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

Answer: -\dfrac{3}{2}


Question: If $P$ is a topological property, is $P$-maximal also a topological property?

Answer: Yes


Question: Let $F$ be an infinite-dimensional complex Hilbert space with inner product $\langle\cdot\;| \;\cdot\rangle$ and norm $\|\cdot\|$. Define the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$. Let $\mathcal{B}(F)$ be the algebra of all bounded linear operators on $F$. Consider a bounded operator $M \in \mathcal{B}(F)$ such that $M \in \mathcal{B}(F)^+$ (i.e., $\langle Mx,x\rangle\geq0$ for all $x\in F$) and $M$ is injective. Define the set $S_M(0,1)=\{x\in F:\;\langle Mx, x\rangle=1\}$. Is $S_M(0,1)$ always homeomorphic to the 1-sphere $S(0,1)$?

Answer: Yes


Question:
Let $[v_1(k) \, v_2(k)]^T\in\mathbb{R}^2$ be a stochastic process that is independent of the process $\beta_i(k)$ for all $i=1,2,3,4$, with $\mathbb{E}[\beta_i(k)]=\bar\beta_i $. Both processes are assumed to be stationary. Determine if the following equality holds:
\[\mathbb{E}\left[[v_1(k) \, v_2(k)] \left[\begin{array}{cc}\beta_1(k) & \beta_2(k)\\ \beta_3(k) & \beta_4(k)\end{array}\right]\left[\begin{array}{c}v_1(k)\\v_2(k)\end{array}\right]\right]=\mathbb{E}\left[[v_1(k) \, v_2(k)] \left[\begin{array}{cc}\bar\beta_1 & \bar\beta_2\\ \bar\beta_3 & \bar\beta_4\end{array}\right]\left[\begin{array}{c}v_1(k)\\v_2(k)\end{array}\right]\right]\]

Answer: Yes


Question: Solve the equation \( 2x - e^{-x} = 0 \).

Answer: 0.3517


Question: Evaluate the line integral \( \int_C xy^4 \, \mathrm{d}s \), where \( C \) is the right half of the circle defined by \( x^2 + y^2 = 16 \).

Answer: \dfrac{8192}{5}


Question: Use Gauss's divergence theorem to calculate the total flux through the solid $V$ enclosed by the set $M=\{(x,y,z)\in \mathbb{R^3}: z=x^2+y^2, 0\leq z\leq 1\}$ and the vector field $X:\mathbb{R^3} \to \mathbb{R^{3\times 1}}, (x,y,z)\to  \begin{bmatrix} xz \\ z \\ -\frac{z^2}{2}\end{bmatrix}$. Compute the divergence of $X$ and evaluate the integral \( \int_V \text{div}(X) \, dV \).

Answer: 0


Question: Given the Hamiltonian \( H = \frac{1}{2}(S_1^2+S_2^2+\beta S_3^2)+R_1 \) and the first integrals \( f_1 = R_1^2+R_2^2+R_3^2 \) and \( f_2 = R_1S_1+R_2S_2+R_3S_3 \), compute the Poisson bracket \( \{f_1, f_2\} \) using the relations \( \{S_i, S_j\} = \epsilon_{ijk}S_k \), \( \{R_i, S_j\} = \epsilon_{ijk}R_k \), and \( \{R_i, R_j\} = 0 \). Verify if \( \{f_1, f_2\} = 0 \).

Answer: 0


Question:
Given the function \( f(x) = \prod_{n=1}^{100} (x-n)^{n(101-n)} \), find the value of \( \frac{f(101)}{f'(101)} \). Choose the correct answer from the following options:

\((A)\) 5050

\((B)\) \(\frac{1}{5050}\)

\((C)\) 10010

\((D)\) \(\frac{1}{10010}\)

Answer: B


Question: For a symmetric function $f(x_1, x_2, \ldots, x_n)$, it is known that \( \int_0^\infty \int_0^\infty f(x_1,x_2) \, dx_1 \, dx_2 = 2 \int_0^\infty \int_0^{x_2} f(x_1,x_2) \, dx_1 \, dx_2 \). Determine if there exists a constant $C$ such that \( \int_0^\infty \cdots \int_0^{\infty} f(x_1, \ldots, x_n) \, dx_1 \cdots \, dx_n = C \int_0^\infty \int_0^{x_n} \cdots \int_0^{x_n} f(x_1, \ldots, x_n) \, dx_1 \cdots \, dx_n \) for any symmetric function $f$ and any integer $n$. If such a $C$ exists, express $C$ in terms of $n$. \(n\) is the number of integrals.

Answer: n


Question: Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$. Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$. Does there exist a natural number $d = d(\overline{X})$ such that for any $k$-form $X$ of $\overline{X}$, the variety $X$ has a $K$-point over some finite field extension $K$ of $k$ with degree $[K:k] \le d$? Answer YES or NO.

Answer: NO


Question: If three prime numbers \(a, b, c > 3\) satisfy the relation \(2a + 5b = c\), what is the maximum integer \(n\) such that \(a + b + c\) is divisible by \(n\)?

Answer: 9


Question: Find the minimum value of the expression \( x^4 + \frac{1}{x^4} + y^4 + \frac{1}{y^4} \) subject to the constraint \( x^2 + y^2 = 4 \).

Answer: \dfrac{17}{2}


Question: Let $A$ be a Lebesgue measurable subset of $\mathbb{R}$ with $m(A) < \infty$, where $m$ denotes the Lebesgue measure. Given $\epsilon > 0$, does there exist an open set $G$ and a closed set $F$ such that $F \subset A \subset G$ and $m(G - F) < \epsilon$?

Answer: Yes


Question: If $P$ is a finite rank projection and $A$ is an infinite dimensional $C^*$ algebra, is $PAP$ necessarily finite dimensional?

Answer: Yes


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

Answer: 2 \ln 2 - 1


Question: What is the coefficient of $x^{19}$ in the expansion of $ \prod \limits_{n=1}^{20} (x+n^2)$?

Answer: 2870


Question: Determine the constant $c$ such that the equation \( \sum_{k=2}^{N}c^{\frac{1}{k\log k}}=N \) holds true for any positive integer \( N \).

Answer: 4


Question:
Determine the radius of convergence for the power series:
\[ \sum_{n=1}^{\infty}\left(\frac{1}{n} - \sin\left(\frac{1}{n}\right)\right)\cdot x^n ,\; x\in\mathbb{R} \]

Answer: 1


Question: Determine the exponent of convergence of the sequence \( \{m+in \ | \ m, \ n \in \mathbb{Z}, (m,n) \neq (0,0) \} \). The exponent of convergence \( \rho \) of a sequence \( \{z_{n}\} \subset \mathbb{C} \) is defined as \[ \rho = \inf \{\lambda \geq 0 \ | \ \sum_{n}\frac{1}{|z_{n}|^{\lambda}} < +\infty\} \]

Answer: 2


Question: Determine \( N(h) \), the maximum number of full subtrees of a full binary tree of height \( h \). For example, \( N(0) = 1 \), \( N(1) = 2 \), \( N(2) = 3 \), and \( N(3) = 4 \). What is the general formula for \( N(h) \)?

Answer: h + 1


Question: Determine all values of \( x \) such that the limit \( \lim_{n \to \infty} T_n = x \), where \( T_n = \sqrt{x \sqrt{x \sqrt{x \dots \sqrt{x}}}} \) (with \( n \) nested square roots) and \( n \in \mathbb{N} \).

Answer: [0, \infty)


Question: Determine the values of \( p \in \mathbb{R} \) for which the integral \( \int_{\mathbb{R}^n} |\log(x)|^p \, dx \) converges.

Answer: \emptyset


Question: Find the radius of convergence for the power series \( \sum_{n=2}^\infty a_n x^n \) given that \( \lim_{n \to \infty} \frac{a_n}{n^3} = 1 \).

Answer: 1


Question: Consider the function \( f = r^2 \) in spherical coordinates. Using the scale factors and the formula for the Laplacian, compute \( \nabla^2 f \). The result should be a number times \( r^{n-2} \). State this number.

Answer: 6


Question: Determine the minimum number of times a coin must be tossed so that the probability that the discrepancy of the relative frequency of heads from 1/2 is less than 0.02 is at least 0.9.

Answer: 1692


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

Answer: 0


Question: Calculate the work done against the force field \( F(x,y) = -\frac{(x,y)}{\sqrt{x^2 + y^2}} \) in moving a particle from \((2a,0)\) to the origin along the top half of the circle \((x−a)^2+y^2=a^2\).

Answer: -2a


Question: Let \(P_1, P_2, P_3, P_4, P_5\) be five distinct planes in \(\mathbb{R}^3\) such that the intersection of any two distinct planes is a line, the intersection of any three distinct planes is a point, and the intersection of any four distinct planes is empty. Let \(X = \bigcup_{i=1}^5 P_i\) with the topology as a subspace of \(\mathbb{R}^3\). Determine the Euler characteristic of \(X\).

Answer: 5


Question: In a circle with diameter $AB$, a chord $PQ$ is drawn perpendicular to $AB$ at point $R$. On the arc $AQ$, a point $U$ is chosen such that line $PU$ intersects $AB$ and $AQ$ at points $S$ and $T$, respectively. Given that $PS \cdot TU = 8 \cdot TS$, find the length of segment $UQ$.

Answer: 8


Question: Let $T$ be an isomorphism acting on a separable Banach space $X$. Determine whether there always exists a countable dense subset $D$ of $X$ such that $T(D) = D$. Provide a proof or counterexample.

Answer: Yes


Question: Determine the number of fixed points of the function \( f(x) = e^x \) where \( f: \mathbb{R} \to \mathbb{R} \).

Answer: 0


Question: Determine the maximum number of connected closed sets \(A_1, A_2, A_3, \ldots\) that can be drawn in \(\mathbb{R}^2\) such that for each pair \(i, j\), the interiors \(\operatorname{int}(A_i)\) and \(\operatorname{int}(A_j)\) are disjoint, but the intersection \(A_i \cap A_j\) is non-empty for all \(i \neq j\).

Answer: 4


Question: Determine the number of six-digit numbers of the form $abcabc$ that have exactly four prime factors, with one of these factors having a power of 3. Express your answer as an integer $n$. Note that $abcabc$ can be expressed as $7 \times 11 \times 13 \times (100a + 10b + c)$. The prime factor with a power of 3 must be $100a + 10b + c$. Find the number of such numbers $n$.

Answer: 1


Question: Evaluate the integral \( \oint_{|z|=2}\frac{\cos(z)}{z^5 -32} \, dz \).

Answer: 0


Question:
Calculate the winding number of the closed curve \( \gamma(t) = \frac{-1}{2} + \cos(3t) + i\sin(4t) \) at the point 0, where \( 0 \leq t \leq 2\pi \). Evaluate the integral:
\[ \int_{0}^{2\pi} \frac{-3\sin(3t) + 4i\cos(4t)}{\frac{-1}{2} + \cos(3t) + i\sin(4t)} \, dt. \]

Answer: 0


Question: Given a square, construct two points $M$ and $N$ such that the perimeter of triangle $MCN$ is equal to half of the perimeter of the square. Find the measure of angle $MAN$. Assume $CM = CN$ and use trigonometry to solve for the angle.

Answer: 45^\circ


Question: Given that \( z = -1 + i \) is a root of the polynomial equation \( z^7 + az^3 + 2b = 0 \), find the value of \( a + b \).

Answer: 4


Question:
Find the determinant of the matrix given by:
\[
\begin{pmatrix}
\cos(\alpha_1 -\beta _1) & \cos(\alpha_1 -\beta _2) & \cdots & \cos(\alpha_1 -\beta _n)\\
\cos(\alpha_2 -\beta _1) & \cos(\alpha_2 -\beta _2)& \cdots & \cos(\alpha_2 -\beta _n)\\
\vdots & \vdots& \ddots& \vdots\\
\cos(\alpha_n -\beta _1)& \cos(\alpha_n -\beta _2)& \cdots & \cos(\alpha_n -\beta _n)
\end{pmatrix}
\]
where \(n \geq 3\).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \rightarrow \infty}\frac{\ln(x+1)-\ln(x)}{x} \] without using L'Hopital's Rule or Taylor Series.

Answer: 0


Question: Given the function \( f(x) = \frac{1}{\sqrt[3]{3x+2}} \), find the Taylor series expansion about \( x = 1 \) and determine the radius of convergence.

Answer: \dfrac{5}{3}


Question: Using Rouche's theorem, determine the number of roots of the polynomial \( p(z) = 2z^4 - 2z^3 + 2z^2 + 7 \) that lie inside the region \( \{z \in \mathbb{C} : |z| < 2\} \).

Answer: 4


Question: A polynomial of degree $2n$ has all coefficients in the range $[100,101]$ and has a real root. Determine the minimum possible value of $n$.

Answer: 100


Question: Find the necessary and sufficient condition for the real numbers $a_1, a_2, \dots, a_n$ such that the expression \( \sum_{k=1}^{n} a_k \cot(kx) \) admits a finite limit as \( x \to 0 \). Compute this limit.

Answer: 0


Question: Evaluate the line integral \( I = \int_\gamma \left(4e^{(y-4x+3)^2}-y\right)dx + \left(-e^{(y-4x+3)^2}+2x\right)dy \), where \( \gamma \) is the curve along the parabola \( y = x^2 \) from \( (1,1) \) to \( (3,9) \).

Answer: 26


Question: Use the fourth order Runge-Kutta method with step size \( h = 1 \) to approximate \( \sqrt{2} \) from the initial value problem given in the previous question. Provide your answer rounded to five decimal places.

Answer: 1.41438


Question: Find the Taylor series of the function \( f(z) = \log(z^2-3z+2) \) at \( z = 0 \) for \( z \in \mathbb{C} \setminus \{1,2\} \). Calculate the radius of convergence for the series.

Answer: 1


Question: Expand the function \( f(z) = \frac{z^2 - 2z + 5}{z - 1} \) into a Laurent series about the singularity at \( z = 1 \). Determine the residue of \( f(z) \) at \( z = 1 \) and use it to calculate the integral of \( f(z) \) along the unit circle centered at \( z = 1 \).

Answer: 8\pi i


Question: Let \( A \subset \mathbb{R}^n \) be a set with finite Lebesgue measure, i.e., \( |A| < \infty \). Show that \( |A \cap B_r(0)^c| \to 0 \) as \( r \to \infty \), where \( B_r(0) \) is the ball of radius \( r \) centered at the origin.

Answer: 0


Question: Determine the possible values of the characteristic $p$ of a field over which the polynomial $x^4 + x + 1$ has multiple roots.

Answer: 229


Question: In triangle \( \triangle ABC \), the sides are given as \( AB = n \), \( AC = n-1 \), and \( BC = n+1 \), where \( n \) is a natural number. The angles opposite these sides are \( A \), \( 2A \), and \( \pi - 3A \) respectively. Determine the value of \( n \).

Answer: 5


Question:
Determine the radius of convergence of the power series solution to the initial value problem \( y' - 2xy = 0, \ y(1) = 2 \), given by the series:
\[ y(x) = 2\left(1 + 2(x-1) + 3(x-1)^2 + \frac{10}{3}(x-1)^3 + \frac{19}{6}(x-1)^4 + \frac{26}{10}(x-1)^5 + \cdots\right) \]
with coefficients generated by the recurrence relation:
\[ a_{n+1} = \frac{2(a_n + a_{n-1})}{n+1}, \quad n > 0. \]

Answer: \infty


Question: Evaluate the limit \( \lim_{x \rightarrow 0} (a^x + b^x - c^x)^\frac{1}{x} \) where \( a, b, c > 0 \).

Answer: \dfrac{ab}{c}


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

Answer: 0


Question: Find the limit of the sequence: \[ \lim_{n\to \infty} \frac{n(n^{1/n}-1)}{\ln n} \]

Answer: 1


Question: Find the critical points of the function \( f(x,y) = (y-x^2)(y-2x^2) \) and determine if they are local extremums.

Answer: (0, 0)


Question: Find the limit of the following function as \( z \to 0 \) in \( \mathbb{C} \): \[ \lim_{z\to0} \left(\frac{1}{e^{z}-1}-\frac{1}{z}\right) \]

Answer: -\dfrac{1}{2}


Question: Find the expectation of the cube of Brownian motion at time \( t \), denoted as \( E(W_t^3) \).

Answer: 0


Question: Let $E$ be a smooth curve, meaning it is a one-dimensional projective variety with a non-singular Jacobian matrix at any point. Suppose there is an isomorphism $\phi$ that maps $E$ to another curve $E'$. Is $E'$ necessarily smooth?

Answer: Yes


Question: Suppose that $A \subseteq \mathbb{R}$ and consider the Lebesgue outer measure $m$ on $\mathbb{R}$. Denote by $A^o$, $\overline{A}$, and $\partial A$ the interior, closure, and boundary of $A$, respectively. Given that $m(A^o) = m(\overline{A})$, is it true that $m(\partial A) = 0$?

Answer: Yes


Question: Find the sum of the infinite series: \[ \sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{8^n} \]

Answer: \dfrac{1}{11}


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

Answer: 1


Question: Find the set of complex numbers \( z \) at which the function \( F: \mathbb{C} \to \mathbb{C} \) defined by \( F(z) = -|3z|^2 \) is complex differentiable.

Answer: \{0\}


Question: Let $V$ be a $\mathbb{C}$-vector space, and let $f_1,\dots,f_n \in S^d(V^*)$ be homogeneous polynomials of degree $d$ such that $V(f_1,\dots, f_n)=\{0\}$. Determine whether there must exist a positive integer $k\geq d$ such that for all $v \in S^k(V^*)$, there exist $g_1,\dots, g_n \in S^{k-d}(V^*)$ for which $v=\sum_{i=1}^n g_i f_i$. Provide a justification for your answer.

Answer: Yes


Question: Find the number of rearrangements of the string "12345" such that none of the sequences "12", "23", "34", or "45" appear consecutively. Use the principle of inclusion-exclusion to solve this problem.

Answer: 53


Question: Evaluate the integral \( \int_{0}^{\infty} \left(\frac{\sin x}{x}\right)^2 dx \) using complex analysis techniques.

Answer: \dfrac{\pi}{2}


Question: Let $V$ be a finite-dimensional $\mathbb{C}$-vector space, and let $\mathscr{S}$ be a collection of rank-1 projectors on $V$ such that $\sum_{p \in \mathscr{S}} pV = V$. Suppose $T$ is an endomorphism of $V$ with $\operatorname{tr} T \neq 0$. Is there necessarily a projector $p \in \mathscr{S}$ such that $pTp \neq 0$?

Answer: Yes


Question: Find the positive integer \( n \) such that \[ \lfloor\log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994. \]

Answer: 312


Question: Evaluate the limit \( \lim_{q\rightarrow 1} \frac{\phi^5(q)_{\infty}}{\phi(q^5)_{\infty}} \), where the Euler function is defined as \( \phi(q)=\prod_{n = 1}^{\infty}(1-q^{n}) \) and \( q=\exp(2\pi i \tau) \) with \( |q|<1 \).

Answer: 0


Question: Find the equation of the plane that is tangent to the surface given by \( z = x^2 + y^2 \) at the point \((1, 2, 5)\).

Answer: z = 2x + 4y - 5


Question: Determine the absolute maximum value of \( f(z) = \left| \frac{5z^4 + 4}{z^2 + 1} \right| \) on the closed disk defined by \( |z - 2i| \leq 0.5 \).

Answer: \dfrac{1063}{28}


Question: What is the largest determinant of a real $3 \times 3$ matrix with entries from the interval $[-1,1]$?

Answer: 4


Question: Evaluate the limit: $$\lim_{x\to 0}(\cos x)^{\dfrac{1}{\ln(\sin^2(x))}}$$ without using L'Hospital's rule.

Answer: 1


Question: Find the sum of all solutions to the equation \( f(x) = |x^2 - 2| \), where \( f(x) = \lim_{n \to \infty} \frac{2}{n^2} \left( \sum_{k=1}^{n} kx \right) \cdot \left( \frac{3^{nx} - 1}{3^{nx} + 1} \right) \).

Answer: 0


Question: Evaluate the integral \( \int_{\vert z \vert = 1} \frac{1}{z^4} \, dz \) over the unit circle.

Answer: 0


Question:
Let $f$ be a function defined for all positive integers $x$ and $y$ satisfying the following conditions:
1. $f(1, 1) = 2$
2. $f(x+1, y) = 2(x+y) + f(x, y)$
3. $f(x, y+1) = 2(x+y-1) + f(x, y)$

Find the value of $x-y$ for which $f(x, y) = 2006$.

Answer: -20


Question: Calculate the flux of the vector field \( H = \frac{-3}{(x^2+y^2+z^2)^{3/2}} \langle x, y, z \rangle \) through the boundary of a solid region \( E \) in \( \mathbb{R}^3 \) with a known volume of 10 cubic units, assuming the origin lies inside \( E \). Use Gauss' Law to find the flux. The given answer is \(-12\pi\).

Answer: -12\pi


Question: Calculate the limit: \[ \lim _{(x,y)\to (1,2)} \frac {\arctan(x+y-3)}{\ln(x+y-2)} \]

Answer: 1


Question: Given a $202 \times 202$ matrix $A$ with $||A||_2 = 100$ and $||A||_F = 101$, find the sharpest lower bound on the 2-norm condition number $\kappa(A) = ||A||_2 \, ||A^{-1}||_2$. Use the theorem that states $||A||_2 = \sigma_1$ (the largest singular value), $||A^{-1}||_2 = \sigma_r^{-1}$ (where $\sigma_r$ is the smallest singular value), and $||A||_F = \sqrt{\sigma_1^2 + \ldots + \sigma_r^2}$, with $r = \text{rank}(A)$. Provide your answer as a single number.

Answer: 100


Question: Verify whether the plane given by the equation \(x + 2y + 3z = 6\) is tangent to the surface defined by \(x^2 + 2y^2 + 3z^2 = 6\) at any point.

Answer: (1, 1, 1)


Question: Is there a finite group $G$ of order $p(p^2-1)$ that has a normal subgroup $N$ such that $G/N \cong PSL(2, p)$?

Answer: Yes


Question: In triangle $ABC$, it is given that $\angle B = 2\angle C$. Point $D$ is located on side $BC$ such that $AD$ bisects $\angle BAC$ and $AB = CD$. Determine the measure of $\angle BAC$. Avoid using trigonometry or constructions in your solution.

Answer: 72^\circ


Question: Consider a matroid $M$ of rank 3 with three bases $E_1, E_2, E_3$. Let $e_{i,j}$ denote the $i$-th element of base $E_j$. Determine if there always exists a permutation $s: \{1,2,3\} \to \{1,2,3\}$ such that $e_{s(1),1}, e_{s(2),2}, e_{s(3),3}$ forms a base of $M$. Provide a proof or counterexample.

Answer: Yes


Question: The probability that a fire will occur is 0.001. If there is a fire, the amount of damage, $X$, follows a Pareto distribution with $P(X>x) = \left(\frac{2(10)^6}{2(10)^6+x} \right)^2$. An insurance policy covers the excess loss over a deductible of $100,000$. The one-time insurance premium is 110% of the expected payment. Calculate the insurance premium.

Answer: 2095.24


Question:
Use the Weierstrass formula for the Gamma function and the reflection formula to find an expression for \( f(z) \) such that:
\[
\sin(z\pi) = f(z)\prod^{\infty}_{n=1} \left(1- \frac{z^2}{n^2}\right)
\]
is valid for any number \( z\not\in \mathbb{Z} \).

Answer: \pi z


Question: Compute the expected value \( E[X_{n+1} | X_0, \dots, X_n] \) given that \( X_n = X_0 e^{\mu S_n} \), where \( X_0 > 0 \), \( S_n \) is a symmetric random walk, and \( \mu > 0 \).

Answer: X_n \cosh(\mu)


Question: Compute the limit: \[ \lim_{ n\rightarrow \infty  }{ { \left (\frac { \sqrt [ n ]{ a } +\sqrt [ n ]{ b } +\sqrt [ n ]{ c } +\sqrt [ n ]{ d }  }{ 4 } \right ) }^{ n } } \] where \(a,b,c,d\geq0\).

Answer: \sqrt[4]{abcd}


Question:
Given a measurable set $A \subset \mathbb{R}$ with Lebesgue measure $\lambda(A) > 0$, determine whether there always exist two measurable sets $K_1$ and $K$ such that:

- $K_1 \subset A$
- $K \subset \mathbb{R}$
- $K_1 \subset K$
- $K$ is compact
- $\lambda(K_1) = \lambda(K)$
- $\lambda(K_1) > 0$

Is it always possible to find such sets $K_1$ and $K$?

Answer: Yes


Question:
Let \( f:\mathbb{R}\to\mathbb{R} \) be a continuous function such that \( f(x)\geq 0 \) for all \( x \) and \( \int_{-\infty}^{\infty}f(x) \, dx = 1 \). For \( r\geq 0 \), define
\[
I_{n}(r) = \int\cdots\int_{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\leq r^{2}}f(x_{1})f(x_{2})\cdots f(x_{n})\,dx_{1}\,dx_{2}\cdots\,dx_{n}
\]
Find \( \lim_{n\to\infty} I_{n}(r) \) for a fixed \( r \).

Answer: 0


Question: Consider a deterministic finite automaton (DFA) with $k$ states, where the language $L(M) = L$ is infinite. Determine whether it is true that there exists a word in $L$ with length at least $k$ and at most $2k-1$. Justify your answer.

Answer: True


Question:
Calculate the sum of all residues at the poles of the function \( f(z) = \frac{\cot(\pi z)}{(z+a)^2} \), where \( a \in \mathbb{Z} \setminus \{0\} \). Choose the correct answer from the following options:

(a) \( \frac{1}{\pi} \sum_{n=-\infty}^{\infty}\frac{1}{(n+a)^2} - \pi \csc^2 \pi a \)

(b) \( -\frac{1}{\pi} \sum_{n=-\infty}^{\infty}\frac{1}{(n+a)^2} - \pi \csc^2 \pi a \)

(c) \( \frac{1}{\pi} \sum_{n=-\infty}^{\infty}\frac{1}{(n+a)^2} + \pi \csc^2 \pi a \)

(d) \( -\frac{1}{\pi} \sum_{n=-\infty}^{\infty}\frac{1}{(n+a)^2} + \pi \csc^2 \pi a \)

Answer: D


Question: Consider the complete bipartite graph $G = K_{3,7}$ with partitions $X = \{x_1, x_2, x_3\}$ and $Y = \{y_1, y_2, y_3, y_4, y_5, y_6, y_7\}$. The genus of $G$ is known to be 2. If we delete the edges $\{x_1y_1\}$, $\{x_2y_2\}$, and $\{x_3y_3\}$, and add the edges $\{x_1x_2\}$, $\{x_2x_3\}$, and $\{x_1x_3\}$, does the genus of the resulting graph remain 2?

Answer: 2


Question: Given the polynomial \( P(x) \) such that \( P(x^4) = ax^8 + (b+1)x^5 + (ab)x^4 + (a-1)x^2 + 2b-2 \), find the remainder when \( P(x^3 - 3) \) is divided by \( x+1 \).

Answer: 16


Question:
Which of the following is the correct interpretation of the statement: "$S_n$ is the sum of $n$ independent observations $X_1,X_2,\ldots,X_n$ on a chance variable $X$"?

1. Let $M$ be a measurable space and let $X:M\to\mathbb{R}$ be a measurable function. Then $S_n:M\times\cdots\times M\to\mathbb{R}$ is the function $S_n(p_1,\ldots,p_n)=X(p_1)+\cdots+X(p_n)$.

2. Let $(M,\mu)$ be a probability space and let $X:M\to\mathbb{R}$ be a measurable function. Let $X_1,\ldots,X_n$ be real-valued measurable functions on $M$ such that $(X_i)_\sharp\mu=X_\sharp\mu$ as measures on $\mathbb{R}$ for each $i$ and such that $(X_1,\ldots,X_n)_\sharp\mu=(X_1)_\sharp\mu\times\cdots\times(X_n)_\sharp\mu$ as measures on $\mathbb{R}^n$. Define $S_n:M\to\mathbb{R}$ by $S_n(p)=X_1(p)+\cdots+X_n(p)$.

3. Something else?

Answer: 2


Question:
Let \( y \) be a polynomial solution of the differential equation \((1-x^2)y''-2xy'+6y=0\). If \( y(1)=2 \), find the value of the integral \( \int_{-1}^{1}y^2\,dx \). Choose the correct answer from the following options:

- \( \frac{1}{5} \)
- \( \frac{2}{5} \)
- \( \frac{4}{5} \)
- \( \frac{8}{5} \)

Answer: D


Question: Determine the mutual information between a tensor \( x \) sampled from its probability distribution in \( \mathbb{R}^{n, m} \) and a constant tensor \( c \) in \( \mathbb{R}^{n, m} \) sampled from a degenerate distribution. What is the value of this mutual information?

Answer: 0


Question: Find the residue at $z = 0$ of the function $\frac{1}{z+z^2}$ by determining the $b_1$ coefficient in its Laurent series expansion.

Answer: 1


Question: In a round-robin tournament with $2n$ teams, each team plays every other team exactly once over $2n-1$ days. On each day, $n$ games are played, with one team winning and one team losing in each game. Is it always possible to select one winning team from each day such that no team is selected more than once?

Answer: Yes


Question: Evaluate the complex integral \( \int_{[-i,i]} \sin(z)\,dz \) and verify if the result is zero.

Answer: 0


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

Answer: \dfrac{1}{2}


Question: Find the largest value of the function \( f(x) = \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} \) for \( x \in \mathbb{R} \).

Answer: \sqrt{10}


Question: Find \( \lim_{n\rightarrow \infty}\frac{(2n-1)!!}{(2n)!!}. \)

Answer: 0


Question: Find the dimension of the complex vector space \( \mathbb{C}[X,Y]/I \), where \( I = (Y^2 - X^2, Y^2 + X^2) \) is an ideal in \( \mathbb{C}[X,Y] \).

Answer: 4


Question: Evaluate the line integral \( \int_C (2xy^3+\cos x)\,dx + (3x^2y^2+5x)\,dy \) where \(C\) is the circle defined by \(x^2+y^2=64\).

Answer: 320\pi


Question: Let $X$ be a topological space such that for any subset $A$ of $X$, if $A$ is compact, then $X \setminus A$ is also compact. Determine whether every subset of $X$ is compact.

Answer: Yes


Question: Determine the convergence radius of the power series \( \sum_{n=1}^{\infty}\frac{\sin n}{n}x^n \).

Answer: 1


Question: Find the expected value of the minimum of two independent and identically distributed random variables drawn from the distribution with cumulative distribution function (CDF) \( F(x) = 1 - \frac{1}{x} \) for \( x \geq 1 \).

Answer: 2


Question: Let \(f: [a, b] \rightarrow \mathbb{R}\) be a convex function such that \(f(x) \geq f(a)\) for all \(x\) in \([a, b]\). Is \(f\) necessarily increasing on \([a, b]\)? Justify your answer.

Answer: Yes


Question: Determine whether the set of singular matrices is negligible with respect to the Lebesgue measure on \( \mathbb{C}^{n^2} \) or \( \mathbb{R}^{n^2} \).

Answer: Yes


Question: Determine the largest number of edges in a graph with 7 vertices and girth 4.

Answer: 12


Question: Suppose there are problems A and B in NP, and problem A polynomially transforms to problem B. If A is in P, does this imply that we learn nothing new about problem B? Answer true or false.

Answer: True


Question: Let $X$ and $Y$ be metric spaces, and let $f: X \rightarrow Y$ be a continuous open mapping. Suppose $F \subseteq X$ is a compact set. Prove that for each $x \in F$ and $r > 0$, there exists $w = w(r) > 0$ such that $B^Y_{w(r)}(f(x)) \subseteq f\left(B^X_r(x)\right)$. Show that $w(r)$ can be chosen independently of $x$.

Answer: w(r)


Question:
Let \( N \) be a Poisson random variable with parameter \( m \). Evaluate the limit:
\[
\lim_{m \to \infty} \frac{E[\log(N+\frac{1}{m})]}{\log(m)}.
\]

Answer: 1


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

Answer: 0


Question: Evaluate the double integral \( \iint_D \arctan e^{xy} \, dy \, dx \), where the region \( D \) is defined by \( \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 4x\} \).

Answer: \pi^2


Question: Determine the median area of a random n-gon formed inside a unit square. The square is bounded by the coordinates (0,0), (0,1), (1,0), and (1,1). Random x and y coordinates are chosen in the interval [0,1] for each of the n points, and these points are randomly connected to form a cycle. What is the median area of this polygon?

Answer: \dfrac{1}{2}


Question: Evaluate the mean value of the expression \( \cos(4\pi \cdot 50t + 8\pi \sin(5 \cdot 10^3 t)) \) over one period.

Answer: 0


Question: Let $V$ be a vector space over a field $F$ with $\dim(V) \ge 2$, and let $T: V \rightarrow V$ be a bijective linear transformation. Does the minimal polynomial of $T$ necessarily have a linear factor? Provide a justification for your answer.

Answer: No


Question: Determine the number of elements in the module \(A = (\mathbb{Z}[i]/I^{2014})\otimes_{\mathbb{Z}[i]}(\mathbb{Z}[i]/J^{2014})\), where \(I\) and \(J\) are the principal ideals of \(\mathbb{Z}[i]\) generated by \(7-i\) and \(6i-7\), respectively.

Answer: 5^{2014}


Question: Evaluate the integral \( \int_{0}^{\pi}\frac{x \, dx}{1+ e \sin x} \) and find the value of \( K \) such that \( \int_{0}^{\pi}\frac{x \, dx}{1+ e \sin x} = K\frac{\arccos e}{\sqrt{1-e^{2}}} \), given that \( e^{2} < 1 \).

Answer: \pi


Question: Evaluate the limit \( \lim_{x \to 0} \frac{e^x-1}{e^{2x}-1} \) using the \( \delta - \varepsilon \) definition of a limit.

Answer: \dfrac{1}{2}


Question: Determine the number of L-structures of size \( n \) for a language \( L \) that contains a single binary function symbol \( + \).

Answer: n^{n^2}


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

Answer: e^{-1/2}


Question: Calculate \( E[Z(Z-1)(Z-2)(Z-3)] \) where \( Z \) is a Poisson random variable with expected value \( E(Z) = 2.5 \).

Answer: \dfrac{625}{16}


Question: Given the function \(f(n) = \left(\frac{5+3\sqrt{5}}{10}\right)\cdot\left(\frac{1+\sqrt{5}}{2}\right)^n + \left(\frac{5-3\sqrt{5}}{10}\right)\cdot\left(\frac{1-\sqrt{5}}{2}\right)^n\), calculate \(f(n+1) - f(n-1)\) based on \(f(n)\).

Answer: f(n)


Question: Determine the number of conjugate subgroups of $S_3$ in $S_4$. Provide a brief explanation for your answer.

Answer: 4


Question: Determine if there exists a positive real number \(a\) such that for the function \(f_a(z) = z + a(1-z)^3\), the inequality \(|f_a(z)| < 1\) holds for all \(z\) in the open unit disk \(B(0,1)\).

Answer: \dfrac{1}{4}


Question: Use the divergence theorem to evaluate \( \iint_S F \cdot n \, dS \) where \( F = y^2x \mathbf{i} + x^2y \mathbf{j} + z^2 \mathbf{k} \) and \( S \) is the complete surface of the region bounded by the cylinder \( x^2 + y^2 = 4 \) and the planes \( z=0 \) and \( z=2 \).

Answer: 32\pi


Question: Does the tensor product \(\otimes\) endow the 2-category \(\operatorname{Mod}_\mathcal{C}\) of \(k\)-linear \((\mathcal{C}, \mathcal{C})\)-bimodule categories with the structure of a monoidal 2-category?

Answer: Yes


Question:
Identify the quotient group $H/K$ in the general linear group $GL_3(\mathbb{R})$, where:
\[ H = \begin{bmatrix} 1 & * & * \\ 0 & 1 & * \\ 0 & 0 & 1 \end{bmatrix} \]
\[ K = \begin{bmatrix} 1 & 0 & * \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
and $*$ represents an arbitrary real number.

Answer: \mathbb{R}^2


Question: Consider a vector space \( V \) of dimension \( n \) over a field \( \mathbb{K} \). Let \( T: V \rightarrow V \) be an affine linear map. Is \( T \) completely specified if it is defined on \( n+1 \) points \( v_1, v_2, \ldots, v_{n+1} \in V \) such that the set \( \{v_2-v_1, v_3-v_2, \ldots, v_{n+1}-v_1\} \) spans \( V \)? Justify your answer.

Answer: Yes


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

Answer: 0


Question: Given the polynomial \( f(x) = ax^2 + bx + c \) with conditions \( f(0) = 0 \) and \( f(2) = 2 \), find the minimum value of \( \int_{0}^{2} |f'(x)| \, dx \).

Answer: 2


Question:
Consider the group $G = \mathrm{SL}(2, \mathbb{R})$. Using the Iwasawa decomposition, a Haar measure on $G$ can be expressed as $\mathrm{d}x = \mathrm{d}a \, \mathrm{d}n \, \mathrm{d}k$. This implies that for a suitable function $f$, the integral over $G$ is given by:
\[ \int_G f(x) \, \mathrm{d}x = \int_K \int_N \int_A f(ank) \, \mathrm{d}a \, \mathrm{d}n \, \mathrm{d}k. \]

Another Haar measure on $G$ can be defined as $\mathrm{d}_1 x = \mathrm{d}k \, \mathrm{d}n \, \mathrm{d}a$, leading to:
\[ \int_G f(x) \, \mathrm{d}_1 x = \int_A \int_N \int_K f(kna) \, \mathrm{d}k \, \mathrm{d}n \, \mathrm{d}a. \]

Determine whether the constant $c$ such that $\mathrm{d}_1 x = c \cdot \mathrm{d}x$ is equal to 1.

Answer: 1


Question:
Calculate the radical of the ideal \((4,x)\) in the ring \(\mathbb{Z}[x]\). Verify if the following calculation is correct:
\[
r(4,x) = r(4) \cap r(x) = r(2^2) \cap r(x) = r(2) \cap r(x) = r(2,x) = (2,x).
\]

Answer: (2, x)


Question: Determine the domain of convergence for the series \( \sum_{n = 1}^{\infty} \frac{\cos(nz)}{n^2} \).

Answer: \mathbb{R}


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function, and define \( g(x) = f(x - x^2) \). Given that \( g(x) \) has a local maximum at \( x = \frac{1}{2} \) but the absolute maximum occurs elsewhere, determine the minimum number of solutions to the equation \( g'(x) = 0 \).

Answer: 3


Question: Determine the number of zeros of the function \( \cos(\pi z) - 100z^{100} \) within the unit disk \( D(0;1) \).

Answer: 100


Question:
Simplify the expression \( \frac{A}{B} \), where:
\[ A = \frac{1}{1 \cdot 2} + \frac{1}{3 \cdot 4} + \frac{1}{5 \cdot 6} + \ldots + \frac{1}{1997 \cdot 1998} \]
\[ B = \frac{1}{1000 \cdot 1998} + \frac{1}{1001 \cdot 1997} + \ldots + \frac{1}{1998 \cdot 1000} \]

Answer: 1499


Question: Find the volume of the solid $E$ bounded by the surfaces $z = 0$, $z = y$, and $y = 1 - x^2$. The region $E$ is defined as $$E = \{(x,y,z) \mid -1 \leq x \leq 1, \ 0 \leq y \leq 1-x^2, \ 0 \leq z \leq y \}.$$

Answer: \dfrac{8}{15}


Question: Let $f \in L^1(\mathbb{R})$ be a Lebesgue integrable function on $\mathbb{R}$ such that $\lim_{|x| \to +\infty} f(x) = 0$. Let $\mu$ be a non-singular Borel probability measure on $\mathbb{R}$. Determine whether the following statement is true: \[ \lim_{|x| \to +\infty} \int_{\mathbb{R}} f(x-y) \, d\mu(y) = 0. \]

Answer: True


Question: Let $k$ be a field and $L$ be the Lawvere theory of $k$-vector spaces, which is the category of finite-dimensional $k$-vector spaces. What is the category of models in finite-dimensional $k$-vector spaces? Is it $L$ itself?

Answer: L


Question: Find the limit \( \lim_{n\to\infty} \frac{x_1 y_n + x_2 y_{n-1} + \cdots + x_n y_1}{n} \) given that \( \lim_{n\to\infty} x_n = a \) and \( \lim_{n\to\infty} y_n = b \).

Answer: ab


Question: Find the area of the closed loop in the region $x \ge 0$ for the Lemniscate of Gerono, described by the equation $x^4 = x^2 - y^2$. Use the parametrization $x(t) = \sin t$, $y(t) = \sin t \cos t$, for $0 \le t \le 2\pi$, and apply Green's theorem with $P(x,y) = 0$ and $Q(x,y) = x$.

Answer: \dfrac{2}{3}


Question: Given the equations \(\frac{2x+3y}{a-2b} = \frac{4y+7z}{3b-c} = \frac{6z+5x}{2c-3a}\), find the value of \(11x+17y+20z\).

Answer: 0


Question: Let $D$ be an integral domain and $S$ be a multiplicatively closed subset of $D$ that does not contain zero. Consider $D$ as a subring of $S^{-1}D$, which is further a subring of the fraction field of $D$. If for every $q \in S^{-1}D$, there exists a monic polynomial $P_q(X) \in D[X]$ such that $P_q(q) = 0$, is it true that every element of $S$ is a unit in $D$?

Answer: Yes


Question: Calculate the dimension of the space of divisors on the Jacobian $J$ of a smooth compact complex curve $C_g$ (of genus $g$) that are linearly equivalent to the theta divisor $\Theta$. Specifically, determine $\dim( H^0(J,\mathcal{O}(\Theta)))$.

Answer: 1


Question: Evaluate the integral \( \int_{\gamma}\frac{1}{4z^2-1}dz \), where \( \gamma \) is the unit circle in the complex plane.

Answer: 0


Question: Let $C \subset \mathbb{CP}^2$ be a nonsingular curve of degree $d$, and let $p_1$, $p_2$, and $q$ be distinct points on $C$. For any complex numbers $a_1$ and $a_2$, determine whether there exists a rational function $f$ on $C$ such that $f(p_1) = a_1$, $f(p_2) = a_2$, and $f$ has a pole of order $2g + 2$ at $q$, where $g$ is the genus of $C$. Provide a justification for your answer.

Answer: Yes


Question:
Consider the sequence of continuous linear operators $T_n: l^2 \to l^2$ defined by \( T_n(x) = (0, 0, \ldots, 0, x_{n+1}, x_{n+2}, \ldots) \) for every \( x = (x_1, x_2, \ldots) \in l^2 \) and \( n \in \mathbb{N} \). For every \( x \neq 0 \) in \( l^2 \), determine which of the following is true:

(a) Both \( \left\lVert T_n \right\rVert \) and \( \left\lVert T_n(x) \right\rVert \) converge to zero.
(b) \( \left\lVert T_n \right\rVert \to 0 \) but not \( \left\lVert T_n(x) \right\rVert \).
(c) \( \left\lVert T_n \right\rVert \) does not converge to zero but \( \left\lVert T_n(x) \right\rVert \to 0 \).
(d) Neither of the two converge to zero.

Select the correct option.

Answer: C


Question: Is it true that for Lebesgue almost every $y$, the conditional law $\nu_y$ is the law of a Brownian bridge starting at $0$ and ending at $y$?

Answer: Yes


Question: Show that \((1+\sqrt{3}i)^9 + (1-\sqrt{3}i)^9 + 2^{10} = 0\). Use the hint: \((1+\sqrt{3}i)^9 = (1-\sqrt{3}i)^9 = (-2)^9\).

Answer: 0


Question: Determine whether the set \( \{ f \in C[a,b] \mid |f(x)| \leq 2+f(x)^2 \} \) is a complete metric space with the metric \( \rho_{\infty} \) (the sup norm metric).

Answer: Yes


Question:
Let \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) be a matrix with \( a, b, c, d \) as positive real numbers such that \( a+b < \frac{1}{2} \) and \( c+d < \frac{1}{2} \). Determine which of the following statements is FALSE:

A. \( I+A \) is invertible.

B. \( I-A \) is invertible.

C. \( I+A^2 \) is invertible.

D. There exists a natural number \( m \) such that \( I+A^m \) is not invertible.

Answer: D


Question: Evaluate the contour integral \( \oint_C \overline{z}^3 \, dz \) where \( C \) is the unit circle \( |z| = 1 \). Use parametrization to perform the integration.

Answer: 0


Question: Compute the total surface area of the remaining part of a solid sphere with radius 11 after a cylindrical hole with radius 8 is drilled through the center of the sphere.

Answer: 76\pi\sqrt{57}


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}\int_{n}^{n+7}\frac{\sin x}{x}\,\mathrm dx \]

Answer: 0


Question: Given positive irrational numbers \(a\) and \(b\) such that \(\lfloor a\lfloor bx\rfloor\rfloor = \lfloor b\lfloor ax\rfloor\rfloor\) for all \(x > 0\), must it be that \(a = b\)? Provide a justification for your answer.

Answer: Yes


Question: Let $G$ be a finite group acting freely on a locally compact Hausdorff space $X$. The canonical quotient map $\pi: X \to X/G$ admits local cross-sections. Is there a finite open cover of $X/G$ consisting of sets where a local cross-section can be defined?

Answer: No


Question: Let $K$ be a number field. Is there always a finite extension $L / K$ such that $L$ has class number 1? Consider the implications of the Golod-Shafarevich theorem and the class field tower problem, particularly for imaginary quadratic fields.

Answer: No


Question: Determine the condition on the constant $a$ such that the integral \( \int_{1}^{2} \frac{1}{x(\ln x)^{a}} \, dx \) is convergent.

Answer: a < 1


Question: Let \( g(a) = \int_{0}^{\pi/2} |\sin 2x - a\cos x| \, dx \) for \( a \in [0, 1] \). If \( L \) and \( M \) are the minimum and maximum values of \( g(a) \) for all \( a \in [0, 1] \), find the value of \( L + M \).

Answer: \dfrac{3}{2}


Question: Given that \( \int_0^1{f'(x)\,dx}=0 \) and \( \int_0^1{x f'(x)\,dx}=-1 \), find the maximum value of \( \int_0^1{|f'(x)|\,dx} \). Assume that \( f(0)=f(1)=0 \) and \( f \) is differentiable.

Answer: 4


Question: Find the sum of the coefficients of all even degree terms in the expansion of \((x + \sqrt{x^3 - 1})^6 + (x - \sqrt{x^3 - 1})^6\), where \(x > 1\).

Answer: 24


Question: Evaluate the expectation \( \mathbb{E}[\operatorname{sign}\langle v,z\rangle] \) where \( v \in \mathbb{R}^n \) is fixed and each \( z_i \sim N(0,1) \) independently for \( i \in [n] \).

Answer: 0


Question:
Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function such that \( f(i) = 0 \) for all \( i \in \mathbb{Z} \). Determine which of the following statements is true:

(A) \( \operatorname{Image}(f) \) is closed in \( \mathbb{R} \).

(B) \( \operatorname{Image}(f) \) is open in \( \mathbb{R} \).

(C) \( f \) is uniformly continuous.

(D) None of the above.

Choose the correct option.

Answer: D


Question: Determine whether the integral of the function \( f(t) = \begin{cases} 1, & \text{if } t \text{ is rational} \\ 0, & \text{otherwise} \end{cases} \) over the interval \((0, 1)\) exists.

Answer: 0


Question: A stick of length 1 is randomly cut into 3 pieces. What is the expected length of the shortest piece?

Answer: \dfrac{1}{9}


Question: Find the residue of the function \( f(z) = \frac{z^2 e^{iz}}{(z^2+1)^2} \) at the double pole \( z = i \).

Answer: 0


Question: Let $X$, $Y$, and $Z$ be the sets of positive divisors of $10^{60}$, $20^{50}$, and $30^{40}$, respectively. Find the number of elements in the union of these sets, $n(X \cup Y \cup Z)$. \( \text{ } \)

Answer: 73001


Question: Let $G$ be a non-abelian group of order $21$. Determine the number of automorphisms of $G$. Provide your answer as a single integer.

Answer: 42


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

Answer: 1


Question: Consider the fiber product \(X \times_Z X\) with two projection arrows \(X \times_Z X \rightrightarrows X\). If we extend this to the fiber product \(X \times_Z X \times_Z X\), how many arrows are associated with this fiber product? Provide the total count of arrows.

Answer: 3


Question: Evaluate the integral \( \int_{-\infty}^\infty \exp(-x^2+2x)\,\mathrm{d}x \) without using a computer package.

Answer: e \sqrt{\pi}


Question:
Find \( \lim_{n \to \infty} x_n \), where:
\[ x_n = \sqrt{n^2+1} + \sqrt{4n^2+2n} - \sqrt[3]{27n^3+n^2}. \]

Answer: \dfrac{25}{54}


Question: Compute the integral \( \int_{|z|=2 } \frac{z^3}{(z+1)^3} dz \) using the Cauchy integral formula.

Answer: -6\pi i


Question: Determine the domain of the real number \(a\) such that the function \(f(x) = |x^2 - 5x| - a(x+4)\) has exactly 4 roots.

Answer: (0, 1)


Question: Verify if the solution $x = 2 \cdot u$ satisfies the equation $\log(x) + \frac{2 \cdot u}{x} = 0$. If it does, provide the solution.

Answer: \dfrac{1}{e}


Question:
Given stochastic processes $X_t$, $\sigma_t$, and a Wiener process $W_t$, where $X_t$ is independent of $\sigma_s$ for all $t, s$, and $W_t$ is independent of $\sigma_s$ for all $t, s$, but $X_t$ and $W_s$ are independent for all $t \neq s$. Also, $\mathbb{E} X_t = 0$. Determine the value of the expression:
\[ \mathbb{E} X_T \int_0^T \sigma_t \, dW_t \]

Answer: 0


Question: Determine the proof-theoretic ordinal of the system $PA+TI(\epsilon_0)$, where $PA+TI(\epsilon_0)$ is Peano arithmetic with transfinite induction up to $\epsilon_0$ added.

Answer: \epsilon_0


Question:
Let $a$ and $b$ be linearly independent vectors in $\mathbb{R}^3$. Consider the linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ defined by:
\[ T(x) = x - (b \cdot x)a, \]
where $b \cdot x$ denotes the dot product. Determine the vectors $a$ and $b$ for which the transformation $T$ is not an isomorphism.

Answer: a \cdot b = 1


Question: Let $G$ be a group acting transitively on the space $X$. Let $x \in X$ and write $X = G \cdot x = G/H$. Let $A$ be a subgroup of $G$ and consider the orbit $Y := A \cdot x$. Suppose that $g \cdot Y \cap Y \neq \varnothing$ for some $g \in G$. Is it necessary that $g \cdot Y = Y$? Provide a justification for your answer.

Answer: No


Question: Let $n = p + q$ and suppose that $f: S^n \to S^p \times S^q$ is a continuous map. Compute the induced map $$f^* : H^n(S^p \times S^q) \to H^n(S^n).$$

Answer: 0


Question: Determine the maximum length of a sequence of consecutive integers such that each integer in the sequence is not relatively prime to its digital sum.

Answer: 8


Question:
Determine the location of all the roots of the equation \(11z^{10}+10iz^9+10iz-11=0\) in relation to the unit circle in the complex plane. Choose the correct option:

(a) inside \(|z|=1\)
(b) on \(|z|=1\)
(c) outside \(|z|=1\)
(d) can't say

Answer: B


Question: Given the probability density function \( f(x) = 3x^2 \) for \( x \in (0,1) \), find the covariance \( \text{Cov}(2X+7, X^2 + 3X - 12) \).

Answer: \dfrac{13}{40}


Question: Find the largest positive integer $k$ such that $S_5 \times S_5$ has an element of order $k$. \(S_5\) is the symmetric group on 5 elements.

Answer: 30


Question: Calculate the determinant of the symmetric matrix \( A \in \mathbb{R}^{82 \times 82} \) where the diagonal entries are \(-6\) and all off-diagonal entries are \(-5\).

Answer: 411


Question: Calculate the residue of the function \( f(z) = \frac{1}{\sin(z)} \) at the points \( z = k\pi \), where \( k = 0, \pm 1, \pm 2, \dots \).

Answer: (-1)^k


Question: Determine whether there exists an embedding \( j: \mathbb{Q}^c \hookrightarrow \mathbb{Q}_p^c \), where \( \mathbb{Q}^c \) is the algebraic closure of \( \mathbb{Q} \) in \( \mathbb{C} \) and \( \mathbb{Q}_p^c \) is the algebraic closure of \( \mathbb{Q}_p \), the \( p \)-adic completion of \( \mathbb{Q} \).

Answer: Yes


Question: A motorist just had an accident. The accident is minor with probability 0.75 and major otherwise. If the accident is minor, the loss amount follows a uniform distribution on the interval \([0, b]\). If the accident is major, the loss amount follows a uniform distribution on the interval \([b, 3b]\). Given that the median loss amount due to this accident is 672, calculate the mean loss amount.

Answer: 882


Question:
Consider all natural numbers whose decimal expansion consists only of the even digits 0, 2, 4, 6, and 8. These numbers are arranged in increasing order. Let \( a_n \) denote the \( n \)-th number in this sequence. Determine the value of the limit \( \lim_{n\to \infty}\frac{\log a_n}{\log n} \). Choose the correct answer from the following options:

(a) 0
(b) \( \log_5 10 \)
(c) \( \log_2 10 \)
(d) 2

Answer: B


Question: If $K$ is a finite field and $F_1$ and $F_2$ are subfields of $K$ such that $|F_1| = |F_2|$, does it follow that $F_1 = F_2$?

Answer: Yes


Question: Suppose $f$ is a $\mathcal{C}^2$ function and $x^*$ is a point in its domain where $\nabla f(x^*)d \geq 0$ and $d^T \nabla^2 f(x^*) d > 0$ for every non-zero feasible direction $d$. Are these conditions sufficient to conclude that $x^*$ is a local minimum of $f$? Provide a justification for your answer.

Answer: Yes


Question: Expand \((2i-2)^{38}\) using de Moivre's formula.

Answer: 2^{57}i


Question: Determine the values of \( x \in \mathbb{R} \) for which the series \( \sum_{k=0}^{\infty} k^3 \cdot 3^{-k} \cdot (x+3)^k \) converges. Use the power series and radius of convergence approach to find the solution.

Answer: (-6, 0)


Question: Calculate the number of 'heavy numbers' between 1002 and 12089. A number is considered 'heavy' if the average of its digits is greater than 7.

Answer: 330


Question: Evaluate the limit: \[ \lim_{n\to \infty} \frac{3}{n}\left(1+\sqrt{\frac{n}{n+3}}+\sqrt{\frac{n}{n+6}}+\cdots+\sqrt{\frac{n}{4n-3}}\right) \]

Answer: 2


Question: Determine the number of subgroups of index \( p \), where \( p \) is a prime, in the additive group \( \mathbb{Z}^2 \).

Answer: p + 1


Question: Find the point \((h, k)\) where the curve \(x + y = e^{xy}\) has a vertical tangent. This occurs when \(\frac{dx}{dy} = 0\).

Answer: (1, 0)


Question: Evaluate the limit: \[ L = \lim_{x \to \infty} x^2 \times \log \left(x \cot^{-1}x\right) \]

Answer: -\dfrac{1}{3}


Question: Determine whether the closure of the set \( \left\{\frac{\tan(n)}{n} \mid n \in \mathbb{N} \right\} \) is equal to \( \mathbb{R} \).

Answer: \mathbb{R}


Question: In ZFC, does there exist a perfect field of characteristic $p$ for any given infinite cardinality, where $p$ is a fixed prime number?

Answer: Yes


Question: Find the exact value of the series \( \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^2} \).

Answer: G


Question: Find all continuous functions \( f(x) \) for \( x > 0 \) that satisfy the functional equation: \[ xf(x^2) = f(x) \]

Answer: \dfrac{C}{x}


Question: Does there exist a closed set $E$ that is a subset of $[a,b]$, with $E \neq [a,b]$, such that the measure of $E$ is $b-a$?

Answer: No


Question: Find the solution set $S_{100}$ for the equation $\sin x \cdot \sin 2x \cdot \ldots \cdot \sin 100x = 1$. What is $S_{100}$?

Answer: \emptyset


Question: Find the residue of \( \frac{1}{\cos z} \) at \( z=\pi/2 \).

Answer: -1


Question: Calculate the limit \( \lim \limits_{x \to 0} x^a \int_x^1 \frac{f(t)}{t^{a+1}} \, dt \), where \( f \) is a continuous function and \( a > 0 \).

Answer: \dfrac{f(0)}{a}


Question: Find the smallest positive value of \( \theta \) such that the polar curves \( R_1 = \cos(2\theta) \) and \( R_2 = \sin(3\theta) \) intersect.

Answer: \dfrac{\pi}{10}


Question: Calculate the variance of the product $Var(AX)$, where $A$ is a unit vector with a uniform random phase between $[0,2\pi]$, and $X$ is a scalar random variable with a uniform distribution on $[-1,1]$. Assume $A$ and $X$ are independent.

Answer: \dfrac{1}{3}


Question: Evaluate the limit: \[ \lim_{n\to\infty} \left( \frac{1^p + 2^p + \cdots + n^p}{n^p} - \frac{n}{p+1} \right) \] where \( p \in \mathbb{N} \).

Answer: \dfrac{1}{2}


Question: Let $p$ be a real number in $[0,1]$. Does there exist a sequence $(x_1, x_2, x_3, \ldots)$ with each $x_i$ being either $0$ or $1$, such that \( \lim_{n\to \infty} \frac{f(n)}{n} = p \), where $f(n)= x_1+x_2+\cdots+x_n$ is the number of times $1$ has appeared in the first $n$ slots?

Answer: Yes


Question: Determine the number of zeros of the function \( f(z) = 10 \sin(z^{2019}) - z \) within the unit disk \( D = \{ z \in \mathbb{C} : |z| < 1 \} \).

Answer: 2019


Question: Find the maximum value of the function \( F(a,b,c,d) = \frac{5}{4}(c-a)^{2} + \frac{3}{4}(d-b)^{2} + 2\sqrt{3}(c-a)(d-b) \) subject to the constraints \( a^{2} + b^{2} \leq 4 \) and \( c^{2} + d^{2} \leq 4 \).

Answer: 44


Question: Is the term \(\frac{\partial^2f}{\partial x^2}d[X]_t\) in Itô's formula a finite variation process?

Answer: Yes


Question:
Determine the point of intersection of the tangents drawn to the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) at the points where it is intersected by the line \(lx+my+n=0\). Choose the correct option:

\((A)\) \(\left(-\frac{a^2l}{n},\frac{b^2m}{n} \right)\)
\((B)\) \(\left(-\frac{a^2l}{m},\frac{b^2n}{m} \right)\)
\((C)\) \(\left(\frac{a^2l}{m},-\frac{b^2n}{m} \right)\)
\((D)\) \(\left(\frac{a^2l}{m},\frac{b^2n}{m} \right)\)

Answer: A


Question: Find the number of distinct Hamiltonian cycles in the complete graph $K_9$ such that no two cycles share an edge.

Answer: 4


Question: Determine the order of the subgroup generated by the matrices \(\begin{pmatrix}2 & 0\\0 & 1\end{pmatrix}\) and \(\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}\) in \(GL_2(\mathbb{F}_5)\).

Answer: 20


Question: Find the value of $x$ such that $x \equiv 2^{-18} \pmod{143}$. This requires determining the modular inverse of $2^{18}$ modulo 143.

Answer: 103


Question: Determine the value of the parameter \( \alpha \) such that the sequence \( a_{n} = \frac{3^n+(-2)^n}{3^{n+1}+(-2)^{n+1}}+\left(\frac{n+1}{n-1}\right)^n+(1+\alpha)\sin\frac{n\pi}{2} \) is convergent.

Answer: -1


Question: Given two dependent random variables $X$ and $Y$, with $E(Y|X=x)=5x$ and $\text{Var}(Y|X=x)=2x+x^2$, and $E(X)=12$ and $\text{Var}(X)=4$, compute the variance of $Y$. Assume the expectation of $Y$ has been computed as $E(Y)=5E(X)$. What is $\text{Var}(Y)$?

Answer: 272


Question: Given two sets of points \( \{A_1, A_2, A_3, A_4\} \) and \( \{B_1, B_2, B_3, B_4\} \) in \( \mathbb{R}^2 \), let \( d_{i,j} \) be the Euclidean distance between \( A_i \) and \( A_j \), and \( d'_{i,j} \) be the Euclidean distance between \( B_i \) and \( B_j \) for \( 1 \le i < j \le 4 \). Assume that \( d_{i,j} - d_{k,l} = d'_{i,j} - d'_{k,l} \) for all \( 1 \le i < j \le 4 \) and \( 1 \le k < l \le 4 \). Does this imply that \( d_{i,j} = d'_{i,j} \) for all \( 1 \le i < j \le 4 \)?

Answer: Yes


Question: Let $X$ and $Y$ be two independent exponentially distributed random variables with parameter $\lambda$. Define $U = \frac{X}{X + Y}$. Find the density function $f_U(u)$ of the random variable $U$. Express your answer in terms of $\lambda$ and $u$.

Answer: 1


Question: Evaluate the complex contour integral \( \int_C \frac{\cosh(z)}{z^4} \, dz \), where \( C \) is the square centered at the origin with side length 4.

Answer: 0


Question:
Determine whether the following equality holds for any function \( f \in C_0^{\infty}(\omega) \), where \( \omega \subset \mathbb{T} \) and \( x_0 \in \mathbb{T} \) on the one-dimensional torus \( \mathbb{T} \):
\[ \int_{\mathbb{T}} f(x) \, dx = \int_{\mathbb{T}} f(x-x_0) \, dx. \]

Answer: Yes


Question: Determine the asymptotic growth rate of the function defined by the recurrence relation \( T(n) = 4T(n/2) + n + \log(n) \) for \( n \ge 2 \). Assume \( n = 2^k \).

Answer: \Theta(n^2)


Question: In a metric space $(X, d)$, let $(a_n)$ be a sequence in $X$ such that $(a_n)$ has no Cauchy subsequence. Define the set $A = \{a_n : n \in \mathbb{N}\}$. Is it true that every Cauchy sequence $(b_n)$ in $A$ has a constant tail, meaning there exists an $N \in \mathbb{N}$ such that $b_n = b_N$ for every $n \geq N$?

Answer: Yes


Question: Solve for \( x \) in the equation: \[ \left(\frac{3}{5}\right)^x + \left(\frac{4}{5}\right)^x = 1 \]

Answer: 2


Question:
Let \( A \) be an \( n \times n \) non-singular matrix with real entries, and let \( B = A^T \) denote the transpose of \( A \). Determine which of the following matrices are positive definite:

1. \( A + B \)
2. \( A^{-1} + B^{-1} \)
3. \( AB \)
4. \( ABA \)

Answer: 3


Question: Evaluate the limit \( \lim_{x \to \infty} 3x^2(e^{-2/x^2} - 1) \) using series expansion.

Answer: -6


Question: Given the quartic polynomial equation \(x^4 - x^3 - 1 = 0\) with roots \(\alpha, \beta, \gamma, \delta\), find the exact value of \(\alpha^6 + \beta^6 + \gamma^6 + \delta^6\).

Answer: 7


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:
Calculate \(\int_{S}\omega\), where \(\omega\) is the 2-form on \(\mathbb{R}^3-\{(1,0,0),(-1,0,0)\}\) given by:
\[
\omega=((x-1)^2+y^2+z^2)^{-3/2}((x-1)dy\wedge dz+ydz\wedge dx+zdx \wedge dy)+
  ((x+1)^2+y^2+z^2)^{-3/2}((x+1)dy\wedge dz+ydz\wedge dx+zdx \wedge dy)
\]
and \(S=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2=5 \}\). The orientation of \(S\) is the natural orientation induced by \(D=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2 \leq 5 \}\).

Answer: 8\pi


Question: How many even 11-digit numbers, with no leading zeros, contain at least three occurrences of the digit 3?

Answer: 3244680630


Question: Determine a bound for the number of generators of prime ideals in the polynomial ring $K[x_1,x_2,...,x_n]$. For instance, consider the case of $K[x,y]$. It is known that maximal ideals of $K[x_1,x_2,...,x_n]$ have $n$ generators.

Answer: n


Question: Calculate the length of the astroid defined by the equation \(x^{2/3} + y^{2/3} = 1\).

Answer: 6


Question: Determine the number of real solutions to the equation: \[ \sqrt{x} + \sqrt{1-x^2} = \sqrt{2-3x-4x^2}. \]

Answer: 1


Question: Determine the order of the subgroup $H$ of the group $G$ of $2 \times 2$ matrices with integer entries modulo a prime $p$, where each matrix in $H$ has a determinant equal to 1. The group operation is matrix multiplication.

Answer: p(p^2 - 1)


Question: Evaluate the limit: \[ L = \lim_{x \to 0}\left(\frac{\arcsin x}{x}\right)^{\frac{1}{x^2}} \]

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


Question:
Find the coefficient of \(x^6\) in the generating function:
\[ f(x) = \frac{1}{x(2x-1)^2} \]

Answer: 1024


Question: Find the limit of \( S_n = \sum_{i=1}^n \left( \cosh\left(\frac{1}{\sqrt{n+i}}\right) - n \right) \) as \( n \to \infty \).

Answer: -\infty


Question: Determine if the expression \((1-5kx^3) \pmod{25}\) takes on the values \(1, 6, 11, 16, 21\) the same number of times for \(0 \leq k \leq 4\) and \(1 \leq x \leq 4\).

Answer: Yes


Question: Consider a nontrivial ring homomorphism from \( \mathbb{C} \) to \( \mathbb{C} \) that is injective and maps 1 to 1 and \( i \) to either \( +i \) or \( -i \). Is such a homomorphism onto?

Answer: Yes


Question: Let $W_t$ be a standard Brownian motion and define the process $$X_t = X_0 e^{t(1-\sigma^2/2) + \sigma W_t}$$ where $\sigma$ has an exponential distribution given by $$ P(\sigma \leq x) = 1-e^{-x}$$ for $x \geq 0$. Determine the probability $$P\left(\lim_{t \to \infty}X_0 e^{t(1-\sigma^2/2) + \sigma W_t}=0\right).$$

Answer: e^{-\sqrt{2}}


Question: Calculate the sum of the series: \[ \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \]

Answer: \dfrac{\pi}{4}


Question: Let $A$ be an integral domain and $K$ its field of fractions. Suppose $p$ is a prime ideal of $A$. If $A$ is a valuation ring of $K$, is it true that any unit of the ring $A/p$ is of the form $x+p$, where $x$ is a unit of $A$?

Answer: Yes


Question: Find the limit of the sequence \((a_n)\), where \(a_n\) is the solution to the equation \(x^n + x - 1 = 0\).

Answer: 1


Question: Find the angle between the tangents to the curve \(3x^2+4xy+5y^2-4=0\) at the points where the lines \(y=(-3/2)x\) and \(y=(-2/5)x\) intersect the curve.

Answer: \dfrac{\pi}{2}


Question: Given the equations \( xw^3 + yz^2 + z^3 = -1 \) and \( zw^3 - xz^3 + y^2w = 1 \), where \( z \) and \( w \) are differentiable functions of \( x \) and \( y \), find \( \frac{\partial z}{\partial x} \) at the point \((x, y, z, w) = (1, -1, -1, 1)\).

Answer: -\dfrac{5}{4}


Question: Given the joint probability density function \( f_{X,Y}(x,y) = 4xy \) for \( (x,y) \in [0,1]^2 \), where \( X \) and \( Y \) are independent random variables, find the probability \( \mathsf{P}(2X > Y) \).

Answer: \dfrac{7}{8}


Question:
Let \( f(x) \) be a polynomial of degree 11 such that \( f(x) = \frac{1}{x+1} \) for \( x = 0, 1, 2, \ldots, 11 \). What is the value of \( f(12) \)?

(A) \( \frac{1}{13} \)
(B) \( \frac{1}{12} \)
(C) 0
(D) \( \frac{1}{7} \)

Answer: C


Question: In triangle $ABC$, an interior point $P$ is defined as "good" if there are exactly 27 rays emanating from $P$ that intersect the sides of the triangle, dividing it into 27 smaller triangles of equal area. Determine the number of good points for a given triangle $ABC$. Provide your answer as a single integer.

Answer: 325


Question: If $X, Y, Z$ are independent standard normal random variables, find the distribution of $X^2 + Y^2 + Z^2$. Use spherical coordinates if necessary.

Answer: \chi^2(3)


Question: Find the smallest real number $a$ such that $e^x > x^5$ for all $x > a$. Provide your answer as a single value of $a$.

Answer: 12.713


Question: Find the exact value of the root of the equation \(x^3 + 5x - 18 = 0\) using the method of expressing \(x\) as \(a + b\) where \(3ab + 5 = 0\). Show that the root is exactly 2.

Answer: 2


Question: Let $f$ be a polynomial of degree $n$ over a finite field $F = \mathbb{F}_q$, and let $E$ be the splitting field of $f$. If $f$ has $n$ simple roots over $E$, determine whether $[E:F]$ divides $n$. Justify your answer.

Answer: No


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{1}{x^4}\left(\frac{6}{6+x^2}-\frac{x}{\sinh(x)}\right). \]

Answer: \dfrac{1}{120}


Question: Let $f(x) = x^3 - 5$. Let $L/\mathbb{Q}$ be the splitting field of $f$ over $\mathbb{Q}$. Compute the degree $[L:\mathbb{Q}]$ and determine the Galois group $G = \text{Aut}_{\mathbb{Q}}(L)$. Provide your answer for the degree as a number and describe the structure of the Galois group.

Answer: 6


Question: Find all functions \( f: \mathbb{C} \to \mathbb{C} \) such that \( f \) is analytic on the open unit disk \( B(0,1) \), \( f(0) = 1 \), and \( |f(z)| \geq 1 \) for all \( |z| < 1 \).

Answer: 1


Question:
Given the random variables \(Y_1 = X_1 + X_2\) and \(Y_2 = X_1 - X_2\), and the joint distribution function \(F_X(\vec x) = (x_1+x_2)/9\) for \(x_1 \in \{-1,0,1\}\) and \(x_2 \in \{1,2\}\), calculate the covariance between \(Y_1\) and \(Y_2\).

Additional information:
- Mean of \(X_1 = 4/9\)
- Variance of \(X_1 = 38/81\)
- Mean of \(X_2 = 5/3\)
- Variance of \(X_2 = 2/9\)
- \(E[Y_1] = 19/9\)
- \(E[Y_2] = -11/9\)

Find the covariance \(\text{Cov}(Y_1, Y_2)\).

Answer: \dfrac{20}{81}


Question: Let $X$ be a locally compact Hausdorff space and $C_{0}(X)$ be the set of all continuous functions vanishing at infinity. If $P \in M_{n}(C_{0}(X))$ is a projection, is $PM_{n}(C_{0}(X))P$ a hereditary subalgebra of $M_{n}(C_{0}(X))$? Recall that a C*-subalgebra $B$ of $A$ is said to be hereditary if for any $a \in A_{+}$ and $b \in B_{+}$, the inequality $a \leq b$ implies that $a \in B$. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Given an infinite geometric sequence with first term \(a_1\) and common ratio \(-1 < r < 1\), the sum of the sequence is given by \(S = \frac{a_1}{1 - r}\). A new sequence is formed by grouping terms as follows: \(a_1 + a_2 + a_3, a_3 + a_4 + a_5, a_5 + a_6 + a_7, \ldots\). The sum of this new sequence is 1.9 times the sum of the original sequence. Determine the common ratio \(r\) of the original sequence.

Answer: -\dfrac{3}{5}


Question: Let \( \alpha \) be a path in the Euclidean plane \( \mathbb{R}^2 \). Let \( A, B \) be distinct connected components of \( \mathbb{R}^2 \setminus \operatorname{Im}\alpha \). Let \( \beta \) be a path in \( B \). Let \( C \) be a connected component of \( \mathbb{R}^2 \setminus \operatorname{Im}\beta \). Is it necessarily the case that either \( A \cap C = \emptyset \) or \( A \subseteq C \)? Justify your answer.

Answer: Yes


Question:
Find the measure of the Cantor set, denoted as \( C \), using the measure function \( l^* : 2^{\mathbb{R}} \to [0,+\infty] \) which satisfies the following properties:

- \( l^*(\emptyset) = 0 \)
- \( l^*(S) \leq l^*(T) \) if \( S \subseteq T \)
- \( l^*(\bigcup_{i=1}^\infty S_i) \leq \sum_{i=1}^\infty l^*(S_i) \) for any \( S_i \subseteq \mathbb{R} \)
- \( l^*([a,b]) = b - a \) for any \( a \leq b \)

The Cantor set is defined as the intersection of the sets \( A_n \), where each \( A_n \) is formed by removing the middle third of each interval from the previous step, starting with \( A_1 = [0, 1/3] \cup [2/3, 1] \). Determine \( l^*(C) \).

Answer: 0


Question: Determine whether the norm of a linear operator is equal to the absolute value of the determinant of its matrix representation. Let $f: \Omega \rightarrow \mathbb{R}^n$ be a linear operator, where $\Omega$ is a bounded open subset of $\mathbb{R}^n$, and $A = \{a_{ij}\}$ is the matrix of the linear operator. Is $||f|| = |\det(A)|$?

Answer: No


Question: Let $X$ be a projective variety over $\mathbf{C}$ of dimension $n$. Consider the blow-up $\pi: Y \to X$ at a smooth point $x \in X$. Find a formula for the intersection number $E^n$, where $E$ is the exceptional divisor.

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


Question: Express the recurrence relation \( t(2^{2^{k}}) = 1 + t(\sqrt{2^{2^k}}) \) in terms of \( k \), i.e., find \( t(k) \).

Answer: k


Question:
A continuous random variable $X$ has the density function given by:
\[ f(x) = 2\phi(x)\Phi(x), \quad x \in \mathbb{R} \]
where \( \Phi(x) \) is the cumulative distribution function and \( \phi(x) \) is the probability density function of the standard normal distribution \( N(0,1) \).

Determine which of the following statements is true:

(A) $E(X) > 0$

(B) $E(X) < 0$

(C) $P(X \leq 0) > 0.5$

(D) $P(X \ge 0) < 0.25$

Answer: A


Question:
Consider a Riemannian manifold $(M,g)$. The Sasaki metric $\\hat{g}$ on the tangent bundle $TM$ and the Sasaki metric $\\bar{g}$ on the cotangent bundle $T^*M$ are defined as follows: for vector fields $X, Y$ on $M$ and one-forms $\\alpha, \\beta$ on $M$, with $\\pi$ as the projection from both bundles onto $M$, and superscripts $h$ and $v$ denoting horizontal and vertical lifts, respectively:

$$
\\hat{g}(X^h,Y^h) = \\hat{g}(X^v,Y^v) = g(X,Y) \\circ \\pi, \\quad \\hat{g}(X^h,Y^v) = 0
$$

and

$$
\\bar{g}(X^h,Y^h) = g(X,Y) \\circ \\pi, \\quad
\\bar{g}(\\alpha^v,\\beta^v) = g^{-1}(\\alpha,\\beta) \\circ \\pi, \\quad
\\bar{g}(X^h,\\alpha^v) = 0.
$$

Is the map $g: (TM, \\hat{g}) \\to (T^*M, \\bar{g})$ an isometry?

Answer: Yes


Question: Given four $m \times n$ matrices $A$, $B$, $A'$, and $B'$, if $AA^\dagger = A'A'^\dagger$, $BB^\dagger = B'B'^\dagger$, and $AB^\dagger = A'B'^\dagger$, does there always exist a unitary matrix $V$ in $U(n)$ such that $A = A'V$ and $B = B'V$?

Answer: Yes


Question: Find the singular integral of the differential equation \((xy^{\prime}-y)^{2}=x^{2}(x^{2}-y^{2})\).

Answer: y^2 = x^2


Question: Does a left-invariant vector field on a complex Lie group preserve holomorphic functions? Let $G$ be a finite-dimensional complex Lie group, and suppose $f : G \to \mathbb{C}$ is holomorphic. Let $X$ be a left-invariant vector field on $G$. Must $Xf$ be holomorphic?

Answer: Yes


Question: Consider a real matrix of size \( P \times Q \) where \( P \geq Q \). This matrix can be viewed as an element of \( \mathbb{R}^{PQ} \). Determine whether the subspace of matrices with repeated singular values has Lebesgue measure zero in this space.

Answer: Yes


Question: Determine the winding number of the function \( f: S^1 \to \mathbb{C} \setminus \{ 0 \} \) given by \( z \mapsto 8z^4+4z^3+2z^2+z^{-1} \) about the origin.

Answer: 4


Question: Determine the number of ways to distribute the elements of the set \([n]\) into four subsets \(A_1, A_2, A_3, A_4\) such that \(A_1 \cap A_2 \cap A_3 \cap A_4 = \emptyset\) and \(A_1 \cup A_2 \cup A_3 \cup A_4 = [n]\).

Answer: 14^n


Question: Let $K$ be a real-closed field. Determine whether $K(i)$, where $i$ is a root of $x^2 + 1$ in an algebraic closure of $K$, is an algebraic closure of $K$.

Answer: Yes


Question:
Consider the unit simplex defined for some $n \in \mathbb{N}$ as \( S \equiv \left\{ \mathbf{x} \in \mathbb{R}_+^n \mid \sum_{i=1}^n x_i = 1 \right\} \). For each \( i \in \{1, \ldots, n\} \), let \( f^i: S \to \mathbb{R}_+ \) be a non-negative continuous function, and \( \alpha^i \geq 0 \) be a constant. Define \( \mathbf{e}^i \in S \) as the vector with the $i$th coordinate equal to 1 and all other coordinates equal to 0. Assume the following conditions hold:

1. \( f^i(\mathbf{e}^i) \geq \alpha^i \) for every \( i \in \{1, \ldots, n\} \).
2. \( \sum_{i=1}^n f^i(\mathbf{x}) \geq \sum_{i=1}^n \alpha^i \) for every \( \mathbf{x} \in S \).

Does there necessarily exist some \( \mathbf{x}^* \in S \) such that \( f^i(\mathbf{x}^*) \geq \alpha^i \) for every \( i \in \{1, \ldots, n\} \)?

Answer: Yes


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

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


Question: Determine the values of \( p \) for which the series \( \sum \frac{1}{n(\log(n))^p} \) converges.

Answer: p > 1


Question: Find the inverse of the element \((x+2) + I\) in the quotient ring \(\mathbb{Z}_5[x]/I\), where \(I\) is the ideal generated by \(b = x^3 + 3x + 2\).

Answer: 3x^2 + 4x + 1


Question: Find the limit using L'Hopital's rule: $$ \lim_{x \to \infty} \frac{1}{x^2} \int_1^x \frac{t^2}{t + |\sin t|} \, dt $$

Answer: \dfrac{1}{2}


Question: A die is thrown repeatedly until the sequence "5 followed by 3" appears. What is the expected number of throws required for this sequence to occur?

Answer: 36


Question: Determine the maximum number of vectors in \(\mathbb{C}^d\) that can have the uniform overlap property, where each vector has unit norm and is defined up to a phase. The uniform overlap property requires that for all distinct vectors \(v_i\) and \(v_j\), the squared magnitude of their inner product is a constant \(C \neq 0\). What is this maximum number of vectors?

Answer: d^2


Question: Determine the cardinality of the set \( A = \left\{ f \in C^1[0,1] : f(0) = 0, \ f(1) = 1, \ \left|f'(t)\right| \le 1 \ \forall \ t \in [0,1] \right\} \).

Answer: 1


Question: Determine all analytic functions \( f(z) \) in the domain \( \{z:|z|<1\} \) such that \( f\left(\frac{1}{n}\right)=\frac{1}{n^3} \) for all positive integers \( n \).

Answer: z^3


Question: Let $T$ be a self-adjoint operator with a bounded spectrum $\sigma(T)$. Does this imply that $T$ is a bounded operator? Justify your answer.

Answer: Yes


Question: Evaluate the integral analytically: $$\int^{\pi/2}_{0}\frac{\sin(2x)^x(\ln(\sin(x))+\ln(\cos(x))-x(\tan(x)-\cot(x)))}{2^x}\,dx$$ and verify that it equals -1.

Answer: -1


Question: Calculate the limit: \[ \lim_{x\rightarrow0}\frac{1-\sqrt{1+x^2}\cos{x}}{x^4} \]

Answer: \dfrac{1}{3}


Question: Evaluate the integral: \[ \int_0^{\pi} \arcsin(\cos^3 x) \, dx \]

Answer: 0


Question:
Consider a random sample of size \(n\) from a distribution with the probability density function given by:
\[
f(x;\theta)=\begin{cases}
 \sqrt{\frac{2}{\pi}}e^{-\frac{1}{2}(x-\theta)^2} & \text{if } x \geq \theta \\
 0 & \text{elsewhere}
\end{cases}
\]
Find the maximum likelihood estimator of \(\theta\).

Answer: X_{(1)}


Question: Let $G$ be a finite group acting regularly and transitively on a set $X$, such that $|G| = |X|$. Can the action of $G$ also be primitive? Provide a justification for your answer.

Answer: Yes


Question: Let \( \mathbb{F} \) be a field with \( 16807 = 7^5 \) elements. Determine the cardinality of the set \( \{a^7 - b^7 : a, b \in \mathbb{F}\} \).

Answer: 16807


Question: Evaluate the triple integral \( \iiint x\, \text{d}x \) over the region bounded by the surfaces \( z = x^2 + y^2 \) and \( z = -4x - 2 \).

Answer: -4\pi


Question: Calculate the integral \( \oint_\gamma f(z)\,dz \) where \( f(z) = \frac{1}{(z-1)(z^2+2)} \) and \( \gamma \) is a circle with center \(-1\), radius \(1\), and positive orientation.

Answer: 0


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

Answer: 1


Question:
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x \) and \( y \), the following equation holds:
\[ f(f(x+y)) = f(x+y) + f(x)f(y) - xy. \]

Answer: f(x) = x


Question: Determine the order of the pole at \( z = 0 \) for the function \( f(z) = \frac{1}{\cos(z^4) - 1} \).

Answer: 8


Question: Determine the fundamental group of the space \( \mathbb{C}^2 \setminus \Delta \), where \( \Delta = \{(z,z) : z \in \mathbb{C}\} \).

Answer: \mathbb{Z}


Question: Let \((a_n)_{n \ge 1}\) be a sequence of positive real numbers such that the sequence \((a_{n+1}-a_n)_{n \ge 1}\) converges to a non-zero real number. Evaluate the limit \[ \lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right)^n. \]

Answer: e


Question: Evaluate the definite integral \( \int_{0}^{2\pi} \frac{dx}{5-3\cos x} \).

Answer: \dfrac{\pi}{2}


Question: Determine whether the unit ball in \(L_2(0,1)\) is closed in \(L_1(0,1)\). Specifically, let \(B\) be the subset of \(L_1(0,1)\) such that a function \(f\) belongs to \(B\) if and only if \(\int_0^1 |f(x)|^2 \, dx \leq 1\). Is \(B\) a closed set in \(L_1(0,1)\)? In other words, if a sequence \(f_n\) converges to \(f\) in \(L_1(0,1)\) and \(\int_0^1 |f_n(x)|^2 \, dx \leq 1\) for all \(n\), does it follow that \(\int_0^1 |f(x)|^2 \, dx \leq 1\)?

Answer: Yes


Question:
Does there exist a commutative ring with unity $R$, a positive integer $n$, and a function $f$ from the set of $n \times n$ matrices over $R$ to $R$ such that:

1. $f$ is linear in each row and each column separately,
2. $f$ of the $n \times n$ identity matrix is $1_R$,
3. for all $n \times n$ matrices $M$, if $M$ is invertible then $f(M)$ is a unit,
4. $f$ is not the restriction of the determinant to $f$'s domain?

Answer: Yes


Question: Prove that \( \frac{\sin 20^\circ \sin 30^\circ \tan 40^\circ}{\sin 20^\circ \sin 30^\circ + \tan 40^\circ \sin 50^\circ} = \tan 10^\circ \).

Answer: \tan 10^\circ


Question: Evaluate the integral \( \oint_{C(0;2)} \frac{e^{2\pi z}-1}{z(z-i)} \, dz \), where \( C(0;2) \) is the circle centered at 0 with radius 2, oriented in the positive (counterclockwise) direction.

Answer: 0


Question: Given a multinomial distribution over an alphabet of 27 symbols parameterized by \(\mathbf{\theta}=(\theta_1, ..., \theta_{27})\), and a dataset \(D = \{x[1],\ldots,x[2000]\}\) where the letter "e" appears 260 times, calculate the predictive probability that the next symbol \(X[2001]\) is "e". Assume a Dirichlet prior over \(\theta\) with parameters \(\mathbf{\alpha}= (\alpha_1, ..., \alpha_{27})\), where each \(\alpha_i = 10\). What is \(P(X[2001] = \text{"e"} \mid D)\)?

Answer: \dfrac{27}{227}


Question: Find the area of the smallest tangential trapezoid with an inscribed circle of radius 8.

Answer: 256


Question: Given three collinear points A, B, and C, construct the fourth point D such that the cross ratio R(ABCD) is \(\frac{1.7}{3}\).

Answer: D


Question: Determine whether the following statement is true: \( \int_{-\infty}^{\infty} f(t) \, dt = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} F(\omega) e^{2i\pi\omega t} \, d\omega \, dt = \int_{-\infty}^{\infty} F^{-1}(t) \, dt \), where \( f(t) \) is an integrable function, \( F(\omega) \) is its Fourier transform, and \( F^{-1}(t) \) is its inverse transform.

Answer: True


Question: Let $k$ be an infinite field with the index of its algebraic closure $\bar{k}$ over $k$ strictly greater than 2. Consider a non-empty open subset $U$ of an affine space over $k$. Does $U$ always contain a $k$-rational point?

Answer: Yes


Question: Find the value of \( k \) if \( k = \lim \limits_{n \to \infty} n[(n^3+3n^2+2n+1)^{1/3}+(n^2-2n+3)^{1/2}-2n] \).

Answer: \dfrac{2}{3}


Question: Evaluate the integral \( \int^{+\infty}_0 \frac{e^{-x^2}}{(x^2+\frac{1}{2})^2} \, dx \).

Answer: \sqrt{\pi}


Question: Let $A \in \mathfrak{B}(\mathbb{R})$ be a Borel set with $\lambda(A) > 0.$ Determine whether there exist $B, C \in \mathfrak{B}(\mathbb{R})$ such that $B \cap C = \emptyset$, $B \cup C = A$, and $\lambda(B), \lambda(C) > 0.$

Answer: Yes


Question: Compute the integral \( \int_{\gamma} \frac{\mathrm{e}^{-z}}{(z+2)^{3}}\mathrm{d}z \) using the Cauchy integral theorem and the Cauchy integral formula, where \( \gamma(t)= 3 \mathrm{e}^{2\pi i t } \) for \( t\in [0,1] \).

Answer: \pi i e^{2}


Question: Given that $f(z) = u(z) + iv(z)$ and $g(z) = u(z) - iv(z)$ are holomorphic functions on the same domain $D$, and $f(1+i) = 2 + 3i$, determine the value of $g(4+3i)$. Is it possible to find $g(4+3i)$?

Answer: 2 - 3i


Question: Find the minimum value of the function \( x^2 + y^2 + z^2 \) subject to the constraints \( x + y + z = 1 \) and \( xyz + 1 = 0 \).

Answer: 3


Question: Consider the integral operator \( K : L^2([0, \infty)) \rightarrow L^2([0, \infty)) \) defined by \( K(f) = \int_0^{\infty} k(x,y)f(y)dy \), where \( k(x,y) \in C^0([0, \infty) \times [0, \infty), \mathbb{R}) \). If this operator is well-defined and bounded, must the kernel \( k(x,y) \) be in \( L^2([0, \infty) \times [0, \infty)) \)? Provide a justification for your answer.

Answer: No


Question: Determine the number of downward-closed subsets of the set \( \mathbb{N}^{<\infty} \), where \( \mathbb{N}^{<\infty} \) is the set of all finite sequences of natural numbers, including the empty sequence, ordered by extension. A subset \( C \subset \mathbb{N}^{<\infty} \) is downward-closed if for any sequence \( t \in C \), all initial segments \( s < t \) are also in \( C \).

Answer: 2^{\aleph_0}


Question:
Consider a sample median $Y_n$ from $n$ independent and identically distributed samples with a population median $m$ and a probability density function $f$. Given the central limit theorem for the sample median:
\[ \sqrt{n}(Y_n - m) \xrightarrow{d} N(0, [2f(m)]^{-2}), \]
as $n \to \infty$, determine if the variances converge such that:
\[ nE[(Y_n-m)^2] \to [2f(m)]^{-2}. \]

Answer: [2f(m)]^{-2}


Question: Determine whether the following statement is true or false: A number $a$ is a primitive root modulo $n$ if and only if the set $\{a, a^2, \ldots, a^{\varphi(n)}\}$ forms a complete set of residues that are prime to $n$. Provide your answer as either "True" or "False."

Answer: True


Question: Given the probability density function \( f(x, \theta) = \frac{\theta}{x^{2}} \) for \( 0 < \theta \le x < \infty \), and an i.i.d. sample \( (X_i)_{i=1}^{n} \), find the sufficient statistic for \( \theta \).

Answer: X_{(1)}


Question: Evaluate the limit \( \lim_{h \to 0} \frac{f(2+2h+h^2) - f(2)}{f(1+h-h^2) - f(1)} \) given that \( f(x) \) is a differentiable function with \( f'(1) = 4 \) and \( f'(2) = 6 \).

Answer: 3


Question: Let $(X,\mathcal{A},\mu)$ be a measure space and consider a sequence of sets $\{A_n\}_{n\in\mathbb{N}} \subset \mathcal{A}$ such that $A_{n+1} \subseteq A_{n}$ and $\mu(A_n) = 1$ for all $n\in \mathbb{N}$. Prove or disprove that $\mu(\bigcap_{n\in\mathbb{N}} A_n) = 1$. Provide a justification for your answer.

Answer: 1


Question: Let $G$ be a finite $p$-group and $(\mathbb{F},\mathcal{O},\mathbb{K})$ a sufficiently large $p$-modular system. Furthermore, let $\mu_{p^\infty} \leq \mathcal{O}^\times$ be the group of roots of unity of $p$-power order. If $\alpha$ is a 2-cocycle with values in $\mu_{p^\infty}$, does there always exist a one-dimensional projective representation $G \to \mathcal{O}^\times$ with cocycle $\alpha$? In other words, does the twisted group algebra $\mathcal{O}_\alpha[G]$ always have a one-dimensional representation $\mathcal{O}_\alpha[G] \to \mathcal{O}$?

Answer: No


Question: Given that \( \mathrm{rank}\begin{bmatrix}\mu\mathbf{I}+\mathbf{A}&\mathbf{B}&\mathbf{E}\\ \mathbf{C}&\mu\mathbf{I}+\mathbf{D}&\mathbf{F}\end{bmatrix}=n \) for every \( \mu\in\mathbb{C} \), can we claim that \( \mathrm{rank}\begin{bmatrix}\mu\mathbf{I}+\mathbf{A}&\mathbf{B}&\mathbf{E}&\vdots&\star\\ \mathbf{C}&\mu\mathbf{I}+\mathbf{D}&\mathbf{F}&\vdots&\star\\ \hdashline \mathbf{O}&\mu\mathbf{I}&\mathbf{O}&\vdots&\mathbf{I}_k\end{bmatrix}=n+k \) for every \( \mu\in\mathbb{C} \)?

Answer: Yes


Question: Determine the condition under which the space \( l^p \) is a subset of the space \( l^q \).

Answer: p \leq q


Question: Let a finite group $G$ act smoothly on a smooth manifold $M$ with one fixed point $p$. By the Slice theorem, there exists a $G$-equivariant neighborhood $U_p$ of $p$ that is $G$-diffeomorphic to the tangent space $T_pM$. Is the tangent space $T_pM$ a faithful $\\mathbb{R}G$-module?

Answer: Yes


Question: Let \( G \) be a finite abelian group with elements \( a_1, a_2, \ldots, a_n \). Suppose \( a_n \neq e \) and \( a_n = a_n^{-1} \). If \( a_i, a_j \in G \setminus \{a_n\} \), is it true that \( a_i a_j \in G \setminus \{a_n\} \)?

Answer: No


Question: Determine whether the set \(X = \{f \in C^\infty[0,1] : f^{(n)}(0)=f^{(n)}(1)=0 \ \forall n\geq0\}\) is dense in \(L^2[0,1]\).

Answer: Yes


Question: Evaluate the definite integral \( \int_0^2{x\int_x^2{\frac{dy}{\sqrt{1+y^3}}}}dx \).

Answer: \dfrac{2}{3}


Question: Given two independent and identically distributed random variables $X$ and $Y$ with moment generating function \( M(t) = \left( \frac{1}{1-1.5t}\right)^2 \) for \( t < \frac{2}{3} \), calculate the standard deviation of the sum $X + Y$. Assume $X$ and $Y$ are Erlang distributed.

Answer: 3


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

Answer: \infty


Question: Calculate the sum of the series \( S = \sum_{k=1}^{2015} (-1)^{\frac{k(k+1)}{2}} k \).

Answer: 0


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

Answer: \infty


Question: Calculate the value of the sum: \[ \sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi i}{2000}\right) \] where \( \gcd \) denotes the greatest common divisor.

Answer: 800


Question: Let \( f: X \to S \) be a holomorphic morphism of Hausdorff complex manifolds, and let \( s \in S \) such that \( f^{-1}(s) \) is compact. Is there an open neighborhood \( U \) of \( s \) in \( S \) such that \( f \) is proper over \( U \)?

Answer: Yes


Question: A student rugby team has 15 members, and they share two identical crates of beer, each containing 24 bottles. How many ways are there to distribute the bottles such that no member has two or more bottles than any other member?

Answer: 455


Question: Let $f: \tilde{M} \to M$ be a $d$-fold covering map, and let $N \subset M$ be a connected submanifold of codimension 2. Determine the number of path-connected components of $f^{-1}(N)$. Assume $N$ is connected.

Answer: d


Question: Assume positive sequences \(\{a_n\}\) and \(\{b_n\}\) such that \(a_n \leq a_{n+1}\) and \(b_n \geq b_{n+1}\), with \(\lim_{n \to \infty} b_n = 0\) and \(\sum_{n=1}^\infty a_n(b_n-b_{n+1})\) convergent. Prove that \(\lim_{n \to \infty} a_n b_n = 0\).

Answer: 0


Question: Let \( \varphi:G\times X\to X \) be a continuous action of a finitely generated group \( G \) on a compact metric space \( X \). Suppose \( K\subseteq X \) is a \( G \)-invariant finite subset of \( X \), meaning for every \( g\in G \), \( \varphi(g, K)=K \). If \( K=\{x, \varphi(g_0, x), \varphi(g_1, x), \ldots, \varphi(g_n, x)\} \), determine whether there exists a subgroup of finite index \( H \) of \( G \) such that \( x \) is a fixed point of the action \( \varphi:H\times X\to X \).

Answer: Yes


Question: Approximate the integral \( \int_{0}^{1}\frac{\sin(x)}{x}dx \) with an error less than \( 10^{-5} \).

Answer: 0.946083


Question: If $U$ and $V$ are homeomorphic subspaces of a topological space $X$, is it true that the complements $X \setminus U$ and $X \setminus V$ are also homeomorphic?

Answer: No


Question: Find \( \lim_{n \to \infty} \sum_{k=0}^{n} \frac{e^{-n}n^k}{k!} \).

Answer: \dfrac{1}{2}


Question: For \( n = 2^k \) where \( k \ge 1 \), find a formula for the number of silver matrices of size \( n \). A silver matrix of size \( n \) is an \( n \times n \) matrix where the \( i \)-th row and \( i \)-th column contain all the numbers in \( \{1, 2, \ldots, 2n-1\} \).

Answer: (2n - 1)!


Question: Compute \((3^{999^{100}} + 7^{960^{961}}) \bmod 225.\)

Answer: 154


Question: Evaluate the line integral \( \int_{\gamma} \frac{e^{iz}}{z^2} \, dz \), where \( \gamma(t) = e^{it} \) for \( 0 \leq t \leq 2\pi \).

Answer: -2\pi


Question:
Consider the set \( S = \{T:\mathbb{R}^3\to\mathbb{R}^3:T\text{ is a linear transformation with }T(1,0,1)=(1,2,3),T(1,2,3)=(1,0,1)\} \). Determine the nature of the set \( S \):

(a) a singleton set
(b) a finite set containing more than one element
(c) a countably finite set
(d) an uncountable set

Answer: D


Question: An unbiased die is cast seven independent times. Calculate the conditional probability that each side appears at least once, given that side 1 appears exactly twice.

Answer: \dfrac{24}{625}


Question: Given that $f$ is a decreasing continuous function satisfying the functional equation $f(x+y) = f(x) + f(y) - f(x)f(y)$ and the derivative condition $f'(0) = -1$, find the value of $\int_{0}^{1} f(x) \, dx$. Choose the correct answer from the following options: \( \text{A}) \ 1, \ \text{B}) \ 1-e, \ \text{C}) \ 2-e. \)

Answer: C


Question: Evaluate the double integral \( \iint_{R} \left| \frac{\sin(x)}{x} \right| \, dA \), where \( R \) is the region defined by \( 0 \leq x < \infty \) and \( 0 \leq y \leq 1 \).

Answer: \infty


Question: Let \( P \) be a plane in \( \mathbb{R}^n \) passing through the origin. Let \( \{e_i\}_{i=1}^n \) be the standard coordinate basis for \( \mathbb{R}^n \) and \( \{e_i^*\}_{i=1}^n \) be their orthogonal projections onto \( P \). For any \( p \in P \), does the expansion \[ p = \sum_{i=1}^n \langle p, e_i^* \rangle e_i^* \] hold, where \( \langle \cdot, \cdot \rangle \) is the Euclidean inner product?

Answer: Yes


Question: Let $P$ be a stochastic matrix, i.e., a real square $n \times n$ non-negative matrix with rows summing to one, $P1 = 1$, where $1$ is the vector of ones. Suppose there exists a row vector $\pi^\top \in \mathbb{R}^n$ such that $\pi P = \pi$, $\|\pi^\top\|_1 = 1$, and $\pi \geq 0$ (entry-wise). Let $\operatorname{diag}(\pi)$ be the $n \times n$ diagonal matrix with entries from $\pi$ on the diagonal and zeros elsewhere. Is it true that $\|\operatorname{diag}(\pi) P\|_2 \leq 1$, where $\| \cdot \|_2$ is the matrix norm induced by $L_2$?

Answer: Yes


Question: Find an integer $n$ such that the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^*$ contains elements of order $4$ and $5$, but no elements of order $3$. What is the value of $n$?

Answer: 25


Question: Evaluate the limit: \[ \lim_{x\rightarrow0} \frac{(4^{\arcsin(x^2)} - 1)(\sqrt[10]{1 - \arctan(3x^2)} - 1)}{(1-\cos\tan6x)\ln(1-\sqrt{\sin x^2})} \]

Answer: 0


Question: Find the argument of the complex number \( z = (1 + i\sqrt{3})^{2i} \).

Answer: 2 \ln 2


Question: Evaluate the limit: \( \lim_{x \to 0} (2^x + \sin (3x)) ^{\cot(3x)} \)

Answer: e \sqrt[3]{2}


Question: Evaluate the integral of the function \( f(x,y) = \frac{1}{x} \) over the region bounded by the polar curve \( r = 2\cos(\theta) \) for \( 0 \leq \theta \leq \frac{\pi}{2} \).

Answer: \pi


Question:
Compute the limit of the sequence \((a_{n})_{n=1}^{\infty}\) defined by
\[a_{n}:=\sum_{k=1}^{n} \frac {n^2} {\sqrt{n^{6}+k}}.\]
Find \(\lim_{n \to \infty} a_n.\)

Answer: 1


Question:
Find a function $f(x)$ such that for any $x_0 > 0$, the integral of $f(x)$ from $x_0$ to $2x_0$ is equal to the integral from $2x_0$ to $4x_0$. Express this condition using the antiderivative $F(x)$, where $\frac{d}{dx} F(x) = f(x)$, and solve the recurrence equation:
\[ F(4x_0) - 2 F(2x_0) + F(x_0) = 0 \]

Answer: \dfrac{C}{x}


Question: Let $V$ be a continuous random variable uniformly distributed on $(0,1)$, and let $Y$ be a discrete random variable with probability mass function given by $P(Y=0) = \frac{1}{4}$ and $P(Y=1) = \frac{3}{4}$. Calculate the expected value $E[VY]$. \(E[VY] = \int_0^1 v \cdot P(Y=1) \, dv = \frac{3}{4} \cdot \int_0^1 v \, dv = \frac{3}{4} \cdot \frac{1}{2} = \frac{3}{8}.\)

Answer: \dfrac{3}{8}


Question: Find the limit: \[ \lim_{n \to \infty} 2^n \left( 1 - \exp\left(-\frac{a}{2^n}\right) \right) \] for some constant \( a > 0 \).

Answer: a


Question: Given three probability groups with proportions \( p_1(\theta):p_2(\theta):p_3(\theta) = (1-\theta):(1+2\theta):(1-\theta) \) and observed counts \( n_1 = 31, n_2 = 47, n_3 = 22 \), find the log-likelihood function and determine the maximum likelihood estimate (MLE) of \( \theta \).

Answer: \dfrac{41}{200}


Question: Find the remainder when $2018^{2017^{16050464}}$ is divided by 1001. Note: Do not use the Chinese Remainder Theorem in your solution.

Answer: 16


Question: Is it possible for \( \pi \) raised to some power \( n \) to be a rational number? Provide a justification for your answer.

Answer: No


Question:
Determine whether the following expression is a valid representation of the Dirac delta function:
\[ \delta(x - t) = \lim_{A \to \infty} \left[A e^{-A (x-t)} \theta(x-t)\right] \]
where $A$, $x$, and $t$ are all positive.

Answer: Yes


Question: In a group of 100 readers, each pair of individuals can start a discussion only if each has read at least one book that the other hasn't. What is the minimum number of distinct books required to ensure that any two members of the group can start a discussion?

Answer: 9


Question: Simplify the fraction: \[ \frac{(8+2\sqrt{15})^{3/2}+(8-2\sqrt{15})^{3/2}}{(12+2\sqrt{35})^{3/2}-(12-2\sqrt{35})^{3/2}} \]

Answer: \dfrac{7}{13}


Question: Find the value of \( z_a \) such that \( \frac{1}{\sqrt{2\pi}} \int_{z_a}^{0} e^{-\frac{z^2}{2}} \, dz = 0.48 \).

Answer: -2.05


Question: For a periodic smooth function $f \in C^\infty$, is the set $\{\max(\text{im}(f^{(n)})): n \in \mathbb{N}\}$ always bounded?

Answer: No


Question: Is it possible to arrange 100 people into 20 different rooms such that no room is empty and one designated person, the president, is alone in a room? Provide a justification for your answer.

Answer: Yes


Question: Assume two random variables X and Y are exponentially distributed with rates p and q, respectively. Given that the random variable X - Y is distributed like X' - Y', where X' and Y' are independent exponential random variables with rates p and q, does it follow that X and Y are independent?

Answer: Yes


Question: Let $G$ be a finite abelian group. Is there a subgroup $H < G$ such that the quotient group $G/H$ has prime order?

Answer: Yes


Question: Determine the last nonzero digit of the factorial of a given number, \( N! \). For example, the last nonzero digit of \( 9! \) and \( 10! \) is 8. Find the last nonzero digit of \( 19! \).

Answer: 2


Question: Calculate the coefficient of the term \(x^6 y^5 z\) in the expansion of \(\left( 2x + xy - z + \frac{1}{xyz} \right)^{10}\).

Answer: 5040


Question: Evaluate the integral: \[ \int_{0}^{1}\frac{x^4+1}{x^6+1} \, dx. \]

Answer: \dfrac{\pi}{3}


Question: Calculate the expected number of cards drawn from a standard 52-card deck until all four kings have been drawn.

Answer: \dfrac{212}{5}


Question: Determine the order of growth for the binomial coefficient \( {}^n C_k \) when approximated by a polynomial. Consider the case where \( n \gg k \) and provide the general order of growth for \( {}^n C_k \).

Answer: \Theta(n^k)


Question: Find the number of ways to distribute 14 balls into 4 different boxes such that the number of balls in the last box is not greater than the sum of the number of balls in the first three boxes.

Answer: 596


Question:
Determine the necessary and sufficient condition for the sequence of functions \( f_n(x) = b_nx + c_nx^2 \) to converge uniformly to 0 on the real line. Choose the correct option:

1. \( \lim_{n\rightarrow\infty}b_n=0 \) and \( \lim_{n\rightarrow\infty}c_n=0 \)
2. \( \sum_{n=1}^{\infty}|b_n|<\infty \) and \( \sum_{n=1}^{\infty}|c_n|<\infty \)
3. There exists a positive integer \( N \) such that \( b_n=0 \) and \( c_n=0 \) for all \( n>N \)
4. \( \lim_{n\rightarrow\infty}c_n=0 \)

Answer: 3


Question: Evaluate the sum \( \sum_{\mathbf{S} \in \{-1,1\}^n} \sum_{i=1}^{n-1} S_i S_{i+1} \), where \( \mathbf{S} = \langle S_1, S_2, \ldots, S_n \rangle \).

Answer: 0


Question: Evaluate the infinite series: \[ \sum\limits_{n=1}^\infty \frac{n^2}{3^n}. \]

Answer: \dfrac{3}{2}


Question: Compute the Jacobi symbol \( \left(\frac{77}{257}\right) \) using the properties of Jacobi symbols. Show all steps in your calculation.

Answer: -1


Question: A bag contains three coins: a fair coin, a biased coin (heads is thrice as likely as tails), and a two-headed coin. One coin is drawn at random and tossed $n$ times. Let $F$ and $B$ denote the events that the chosen coin is fair and biased, respectively. Assume each coin is equally likely to be drawn and tosses are independent. Let $A_n$ be the event that all $n$ tosses result in heads. Find \( \lim_{n\to\infty} \frac{P(B|A_n)}{P(F|A_n)} \).

Answer: \infty


Question: Determine the number of zeros, including multiplicities, of the polynomial \( f(z) = z^7 + 9z^4 - 7z + 3 \) that lie outside the disk \( \{ z : |z| \leq 2 \} \).

Answer: 3


Question: Let $G$ be a group of order 12 such that the center $Z(G)$ has no element of order 2. Prove that $G \cong A_4$. \(A_4\) is the alternating group on 4 elements.

Answer: A_4


Question: Determine the radius of convergence of the series \( \sum_{n=0}^\infty n!x^{n^2} \).

Answer: 1


Question:
Let \( G \) be a finite abelian group such that for any \( x \in G \), \( x + x = 0 \), i.e., \( G = \mathbb{Z}_{2}^{\oplus k} \) for some \( k \in \mathbb{N} \). Consider a symmetrical bilinear form \((\cdot, \cdot): G \times G \to \mathbb{Z}_{2}\). Given the conditions:
\[(a, m) = 0,\]
\[(a, p) = 1,\]
\[(b, m) = 1,\]
\[(b, p) = 0,\]
determine if it is true that \((a, b) = 1\).

Answer: 0


Question: Given three positive integers $a$, $b$, and $c$, each greater than 1, if $abc$ divides $(ab - 1)(bc - 1)(ca - 1)$, find the value of $abc$. \( \boxed{} \)

Answer: 30


Question: Let \( \alpha = 2^{1/5} \) and \( \zeta = e^{2i\pi/5} \). Define the field \( K = \mathbb{Q}(\alpha\zeta) \). How many field automorphisms does \( K \) have?

Answer: 1


Question:
Find the determinant of the following NxN matrix:
\[ \begin{pmatrix} 3 & 2 & \cdots & 2 \\ 2 & 3 & \cdots & 2 \\ 2 & 2 & 3 & \cdots \\ 2 & 2 & \cdots & 3 \end{pmatrix} \]

Answer: 2N + 1


Question: Calculate the limit: \[ \lim_{n\to\infty}\prod_{i=0}^n \left(1 - \frac{1}{3i+2}\right). \]

Answer: 0


Question: Consider a length space $M$ and a finite set of geodesics $\Pi$ in this space, where each geodesic is the unique path between its endpoints. Suppose that any two distinct geodesics $\pi_1, \pi_2 \in \Pi$ intersect at most at a single point. Is it possible to assign a direction to each geodesic in $\Pi$ such that the resulting system of paths is acyclic, meaning there is no closed path in $\bigcup_{\pi \in \Pi} \pi$ that only travels along any given path $\pi$ in its chosen direction?

Answer: Yes


Question: Evaluate the expression: \[ \sum_{a=2}^\infty \sum_{b=1}^\infty \int_{0}^\infty \frac{x^{b}}{e^{ax} \ b!} \ dx .\]

Answer: 1


Question:
Determine the coefficient $a_1$ in the series expansion of the Bessel function of order $n$, given by $J_n(x) = x^n \sum_{k=0}^{\infty} a_k x^k$, by substituting this series into the differential equation:
\[ y'' + \frac{1}{x}y' + \left(1 - \frac{n^2}{x^2}\right)y = 0. \]

Answer: 0


Question: The temperature $T$ at any point $P(x,y,z)$ in space is inversely proportional to the square of the distance of $P$ from the origin. Given that the temperature at the point $R(0,0,1)$ is $\sqrt{3}$, find the rate of change of $T$ at the point $Q(1,1,2)$ in the direction of the vector $\vec{QR}$. Round your answer to two decimal places.

Answer: 0.22


Question: Determine whether there exists a nonzero continuous function \(f(x)\) on the interval \([0, 1]\) such that \(\int_0^1 \sqrt{x} \ f(x) \, dx = 0\).

Answer: Yes


Question: Find the McLaurin series of \( f(x) = (1+x)^\alpha \) and determine the radius of convergence using the ratio test. Assume \( \alpha \) is not necessarily an integer.

Answer: 1


Question: Determine the number of elements in \( \mathbb{Z}/p^2\mathbb{Z} \) that are p-th powers, where \( p \) is an odd prime greater than 11.

Answer: p


Question: Evaluate the integral \( \int_{-5}^{+5} \frac{dx}{1+f(x)} \) given that \( f(-x)f'(x)-f(x)f'(-x)=0 \) and \( f(0)=1 \).

Answer: 5


Question: Find the function $f(x)$ that is differentiable for $x > 0$ and satisfies the equation: \[ f(x) = 1 + \frac{1}{x} \int_{1}^{x} f(t) \, dt. \]

Answer: 1 + \ln x


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\sqrt{n^3}(\sqrt{n+1}+\sqrt{n+2}-\sqrt{n}-\sqrt{n+3}) \]

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$\lim\limits_{x\to0}\frac{(1+x\cdot2^x)^{\frac{1}{x^2}}}{(1+x\cdot3^x)^{\frac{1}{x^2}}}$$

Answer: \dfrac{2}{3}


Question: Let \( G \) be a simple group of order 168. Determine the number of subgroups of \( G \) of order 7.

Answer: 8


Question: Given integers \(a, b, c,\) and \(d\) such that \(ab + cd = 44\) and \(ad - bc = 9\), find the minimum possible value of \(a^2 + b^2 + c^2 + d^2\).

Answer: 2018


Question: Determine the smallest non-commutative ring with unity, where "smallest" refers to the ring having the least cardinality.

Answer: 8


Question: Determine the number of connected components of the set \( S = \{ A \in SL(2, \mathbb{R}) \mid A^T \operatorname{diag}(1, -1) A = \operatorname{diag}(1, -1) \} \) in the induced topology, where \( SL(2, \mathbb{R}) \) is the set of real \( 2 \times 2 \) matrices with determinant 1.

Answer: 2


Question: Let \( p(x) \) be a fifth-degree polynomial such that \( p(x) + 1 \) is divisible by \((x-1)^3\) and \( p(x) - 1 \) is divisible by \((x+1)^3\). Find the value of the definite integral \( \int_{-10}^{10} p(x) \, dx \).

Answer: 0


Question: Find the third digit in the product of all odd numbers from 1 to 1,000,000.

Answer: 6


Question: Find \( \lim_{x\to\infty} \left(\cos\sqrt{\frac{2a}{x}}\right)^{x} \) where \( a>0 \).

Answer: e^{-a}


Question: Find the number of ways to arrange 6 people at three identical round tables such that each table is occupied.

Answer: 225


Question: Evaluate the triple integral \( \iiint (xyz) \, dx\,dy\,dz \) over the region \( D:=\left \{ (x,y,z) \in\mathbb{R}^3 :x^2+y^2+z^2 \leq 4 \wedge y > x\right \} \). Determine if the integral evaluates to zero.

Answer: 0


Question: Determine whether there exists a power of 2 that begins with the digits 999 in its decimal representation.

Answer: Yes


Question: Evaluate the surface integral \( \iint A \cdot dS \) for the vector field \( A = 6z\hat{i} + (2x+y)\hat{j} - x\hat{k} \) over the entire surface \( S \) of the region bounded by the cylinder \( x^2 + z^2 = 9 \), and the planes \( x = 0 \), \( y = 0 \), \( z = 0 \), and \( y = 8 \).

Answer: 18\pi


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

Answer: \infty


Question:
Determine the number of solutions in positive integers \((x, y)\) for the equation:
\[ 37 + \frac{3(x-1)x}{2} + 3^{x+2} = y^2 \]

Answer: 4


Question: Evaluate the integral: \[ \int_{-\infty}^{\infty} \mathrm{e}^{-\frac{x^2}{2}} \, \mathrm{d}x \]

Answer: \sqrt{2\pi}


Question: Consider the sequence \(a_n\) defined by \(a_1 = 1\) and the recursive formula \(a_{n+1} = \sqrt{1 + a_n^{\delta}}\), where \(\delta > 0\) is a real number. Determine the condition on \(\delta\) for which the sequence \(a_n\) converges.

Answer: \delta < 2


Question: Solve the equation \( \lfloor \frac{2x-1}{3} \rfloor + \lfloor \frac{4x+1}{6} \rfloor = 5x-4 \).

Answer: \dfrac{4}{5}


Question:
Let \( A \) be an invertible matrix and \( \lambda \in \mathbb{R} \) such that \( \operatorname{det}(A - \lambda I) = 0 \), where \( I \) is the identity matrix. Prove that:
\[
\operatorname{det}\left(A^{-1} - \frac{1}{\lambda} I\right) = 0
\]

Answer: \operatorname{det}\left(A^{-1} - \frac{1}{\lambda} I\right) = 0


Question: Compute \( \mathbb{E}(2X_1+3X_2\mid X_1+3X_2-X_3=4) \) where \( X_1, X_2, \) and \( X_3 \) are independent and identically distributed standard normal random variables.

Answer: 4


Question: Determine the number of 10-digit ternary sequences (using digits 0, 1, and 2) such that the digit 2 occurs at least once, and the digit 0 occurs an even number of times. Use generating functions to solve this problem.

Answer: 29013


Question: Find the independent coefficient in the inverse of the polynomial $2x^2+2$ in the ring $\mathbb{Z}_3[x]/(x^3+2x^2+2)$. Provide your answer as a single integer.

Answer: 2