Question: Consider a smooth manifold $M$ and a curve $c: [0,1] \to M$. Is it always possible to construct a smooth vector field $V$ along the curve $c$ such that $V(0) = 0$ and $V(1) = w$ for some fixed vector $w$?

Answer: Yes


Question: Evaluate the integral \( \int_0^{2\pi} e^{e^{i\theta}} \, d\theta \).

Answer: 2\pi


Question: Do there exist real non-diagonalizable matrices with purely imaginary eigenvalues?

Answer: Yes


Question: Find the radius of convergence of the series \( \sum_{n=0}^{\infty} \frac{z^{2n}}{(2+i)^n} \).

Answer: \sqrt[4]{5}


Question: Calculate the limit: \[ \lim_{x \to 0} [(x+1)^x-1]^x \]

Answer: 1


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

Answer: \infty


Question: Given that $X_1, X_2, X_3$ are independent random variables each following a uniform distribution on $[A, 2A]$, where $A > 0$, find the constant $k$ such that $E(kA_{ML}) = A$, where $A_{ML} = \frac{1}{2} \max \{X_1, X_2, X_3\}$. Assume independence of $X_1, X_2, X_3$.

Answer: \dfrac{8}{7}


Question: Let \( X_1, X_2, \ldots \) be independent random variables such that \( P(X_n = 1) = P(X_n = -1) = \frac{1}{2} \). Define \( t = \inf \{ n : X_1 + X_2 + \cdots + X_n = 1 \} \). Find the expected value \( E\left(\left(\frac{1}{3}\right)^t\right) \).

Answer: 3 - 2\sqrt{2}


Question: Let $X$ be a locally connected, locally compact, separable metric space. Determine whether there exists a countable collection $\mathcal{B}$ such that every member of $\mathcal{B}$ is a nonempty Peano subspace of $X$ and $\cup \mathcal{B} = X$. Answer with 'Yes' or 'No'.

Answer: Yes


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

Answer: -1


Question: Let $F$ be a field and $K$ a subfield of $F$. Consider a matrix $A \in M(n,K)$. If there exists a non-zero vector $x \in F^n$ such that $Ax = 0$, does there necessarily exist a non-zero vector $y \in K^n$ such that $Ay = 0$? Provide a justification for your answer.

Answer: Yes


Question: Let $L/K$ and $M/K$ be two finite extensions of henselian fields. Let $T$ and $U$ be the maximal unramified subextensions of $L/K$ and $M/K$, respectively. Determine if the maximal unramified subextension of $LM/K$ is $TU$. Provide a justification for your answer.

Answer: TU


Question: The surface defined by \( z = x^2 - y^2 \) is intersected by the plane \( y = 3x \), forming a curve. Determine the slope of this curve at the point \((1, 3, -8)\).

Answer: -16


Question: Compute the area of the region on the cylinder $x^2+y^2=R^2$ bounded by the planes $z=mx$ and $z=nx$, where $z \ge 0$, $m > n > 0$. Express your answer in terms of $R$, $m$, and $n$.

Answer: 2R^2(m - n)


Question: What is the probability that a positive divisor of $8748$ million is the product of exactly $20$ non-distinct primes?

Answer: \dfrac{1}{168}


Question: Determine the interior points of the set \( \mathbb{Q} \cap [0,1[ \).

Answer: \emptyset


Question: Evaluate the limit: \[ \lim_{x\rightarrow\pm\infty}\left(e^{x/(x+1)}(1-x)+e\cdot x\right) \] without using L'Hôpital's rule.

Answer: 2e


Question: Given a parabola with equation \( y^2 = 4ax \), the distances of two points \( P \) and \( Q \) from the focus are 4 and 9, respectively. Find the distance from the focus to the point of intersection of the tangents at \( P \) and \( Q \).

Answer: 6


Question: Suppose $A$ and $B$ are two positive semidefinite matrices of size $N \times N$, satisfying \( \ker(A) \subseteq \ker(B) \Rightarrow \textrm{Im}(B) \subseteq \textrm{Im}(A) \). Further assume $A$ has rank $P \leq N$, hence admitting the decomposition $A=VV^T$, where $V$ is a matrix of size $N \times P$. Does $B$ always admit the decomposition $B = VXV^T$ where $X$ is a positive semidefinite matrix of size $P \times P$?

Answer: Yes


Question:
Determine whether the following two linear programs are equivalent:

1. \( \min\limits_{a\in A} \left( \max\limits_{b\in B} f(a, b) + \max\limits_{c\in C} f(a, c) \right) \)

2. \( \begin{align} \min\limits_{a\in A} & \quad (z_1 + z_2) \\
\text{subject to}& \quad z_1\geq f(a,b) &\forall b\in B\\
& \quad z_2 \geq f(a,c) &\forall c\in C  \end{align} \)

Answer: Yes


Question: Find the Nyquist frequency for the signal \( x(t) = \sum_{n=1}^{3} \sin\left(2\pi \frac{n}{8}\frac{t}{T}\right) \), where \( T > 0 \) is a positive number. Determine the smallest frequency \( u_0 \) such that for all \( u > u_0 \), the Fourier transform \( \hat{x}(u) = 0 \).

Answer: \dfrac{3}{8T}


Question: Determine whether the limit \( \lim_{n \to \infty} \sum_{k=1}^n \ln\left(1-\frac{x^2\sin^2k}{2n}\right) \) exists for \( x \in \mathbb{R} \). If it exists, find its value.

Answer: -\dfrac{x^2}{4}


Question: Given the vector field \( \vec{F}(x,y) = (y + xg(x), y^2) \) and the condition \( \vec{F}(1,1) = (3,1) \), find the function \( g(x) \) such that the partial derivatives \( \vec{F}_x \) and \( \vec{F}_y \) are orthogonal. Determine \( g(x) \).

Answer: \dfrac{2}{x}


Question: Describe the connected components of the topological space \((\Bbb R, \mathcal T_{\text{lower limit}})\), where \(\Bbb R\) is equipped with the lower limit topology.

Answer: \{x\}


Question: Determine the value of the constant \( c \) such that the function \( f(x,y) = c\frac{2^{x+y}}{x!y!} \) is a valid probability density function for the discrete random vector \((X,Y)\) over \( \mathbb{N}_0 \times \mathbb{N}_0 \).

Answer: e^{-4}


Question: Use Stokes’ Theorem to evaluate \( \iint_S \text{curl } \mathbf{F} \cdot d\mathbf{S} \), where \( \mathbf{F}(x,y,z)=e^{xy}\cos z \ \mathbf{i}+x^2z \ \mathbf{j}+xy \ \mathbf{k} \). The surface \( S \) is the hemisphere \( x=\sqrt{1-y^2-z^2} \), oriented in the direction of the positive x-axis.

Answer: 0


Question: Determine whether the following statement is true or false: \( \int_{\mathbb{R}} f(t)g(t) \, dt = \int_{\mathbb{R}} f'(t) \, dg(t) \), where \( f : \mathbb{R} \to \mathbb{C} \) is a \( C^\infty \) function, and \( g \) is continuous and of bounded variation. The left-hand side is a Riemann integral, and the right-hand side is a Riemann-Stieltjes integral.

Answer: False


Question: Find the point $P$ on the $x$-axis such that its orthogonal projection onto the line $r: x + y = z - 1 = 0$ is the point $M = (1, -1, 1)$. What are the coordinates of point $P$?

Answer: (2, 0, 0)


Question:
Evaluate the flux through a sphere of radius $R$ centered at the origin, due to the field given by \( \overline{E} = \frac{ax^2\vec{i} + by^2\vec{j} + cz^2\vec{k}}{ax^3 + by^3 + cz^3} \), where $a$, $b$, and $c$ are non-zero constants. The flux is defined as \( \Phi = \oint \overline{E} \cdot \mathrm{d}\overline{A} \). Using Gauss' law in its differential form, the charge density \( \rho(x,y,z) \) is given by:

\[ \rho(x,y,z) = \frac{2(ax+by+cz)}{ax^3+by^3+cz^3} - \frac{3(a^2x^4+b^2y^4+c^2z^4)}{(ax^3+by^3+cz^3)^2} \]

Calculate the integral of \( \rho \mathrm{d}V \) over the sphere \( \Gamma : x^2 + y^2 + z^2 = R^2 \):

\[ \iiint_{\Gamma} \left( \frac{2(ax+by+cz)}{ax^3+by^3+cz^3} - \frac{3(a^2x^4+b^2y^4+c^2z^4)}{(ax^3+by^3+cz^3)^2} \right) \mathrm{d}x \mathrm{d}y \mathrm{d}z \]

Answer: 4\pi R


Question: Determine the number of cyclic subgroups of order 15 in the group \( \mathbb{Z}_{30} \oplus \mathbb{Z}_{20} \).

Answer: 6


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

Answer: \sqrt{e}


Question: Given n circles and 1 straight line on a plane, the plane can be divided into a maximum of 44 regions. Determine the value of n.

Answer: 6


Question:
Differentiate the following integral with respect to the function $f(t)$, where the upper bound of the integral is $t$:
\[
\frac{\partial}{\partial f(t)} \int_0^t f^3(\tau) \cdot g(\tau) \, d\tau
\]

Answer: 3 f(t)^2 g(t)


Question: Find a basis for the vector space \( \mathbb{W} = \{(a,b) \in \mathbb{R}^2 \mid b > 0\} \) with the operations defined as \((a,b) + (c,d) = (ad + bc, bd)\) and scalar multiplication \(k * (a,b) = (kab^{k-1}, b^k)\). Determine the dimension of \(\mathbb{W}\).

Answer: 2


Question: Find the series expansion or limit of the function \( f(x) = \frac{1}{x} \exp \left[-a \left( \frac{b-x}{x} \right)^2 \right] \) as \( x \to 0 \).

Answer: 0


Question: Compute the limit: $$\lim_{x \to 0^+} \frac{\sqrt{e^x-1}-e^x\sqrt{x}}{x}.$$

Answer: 0


Question: Evaluate the definite integral: \[ \int_0^{2\pi}(t-\sin t)(1-\cos t)^2\,\mathrm{d}t. \]

Answer: 3\pi^2


Question: Find the integer $k$ such that $27^k \equiv 2 \pmod{2021}$, given that $2^{11} \equiv 27 \pmod{2021}$ and $2021 = 43 \cdot 47$. Use the properties of modular arithmetic to solve for $k$.

Answer: 205


Question: Must all inner-product preserving maps be linear?

Answer: Yes


Question: Find the partial derivative \( f_x(0,0) \) of the function \( f(x, y) = \frac{5x^2y}{x^4+y^2} \), given that \( f(0,0) = 0 \).

Answer: 0


Question: Find the radius of convergence of the power series \( \sum_{n=1}^{\infty} n^{-\sqrt{n}} z^n \).

Answer: 1


Question: Evaluate the double integral: \[ \int_{-\pi/2}^{\pi/2}\int_{-\pi/2}^{\pi/2} \sin\left(\,\left\lvert\, {x - y\,}\right\rvert\,\right)\, \mathrm{d}x\,\mathrm{d}y \]

Answer: 2\pi


Question: Consider the function $u(x,t)$ that satisfies the heat equation $$u_{t}=u_{xx}, \ x \in\mathbb{R}, \ t >0$$ with the initial condition $$u(x,0)= \begin{cases} 1, & x \in [0,1] \\ 0, & \text{otherwise} \end{cases}$$ Determine the value of $ \lim_{t \to 0^{+}} u(1,t)$.

Answer: \dfrac{1}{2}


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

Answer: \dfrac{1}{6}


Question: Evaluate \( \lim_{x\to \infty} \frac{\{x\} \ln x^n}{[x]} \), where \( \{x\} \) is the fractional part of \( x \) and \( [x] \) is the integral part of \( x \).

Answer: 0


Question: Given the Sobolev norm \( \| \cdot \|_{H^s(\mathbb{R})} \) in \( \mathbb{R} \) and \( r > 0 \), if the inequality \( \|f\|_{L^\infty(\mathbb{R})} \le \| f\|_{H^k(\mathbb{R})} \) holds for all \( k > r \), does the inequality \( \|f\|_{L^\infty(\mathbb{R})} \le \| f\|_{H^r(\mathbb{R})} \) also hold?

Answer: Yes


Question: Evaluate the integral \( \int_0^{\pi/2} \frac{\cos^3x}{\sin^2x + \cos^3x} \, dx \).

Answer: \dfrac{\pi}{4}


Question: Let \( \{z_1, z_2, z_3, z_4, z_5\} \) be the solutions of the equation \( z^5 = 2i \). Calculate the value of \( x^2 = \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} + \frac{1}{z_4} + \frac{1}{z_5} + i \).

Answer: i


Question: Determine the dimension of \( \frac{m}{m^2} \) over \( k \), where \( m = m_P(V) \) and \( V = V(x^2-y^3, y^2-z^3) \).

Answer: 3


Question: Two robots start at position X=0 on the X-axis. Each turn, each robot moves one unit forward or backward with equal probability (0.5), independently of each other and of previous turns. After 450 turns, let D be the distance between the two robots. Calculate approximately the probability that D is less than or equal to 6, using the Central Limit Theorem.

Answer: 0.159


Question: Let $(R,m)$ be a noetherian local ring with dimension 1 and a principal maximal ideal. Does it follow that $R$ is a domain? Justify your answer.

Answer: Yes


Question: Determine whether there exists a $3 \times 3$ real orthogonal matrix with all non-zero entries. Justify your answer.

Answer: Yes


Question: Find the integer \( n \) such that \( \left|\sum_{r=0}^{3n-1}\beta^{2^r}\right| = 4\sqrt{2} \), where \( \beta = \exp(i2\pi/7) \).

Answer: 4


Question: Determine the cardinality of the set \( S = \{ A \in M_3(\mathbb{Z}) : A A^{\top} = I \} \).

Answer: 48


Question: Evaluate the limit: \[ \lim_{x \to +\infty} \frac{\left(\int^x_0 e^{t^2} \, dt\right)^2}{\int_0^x e^{2t^2} \, dt} \]

Answer: 0


Question: Calculate \(\int_{\gamma}(z^2-4)^{-1}dz\) where \(\gamma\) is the unit circle traversed positively once. Determine whether the integral is zero by considering the analyticity of the function \((z^2-4)^{-1}\) on the unit circle.

Answer: 0


Question: Evaluate the limit: \[ \lim _{x\to 0}\left\lfloor\frac{\tan x \sin x}{x^2}\right\rfloor \] where \(\lfloor\cdot\rfloor\) denotes the greatest integer function.

Answer: 1


Question: Given random variables $X$ and $Y$ with joint distribution $f(x,y) = 1$ for $0 \leq x, y \leq 1$ and 0 elsewhere, find the conditional expectation $E[X | Y > X]$. Provide your answer as a single value.

Answer: \dfrac{1}{3}


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\sin^3 x - x^3 \operatorname{sgn}\left(1 - \left[\frac{x}{\sin^{-1} x}\right]\right)}{x \tan^2 x \sin(\pi \cos x)} \]

Answer: -\dfrac{1}{\pi}


Question: Consider the measure \( \nu \) on \( \mathbb{R} \) defined by \( \nu((-\infty,0)) = 0 \) and for every Borel set \( A \subseteq [0,\infty) \), \( \nu(A) = \lambda(f^{-1}[A]) \), where \( f(x) = x^2 \) and \( \lambda \) is the Lebesgue measure on \( \mathbb{R} \). Is the measure \( \nu \) absolutely continuous with respect to the Lebesgue measure \( \lambda \)?

Answer: Yes


Question: Calculate the sum \( S = \lim_{N\rightarrow \infty} \sum_{n=-N}^N \frac{1}{n+e} \).

Answer: \pi \cot(\pi e)


Question: Given a polynomial \( p(x) \) that outputs an integer for every integer input \( x \), is it necessary for \( p(x) \) to have rational coefficients? Answer yes or no and justify your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{-1}^{1} x^{2} \delta(x^3) \, dx \), where \( \delta(x) \) is the Dirac delta function.

Answer: \dfrac{1}{3}


Question:
Given a Gaussian random vector $X = (Y, X, Z)$ with mean $\mu = (1, 2, 4)^T$ and covariance matrix:

$$\Sigma = \begin{pmatrix} 2 & 3 & 1\\ 3 & 5 & 2\\ 1 & 2 & 6 \end{pmatrix}$$

Calculate the conditional variance $\operatorname{Var}(Y \mid X, Z)$. Use the formula for conditional variance in a multivariate Gaussian distribution.

Answer: \dfrac{5}{26}


Question: Evaluate the limit: \[ \lim_{x \to 0} \left\lfloor \frac{\tan^{98}x - \sin^{98} x}{x^{100}} \right\rfloor \] where \( \lfloor x \rfloor \) denotes the floor function.

Answer: 49


Question: Find the Sylow subgroups of order \(2^3\) in the group \(G = \text{Aut}(\mathbb{Z}_{276})\).

Answer: \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2


Question:
Consider a triangular array of nonnegative numbers given by:
\[x_{1,1} \\
x_{2,1} \ x_{2,2} \\
x_{3,1} \ x_{3,2} \ x_{3,3} \\
\ldots \]
If the maximum of each row converges to zero, \(\max_{1 \le k \le n} x_{n,k} \to 0\), and the sequence of row sums converges to \(\lambda > 0\), \(\sum_{k=1}^n x_{n,k} \to \lambda\) as \(n \to \infty\), does it follow that
\[\prod_{k=1}^n (1 + x_{n,k}) \to e^{\lambda}?\]

Answer: e^{\lambda}


Question: Let \( p \equiv 3 \, (\text{mod 4}) \) be a prime and let \( D \in \mathbb{F}_p^{\times}. \) Show that the equation \( C: v^2 = u^4 - 4D \) has \( p-1 \) solutions \((u,v) \in \mathbb{F}_p \times \mathbb{F}_p.\)

Answer: p - 1


Question: Prove that for any nonempty set \( A \), the space \( l^2(A) \) is complete.

Answer: l^2(A)


Question:
Determine which of the following pairs of sets are homeomorphic:

Let $X=[-1,1] \times [-1,1]$, and consider the following subsets of $X$:
- $A=\{(x,y) \in X: x^2+y^2 =1\}$
- $B=\{(x,y) \in X:|x|+|y|=1\}$
- $C=\{(x,y) \in X: xy=0\}$
- $D=\{(x,y) \in X: x=\pm y\}$

Which of the following statements are true?

A. $A$ is homeomorphic to $B$
B. $B$ is homeomorphic to $C$
C. $C$ is homeomorphic to $D$
D. $D$ is homeomorphic to $A$

Answer: AC


Question: Find the last three digits of the product \(1 \times 3 \times 5 \times \cdots \times 2017\).

Answer: 625


Question: Simplify the expression: $$\frac{d}{dx}\left(\int_{-(x^2)}^{x^3}\left(\frac{\sin(t)}{1+\cos(t)}+1\right)dt+\int_{x^3}^{x^2}\left(\frac{\sin(t)}{1+\cos(t)}-1\right)dt\right)$$ for \(-1 \le x \le 1\).

Answer: 6x^2


Question: Find the angle between the vector \( \vec{a} \) and the vector \( \hat{i} - 2\hat{j} + 2\hat{k} \). The vector \( \vec{a} \) is parallel to the line of intersection of the plane determined by the vectors \( \hat{i} \) and \( \hat{i} + \hat{j} \), and the plane determined by the vectors \( \hat{i} - \hat{j} \) and \( \hat{i} + \hat{k} \).

Answer: \dfrac{\pi}{4}


Question: Given a compound Poisson claims distribution with individual claim amounts of $1$, $2$, or $3$, an expected value $E(S) = 56$, variance $Var(S) = 126$, and rate $\lambda = 29$, determine the expected number of claims of size 2.

Answer: 11


Question:
Let \( f(x) = \sin^{23}x - \cos^{22}x \) and \( g(x) = 1 + \frac{1}{2}\arctan|x| \). Determine the number of values of \( x \) in the interval \([-10\pi, 20\pi]\) that satisfy the equation \( f(x) = \text{sgn}(g(x)) \). Choose the correct answer from the following options:

(A) 6 \hspace{1cm} (B) 10 \hspace{1cm} (C) 15 \hspace{1cm} (D) 20

Answer: C


Question: Given a matrix $A$ with a null trace, what is the minimum number of linear measurements required to determine $A$? A linear measurement is defined as knowing the quantities $A\vec{x}_i \cdot \vec{h}_j$ for given vectors $\vec{x}_i$ and $\vec{h}_j$.

Answer: n^2 - 1


Question: Compute the limit: \( \lim_{x\to 0} (x^{-2}-\sin(x)^{-2}) \).

Answer: -\dfrac{1}{3}


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

Answer: 6


Question: Find the expectation of the product of three Brownian motions: \( E(B(t_1)B(t_2)B(t_3)) \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{1}{x^3}\int_{0}^{x} \sin^{2}(3t)\,dt \]

Answer: 3


Question: Let \( f \) be a function that is analytic in a punctured disk and has a bounded derivative \( f' \). Is \( f \) necessarily bounded in that disk?

Answer: Yes


Question: Compute the limit: $$\lim_{x\to 0^+} \frac{\sqrt{1+\frac{1}{x}}-\sqrt{x^2+2x}}{x}$$

Answer: \infty


Question: Evaluate the integral \( \int_C \frac{x\mathrm{d}y-y\mathrm{d}x}{x^2+y^2} \), where \( C \) is the curve defined by the equation \( x^{48} + y^{76} = 987654321^{123456789} \).

Answer: 2\pi


Question: Find the smallest divisor of the expression $12! + 6! + 12! \times 6! + 1$.

Answer: 7


Question: In a cyclic pentagon ABCDE, it is given that ∠ABD = 90°, BC = CD, and AE is parallel to BC. If AB = 8 and BD = 6, find the value of AE^2.

Answer: \dfrac{338}{5}


Question: Evaluate the limit: \[ \lim_{(x,y)\to(0,0)}\dfrac{x^2+y^2}{x^4+y^4} \]

Answer: \infty


Question: Let $\Sigma$ be an Alexandrov space of curvature $\geq 1$ without boundary, and let $X$ be the cone over $\Sigma$. Consider a function $f\colon X\to \mathbb{R}$ that is concave, 1-homogeneous, and satisfies $f\geq 0$. Does it necessarily follow that $f\equiv 0$?

Answer: 0


Question: Is there an algorithm to decide whether the intersection of a context-free language $L_1$ and a regular language $L_2$ is empty?

Answer: Yes


Question: Let $A$ be a Lebesgue-measurable subset of $\mathbb{R}_{>0}$ such that for every $x \in \mathbb{R}_{>0}$, there exists an $n \in \mathbb{N}_{>0}$ such that $nx \in A$. In other words, $\mathbb{R}_{>0} = \bigcup_{n \in \mathbb{N}_{>0}} \frac{1}{n} A$. Can $A$ have finite measure?

Answer: No


Question: Find the distributional derivative of the delta function \( \delta(x) \).

Answer: \delta'(x)


Question: Compute the residue of the function \( f(z) = \frac{z - \pi}{\sin^2 z} \) at \( z = \pi \).

Answer: 1


Question: Calculate \( 1819^{13} \pmod{2537} \) using Fermat's Little Theorem.

Answer: 2081


Question: Determine the radius of convergence for the series expansion of \((1-x)^{1/4}\).

Answer: 1


Question: Evaluate the integral \[ \frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^y e^{-\frac{1}{2}(x^2 + y^2)}\,dx\,dy \] for a bivariate normal distribution with parameters (0, 0, 1, 1, cor=0).

Answer: \dfrac{1}{2}


Question: Determine the value of the parameter \(a\) such that \(\lim_{x\rightarrow \infty} \left(\sqrt[3]{x^3-ax^2+1}-\sqrt[3]{x^3-x^2+1}\right) = \frac{1}{3}.\)

Answer: 0


Question: Evaluate the limit of the multiple integral: $$\lim_{n\to \infty} \int_0^1 \int_0^1...\int_0^1 \cos^2\left(\frac{\pi}{2n}(x_1+x_2+...+x_n)\right)dx_1 dx_2...dx_n.$$

Answer: \dfrac{1}{2}


Question: Let $T$ be a linear transformation from $M(n, \mathbb{R})$ to $M(n, \mathbb{R})$ defined by $T(X) = (A - I)X$, where $A$ is a non-zero nilpotent matrix of order $n \times n$. Determine the trace of the transformation $T$. \( \text{(Express your answer as a single number.)} \)

Answer: -n^2


Question: Let \( n \) be an even number, and let \( \alpha \) and \( \beta \) be the roots of the equation \( x^2 + px + q = 0 \) and also of the equation \( x^{2n} + p^n x^n + q^n = 0 \). Given the function \( f(x) = \frac{(1+x)^n}{1+x^n} \), find the value of \( f\left(\frac{\alpha}{\beta}\right) \) under the conditions \( \alpha^n + \beta^n \neq 0 \) and \( p \neq 0 \).

Answer: -1


Question: Using only one circle and four lines, construct the center of a given circle using a straight-edge and compass. No tangents are allowed. This task is based on Euclidean Geometry principles.

Answer: O


Question: Let $K$ be a field and $A$ be a division Albert algebra over $K$. Is it true that for all nonzero elements $x, y \in A$, the equation $(xy) y^{-1} = x$ holds?

Answer: No


Question: Find the radius of convergence of the series \( \sum \left(\frac{n+1}{n}\right)^{n^2} z^{kn} \), where \( k \) is a constant.

Answer: e^{-1/k}


Question: Find the value of \(a+b\) if \[ \lim _{x\to 0}\left(\frac{\tan\left(2x\right)\:-2\sin\left(ax\right)}{x\left(1-\cos\left(2x\right)\right)}\right)=b. \]

Answer: \dfrac{5}{2}


Question: Determine if the function $g(x) = f(2^x, 0)$ is periodic. If it is periodic, find its period, given that $f(x, y)$ satisfies the condition $f(x, y) = f(2x + 2y, 2y - 2x)$ for all $x, y \in \mathbb{R}$. Provide a justification for your answer.

Answer: 12


Question: Determine the values of \( \alpha \) for which the integral \( \int_{1}^{\infty}\left(\sum_{j = 1}^{10}x^{j}\right)^{-(n +1)/10}\,\frac{-n + 1}{10}\,x^{\alpha}\,\mathrm{d}x \) is convergent, given that \( n > 0 \) and \( \alpha \) is a real number.

Answer: \alpha < n


Question: Is the absolute value of $x$ a sufficient statistic for a continuous distribution that is symmetric about the y-axis, such as the standard normal distribution?

Answer: Yes


Question: Evaluate the limit: $$ \lim_{x \to \infty} \frac{x^x}{x!} $$

Answer: \infty


Question: Construct a non-geometric $C^\infty(M)$-module, where $M$ is a smooth manifold.

Answer: C^\infty(M)/I_p


Question:
Find the limit of the sequence given by:
\[
a_n = (n^2 + 1)\sin\left(\frac{1}{n^3+1}\right) + (n^2 + 2)\sin\left(\frac{1}{n^3+2}\right) + \dots + (n^2 + n)\sin\left(\frac{1}{n^3+n}\right)
\]

Answer: 1


Question: A hotel has 300 rooms and follows an overbooking policy to maximize room usage. Each reservation holder has a 20% chance of canceling or not showing up. Determine the largest number of reservations the hotel can accept while ensuring at least a 95% probability that all guests who arrive will have a room available.

Answer: 360


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

Answer: \sqrt{6}


Question: Find the minimum value of $t$ such that $x_1^3 + x_2^3 + \ldots + x_t^3 = 2002^{2002}$ can be satisfied by some natural numbers $x_i$. \(t\) is a natural number.

Answer: 4


Question: In a topological vector space $X$, if $K$ and $C$ are closed subsets such that $K \cap C = \emptyset$, does there exist a neighborhood $V$ of $0$ such that $$(K+V) \cap (C+V) = \emptyset?$$

Answer: No


Question: Let $f$ be a non-negative real-valued function defined on $[0,1]$. Determine whether the following statement is true or false: $f$ is measurable if and only if there exists a sequence $\{E_n\}$ (finite or infinite) of measurable subsets of $[0,1]$ and a sequence of non-negative constants $\{c_n\}$ such that $f(x) = \sum_n c_n \chi_{E_n}(x)$ for every $x\in[0,1]$, where $\chi_{E_n}$ is the characteristic function of $E_n$ for each $n$. Provide a justification for your answer.

Answer: A


Question:
Let \( G \) be a continuously differentiable cumulative distribution function (CDF) over the interval \([a, b]\). Determine whether the following inequality holds for any natural number \( n \):
\[
n\int_{a}^{b}G(x)^n\,dx \leq (n+1)\int_{a}^{b}G(x)^{n+1}\,dx.\]

Answer: Yes


Answer: 2


Question: Can a finite distributive lattice of length \( n \) be embedded into a boolean lattice of rank \( n \)?

Answer: Yes


Question: Consider a surface in \(\mathbb{R}^3\) with the first fundamental form given by \(ds^2=E(u)du^2+G(v)dv^2\). Determine whether the Gaussian curvature of this surface vanishes.

Answer: 0


Question: Find the order of the entire function \( f_k(z) = \prod\limits_{n=1}^{\infty} \left(1-\frac{z}{n^k}\right) \). The order \( \lambda \) is defined as: \[ \lambda = \limsup_{r\to \infty} \frac{\log \log M(r)}{\log r}, \] where \( M(r) = \max \{|f_k(z)|:|z| = r\} \).

Answer: \dfrac{1}{k}


Question: Find the shortest distance from the origin to the quadric surface given by the equation \(3x^2 + y^2 - 4xz = 4\).

Answer: 1


Question: Evaluate the integral \( \frac{1}{2\pi i}\oint_C \frac{e^{zt}}{z(z^2+1)}dz \), where \( t > 0 \) and \( C \) is the rectangle with vertices at \( 2+2i \), \( -2+2i \), \( -2-2i \), and \( 2-2i \). Use the residue theorem to find the value of the integral, which is given as \( 1 - \cos t \).

Answer: 1 - \cos t


Question: Determine the number of distinct terms in the expansion of \( \left( x + \frac{1}{x} + x^2 + \frac{1}{x^2} \right)^{15} \).

Answer: 61


Question: Let $G$ be the special linear group $SL(2,F_q)$ where $F_q$ is a finite field of order $q$. Determine whether all matrices of the upper triangular group (or a conjugate of it) of $G$ have a common eigenvector.

Answer: Yes


Question: Find the limit: \( \lim_{n \to \infty}\frac{a^n}{n^{n/2}} \)

Answer: 0


Question: Assume that \( \left|\sum_{n=0}^\infty a_n\frac{\lambda^n}{n!}e^{-\lambda}\right|\leq1 \) holds for all \( \lambda\geq 0 \). Do the coefficients \( a_n \in \mathbb{R} \) need to be bounded, such as \( |a_n|\leq 1 \)?

Answer: No


Question: Find \( g'\left(\frac{1}{2}\right) \) where \( g(t) = \int_{0}^{9\sin(\pi t)} e^{-s^2} \, ds \).

Answer: 0


Question: Determine the order of the group $G = \{A \in M_2(\mathbb{Z}_p): \mathrm{det}A = \pm 1 \}$, where $p > 2$ is a prime number.

Answer: 2p(p^2 - 1)


Question: Find and classify the critical points of the function \( f(x,y) = x^y - xy \) in the domain \( x > 0, y > 0 \).

Answer: (e, 1)


Question: Find the values of \( m \) such that the inequality \( \left(\frac{9}{25}\right)^x - m\left(\frac{3}{5}\right)^x + 1 > 0 \) holds for all \( x < 0 \).

Answer: (-\infty, 2]


Question: Evaluate the Fourier series \( 7 + \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \cdot 80}{n \pi} \sin \left(\frac{n \pi x}{8}\right) \) at \( x = 5 \).

Answer: 32


Question: Let $u \in \mathcal{D}'(\mathbb{R}^d)$ be a distribution and $f \in C_c^\infty(\mathbb{R}^d)$ be a smooth function such that $f(x) = 1$ for $|x| \leq 1$ and $f(x) = 0$ for $|x| > 2$. Determine whether there exists a distribution $v \in \mathcal{D}'(\mathbb{R}^d)$ such that for any test function $\phi \in \mathcal{D}(\mathbb{R}^d)$, the following limit holds: \[ \lim_{n\to\infty} \langle u \ f(n\cdot),\phi\rangle = \langle v,\phi\rangle. \]

Answer: Yes


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function satisfying the equation \( f\left(\frac{x+y}{3}\right) = \frac{2 + f(x) + f(y)}{3} \) for all real numbers \( x \) and \( y \). Given that \( f'(2) = 2 \), determine the range of the function \( f(|x|) \).

Answer: [2, \infty)


Question:
Determine whether the following statement is true or false:

Let \( f:\mathbb{R}\to\mathbb{R} \) be a continuous function. For every non-negative sequence of real numbers \( x_n \) such that \( \lim_{n\to\infty}x_n=\infty \), the sequence \( f(x_n) \) has a convergent subsequence that converges to a real number. Then \( \lim_{x\to\infty}f(x)=L \in \mathbb{R} \).

Is this statement true or false?

Answer: B


Question: What is the remainder when \(45!\) is divided by \(47\)?

Answer: 1


Question: Using generating functions, determine the number of ways to choose $3n$ letters from the set \( \{a, b, c\} \) such that each letter is chosen at most $2n$ times.

Answer: 3n^2 + 3n + 1


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

Answer: 0


Question: Evaluate the limit of the recurrence relation defined by \( a_{n+1} = 1 + \frac{1}{1 + a_n} \) with initial condition \( a_1 = 1 \). Show that \( \lim_{n \to \infty} a_n = \sqrt{2} \).

Answer: \sqrt{2}


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{e^{2\sin x}\cosh x-1}{\sqrt[3]{x(\cosh x-\cos x)}}$$ without using L'Hospital's Rule.

Answer: 2


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0} \frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2} \]

Answer: \dfrac{11e}{24}


Question: Let X and Y be independent random variables, each uniformly distributed on the interval [0,1]. Find the probability density function of the random variable Z = XY.

Answer: -\ln z


Question: Evaluate the limit: \[ \lim_{n\to\infty} \int_0^1 \frac{nx^{n-1}}{x+1} \, dx \]

Answer: \dfrac{1}{2}


Question: Compute the limit: \[ \lim_{n\rightarrow\infty}e^{-n}\left(1+n+\frac{n^2}{2!}+\cdots+\frac{n^n}{n!}\right) \]

Answer: \dfrac{1}{2}


Question: Find the value of \( A = \lim_{x\to +\infty}\left(\frac{x}{x+1}\right)^x + \lim_{x\to +\infty}\left[\frac{x}{x+1}\right]^x \), where \([x]\) is the floor function of \( x \).

Answer: \dfrac{1}{e}


Question: Determine if there exists any integer \( n > 2 \) such that the Mertens function \( M(n!) = 0 \).

Answer: No


Question: Determine whether the following statement is true or false: There exists an infinite subset $S \subseteq \mathbb{R}^3$ such that any three vectors in $S$ are linearly independent.

Answer: True


Question: Determine the number of unique factorizations of the number 10010 into integers greater than 1. A unique factorization is defined as an unordered multiset of integers such that their product is 10010. For example, {10, 1001} is one such factorization. How many such unique factorizations exist?

Answer: 52


Question: Is it correct to say that a linear operator $T:V\to V$, where $\dim V=n$, is a rotational operator if there exists a basis $B$ of $V$ such that the matrix of $T$ in basis $B$, denoted as $R$, satisfies the conditions $R \cdot R^T = I_n$ and $\det(R) = 1$?

Answer: No


Question: Find the probability density function of the product of two independent random variables $X$ and $Y$, each with the density function $f(x) = 5x^4$ for $0 \leq x \leq 1$. Provide your answer in terms of the variable $z = XY$.

Answer: -25 z^4 \ln z


Question: If a set of natural numbers $A \subseteq \omega$ has positive upper Banach density, does it contain arithmetic sequences of arbitrary finite length?

Answer: Yes


Question: Calculate the pullback of the 1-form \( \omega = -\frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy \) under the transformation \( f: U \to \mathbb{R}^{2} \) given by \( f(r,\theta) = (r\cos\theta, r\sin\theta) \), where \( U = \{(r,\theta) \mid r>0,0<\theta<2\pi\} \). Express the result in terms of \( r \) and \( \theta \).

Answer: d\theta


Question: Given a spherical planet X, the angle $A$ from the center of the planet to the outer edge is 3 degrees, and the distance $d$ from your spaceship to the nearest surface of the planet is 100,000 miles. The radius $r$ of the planet is calculated as 5,522.6 miles. After a mishap, your spaceship travels directly away from planet X, and 5 hours later, the angle $B$ is measured as 2 degrees. Write an equation for $V$, the average velocity away from planet X, in terms of $r$, $t$, $A$, and $B$. Verify that $V = 10,544$ miles per hour using $r = 5522.6$, $t = 5$, $A = 3$, and $B = 2$. The equation is: $$V = \frac {r(\sin(A)-\sin(B))}{t\sin(A)\sin(B)}$$

Answer: 10544


Question: For which real number $a$ does the series \( \sum_{n=0}^\infty \frac{1}{u_n} \) converge, where the sequence \( \{u_n\} \) is defined recursively by \( u_0 = 2 \) and \( \int_{u_n}^{u_{n+1}} (\ln u)^a \, du = 1 \)?

Answer: a < -1


Question: Evaluate the integral \( \int_0^\infty \lim_{n \to \infty} \frac{\cos x}{1+(\arctan x)^n} \, dx. \)

Answer: \sin(\tan 1)


Question: Let \( \mathcal{H} = \mathbb{C}^2 \) and \( \mathcal{M}_1 = \mathbb{C}|0\rangle \). Given \( |\psi\rangle = \alpha |0\rangle + \beta|1\rangle \), show that the probability \( Pr(\mathcal{M}_1) = |\alpha|^2 \).

Answer: |\alpha|^2


Question: Consider a symmetric random walk on a regular hexagon with vertices labeled as $A, B, C, D, E, F$ and a center point $O$. The walk starts at vertex $A$. At each step, if the walk is at a vertex, it moves to one of its two neighboring vertices or the center $O$ with equal probability of $\frac{1}{3}$. Calculate the probability that the walk returns to vertex $A$ without passing through the center $O$ before its first return to $A$. Express your answer as a probability.

Answer: \dfrac{7}{27}


Question: Verify that the number of lines on a general quintic threefold $Y \subset \mathbb{P}^4$, which is classically known to be 2875, can be obtained as the Donaldson-Thomas invariant of the moduli space of lines in $Y$. Specifically, show that the integral of the virtual fundamental class of the moduli space $M(1,0,-1,c_3)$ of torsion-free sheaves with rank 1, trivial determinant, $c_2 = -1$, and $c_3 = \chi(\mathscr{O}_C)$ is 2875.

Answer: 2875


Question: Determine the number of solutions to the equation \(f(x) = x\), where \(f(x) = \left[\cos x \cdot \cos(x+2) - \cos^2(x+1) \right]\) and \([.]\) denotes the greatest integer function.

Answer: 1


Question: Consider a collection of 12 sets, each containing 8 elements. Every pair of sets shares 6 elements, any collection of 3 sets shares 4 elements, and no collection of 4 sets shares any elements. Determine the total number of distinct elements in this collection.

Answer: 580


Question: Identify the tensor product $H \otimes_{\mathbb{R}} \mathbb{C}$, where $H$ is the algebra of Hamiltonian quaternions, $\mathbb{C}$ is the field of complex numbers, and $\mathbb{R}$ is the field of real numbers. Express your answer in familiar mathematical terms.

Answer: M_2(\mathbb{C})


Question: Evaluate the integral \( \int_{0}^{e} f(x) \, dx \) given that \( f(x) e^{f(x)} = x \) for a differentiable real function \( f \).

Answer: e - 1


Question: Consider a $2 \times n$ table. Determine the number of ways to fill this table with numbers from 1 to $2n$ such that the numbers increase from left to right and from bottom to top. Express your answer in terms of the nth Catalan number.

Answer: C_n


Question: Determine whether the following statement is true or false: Peano Arithmetic (PA) has a model of order type \(\alpha\) for every ordinal \(\alpha\).

Answer: False


Question:
Let $Z \sim N(0,1)$ be a standard normal random variable. Calculate $E[\max\{Z,0\}]$. Choose the correct answer from the following options:

- $\frac{1}{\sqrt{\pi}}$
- $\sqrt{\frac{2}{\pi}}$
- $\frac{1}{\sqrt{2\pi}}$
- $\frac{1}{\pi}$

Answer: C


Question: Determine the exact interval of convergence of the power series \( \sum x^{n!} \).

Answer: (-1, 1)


Question: Determine the number of distinct terms in the expansion of \((1+x)^{100} + (1+x^2)^{100} + (1+x^3)^{100}\).

Answer: 201


Question: Evaluate the expression \( \Delta^3 0^6 \) using the forward difference operator \( \Delta \). Choose the correct answer from the following options: (a) 500  (b) 515 (c) 530 (d) 540.

Answer: D


Question: Determine the minimum number of times, \( n \), you need to toss a biased coin to ensure that the probability \( P(0.95p \leq \hat{p} \leq 1.05p) \geq 0.95 \), where \( \hat{p} = \frac{\hat{E}}{n} \) is the estimated probability of heads, and \( 0.5 \leq p \leq 0.6 \). Justify your answer.

Answer: 1537


Question: Find the volume of the finite region enclosed between the surface \( y = 1 - x^2 - 4z^2 \) and the plane \( y = 0 \).

Answer: \dfrac{\pi}{4}


Question: Suppose \( \pi:E\to M \) is a smooth \( S^{k-1} \)-bundle with structure group \( O(k) \). Is it true that there exists a vector bundle \( E'\to M \) with a Riemannian metric such that the unit sphere bundle of \( E' \) is \( E \)?

Answer: Yes


Question: Determine the number of solutions to the equation $x^2 - x\sin x - \cos x = 0$ for $x \in (-\infty, +\infty)$. Provide your answer as a single integer.

Answer: 2


Question:
Two guerilla forces, with troop strengths $x(t)$ and $y(t)$, are in combat without reinforcement. The combat loss rate for the $x$-force is proportional to $xy$, and the loss rate for the $y$-force is also proportional to $xy$. The differential equations governing the troop strengths are given by:
\[ \frac{dx}{dt} = -axy \]
\[ \frac{dy}{dt} = -bxy \]
where $a$ is the combat effectiveness of the $y$-force and $b$ is that of the $x$-force. Initially, $x_0$ is three times $y_0$. Determine the effectiveness ratio $\frac{a}{b}$ required for the $y$-force to stalemate its enemy.

Answer: 3


Question: Find an integer $x$ such that $x^2 \mod 1799832043$ is divisible by $67610791$. Use the values $n = 1799832043$ and $p = 67610791$.

Answer: 0


Question: Given the equation \(1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{23} = \frac{a}{23!}\), find the value of \(a \pmod{13}\).

Answer: 7


Question: Show that for Brownian motion $B(t)$, the expectation $E(B(t)-B(s))^4 = 3(t-s)^2$.

Answer: 3(t - s)^2


Question: Determine whether every homomorphism from the group \( \langle(0,+\infty),\,\cdot\,,{}^{-1}\rangle \) to the group \( \langle\{-1,1\},\,\cdot\,,{}^{-1}\rangle \) is a constant function. Specifically, must such a homomorphism \( f \) satisfy \( f(x) = 1 \) for all \( x \)?

Answer: Yes


Question: Let $U = [u_1 \ u_2 \ \cdots \ u_n]$ be an orthogonal matrix, where $u_i$ is the $i$-th column of $U$. Given a unit vector $x \in \mathbb{R}^n$, determine if the sum $\sum_{i=1}^n (x^T u_i)^2$ equals 1.

Answer: 1


Question: Calculate the surface area of the portion of the cone defined by $x^2 + y^2 = z^2$ that lies inside the sphere $x^2 + y^2 + z^2 = 6z$. What is the surface area of this ice cream cone?

Answer: 9\sqrt{2}\pi


Question: Let $X$ be a Banach space and $P: X \rightarrow X^{*}$ be a coercive function, meaning there exists a function $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $\lim\limits_{s \rightarrow +\infty}f(s) = +\infty$ and $\langle P(u),u \rangle \geq f(\Vert u \Vert)\Vert u \Vert$. Consider the translation of $P$ defined as $P_{T}(u) := P(u+w)$ for some $w \in X$. Is $P_{T}$ also coercive?

Answer: Yes


Question: Let $F_1$ and $F_2$ be reflections about two lines in the plane, with the angle between the lines being $\theta$. Determine the angle of rotation of the composition $F_1F_2$. Express your answer in terms of $\theta$. \( \boxed{} \)

Answer: 2\theta


Question:
Is the following affirmation correct based on Lemmas 3.7, 3.8, and Corollary 3.9 from Rick Miranda's "Algebraic Curves and Riemann Surfaces"?

For a point $p$ in a nondegenerate smooth curve $X \subset \mathbb{P}^n$ and a tangent hyperplane $H$ to $X$ at $p$, the hyperplane divisor $\operatorname{div}(H)$ can be:

1. $\operatorname{div}(H)=2\cdot p + q_3 + \cdots + q_d$ with all $q_i$ distinct and unequal to $p$, if $H$ is a general tangent;
2. $\operatorname{div}(H)=k\cdot p + q_{k+1} + \cdots + q_d$ with all $q_i$ distinct and unequal to $p$, if $p \in X$ is a flex point, i.e., $k \ge 3$;
3. $\operatorname{div}(H)=k_1\cdot p_1+ k_2\cdot p_2 + q_{k_1 + k_2 +1} + \cdots + q_d$ with all $q_i$ distinct and unequal to $p_1$ and $p_2$, if $H$ is bitangent to $X$.

Is this affirmation true?

Answer: True


Question: In the 22nd century (the years 2101 through 2200, inclusive), find the expected number of years that have either a record low or a record high rainfall, assuming that annual rainfalls are i.i.d. draws from a continuous distribution.

Answer: 2H_{100} - 1


Question: Consider a finitely generated Fuchsian group \( \Gamma \) of the first kind, which is a discrete subgroup of \( PSL_2(\mathbf{R}) \) acting on the upper half-plane and admitting a fundamental domain of finite hyperbolic area. Two fundamental domains \( D_1 \) and \( D_2 \) are said to be comparable if each is contained in a finite union of \( \Gamma \)-translates of the other. Is it true that any two Dirichlet domains for \( \Gamma \) are comparable?

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \frac{4k^3 e^{\frac{k^4}{n^4}}}{n^3} \).

Answer: e - 1


Question: Let \( \{A_i\}_{i\in S} \) be an open cover for a regular topological space \( X \). The family \( \{A_i\}_{i\in S} \) is called locally finite if for every point \( x\in X \) there exists a neighborhood \( U \) of \( x \) such that the set \( \{s\in S : U \cap A_s\neq \emptyset\} \) is finite. Suppose the family \( \{A_i\}_{i\in S} \) has the property that every point of \( X \) is contained in only finitely many of the \( A_i \)'s. Can we deduce that the family \( \{A_i\}_{i\in S} \) is locally finite?

Answer: No


Question: Given the equations \((x+y)(x+1)(y+1) = 3\) and \(x^3 + y^3 = \frac{45}{8}\), find the value of \(xy\).

Answer: -\dfrac{1}{2}


Question: Calculate the expectation \( \text{E}[X(X - 1) \ldots (X - k + 1)] \) where \( X \) is a Poisson-distributed random variable with parameter \( \lambda \), and \( k \in \mathbb{N} \) is fixed.

Answer: \lambda^k


Question: Given triangle \( \triangle ABC \) with vertex \( A(3, -1) \), the equation of median \( BE \) is \( 6x + 10y - 59 = 0 \), and the equation of angle bisector \( CF \) is \( x - 4y + 10 = 0 \). Determine the slope of line \( BC \).

Answer: -\dfrac{2}{9}


Question: Evaluate the improper integral: \[ \int_{-\infty}^{+\infty} \sin(2\pi x) e^{-x^2/2} \, dx \]

Answer: 0


Question: Let $X$ and $Y$ be smooth affine varieties over $\mathbb{C}$. Consider a dominant quasi-finite morphism $T: X \rightarrow Y$ with the induced map on coordinate rings $T^\#: \mathbb{C}[Y] \rightarrow \mathbb{C}[X]$. Let $D(X)$ denote the ring of differential operators on $X$. Determine whether the following statement is true: If $d \in D(X)$ satisfies $d(T^\#(f)) = 0$ for all $f \in \mathbb{C}[Y]$, then $d = 0$. Provide a justification for your answer.

Answer: True


Question: Let $M$ be a Poisson distributed random variable with mean 4, and let $X_1, X_2, \ldots$ be independent identically distributed copies of a Bernoulli random variable $X$ with $\mathbb{P}(X=1)=\frac{3}{4}$. Determine the generating function $\mathbb{E}\{z^{S}\}$ of $S=2\sum_{i=1}^{M}X_{i}$.

Answer: e^{3(z^2 - 1)}


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

Answer: \dfrac{1}{2}


Question: Compute the integral \( \int_\gamma \frac{dz}{z} \) where \( \gamma \) is the unit circle oriented counterclockwise.

Answer: 2\pi i


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{\sqrt[3]{n+1}-\sqrt[3]{n+\cos\frac{3}{n}}}{\sqrt[6]{n^2+\sin\frac{2}{n}}-\sqrt[3]{n}}. \]

Answer: \dfrac{9}{2}


Question: Let \( T: \mathbb{C} \to \mathbb{C} \) be a transformation defined by \( T(z) = az + b\bar{z} \), where \( a, b \in \mathbb{C} \). Prove that if \( |T(z)| = |z| \) for all \( z \in \mathbb{C} \), then \( ab = 0 \).

Answer: ab = 0


Question: If $f(x)$ is a smooth function from $\mathbb{R}$ to $\mathbb{R}$ and $f(0)=0$, is it true that $f(x)/x$ is smooth? Justify your answer.

Answer: Yes


Question: For any \(\varepsilon > 0\), does there exist a countable \(\varepsilon\)-net for the family of all Lebesgue measurable sets of the interval \([0,1]\) with respect to the metric \(d(A_1,A_2) = \int |1_{A_1} - 1_{A_2}|\)?

Answer: Yes


Question: Evaluate the contour integral \( \oint_{|z-\frac{1}{3}|=3} z \text{Im}(z)\text{d}z \).

Answer: -3\pi


Question: Evaluate the limit: \[ \lim_{n \to +\infty} \prod_{k=1}^{n} \frac{2k}{2k+1} \]

Answer: 0


Question: Let $\mathscr{B}$ be an open cover of a connected space $X$, where $\varnothing \notin \mathscr{B}$. For any $U, V \in \mathscr{B}$, does there exist a finite sequence $U, A_2, A_3, \ldots, A_{n-1}, V$ in $\mathscr{B}$ such that each pair of adjacent open sets in the sequence intersects?

Answer: Yes


Question:
Find the sum of the infinite series:
\[ 1 + \frac{1}{4} + \frac{1 \times 3}{4 \times 8} + \frac{1 \times 3 \times 5}{4 \times 8 \times 12} + \cdots \]

Answer: \sqrt{2}


Question: Compute the fundamental group of the quotient space $G/G_x$, where $G$ is a compact, connected, and simply connected Lie group, $\mathfrak{g}$ is its Lie algebra, and $x \in \mathfrak{g}^*$. Here, $G_x$ denotes the isotropy group of $G$. What is $\pi_1(G/G_x)$?

Answer: 0


Question:
Calculate the ratio \( \frac{I_1}{I_2} \) where:
\[ I_1 = \int_0^1 \frac{dt}{\sqrt{1-t^4}} \]
\[ I_2 = \int_0^1 \frac{dt}{\sqrt{1+t^4}} \]

Answer: \sqrt{2}


Question: Let $P(x) \in \mathbb{Z}[x]$ be a monic polynomial, and suppose $P(x) = Q(x)R(x)$ where $Q(x), R(x) \in \mathbb{Q}[x]$ are also monic polynomials. Is it true that $Q(x), R(x) \in \mathbb{Z}[x]$? Justify your answer.

Answer: Yes


Question: Let $ABC$ be a triangle with inradius $r$ and circumradius $R$. Construct triangle $A'B'C'$ such that $A'B'$ is perpendicular to $OC$ through $C$, and similarly for the other sides. Let $r_1$, $r_2$, and $r_3$ be the inradii of triangles $A'BC$, $AB'C$, and $ABC'$ respectively. Find the sum $r_1 + r_2 + r_3$ in terms of $r$ and $R$. Express your answer as a single expression.

Answer: 2R - r


Question: Find the generating function for the sequence \( \{c_n : n \geq 0\} \) where \( c_n \) is the number of solutions to the equation \( x_1 + x_2 + x_3 + x_4 = n \) with the constraints: \( x_1, x_2, x_3, x_4 \geq 0 \), \( x_2 \geq 2 \), \( x_3 \) is a multiple of 4, and \( 1 \leq x_4 \leq 3 \). Use the generating function to find a closed formula for \( c_2 \).

Answer: 0


Question:
Evaluate the limit: \[ \lim_{x \to 0} \frac{1}{x} \int_x^{2x} e^{-t^2} \, dt \] Choose the correct answer from the following options:

- Does not exist.
- Is infinite.
- Exists and equals to 1.
- Exists and equals to 0.

Answer: C


Question: Evaluate the integral \( \int_\gamma \frac{1}{(z-4)^2} \) where \( \gamma \) is the circle of radius 2 centered at \( 4+i \).

Answer: 0


Question: A bridge can tolerate a mass without structural damage that follows a normal distribution with mean 200,000 kg and standard deviation 20,000 kg. Car weights follow a normal distribution with mean 1,300 kg and standard deviation 130 kg. Determine the minimum number of cars needed on the bridge to cause structural damage with a probability greater than 0.1.

Answer: 135


Question: Evaluate the limit: $$\lim\limits_{h\to0} \frac{\tan(a+2h)-2\tan(a+h)+\tan a}{h}.$$

Answer: 0


Question: Let $k$ be an algebraically closed field, $C$ a non-singular projective curve over $k$ of genus at least $2$, and $\mathcal{F}$ a locally free sheaf on $C$. Let $r,d$ be two integers satisfying $\mathrm{gcd}(r,d)=1$ and $r>0$. Denote by $Q$ the Quot-scheme parametrizing quotients of $\mathcal{F}$ of rank $r$ and degree $d$. Is every irreducible component of $Q$ generically reduced?

Answer: Yes


Question: Determine the set of rational numbers \( \{y \in \mathbb{Q} \mid y = \cos(x), \ x \in [0, 2\pi] \cap \mathbb{Q} \} \).

Answer: \{1\}


Question: Find the last two digits of \(1717^{1717}\).

Answer: 77


Question:
Consider a $5\times5$ matrix $A$ with real entries and a $5\times1$ vector $x$. Determine the correct statement about the set $\{Ax, A^2x, A^3x, A^4x, A^5x\}$. Choose the correct option:

(a) Linear dependence/independence cannot be determined without knowing the whole matrix $A$.
(b) The vectors are linearly dependent.
(c) The vectors are linearly independent if and only if $A$ is symmetric.
(d) The vectors are linearly independent if and only if $A$ is positive definite.
(e) The vectors are linearly independent.

Answer: B


Question: Find the minimum value of \( \overline{AP} + \overline{AQ} \), where \( A(-3,2) \) is a point on the coordinate plane. A line \( L \) is perpendicular to the line \( x = y \). Line \( L \) intersects the curve \( y = 2^x \) at point \( P \) and the curve \( y = \log_2 x \) at point \( Q \). Determine the minimum value of the sum of the distances from \( A \) to \( P \) and \( Q \).

Answer: 5\sqrt{2}


Question: Determine the number of solutions to the equation \( \sin^5 x + \cos^3 x = 1 \) in the interval \((0, 2\pi]\).

Answer: 2


Question: Evaluate the line integral \( \oint_C u \, \mathrm{d}y + v \, \mathrm{d}x \) over the simple closed curve \( |z| = 3 \), where \( u \) and \( v \) are the real and imaginary parts of the complex function \( \frac{1}{z^2 - 6z + 8} \) with \( z = x + iy \).

Answer: -\pi


Question: Determine whether an isogeny between two principally polarized abelian varieties that respects the polarization is necessarily an isomorphism.

Answer: Yes


Question: Determine all complex numbers \(z\) that satisfy the equation \(\operatorname{Log}(z^2) = \operatorname{Log}(z)\).

Answer: 1


Question:
Determine the number of rational non-integer solutions to the equation \(x + \frac{87}{x} = \lfloor x \rfloor + \frac{87}{\lfloor x \rfloor}\). Choose the correct answer from the following options:

1) Zero
2) One
3) Two
4) Eighty-seven
5) Infinity

Answer: B


Question: Calculate the line integral \( \int_{\gamma} |z|^2 \, dz \) over the ellipse defined by \( \gamma: [0,2\pi] \rightarrow \mathbb{C}, \, t \mapsto a\cos(t) + ib\sin(t) \), where \( a, b \in \mathbb{R}_{>0} \).

Answer: 0


Question: Calculate the limit: $$ \lim_{x \to 2} \frac{1-\sqrt{1-\sqrt{4x-8}}}{1-\sqrt{1-\sqrt{\frac{x-2}{x+2}}}} $$ without using L'Hopital's rule.

Answer: 4


Question: Let $E$ be an elliptic curve with complex multiplication (CM) by an order $\mathcal{O}$ in an imaginary quadratic field $K$. Choose a basis for $E[N]$ to obtain an isomorphism $\operatorname{Aut}(E[N]) \cong \operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$. The complex multiplication on $E$ induces a homomorphism $(\mathcal{O}/N\mathcal{O})^\times \rightarrow \operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$, and let $C_N$ be the image. Determine if $C_N$ is equal to its centralizer in $\operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$. Provide a justification for your answer.

Answer: Yes


Question: If \( p \) is a prime number, \( m \) is a composite and deficient number, and \( I(p) < I(m) \) with \( p \nmid m \), does it follow that \( m < p \), where \( I(x) = \frac{\sigma(x)}{x} \) for \( x \in \mathbb{N} \)?

Answer: No


Question:
Let $G$ be a permutation group on a finite set $\Omega$ with orbits $\Omega_1,\ldots,\Omega_k$. A transversal is a set $\{\omega_1,\ldots,\omega_k\}$ with $\omega_j\in\Omega_j$ for each $j$. For a transversal $\tau=\{\omega_1,\ldots,\omega_k\}$ and subset $H\subseteq G$, define
\[H(\tau) = \bigcup_{h\in H} \{\omega_1^h,\ldots,\omega_k^h\}. \]
Let $N(G)$ be the smallest integer such that for some transversal $\tau$ and some $H\subseteq G$ with $|H|=N$, we have $H(\tau)=\Omega$. Given $n=|\Omega|$ and $m=\max_{i=1}^k |\Omega_i|$, determine $N(G)$. It is known that $N(G)\ge m$ and $N(G)\le m\log n$. What is $N(G)$?

Answer: m


Question: Find the value of \( L = \lim_{x \to 0}\frac{1 - \cos x \cos 2x \cos 3x}{x^2} \) if it exists.

Answer: 7


Question: In a right pyramid with all edges equal and a height that intersects the center of its base, if the base is an isosceles triangle, does the height of the pyramid intersect the base at the point where the perpendicular bisectors of the triangle meet?

Answer: Yes


Question: Determine whether the inequality \( {\left| {{{(Ax)}^*}(By)} \right|^2} \le {(Ax)^*}(Ax)\cdot{(By)^*}(By) \) holds for all matrices \( A, B \in M_n \) and vectors \( x, y \in \mathbb{C}^n \).

Answer: Yes


Question: Let $S$ be a set such that for every countable family \(\{U_{i}\}_{i\in \mathbb{N}}\) of subsets of $S$ where \(S=\bigcup_{i\in \mathbb{N}}U_{i}\), there exists a finite set \(I\subset \mathbb{N}\) such that \(S=\bigcup_{i \in I}U_{i}\). Does it follow that $S$ is finite?

Answer: Yes


Question: Determine the number of possible Beta-reductions for the lambda calculus expression \((\lambda x.\lambda y.y)(\lambda x.x)((\lambda x.x)(\lambda y.y))\). Consider the order of evaluation in your calculation.

Answer: 3


Question: Find a constant \( C > 0 \) such that for Hardy's integral operator \( T \) defined on \( L^1(0, +\infty) \) by \( Tf(x) = \frac{1}{x}\int_0^x f(t) \, dt \), the inequality \( \lambda m(|Tf| > \lambda) \leq C{\Vert f \Vert}_1 \) holds for all \( \lambda > 0 \) and \( f \in L^1(0, +\infty) \), where \( m \) denotes the Lebesgue measure.

Answer: 1


Question: Find the scalar value of \( a \) such that the vector set \( \{u_1+au_2, u_2+au_3, u_3+au_4, u_4+au_1\} \) is linearly independent, given that the vector set \( \{u_1, u_2, u_3, u_4\} \) is linearly independent.

Answer: a \neq \pm 1


Question:
Consider the Cauchy problem given by the partial differential equation \( au_x + bu_y = e^{x+y} \) for \( x, y \in \mathbb{R} \), with the initial condition \( u(x,y) = 0 \) on the line \( cx + dy = 0 \), where \( a, b, c, d \in \mathbb{R} \) and \( c^2 + d^2 \neq 0 \). Determine the condition under which this problem has a unique solution:

(A) \( ac + bd \neq 0 \).
(B) \( ad - bc \neq 0 \).
(C) \( ac - bd \neq 0 \).
(D) \( ad + bc \neq 0 \).

Using Lagrange's equations, we find that \( bx - ay = C_1 \) and \( u - \frac{a}{a+b}e^{x+y} = C_2 \). The solution is \( u(x,y) = \frac{a}{a+b}e^{x+y} + \phi(bx-ay) \). Determine the correct condition for uniqueness.

Answer: A


Question:
Consider the function \( f(x,y) = \begin{cases} 1, & \text{if } xy = 0 \\ xy, & \text{if } xy \neq 0 \end{cases} \). Determine which of the following statements is true:

(A) \( f \) is continuous at \((0,0)\) and \( \frac{\partial f}{\partial x}(0,0) \) exists.

(B) \( f \) is not continuous at \((0,0)\) and \( \frac{\partial f}{\partial x}(0,0) \) exists.

(C) \( f \) is continuous at \((0,0)\) and \( \frac{\partial f}{\partial x}(0,0) \) does not exist.

(D) \( f \) is not continuous at \((0,0)\) and \( \frac{\partial f}{\partial x}(0,0) \) does not exist.

Answer: B


Question: Find the variance of the stochastic process \(X_{t} = \int_{0}^{t} W^{+}_{s}dW_{s}\), where \(W_{t}\) is a Wiener process and \(a^{+} = \max(a, 0)\).

Answer: \dfrac{t^2}{4}


Question: Find the spectrum \( \sigma(A) \) of the operator \( A \) defined on the Banach space \( X = C([0,1],\mathbb{R}) \) of continuous real functions on \([0,1]\) with the supremum norm, where for each \( x \in X \), \((Ax)(t) = \int_0^t x(s) \, ds\) for all \( t \in [0,1] \).

Answer: \{0\}


Question: Determine the values of \(\alpha\) for which the integral \(\int_0^1(-\ln x)^\alpha \, dx\) converges.

Answer: \alpha > -1


Question:
Determine the nature of the ring $R = \mathbb{R}[x]/\langle x^4+4 \rangle$. Choose the correct option:
(a) $R$ is a field.
(b) $R$ is an integral domain, but not a field.
(c) $R$ is not an integral domain, but has $0$ as the only nilpotent element.
(d) $R$ is a ring which contains non-zero nilpotent elements.

Answer: C


Question: Let $B \subset A$ and $f: A \rightarrow B$ be an injective function. Define the set \( C = \bigcup_{n \geq 0} f^n(A-B) \), where \( f^0 = \mathrm{id}_A \) and \( f^n(x) = f(f^{n-1}(x)) \). Is it true that \( C = A \)? Justify your answer.

Answer: No


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

Answer: S_3


Question: Find the smallest integer value of \(a\) such that the function \(f(x) = \int_{0}^{x} |\log_{2}(\log_{3}(\log_{4}(\cos t + a)))| \, dt\) is increasing for all \(x \in \mathbb{R}\).

Answer: 6


Question:
Given the transfer function of a system \( \frac{V_o(s)}{V_i(s)} = \frac{1-s}{1+s} \), and the output \( v_o(t) = V_m \sin(\omega t + \phi) \) for the input \( v_i(t) = V_m \sin(\omega t) \), determine the minimum and maximum values of \( \phi \) (in radians). Choose the correct option:

- \( \frac{-\pi}{2} \) and \( \frac{\pi}{2} \)
- \( \frac{-\pi}{2} \) and \( 0 \)
- \( 0 \) and \( \frac{\pi}{2} \)
- \( -\pi \) and \( 0 \)

Answer: D


Question: Consider two smooth Morse functions \( \varphi_{1}, \varphi_{2}: \mathbb{S}^{1} \rightarrow \mathbb{R} \) with the same set of critical points \( \{ p_{1}, ..., p_{n} \} \subset \mathbb{S}^{1} \), where \( n \) is even, and both have a local maximum at \( p_{1} \). These functions are similar in the sense that for any \( i \neq j \), \( (\varphi_{1}(p_{i}) - \varphi_{1}(p_{j}))(\varphi_{2}(p_{i}) - \varphi_{2}(p_{j})) > 0 \). Consider the Dirichlet problems \( \Delta u = 0 \) with boundary conditions \( u|_{\mathbb{S}^{1}} = \varphi_{i} \) for \( i = 1, 2 \), resulting in harmonic solutions \( u_{1}, u_{2}: \mathbb{B}^{2} \rightarrow \mathbb{R} \). Is it true that the level lines of \( u_{1} \) and \( u_{2} \) are topologically equivalent, i.e., there exists a homeomorphism \( h: \mathbb{B}^{2} \rightarrow \mathbb{B}^{2} \) fixing all \( p_{i} \) and mapping the level lines of \( u_{1} \) onto those of \( u_{2} \)?

Answer: Yes


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

Answer: \dfrac{\pi}{4}


Question: Determine the number of Cayley tables that can be constructed for the set \( \{0, 1, \ldots, n-1\} \).

Answer: n^{n^2}


Question: Evaluate the Legendre symbol \( \left( \frac{2017}{5003} \right) \).

Answer: 1


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

Answer: |z| < 1


Question: If $R$ is a finite integral domain, is it necessarily commutative?

Answer: Yes


Question: Find the volume of the solid obtained by revolving the region bounded by the curve \( f(x) = \sqrt{\frac{(x+1)}{x^3}} \) and the \( x \)-axis over the interval \([1, \infty)\) about the \( x \)-axis, or state that the volume does not exist.

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


Question: Given the functions \( f, g : \mathbb{R} \to \mathbb{R} \) such that \( f(x + g(y)) = -x + y + 1 \) for each pair of real numbers \( x \) and \( y \), find the value of \( g(x + f(y)) \).

Answer: -x + y - 1


Question: Provide an example of an ideal in the ring \(\mathbb{Z}[x]\) that is not principal.

Answer: (2, x)


Question: Let \( P \in \mathbb{R}[X_1,\dots,X_n] \) be a non-constant multivariate polynomial with real coefficients. Consider the map \( P: \mathbb{R}^n \to \mathbb{R} \). Prove or disprove that the Lebesgue measure of the set \( P^{-1}(\{0\}) \) is zero.

Answer: 0


Question: Evaluate the double integral \( \iint\limits_D (10x^2+39xy+14y^2)^2 \, dx \, dy \) over the region \( D \) bounded by the lines \( 2x+7y=1 \), \( 2x+7y=-1 \), \( 5x+2y=3 \), and \( 5x+2y=-3 \).

Answer: \dfrac{12}{31}


Question: For any real number \( t \) between -1 and 1, does there exist an infinite sequence of positive integers \( x_n \) such that \( \lim_{n \to \infty} \sin(x_n) = t \)?

Answer: Yes


Question: Determine the independence number of the Hamming graph \( H(d, q) \), which is the Cartesian product of \( d \) complete graphs \( K_q \).

Answer: q^{d-1}


Question: Evaluate the infinite series: \[ \sum_{r=1}^{\infty} \frac{r^2}{(-5)^{r-1}} \]

Answer: \dfrac{25}{54}


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

Answer: 2


Question: Determine the cardinality of the image of the group homomorphism \( \varphi : SL_2 (\Bbb Z) \longrightarrow SL_2 (\Bbb Z/ 3 \Bbb Z) \) defined by \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \begin{pmatrix} \overline {a} & \overline {b} \\ \overline {c} & \overline {d} \end{pmatrix} \).

Answer: 24


Question: Find the general formula for the number of subwords of the form \((ba)^m\) in a word of the form \((ba)^n\).

Answer: n - m + 1


Question: Let $\phi$ be a non-negative test function with total integral equal to 1 and support in the ball of radius 1 centered at the origin in $\mathbb{R}^{n}$. Define $\phi_{t}(x) = t^{-n}\phi(x/t)$. Determine if the limit superior \(\limsup_{t\rightarrow0}\int_{\mathbb{R}^{n}}|\Delta\phi_{t}(y)|dy\) is bounded, where \(\Delta\) denotes the Laplacian.

Answer: \infty


Question: Solve the equation \( \ln x + x = 1 \) for \( x \).

Answer: 1


Question: Solve the equation \( \lfloor x \rfloor ^2 + 9 \{x\}^2 = 6x - 10 \) for all \( x \in \mathbb{R} \).

Answer: \dfrac{10}{3}


Question: If $U(ij)$ is isomorphic to $U(i) \oplus U(j)$, does this imply that $i$ and $j$ are relatively prime?

Answer: Yes


Question: Determine whether the quotient field of $R[a]$ is equal to $R(a)$, given the definitions: $R[x]$ is the polynomial ring with coefficients from $R$, $R[a] = \{f(a): f \in R[x]\}$ for $a \in S$, and $R(a)$ is the smallest field containing $a$ and $R$. Assume $S$ is a field.

Answer: Yes


Question: Determine the conditions on \(z \in \mathbb{C}\) under which the product \(|\prod_{k=1}^n(1-z^k)|\) tends to infinity as \(n\) approaches infinity.

Answer: |z| > 1


Question: Compute the limit without using L'Hospital's rule or series expansion: \[ \lim_{x\to 0}\frac{e^x -1-x-\frac{x^2}{2}}{x^3} \]

Answer: \dfrac{1}{6}


Question: Find the exact solution for the integral equation: \[ u(x) + \int_0^{2\pi} \cos(x+t)u(t)\,\mathrm{d}t = (\pi+1)\cos x. \]

Answer: \cos x


Question: Given the solution to the differential equation \(x(t) = Ae^{k_1t} + Be^{-k_2t}\), where \(k_1 > 0\) and \(-k_2 < 0\), find the relationship between the initial conditions \(x(0) = x_0\) and \(\dot{x}(0) = y_0\) that ensures \(x(t) \to 0\) as \(t \to \infty\).

Answer: y_0 = -k_2 x_0


Question: If every closed $G_\delta$-set in a topological space $X$ is a zero-set, does it follow that $X$ is a normal topological space?

Answer: No


Question: Calculate the average value of the function \( f(x,y,z) = x^2 + y^2 + z^2 \) over the region defined by \( |x| + |y| + |z| \leq 1 \).

Answer: \dfrac{3}{10}


Question: Find an integer $x$ such that $x^{677} \equiv 3 \pmod{2020}$. Provide your answer as the smallest non-negative integer that satisfies this congruence.

Answer: 543


Question: Find the remainder when \(2012^{2013} + 2013^{2012}\) is divided by \(2012 \times 2013\).

Answer: 4025


Question: Find an explicit formula for the recurrence relation \(a_n = n(a_{n-1} + a_{n-2})\) with initial conditions \(a_0 = 1\) and \(a_1 = 2\).

Answer: (n + 1)!


Question: Calculate the limit: \[ \lim_{n\to\infty}\int_{0}^{\infty} \frac{\sqrt x}{1+ x^{2n}} \, dx \]

Answer: \dfrac{2}{3}


Question: Does an automorphism of a group always map subgroups to subgroups of the same order?

Answer: Yes


Question: Find the residue of the function \( f(z) = \frac{1}{z^2(e^z-1)} \) at \( z = 0 \).

Answer: \dfrac{1}{12}


Question: Find the minimum value of \( P = x^2 + y^2 + \frac{x^2y^2}{(4xy-x-y)^2} \) given that \( \frac{1}{3} < x \le \frac{1}{2} \) and \( y \ge 1 \).

Answer: \dfrac{9}{4}


Question: Find the remainder when \(12^{39} + 14^{39}\) is divided by 676.

Answer: 0


Question:
Find the point-wise limit of the sequence of functions \( f_n(x) \) defined on \([0,1]\) by
\[
f_n(x)=\begin{cases}
n^2x,  & \text{if } 0 \leq x \leq \frac{1}{n} \\
-n^2x+2n,  & \text{if } \frac{1}{n} \leq x \leq \frac{2}{n} \\
0, & \text{if } \frac{2}{n} \leq x \leq 1
\end{cases}
\]
as \( n \to \infty \).

Answer: 0


Question: Let $T: X \to Y$ be a linear operator between normed spaces $X$ and $Y$. If $T$ is bounded on the unit sphere $S[0,1] = \{x \in X : \|x\| = 1\}$, does it follow that $T$ is also bounded on the closed unit ball $B[0,1] = \{x \in X : \|x\| \le 1\}$? Justify your answer.

Answer: Yes


Question: Let $a, b \in \mathbb{R}$ such that $a/b$ is irrational. Consider the continuous function $f : \mathbb{R} \to S^1 \times S^1$ defined by $f(t) = (e^{2\pi iat}, e^{2\pi ibt})$ for all $t \in \mathbb{R}$. Is it true that $f(\mathbb{R})$ is dense in $S^1 \times S^1$?

Answer: Yes


Question: Find the sum of the series: \( \sum_{x=0}^{\infty}\frac{x^2(1/2)^xe^{-1/2}}{x!} \).

Answer: \dfrac{3}{4}


Question: Two particles start at the $x$-intercepts of a circle with a circumference of $12$ units, centered at the origin. Each second, both particles move independently one unit either clockwise or counterclockwise with equal probability. What is the expected number of seconds until the two particles simultaneously occupy the same point on the circle for the first time?

Answer: 18


Question:
Determine the spectrum of the weighted right shift operator \(T\) on \(\ell^{\infty}(\mathbb{N})\), defined by:
\[ T(a_1,a_2,a_3,...) = \left(0, \frac{a_1}{1}, \frac{a_2}{2}, \frac{a_3}{3}, ...\right) \]

Answer: \{0\}


Question: Consider a tetrahedron ABCD with the areas of triangles ABC, BCD, ACD, and ABD being $k, l, m, n$ respectively. The volume of the tetrahedron is \( \frac{1}{6} \) cubic units. If each face ACB, ACD, and BCD of the tetrahedron subtends a right angle at C, find the geometric mean of $k, l, m$. \( \frac{1}{3}(\text{base})(\text{height})=\frac{1}{6} \).

Answer: \dfrac{1}{2}


Question: Determine the completion of the function field $F_q(t)$ with respect to the degree valuation, where the valuation is given by $v(f/g) = \deg(f) - \deg(g)$. What is this completion?

Answer: \mathbb{F}_q((t^{-1}))


Question: Let \( A = \{ z \in \mathbb{C} : z^{18} = 1 \} \) and \( B = \{ z \in \mathbb{C} : z^{48} = 1 \} \). Find the number of distinct elements in the set \( C = \{ x \times w : x \in A \text{ and } w \in B \} \).

Answer: 144


Question: Determine whether the following statement is true or false: A group $G$ is finite if and only if the order of any element divides a fixed finite number $n$. Justify your answer.

Answer: False


Question: Evaluate the limit: $$ \lim_{n\to\infty}\frac{3^{3n} \binom{2n}{n}^2}{\binom{3n}{n} \binom{6n}{3n}} $$

Answer: 2


Question: Find all positive real numbers \(x_1, x_2, x_3, \ldots, x_q\) for integers \(q \geq 4\) such that the equation \(x_n^2 = 9x_{n+1} + 10x_{n+2} + 11x_{n+3}\) holds for all \(n = 1, 2, 3, \ldots, q\), with indices taken modulo \(q\).

Answer: 30


Question: Determine the order of the pole of the function \( f(z) = \frac{e^z}{(z+1)^3(z-2)} \) at \( z_0 = 2 \).

Answer: 1


Question: Let $a$ and $b$ be real numbers satisfying $a^3 - 3ab^2 = 47$ and $b^3 - 3a^2 b = 52$. Find the value of $a^2 + b^2$. \( \)

Answer: 17


Question: In how many ways can a coin be flipped 25 times in a row such that exactly 5 heads occur and no more than 7 tails occur consecutively?

Answer: 17892


Question: Find the limit: \( \lim _{x \to 0} \frac {x \cos x - \sin x} {x^2 \sin x} \) without using l'Hopital's rule or Taylor expansion.

Answer: -\dfrac{1}{3}


Question: Determine the values of \(x\) and \(y\) for which the series \(\sum_{k=1}^{\infty}\frac{k^x}{\ln(1+k^y)\ln(1+k^{-y})}\) converges.

Answer: x + |y| < -1


Question: Find all the Fritz-John points for the optimization problem: \( \min x+y^2 \) subject to \( x^2-4x+4 \leq 0 \).

Answer: (2, y)


Question: Evaluate \( \iint \text{curl} (y\,\mathbf{i}+2\,\mathbf{j})\cdot \mathbf{n}\; d\sigma \) using Stokes' Theorem, where \( \sigma \) is the surface in the first octant made up of the plane \( 2x+3y+4z=12 \), and the triangles in the \((x,z)\) and the \((y,z)\) planes.

Answer: -12


Question: Evaluate the integral \( \iint_D \sin(x^2+y^2)~dA \) by transforming it to polar coordinates, where \( D \) is the region in the first quadrant bounded by the lines \( x=0 \) and \( y=\sqrt{3}x \) and the circles \( x^2+y^2=2\pi \) and \( x^2+y^2=3\pi \).

Answer: \dfrac{\pi}{6}


Question: Let $f:X\rightarrow X$ be a contractive mapping of a complete metric space satisfying \[d(f(x),f(y))\leq\alpha(d(x,y))d(x,y)\] where $\alpha:\mathbf{R}^+\rightarrow [0,1)$, and $\alpha(t_n)\rightarrow 1$ implies $t_n\rightarrow 0$. Does $f$ have a fixed point?

Answer: Yes


Question: Suppose \( u_n \) is a sequence of harmonic functions that converges weakly to \( u \) in \( L^2(\Omega) \). Prove that \( \Delta u = 0 \) in \( \Omega \).

Answer: \Delta u = 0


Question: A circular sector with radius \( R \) and an opening angle of \( \frac{\pi}{2} \) is rotated around its axis of symmetry (the x-axis) to form a homogeneous body. Determine the position of the center of gravity of this body.

Answer: \dfrac{3R}{8}


Question: Given that \( \cos (A-B) + \cos (B-C) + \cos (C-A) = -\frac{3}{2} \), find the value of \( \cos A + \cos B + \cos C \).

Answer: 0


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a convex function. Prove that there exists a linear function $g(y) = ay + b$ such that $f(y) \geq g(y)$ for all $y \in \mathbb{R}$ and $f(x) = g(x)$ for some $x \in \mathbb{R}$. Consider the case where $f$ is not differentiable.

Answer: g(y) = ay + b


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{x_n^2y_n}{3x_n^2-2x_ny_n+y_n^2} \] where \((x_{n})_{n\ge1}\) and \((y_{n})_{n\ge1}\) are sequences of real numbers that both converge to 0.

Answer: 0


Question: Determine the number of cyclic subgroups of order 12 in the group \( \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z} \).

Answer: 8


Question: Determine the number of ways to distribute 17 red balls and 8 blue balls into up to 5 groups such that each group contains more red balls than blue balls.

Answer: 45223


Question: Determine the structure of the quotient group \((\mathbb{Z}\times\mathbb{Z})/\langle (0,1)\rangle\). Prove that this quotient is isomorphic to \(\mathbb{Z}\).

Answer: \mathbb{Z}


Question: Consider a subset $S$ of a group $(G, \cdot)$. For any $g \in G$, the coset $gS$ is defined as $gS = \{g \cdot s \mid s \in S\}$. Let $C(S)$ be the collection of all cosets of $S$. If the binary operation $\times : C(S) \times C(S) \to C(S)$ is defined by $aS \times bS = (a \cdot b)S$, determine whether $S$ must be a subgroup of $G$ for this operation to be well-defined when $S$ has more than one element.

Answer: Yes


Question:
Given a function $f(n)$ that satisfies the following conditions:
1. $f(ab) = f(a) + f(b)$ for all positive integers $a$ and $b$.
2. $f(n) = 0$ if $n$ is a prime greater than 10.
3. $f(1) < f(243) < f(2) < 11$.
4. $f(2106) < 11$.

Determine the value of $f(96)$. Note that $f(243) = f(3^5) = 5f(3)$ and $2106 = 2 \cdot 3^4 \cdot 13$. Use these properties to find $f(96)$.

Express your answer as a single integer.

Answer: 31


Question: Given that \( \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \), find the value of \( \lambda \) such that \( \int_{0}^{1} \frac{\ln x}{1-x^2} \, dx = -\frac{\pi^2}{\lambda} \).

Answer: 8


Question: Find the power series representation of the function \( f(z) = \frac{3iz - 6i}{z - 3} \) about \( z_0 = 1 \). Determine the radius of convergence of this series.

Answer: 2


Question:
Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function and let \( S \) be a non-empty proper subset of \( \mathbb{R} \). Which one of the following statements is always true? (Here \( \overline{A} \) denotes the closure of \( A \) and \( \operatorname{int}A \) denotes the interior of \( A \).)

A. \( \operatorname{int}f(S) \subseteq f(\operatorname{int}S) \)
B. \( f(\overline{S}) \subseteq \overline{f(S)} \)
C. \( f(\overline{S}) \supseteq \overline{f(S)} \)
D. \( \operatorname{int}f(S) \supseteq f(\operatorname{int}S) \)

Select the correct statement.

Answer: B


Question: Evaluate the integral \( \int_{\gamma} \frac{1}{z} \, dz \), where \( \gamma \) is the ellipse \( x^2 + 4y^2 = 1 \) traversed once with positive orientation.

Answer: 2\pi i


Question: Determine the maximum number of negative eigenvalues that the matrix $AB + BA$ can have, where $A$ and $B$ are symmetric positive definite matrices in $\mathbb{R}^{n \times n}$. Can this number exceed $\lfloor n/2 \rfloor$?

Answer: \lfloor n/2 \rfloor


Question: Evaluate the integral \( \int_0^{2\pi} \ln(2 R \sin(\frac{\theta}{2})) \, d\theta \) using contour integration, where \( R > 0 \).

Answer: 2\pi \ln R


Question:
An insurance product with a per loss limit of 200 covers losses from an exponential distribution with parameter $\theta$. Based on the following table, the maximum likelihood estimate of $\theta$ is 168.

$$\begin{array}{|c | c | c |}\hline
\text{Size of loss} & \text{Number of claims} & \text{Sum of losses} \\ \hline
\text{Less than  } 200 & 1,114 & 142,752\\ \hline
\text{At least  } 200 & N & 200N \\ \hline
\end{array}$$

Calculate $N$ for the table above.

Answer: 222


Question: Find the limit: \[ \lim_{x \to \infty} \frac{(x+3)^x}{\sum_{n=3}^{x+2}n^x} \]

Answer: e - 1


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

Answer: 2


Question: If $A$ is a convex body and $A$ is not a convex polytope, does $A$ have infinitely many extreme points?

Answer: Yes


Question: Is the area of the fundamental parallelogram of the lattice generated by \((n, n\alpha \mod 1)\) always 1, even if \(\alpha = \sqrt{3}\) or another number?

Answer: 1


Question: Consider the heat operator $H_t = e^{-t\Delta}$, where $\Delta$ is the Laplacian on $\mathbb{R}^n$. Determine if there exists an eigenfunction of $H_t$ that is not an eigenfunction of $\Delta$. Provide a yes or no answer.

Answer: No


Question: Find \( \lim \int f_n \, d\lambda \) where \( f_n = (\frac{1}{n})\chi_{[n, +\infty)} \), \( X = \mathbb{R} \), \( \textbf{X} = \textbf{B} \), and \( \lambda \) is the Lebesgue measure on \( \textbf{X} \).

Answer: \infty


Question: Determine the order of the pole at \( z = \frac{\pi}{2} \) for the function \( f(z) = \frac{1}{(2\log(z))(1-\sin(z))} \).

Answer: 2


Question: Let \( f \in L^2[0,1] \) and define \( g:[0,1]\to[0,1] \) by \( g(x)=\sqrt{x} \). Is the composition \( f \circ g \) necessarily in \( L^2[0,1] \)?

Answer: Yes


Question: Calculate the Lebesgue integral of the function \( f(x) = x^2 \) over the interval \([0, 1]\).

Answer: \dfrac{1}{3}


Question: Find a bijection \( \phi: [n] \times S_{n-1} \to S_n \) and conclude that \( |S_n| = n! \), where \([n] = \{1, 2, \ldots, n\}\).

Answer: n!


Question: Determine whether the statement "A normal domain is equivalent to an integrally closed domain" is true or false.

Answer: A


Question: Find all positive integer pairs \((m, r)\) that satisfy the equation \[ m = \dfrac{1}{18}\left(\sqrt{48r^2+1}-1\right). \]

Answer: (75, 195)


Question: Given that $(X,Y)\sim N(\mu_x=1,\mu_y=1,\sigma^2_x=4,\sigma^2_y=1,\rho=1/2)$, compute the probability $P(X+2Y\leq 4)$. Provide your answer to four decimal places.

Answer: 0.6136


Question: Find the integrating factor for the differential equation: \[ xdy - 3ydx = \frac{x^4}{y}dy \]

Answer: \dfrac{1}{x^4}


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

Answer: 0


Question: Find the sum of the series \( \sum_{n=1}^\infty (-1)^n \frac{n^2}{3^n} \).

Answer: -\dfrac{3}{32}


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

Answer: \tau(m)


Question:
Find the coefficient of \(x^{25}\) in the expansion of the generating function:
\[
({{x}^{48}}-6{{x}^{40}}+15{{x}^{32}}-20{{x}^{24}}+15{{x}^{16}}-6{{x}^{8}}+1)\cdot {{\left( \frac{1}{1-x} \right)}^{6}}
\]

Answer: 14412


Question: Given a subset of $n$ vectors in the vector space $\mathbb{C}^n$, form an $n \times n$ matrix $A$. If $\det(A) = 0 + bi$, where $b \in \mathbb{R} \setminus \{0\}$, does this imply that the $n$ vectors are linearly independent when considering $\mathbb{C}^n$ as a vector space over the field $\mathbb{R}$?

Answer: Yes


Question: Let $A$ be a Hermitian matrix and $B$ be a skew-Hermitian matrix. Prove that the trace of the product $AB$ is zero.

Answer: 0


Question: Evaluate the surface integral \( \iint_S \underline{F} \cdot d \underline{a} \) for the vector field \( \underline{F} = (y+z, x+z, y+x) \). The surface \( S \) is the boundary of the cube defined by the planes \( x=0, y=0, z=0, x=1, y=1, z=1 \) with outward normal. Compute the integral by considering the contributions from each face of the cube.

Answer: 0


Question: Is the norm on a unital semi-simple commutative Banach algebra with \(\|I\|=1\) unique?

Answer: Yes


Question: Determine the greatest possible value of $a + b + c$ for which there exist at least two values of $n$ such that $C_n - B_n = A_n^2$. Here, for a positive integer $n$ and nonzero digits $a$, $b$, and $c$, $A_n$ is the $n$-digit integer with each digit equal to $a$, $B_n$ is the $n$-digit integer with each digit equal to $b$, and $C_n$ is the $2n$-digit integer with each digit equal to $c$. What is the greatest possible value of $a + b + c$?

Answer: 18


Question: Given linear operators $X$ and $P$ on a complex vector space $V$, suppose the commutator $[X,P] = ciI$ for some real, positive constant $c$. What can be concluded about the dimension of $V$?

Answer: \infty


Question: Suppose $G$ is a graph where every vertex has degree 3, and there is no single edge which separates the graph. Must $G$ necessarily have a perfect matching? Provide a justification for your answer.

Answer: Yes


Question: Let $R$ be a right Noetherian ring, and $S$ a multiplicative set consisting of regular elements where $1\in S$ and $0\not\in S$. Does the right ring of fractions $RS^{-1}$ exist?

Answer: Yes


Question: Let the function $f: \mathbb{R} \to \mathbb{R}$ satisfy the condition \( \lim_{(y,z)\rightarrow (x,x) \atop y\neq z} \frac{f(y)-f(z)}{y-z}=0. \) Does this imply that \( f'(x)=0 \)? Provide a justification for your answer.

Answer: Yes


Question: Two right circular cones and one right circular cylinder interpenetrate, with the base of each cone and both bases of the cylinder sealed by precisely fitting flat circular surfaces. What is the maximum number of completely bounded volumes that can be formed by the surfaces of these three figures?

Answer: 7


Question: Maximize \( \int_{0}^{3} f(x) \, dx \) where \( f \) is a differentiable function such that \( f(3) = 7 \) and \( f'(x) \geq x \) for all \( x > 0 \). What is the maximum possible value of \( \int_{0}^{3} f(x) \, dx \)?

Answer: 12


Question: Does any subset of the metric space \( \mathbb{Q} \times \mathbb{Q} \) have a condensation point? Justify your answer.

Answer: No


Question: Consider an infinite sequence of coin tosses with a coin that has a probability \( p \) of landing on 1, where \( p \in (0,1) \). The sequence is divided into consecutive blocks of increasing length: 2, 4, 6, and so on. Determine the probability that in infinitely many of these blocks, the first half of the terms are either all 0s or all 1s.

Answer: 0


Question: A fifth degree polynomial $P(x)$ with integral coefficients takes on values $0, 1, 2, 3, 4$ at $x = 0, 1, 2, 3, 4$, respectively. Which of the following is a possible value for $P(5)$? A) $5$ B) $24$ C) $125$ D) None of the above

Answer: C


Question: Is a smooth $k$-cell in a CW complex diffeomorphic to the open $k$-ball?

Answer: Yes


Question: If $A$ is a real normal matrix with real eigenvalues, is it true that $A$ is diagonalizable over $\mathbb{R}$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\left(\frac{n!}{n^n}\right)^{1/n} \]

Answer: \dfrac{1}{e}


Question: Is $GL_2(\mathbb{Z}_p)$ a maximal compact subgroup of $GL_2(\mathbb{Q}_p)$, where $\mathbb{Q}_p$ is the set of all p-adic numbers and $\mathbb{Z}_p$ is the set of all p-adic integers?

Answer: Yes


Question: Evaluate the integral: \[ I = \int_{-1}^1 \frac{d}{dx}\left( \tan^{-1} \left( \frac{1}{x} \right) \right)\, dx \]

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


Question: If a symmetric matrix $A$ commutes with every other symmetric matrix, does it follow that $A = \lambda I$ for some $\lambda \in \mathbb{R}$?

Answer: A = \lambda I


Question: Find the minimal polynomial of the linear transformation $T_p: M_{2\times 2}^{\mathbb{C}} \to M_{2\times 2}^{\mathbb{C}}$ defined by $T_p(X) = P^{-1}XP$, where $P = \begin{pmatrix} i & 2 \\ -1 & -i \end{pmatrix}$. Determine the minimal polynomial of $T_p$.

Answer: x^2 - 1


Question: Evaluate the triple integral: \[ \int_{0}^{3} \int_{0}^{\frac{6-2x}{3}} \int_{0}^{\frac{6-3y-2x}{2}} xyz \, dz \, dy \, dx \]

Answer: \dfrac{9}{20}


Question: In a triangulation of a polygonal domain $\Omega$ in $\mathbb{R}^3$, suppose $z$ is an interior vertex shared by $m$ tetrahedra. Determine the number of interior faces adjacent to $z$ in terms of $m$. Express your answer using $m$.

Answer: \dfrac{3m}{2}


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

Answer: 3


Question: Determine the order of the group $G$ with the presentation $\langle a,b \mid a^{16}=b^6=1,bab^{-1}=a^3\rangle$. Provide your answer as a single integer.

Answer: 48


Question: Let $x$ and $y$ be independent random variables, each following a standard normal distribution $N(0,1)$. Calculate the conditional probability $P(x > 0 \mid x > y)$. Provide your answer as a decimal rounded to four decimal places.

Answer: 0.7500


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

Answer: -\dfrac{1}{4}


Question: Determine whether the converse holds: If $D \in L(l^2(\mathbb{N}))$, does it imply that $(a_n)_{n\in\mathbb{N}} \in l^\infty(\mathbb{N})$?

Answer: Yes


Question: Find the number x such that \( 579^{65} \equiv x \pmod{679} \) and \( x < 676 \).

Answer: 108


Question: In a test with \( n \) questions, \( 2^{n-i} \) students gave wrong answers to at least \( i \) questions for \( i = 1, 2, 3, \ldots, n \). If the total number of wrong answers given is 2047, find the value of \( n \).

Answer: 11


Question: Is the point-wise limit of a sequence of simple functions a Borel measurable function?

Answer: Yes


Question: Given finite-dimensional vector spaces $L$ and $M$, and a linear map $g: L \to M$, determine the space to which $\operatorname{coker} g^*$ is canonically isomorphic: $\ker g$ or $(\ker g)^*$. Provide a justification for your answer.

Answer: (\ker g)^*


Question: Consider a sequence of independent random variables \((X_n)_{n\ge 1}\) such that \(X_n=\frac{1}{n}\) with probability \(\frac{3}{4}\) and \(X_n=-\frac{1}{n}\) with probability \(\frac{1}{4}\) for every \(n\ge 1\). Determine whether the sequence \(Y_n=\frac{1}{n^\alpha}\sum_{k=1}^n kX_k\) converges in probability to a non-zero random variable. Justify your answer.

Answer: 1


Question: Suppose $p$ and $q$ are two equivalent projections in $B(H)$. Do $p(H)$ and $q(H)$ have the same dimension?

Answer: Yes


Question: Find all positive integer solutions \((n, m)\) to the equation \(n^5 + n^4 = 7^m - 1\).

Answer: (2, 2)


Question: Find the value of \( k \) such that \( \sin {1^\circ}\sin {3^\circ}\ldots\sin {179^\circ} = \frac{1}{{2^k}} \).

Answer: 89


Question: Consider the expression \((n_1+n_2+n_3+...+n_k)^2\). When expanded, it includes terms like \(n_1^2+n_1n_2+n_2^2+...\). If each term of this expansion is multiplied out as \((m_1n_1^2+c_1)(m_2n_1n_2+c_2)(m_3n_2^2+c_3)...\) and then expanded, determine if the number of resulting terms is \(O(2^{k^2})\).

Answer: O(2^{k^2})


Question:
Let $(x_1,...,x_n)$ and $(y_1,...,y_n)$ be two different tuples of positive real numbers such that $x_1 \times \dots \times x_n = y_1 \times \dots \times y_n = c$. Determine if the following inequality is true:
\[ \left(\frac{x_1+y_1}{2}\right)\times\cdots\times \left(\frac{x_n+y_n}{2}\right) > c. \]

Answer: True


Question: Find the number of spanning trees of the complete bipartite graph $K(3,5)$. Express your answer as a single number.

Answer: 2025


Question: Determine the number of group homomorphisms from $\Bbb Z_p \oplus \Bbb Z_p$ to $\Bbb Z_p$, where $p$ is a prime number.

Answer: p^2


Question: Evaluate the double integral \( \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} F(x, y) \,dx \,dy \) where the function \( F(x,y) \) is defined as \( F(x,y) = x+y \) for \( 0 < x + y < 1 \) and \( F(x,y) = 0 \) elsewhere.

Answer: \infty


Question: In a completely regular space $X$, for every closed set $F \subseteq X$, does there exist a nonzero continuous function $f: X \to [0,1]$ such that $f = 0$ outside $F$?

Answer: No


Question: Five bags, each of a distinct color (Blue, Green, Pink, Red, and Yellow), contain 5 balls of the same color as the bag. A ball is drawn from each bag and transferred to one of the other bags, ensuring that each bag still contains 5 balls at the end. Calculate the probability that, after all transfers, each bag contains 5 balls of its original color.

Answer: 0


Question: Find the equation of the tangent plane to the surface parametrized by \( x = u^2-v^2, \ y = u + v, \ z = u^2 + 4v \) at the point \( \left(\frac{-1}{4},\frac{1}{2},2\right) \).

Answer: 4x + z = 1


Question: Calculate the value of the integral \( \int_{0}^{0.5} \frac{\ln(1+x)}{x} \, dx \) with at least two decimal places of precision.

Answer: 0.45


Question: Evaluate whether the integral \( p_1 = \int_{-k_{\max}}^{k_{\max}} \frac{u}{k}(e^{jv_1k} - e^{jv_2k}) \, dk \) is equal to zero, given that \( j = \sqrt{-1} \), and \( u, v_1, v_2, \) and \( k_{\max} \) are constants. It is known that \( p_2 = \int_{-k_{\max}}^{k_{\max}} \frac{u}{k}e^{jvk} \, dk = 2juSi(k_{\max}) \) by principal value, and \( p_2 \) does not depend on \( v \).

Answer: 0


Question: Given the discrete joint probability function \( P_{X,Y}(n,m) = c e^{-m} \frac{ m^n}{ n!} \binom{146}{m} \) for all \( n \ge 0, 0 \le m \le 146 \) and a constant \( c > 0 \), find the expected value \( \mathbb{E}[X] \).

Answer: 73


Question: Evaluate the expression \( \frac{bc}{a^{5}} + \frac{ac}{b^{5}} + \frac{ab}{c^{5}} \) in terms of \( p \), where \( a, b, c \) are the roots of the polynomial \( x^3 - px + 1 = 0 \).

Answer: -p^6 + 6p^3 - 3


Question: Maximize $xy^4$ subject to $xe^y \leq 3e^2$, $0 \geq y$, $x \geq 0$, and $y \geq 0$. Determine the maximum value of $xy^4$.

Answer: 0


Question: Find the norm of the operator \( A: \ell_1 \to \ell_1 \) defined by \( Ax = (x_1 + x_2, x_1 - x_2, x_3, x_4, \ldots) \).

Answer: 2


Question: Simplify the expression \( \frac{x^7+y^7+z^7}{xyz(x^4+y^4+z^4)} \) given that \( x+y+z=0 \).

Answer: \dfrac{7}{2}


Question: Find the residue of the function \( f(z) = z\sin\left(z + \frac{1}{z}\right) \) at the point \( z_0 = 0 \).

Answer: 0


Question: Determine the image of the set \( \mathcal{F} \) under the map that takes a function \( f(x) \) in \( \mathcal{F} \) to \( \int_{-1}^{1} f(x) \, dx \). Here, \( \mathcal{F} \) is the subset of real-valued continuous functions on \([-1, 1]\) such that \( f(x) \geq 0 \) for all \( x \) in \([-1, 1]\) and \( \limsup_{n\rightarrow \infty}\int_{-1}^{1}f(x)^{n}dx <\infty \).

Answer: [0, 2]


Question: Evaluate the integral \( \int_{|z|=3}\frac{e^{zt}}{z^2+4} \, dz \) using the Cauchy integral formula, where the contour \(|z| = 3\) is traversed in the counterclockwise direction. Verify that the result is \( \pi i \sin(2t) \).

Answer: \pi i \sin(2t)


Question: Compute the limit: \[ \lim_{n\to \infty} n!e - \lfloor n!e \rfloor \]

Answer: 0


Question: Consider a random variable \( X \) with probability density function \( f(x) = \frac{k(p)}{x^p} \) for \( x > 1 \), where \( p > 0 \) and \( k(p) \) is a positive constant. Determine the set of values for \( p \) such that the variance of \( X \) is finite, but the fourth moment does not exist.

Answer: (3, 5]


Question: Evaluate the sum: \[ \sum_{r=0}^{15} \binom{15}{r} (15-r)^{15} (-1)^r \]

Answer: 15!


Question: Determine the number of non-equivalent formulas that can be constructed using the propositions \( p_1, p_2, \ldots, p_n \).

Answer: 2^{2^n}


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{1}{x} - \frac{2}{e^{2x}-1} \]

Answer: 1


Question: Let $X$ be a set and $F(X,\mathbb{R})$ be the set of real-valued functions on $X$. Consider the subbase \( S = \{V_{x,y,\epsilon}\}_{x\in X,y\in \mathbb{R}, \epsilon >0} \) where \( V_{x,y,\epsilon} = \{f\in F(X,\mathbb{R}): |f(x)-y| < \epsilon\} \). The collection \( B \) of finite intersections of these sets forms a base for the topology of pointwise convergence on \( F(X,\mathbb{R}) \). Let \( B' \) be the subset of \( B \) where \( \epsilon = \frac{1}{n} \) for some \( n\in \mathbb{Z}_+ \). Does \( B' \) still form a base for this topology?

Answer: Yes


Question: Calculate the volume of the region $V \subset \mathbb{R}^3$ that is bounded by the paraboloid $z = 1 - x^2 - y^2$ and the plane $z = 1 - y$, for $z \geq 0$. Determine the region of integration $D$ in the $xy$-plane and evaluate the integral to find the volume.

Answer: \dfrac{\pi}{32}


Question: The complex equation \(x^3 = 9 + 46i\) has a solution of the form \(a + bi\) where \(a, b \in \mathbb{Z}\). Find the value of \(a^3 + b^3\).

Answer: -19


Question: Given that the ABC stock pays no dividend and is currently priced at $S_0 = \$10$, with the stock price at expiry $T$ going up to $uS_0$ with probability $0 < p < 1$ and down to $dS_0$ with probability $1 - p$, where $d < 1 < u$. Assume no arbitrage and a zero interest rate. A European put option with a strike price of $\$9$ is priced at $\$14/9$, and a European put option with a strike price of $\$8$ is priced at $\$8/9$. What is the fair value of a European call option with a strike price of $\$7$?

Answer: \dfrac{29}{9}


Question: Is it true that any set \( A \subset [T_0, T] \), where \( 0 \leq T_0 < T \), can be written as a countable union of disjoint subintervals, which are not necessarily open or closed?

Answer: No


Question: Determine the prime \( p \) such that the number of groups of order \( 31p^2 \) is maximized. Consider the cases where \( p = 31 \) and \( p \neq 31 \).

Answer: 31


Question: Find $G'(0)$ where \( G(x) = \int_0^{x+\pi} \frac{\sin(x+t)}{1+(x+t)^2} \, dt \).

Answer: 0


Question: Does there exist a function \( f: \mathbb{R} \to \mathbb{R} \) that is strongly discontinuous at some point? A function \( f: \mathbb{R} \to \mathbb{R} \) is strongly discontinuous at \( x_0 \in \mathbb{R} \) if for every injective sequence \( a_n \to x_0 \), \( f(a_n) \) diverges.

Answer: Yes


Question: Consider a uniform distribution on the interval [0, 1]. You draw seven independent samples, denoted as \(X_1, X_2, \ldots, X_7\), and two additional independent samples, denoted as \(Y_1, Y_2\), from this distribution. What is the probability that the third smallest value among the \(X\) samples is greater than the largest value among the \(Y\) samples? Calculate \(P(X_{(3)} > Y_{(2)})\).

Answer: \dfrac{1}{6}


Question: Consider a sequence of points \(\{(x_j, y_j)\}\) where neighboring points \((x_j, y_j)\) and \((x_{j + 1}, y_{j + 1})\) are connected with edges of length 1. The direction of each edge is random, with angles uniformly distributed on \([0, 2\pi)\). Find the square root of the expected value of the squared distance between \((x_0, y_0)\) and \((x_n, y_n)\).

Answer: \sqrt{n}


Question: Given that \( f_1, \ldots, f_n \in L_1(\mu) \) form an Auerbach basis for an \( n \)-dimensional Banach space \( X \), with \( \| f_i \| = 1 \) and linear functionals \( e_1, \ldots, e_n \) such that \( \| e_i \| = 1 \) and \( e_i(f_j) = \delta_{ij} \), choose simple functions \( g_1, \ldots, g_n \) such that \( \| f_i - g_i \|_1 < \frac{1}{n^2} \) for each \( i \). Can you conclude that the functions \( g_1, \ldots, g_n \) are linearly independent?

Answer: Yes


Question: Let \( A \) be a \( 7 \times 7 \) matrix satisfying \( 2A^2 - A^4 = I \). If \( A \) has two distinct eigenvalues, each with geometric multiplicity 3, determine the number of non-zero entries in the Jordan Canonical Form of \( A \).

Answer: 8


Question: Find the minimum value of the function \( f(n) = \lceil a \cdot n \rceil - a \cdot n \) for \( n \in \mathbb{N} \), where \( a \) is an irrational number.

Answer: 0


Question: Find the sum of all Fourier coefficients of the function \( g(x) = \cos(\pi (x + 2010)) \) from \(-\infty\) to \(\infty\).

Answer: 1


Question: Evaluate the limit: \[ \lim_{x\to 0^+} \ln x \cdot \ln(1-x) \]

Answer: 0


Question:
Determine whether the following claim is true:

Let $f_1,..., f_N:X\longrightarrow \mathbb{R}^+$ be a set of bounded functions and suppose that for each $i=1,...,N$, there exists a sequence of functions $\{g_{n,i}\}_{n\in\mathbb{N}}$ such that $g_{n,i}\downarrow f_i$ pointwise. Define the sequence $\{g_n\}_{n\in\mathbb{N}}$ of functions by:
\begin{equation*}
g_n:= \max_{i\le N} g_{n,i} \qquad \forall n\in\mathbb{N}
\end{equation*}
Then $g_{n}\downarrow \max_{i\le N} f_i$ pointwise.

Answer: True


Question: In a complete graph with 10 vertices, every pair of distinct vertices is connected by an edge. Let $u$, $v$, and $w$ be three distinct vertices in this graph. How many simple paths are there from $u$ to $v$ that pass through $w$?

Answer: 95901


Question: Find the value of the complex number $z$ such that the real part of \( \frac{z+1}{z-1} \) is zero.

Answer: |z| = 1


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

Answer: -\dfrac{1}{2e}


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

Answer: 1


Question: Find the sum of the infinite series: \[ S = \frac{3}{6} + \frac{3 \cdot 5}{6 \cdot 9} + \frac{3 \cdot 5 \cdot 7}{6 \cdot 9 \cdot 12} + \cdots \]

Answer: 3\sqrt{3} - 4


Question: Find the Weierstrass normal form of the elliptic curve given by the homogeneous equation $X^3 + Y^3 + Z^3 = 0$ over a field $K$.

Answer: y^2 = x^3 - 432


Question: Suppose that $X$ and $Y$ are independent Poisson random variables with parameters 1 and 2, respectively. Calculate $P(X=1|X+Y=4)$. Provide your answer to six decimal places.

Answer: 0.395062


Question: Does the product functor \(X \times - : \mathbf{FSet} \to \mathbf{FSet}\) have a left adjoint for every finite set \(X\)?

Answer: No


Question: Prove that if \( C \) is a matrix of full column rank, then \( CA = CB \) if and only if \( A = B \).

Answer: A = B


Question: The number of new cars sold by a dealer each week follows a Poisson distribution with a mean of 10. If 80% of the customers need a loan to pay for the car, and 62.5% of these customers take the loan from the dealer, with loan amounts averaging $22,000 and a standard deviation of $5,000, find the approximate probability that the total new car loan issued by the dealer in one week will exceed $600,000.

Answer: 0


Question: Determine whether the matrix of a non-orthogonal projection is idempotent.

Answer: Yes


Question: Compute the value of the polynomial \( \left(\frac{2}{z}+\frac{z}{2}\right)^2+2 \) where \( z = -1 + \sqrt{3}i \).

Answer: 3


Question: Calculate the remainder when \( 37^{877} \) is divided by 323.

Answer: 284


Question: Find the greatest lower bound of the set \( \{(e^n + 2^n)^{\frac{1}{n}} \mid n \in \mathbb{N}\} \).

Answer: e


Question: Evaluate the limit: \[ \lim_{x\to \pi/2} (\tan x)^{1/(\pi/2-x)} \]

Answer: \infty


Question: Determine the number of solutions to the equation \( e^{f(x)} = f(x) + 2 \), given that \( f(x) \) is an everywhere differentiable function with a unique solution for \( f(x) = 0 \) and no local extreme points.

Answer: 2


Question: Let $G$ be a finite simple group and $H, K$ be subgroups of $G$ with prime index. Is it true that $|H| = |K|$?

Answer: Yes


Question: Evaluate the contour integral \( \int_C \frac{z}{(z+2)(z-1)} \, dz \), where \( C = \{|z| = 4\} \) is the circle of radius 4 centered at the origin, oriented anticlockwise.

Answer: 2\pi i


Question:
Let $A$ and $B$ be $n \times n$ matrices such that $BA + B^2 = I - BA^2$, where $I$ is the $n \times n$ identity matrix. Determine which of the following statements is always true:

1) $A$ is non-singular
2) $B$ is non-singular
3) $A + B$ is non-singular
4) $AB$ is non-singular

Justify your answer.

Answer: 2


Question: Evaluate the integral \( \int_0^1 \cos(\ln(x)) \, dx \).

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_0^\infty xe^{-ax^2} \, dx \) and show that it equals \( \frac{1}{2a} \).

Answer: \dfrac{1}{2a}


Question: Let $X$ be a random variable with mean $\mu$ and variance $\sigma^2$. Determine the upper bound on the variance of $Y = |X|$. Provide a justification for your answer.

Answer: \sigma^2


Question: Compute the limit: \[ \lim_{N\rightarrow \infty} \sum_{n=N}^\infty e^{-n^2} \]

Answer: 0


Question: Does there exist a $C^*$-algebra $A$ such that the center of $A$ is $0$ and $A$ also has a tracial state?

Answer: Yes


Question: Evaluate the integral \( \int \frac{1}{z-2r}\ dz \) over the contour \( \alpha(t) = re^{it}, t \in [0, 2\pi] \). Determine if the integral is zero by considering the domain of the antiderivative and the properties of the contour.

Answer: 0


Question: You have a bag containing 90 marbles, with 30 unique colors, and each color is represented by 3 marbles. You pick one marble at a time without replacement. How many marbles do you need to pick to be 80-90% confident that you have picked at least one marble of each color?

Answer: 75


Question: A single fair die is rolled repeatedly. What is the probability that two sixes in a row will appear before a five followed by a six?

Answer: \dfrac{5}{12}


Question: Find the first odd abundant number less than 1000.

Answer: 945


Question: Let \(A\) and \(B\) be \(n \times n\) matrices. Define \(X = AB - BA\). Prove that the trace of \(X\), denoted as \(\text{tr}(X)\), is zero.

Answer: 0


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

Answer: 2


Question: Given a twice differentiable function \( f(x) \) such that \( f(a)=0, f(b)=2, f(c)=-1, f(d)=2, f(e)=0 \) where \( a < b < c < d < e \), find the minimum number of zeros of the function \( g(x) = (f'(x))^2 + f(x)f''(x) \) in the interval \([a, e]\).

Answer: 6


Question: Evaluate the integral \( \int_{|z-i|=1} \frac{1}{4z^2+1}dz \) using Cauchy's integral formula.

Answer: \dfrac{\pi}{2}


Question: Determine the Galois group of the polynomial \( f(X) = X^4 - 6X^2 - 2 \) over \( \mathbb{Q} \).

Answer: D_4


Question: Find the power series representation of \( f(x) = \frac{7}{8-x} + \frac{8}{x-9} \) at \( x = 0 \) and determine the radius of convergence of this series.

Answer: 8


Question: Determine the order of an orthogonal matrix composed only of the elements $+\frac{1}{4}$ and $-\frac{1}{4}$. The matrix satisfies the condition $A A^{T} = I$, where $I$ is the identity matrix.

Answer: 16


Question: Find the area enclosed by the ellipse given by the equation \(5x^2 + 6xy + 2y^2 + 7x + 6y + 6 = 0\).

Answer: \dfrac{\pi}{2}


Question: Let $X$ be a uniformly chosen number in the interval $(0, 1)$. Choose a point $P$ uniformly from the triangle with vertices at $(X,0)$, $(0,X)$, and $(0,0)$. Let $Y$ be the $y$-coordinate of the point $P$. Compute $E(Y)$. Provide your answer as a single numerical value.

Answer: \dfrac{1}{6}


Question: Determine $d(A)$ for the set $A = \{f_n:[0,1] \to \mathbb{R} \mid f_n(x) = x^n, n \in \mathbb{N} \}$, where $E = ([0, 1], \mathbb{R})$ is the set of bounded functions equipped with the sup norm and the metric it creates.

Answer: 1