. Determine the maximum possible area of the pentagon $OABCD$, where $O$ is the origin.

Answer: 12


Question: Determine the rank of the linear operator \( T \) on \( M_{3 \times 3}(\mathbb{R}) \) defined by \( T(P) = QP \), where \( Q \) is a 3x3 matrix with rank 2.

Answer: 6


Question: Determine the values of the parameter \( p \) for which the series \( \sum_{n=2}^{\infty} \frac{1}{n \ln^p n} \) is convergent.

Answer: p > 1


Question: Compute the value of the 30th derivative of the function \( g(x) = \sinh{\left(x^4\right)} \) at the origin, i.e., find \( g^{(30)}(0) \).

Answer: 0


Question: Suppose $\mathbf{Y} \sim N_3(0,\,\sigma^2\mathbf{I}_3)$ and $Y_0 \sim N(0,\,\sigma_0^2)$ are independent random variables. Does the vector $(\mathbf{Y}, Y_0)$ have a multivariate normal distribution?

Answer: Yes


Question: In the triangular pyramid $VABC$, let $G$ be the centroid of the pyramid. Prove that \( \frac{MA}{MV} + \frac{NB}{NV} + \frac{PC}{PV} = 1 \).

Answer: 1


Question: Determine whether the series \( \sum_{n=1}^\infty \frac{\log n}{n^q+1} \) converges for different values of \( q \).

Answer: q > 1


Question: Given the elliptic curve defined by the equation \( y^2 = x^3 - 15x + 22 \) and a point \( P = (-1,6) \) on this curve, calculate the point \( 2P \). It is known that \( 2P = (3,-2) \). Verify this calculation and then find the point \( 3P \).

Answer: (2, 0)


Question:
Given the joint probability density function of continuous random variables $X$ and $Y$:
\[ f(x,y) = \begin{cases}2x & \text{if } x \in [0,1], y \in [0,1] \\
0 & \text{otherwise}\end{cases}\]
calculate the correlation coefficient between $X$ and $Y$. Use the formula:
\[ \mathsf{Corr}(X,Y) = \frac{\mathsf{Cov}(X,Y)}{\sqrt{\mathsf{Var}(X)\,\mathsf{Var}(Y)}} \]

Answer: 0


Question: Determine the smallest integer \( n \) such that there exists a set \( \{g_1, g_2, \ldots, g_n\} \subset SO_4(\mathbb{R}) \) for which the subgroup generated by this set is dense in \( SO_4(\mathbb{R}) \).

Answer: 2


Question: Using Fubini's theorem, evaluate the limit \( \lim_{n \rightarrow \infty}\int_{0}^{n}\frac{\sin x}{x}dx \). Given that \( \int_{0}^{\infty}e^{-tx}dx = \frac{1}{t} \) for \( t > 0 \), show that this limit equals \( \frac{\pi}{2} \).

Answer: \dfrac{\pi}{2}


Question: Determine the minimum value of $n$ such that the number of iterations of the function $p(n)$, which reduces $n$ to 1, equals $k$. For example, $m(1)=2$, $m(2)=4$, and $m(3)=16$. Find $m(4)$. Is $m(k)$ always a power of 2?

Answer: 256


Question:
Evaluate the following expression without using a calculator:
\[ \frac{1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+...+\frac{1}{997^2}+\frac{1}{999^2}-\frac{1}{1002^2}-\frac{1}{1004^2}-\frac{1}{1006^2}-...-\frac{1}{1998^2}-\frac{1}{2000^2}}{1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{999^2}+\frac{1}{1000^2}} \]

Answer: \dfrac{3}{4}


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

Answer: 1


Question: Find the coefficient of \(x^{21}\) in the expansion of \((x^{2} + x^{3} + x^{4} + x^{5} + x^{6})^{8}\).

Answer: 784


Question: Evaluate the limit: \[ \lim_{(x,y,z)\rightarrow(0,0,0)}\log\left(\frac{x}{yz}\right) \] Choose the correct answer from the following options: (a) 0, (b) 2, (c) 4, (d) does not exist.

Answer: D


Question: For every positive integer \( n \), does there exist a sum-balanceable graph \( G \) such that the chromatic number \( \chi(G) = n \)?

Answer: Yes


Question: Determine the value of the infinite product \( \prod_{n=1}^{\infty}\left(1 - \frac{1}{3n + 2}\right) \).

Answer: 0


Question: Find the sum of all integers \( n \) such that \( n^2+2n+2 \) divides \( n^3 + 4n^2 + 4n - 14 \).

Answer: -11


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\ln\left[\frac{(1-3x)(1+x)^3}{(1+3x)(1-x)^3}\right]}{x^3} \] without using L'Hôpital's rule, Taylor series, differentiation, or integration.

Answer: -16


Question: Find the product \(ab\) if \( \int \frac{15x^2}{\sqrt{2x-1}} \, dx = (ax^2 + bx + b)\sqrt{2x-1} + C \). Express your answer as an integer.

Answer: 6


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int_0^\frac{\pi}{2} \frac{\sin^n x}{\sqrt{1+x}}\, dx. \]

Answer: 0


Question: Let \(X\) and \(Y\) be independent and identically distributed normal random variables with mean \(a = 0\) and variance \(b^2 = 6\). Calculate the variance of the sum of their squares, \(\text{var}(X^2 + Y^2)\).

Answer: 144


Question: Solve the equation: \(5x = 0.01^x\)

Answer: 0.1168


Question: Find the residues of the function \( f(z) = \frac{1}{(e^{z} - 1)^{2}} \) at its poles.

Answer: -1


Question:
Consider a bounded operator \( T: c_0 \to c_0 \) represented by a double-infinite matrix \([T_{mn}]_{m,n \geq 1}\). This operator acts on a sequence \( a = [a_1, a_2, a_3, \ldots] \in c_0 \) as follows:
\[ Ta = [T_{mn}][a_n] = \left[ \sum_{n=1}^\infty T_{mn}a_n \right] \]
Suppose \( a = [a_1, a_2, a_3, \ldots] \in \ell^\infty = (c_0)^{**} \). Does the formula
\[ T^{**}a = [T_{mn}][a_n] = \left[ \sum_{n=1}^\infty T_{mn}a_n \right] \]
still hold?

Answer: Yes


Question: Determine the value of \( b \) such that the initial value problem \( y' + \frac{y}{\tanh(x)} = 2 \cdot \frac{\cosh(x)}{\sinh(1)} \) with \( y(1) = b \) has a solution where the limit \( \lim_{x \to 0^+} y(x) \) exists.

Answer: 1


Question: In an algebraic extension field $E = k(\alpha_i)_{i \in I}$, where $k$ is a field and each $\alpha_i$ is an algebraic element over $k$, does every element of $E$ lie in a finitely generated subextension $k(\alpha_{i(1)}, \ldots, \alpha_{i(n)})$ when the index set $I$ is infinite?

Answer: Yes


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables from a uniform distribution on $[0, \beta]$. Consider the estimators $\hat{\beta_1} = (n+1)X_{\min}$ and $\hat{\beta_2} = \frac{n+1}{n} X_{\max}$. Compute the ratio of the variances $\frac{\operatorname{Var}[\hat{\beta_1}]}{\operatorname{Var}[\hat{\beta_2}]}$. What conclusion can be drawn from the result?

Answer: n^2


Question: Find all real numbers \(x\) such that \(2^x + 3^x - 4^x + 6^x - 9^x = 1\).

Answer: 0


Question: In rectangle $ABCD$, a circle is drawn passing through point $C$ and touching the sides $AB$ and $AD$ at points $M$ and $N$, respectively. The length of the altitude from $C$ to $MN$ is 5 units. Find the area of the rectangle $ABCD$.

Answer: 25


Question: Given n coins, each with a probability p of landing heads, all n coins are flipped. If there are x successes (heads), the n - x coins are re-flipped to achieve a total of exactly k successes. Calculate the probability of having exactly k successes after this process. Assume n = 3, k = 3, and p = 1/2.

Answer: \dfrac{27}{64}


Question: Suppose $E_1$ and $E_2$ are elliptic curves given by $y^2 = 4x^3 + A_ix + B_i$ for $i=1,2$, where $A_i, B_i \in \mathbb{C}$ are algebraic over $\mathbb{Q}$. Let $\Lambda_1$ and $\Lambda_2$ be the unique lattices parametrizing $E_1$ and $E_2$, respectively, such that $g_2(\Lambda_i) = A_i$ and $g_3(\Lambda_i) = B_i$. If $\psi: E_1 \to E_2$ is an isogeny, it corresponds to a complex number $\alpha$ such that $\alpha \Lambda_1 \subseteq \Lambda_2$. Does $\alpha$ need to be algebraic over $\mathbb{Q}$?

Answer: Yes


Question: Determine the Laurent series expansion of \( f(z)=\frac{1}{1+z^2} \) about \( z_0=i \), valid in the region \( 0<|z-i|<2 \). Then, evaluate the integral \( \int_\gamma \frac{1}{1+z^2}dz \), where \( \gamma(t)=i+e^{it}, 0\leq t\leq2\pi \).

Answer: \pi


Question: Calculate the limit: \[ \lim_{x\rightarrow \frac{\pi }{4}}\frac{\sin2x-\cos^{2}2x-1}{\cos^{2}2x+2\cos^{2}x-1} \]

Answer: 0


Question: Find the coefficient of \(x^{18}\) in the expansion of \((1+x^5+x^7)^{20}\).

Answer: 0


Question: Identify the discrete valuation ring corresponding to the stalk at the point at infinity on the projective line $P^1$ over an algebraically closed field $k$, given that the affine line $A^1$ is embedded into $P^1$ by mapping $x$ to $(x,1)$.

Answer: k\left[\frac{1}{x}\right]_{\left(\frac{1}{x}\right)}


Question: Provide an example of a field $K$ other than $F_p$ that can be the endomorphism ring of an abelian group $G$. Justify your example.

Answer: \mathbb{Q}


Question: Evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \frac{\cos(x)^{\sin(x)}}{(\cos x)^{\sin(x)}+(\sin x)^{\cos(x)}} \, dx \).

Answer: \dfrac{\pi}{4}


Question:
Consider a directed graph where each node has an out-degree of 2 and is colored either blue or red. A random walk on this graph, where each out-edge is chosen with equal probability, generates a stochastic process \((X_i)\) with values in \(\{\text{blue}, \text{red}\}\). Determine the entropy rate \(H(X)\) of this process, defined as:
\[
H(X) := \lim_{n\to\infty} \frac{H(X_1,\ldots,X_n)}{n},
\]
where \(H(X_1,\ldots,X_n)\) is the entropy of the sequence \((X_1, \ldots, X_n)\).

Answer: 1


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

Answer: 1


Question: Compute the limit \( \lim_{n\rightarrow \infty} \{(2+\sqrt{3})^{n}\} \), where \( \{x\} \) denotes the fractional part of \( x \).

Answer: 1


Question:
Suppose that the bivariate random variable \((X,Y)\) has the joint density function:
\[
f_{X,Y}(x,y)=\left\{
 \begin{array}{ll}
  1  & \mbox{when } y > 0 , x+y<1 \text{ and } y<x+1 \\
  0 & \mbox{otherwise }
 \end{array}
\right.
\]
Find \(\operatorname{Cov}(X,Y)\).

Answer: 0


Question: Is there a graph $G$ such that the chromatic number $\chi(G)$ is finite, but the total chromatic number is infinite?

Answer: Yes


Question: Find the limit of the sequence \( a_n = \frac{2n^{2n}}{(2n)!} \) as \( n \to \infty \).

Answer: \infty


Question: Compute the value of the sum \( \sum_{k=0}^{1006} \binom{2013}{2k} \).

Answer: 2^{2012}


Question: Determine the values of \( \alpha > 0 \) for which the series \( \sum_{n=1}^{\infty} \frac{n!}{\prod_{k=1}^n (\alpha+k)} \) is convergent.

Answer: \alpha > 1


Question: Evaluate the limit using Taylor series expansion with remainder: \[ \lim \limits_{x \to 1} \frac{\ln x}{x^2+x-2} \] without using L'Hopital's Rule.

Answer: \dfrac{1}{3}


Question: For every integer \( k > 1 \), is it true that there exist natural numbers \( w_1, w_2, \ldots, w_k \) (all different from 1) such that there is a prime \( p \neq 2 \) that divides \( \sum_{i=1}^k w_i^2 \) but does not divide any of the numbers \( w_1, w_2, \ldots, w_k \)?

Answer: Yes


Question:
Let $X, Y, Z$ be CW complexes, and let $f : X \rightarrow Z$ and $g : Y \rightarrow Z$ be continuous maps. Consider ordinary singular cohomology with $\mathbb{Z}$ coefficients, denoted by $H^\bullet$. Let $j_0 \geq 0$ be an integer. Suppose the following conditions hold:

1. For every integer $k \geq 0$ with $k \neq j_0$, the induced morphism $H^k(g) : H^k(Z) \rightarrow H^k(Y)$ is an isomorphism.
2. The induced morphisms $H^{j_0}(g) : H^{j_0}(Z) \rightarrow H^{j_0}(Y)$ and $H^{j_0}(f) : H^{j_0}(Z) \rightarrow H^{j_0}(X)$ are both zero maps.

Does there exist a continuous map $h : X \rightarrow Y$ such that $g \circ h = f$?

Answer: Yes


Question:
Let \( f:[0,1]\rightarrow \mathbb{R} \) be a continuous function such that \( f(0)=2016 \). Evaluate the limit:
\[ \lim_{n\to \infty}\int_{0}^{1}f(x^n)dx. \]

Answer: 2016


Question: Find the real number \( t \) such that the sequence of integers not in the sequence of floors of multiples of \( \pi \) is given by the floor of \( t \cdot n \). Provide \( t \) to the nearest thousandth.

Answer: 1.467


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

Answer: \dfrac{1}{e}


Question: Let $f(x)$ be a function satisfying $f(x) \ge |x|^{\alpha}$ for $\frac{1}{2} < \alpha < 1$ and $f(x)f(2x) \le |x|$ for all $x$ in the deleted neighborhood of zero. Determine $\lim_{x \to 0} f(x)$. Choose from the following options: $1/\alpha$, $0$, or "Does not exist".

Answer: 0


Question: Evaluate the sum: \[ f\left(\frac{1}{2011}\right) + f\left(\frac{2}{2011}\right) + f\left(\frac{3}{2011}\right) + \dotsb + f\left(\frac{2010}{2011}\right) \] where \( f(x) = \frac{9^x}{9^x + 3} \).

Answer: 1005


Question: Calculate the limit \( \lim_{n\to\infty}P\left(\sum^n_{i=1}X_i\ge\sum^n_{i=1}Y_i\right) \), where \( X_1, X_2, \ldots \) are independent and identically distributed with a uniform distribution over \((0,1)\), and \( Y_1, Y_2, \ldots \) are independent and identically distributed with density \( e^{-x}\cdot\mathbb{I}\{x\ge0\}\).

Answer: 0


Question: Given that \( \lim_{x\to2}\frac{\ xf(x)-4}{x-2}=3, \) find the value of \( \lim _{ x\rightarrow 2 }{ \frac { f\left( x \right) -2 }{ \sqrt { x-1 } -1 }  }. \)

Answer: 1


Question: Consider the limit comparison test for positive continuous functions \(f\) and \(g\) on \([a, \infty)\): if \(\lim_{x\rightarrow \infty }\frac{f(x)}{g(x)}=L\) where \(0< L< \infty\), then \(\int_{a}^{\infty }f(x)\) and \(\int_{a}^{\infty }g(x)\) both converge or both diverge. Now, assume \(L = 0\) or \(L = \infty\). Can one of \(\int_{a}^{\infty }f(x)\) and \(\int_{a}^{\infty }g(x)\) converge while the other diverges?

Answer: Yes


Question: A polynomial of 6th degree, \( f(x) \), satisfies \( f(x) = f(2-x) \) for all \( x \in \mathbb{R} \). If \( f(x) = 0 \) has 4 distinct roots and two equal roots, find the sum of the roots of \( f(x) = 0 \).

Answer: 6


Question: Let \( \phi: G \rightarrow H \) be a surjective homomorphism, where \( G \) is a finite group. Given that there exists an element \( a \in H \) such that \( |a| = 5 \), prove that there exists an element \( x \in G \) such that \( |x| = 5 \).

Answer: 5


Question: Let $k$ be an integer at least 4 and let $[k] = \{1,2,\ldots,k\}$. Define a function $f:[k]^4 \to\{0,1\}$ by \[f(y_1,y_2,y_3,y_4) = 1 \text{ if and only if the } y_i\text{'s are all distinct.}\] For each choice $\mathbf{z}=(z_1,z_2,z_3)\in[k]^3$, define $g_\mathbf{z}:[k]\to\{0,1\}$ by $g_\mathbf{z}(Y)=f(Y,z_1,z_2,z_3)$. Let $N$ be the number of distinct functions $g_\mathbf{z}$ as $\mathbf{z}$ varies in $\{1,2,\ldots,k\}^3$. What is $N$? Choose from the following options: \(i)\ k^3+1 \ \ \ \ \ \ \ \ (ii)\ 2^{k\choose3} \ \ \ \ \ \ \ \ (iii)\ {k\choose3} \ \ \ \ \ \ \ \ (iv)\ {k\choose3}+1 \ \ \ \ \ \ \ \ (v)\ 4\cdot{k\choose3}.

Answer: iv


Question: Determine the maximum number of $1\times1\times1$ generator blocks that can be placed inside a $3\times3\times3$ cube, such that each generator is connected to an outlet through adjacent wire blocks. The outlet is part of the room's wall and does not occupy any space within the cube. Diagonal connections are not allowed. What is the maximum number of generators that can be powered?

Answer: 20


Question: Let $C$ be the curve of intersection of the cylinder $x^2 + y^2 = 1$ and the surface $z = xy$, oriented counterclockwise around the cylinder. Compute the line integral $\int_C y\,dx + z\,dy + x\,dz$. Provide your answer as a single numerical value.

Answer: -\pi


Question: Find the minimum value of the expression \(ab + bc + ca\) given that \(a^2 + b^2 + c^2 = 1\).

Answer: -\dfrac{1}{2}


Question: Does an embedding always exist between a finite poset \( X \) and \( \mathbb{R}^3 \)?

Answer: No


Question: Determine whether the limit \( \lim_{r\to0}\frac{r(\cos^2\theta-\sin^2\theta) + i|r|\sqrt{|\cos\theta\sin\theta|}}{e^{i\theta}} \) exists. If it does, what is the limit?

Answer: 0


Question: Let \( A = (\sqrt{3}+\sqrt{2})^{2016} \). When \( A \) is written in decimal form, what is its \( 31^{st} \) digit after the decimal point?

Answer: 9


Question: Solve the recurrence relation using the Divide and Conquer method: \[ T(n) = T\left(\frac{n}{7}\right) + T\left(\frac{11n}{14}\right) + n \]

Answer: \Theta(n)


Question: Let $X(t)$ be a Brownian motion. Compute the expected value $E[X(1)X(5)X(7)]$. Use the properties of Brownian motion, such as independent increments, to find the solution.

Answer: 0


Question: Consider a finite connected simplicial complex G. If the fundamental group of G is finite, does this imply that the universal cover of G is compact?

Answer: Yes


Question: Find the radius of convergence of the power series \(\sum_{n=1}^{\infty}\frac{n!}{n^n}(x+3)^n\).

Answer: e


Question: Determine whether the function \( f(x) = \begin{cases} \frac{1}{m^{2} n^{2}}, & \text{if } x = \frac{m}{n} \text{ is a rational number in reduced form} \\ 0, & \text{otherwise} \end{cases} \) has bounded variation on the interval \((0, 1)\).

Answer: Yes


Question: Evaluate the integral \( \int_C \frac{\log z}{(z-i)^2} \, dz \), where \( C \) is the contour defined by \( |z-i|=\alpha \) with \( 0<\alpha<1 \).

Answer: 2\pi


Question: A mouse enters a box with 9 spaces. The probability of moving from one space to another is equal, depending on the current space. Space 9 contains a cat, and space 1 contains cheese. What is the probability that the mouse finds the cheese before encountering the cat?

Answer: \dfrac{1}{2}


Question: Let $T: X \to Y$ be a linear and bounded operator such that there exists a constant $C > 0$ satisfying $\|Tx\|_Y \ge C\|x\|_X$ for all $x \in X$. Is the inverse operator $T^{-1}$ linear and bounded?

Answer: Yes


Question: Determine whether the soluble Baumslag-Solitar groups, given by the presentation \(\mathrm{BS}(1,m) = \langle a,b \mid a^{-1}ba = b^m\rangle\), have finite commutator width.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{1}\int_{x}^{1} y^2 \sin\left(2\pi \frac{x}{y}\right)\,dy\,dx \).

Answer: 0


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

Answer: e


Question:
Determine the number of digits in the sum \(1! + 2! + \cdots + 100!\). Choose the correct option:

- \(137\)
- \(283\)
- \(314\)
- \(189\)
- None of these

Answer: E


Question: Let $R$ be a ring and $I$ an ideal of $R$. Suppose $n \in I$ is a unit in $R$. Prove that $R = I$.

Answer: R = I


Question: Consider a two-dimensional sphere. If a curve \( C \) on the sphere has geodesics that are normal to it, do all these geodesics intersect at a single point? Answer "True" or "False".

Answer: B


Question: Let $A$ be a subset of a topological space $X$ with a topology $T$. For every $x \in A$, there exists an open set $U$ containing $x$ such that $U \subseteq A$. Is it possible to construct a function $f: A \to B$, where $B = \{ U \in T : U \subseteq A \}$?

Answer: Yes


Question:
Given that \( w \) is a complex number satisfying the equation:
\[ 5w^3 - 3i|w|^2 - 2i = 0 \]
find the modulus \( |w| \) without explicitly solving for \( w \).

Answer: 1


Question: Evaluate the sum \( l + m \), where \( l = \lim_{x\rightarrow 0}\int_{0}^{x}\frac{(1+\cos t)^2}{x}dt \) and \( m = \lim_{x\rightarrow \infty}\int_{0}^{x}\frac{(1+\cos t)^2}{x}dt \).

Answer: \dfrac{11}{2}


Question: Determine whether the following statement is true or false: For two skew-adjoint matrices $A$ and $A'$, where $A^*=-A$ and $A'^*=-A'$, is it true that $\sup_{t \in \mathbb{R}} \Vert e^{-tA}-e^{-tA'} \Vert = 2(1-\delta_{A,A'})$?

Answer: True


Question: Given that $T_n$ is a continuous random variable with the probability density function $f_{T_n}(t) = 8n(1-nt) I_{(\frac{1}{2n},\frac{1}{n})}(t)$ for $n \in \mathbb{N}$, determine the probability density function of the limiting distribution for the sequence $T_1, T_2, \ldots$.

Answer: \delta(t)


Question: Evaluate the integral \( \int_0^{\frac{\pi}{2}} g(x) \, dx \), where \( g(x) = \sum_{n=1}^{\infty} \frac{\cos(2nx)}{n^3} \).

Answer: 0


Question: Determine whether the following statement is true or false: Let $v_1, v_2, v_3, v_4, w \in \mathbb{R}^7$. If $\text{Span}(v_1, v_2, w) = \text{Span}(v_3, v_4, w)$, then $\text{Span}(v_1, v_2) = \text{Span}(v_3, v_4)$. Assume that $\{v_1, v_2\}$ and $\{v_3, v_4\}$ are linearly independent sets.

Answer: B


Question: Let $P \subset \mathbb{R}$ be a perfect set. For each nonzero $r \in \mathbb{R}$, define $D_r = r \cdot P$. Is $D_r$ still a perfect set? Justify your answer.

Answer: Yes


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function with the following properties: \( f(0) = 0 = f(b) \), \( f'(0) = 1 \), \( f'(b) < 0 \), and \( f > 0 \) on \((0, b)\). Given that \( b > 0 \), determine if there exists a constant \( C > 0 \) such that \( f(x) \geq C \min \{ x, b-x \} \) for all \( x \in (0, b) \).

Answer: Yes


Question: Let X be a continuous uniform random variable on the interval [0, 50]. The conditional distribution of Y given X = x is uniform on the interval [0, 15x]. Calculate the expected value of Y, E[Y].

Answer: 187.5


Question: Let $R$ be a commutative ring isomorphic to the product of a finite number of fields, each containing a copy of the real numbers, i.e., $R \cong \prod_{n=1}^k F_n$ and for each $n$, $\mathbb{R} \subseteq F_n$. Determine if there exists a surjective ring homomorphism from the polynomial ring $S = F[x_1, \ldots, x_n]$ onto $R$.

Answer: Yes


Question: Given integers \(a, b, c, d > 0\) such that \(a + b + c + d = 40\), find the maximum value of \(ab + bc + cd\).

Answer: 399


Question: Consider a noncompact symmetric space $X=G/K$, where $G$ is the isometry group acting on $X$ and $K$ is the stabilizer of a fixed point $x_0 \in X$. Let $n=\dim X$. For every $x \in X$, for a small radius $r>0$, the volume of a geodesic ball $B_X(x,r)$ satisfies $|B_X(x,r)| \sim r^n$. Now, for a discrete, torsion-free subgroup $\Gamma$ of $G$, the quotient $M=\Gamma \backslash G/K$ is a locally symmetric space. Can we claim that for every $\tilde{x} \in M$ and a sufficiently small radius $r>0$, $|B_M(\tilde{x},r)| \sim r^n$?

Answer: Yes


Question: In a square room with mirrored walls, you are a fixed point, and there is another fixed point representing a man with a laser gun. The laser beam reflects off the walls at angles equal to its angles of incidence. Can you arrange a finite number of bodyguards in the room to completely protect yourself from the laser beam? If so, what is the maximum number of bodyguards needed to ensure protection regardless of the positions of you and the gunman in the room?

Answer: 4


Question: Determine whether the set of all orthogonal \(m \times m\) matrices, considered as points in \(\mathbb{R}^{m^2}\), is locally compact in the subspace topology.

Answer: Yes


Question: Find the rank of the linear transformation \( T: M_{8\times 8}(F) \to M_{8\times 8}(F) \) defined by \( T(A) = A - A^T \).

Answer: 28


Question:
Given the sequences \(a_n=\int_0^1 (1-x^2)^n \,dx\) and \(b_n=\int_0^1 (1-x^3)^n \,dx\) where \(n \in \mathbb{N}\), find the limit:
\[ \lim_{n\to \infty}(10\sqrt[n]{a_n}+5\sqrt[n]{b_n}). \]

Answer: 15


Question: Find the smallest positive solution of the equation \(\sum_{n=0}^{\infty}\frac{(-x)^n}{(n!)^2}=0\) and provide the answer with four digits of the fractional part.

Answer: 1.4458


Question: For compact probability measure-preserving (pmp) actions of a group $G$ on a measure space $(X, \mu)$, consider a subalgebra $B$ such that $L(G) \subset B \subset L^{\infty}(X) \rtimes G$. Is it always true that $B$ can be expressed as a crossed product with an action of $G$?

Answer: Yes