Question: Prove that \( \lim_{n\to\infty} \left(1+\frac{z}{n}\right)^{n}=e^{z} \) for any complex number \( z \in \mathbb{C} \), where \( e^z \) is defined by its power series.

Answer: e^z


Question: Let $K$ be a field and $\alpha_i$ ($i=1,\dots,n$) be Galois conjugates. Let $L=K(\alpha_1,\dots,\alpha_n)$ and $G=\text{Gal}(L/K)$. We embed $G$ in $S_n$ by its action on $\alpha_i$. Let $H$ be another subgroup of $S_n$, acting on the variables $X_i$ ($i=1,\dots,n$). Let $R=K[X_1,\dots,X_n]^H$ be the ring of invariant polynomials under $H$. Suppose for all $f\in R$, $f(\alpha_1,\dots,\alpha_n)\in K$. Does it follow that $G$ is contained in $H$?

Answer: Yes


Question: Can a Euclidean norm be defined on an infinite-dimensional vector space $V$ such that for every vector $v \in V$, represented as $v = \sum_{i=1}^{n(v)} a_i e_i$ with $e_i \in B$, the norm is given by $\| v \| = \sqrt{\sum_{i=1}^{n(v)} |a_i|^2}$?

Answer: Yes


Question: Determine whether cardinal exponentiation is strictly monotone in the exponent. Specifically, for a cardinal \( \kappa \), does the inequality \( 2^\kappa < 2^{\kappa^+} \) always hold, even without assuming the generalized continuum hypothesis?

Answer: No


Question: Evaluate the double integral \( \iint_D x^2 \, dA \), where \( D = \{(x,y) \in \mathbb{R}^2 \mid \frac{x^2}{4} + \frac{y^2}{9} \le 1 \} \).

Answer: 6\pi


Question: Provide an example of a function that is absolutely continuous on \(\mathbb{R}\) but not of bounded variation on \(\mathbb{R}\).

Answer: f(x) = x


Question: Calculate the number of even permutations in the subgroup \( S_3 \times S_3 \) of \( S_6 \).

Answer: 18


Question: Given that \(\beta_n\) is a sequence of positive real numbers such that \(\beta_n = O(1/n)\) and \(\beta_n \neq o(1/n)\), determine the behavior of the series \(\sum_{k=0}^n \beta_k\).

Answer: \Theta(\log n)


Question: Show that the sum of squared residuals is zero for the model \( y_i = B_0 + \sum\limits_{k=0}^p B_k X_{ik} + e_i \) when \( p = n-1 \).

Answer: 0


Question: Evaluate the double integral \( \iint \frac{1}{(x+y)^2} \, dA \) using a change of variables over the region bounded by the lines \( x+y=1 \), \( x+y=4 \), and the \( x \) and \( y \) axes.

Answer: \ln 4


Question: Given the functional equation \( f(x+1) = f(x) \cdot f(y) \) for all \( x, y \in \mathbb{R} \) and the condition \( f(1) = 8 \), find the value of \( f\left(\frac{2}{3}\right) \).

Answer: 4


Question: Let $G$ be a compact group and $u: G \to B(H)$ be a strongly continuous unitary representation on the Hilbert space $H$. Is $u: G \to B(H)$ strictly continuous? Consider $B(H)$ with the topology induced by the $*$-isomorphism $M(B_0(H)) \cong B(H)$. A net $(x_i)$ in $B(H)$ converges strictly to $x$ if $\|x_i y - x y\| \to 0$ and $\|y x_i - y x\| \to 0$ for all compact operators $y \in B_0(H)$. On bounded subsets, this agrees with the $*$-strong topology.

Answer: Yes


Question:
Suppose \( y_{p}=x\cos(2x) \) is a particular solution of the differential equation \( y^{n}+\alpha y=-4\sin(2x) \), where \( y^{n} \) denotes the \( n \)-th derivative of \( y \). Determine the value of the constant \( \alpha \).

(a) \(-4\)
(b) \(-2\)
(c) \(2\)
(d) \(4\)

Answer: D


Question: Evaluate the limit \( \lim_{x \to 0^-} \int_{1}^x f(t) \, dt \), where \( f(t) = \begin{cases} t^3 \ln{t} & \text{for } t > 0 \\ \arctan{t} & \text{for } t \leq 0 \end{cases} \). Determine if this limit equals \( \frac{1}{16} \).

Answer: \dfrac{1}{16}


Question: Evaluate the limit: \[ \lim _{x\to 0}\left(2-e^{\arcsin^{2}\left(\sqrt{x}\right)}\right)^{\frac{3}{x}} \]

Answer: e^{-3}


Question: Determine whether the sequence \((a_n)_{n \in \mathbb{N}}\) defined by \(a_n = \sqrt[n]{|\sin(n)|}\) converges to 1.

Answer: 1


Question: In a triangle, determine the ratio of the distance between a vertex and the orthocenter to the distance from the circumcenter to the side opposite that vertex.

Answer: 2


Question: Let $L$ be a first order language, and let $M$ and $N$ be $L$-structures such that $N$ is finite. Suppose $h: N \to M$ is an elementary embedding. Is $h$ an isomorphism of $L$-structures?

Answer: Yes


Question: Evaluate the limit: \( \lim_{x \rightarrow 0}(1-\cos x)^{\tan x} \)

Answer: 1


Question: Let $X$ be a binomial($n$, $p$) random variable with $p<1$ and expected value $E(X)=np\geq 1$. Prove or disprove that $P(X>E(X))\geq 1/4$. Consider the case where $n=2$ and $p=1/2$ to determine if this bound is the best possible.

Answer: \dfrac{1}{4}


Question: Find the maximum value of the function \( f(x) = (2\sin A + 3\cos A + 4)^2 \times (6-2\sin A-3\cos A)^3 \) as \( A \) ranges over all real numbers.

Answer: 3456


Question: Compute \(\int_{-2}^{2}(x-3)\sqrt{4-x^2}dx\) in terms of \(\pi\), using the properties of integrals such as linearity, additivity, and invariance under translation.

Answer: -6\pi


Question: Does there exist an open cover of the set $[0,1] \cap \mathbb{Q}$ in the real numbers with the standard metric that does not have a finite subcover?

Answer: Yes


Question: Let \( E \subset [0, 2\pi] \) be a set with positive measure. For any sequence \( t_n \) of real numbers, determine whether the following limit holds: \[ \lim_{n \to \infty} \int_E \cos(n x + t_n)\,dx = 0. \]

Answer: 0


Question: Evaluate the limit \( \lim_{n\to\infty} (n\sin(\pi/n))^n \).

Answer: \infty


Question: Let \((R, m)\) be a commutative regular local Noetherian ring. Determine whether it is true that for every prime \(p \in \text{Spec}(R)\), the factor ring \(R/p\) has a maximal Cohen-Macaulay \(R/p\)-module.

Answer: True


Question: Given the polynomial equation \(x^4 - x^3 + 2x^2 + x + 1 = 0\) with roots \(a, b, c, d\), find the value of \((1+a^3)(1+b^3)(1+c^3)(1+d^3)\).

Answer: 16


Question: Determine the intervals on which the function \( y = \frac{\ln{(\pi+x)}}{\ln{(e+x)}} \) is increasing or decreasing for \( x > 0 \).

Answer: (0, \infty)


Question: Determine whether the following statement is true or false: Every normal matrix over \( \mathbb{C} \) can be diagonalized to an invertible matrix.

Answer: B


Question: Evaluate the limit \( I = \lim_{n \to \infty} \sum_{k=0}^{n} \frac{\cos{\sqrt{\frac{k}{n}}}}{2^k} \).

Answer: 2


Question: Let ${\bf A}$ be an $n \times n$ matrix where $a_{ij} = n$ for $i \neq j$ and $a_{ii} = i$. Find the determinant of ${\bf A}$. \( \det({\bf A}) = ? \)

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


Question: Let $X$ and $Y$ be random subsets of $\{1, 2, \dots, k\}$, each picked uniformly at random from all $2^k$ subsets, and independent of each other. Determine the variance of the random variable $|X \cup Y|$. \( \text{Express your answer in terms of } k. \)

Answer: \dfrac{3k}{16}


Question: Consider length spaces $X$ and $Y$. A map $f: X \to Y$ is called an arcwise isometry if $L(f(\gamma)) = L(\gamma)$ for every path $\gamma$. A map $f: X \to Y$ is a local isometry at $x \in X$ if there exists a neighborhood $U$ of $x$ such that the restriction $f|_U$ maps $U$ isometrically onto an open set $V$ in $Y$. Is it true that an arcwise isometry that is also a local homeomorphism is a local isometry? Justify your answer.

Answer: Yes


Question: Evaluate the sum \( \sum_{i=1}^{1995} \frac{9^{i/1996}}{9^{i/1996} + 3} \).

Answer: \dfrac{1995}{2}


Question: An urn contains 4 red balls and 3 blue balls. Picking a red ball earns you $1, while picking a blue ball costs you $1. You may stop picking balls at any time. What is the expected value of this game if you play optimally?

Answer: \dfrac{58}{35}


Question: Let \( f:[a,b]\to \mathbb{R} \) be a function that is continuous on \([a,b]\), differentiable on \((a,b)\setminus\{x\}\), and suppose \( \lim_{t \to x} f'(t) \) exists. Prove or disprove that \( f \) is differentiable at \( x \) and that \( f'(x)= \lim_{t \to x} f'(t) \).

Answer: f'(x) = \lim_{t \to x} f'(t)


Question: Determine whether there exists a Hamiltonian group of order \( p^4 \) for any prime \( p \geq 3 \).

Answer: No


Question: On a $2020 \times 2020$ chessboard, what is the maximum number of warriors you can place such that no two warriors attack each other? A warrior can move either 3 steps forward and 1 step sideways in any direction, or 2 steps forward and 2 steps sideways in any direction. Determine the maximum number of warriors that can be placed on the board.

Answer: 1020100


Question:
Find all positive integer triples \((a, b, c)\) such that:
\[ ab + c = \gcd(a^2, b^2) + \gcd(a, bc) + \gcd(b, ac) + \gcd(c, ab) = 239^2 \]

Answer: (238, 238, 477)


Question: Determine the order of the pole of the function \( f(z) = \frac{e^{bz}}{z\sinh(az)} \).

Answer: 2


Question: Determine whether there exists a subset $A$ of the real numbers $\mathbb{R}$ such that $A \cap A' = \emptyset$ and the closure $\overline{A}$ is uncountable.

Answer: Yes


Question: Calculate the expected value \( \mathbb{E}[X|X|] \) where \( X \sim \mathcal{N}(0,1) \).

Answer: 0


Question: Suppose $G$ is a group with an operation $\cdot$ and is also a group with another operation $*$. For each $a, b \in G$ with $a \neq b$, we have $a \cdot b = a * b$. Can the condition $a \neq b$ always be dropped?

Answer: Yes


Question: Determine the cardinality of $\mathbb{R}^\infty$. Is it the same as $\beth_1$?

Answer: \beth_1


Question: Verify the Divergence Theorem for the vector field \( F(x,y,z) = (xz, 0, 0) \) over the solid \( S \) bounded by the paraboloid \( z = x^2 + y^2 \) and the plane \( z = 1 \).

Answer: \dfrac{\pi}{3}


Question: Does there exist a countable non-standard model of Peano Arithmetic where every element is definable, similar to the standard model?

Answer: No


Question: Given a discrete-time Markov chain (DTMC) with the transition matrix \( \textbf{P} = \begin{pmatrix}0 & 1\\ \frac{1}{2} & \frac{1}{2} \end{pmatrix} \) and the initial distribution \( X_1 = \left(\frac{1}{4}, \frac{3}{4}\right) \), calculate the conditional entropy \( H_2\left(X_2|X_1\right) \).

Answer: \dfrac{3}{4}


Question: Determine the number of distinct subgroups of order 10 in a non-cyclic abelian group of order 20.

Answer: 3


Question: Calculate the expected value of the random variable $Y = \min(X, 3)$, where $X$ is a random variable with the density function \( g(x) = \begin{cases} 0 & \text{if } x < 0 \\ e^{-x} & \text{if } x \geq 0 \end{cases} \).

Answer: 1 - e^{-3}


Question: Find the exact value of the convergent improper integral: \[ \int_{0}^{\infty}\frac{\ln x\,dx}{\sqrt{x}(1+x) } \]

Answer: 0


Question: Evaluate the infinite series \( S = \sum_{n=1}^{\infty} \frac{H_n}{n^2} \) using complex analysis techniques, such as residues.

Answer: 2\zeta(3)


Question: Evaluate the integral \( \int_0^{\pi} [\cot(x)] \, dx \), where \([\cdot]\) denotes the greatest integer function.

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


Question: Evaluate whether the integral \( \int_{\Omega}(\partial_x^2-\partial_y^2)u\phi \, d\omega = 0 \) is zero, given that \( \phi \in C_0^{\infty}(\Omega) \) with \( \Omega = (0,1)\times(0,1) \) and \( u \in L_2(\Omega) \) is defined by \( u(x,y) = 1 \) for \( x>y \) and \( u(x,y) = 0 \) for \( x\leq y \).

Answer: 0


Question: Let \( T: C[0,1] \rightarrow C[0,1] \) be defined by \((Tf)(x) = \int_{0}^{1} xyf(y) \,\mathrm{d}y.\) Show that \( T \) is a bounded linear operator and calculate its norm.

Answer: \dfrac{1}{2}


Question: Use Green's Theorem to evaluate the line integral \( \int_C (y-x) \, dx + (2x-y) \, dy \), where \( C \) is the path given by the parametric equations \( x = 2\cos(\theta) \) and \( y = \sin(\theta) \) for \( 0 \leq \theta \leq 2\pi \).

Answer: 2\pi


Question: Evaluate the integral \( \int_{-1}^{1} \frac{t^{2}}{1+\exp(t)} \, dt \).

Answer: \dfrac{1}{3}


Question: Let $Q$ be a point on or inside the boundary of a square $ABCD$. Determine the minimum value of the sum $\angle QAB + \angle QBC + \angle QCD + \angle QDA$. Provide your answer in radians.

Answer: \pi


Question: Let \( p(x) \) be a monic polynomial of degree \( n \). Can there always be found an \( n \times n \) matrix \( A \) such that \( p(x) = \det(xI - A) \), where \( \det(xI - A) \) is the characteristic polynomial of \( A \)?

Answer: Yes


Question: Find the number of matrices in \(GL_{3}(\mathbb{Z}_{3})\) such that \([1 \ 1 \ 1]^{T}\) is an eigenvector.

Answer: 864


Question: If $R$ is a commutative noetherian ring, $I$ is an ideal of $R$, and $M$ is an $R$-module, does the condition that $Tor_i^R(R/I, M)$ is finitely generated for $i \ge 0$ imply that $Ext^i_R(R/I, M)$ is finitely generated for $i \ge 0$?

Answer: Yes


Question:
Identify the imaginary part of the analytic function $f(z)$ given that $\Re(f'(z)) = 2\Im(z)$ and $f(1+i) = 2$. Choose the correct option from the following:

(1) $-2xy$
(2) $x^2-y^2$
(3) $2xy$
(4) $y^2-x^2$

Answer: D


Question: Let \( E = (0,1) \times (0,1) \subset \mathbb{R}^2 \). The function \( \Phi : E \to \mathbb{R}^2 \) is defined by \( \Phi(x,y) = (e^{x+y}, e^{x-y}) \). Determine the area of the region \( \Phi(E) \).

Answer: e^2 - 1


Question: Let \( I = [t_{0}, t_{1}] \) be a bounded interval of \( \mathbb{R} \) and suppose \( f: I \to \mathbb{R} \) is a \( C^{k}(I) \) function for some \( k \ge 2 \). Suppose that \( \int_{t_{0}}^{t_{1}} f(t)g(t) \, dt = 0 \) for every \( C^{k}(I) \) function \( g: I \to \mathbb{R} \) such that \( g(t_{0}) = g(t_{1}) = 0 \). Does it follow that \( f(t) \equiv 0 \)? Provide a justification for your answer.

Answer: Yes


Question:
Let \( z = \cos(2^\circ) + i \sin(2^\circ) \). Find the value of \( \sum_{m=1}^{15} \Im(z^{2m-1}) \). Choose the correct answer from the options below:

\[\begin{align*}
(a) & \ \frac{1}{\sin 2^\circ} \\
(b) & \ \frac{1}{3\sin 2^\circ} \\
(c) & \ \frac{1}{2\sin 2^\circ} \\
(d) & \ \frac{1}{4\sin 2^\circ}
\end{align*}\]

Answer: D


Question: Given that \(a\) and \(b\) are two positive, co-prime integers, find the value of \(a + b\) if \(\lim _{n \rightarrow \infty}\left(\frac{^{3n}C_n}{^{2n}C_n}\right)^\frac{1}{n}=\frac{a}{b}\).

Answer: 43


Question: Let $F \subset \mathbb{R}^d$ be a set of positive measure. Define $F_x = \{y \in \mathbb{R}^{d-1} \mid (x,y) \in F \}$. By Fubini's theorem, we have $|F| = \int_{\mathbb{R}} |F_x| \, dx$. Let $X = \{ x \in \mathbb{R} \mid F_x \neq \emptyset\}$. Is it true that for almost every $x \in X$, $|F_x| > 0$?

Answer: No


Question: Let \( G \) be a finite group of order \( n \) and let \( \phi \) be the Cayley (right) action \( \phi: G \hookrightarrow S_G \cong S_n \) defined by \( \phi(g)=[x \mapsto xg \quad \forall x \in G] \). Prove that if \( g \) is an element of \( G \) of order 2, then \( \phi(g) \in A_n \) if and only if \( 4 \mid n \).

Answer: 4 \mid n


Question: A point particle moves along a line with its speed described by a function: at irrational times, the speed is 0 meters per second, and at rational times, the speed is 1 meter per second. How far does the particle travel in 10 seconds?

Answer: 0


Question: Given the matrices \( A = \begin{bmatrix} 2 & 0 & 3 \\ -1 & -2 & 1 \\ 2 & 0 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & -1 & 0 \\ 1 & 0 & 1 \\ -1 & 1 & \frac{1}{3} \end{bmatrix} \), calculate the determinant of the matrix expression \( \det(\det(A)B[\det(B)A^{-1}]) \).

Answer: \dfrac{64}{81}


Question: Evaluate the limit: \[ \lim_{x\to0}\int_{0}^{x}\frac{\cos(t^3)}{t+x}dt \]

Answer: \ln 2


Question: Let $F = \mathbb{Z}_2$ and consider the polynomial $f(x) = x^3 + x + 1 \in F[x]$. Suppose $a$ is a root of $f(x)$ in some extension of $F$. Determine the elements of the field $F(a)$ in terms of $a$. How many elements does $F(a)$ have?

Answer: 8


Question: Determine the number of zeros of the function \( g(z) = z^4 + iz^3 + 1 \) in the region \( \{ z \in \mathbb{C} : \text{Re}(z) > 0, \text{Im}(z) > 0 \} \).

Answer: 1


Question: Let $a$ be a 28-digit palindromic number that is a multiple of 13. All the digits of $a$ are 1, except for the 13th, 14th, 15th, and 16th digits, which are $A$, $B$, $C$, and $D$ respectively. Determine the minimum possible value of $A + B + C + D$.

Answer: 0


Question: For any Banach space $X$, does there exist a Banach space $Y$ and an injective compact operator $T: X \to Y$?

Answer: No


Question: Evaluate the integral \( \int_{S} (x + 2y - z) \, dV \), where \( S \) is the tetrahedron in \( \mathbb{R}^{3} \) with vertices \((0,0,0), (1,2,3), (0,1,2), (-1,1,1)\).

Answer: \dfrac{1}{6}


Question: Let $G$ be a group that acts on an ordered set and preserves its order, meaning if $a < b$, then $g(a) < g(b)$ for any $g \in G$. Does this imply that there is a left order on $G$, such that if $f < g$, then $fh < gh$ for any $h \in G$?

Answer: No


Question: Evaluate the integral \( \int_0^\infty\int_x^\infty\left(\frac{1}{y}\right)e^{-y/2}dydx \).

Answer: 2


Question: Calculate the expected value \( \mathbb{E}\left(\frac{X}{X+Y}\right) \) where \( X \) and \( Y \) are independent random variables with exponential distributions such that \( \mathbb{E}X=1 \) and \( \mathbb{E}Y=\frac{1}{2} \).

Answer: 2 - 2\ln 2


Question: Calculate the variance of the expression \(e^U + e^{1-U}\) where \(U\) is a uniformly distributed random variable on the interval \([0, 1]\).

Answer: -3e^2 + 10e - 5


Question: Seventy-five percent of claims follow a normal distribution with a mean of 3,000 and a variance of 1,000,000. The remaining 25% of claims follow a normal distribution with a mean of 4,000 and a variance of 1,000,000. Calculate the probability that a randomly selected claim exceeds 5,000.

Answer: 0.0568


Question: Calculate the Legendre symbol \(\left(\frac{97}{131}\right)\) and determine its value.

Answer: -1


Question: Determine the cardinality of the ring \( R = \frac{\mathbb{Z}[x]}{((x^{2}+x+1)(x^{3}+x+1))} \), where \( I \) is the ideal generated by 2 in \( R \).

Answer: 32


Question: Determine the fundamental period of the signal \( x(t) = \sum_{k=-\infty}^{\infty}\operatorname{rect}\left(\frac{t+k}{2c}\right) \).

Answer: 1


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\sin x}{x} \, \mathrm{d}x \).

Answer: \pi


Question: Determine the maximum number of elements of order 5 in a group of order 80 using Sylow's theorem.

Answer: 64


Question: Determine the number of pairs \((m, n)\) of non-zero integers such that the equation \(n^{2}x^{3} + 2020x^{2} + mx = 0\) has exactly two real solutions.

Answer: 16


Question: Compute the integral \( \oint_{|z|=1} \frac{\cos z}{z^3} \, dz \).

Answer: -\pi i


Question: Determine whether the limit \( \lim_{(x,y) \to (0,0)} xy \log(x^2+y^2) \) exists, and if so, find its value.

Answer: 0


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

Answer: \infty


Question: Let \( f: [a,b] \rightarrow I \subset \mathbb{R}^n \) be a homeomorphism, where \([a,b] \subset \mathbb{R}\) and \(n > 1\). Can the n-dimensional Lebesgue measure of \(I\) be positive?

Answer: No


Question: Determine the number of prime numbers \( p \) for which the expression \( p^3 + p^2 + 11p + 2 \) is also a prime number.

Answer: 1


Question: Find the minimum possible value of \( f \) such that \( \sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d} + \sqrt{e} = \sqrt{f} \), where \( a, b, c, d, \) and \( e \) are distinct positive integers.

Answer: 225


Question: Find the value of $u(e,1)$, where $u(x,y)$ is the solution to the Cauchy problem given by the partial differential equation $$xu_{x}+u_{y}=1$$ with the initial condition $u(x,0)=2\ln(x)$ for $x>1$. What is $u(e,1)$?

Answer: 1


Question: Find all natural numbers \( n \) such that \( n! + n^2 + 2 \) is a prime number.

Answer: 3


Question: Determine the number of integral solutions for the equation \(x^4 - ax^3 - bx^2 - cx - d = 0\), where \(a \ge b \ge c \ge d\) and \(a, b, c, d\) are natural numbers.

Answer: 0


Question:
Given the system of equations:
\[\begin{cases} ab+c+d=15 \\ bc+d+a=24 \\ cd+a+b=42 \\ da+b+c=13 \end{cases}\]
where $a, b, c, d \in \mathbb{R}$, calculate the value of $a+b+c+d$. The solution using Gröbner basis gives $a+b+c+d = \frac{169}{10}$. Find this value using high school mathematics.

Answer: \dfrac{169}{10}


Question: A grasshopper sits on a square of a 5x4 rectangular board. Each minute, it jumps at random to a neighboring square, with 2, 3, or 4 possible moves depending on its position (2 at corners, 3 at edges, 4 at the center), all equally likely. The grasshopper always remains on the board. What is the long-run average probability that the grasshopper will be on a corner square of the board after many jumps?

Answer: \dfrac{4}{31}


Question: Is it possible to find complex numbers $z$ and $w$, and a complex exponent $\alpha$, such that the principal values of $z^\alpha w^\alpha$ and $(zw)^\alpha$ are different?

Answer: Yes


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function such that the equation \( f(x) = x \) has no real solution. Is it true that the equation \( f(f(x)) = x \) also has no real solution?

Answer: True


Question:
Determine whether there exists a function \( f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) that can be expressed both as a product and as a sum of single-variable functions. Specifically, find if there exist functions \( g, h, w, z: \mathbb{R} \rightarrow \mathbb{R} \) such that:
\[ f(x, y) = g(x)h(y) \]
and
\[ f(x, y) = w(x) + z(y) \]
for all \( x, y \in \mathbb{R} \).

Answer: Yes


Question: Given a sequence \( \beta_n \) of positive real numbers such that \( \sum_{n=1}^\infty n\beta_n < \infty \), determine if there exists a sequence \( \alpha_n \) of positive numbers with \( \alpha_n \to \infty \) such that \( \sum_{n=1}^\infty n\alpha_n\beta_n < \infty \).

Answer: Yes


Question: Find the integer $a$ such that $539$ divides $a \cdot 3^{253} + 5^{44}$. Express your answer as a congruence modulo 539.

Answer: 106


Question:
Calculate the volume of the top half of a pseudosphere, which is parametrized by \( \Phi(t,\theta) = \Big ( \frac{\cos(\theta)}{\cosh(t)}, \frac{\sin(\theta)}{\cosh(t)}, t-\tanh(t)\Big) \) with \(0\le t\) and \(0\le \theta < 2\pi\). Use the integral formula for the volume of a region \(V\) in \(\mathbb{R}^3\):
\[ \iiint_V r\tanh^2(t) \, \mathrm{d}r \, \mathrm{d}\theta \, \mathrm{d}t \]
where the change of variables is \(x = r\cos(\theta)\), \(y = r\sin(\theta)\), and \(z = t - \tanh(t)\). Determine the correct volume of the top half of the pseudosphere.

Answer: \dfrac{\pi}{3}


Question: In a binomial random graph $G_{n,p}$ with $n$ vertices, where each edge is present with probability $p$, consider the event that $k$ distinct sets of 3 vertices each form a triangle. Express the probability of this event, $\mathbb{P}\left(\bigcap_{j=1}^k \mathcal{E}_j\right)$, in terms of $p$. Here, $\mathcal{E}_j$ denotes the event that a set of 3 vertices forms a triangle.

Answer: p^{3k}


Question: Consider a measure \( \mu \) with a positive singular part in a complete separable metric space. Determine whether there can exist a compact set \( K \) such that the Lebesgue measure of \( K \) is zero, but \( \mu(K) > 0 \).

Answer: Yes


Question: Given the sequence defined by \( x_0 = 2\sqrt{2} \) and \( x_{n+1} = x_n^2 - 2 \) for \( n \geq 1 \), show that \( \lim_{n \to \infty} \frac{x_n}{x_0 x_1 \cdots x_{n-1}} = 2 \).

Answer: 2


Question: Let $K \subseteq \mathbb{R}^n$ be a compact subset, and let $\{U_i\}_{i\in I}$ be a locally finite collection of open subsets $U_i \subseteq \mathbb{R}^n$, each disjoint from $K$. For each $i \in I$, let $K_i \subseteq U_i$ be a compact subset. Since $K$ is disjoint from each $U_i$, it is also disjoint from each $K_i$. Does there exist a neighborhood of $K$ that is disjoint from all the $K_i$?

Answer: Yes


Question: Evaluate the line integral \( \int_{\gamma} x \, dx + (x+y) \, dy + (x+y+z) \, dz \) where \( \gamma \) is the curve of intersection between the surfaces \( x^2 + y^2 = 1 \) and \( z = x + y \). Use the parametrization \( x(t) = \cos t, \; y(t) = \sin t, \; 0 \leq t \leq 2\pi \) to find the value of the integral.

Answer: \pi


Question: Find the limit of \(\exp\left(\frac{|x-2y|}{(x-2y)^2}\right)\) as \((x, y)\) approaches \((2y, y)\).

Answer: \infty


Question: Find the minimum value of \( P_{2000} \) in a positive integer sequence \( P_n \) that satisfies the following conditions: for any positive integers \( m \) and \( n \), if \( m|n \) and \( m<n \), then \( P_m|P_n \) and \( P_m<P_n \). Assume \( P_1 = 1 \) and for any prime \( p \), \( P_p = 2 \).

Answer: 128


Question:
In a Cartan-Hadamard 3-space, which is a complete simply connected 3-dimensional Riemannian manifold with nonpositive sectional curvature, consider a smooth convex surface \( \Gamma \subset M \) that is an embedded topological sphere with a nonnegative second fundamental form \( \mathrm{I\!I} \). The total Gauss-Kronecker curvature of \( \Gamma \) is given by:
\[
\mathcal{G}(\Gamma):=\int_\Gamma\det(\mathrm{I\!I}).
\]
Given that \( \mathcal{G}(\Gamma) = 4\pi \), does it follow that the compact region of \( M \) bounded by \( \Gamma \) is Euclidean, meaning all its sectional curvatures are zero?

Answer: Yes


Question:
Solve the partial differential equation using the method of separation of variables:
\[ u_t = 9u_{xx}, \quad 0<x<1,\quad t>0 \]
with the boundary conditions:
\[ u_x(0,t) = 0 \]
\[ u(1,t)=u(0,t) \]
and the initial condition:
\[ u(x,0) = \sin\pi x, \quad 0 \leq x \le 1 \]
Find an approximation for \( u(x,t) \) for large \( t \).

Answer: \dfrac{2}{\pi}


Question: Let $X$ and $Y$ be locally convex spaces. Consider the biduals $X^{\prime \prime}$ and $Y^{\prime \prime}$ with the strong topologies $\beta(X^{\prime \prime}, X^{\prime})$ and $\beta(Y^{\prime \prime}, Y^{\prime})$, respectively. Suppose $A: (X^{\prime \prime}, \beta(X^{\prime \prime}, X^{\prime})) \rightarrow (Y^{\prime \prime}, \beta(Y^{\prime \prime}, Y^{\prime}))$ is a linear continuous map. Does this imply that $A : (X^{\prime \prime}, \tau_N(X^{\prime \prime}, X^{\prime})) \rightarrow (Y^{\prime \prime}, \tau_N(Y^{\prime \prime}, Y^{\prime}))$ is also continuous?

Answer: No


Question: Let \( f(x) = e^{\{x\}^2}(1+2\{x\}^2) \), where \( \{x\} \) denotes the fractional part of \( x \). A new periodic function \( g(x) \) is obtained by compressing \( f(x) \) by \( 1/k_1 \) times along the x-axis and magnifying it by \( k_2 \) times along the y-axis. Given that \( k_1 \) and \( k_2 \) satisfy the equation \( k_1^2 - 4k_1 + k_2^2 - 6k_2 + 13 = 0 \), find the value of \( \int_0^{500} g(x) \, dx \).

Answer: 1500e


Question: Suppose $\Omega_3\subset \mathbb{C}^3$ and $\Omega_2\subset \mathbb{C}^2$ are two domains. Let $g:\Omega_3\to\Omega_2$ be a surjective analytic function and $h:\Omega_2\to\mathbb{C}^2$ be any function. If the composition $f:=h\circ g:\Omega_3\to\mathbb{C}^2$ is analytic, does this imply that $h$ is analytic?

Answer: Yes


Question: Solve the congruence equation: \(x^{257} \equiv 5 \pmod{2016}\).

Answer: 101


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

Answer: -\dfrac{1}{2}


Question: Let $X_1, X_2, X_3, X_4, X_5$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Consider the statistics $T_1 = \frac{X_1 − X_2 + X_3 + X_4 + X_5}{3}$ and $T_2 = \frac{X_1 + X_2 + 2X_3 + X_4 + X_5}{6}$ as unbiased estimators for $\mu$. Determine which statistic has the least variance.

Answer: T_2


Question:
Determine the convergence radius of the series:
\[-W(-x) = \frac{1^0}{1!}x^1 + \frac{2^1}{2!}x^2 + \frac{3^2}{3!}x^3 + \frac{4^3}{4!}x^4 + \dots\]
Is it true that the convergence radius is $\frac{1}{e}$?

Answer: \dfrac{1}{e}


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{\sin x^4-x^4\cos x^4+x^{20}}{x^4(e^{2x^4}-1-2x^4)}$$

Answer: \dfrac{1}{6}


Question: Evaluate the integral \( \iiint\limits_\Omega (x+y+z)^2 \, dv \), where \( \Omega \) is the region defined by \( x^2+y^2+z^2 \le 1 \).

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


Question: Determine whether the following statement is true or false: If a linear algebraic group defined over \(\mathbb{R}\) has a faithful irreducible representation, then it is reductive.

Answer: True


Question: Calculate the limit: \[ \lim_{z\to0} \frac{(z\overline{z})^{2}}{z^{2}} \] where \( z \) is a complex variable and \( \overline{z} \) denotes its complex conjugate.

Answer: 0


Question: Is there a bounded linear operator \( \phi: B(H) \to H \) such that \( \ker \phi \) is a Lie subalgebra, but \( \phi \) is not of the form \( \phi(T) = Th \) for some \( h \in H \)?

Answer: Yes


Question: Determine the number of spanning trees in the complete graph $K_9$ such that vertex 1 has degree 4.

Answer: 143360


Question: Given that $f(x+2) - f(x) = (6x+4)^2$ and $f(0) = -16$, find the value of $f(5)$.

Answer: 564


Question: Evaluate the sum \( \tan^4{10°} + \tan^4{50°} + \tan^4{70°} \).

Answer: 59


Question: Evaluate the integral \( \int_c \frac{z}{z^2 + 2z + 5} \, dz \), where \( c \) is the unit circle.

Answer: 0


Question: Calculate the expectation \( E\left(\frac{4}{X_1 + X_2 + X_3 + X_4}\right) \), where \( X_1, X_2, X_3, \) and \( X_4 \) are independent exponential random variables with parameter 3.

Answer: 4


Question:
Let $A \in M_{10}(\mathbb{C})$ be a matrix such that $\det(A) < 0$ and $AA^{t} = I$. Determine which of the following statements is true about $\det(A+I)$:

(a) $\det(A+I) = -1$

(b) $\det(A+I) = 0$

(c) $\det(A+I) > 1$

(d) $\det(A+I) = 1$

Answer: B


Question: In the ring \( \mathbb{Z}_5[x] \) with the ideal \( I = (x^2 + 3) \), consider the polynomial \( f(x) = 14x^2 + k \). For which value of \( k \in \{0, 1, 2, 3, 4\} \) does it hold that \( f(t) = 1 \) in \( \mathbb{Z}_5[x]/I \), where \( t \) is the class of \( x \) in \( \mathbb{Z}_5[x]/I \)?

Answer: 3


Question:
Consider a homogeneous manifold $X = G/H$, where $G$ and $H$ are connected Lie groups, and there is a $G$-invariant Riemannian metric on $X$. Let $B(R)$ be the closed ball of radius $R > 0$ around the base point $eH$, and let $b(R)$ denote its volume. Determine if the following limit holds:
\[
\lim_{\varepsilon\to 0}\ \limsup_{R\to\infty}\ \frac{b(R+\varepsilon)}{b(R)}=1.
\]

Answer: 1


Question: If $X$ is a proper scheme over $k$, is the composition of morphisms $X \rightarrow \text{Spec}k \rightarrow \mathbb{P}_k^1$ proper?

Answer: Yes


Question: Evaluate the line integral of the vector field \( f(x,y) = \langle y - \cos y, x \sin y \rangle \) over the circle \( C \) defined by \((x-3)^2 + (y+4)^2 = 4\), oriented clockwise, using Green's theorem.

Answer: 4\pi


Question: Evaluate the integral \( \int_0^\infty \frac{\sqrt{x}}{x^3+1}\ \mathrm{d}x \) using residue theory with a keyhole contour to exclude the branch cut.

Answer: \dfrac{\pi}{3}


Question: Determine whether the following statement is true: If $A \times B$ is an element of a product $\sigma$-algebra $\mathcal{A} \times \mathcal{B}$, then $A \in \mathcal{A}$ and $B \in \mathcal{B}$. Justify your answer.

Answer: True


Question: Let $T: \mathbb{R}^n \to \mathbb{R}^m$ be a linear map where $n > m$. Is it true that for any Borel set $A \subseteq \mathbb{R}^n$, the set $T(A)$ is Lebesgue measurable in $\mathbb{R}^m$?

Answer: Yes


Question: Evaluate the line integral \( \oint_c (x^2 + iy^2) \, ds \) where \( c \) is the circle defined by \( |z| = 2 \) and \( s \) is the arc length. Express your answer in the form of a complex number.

Answer: 8\pi(1 + i)


Question:
Evaluate the integral \( \int_{-a}^{a} \sqrt{\frac{a-x}{a+x}}\,\mathrm{d}x \). Choose the correct answer from the options below:

(A) \( a \)

(B) \( \pi a \)

(C) \( \frac{a}{2} \)

Answer: B


Question: Given three integers $N$, $K$, and $M$, calculate the sum of the K'th powers of all positive integers up to $N$ that are coprime with $N$, and find the result modulo $M$. For example, if $N=5$, $K=2$, and $M=100$, the answer is $(1^2 + 2^2 + 3^2 + 4^2) \bmod 100 = 30$. Compute this value for the given $N$, $K$, and $M$.

Answer: 30


Question: Use the zeroes of the Chebyshev polynomial \( T_3(x) = 4x^3 - 3x \) to construct an interpolating polynomial of degree two for the function \( x^3 \) on the interval \([-1, 1]\).

Answer: \dfrac{3}{4}x


Question: Let $M_1$ and $M_2$ be two simply connected, connected, compact smooth manifolds without boundary and of the same dimension. Assume that the Lie algebras of vector fields $\mathfrak{X}(M_1)$ and $\mathfrak{X}(M_2)$ are isomorphic. Are $M_1$ and $M_2$ diffeomorphic?

Answer: Yes


Question: Let \( n_1, n_2, \ldots, n_k \) be a sequence of \( k \) consecutive odd integers. If \( n_1 + n_2 + n_3 = p^3 \) and \( n_k + n_{k-1} + n_{k-2} + n_{k-3} + n_{k-4} = q^4 \), where both \( p \) and \( q \) are prime numbers, determine the value of \( k \).

Answer: 62


Question:
Determine the value of gain \( K \) at which the root locus of the closed-loop system \( T(s) = \frac{-K(s+1)^2}{(1-K)s^2 + 2(1-K)s + (2-K)} \) intersects with the imaginary axis. Use the provided Routh table:
\[
\begin{bmatrix}
s^2 & (1-K) & (2-K) \\
s^1 & 2(1-K) & 0 \\
s^0 &(2-K) & 0
\end{bmatrix}
\]

Answer: 2


Question: Determine the conditions under which $2^x + 3^y$ is a perfect square, where $x$ and $y$ are positive integers.

Answer: (4, 2)


Question: Calculate the limit of the sequence defined by \( a_n = \frac{1}{\sqrt{n}}\sum_{k=0}^n\sqrt{C_n^kp^k(1-p)^{n-k}} \), where \( C_n^k = \frac{n!}{k!(n-k)!} \) and \( p \in (0,1) \). Find \( \lim_{n \to \infty} a_n \).

Answer: 0


Question:
Calculate the following double limit:
\[ \lim_{v\to c}\lim_{V\to c}\frac{v - V}{1 - vV/c^2}. \] Here, \( c \) is a positive real constant.

Answer: -c


Question: Let $A$ be a bounded Borel set in $\mathbb{R}^n$. Consider its projection $A_1$ onto $\mathbb{R}^{n-1}$. Let $\mu$ be a given nonnegative, finite Borel measure on $\mathbb{R}^{n-1}$. Does $A_1$ have the following property: For every $\epsilon > 0$, there exist Borel sets $B, C \subset \mathbb{R}^{n-1}$ such that $B \subset A_1 \subset C$ and $\mu(C \setminus B) < \epsilon$?

Answer: Yes


Question: Determine the total number of 7-bit strings such that any pair of these strings differ in exactly 4 bits.

Answer: 8


Question: Let \( f:(0,1) \to \mathbb{R}^n \) be a continuous injective function such that \( \lim_{x \to 0} f(x) \) and \( \lim_{x \to 1} f(x) \) do not exist. Is it true that \( f((0,1)) \) is contractible?

Answer: Yes


Question: Suppose there are ten genes that determine tallness, each represented by two bits (either 0 or 1). Each gene can be 00, 01, 10, or 11, and a person's height is determined by the number of '11' genes they possess. If both parents have a height of 7, meaning they each have seven '11' genes, what is the expected height of their offspring, assuming the bits of the three remaining non-'11' genes are uniformly distributed? Inheritance is such that for each gene, one bit is randomly chosen from the male and the other from the female.

Answer: \dfrac{22}{3}


Question: Determine if $d = 2$ is the only square-free natural number such that the fundamental unit of the quadratic field $\mathbb{Q}(\sqrt{d})$ is of the form $1 + \sqrt{d}$. If not, identify any other such values of $d$.

Answer: 2


Question: Assume that a positive series \(\sum_{k=1}^{\infty} a_k\) converges. Is it always true that \(\sum_{k = 1}^n \sqrt{a_k} = o(\sqrt{n})\)? Justify your answer.

Answer: Yes


Question: Find the degree of the field extension \([\mathbb{Q}(2^{1/5}, e^{i \pi/5}) : \mathbb{Q}]\).

Answer: 20


Question: Let $[a,b]$ be a closed interval in $\mathbb{R}$ and $f:[a,b] \rightarrow \mathbb{R}$ a non-decreasing function. Determine whether there exists a sequence of continuous non-decreasing functions $f_{n}:[a,b] \rightarrow \mathbb{R}$ such that $\{ f_{n} \}$ converges to $f$ in $L^{1}([a,b])$. Provide a justification for your answer.

Answer: Yes


Question: Let \( S \subsetneq \mathbb{Z} \) be an infinite set. Does there always exist a prime \( p \) such that the closure of \( S \) in the \( p \)-adic integers, \( \mathbb{Z}_p \), contains a rational integer \( n \notin S \)? In other words, does there exist a prime \( p \) and \( n \in \mathbb{Z} \setminus S \) such that for all \( k \), there is an \( s \in S \) with \( p^k \mid n-s \)?

Answer: Yes


Question: Let $(x_n)_{n \ge 0}$ be a sequence of real numbers defined by $x_0 > 0$ and $x_{n+1} = x_n + \frac{1}{\sqrt{x_n}}$. Determine the limit \( L = \lim_{n \to \infty} \frac{x_n^3}{n^2} \).

Answer: \dfrac{9}{4}


Question:
Let $f(r)$ be the number of integral points inside a circle of radius $r$ and center at the origin. Determine the value of \( \lim_{r\to\infty}\frac{f(r)}{r^2} \). Choose the correct answer from the following options:

(A) $1$

(B) $\pi$

(C) $2\pi$

(D) $\pi/2$

Answer: B


Question: Evaluate the integral \( \iint_R \frac{\sin(x-y)}{x^2+y^2+1} \, dA \), where \( R = \{(x,y): -a \le x \le a, -b \le y \le b\} \) and \( a \) and \( b \) are positive real numbers, using symmetry arguments.

Answer: 0


Question: Given the definition of the less-than relation in the construction of real numbers, where \( a < b \) means that \( a \) is contained in \( b \) and \( a \neq b \), does this imply that all negative numbers are contained in any positive number?

Answer: Yes


Question: Suppose that $P$ is the polynomial of least degree with integer coefficients such that \(P(\sqrt{7} + \sqrt{5}) = 2(\sqrt{7} - \sqrt{5})\). Find \(P(2)\).

Answer: 40


Question: Let $a$ be the solution of the equation $xe^x = e^2$ and $b$ be the solution of the equation $x\ln x = e^2$. Find the value of $ab$.

Answer: e^2


Question:
Determine whether the following equality holds for an integer $n \ge 1$ and interval $[a,b]$:
\[ \inf_{x_0,x_1,...,x_n\in[a,b], x_0<x_1<...<x_n} \sup_{x\in [a,b]} \prod_{i=0}^n |(x-x_i)| = \inf_{P\in \mathcal P_n } \sup_{x\in [a,b]} |x^{n+1}-P(x)|. \]

Answer: Yes


Question: Find all values of \(a\) for which the image of the function \(y=\frac{\sqrt{a}-2\cos x+1}{\sin^2x+a+2\sqrt{a}+1}\) contains the interval \([2, 3]\).

Answer: 0


Question: If $\Omega$ is a simply-connected domain in $\mathbb{R}^n$ and $f$ is an injective continuous map from $\Omega$ to $\mathbb{R}^n$, is it necessary that $f(\Omega)$ is a simply-connected domain?

Answer: Yes


Question: Let $p(z)$ be a polynomial of degree $n > 1$. Determine the condition under which $p(\overline{z})$ is complex differentiable at $z = 0$. Specifically, show that $p(\overline{z})$ is complex differentiable at $z = 0$ if and only if the coefficient $a_1 = 0$.

Answer: a_1 = 0


Question: What is the probability of randomly selecting the number 13 from the natural numbers \( \mathbb{N} \) if the probability of selecting any specific number \( k \) is given by \( P(\{k\}) = \frac{1}{2^k} \)?

Answer: \dfrac{1}{8192}


Question:
Determine which of the following expressions correctly represents the tangent line to a curve in projective space at the point $(a:b:c)$:
1. $\left( \frac{\partial P}{\partial x} (a,b,c) \right )x + \dots =0$
2. $\left( \frac{\partial P}{\partial x} (a,b,c) \right )(x-a) + \dots =0$

Answer: 1


Question: Find the minimum value of \[ A = \frac{1}{\sqrt{a^2+b^2+c^2+1}} - \frac{2}{(a+1)(b+1)(c+1)} \] for \( a, b, c > 0 \).

Answer: -1


Question: Determine the number of distinct patterns that can be created by filling a 5x5 grid using 3 colors, considering symmetry. You do not have to use all three colors in each pattern.

Answer: 105918450471


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

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


Question: Find the area of the region in the XY plane defined by the set \(S = \{(x, y) : x^2 + y^2 \leq 144 \text{ and } \sin(2x + 3y) \leq 0\}\).

Answer: 72\pi


Question: A toll bridge has a probability density function for the time in minutes needed to collect tolls from motorists given by \( f(x) = 2\exp(-2x) \) for \( x \in [0, \infty) \). A motorist approaches the bridge and observes 50 vehicles waiting in a queue at a single toll booth. Using the central limit theorem, calculate the approximate probability that the motorist will have to wait more than 25 minutes before reaching the front of the queue.

Answer: 0.5


Question: Evaluate the contour integral \( \oint_{|z|=2} \frac{2z-1}{z(z-1)} \, dz \) taken counterclockwise around the circle \(|z|=2\).

Answer: 4\pi i


Question:
Consider the initial value problem (IVP) given by the differential equation \( \frac{dy}{dx} = \lambda \sin(x+y(x)) \) with the initial condition \( y(0) = 1 \), where \( \lambda \in \mathbb{R} \). Determine which of the following statements is true:

1. No solution exists in any neighborhood of 0.
2. No solution exists in \( \mathbb{R} \) if \( |\lambda| < 1 \).
3. A solution exists in a neighborhood of 0.
4. A solution exists in \( \mathbb{R} \) only if \( |\lambda| > 1 \).

Select the correct statement.

Answer: 3


Question: Evaluate the contour integral \( \oint_{|z-1/2|=3/2} \frac{\tan(z)}{z} \, dz \) using the residue theorem.

Answer: -4i


Question: Determine whether the limit \( \lim_{k\rightarrow\infty} \int_0^1 |\cos{(kx)}|\,dx \) exists.

Answer: \dfrac{2}{\pi}


Question: Find the maximum and minimum values of the function \( f(x,y) = e^{x^2+y^2} \) on the region defined by \( x^4+2x^2y^2+y^4=4 \).

Answer: e^2


Question: Suppose $U$ is an open set in $\Bbb R^n$, and $f\colon U\to \Bbb R$ is a continuous function that is uniformly continuous on every bounded subset of $U$. Can $f$ be continuously extended to the closure of $U$ in $\Bbb R^n$?

Answer: Yes


Question:
Given two noncollinear and nonzero vectors \( \bar{q} \) and \( \bar{p} \), and the equation \((b-c)\bar{p} \times \bar{q} + (c-a)\bar{p} + (a-b)\bar{q} = 0\), where \(a\), \(b\), and \(c\) are the lengths of the sides of a triangle, determine the type of triangle. Choose from the following options:

(A) Right-angled
(B) Obtuse-angled
(C) Equilateral
(D) Isosceles

Answer: C


Question: Find the coefficient of \(x^{16}\) in the expansion of \((x^2+5x+6)^4(x^3+6x^2+11x+6)^3\).

Answer: 38


Question: Given that $A(z_1)$, $B(z_2)$, and $C(z_3)$ are the vertices of a triangle $\triangle ABC$ with $|z_1| = |z_2| = |z_3| = 1$, and there exists an $\alpha \in \left(0, \frac{\pi}{2}\right)$ such that $z_1 + z_2 \cos \alpha + z_3 \sin \alpha = 0$, find the value of $\bar{z}_2 z_3 + z_2 \bar{z}_3$. \[ \]

Answer: 0


Question: In the symmetric group S8, consider the permutations \( a = (147)(258) \) and \( b = (14)(2578) \). Determine the number of permutations \( s \) such that \( as = sb \).

Answer: 0


Question: Determine whether there exists a positive integer \( k \) such that both \( 4k+1 \) and \( 9k+1 \) are perfect squares.

Answer: No


Question: Calculate the integral \( \int_{C}{xy^{2}dy-yx^{2}dx} \) over the circle \( C: x^{2}+(y-1)^{2}=1 \) using Green's theorem.

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


Question: Evaluate the limit \( \lim_{n\to\infty} \frac{S_1(n)S_5(n)-S_3(n)^2}{S_7(n)} \), where \( S_k(n) = \sum_{r=1}^n r^k \).

Answer: \dfrac{1}{6}


Question: Given the equations \( \cos x + 2 \cos y + 3 \cos z = 0 \), \( \sin x + 2 \sin y + 3 \sin z = 0 \), and \( x + y + z = \pi \), find the value of \( \sin 3x + 8 \sin 3y + 27 \sin 3z \).

Answer: 0


Question: Find the value of \( g\left(\frac{1}{2}\right) \) where \( g(a) = \lim_{h \to 0^+} \int_h^{1-h} t^{-a} (1-t)^{a-1} \, dt \) for \( a \in (0, 1) \), and it is given that \( g(a) \) is differentiable on \( (0, 1) \).

Answer: \pi


Question: Find the splitting field \(E\) of the polynomial \(f(x) = x^3 + 1\) over the field \(F = \mathbb{Z}_2\), and compute the degree \([E:F]\).

Answer: 2


Question: Determine whether there exists a normal proper subgroup of the symmetric group $S_n$. If such a subgroup exists, provide an example.

Answer: A_n


Question: Evaluate the sum \( \sum^{\infty}_{n=1} \frac{n^2}{6^n} \).

Answer: \dfrac{42}{125}


Question: In a social party, there are 8 girls and 7 boys sitting around a round table. Calculate the average number of pairs of seats that are adjacent and consist of one girl and one boy.

Answer: 8


Question: Let \(X_1, X_2, \ldots, X_n\) be independent and identically distributed random variables following a standard normal distribution \(N(0,1)\). Determine whether the following statement is true: \[ \frac{1}{\sqrt{n}}\sum_{i=1}^n\mathbf{1}\left(0 < X_i \leq \frac{1}{\sqrt{n}}\right) \xrightarrow{P} 0 \]

Answer: False


Question: Find the splitting field of the polynomial \( f(x) = x^4 + 2 \) over the field of rational numbers \( \mathbb{Q} \), and determine the degree of this splitting field over \( \mathbb{Q} \).

Answer: 8


Question: Evaluate the surface integral \( \int F \cdot \text{d}S \) for the vector field \( F = [2y + 3, xz, yz - x] \), where \( S \) is the surface of a cylinder with unit radius centered around the origin, and \( -1 < z < 1 \).

Answer: 0


Question: Find all nilpotent elements in the ring \( \mathbb{Z}_3[x]/(x^2-1) \).

Answer: 0


Question: Calculate the value of \(X^2 + Y^2\) where \(X = \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{4\pi}{7}\right) + \sin\left(\frac{8\pi}{7}\right)\) and \(Y = \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{8\pi}{7}\right)\). Use complex numbers to find the solution.

Answer: 2


Question: Evaluate the limit: \[ \lim_{x\rightarrow \infty}\frac{\left(2^{x^n}\right)^{\frac{1}{e^x}}-\left(3^{x^n}\right)^{\frac{1}{e^x}}}{x^n} \] where \( n \in \mathbb{N} \).

Answer: 0


Question:
Given the context-free grammar with the following productions:

- S → aS | A
- A → aAb | bAa | ε

Which of the following strings is generated by this grammar?

1. aabbaba
2. aabaaba
3. abababb
4. aabbaab

Select the correct option.

Answer: 4


Question: Compute the value of \( \sum_{n=0}^{\infty}\int_{0}^{\pi/2}(1-\sqrt{\sin(x)})^{n}\cos(x)dx \). Justify the use of the Lebesgue Dominated Convergence Theorem to interchange the order of summation and integration.

Answer: 2


Question: Find the minimum value of \( P = \frac{x+1}{1+y^2} + \frac{y+1}{1+z^2} + \frac{z+1}{1+x^2} \) given that \( x, y, z > 0 \) and \( x+y+z=3 \).

Answer: 3


Question: Determine whether the intersection of the fields \(\mathbb{Q}(\sqrt[3]{2})\) and \(\mathbb{Q}(\sqrt[3]{5})\) is equal to \(\mathbb{Q}\). Prove your answer.

Answer: \mathbb{Q}


Question: Find the volume of the intersection of the two cylinders defined by the inequalities: \(0 \le x^2 + z^2 \le 1\) and \(0 \le y^2 + z^2 \le 1\).

Answer: \dfrac{16}{3}


Question: Given the endomorphism \( \varphi: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) defined by \( \varphi(0,1,1) = (0,1,1) \), \( \varphi(2,2,0) = (0,0,0) \), and \( \varphi(1,0,0) = (-1,0,0) \), calculate \( \varphi^{106}(1,1,4) \).

Answer: (4, 4, 4)


Question: Evaluate the integral \( \int_0^1 \frac{\sin(x)}{x} \, dx \) with an accuracy better than \(10^{-4}\). Use the Taylor polynomial of order \(2n\) for \(\sin(x)\) and evaluate the remainder \(R_n(x)\) to achieve the desired accuracy.

Answer: 0.9461


Question: Let $X$, $Y$, and $Z$ be independent exponentially distributed random variables with means $1/\lambda_X$, $1/\lambda_Y$, and $1/\lambda_Z$, respectively. Calculate the probability $P(X > \max\{Y, Z\})$. Additionally, consider a queueing system with three cash boxes, each with independent exponential service times with a mean of 5. Determine the probability that the last cash box to finish serving is the one that started last. Express your answer as a fraction.

Answer: \dfrac{1}{3}


Question: Find the value of \( a \) such that the point on the curve \( f(x) = e^x \) closest to the line \( y = \frac{x}{a} - 1 \) has a \( y \)-coordinate of \( \frac{1}{e} \).

Answer: e


Question: Find the value of \( p \) such that the set \( \mathbb{R} \setminus \{p\} \) forms a group under the operation \( a \circ b = a + b - \frac{5ab}{3} \).

Answer: \dfrac{3}{5}


Question: Determine the integer value of \( a \) such that \( x^2 - x + a \) divides the polynomial \( x^{13} + x + 90 \).

Answer: 2


Question: Consider the infinite sum \( S = \sum_{k=1}^\infty a_k \), where each \( a_k \) is a positive irrational number. Is it possible for \( S \) to be rational, given that none of the \( a_k \)'s is a linear combination of the others? Here, a linear combination means there exist rational numbers \( u, v \) such that \( a_i = ua_j + v \). Provide a justification for your answer.

Answer: Yes


Question: Determine the values of $x \in \mathbb{R}$ for which the series \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \cos(nx) \) converges.

Answer: \mathbb{R}


Question: Evaluate the improper Lebesgue integral: \[ \int_0^\infty \cos(2x)e^{-x}\,d\lambda(x) \] where \( \lambda \) denotes the Lebesgue measure.

Answer: \dfrac{1}{5}


Question: Sidney Crosby is standing on the goal line, three feet outside one of the goal posts of a hockey goal that is six feet wide. How far should he move out perpendicular to the goal line to maximize the angle at which he can shoot at the goal?

Answer: 3\sqrt{3}


Question: Find the largest positive integer \( n \) such that \( 2^n \) divides \( 3^{4096} - 1 \).

Answer: 14


Question: Evaluate the sum of the series: \[ \sum_{n=0}^{\infty} \frac{(-1)^n (n+1)}{n!} \]

Answer: 0


Question:
Determine whether the following proposition holds:

Let $a < b$ and $y:(a,b) \to \mathbb{R}$ be a function. Furthermore, let $F\colon (a,b) \times \mathbb{R} \to \mathbb{R}$ be a continuously differentiable function such that:
\[ \forall x \in (a,b): F(x,y(x)) = 0 \]
\[ \forall (x,y) \in (a,b) \times \mathbb{R}: \frac{\partial}{\partial y}F(x,y) > 0. \]
Then, is $y$ continuously differentiable?

Answer: Yes


Question: For a village with $M$ villagers, a virus is introduced by a wanderer. Each week, every infected villager coughs on $n$ random villagers, infecting them if they are not immune, and then becomes immune. Determine the value of $n$ that maximizes the expected survival time of the virus.

Answer: 1


Question: Find the residue of the function \( \cot z \) at the point \( z = n\pi \), where \( n \) is an integer.

Answer: 1


Question: Determine the order of the pole of the function \( f(z) = \frac{\cos z - 1}{z^3} \) at \( z = 0 \).

Answer: 1


Question: Determine whether the equation \( \sum_{i=1}^n (i^5 + i^7) = 2\left( \sum_{i=1}^n i \right)^4 \) holds for all natural numbers \( n \).

Answer: Yes


Question: Determine whether the following statement is true or false: If $m \ge n$, $U \in M_{mn}$ such that $U^TU = \mathrm{id}_n$, then for any $A \in M_{mm}$, \[ \det\left(U^TAU\right) = \det\left(A\right). \]

Answer: False


Question: Compute the area of the image of the unit disc \(D = \{z \mid |z| < 1\}\) under the map \(f(z) = z + \frac{z^2}{2}\).

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


Question:
Determine the radius of convergence \( R \) of the power series \( \sum_{n=1}^{\infty}{\frac{\sin n!}{n!}} {x^n} \). Which of the following statements is true?

- \( R \geq 1 \)
- \( R \geq e \)
- \( R \geq 2e \)
- All are correct

Answer: D


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

Answer: \infty


Question: Find all natural numbers \( n \) such that \( \sqrt{1+5^n+6^n+11^n} \) is a natural number.

Answer: 0


Question: Calculate the volume of the tetrahedron in \(\mathbb{R}^4\) with vertices \((1,0,0,1), (-1,2,0,1), (3,0,1,1), (-1,4,0,1)\).

Answer: \dfrac{2}{3}


Question: Let \(X_1, X_2,\) and \(X_3\) be independent and identically distributed random variables, each following a uniform distribution on the interval \((0, 1)\). Calculate the probability \(P(X_1 > X_2 + X_3)\).

Answer: \dfrac{1}{6}


Question: Consider a countably infinite set $A$ and a relation $r$ on $A$. The relation $r$ is complete, meaning for any $a, b \in A$, either $a r b$, $b r a$, or both hold. The relation $r$ is also transitive, meaning for any $a, b, c \in A$, if $a r b$ and $b r c$, then $a r c$. Must there exist a function $f: A \rightarrow \mathbb{R}$ such that for any $a, b \in A$, $a r b$ if and only if $f(a) \geq f(b)$? Provide a justification for your answer.

Answer: Yes


Question: Let $E$ be a simple $R$-module. Suppose there exist positive integers $n$ and $m$ such that $E^{(n)} \cong E^{(m)}$ as $R$-modules. Prove that $n = m$. Use the fact that $\text{End}_{R}(E)$ is a division ring and consider the dimensions of the endomorphism rings.

Answer: n = m


Question: Let $M$ be a von Neumann algebra. Suppose that $a,b \in M_+$, $x \in M$, and that the matrix \( \begin{pmatrix} a & x \\ x^* & b \end{pmatrix} \) is positive. Is there a $y \in M$ with $\|y\| \leq 1$ such that \( x = a^{1/2} y b^{1/2} \)?

Answer: Yes


Question: Let \( a, b \in \mathbb{Z}^+ \) such that \( a \equiv b \pmod{49} \) and \( \gcd(a, 49) = 1 \). Find a positive integer \( n > 1 \) such that \( b^n \equiv a \pmod{49} \).

Answer: 43


Question: Use Stokes' Theorem to evaluate the integral \( \iint_S \operatorname{curl} F \cdot \hat{n}\, dS \), where \( F =\langle xyz, x, e^{xy} \cos(z)\rangle \) and \( S \) is the hemisphere \( x^2+y^2+z^2=25 \) for \( z \geq 0 \) oriented upward.

Answer: 25\pi


Question: If $f$ is continuous on $[0,1]$, compute \( \lim\limits_{x\to 0^+} x\int_x^1 \frac{f(t)}{t}dt \).

Answer: 0


Question: In an urn containing 2 red balls, 1 blue ball, and 1 green ball, John draws three balls, one at a time, with replacement. Let $B$, $R$, and $G$ represent the number of blue, red, and green balls drawn, respectively. Calculate the variance of the difference between the number of blue and red balls drawn, $\operatorname{Var}(B-R)$. Express your answer in terms of expectations: $\operatorname{Var}(B-R) = E[(B-R)^2] - (E[B-R])^2$. Compute this variance.

Answer: \dfrac{33}{16}


Question: Given the elliptic curve $y^2 = x^3 + 2x + 2$ over $\mathbb{Z}_{17}$, with order 19, calculate the result of the point addition $6\cdot(5, 1) + 6\cdot(0,6)$. Verify if the result is $(7, 11)$. Provide the correct result if it differs.

Answer: (7, 11)


Question:
Let $f$ be a polynomial satisfying $f(x^2) - x f(x) = x^4(x^2 - 1)$ for $x \in \mathbb{R}^+$. Determine which of the following statements is correct:

A) $f$ is an even function
B) $f$ is an odd function
C) $\displaystyle\lim_{x\to \infty} \frac{f(x)}{x^3}=1$
D) $\displaystyle\lim_{x\to \infty} \left(\frac{f(x)}{x^2}-x \right)$ exists and is equal to a non-zero quantity.

Answer: BC


Question: Determine whether there exists a sequence of positive continuous functions $f_n: E \to \mathbb{R}$, where $E \subset \mathbb{R}$ is compact, such that for every $x \in E$, $f_n(x) \to f(x)$ for some $f \in L^\infty(E)$, and $\int_E f_n(x) \, dx \to C < \infty$, but $C \neq \int_E f(x) \, dx$. All integrals are Lebesgue integrals.

Answer: Yes


Question: Given the matrix \( \Delta = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \), where \( D_1, D_2, \) and \( D_3 \) are the co-factors of \( c_1, c_2, \) and \( c_3 \) respectively, such that \( D_1^2 + D_2^2 + D_3^2 = 16 \) and \( c_1^2 + c_2^2 + c_3^2 = 4 \), find the maximum value of \( \Delta \).

Answer: 8


Question: Determine the automorphism group of the group \(\mathbb{Z}_2 \oplus \mathbb{Z}_2\).

Answer: S_3


Question: Compute the limit: \[ \lim_{n\to \infty} \int_0^\infty \int_0^\infty \frac{n}{x} \sin\left( \frac{x}{ny} \right) e^{-\frac{x}{y} - y}\ dx\ dy. \]

Answer: 1


Question: Consider the polynomial \( p(z) = 1 + z + \frac{z^2}{2} + \cdots + \frac{z^m}{2^{m-1}} + 3z^n \). Determine the number of zeroes of \( p(z) \), counting multiplicities, that lie within the unit disk.

Answer: n


Question: Evaluate the line integral \( \int_C y^2 \,dx + 2xy\,dy \) where \( C \) is the curve parameterized by \( (t^8, \sin^7(\frac{\pi t}{2})) \) for \( 0 \leq t \leq 1 \).

Answer: 1


Question:
Given that $A$ and $B$ are positive acute angles satisfying the equations $3\cos^2 A + 2\cos^2 B = 4$ and $\frac{3\sin A}{\sin B} = \frac{2\cos B}{\cos A}$, find the value of $A + 2B$. Choose the correct answer from the following options:

1. $\frac{\pi}{4}$
2. $\frac{\pi}{3}$
3. $\frac{\pi}{6}$
4. $\frac{\pi}{2}$

Answer: D


Question:
Determine which of the following conditions can be satisfied by a non-constant analytic function $f$ on the unit disk $D = \{z \in \mathbb{C} : |z| < 1\}$. For each $n = 2, 3, 4, \ldots$, the function $f$ satisfies:

(a) $f\left(\frac{i}{n}\right) = 0$.

(b) $f\left(\frac{1}{n}\right) = 0$.

(c) $f\left(1 - \frac{1}{n}\right) = 0$.

(d) $f\left(\frac{1}{2} - \frac{1}{n}\right) = 0$.

Which condition(s) can be satisfied by such a function?

Answer: C


Question: Let $X$ be a holomorphic vector bundle over a complex manifold $Y$. Determine whether there exists an analytic subset $Z$ of $Y$ such that the restriction of $X$ to $Y \setminus Z$ is a trivial vector bundle.

Answer: No


Question:
A Markov chain $X_0,X_1,X_2,\ldots$ has the transition probability matrix:
$$P=\begin{Vmatrix}
0.3 & 0.2 & 0.5\\
0.5 & 0.1 & 0.4 \\
0 & 0 & 1
\end{Vmatrix}$$
and starts in state $X_0=0$. Eventually, the process will end up in state 2. What is the probability that the time $T=\text{min}\{n\geq 0;X_n=2\}$ is an odd number?

Answer: \dfrac{90}{133}


Question: Determine the number of \(2 \times 2\) matrices \(A\) with elements from the set \(\{-1, 0, 1\}\) such that \(A^2 = I\), where \(I\) is the identity matrix of order 2.

Answer: 14


Question: Is the size of the Galois group of a polynomial always equal to the factorial of the degree of the polynomial? Justify your answer.

Answer: No


Question: Let $f,g \in \mathbb{Q}[x]$ be polynomials such that the set of values $\{f(a) : a \in \mathbb{Q}\}$ is a subset of the set $\{g(a) : a \in \mathbb{Q} \}$. Determine whether there must exist a polynomial $h \in \mathbb{Q}[x]$ such that $f(x) = g(h(x))$ for all $x \in \mathbb{Q}$. Provide a justification for your answer.

Answer: Yes


Question:
Given an ellipse with a point \((a \cos\theta, b \sin\theta)\) on it, the tangent at this point meets the auxiliary circle in two points. The chord joining these points subtends a right angle at the center of the ellipse. Determine the eccentricity of the ellipse from the following options:

A) \((1+(\cos\theta)^2)^{-\frac{1}{2}}\)

B) \(1+(\sin\theta)^2\)

C) \((1+(\sin\theta)^2)^{-\frac{1}{2}}\)

D) \(1+(\cos\theta)^2\)

Answer: C


Question: Evaluate the limit \( \lim \limits_{ x \to \infty} \left((x+1)^k - x^k\right) \), where \( 0 < k < 1 \).

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\frac{\cos x}{1+x^2}dx \) using methods from complex analysis.

Answer: \dfrac{\pi}{e}


Question: Determine a Lipschitz constant for the function \( f(t,x) = x(\sin(t) - x^2) \) in the region where \( 0 \le t \le 2\pi \) and \( |x| \le M \).

Answer: 1 + 3M^2


Question: Find the spectrum of the operator $T = S + S^*$, where $S$ is the unilateral shift operator on $\mathcal{l}^2$ and $S^*$ is its adjoint, the backward shift operator.

Answer: [-2, 2]


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

Answer: 0


Question: Calculate the limit: \[ \lim_{n\to\infty}\frac{(-3)^{n+1}-4^n+5}{3^{n+2}+2^n-5}. \]

Answer: -\infty


Question: Find the unit digit of the expression: \[ \left\lfloor \frac{10^{20000}}{100^{100} + 3} \right\rfloor \]

Answer: 3


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

Answer: 1


Question: Let $X$ and $Y$ be independent and identically distributed random variables with an exponential distribution of parameter 2. Define $Z = \min \{X, Y\}$. Calculate the probability $P(Z \leq 4 \mid X > 3)$. Provide your answer as a single numerical value.

Answer: 1 - e^{-10}


Question: Let \( h \in L_1(\mathbb{R}) \) and define the function \( g(t) = \int_{t}^{\infty} h(\tau) \, \mathrm{d}\tau \). Prove that \( \lim_{t \to \infty} g(t) = 0 \).

Answer: 0


Question: Evaluate the double integral \( \int_0^a \int_0^a \frac{dx \, dy}{(x^2+y^2+a^2)^{\frac{3}{2}}} \).

Answer: \dfrac{\pi}{6a}


Question: Given the function \( S_n(t) = \sum_{k=1}^n a_k \sin((k-1/2)\pi t) \), which is periodic with period 4, determine if it is possible to deduce the coefficients \( a_k \) by observing \( S_n \) only on the interval \([0,1]\).

Answer: Yes


Question: Find the probability that for random variables \(a, b, c,\) and \(d\) each independently and uniformly distributed over \([-1, 1]\), the inequality \(ad - bc > 0\) holds.

Answer: \dfrac{1}{2}


Question: Find the smallest positive integer \( n \) such that there exists a complex number \( A \) for which \( A^n = 1 \) and \( (A+1)^n = 1 \).

Answer: 6


Question: Let $k$ be a field and $G$ a finite group. Determine whether every sub-Hopf algebra over $k$ of the group algebra $k[G]$ is of the form $k[U]$ for a subgroup $U$ of $G$. Provide a justification for your answer.

Answer: Yes


Question: The slope of the hyperbola \(b^2 x^2 - a^2y^2 = a^2 b^2\) at the upper end of its right-hand latus rectum is \(\frac{4}{3}\). Determine the eccentricity of the hyperbola.

Answer: \dfrac{4}{3}


Question: Find all pairs of positive integers \((x, y)\) greater than 1 such that \(x^2 - y^3 = 1\).

Answer: (3, 2)


Question: Let $X$ be a random variable with probability density function $f(x) = xe^{x-2}$ for $1 < x < 2$ and $0$ elsewhere. Let $F(x)$ be the cumulative distribution function of $X$. Determine the probability density function of the random variable $Y = F(X)$. Provide your answer in terms of $y$, the value of the cumulative distribution function.

Answer: 1


Question: If a random variable X follows a t-distribution with n degrees of freedom, what is the probability distribution of the random variable X^2?

Answer: F(1, n)


Question: A hoard of gold pieces is to be divided among a band of 15 pirates. When they attempt to divide the coins, 3 coins are left over. After a disagreement, only 7 pirates remain to claim the hoard, and dividing the coins among them leaves 2 coins. Following another disagreement, 4 pirates remain, and they are able to divide the hoard evenly. What is the minimum number of gold pieces in the hoard?

Answer: 408


Question: Compute the product \((a^4 + a^2)(a^3 + a + 1)\) in the finite field \(\Bbb F_{32} = \Bbb F_2[X]/(X^5 + X^2 + 1)\), where \(a = [X]\) is the congruence class of \([X]\). Reduce the result to a polynomial of degree less than 5.

Answer: a^3


Question: Evaluate the integral \( \int_{0}^{\pi/2} \sqrt{\sin{2x}} \cdot \sin{x} \, dx \).

Answer: \dfrac{\pi}{4}


Question: If $A$ and $B$ are subsets of $\mathbb{N}$ with defined asymptotic densities, and $A \cap B = \emptyset$, is the asymptotic density of $A \cup B$ also defined? Justify your answer.

Answer: Yes


Question: Does there exist a surface defined by a continuous function $R^2 \rightarrow R$ such that any two different triangles on it, formed by three non-collinear points connected by geodesics, are not congruent?

Answer: Yes


Question: Determine the topological entropy of the dyadic Toeplitz system, which is a subshift of \(\{0,1\}^{\mathbb{Z}}\). The system is defined as the orbit closure of a point \(x\) constructed by placing zero on every second coordinate, resulting in the sequence \((...*0*0*0*0*...)\), and then placing one instead of every second \(*\), resulting in \((...010*010*010...)\). In this system, the coordinates of element \(x\) can be decomposed into arithmetic progressions, on which \(x\) is constant.

Answer: 0


Question: Given two distributions \( p \) and \( q \) over the same finite set \( X \) with a total variation distance \( TV(p,q) = \epsilon \), and a function \( f: X \to \mathbb{R} \) such that \( 0 \leq \sum_{x \in X} f(x) p(x) \leq 1 \) and \( 0 \leq \sum_{x \in X} f(x) q(x) \leq 1 \), determine the best upper bound for \( \Big|\sum_x f(x)(p(x)-q(x))\Big| \) as a function of \( \epsilon \).

Answer: 1


Question:
Let \(X_1, X_2, \ldots\) be an i.i.d. sequence of random variables with \(E[X_1] = 1/2\) and \(\operatorname{Var}[X_i] = 2\). Compute the probability:
\[ P\left(\lim_{n \to \infty} \frac{X_1 + X_2 + \cdots + X_n}{n} > 1\right) \]

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 1^+}\frac{\sqrt{x+1}+\sqrt{x^2 -1}-\sqrt{x^3+1}}{\sqrt{x-1}+\sqrt{x^2 +1}-\sqrt{x^4+1}}.$$

Answer: \sqrt{2}


Question: Determine the number of surjective functions from the set \( N_{10} = \{1, 2, \ldots, 10\} \) to the set \( N_4 = \{1, 2, 3, 4\} \) such that each element in \( N_4 \) has exactly 2 pre-images. Provide your answer as a single number.

Answer: 0


Question: Consider a random point $X = (X_1, \ldots, X_d)$ uniformly distributed on the sphere of radius $\sqrt{d}$ in $\mathbb{R}^d$. It is known that as $d \to \infty$, the marginal distribution of $X_1$ converges weakly to the standard normal distribution $N(0,1)$. For a fixed integer $k \ge 2$, does the marginal distribution of $(X_1, \ldots, X_k)$ converge weakly to the multivariate normal distribution $N(0, I_k)$ as $d \to \infty$?

Answer: Yes


Question: Let $S$ be a $*$-algebra with two $C^*$-norms, $\left\Vert \cdot \right\Vert _{1}$ and $\left\Vert \cdot \right\Vert _{2}$, such that $\left\Vert \cdot \right\Vert _{1} \leq \left\Vert \cdot \right\Vert _{2}$. Denote by $A$ and $B$ the $C^*$-algebras generated by these norms, respectively. Consider a sequence $(x_n)_{n \in \mathbb{N}}$ that converges to $0$ in the norm $\left\Vert \cdot \right\Vert _{1}$ and to $b$ in the norm $\left\Vert \cdot \right\Vert _{2}$. Does this imply that $b = 0$?

Answer: No


Question: Is there a set of edges \(E\subseteq\big\{\{m,n\}: m,n \in \mathbb{N} \text{ and } m\neq n\big\}\) such that the graph \((\mathbb{N}, E)\) is connected and every vertex has degree 2?

Answer: Yes


Question:
Let \(f(x) = x^3 + ax^2 + bx + c\) and \(g(x) = x^3 + bx^2 + cx + a\), where \(a, b, c\) are integers with \(c \neq 0\). Suppose that the following conditions hold:

1. \(f(1) = 0\)
2. The roots of \(g(x) = 0\) are the squares of the roots of \(f(x) = 0\).

Find the value of \(a^{2013} + b^{2013} + c^{2013}\).

Answer: -1


Question:
Consider the heat equation given by \( \frac{\partial{^2u}}{\partial{x^2}}=\frac{\partial{u}}{\partial{t}} \) with the initial condition \( u(x,0) = \cos(x) \) and the condition that \( u(x,t) \to 0 \) as \( t \to \infty \). Determine which of the following expressions correctly represents the solution \( u(x,t) \):

1. \( u=\sum^\infty_{n=1}a_n\sin(nx+b_n)e^{-nt} \), where \( a_n,b_n \) are arbitrary constants.
2. \( u=\sum^\infty_{n=1}a_n\sin(nx+b_n)e^{-n^2t} \), where \( a_n,b_n \) are arbitrary constants.
3. \( u=\sum^\infty_{n=1}a_n\cos(nx+b_n)e^{-nt} \), where \( a_n \) are not all zeros and \( b_n=0 \).
4. \( u=\sum^\infty_{n=1}a_n\cos(nx+b_n)e^{-n^2t} \), where \( a_1\neq0,\;a_n=0 \text{ for } n>1,b_n=0 \text{ for } n\geq1 \).

Answer: 4


Question: Let \( A \in M_2 \) be a matrix with characteristic polynomial \( p(x) = x^2 - 3x - 5 \). Find the values of \( a \) and \( b \) such that \( -2A^2 + 4A - 3I_2 = aA + bI_2 \). What is \( a + b \)?

Answer: -15


Question: Determine whether the following proposition is true: For a finite field extension $K/k$, the extension contains a primitive element (simple extension) if and only if the number of intermediate fields between $k$ and $K$ is finite.

Answer: True


Question: Determine the number of real solutions to the equation \(f(f(f(f(x)))) = 1\), where \(f(x) = x - \frac{1}{x}\).

Answer: 16


Question: Let \(( \sqrt{2} + 1)^{1000} = a + b \sqrt{2}\), where \(a\) and \(b\) are integers. Find the greatest common factor of \(b\) and 81.

Answer: 3


Question: Let $V$ be the vector space of all polynomials over $\mathbb{R}$, and let $W$ be the subspace of polynomials divisible by $x^4$. Find the dimension of the quotient space $V/W$. \( \text{Provide your answer as a single integer.} \)

Answer: 4


Question: Evaluate the limit: \[ \lim_{x\rightarrow0}\frac{\log(1+\sin{x})-\log(1+x)}{x-\tan{x}} \]

Answer: \dfrac{1}{2}


Question: Let X and Y have a joint uniform distribution on the triangle with vertices at $(0, 2)$, $(2, 0)$, and the origin. Find the expected value $E(Y | X = 1/2)$. The joint density function is $f(x,y) = \frac{1}{2}$ on the region of the triangle.

Answer: \dfrac{3}{4}


Question: Determine the number of countable partitions of the set of natural numbers \( \mathbb{N} \) into countable sets.

Answer: 2^{\aleph_0}


Question: Calculate the limit: \[ \lim_{n \to \infty} \sum_{k=n}^{5n} \binom{k-1}{n-1} \left(\frac{1}{5}\right)^{n} \left(\frac{4}{5}\right)^{k-n} \]

Answer: \dfrac{1}{2}


Question: Evaluate the integral: \[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt[3]{\sin{x}}}{\sqrt[3]{\sin{x}}+\sqrt[3]{\cos{x}}} \, dx \]

Answer: \dfrac{\pi}{12}


Question: Determine whether every regular Noetherian domain has finite Krull dimension. Provide a justification for your answer.

Answer: No


Question:
Which Möbius transformation maps the unit disc onto the right half-plane?

(a) \( f(z) = \frac{z-i}{z+i} \)
(b) \( f(z) = \frac{z-1}{z+1} \)
(c) \( f(z) = \frac{1+z}{1-z} \)
(d) \( f(z) = i\left(\frac{1+z}{1-z}\right) \)

Select the correct option.

Answer: C


Question: Using Rouché's theorem, determine how many roots the polynomial \( p(z) = z^4 - 5z + 1 \) has in the region \( |z| > 2 \).

Answer: 0


Question: Given that $a$ and $b$ are roots of the equation $x^2 - 3cx - 8d = 0$, and $c$ and $d$ are roots of the equation $x^2 - 3ax - 8b = 0$, find the value of $a + b + c + d$.

Answer: 96


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

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


Question: If $G$ is a non-abelian group of order 27, what is the order of its commutator subgroup $G'$?

Answer: 3


Question: Given a Cauchy matrix \(C\) of size \(n \times n\) defined by \(C[i,j] = \frac{1}{x_i + y_j}\), and a vector \(a = [a_1, ..., a_n]\), determine if the matrix-vector product \(b = C \times a\) can be computed with a complexity less than \(O(n^2)\).

Answer: Yes


Question: Let $X = (0, \infty)$ be the set of positive real numbers, and let $G = \mathbb{Q} \setminus \{0\}$ be the multiplicative group of non-zero rational numbers. The group $G$ acts on $X$ by exponentiation: for $x \in X$ and $r \in G$, the action is given by $r \cdot x = x^r$. Determine whether the cardinality of the orbit space $X/G$ is equal to the cardinality of the continuum, i.e., $|X/G| = |X|$. Provide a justification for your answer.

Answer: Yes


Question:
Given the following conditions:

- \( \sum_{i \in I} p_i = 1 \) with \( 0 \le p_i \le 1 \) for all \( i \in I \)
- \( \sum_{i \in I} q_i = 1 \) with \( 0 \le q_i \le 1 \) for all \( i \in I \)
- \( 0 \le a_i, b_i, c_i, d_i \le 1 \) for all \( i \in I \)

If \( \sum_{i \in I} p_i a_i \le \sum_{i \in I} q_i b_i \) and \( \sum_{i \in I} p_i c_i \le \sum_{i \in I} q_i d_i \), does it follow that \( \sum_{i \in I} p_i a_i c_i \le \sum_{i \in I} q_i b_i d_i \)?

Determine whether this statement is true or false.

Answer: False


Question: Suppose $t_{n}$ is a sequence of positive real numbers such that $c_{1} \geq \lim \sup_{n\to \infty}t_{n}/n \geq \lim \inf_{n\to \infty}t_{n}/n \geq c_{2} > 0$, where $c_{1} \geq c_{2} > 0$ are positive constants. Does it follow that $\lim \inf_{N\to \infty}\dfrac{m(\cup^{N}_{n=1}[t_{n}-1,t_{n}+1])}{N} > 0$, where $m$ is the Lebesgue measure?

Answer: Yes


Question: Let \( \sigma : \mathbb{N} \rightarrow \mathbb{R} \) be an injective sequence of real numbers. Does there exist an infinite set \( A = \{ a_1, a_2, \ldots, a_n, \ldots \} \subset \mathbb{N} \) such that \( \sigma_{|A} \) is monotone and \( a_n = O(n^2) \)?

Answer: Yes


Question: Given a homogeneous function of degree 3, $f(x,y)$, with $f(6,9)=54$, find the value of $g'(2)$ where $g(t)=f(0.5t^2, 0.25t^3+1)$. Use the chain rule to determine the derivative.

Answer: 6


Question: What is the probability that a randomly chosen number from the interval [0, 1] is rational?

Answer: 0


Question: Let $A$ be the set of twice continuously differentiable functions on the interval $[0, 1]$ and let $B = \{f \in A : f(0) = f(1) = 0, f'(0) = 2\}$. Find the minimum value of $$\int_0^1 (f''(x))^2dx$$ for $f \in B$. Provide your answer as a single number.

Answer: 12


Question:
Evaluate the double integral in polar coordinates to find the volume of the 3D-body defined by \( z = y + \sqrt{4 - x^2} \) over the region where \( x \leq 0 \) and within the circle of radius 2:
\[
\int_{\pi}^{2\pi}\int_{0}^{2} \left( r\sin \theta + \sqrt{4-r^2\cos^2\theta} \right) r\,drd\theta
\]

Answer: \dfrac{16}{3}


Question: Determine whether the stabilizer in $O(n)$ of a rank $k$ matrix is isomorphic to $O(n-k)$. Let $X \in M_{n, k}(\mathbb{R})$ such that $\text{rank}(X) = k$. Consider the subgroup $O(n)_X = \{A \in O(n) : AX = X\}$. Is it true that $O(n-k) \cong O(n)_X$?

Answer: Yes


Question: Evaluate the limit \( \lim_{(x,y)\to(0,0)} \frac{yx^n}{x^6+y^2} \) for all \( n \ge 4 \).

Answer: 0


Question: Evaluate the limit of the parametric integral: \[ \lim_{x \to 0} \int_0^1 \frac{1}{x} \left( e^{-x^2(1+t)}(1+t)^{-1} - (1+t)^{-1} \right) dt. \]

Answer: 0


Question: Let \((y_1, y_2, \cdots, y_n, \cdots)\) be an element of \(\ell_2\). For any complex number \(\lambda\), consider the sequence defined recursively by \(x_1 = \lambda y_1\) and \(x_n = \lambda\left(y_n + \frac{x_{n-1}}{n-1}\right)\) for \(n = 2, 3, \cdots\). Is this sequence also in \(\ell_2\)?

Answer: Yes


Question: Given a sample of 100 units from an integrated-circuit production line, where 38 units are defective, find the mean of the posterior distribution of the proportion defective, \( \theta \), assuming a Beta prior distribution. The process has a long-term yield of 80% with a standard deviation of 0.04 for the proportion defective. Use this information to determine the parameters of the posterior distribution and calculate its mean.

Answer: 0.2905


Question: Consider a sequence \( \{x_n\} \) of real numbers. Suppose there exist two subsequences \( \{x_{n_k}\} \) and \( \{x_{m_j}\} \) that both converge to the same limit \( x \), with \( \{x_{n_k}\} \) being monotonically increasing and \( \{x_{m_j}\} \) being monotonically decreasing. Does the sequence \( \{x_n\} \) converge to \( x \)?

Answer: No


Question: Let \( \{ N(t) \}_{t \geq 0} \) be a Poisson process with intensity \( \lambda = 3 \). Compute the probability \( P\left[N(6) = 2 \,|\, N(8) = 4, N(3) = 1\right] \).

Answer: \dfrac{36}{125}


Question: Determine the values of \( a \) for which the improper integral \( \int_0^{\frac{1}{2}} \frac{1}{x|\log(x)|^a} \, dx \) is finite.

Answer: a > 1


Question: Evaluate the definite integral \( \int_0^{\pi} \frac{\sin 5x}{\sin x} \, dx \).

Answer: \pi


Question: If $C(K)$ is isomorphic to $C(H)$, are the compact Hausdorff spaces $K$ and $H$ necessarily homeomorphic?

Answer: No


Question: Evaluate the contour integral \( \int_{\delta B(0,2)} \frac{1}{1+z^2} \, dz \), where \( \delta B(0,2) \) is the boundary of the disk centered at the origin with radius 2.

Answer: 0


Question:
Determine all natural numbers \( m \) such that the sequence \( \langle b_n \rangle \) converges to a positive real number, where
\[
a_k = \frac{k^2 + mk}{(k+5)^2} \quad \text{and} \quad b_n = a_1 a_2 a_3 \ldots a_n.
\]

Answer: 10


Question: Determine the set of values of $x$ for which the series \( \sum^{\infty}_{n=1} \frac{e^n+1}{e^{2n}+n} x^n \) converges.

Answer: (-e, e)


Question: Find the volume of the region bounded by the paraboloid \(x^2 + y^2 + z = 7\) and the plane \(z = 1\).

Answer: 18\pi


Question: Find the eighth positive integer \( n \) for which the Catalan number \( C_n \) is odd, where the \( n^{\text{th}} \) Catalan number is given by \( C_n = \frac{1}{n+1}\binom{2n}{n} \).

Answer: 255


Question: Find the constant \( c \) such that the function \( f_{X,Y}(x,y) = \begin{cases} cx, & \text{if } x > 0, y > 0, 1 \leq x+y \leq 2, \\ 0, & \text{elsewhere} \end{cases} \) is a valid probability density function.

Answer: \dfrac{6}{7}


Question: Let \(x_1, x_2, \ldots, x_9\) be the roots of the polynomial \(x^9 + 7x - 2 = 0\). Find the value of \((x_1)^9 + (x_2)^9 + \cdots + (x_9)^9\).

Answer: 18


Question: Let $G$ be an $r$-regular graph with $n$ vertices and diameter 2. Find an upper bound for $n$ in terms of $r$. Justify your answer.

Answer: r^2 + 1


Question: Determine whether the matrix \( R = \begin{pmatrix} i & 0 & 2 \\ 2i & 0 & -1 \\ 0 & -1 -2i & 0 \end{pmatrix} \) can be represented as \( \lambda U \) where \( U \) is a unitary matrix and \( \lambda \) is a complex number such that \( \overline{\lambda} = -\lambda \).

Answer: Yes


Question:
Determine all values of $m$ such that the equation \(3^{x^2 + 2mx + 4m - 3} - 2 = \left|\dfrac{m - 2}{x + m}\right|\) has two distinct roots on the interval \([-4, 0]\). Choose the correct option:

\(\begin{aligned} &&A. \, m \in [1, 3] &&B. \, m \in (1, 3)\\ &&C. \, m \in [1, 3] \setminus \{2\} &&D. \, m \in (-\infty, 1] \cup [3, +\infty) \end{aligned}\)

Answer: C


Question: Determine whether the sequence \( a_n = \frac{\sum_{k=1}^{n} \frac{1}{k}}{\log n} \) is convergent or divergent.

Answer: 1


Question: Find the antiderivative $F$ of the function \( f(x) = \begin{cases} x^{-1} & x \ge 1 \\ x^{-2} & 0 < x < 1 \end{cases} \) such that \( F(e) = 1 \). Ensure that \( F(x) \) is continuous at \( x = 1 \). Then calculate the expression \( F(e^2) + F(e^{-1}) + F(e) + F'(0.5) + F'(2) \) and choose the correct answer from the following options: 1) \( 10.5-e \) 2) \( 9.5 + e \) 3) \( 9-e \) 4) \( 8.5 + 2e \).

Answer: 3


Question: Evaluate the integral \( \int_{\mathbb{R}^3} x_3^2 e^{-\lVert x \rVert_2} \lambda_3(dx) \), where \( x = (x_1, x_2, x_3) \).

Answer: 32\pi


Question: Simplify the matrix expression: \[ B = (Z^TZ)^{-1} + (Z^TZ)^{-1}Z^T( I - P_z)^{-1}Z(Z^TZ)^{-1} \] where \( Z \) is an \( n \times 1 \) vector, \( I \) is the identity matrix, and \( P_z = Z(Z^TZ)^{-1}Z^T \) is the projection matrix of \( Z \).

Answer: (Z^T Z)^{-1}


Question: Consider a simple, locally finite, acyclic graph \( \Gamma \) with a vertex \( v_0 \). Let \( X_n \) be a simple random walk on \( \Gamma \) starting at \( X_0 = v_0 \). If \( \limsup_{n\to\infty} \frac{d(X_0,X_n)}{n}=0 \) almost surely, does it follow that \( X_t = X_0 \) for some \( t > 0 \) almost surely?

Answer: No


Question: What is the sum of the digits of the number C, where C is defined as follows: Let A be the sum of the digits of the number 7^(7^(7^7)), B be the sum of the digits of A, and C be the sum of the digits of B?

Answer: 7


Question: Does the inequality \( \text{KL}(f,g) \le \int \frac{(\sqrt{f}-\sqrt{g})^2}{f}\,dx \) hold for general probability density functions \( f \) and \( g \)?

Answer: No


Question: Evaluate the integral \( \int_0^1 f(x) \, dx \), where the function \( f: [0,1] \rightarrow \mathbb{R} \) is defined by \( f(x) = 1 \) if \( x \) is algebraic and \( f(x) = 0 \) otherwise.

Answer: 0


Question:
Find the value of $x$ if $x^{x^4} = 4$. Choose the correct option:

1. $2^{1/2}$
2. $-2^{1/2}$
3. Both 1 and 2
4. None of the above

Answer: 3


Question: The time taken for a secretary to process a memo is modeled as an exponential random variable with the probability density function (PDF) \( f_{T}(t) = \frac{1}{2}e^{-\frac{t}{2}} \). The number of memos assigned to the secretary daily follows a Poisson distribution with the probability mass function (PMF) \( P_{N}(k) = \frac{L^k}{k!}e^{-L} \) for all integers \( k \ge 0 \), where \( L \) is a positive constant. Calculate the total expected amount of time the secretary spends on memos per day.

Answer: 2L


Question: Determine the Krull dimension of the ring \( \mathbb{Q}[x_1,x_2,x_3]/(x_1^3 + x_2x_3^2) \).

Answer: 2


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

Answer: 2\sqrt{6}


Question: Let $p$ be a prime number and $x, y, z \in \mathbb{N}$ such that $0 < x < y < z < p$ and $x^2, y^2, z^2$ give the same remainder when divided by $p$. Determine if $x^2 + y^2 + z^2$ is divisible by $x + y + z$.

Answer: Yes


Question: Does there exist a holomorphic function with a multiple root such that the fixed point of the corresponding Newton map is not indifferent? Specifically, is there an analytic function \( f(t) \) with a root \( \alpha \) of multiplicity greater than 1, where the multiplier \( |N'_f(\alpha)| \) of the Newton map \( N_f(t) = t - \frac{f(t)}{f'(t)} \) is not equal to 1?

Answer: Yes


Question: Given that $x, y, z > 0$ and \( \frac{2}{3x+2y+z+1} + \frac{2}{3x+2z+y+1} = (x+y)(x+z) \), find the minimum value of \( P = \frac{2(x+3)^2 + y^2 + z^2 - 16}{2x^2 + y^2 + z^2} \).

Answer: 2


Question: Given the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfying the equations \( f(x+1) + f(x+2) = f(x-1) + f(x) \) and \( f(-x) = -f(x) \), find the value of \( f(2) \).

Answer: 0


Question: Determine the minimal constant $c$ such that the expression $n^{cn}$ eventually outgrows $n!$ as $n$ approaches infinity.

Answer: 1


Question: Solve the functional equation \( 2f(x) + f(1-2x) = 1 \) for \( x \in [0, 1/2] \), given that \( f(1/2) = 1/2 \).

Answer: f(x) = x


Question: Determine the degree of the field extension \([\mathbb{Q}(\mu):\mathbb{Q}(\mu + \mu^{-1})]\), where \(\mu\) is a primitive \(n\)th root of unity.

Answer: 2


Question: Is the statement "In an infinite group, the number of elements of order $n$ is divisible by the Euler's totient function $\varphi(n)$" correct?

Answer: Yes


Question: Find the minimum value of \( \left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)\left(\sqrt{x} + \sqrt{y} + \sqrt{z}\right)^2 \) for \( x, y, z \in \mathbb{R}_{>0} \).

Answer: 27


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

Answer: \dfrac{\pi}{4}


Question: Given the matrix \( A = \begin{bmatrix}-3 & 2 \\ -1 & 0\end{bmatrix} \), calculate \( e^{\text{tr}A} \).

Answer: e^{-3}


Question: Find a composite number \( n \) such that \( n \mid a^n - a \) for every positive integer \( a \).

Answer: 561


Question: Determine whether the following statement is true: For any natural number $k$, there exists a natural number $n$ such that all numbers $n, 2n, \dots, kn$ have an odd sum of digits.

Answer: True


Question: Determine the number of subgroups of order $p^2$ in the group $\mathbb{Z}_p \times \mathbb{Z}_p \times \mathbb{Z}_p$, where $p$ is a prime number.

Answer: p^2 + p + 1


Question: Given a hyperbola with one focus at $(0,-5)$ and the nearest directrix $d_1: x-3y-4=0$, and an eccentricity of $\sqrt{10}$, find the equation of the second directrix of the hyperbola.

Answer: 9x - 27y - 14 = 0


Question: Find the minimum value of \( m \) such that the expression \( mx - 1 + \frac{1}{x} \) is non-negative for all positive real \( x \).

Answer: \dfrac{1}{4}


Question: Determine the Big-O notation for the expression \( \prod_{i=2}^n \log(i) \).

Answer: O((\log n)^n)


Question: Determine whether there exist sets $A, B \subseteq \mathbb{N}$, each with lower density 0, such that the set $A+B = \{a+b: a \in A, b \in B\}$ does not have lower density 0.

Answer: Yes


Question:
Determine the range of \(a\) such that the equation \(e^{|\sin x|} + e^{- |\sin x|} + 4a = 0\) has exactly four different solutions in the interval \([0, 2\pi]\). Choose the correct option:

(A) \(a \in \left[\frac{-e}{4}, \frac{-1}{4}\right]\)
(B) \(a \in \left[\frac{-1-e^2}{4e}, \infty\right)\)
(C) \(a \in \mathbb{R}\)
(D) None of these

Answer: D


Question: Let $A \subset [a,b]$ with $m(A) > \frac{1}{2}(b-a)$, where $m$ is the Lebesgue measure. Determine whether $A$ contains a subset of positive measure that is symmetric about $(a+b)/2$. A subset $S$ of the real line is symmetric about a real number $c$ if $c+x \in S \iff c-x \in S$. Is it true that such a subset exists?

Answer: Yes


Question: Let \( R(x) = \frac{f(x)}{g(x)} \) be a rational function where the coefficients of \( f(x) \) and \( g(x) \) are in \( \mathbb{C} \). Suppose \( R(n) \in \mathbb{Z} \) for all \( n \in \mathbb{Z} \). Is \( R(x) \) necessarily a polynomial? Justify your answer.

Answer: Yes


Question: Evaluate the integral \( \int_0^1 x^{k - 1} e^{-x} \, dx \).

Answer: \gamma(k, 1)


Question: Let \( A \) be an \( n \times n \) matrix. Determine if the dimension of the vector space generated by \( \{I, A, A^2, \dots, A^{2n}\} \) is at most \( n \). Justify your answer.

Answer: Yes


Question: Calculate the sum \( \sum \gcd(x,y) \) where \( x = 3^{a_1}5^{a_2}7^{a_3} \) and \( y = 3^{b_1}5^{b_2}7^{b_3} \), with nonnegative integers \( a_i, b_i \) such that \( a_k + b_k \le 5 \) for all \( k \). The sum is over all possible pairs \( (x, y) \).

Answer: 1477773


Question: Solve the differential equation \( x y' = 3y + 3 \) using a power series method. Express the solution as a power series \( y = \sum_{m=0}^{\infty} a_m x^m \).

Answer: -1 + C x^3


Question: Given that \( \int_0^4 e^{(x-2)^4} \, dx = A \), find the value of \( \int_0^4 x e^{(x-2)^4} \, dx \).

Answer: 2A


Question: Find the equation of the line tangent to the curve formed by the intersection of the cylinder $S = \{ (x,y,z) | x^2 + y^2 = 2 \}$ and the graph of the function $z = f(x,y)$ at the point $M(1,1,2)$. Given that $f(x,y)$ is differentiable everywhere and $\nabla f(1,1) = (1,2)$, determine the equation of the tangent line at $M$. Assume the tangent plane to $f$ at $(1,1,2)$ is $z = x + 2y - 1$. Provide the direction vector of the tangent line and use it to find the equation of the line.

Answer: \begin{cases} x = 1 - t \\ y = 1 + t \\ z = 2 + t \end{cases}


Question: Evaluate the limit: \[ \lim_{x\to 0}\sum_{i=1}^{10}x\left\lfloor\frac{i}{x}\right\rfloor \] using the fact that \( \lim_{x\to 0} x\left\lfloor\frac{1}{x}\right\rfloor = 1 \). Is the limit equal to 55?

Answer: 55


Question: Determine whether the following statement is true or false: In a metric space, every open set is the union of countably many closed sets.

Answer: A


Question: Let \( f(x) = \left( \int_{0}^{x} e^{-t^{2}} \, dt \right)^{2} \) and \( g(x) = \int_{0}^{1} \frac{e^{-x^{2(1+t^{2})}}}{1+t^{2}} \, dt \). Find the value of \( f'(\sqrt{\pi}) + g'(\sqrt{\pi}) \).

Answer: 0


Question: Find the number of integer values of \( n \) such that \( \frac{1}{\sin \frac{3\pi}{n}} = \frac{1}{\sin \frac{5\pi}{n}} \).

Answer: 2


Question: Determine the number of zeroes of the function \( f(x) = \exp(x) - 3x^2 \) in \( \mathbb{R} \).

Answer: 3


Question: Find the conditional extremum of the function \(f(x, y) = x^4 + y^4\) subject to the constraint \((x - 1)^3 - y^2 = 0\).

Answer: 1


Question: In triangle \( \triangle ABC \), points \( D, E, F \) are the midpoints of sides \( BC, CA, \) and \( AB \) respectively. \( G \) is the centroid of \( \triangle ABC \). Given that the area of \( \triangle AFG \) is 14, \( EC = \frac{15}{2} \), and the perpendicular distance from \( F \) to \( BC \) is 6, find \( BC^2 - AB^2 \).

Answer: 27


Question: Consider the vector space $V = \{p(x) \in F[x] \mid \deg(p(x)) \le n\}$, where $F$ is a field. Let $T : V \to V$ be the differentiation operator defined by $T(p(x)) = p'(x)$. Determine whether the only eigenvalue of $T$ is $\lambda = 0$.

Answer: 0


Question: Given a positive sequence \( \{a_n\} \) that satisfies \( S_{n} = \frac{1}{2}(a_{n}+\frac{1}{a_{n}}) \) where \( S_n = a_1+a_2+\ldots+a_n \), find \( \lim_{n\to \infty} S_{n+1} \times (S_{n}-S_{n-1}) \).

Answer: \dfrac{1}{2}


Question: Determine whether the set \( \{x^2+y^2: x,y \in \mathbb{N}\} \) has positive lower density, where a set \( A \subseteq \mathbb{N} \) has positive lower density if \( \liminf_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} > 0. \)

Answer: No


Question: In a right Artinian semiprimitive ring $R$, is there a finite number of maximal right ideals $M_1, \dots, M_n$ such that $M_1 \cap \dots \cap M_n = (0)$?

Answer: Yes


Question: Let $(X_n:n=1,2,\ldots)$ be an independent sequence of random variables, where, for each $n$, $X_n$ is uniformly distributed on $[0,n]$. Calculate $P(\{\omega:X_n(\omega)\to \infty \text{ as } n\to\infty\})$. Provide your answer as a probability value.

Answer: 0


Question: Given that \( \int_{0}^{2} p(x) \, dx = p(\alpha) + p(\beta) \) holds for all polynomials \( p(x) \) of degree at most 3, find the value of \( 3(\alpha - \beta)^2 \).

Answer: 4


Question: Determine whether an equivalence of categories $F: \mathcal{C} \rightarrow \mathcal{D}$ preserves colimits. Is it true that if $F$ is full, faithful, and essentially surjective on objects, then $F$ preserves colimits?

Answer: Yes


Question: Compare the values of $\gamma^e$ and $e^\gamma$, where $\gamma$ is the Euler-Mascheroni constant. Determine which is larger without using a calculator.

Answer: e^\gamma


Question: Determine the class group of the number field $\mathbb{Q}(\sqrt{7})$. Use the hint that $2=(3+\sqrt{7})(3-\sqrt{7})$ and $-1+\sqrt{7}=(2+\sqrt{7})(3-\sqrt{7})$ to assist in your calculations.

Answer: 1


Question: Using Green's theorem, evaluate the line integral \( \oint_C (-x^2+x)\, dy \) where \( C \) is the curve enclosed by the parabola \( x=2y^2 \) and the line \( y=2x \).

Answer: \dfrac{3}{640}


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)}. \]

Answer: e - 2


Question: Find the extreme value of the function \( f(x, y) = x^2 + y^2 + xy + \frac{1}{x} + \frac{1}{y} \).

Answer: 3^{\frac{4}{3}}


Question: Let $(R,m)$ be a local ring of Krull dimension $0$ with a finite residue field $R/m$. Determine whether there exists a polynomial $f(X) \in R[X]$ such that $f(R) = \{0,1\}$. Assume $m^n = 0$ for some $n > 0$. Is the claim true?

Answer: Yes


Question: In a group $G$ of order 26, let $x$ and $y$ be two distinct elements of order 2. Determine the possible order of the subgroup $\langle x, y \rangle$.

Answer: 26


Question: Determine whether the statement \( f_M = g_M - h_M \) is true for functions \( g: \mathbb{R}^n \to [0,\infty] \) and \( h: \mathbb{R}^n \to [0,\infty) \), where \( f = g - h \) and for any function \( j: \mathbb{R}^n \to [0,\infty] \), \( j_M(x) = \min\{j(x),M\} \) for \( M \in \mathbb{N} \).

Answer: False


Question: Given the probability density function (pdf) \( f(x) = 10x^9 \) for \( 0 < x < 1 \), and assuming independence, let \( Y = \max(X_1, X_2, \ldots, X_8) \). Find the cumulative distribution function (CDF) for \( Y \) and calculate the probability \( P(0.9999 < Y < 1) \).

Answer: 0.007968


Question: Evaluate the contour integral \( \int_{c}(1-z)^2dz \), where the contour \( c \) is defined by the parametric equations \( x=3\cos\theta \) and \( y=3+3\sin\theta \) for \( \theta \) ranging from \( 0 \) to \( \pi/2 \).

Answer: 51 - 69i


Question: Solve the equation \( \dfrac{3x^2 + 4x + 5}{\sqrt{5x^2 + 4x +3}}+\dfrac{8x^2 + 9x + 10}{\sqrt{10x^2 + 9x +8}} = 5 \) for \( x \in \mathbb{R} \). It is given that \( x = -1 \) is a solution.

Answer: -1


Question: Does every nonorientable surface have a degree 2 orientable cover?

Answer: Yes


Question: Let $f$ be a strictly positive function such that $\int_{0}^{\infty}f(x)dx=\int_{0}^{\infty}xf(x)dx=1$. Let $g$ be a nonnegative nonconstant function satisfying $\int_{0}^{\infty}g(ax)f(x)dx=a$ for all $a>0$. Does this imply that $g(x)=x$ almost everywhere?

Answer: g(x) = x


Question: Factor the expression: \[ a^{3}\left(\frac{a^{3}-2b^{3}}{a^{3}+b^{3}}\right)^{3} + b^{3}\left(\frac{2a^{3}-b^{3}}{a^{3}+b^{3}}\right)^{3} \] and find the factor \( a^{3} - b^{3} \).

Answer: a^3 - b^3


Question: In circle $\Gamma_1$, let $CD$ be a chord and $AB$ be a diameter perpendicular to $CD$ at point $N$, with $AN > NB$. A second circle $\Gamma_2$ is centered at $C$ with radius $CN$ and intersects $\Gamma_1$ at points $P$ and $Q$. The segments $PQ$ and $CD$ intersect at point $M$. Given that the radii of $\Gamma_1$ and $\Gamma_2$ are $61$ and $60$ respectively, find the length of $AM$. \( \boxed{} \)

Answer: 78


Question: Let $X$ and $Y$ be independent random variables with standard normal distributions. Define $Z = X^2 + Y$. Find the expected value $E(Z/X)$. Express your answer in terms of known constants or functions.

Answer: 0


Question: Compute the integral \( \int_{|z|=2} \frac{f'_n(z)}{f_n(z)} \, dz \) for the function \( f_n(z) = 1 + \frac{1}{z} + \frac{1}{2!z^2} + \cdots + \frac{1}{n!z^n} \).

Answer: 0


Question: Is it possible to partition the set of natural numbers \( \mathbb{N} \) into an infinite number of infinite sets \( N_k = \{ n_{mk} : m \in \mathbb{N} \} \) for \( k \in \mathbb{N} \) such that the series \( S_w = \sum_{v=1}^{\infty} \frac{1}{n_{vw}} \) diverges for each \( w \in \mathbb{N} \)? Assume that no element is repeated in any of the sets.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\int_0^n e^{-x^n} \, dx \]

Answer: 1


Question: Determine the character of the topological space $X = (0,1) \times (0,1)$ with the anti-lexicographic order topology, where the order is defined as $(x_1,y_1) < (x_2,y_2)$ if and only if $y_1 < y_2$ or $(y_1 = y_2$ and $x_1 < x_2)$. The character of a point in a topological space is the minimum cardinality of a local base at that point.

Answer: \aleph_0


Question:
Determine which of the following statements about the properties of curl is true:
1) \(\text{curl}~{\vec{F}}=0 \implies \vec{F} ~\text{is conservative.}\)
2) \(\text{curl}~{\vec{F}}=0 \impliedby \vec{F} ~\text{is conservative.}\)
3) \(\text{curl}~{\vec{F}}=0 \iff\vec{F} ~\text{is conservative.}\)

Answer: 2


Question:
Determine the correct classification of the partial differential equation $y^{3}u_{xx}-(x^{2}-1)u_{yy}=0$ based on the following options:

(a) Parabolic in $\{(x,y):x<0\}$,
(b) Hyperbolic in $\{(x,y):y>0\}$,
(c) Elliptic in $\mathbb{R}^{2}$,
(d) Parabolic in $\{(x,y):x>0\}$.

Use the condition $B^2 - 4AC = y^3(x^2 - 1)$ to determine the classification.

Answer: B


Question: A function $y(x)$ satisfies the differential equation $y^{\prime}=4\sqrt{y-x^2}$. Given that $y(1)=2$, find $y(3)$.

Answer: 18


Question: Find the gradient \( \nabla f \) of the function \( f(\mathbf{x}) = \frac{1}{2}(A \mathbf{x}) \cdot \mathbf{x} \), where \( A \) is a nonzero symmetric \( 3 \times 3 \) matrix.

Answer: A \mathbf{x}


Question: Evaluate the integral \( \int_0^{\pi/2} f(x) \, dx \), where \( f(x) = \sin^2(x) \) almost everywhere on \([0, \pi/2]\).

Answer: \dfrac{\pi}{4}


Question: Let $R$ be the radius of convergence for the power series $\sum^{\infty}_{n=0} a_n z^n$. For given $k \in \mathbb{N}$ and $l \in \mathbb{N}_0$, determine the radius of convergence for the series $\sum^{\infty}_{n=0} a_n z^{kn+l}$ in terms of $R$. Provide your answer as an expression involving $R$. \(R \in [0,\infty)\)

Answer: R^{1/k}


Question: Given two independent random variables $X$ and $Y$, where $X$ is normally distributed with $E[X] = -1$ and $Var(X) = 4$, and $Y$ is exponentially distributed with $E[Y] = 1$, find the variance of the random variable $T = pXY + q$, where $p, q \in \mathbb{R}$. Express $Var(T)$ in terms of $p$ and $q$.

Answer: 9p^2


Question: Determine the asymptotic growth rate, in big O notation, of the divisor function \( \sigma_{-1}(n) \).

Answer: O(\log \log n)


Question: Is a field necessarily a flat \( \mathbb{Z} \)-module?

Answer: No


Question: Find the Cramer-Rao bound for an unbiased estimator of \( b^2 \) given that \( X \) is a random variable with the probability density function \( f(x,b) = \frac{x}{b^2} \exp \left\{-\frac{x^2}{2b^2} \right\} \). Use the maximum likelihood estimator \( \hat{b^2} = \frac{1}{n} \sum_{i=1}^n X_i^2 \). The Fisher information is given by \( J(b) = \frac{4}{b^2} \). Calculate the Cramer-Rao bound using the formula \( \mathbb{E}[T(X)-g(b)]^2 \ge \frac{(g'(b))^2}{J(b)} \), where \( g(b) = \mathbb{E}T(X) \).

Answer: b^4


Question: Let $R$ be a commutative ring and $P$ a prime ideal of $R$. Prove or disprove that $\sqrt{P^{n}} = P$ for all $n \in \mathbb{N}$.

Answer: \sqrt{P^{n}} = P


Question:
Determine which of the following statements is false:

1. Any abelian group of order 27 is cyclic.
2. Any abelian group of order 14 is cyclic.
3. Any abelian group of order 21 is cyclic.
4. Any abelian group of order 30 is cyclic.

Answer: 1


Question: Calculate the value of \(2^{2^{517}} \mod 23\).

Answer: 13


Question: Consider the set of integers, \(\Bbb{Z}\). As we divide each integer by increasing divisors \(2, 3, 4, \ldots\), the elements of the resulting sets get closer together. What is the limiting set as the divisor approaches infinity?

Answer: \mathbb{R}


Question: Evaluate the integral: \[ \int^{2 \pi}_{0} \sin(x - \sin x) \,dx \]

Answer: 0


Question: Determine whether the expression \( \sum_{i=0}^{a-1}\binom{a}{i}\binom{2a-i}{a} \) is even.

Answer: Even


Question: Determine the number of solutions to the equation \( x^{10} = \begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix} \) in the space of 2x2 real matrices, \( M_{2}(\mathbb{R}) \).

Answer: \infty


Question: Let \( f \in L^p(X) \). Prove that \( \lim_{n\to 0} n^p\mu(\{x\in X:|f(x)|>n\})=0 \).

Answer: 0


Question: A lottery has $N$ tickets numbered $1, 2, \dots, N$, where $N$ is unknown. A random sample of 5 winning tickets is drawn, and their numbers are $\{a, b, c, d, e\}$. Determine the maximum likelihood estimator for $N$ based on this sample.

Answer: \max\{a, b, c, d, e\}


Question: Evaluate the integral \( \int_0^{2\pi} \frac{1}{1+3\cos^2(\theta)} \, d\theta \).

Answer: \pi


Question: Calculate the gradient with respect to the matrix $A$ of the function $f(A) = \log(\det(AX))$, where $X$ and $A$ are positive definite matrices.

Answer: A^{-1}


Question: Evaluate the integral: \[ \int_{0}^{\infty} \frac{\cos(2x) - 1}{x^2} \, dx \]

Answer: -\pi


Question: Calculate the line integral \( I = \int \limits_{(-1,2)}^{(1,-2)} (3x^2-2xy+y^2)dx + (2xy-x^2-3y^2)dy \).

Answer: 30


Question: Consider the non-homogeneous differential equation \(ay''(x)+by'(x)+cy(x)=\sum_{i=1}^n s_i e^{r_ix}\). Is the particular solution of the form \(y_p=x^k\sum_{i=1}^n h_ie^{r_ix}\) where \(k\) is the multiplicity of the eigenvalue \(w=r_i\)?

Answer: No


Question: Evaluate the improper double integral \( \iint_D (1-x+y)e^{(-2x+y)}\,dx\,dy \) where \( D \) is the region defined by \( 0 \le y \le x \).

Answer: \dfrac{1}{4}


Question: Determine the degree of the field extension \([\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]\).

Answer: 3


Question:
Determine the uniform convergence of the series \( \sum_{n=1}^{\infty} (-1)^n \frac{\sin nx}{n^{\ln n}} \) for \( x \in \mathbb{R} \). Choose the correct option:

A) The series converges only for \( x=0 \).
B) The series converges uniformly only for \( x \in [-\pi, \pi] \).
C) The series converges uniformly for \( x \in \mathbb{R} \setminus \{n\pi : n \in \mathbb{Z}\} \).
D) The series converges uniformly for all \( x \in \mathbb{R} \).

Answer: D


Question: Let $X$ and $Y$ be two independent and identically distributed uniform random variables on the interval $(0,1)$. Define $A = \min(X,Y)$ and $B = \max(X,Y)$. Determine the correlation between $A$ and $B$. \( \text{Express your answer as a single numerical value.} \)

Answer: \dfrac{1}{2}


Question: Determine the Galois group of the polynomial $X^8 - 1$ over the finite field $\mathbb{F}_{11}$. Provide a detailed explanation of your reasoning.

Answer: \mathbb{Z}/2\mathbb{Z}


Question: Use Stirling's formula to find \( \lim_{n\to\infty} \frac{a_{n}}{\left(\frac{n}{e}\right)^{n} 4^{n} \sqrt{2}} \), where \( a_{n} = \prod^{n}_{k=1} (2k-1) \).

Answer: 0