Question: Determine the order of the center of the group $G = \{e, x, x^2, x^3, y, xy, x^2y, x^3y\}$, where $O(x)=4$, $O(y)=2$, and $xy=yx^3$. Express your answer as an integer.

Answer: 2


Question: Determine the possible values for the number of Sylow 2-subgroups of a noncommutative group G of order 28, given that it has exactly one Sylow 7-subgroup.

Answer: 7


Question: Given the equations $x^2 y + y^2 z + z^2 x = 2186$ and $xy^2 + yz^2 + zx^2 = 2188$, where $x, y, z$ are integers, find the value of $x^2 + y^2 + z^2$. Express your answer as an integer.

Answer: 245


Question: Determine whether the following statement is true or false: Let $X$ be a normed linear space, $x_k \in X$, $k \in \mathbb{N}$, and suppose $\sum_{k=0}^\infty \lVert x_k\rVert$ is convergent. Then $\sum_{k=0}^\infty x_k$ is also convergent.

Answer: B


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

Answer: e^4


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

Answer: \sqrt{e}


Question: Determine the maximum number of students who can participate in a math competition where each student solves 3 problems, each scored from 0 to 7, such that no student scores equal to or higher than another student on all three problems.

Answer: 48


Question: Simplify the expression: $$\sum_{k=0}^n\binom{n}{k}(-1)^k\left(\sum_{j=0}^k\binom{k}{j}3^j\right)$$

Answer: (-3)^n


Question: Find the infimum of the expression \(\sqrt{(1+k^{2n})^2 + (1+k^{2n-1})^2}\) for all \(n \in \mathbb{N}\), given that \(k \in [0,1)\).

Answer: \sqrt{2}


Question: Determine the greatest possible value of \( \frac{P}{20!} \), where \( P \) is the product of the numbers in a list of 20 numbers. For each positive integer \( n \) from 1 to 20, the \( n \)th number in the list is either \( \frac{1}{2}n \) or \( 2n \). The mean of the numbers in the list is exactly 12.

Answer: 16


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

Answer: [0, \infty)


Question: Is the set $B_0$, consisting of all finite unions of intervals contained in $[0,1]$, closed under countable intersections?

Answer: No


Question: Compute the limit \(\lim_{n \rightarrow \infty} \left( \frac{1}{n+2}-\frac{1}{n+3}+\cdots+\frac{(-1)^{n}}{2n+1} \right).\)

Answer: 0


Question: Find the exponential generating function of the sequence $(a_n)_n$ defined by $a_0=1$ and $a_n=\alpha(\alpha-1)\cdots(\alpha - (n-1))$ for $n\geq 1$, where $\alpha$ is a real number.

Answer: (1 + x)^\alpha


Question: Find the smallest integer \( n \) such that there are no twin primes between \( n \) and \( 2n \).

Answer: 6


Question: Let $F$ be a field of order 64. How many solutions $(x,y) \in F \times F$ satisfy the equation $x^3 - y^3 = 0$?

Answer: 190


Question: Does the supporting hyperplane theorem imply that the closure of any convex set can be expressed as the intersection of halfspaces?

Answer: Yes


Question: Determine whether there exists a pair of consecutive positive integers, both of which are super exponential numbers. A number is super exponential if its prime factorization includes at least one prime raised to the power of 1000 or higher.

Answer: Yes


Question: Find the derivative of the scalar function \( f(A) = x^T A x \) with respect to the matrix \( A \), where \( x \) is a column vector.

Answer: x x^T


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

Answer: 0


Question: Evaluate the limit: $$\lim_{n\to\infty}\dfrac{|\cos{1}|+|\cos{2}|+|\cos{3}|+\cdots+|\cos{n}|}{n}$$

Answer: \dfrac{2}{\pi}


Question: Determine whether the first homology group of the space $D^2\setminus(S^0 * D^2)^n$ is $\mathbb{Z}^n$. Provide a justification for your answer.

Answer: \mathbb{Z}^n


Question: Determine whether the set of points within a given distance of an algebraic curve with no asymptotic branches can contain a line segment.

Answer: Yes


Question: Find the kernel of the homomorphism $D : F[x] \to F[x]$ defined by $D(f(x)) = f'(x)$, where the characteristic of the field $F$ is a prime number $p$. What is the set of polynomials that form this kernel?

Answer: F[x^p]


Question: Is it possible to endow the cube \(C=\{x\in \mathbb{R}^{n}; \max\{|x_{i}|; 1\leq i \leq n\} = 1\}\) with a differentiable structure that is not inherited from \(\mathbb{R}^n\)?

Answer: Yes


Question: Find the equation of a surface that contains circles with a curvature of \( \frac{1}{4} \) and intersects planes parallel to the xy-plane in circles.

Answer: x^2 + y^2 = 16


Question: Calculate the integral \( \int_{-1}^1 \sin x \cos \sqrt{1-x^2} \, dx \) without using the mean value theorem.

Answer: 0


Question: A group of 20 students with varying abilities needs to be assigned to 15 different classes of varying difficulty. Each student can only take one class, and students with higher abilities must take classes that are at least as difficult as those taken by students with lower abilities. Determine the number of ways the students can be assigned to the classes.

Answer: \dbinom{34}{14}


Question: Suppose we have a sequence of functions $f_n$ defined on a set $S$ that converges uniformly to a function $f$. If $x_n$ is any sequence in the interval, does $f_n(x_n) - f(x_n)$ converge to zero?

Answer: Yes


Question: Find the limit of the series as \( x \) approaches 0: \[ \lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2} \]

Answer: \dfrac{\pi}{4}


Question: Let $X$ be a locally compact Hausdorff space. Suppose $U$ is an open subset of $X$ and $X-U$ is non-empty. Determine whether it is always possible to find a compact neighborhood of a point $x \in U$ that is contained entirely within $U$. Provide a justification for your answer.

Answer: Yes


Question: Determine the condition on the parameters $m$ and $n$ such that the line $mx + ny = 1$ is tangent to the curve $x^{3/2} + y^{3/2} = 1$.

Answer: m^3 + n^3 = 1


Question: Suppose $(\Omega,\mathcal{F},P)$ is a probability space where $\Omega$ is the set of sequences of natural numbers. With $s_n$ denoting the $n$th element of sequence $s$, let \(p:=P\{s\in\Omega: s_n=0\text{ for some }n\}\), and for each $k\ge 1$ let \(q(k):=P\{s\in\Omega: s_n=0\text{ for some }n\le k\}\). Determine if it is necessarily the case that \(\lim_{k\to\infty}q(k)=p\).

Answer: Yes


Question: Determine if any general symmetric \((n \times n)\) matrix can be expressed as a sum of products of the form \(M=\sum_{i=1}^k j_iv_iv_i^T\), where \(j_i\) are scalars and \(v_i\) are \((n \times 1)\) vectors. If possible, find the minimum number of vectors \(k\) required.

Answer: n


Question:
Let $g$ be a real continuous function satisfying $g(1)=1$ and for each natural number $n$:

$$(n+1)^2\int_{x}^{x+\frac{1}{n+1}} g(t)\;\text{d}t = ng(x) + 1$$

Find the value of $g(1001)$. \(\boxed{\text{Your answer here}}\)

Answer: 1


Question: Determine whether any curve of constant width can only be a finite union of arcs of circles. Answer with True or False.

Answer: False


Question:
Determine the Big O upper bound of the function \(T(n) = T(\sqrt{n}) + n\), where \(T(1) = 1\), using the inequality:

\[n\leq 1 + \sum_{k = 1}^{\log(\log(n))} n^{\frac{1}{2^{k-1}}}\]

Answer: O(n)


Question: Determine whether the sequence defined recursively by $y_1=1$ and $y_{n+1}=2^{y_n}$ converges, and if so, find its limit.

Answer: \infty


Question: Determine the value of \( c \) such that the function \( f(x) = \begin{cases} \left(\sin\left(\frac{\pi}{2+x}\right)\right)^{\frac{1}{x^2}}, & x \neq 0 \\ c, & x = 0 \end{cases} \) is continuous at \( x = 0 \).

Answer: e^{-\pi^2/32}


Question: Determine whether the following statement is true or false: Suppose $A \subset C$ and $B \subset C$. Assume $A$ and $B$ are conjugate subgroups, meaning $cAc^{-1}=B$ for some $c \in C$. Then $A=B$ if and only if $A$ and $B$ are normal subgroups of $C$. Provide a proof or counterexample.

Answer: False


Question: Find the values of \(\beta\) for which the function \(g(x) = |x|^\beta\) is weakly differentiable on the unit ball in \(\mathbb{R}^m\).

Answer: \beta > 1 - m


Question: Let T be a linear transformation from $M_{2\times2}$ to $M_{2\times2}$ defined by $T(A)=BA$, where $B$ is a fixed $2\times2$ matrix with $\det(B) \neq 0$. If the nullity of $T$ is 2, find the rank of $B$. \(\boxed{2}\)

Answer: 2


Question: Find the minimum value of the integral \( \int_0^1 f(x)^2 \, dx \) given the constraints \( \int_0^1 f(x) \, dx = 0 \) and \( \int_0^1 x f(x) \, dx = 1 \).

Answer: 12


Question: Find the maximum value of the function \( M = \sqrt{x^2 + y^2 - 2xy} + \sqrt{y^2 + z^2 - 2yz} + \sqrt{z^2 + x^2 - 2xz} \) given that \( 0 \leq x, y, z \leq 3 \).

Answer: 6


Question: Find the moment generating function of the probability density function \( p(x) = xe^{-x} \) for \( x \ge 0 \) and use it to calculate the variance.

Answer: 2


Question: Determine the one-point compactification of the product space $S^n \times \mathbb{R}$, where $S^n$ is an n-dimensional sphere.

Answer: S^{n+1}


Question: Find the coefficient of \(x^{103}\) in the expansion of \((1+x+x^2+x^3+x^4)^{199}(x-1)^{201}\). Express your answer as an integer between 0 and 9.

Answer: 0


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{\sin^2 x}{x^2} \, \mathrm{d}x \) using complex analysis.

Answer: \dfrac{\pi}{2}


Question: Given a Jordan curve $J$ in the plane, which is homeomorphic to the unit circle $T$, and another Jordan curve $K$ contained within the $\epsilon$-neighborhood $V$ of $J$ for some $\epsilon > 0$, determine if there exist parametrizations $j: T \to J$ and $k: T \to K$ such that $|j(\xi) - k(\xi)| < C\epsilon$ for all $\xi \in T$.

Answer: Yes


Question: Calculate the residue of the function \( f(z) = \frac{1}{z(z^2+4)(z+2i)} \) at \( z = -2i \).

Answer: \dfrac{3i}{32}


Question: Evaluate the limit \(\lim_{x \to 0} \frac{x-\sin x}{x^3}\) without using de l'Hôpital's Rule or Taylor series.

Answer: \dfrac{1}{6}


Question: Find the function \( y(x) \) that satisfies the integral equation: \[ y(x) = \int_x^1 \left(\frac{y(t)}{t+1}\right)^2 \, \mathrm{d}t \]

Answer: 0


Question: Calculate the limit \(\lim_{N\to\infty} \frac{N!}{(N-k)!N^{k}}\) for any integer \(k\) between \(0\) and \(N\).

Answer: 1


Question: Evaluate the limit: $$\lim_{x\to0}\left(\frac {\csc(x)}{x^3} - \frac{\sinh(x)}{x^5}\right)$$ using L'Hopital's rule.

Answer: \dfrac{1}{90}


Question: Evaluate \( \lim _{x\to -\infty} \left(\frac{x^2-2}{x^2-1}\right)^{3x^2-\frac{1}{x}} \) using the fact that \( \lim _{x\to \infty} \left(1+\frac{1}{x}\right)^x=e \).

Answer: e^{-3}


Question: Calculate the sum of the series: \( \sin^2(2^\circ)\cos^2(2^\circ) + \sin^2(4^\circ)\cos^2(4^\circ) + \cdots + \sin^2(88^\circ)\cos^2(88^\circ) \).

Answer: \dfrac{45}{8}


Question: Find the equation of the tangent line to the curve given by $$(x^2 + y^2 - 3)^3 = (xy)^3 - x^2 - y^2$$ at the point $(1, 1)$.

Answer: y = -x + 2


Question:
Find all real values of $x_i$ that satisfy the following equations:

$$x_1 + x_2 + \ldots + x_{1994} = 1994$$

$$x_1^3 + x_2^3 + \ldots + x_{1994}^3 = x_1^4 + x_2^4 + \ldots + x_{1994}^4$$

Answer: 1


Question: Determine the number of distinct prime ideals in the ring \( \mathbb{Q}[x]/(x^5 - 1) \).

Answer: 2


Question: Find the remainder when \(30^{30}\) is divided by 3030.

Answer: 600


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

Answer: e


Question: Determine whether the adjoint of a normal operator \( A \) can be expressed as a polynomial in \( A \).

Answer: Yes


Question: Determine the largest integer $k$ such that there exists a perfect square, not divisible by 10, whose last $k$ digits are all the same.

Answer: 3


Question: Determine whether the following statement is true or false: If you run at a constant pace along any closed path, starting and ending at the same spot, and the sun is assumed to be stationary, you will receive the same amount of sunlight on the front and back of your body.

Answer: True


Question: Provide a specific example of a non-commutative ring with a multiplicative identity that lacks non-trivial maximal ideals.

Answer: M_2(\mathbb{R})


Question: Given that $a + b + c = 0$, evaluate the expression: $$\frac{a^2}{2a^2 + bc} + \frac{b^2}{2b^2 + ac} + \frac{c^2}{2c^2 + ab}$$

Answer: 1


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{x^{2n}}{(2n+1)!!}\prod_{i=0}^{2n} \frac{x}{(2n+1-2i)}$$

Answer: 0


Question: Determine whether the evaluation homomorphism from the affine space $A^n$ over $\mathbb{C}$ to the set of ring homomorphisms from the polynomial ring $\mathbb{C}[X_1, \ldots, X_n]$ to $\mathbb{C}$ is surjective.

Answer: Yes


Question: Determine the value of \(m \in \mathbb{R}\) for which the limit \[ l = \lim_{x \rightarrow 0} \frac{x - \sin x}{(1 - \cos x)^m} \] is finite and different from zero.

Answer: \dfrac{3}{2}


Question: Let $X$ be a full rank $n \times p$ matrix with $n > p$. Given the skinny SVD $X = UDV^T$ and the skinny QR decomposition $X = QR$, prove that $U^TQQ^TU = Q^TUU^TQ = I$. Consider the relationship between these decompositions to show the required equalities.

Answer: I


Question: Find the joint density function $g(u, v)$ for the transformed random variables $U = \exp(X) + 2\exp(Y)$ and $V = 2\exp(X^2) + \exp(Y^2)$, where $X$ and $Y$ are independent Exponential random variables with mean 1. Then, evaluate $g(3e, 3e)$. Provide your answer as a numerical value.

Answer: \dfrac{1}{6e^4}


Question: Let $p$ be a prime number with $p \geq 3$, and let $n$ be an integer such that $1 \leq n \leq p$. Determine a general formula for the cardinality of the set $S(n,p) = \{a \in \mathbb{Z}_p : a^n \equiv 1 \pmod{p}\}$.

Answer: \gcd(n, p-1)


Question: Evaluate the limit \( \displaystyle \lim_{x \to 0^+} \frac{(\sin x)^\frac{1}{\ln x}}{x^\frac{1}{\ln x}} \).

Answer: 1


Question: Determine the number of connected components of the set \(T=\{(a,b,c)\in\mathbb{R}^3: a^2b^2c^2=4\}\).

Answer: 8


Question: In a game where two players, A and B, alternately place dots of their respective colors (purple for A and green for B) on a 2D plane, can one of the players force a win by forming a triangle of their own color without any dots of the other color inside it? Provide a justification for your answer.

Answer: Yes


Question: Determine the limit of the series \( x_n = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots + \frac{(-1)^{n-1}}{n} \) as \( n \to \infty \).

Answer: \ln 2


Question: Determine the number of polynomials \( P(x) \) of the form \( P(x)=\sum_{i=0}^{12} b_ix^i \) with integer coefficients such that \( b_{12}=5 \), \( b_{11}=-21 \), and all the zeroes of \( P(x) \) are negative integers.

Answer: 0


Question:
Calculate the value of the following summation for a non-negative integer $n$:
\[ \sum_{k=0}^{n} \binom{2n-k} {n} 2^{k} \]

Answer: 4^n


Question: Determine whether the statement "Any Cauchy net in a metric space is convergent" is true or false.

Answer: B


Question: Determine whether it is possible to construct a subset of the plane that has no area and contains no three collinear points.

Answer: Yes


Question: Determine whether a $\sigma$-finite measure space, where every set of strictly positive measure contains an atom, is equivalent to a space that can be partitioned into disjoint atoms.

Answer: Yes


Question: Find the minimum value of \( \frac{1}{2} x^T P x \) subject to the constraint \( a^T x = b \), where \( P \) is a symmetric negative definite matrix.

Answer: -\infty


Question: Evaluate the line integral \( \int (u \, dy + v \, dx) \) over the circle \(|z|=3\) counterclockwise, where \( f(z) = \frac{1}{z^2 - 6z + 8} \) and \( u \) and \( v \) are the real and imaginary parts of \( f(z) \), respectively.

Answer: -\pi


Question: Consider a complex matrix $A$ with a single eigenvalue $\lambda$. If $\lVert A^k \rVert \leq C|\lambda|^k$ for some constant $C$ and all $k \geq 1$, is $A$ necessarily diagonalizable? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the line integral of the vector field \(\vec F (x,y,z) = (6xy + 4xz)\vec i + (3x^2 + 2yz)\vec j + (2x^2 + y^2)\vec k\) along the path \(\Gamma\) defined by \(y = x^2, z = 0\) from \((0,0,0)\) to \((2,4,0)\), followed by a line segment from \((2,4,0)\) to \((1,1,2)\). Given that \(\text{curl}\, \vec F = 0\), compute the integral using a potential function.

Answer: 9


Question: Calculate the line integral \(\int_C(x-y)dz\) where the curve \(C\) is the intersection of the paraboloid \(z=x^2+y^2\) and the plane \(z-2x-2y=1\), oriented counterclockwise when viewed from the point \((0,0,10)\).

Answer: 12\pi


Question: Determine the leading order term of the asymptotic expansion for \( \int_{1/x}^1 e^{-t} \, dt \) as \( x \to \infty \).

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


Question: Find the maximum value of \(a^3 + b^3 + c^3 + 4abc\) for all tuples \((a, b, c) \in \mathbb{R}^3\) satisfying \(0 \leq a, b, c \leq \frac{3}{2}\) and \(a + b + c = 3\).

Answer: \dfrac{243}{32}


Question: Determine whether the subspace of $L^p(I^2)$ consisting of functions that depend only on one variable is a closed subspace under the $L^p$ norm.

Answer: Yes


Question: Evaluate the series: \[ \sum_{n\ge 0}\frac{1}{4^n(n+1)}\binom{2n}{n}. \] Express your answer in closed form.

Answer: 2


Question: Compute \( \mathbb{E} \left( \int_0^1 \left| B_s \right| dB_s \right)^2 \) using Ito's formula, where \(B_s\) is a Brownian motion.

Answer: \dfrac{1}{2}


Question: You have 100 chips and are offered a game with a winning probability of \( p = 0.66 \) and a payout of 1:1. Determine the optimal number of chips to bet in this game.

Answer: 32


Question: Is the center of the Heisenberg group also its commutator subgroup?

Answer: Yes


Question: Find the splitting field of the polynomial \( \varphi = X^5+2X^4+X^3+X^2+X+2 \) over \( \mathbb{F}_3 \) and determine its cardinality.

Answer: 243


Question: Find the exact value of \( C \) such that the sequence defined by the recursive relation \( x_{i} = \frac{x_{i-1} + x_{i-2}}{Cx_{i-1}} \), with initial conditions \( x_1 = 1 \) and \( x_2 = 2 \), converges to \( \frac{2}{C} \) as \( i \) approaches infinity.

Answer: \dfrac{2}{3}


Question: Find the smallest positive integer $n$ such that the symmetric group $S_n$ contains an element whose order is a multiple of 30.

Answer: 10


Question: Find the value of \( b \) that minimizes the work required to move a particle from the point \((1,0)\) to the point \((-1,0)\) along the ellipse \(x^2+\frac{y^2}{b^2}=1\) against the force \(F(x,y)=(3y^2+2,16x)\).

Answer: \pi


Question: Find the coefficient of \(x^{25}\) in the expansion of \((1 + x^2 + x^4 + x^7)^{10}\).

Answer: 34750


Question: Determine whether the sequence \( \left( x_n \right)_{n \in \mathbb{N}} \), defined as \( x_n = \frac{1}{\sqrt{n}} \left( \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \right) \), converges. If it does, find its limit.

Answer: 2


Question: Determine if the sequence \( \left\{ \left(\frac{\ln (n+2)}{\ln (n+1)}\right)^{n+1} \right\} \) is convergent. If it converges, find its limit.

Answer: 1


Question: Determine the number of solutions to the equation \(x^3 + y^3 + 1 = 0\) over the finite field \(\mathbb{F}_8\).

Answer: 8


Question: Determine the set of real numbers \(\alpha\) for which the function \(f_\alpha(x) = \sum_{n=1}^\infty n^\alpha x e^{-nx}\) is continuous on \([0, \infty)\) and integrable with respect to the Lebesgue measure on \([0, \infty)\).

Answer: (-\infty, 0)


Question: Determine whether a graph in which any two vertices are joined by a unique walk is necessarily a tree. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the contour integral \( \oint_{\gamma} \frac{1}{z} \, dz \) where \( \gamma \) is a closed curve lying entirely within the open right half-plane \( A = \mathbb{C} \setminus \{z \in \mathbb{C}: \text{Re}(z) \leq 0\} \).

Answer: 0


Question: Evaluate the infinite product: $$ \prod_{n=3}^\infty \left ( 1 - \frac{1}{\binom{n}{2}} \right) $$.

Answer: \dfrac{1}{3}


Question: Determine the maximum value of the infinite power tower \(x^{x^{x^{\cdots}}}\).

Answer: e


Question: Evaluate the limit: \( \lim_{n\to\infty}\frac{e^{n^{2}}}{(2n)!} \).

Answer: \infty


Question: Find the Taylor series for the function \( f(x) = x^3 \cdot \ln{\sqrt{x}} \) centered at \( x = 1 \). Determine the radius of convergence of this series.

Answer: 1


Question: Determine the maximum value of \(|f(z)|\) for \(z\) within the unit disc, where \(f(z) = \frac{z^2}{z + 2}\).

Answer: 1


Question: Evaluate the limit: $$\lim_{n\to \infty} \left(1 + \frac{1}{n^2 + n}\right)^{n^2 + n^{1/2}}$$

Answer: e


Question:
Find the limit of the expression as \( x \to \infty \):
\[ \lim_{x\to\infty} (xe^{1/x} - x)^x \]

Answer: \sqrt{e}


Question: Find the value of $x$ that minimizes the function $f(x^2)$, where $f(x) = x^5 - x^3 + x$. Express your answer as a single value of $x$.

Answer: 0


Question: Does the free group on four generators, denoted by \( \mathbb{Z} \star \mathbb{Z} \star \mathbb{Z} \star \mathbb{Z} \), have a subgroup isomorphic to \( \mathbb{Z} \)?

Answer: Yes


Question:
Solve the following ordinary differential equation with the initial condition $y(\pi)=0$:
$$ \cos(x) \, dx + (4ye^{-y} + \sin(x)) \, dy = 0 $$

Answer: e^{y} \sin(x) + 2y^2 = 0


Question: Determine whether the following limit holds: \[ \lim_{c\to\infty}\sum_{n\in\mathbb{N}\setminus\{0\}}ce^{-n-\frac{c}{n}}=0 \] Or equivalently, \[ \lim_{c\to\infty}\int_1^\infty ce^{-x-\frac{c}{x}}\mathrm{d}x=0 \]

Answer: 0


Question: Evaluate the limit of the Riemann sum as \( n \to \infty \): $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\cos\left(\frac{2i\pi}{n}\right)$$

Answer: 0


Question: Determine if there exist any prime numbers $x$ and $y$ such that $x^2 + y^2 + 2023$ is a perfect square.

Answer: Yes


Question: Find the closed form of the function \( f(x) = \sqrt[k]{(1+x)^k-(1+x)+\sqrt[k]{(1+x)^k-(1+x)+\sqrt[k]{\cdots}}} \). Determine the values of \( x \in \mathbb{C} \) for which it converges.

Answer: 1 + x


Question: Solve the equation: \[ 9^x + 3^{2x-1} = 2^{x+3.5} + 2^{x+0.5} \]

Answer: \dfrac{3}{2}


Question: Determine if there are any functions other than \( f(n) = n + 1 \) that satisfy the functional equation \( f(n) + 2f(f(n)) = 3n + 5 \) for all positive integers \( n \).

Answer: f(n) = n + 1


Question: Let $X$ be a random variable with $X \sim \text{Geom}(1/3)$. Let $Y$ be another random variable which has a $\text{Binomial}(N,1/4)$ distribution where $N$ is the value taken by the random variable $X$. Using the law of total probability, compute the expectation value of the product $XY$. Express your answer in terms of a sum involving $N$. What is $E(XY)$?

Answer: \dfrac{15}{4}


Question: Find a Gröbner basis for the ideal $I \cap \mathbb{Q}[y]$ in the polynomial ring $\mathbb{Q}[x,y]$, where $I=\langle x^2-y,y^2-x,x^5-x^2\rangle$, using the lexicographic ordering $y > x$. Provide your answer as a set of polynomials.

Answer: \{y^4 - y\}


Question: Find the value of \(18! \mod 437\) given that \(437 = 19 \cdot 23\), \(18!\equiv -1 \pmod {19}\), and \(22!\equiv -1 \pmod {23}\).

Answer: 436


Question: Consider the set $A = \{x \in \mathbb{N} \mid x \le 100\}$. Determine the number of relations $R \subseteq A \times A$ that are reflexive, symmetric, and satisfy the condition that $(2,x) \in R$ if and only if $x$ is an even number.

Answer: 2^{4851}


Question: Let $f(x)$ be a polynomial function such that $f(x)f(\frac{1}{x}) = f(x) + f(\frac{1}{x})$ for all $x \neq 0$. Given that $f(4) = 65$, determine the value of $f(6)$. \( \boxed{\text{Your answer here}} \)

Answer: 217


Question: Evaluate the double integral \(\int_{0}^{2}\int_{0}^{2-x}|x-y|\, {\rm d}y\, {\rm d}x\) by splitting the region of integration into two parts.

Answer: \dfrac{4}{3}


Question: Determine whether it is possible to construct a linear mapping from the vector space of real polynomials to itself that swaps the basis elements $x$ and $x^2$ while leaving all other basis elements unchanged.

Answer: Yes


Question: Determine whether the following statement is true or false: If every bounded subsequence of a sequence converges, then the entire sequence converges.

Answer: B


Question: In a right triangle $ABC$ with $	riangle CAB = 90^\circ$, points $M$ and $N$ are located on the hypotenuse $BC$. Given that $BM^2 + CN^2 = MN^2$, $AN = \frac{\sqrt{90}}{12}$, and $AM = \frac{\sqrt{80}}{12}$, find the measure of $\angle MAN$. \(\text{Express your answer in degrees.}\)

Answer: 45


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty}\left \{ \frac{\left ( n+1 \right )\left ( n+2 \right )...3n}{n^{2n}} \right \}^{\frac{1}{n}}$$

Answer: \dfrac{27}{e^2}


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

Answer: -\dfrac{7}{5}


Question: Find the minimum value of the function \( f(x) = \sqrt{x + 25} + \sqrt{36 - x} + \sqrt{x} \) without using a calculator or calculus.

Answer: 11


Question: Determine the set of points in the complex plane where the function \( f(z) = \sum_{n=0}^{\infty} \frac{z^{2n}}{(5n)^n} \) is holomorphic.

Answer: \mathbb{C}


Question: Calculate the limit \[ \lim_{n\to\infty} P\left(-\ln(1-X_1) - \ln(1-X_2) - \cdots - \ln(1-X_n) \geq n\right) \] where \(X_1, X_2, X_3, \ldots\) are independent and identically distributed (i.i.d.) uniform random variables on the interval \((0,1)\).

Answer: \dfrac{1}{2}


Question: Consider the set $A$ of all $3 \times 3$ determinants with entries either $+1$ or $-1$. If a determinant $B$ from set $A$ is chosen randomly, what is the probability that the product of the elements in any row or any column of $B$ is $-1$?

Answer: \dfrac{1}{32}


Question: Consider the sequence of functions \( f_n:[1,\infty) \rightarrow \mathbb{R} \) defined by \( f_n(x)=\frac{1}{x}\chi_{[n,\infty)}(x) \). Determine whether \( \int f_n \to \int f \) when \( f(x) = \frac{1}{x^2} \).

Answer: No


Question: Do totally unimodular matrices have the property that the determinant of every square submatrix is either 0, 1, or -1?

Answer: Yes


Question: Transform the integral \( \int_0^\infty \frac{\sin^2 x}{x^2} \, dx \) into another form.

Answer: \dfrac{\pi}{2}


Question: Let $p$ be a polynomial function on $\mathbb{C}$ with no roots on the unit circle $S^1$. Determine if the number of roots, up to multiplicity, of $p(z) = 0$ with $|z| < 1$ necessarily equals the degree of the map $\widehat{p}: S^1 \to S^1$ specified by $\widehat{p}(z) = p(z)/|p(z)|$. Provide a justification for your answer.

Answer: Yes


Question: Given a fixed vector $a > 0$ and vectors $a, b, x \in \mathbb{R}^n$, determine if the following statement is true: For any $x > 0$, if $a^T x \leq 1$ implies $b^T x \leq 1$, then $a \geq b$. Is this statement true?

Answer: True


Question: Calculate the surface integral of the vector field \( \vec{F} = [x^2, z, -y]^T \) over the unit sphere \( S = \{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 = 1\} \): $$\iint_S \vec{F} \cdot \vec{n} \, dA$$

Answer: 0


Question: Is it possible to find two strictly increasing functions $f$ and $g$ defined on the real numbers such that their product $f \cdot g$ is also strictly increasing?

Answer: Yes


Question: Suppose the sequence \(\{a_n\}\) converges to zero. Determine \(\lim_{n\to\infty} a_n^n\).

Answer: 0


Question:
Consider the system of ordinary differential equations:
$$\begin{cases}\frac{dx}{dt}=4x^3y^2-x^5y^4\\ \frac{dy}{dt}=x^4y^5+2x^2y^3\end{cases}$$
Determine which of the following statements about closed paths in the phase portrait is true:

1. There exists a closed path within the region $\{(x,y) \in \mathbb{R^2} | x^2+y^2 \leq 5\}$.
2. There exists a closed path within the region $\{(x,y) \in \mathbb{R^2} | 5 < x^2+y^2 \leq 10\}$.
3. There exists a closed path within the region $\{(x,y) \in \mathbb{R^2} | x^2+y^2 > 10\}$.
4. There is no closed path in $\mathbb{R^2}$.

Select the correct statement.

Answer: 4


Question: Find all integer pairs \((j, k)\) such that \(2^j + 3 = 7^k\).

Answer: (2, 1)


Question: Determine the value of \( c \in \mathbb{R} \) such that the limit \( \lim_{(x,y) \to (0,0)} f(x,y) \) exists for the function \[ f(x,y) = \begin{cases} \frac{1-\cos(2xy)}{x^2y\sin(\pi y)} & \text{if } x \neq 0, y \neq 0 \\ c & \text{if } x = 0 \text{ or } y = 0 \end{cases} \]

Answer: \dfrac{2}{\pi}


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

Answer: 0


Question: Given real numbers $a, b, c, d$ such that $a^2 + b^2 = c^2 + d^2 = 1$, find the maximum value of the expression \((1 - a)(1 - c) + (1 - b)(1 - d)\). State the conditions under which this maximum value is achieved.

Answer: 3 + 2\sqrt{2}


Question: Determine the total number of permutations of the numbers \(1, 2, 3, \ldots, 12\) that satisfy the conditions \(a_1 > a_2 > a_3 > a_4 > a_5 > a_6\) and \(a_6 < a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12}\).

Answer: 462


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

Answer: e^2


Question: Does there exist a Hilbert space $\mathcal{H}$ such that a sequence $\{ x_n \} \subset \mathcal{H} $ is weakly convergent, but not convergent in the Hilbert space norm?

Answer: Yes


Question: Calculate the fourth-degree Taylor term at $x = 0$ for the function $$f(x) = \int_0^x e^{-t^2} dt.$$

Answer: 0


Question: Calculate the limit as \( n \) approaches infinity: \[ \lim_{n\to\infty}{\dfrac{1}{n}\sum_{k=1}^{n}{\left(\dfrac{n}{k}-\left\lfloor\dfrac{n}{k}\right\rfloor\right)}} \]

Answer: 1 - \gamma


Question: Determine the number of distinct pairs of (grade 4 nodes, grade 5 nodes) that can be used to construct a tree with 14 one-degree nodes.

Answer: 3


Question: Find the ratio of the circumradius to the inradius of a regular tetrahedron.

Answer: 3


Question: Determine the coefficient of \(x^6\) in the expansion of the product \((1+x)(1+x^2)^2 \cdots (1+x^n)^n\).

Answer: 28


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

Answer: 1


Question: Given a skew-symmetric matrix $M_{ij}$ constructed from $n$ random points in three dimensions $(x_i, y_i, z_i)$, determine the number of purely complex eigenvalues of the matrix.

Answer: 2


Question: Determine whether the characteristic function of a normally distributed random variable is an analytic function on the complex plane.

Answer: Yes


Question: Find the limit of the sequence \( \lim_{n \to \infty}\left[\frac{(n + 3)(n + 4)\cdots(n + n+2)}{(n+2)^{n}}\right]^{1/n} \) using Cauchy's second limit theorem.

Answer: \dfrac{4}{e}


Question: Determine whether it is true or false that no subset of the real numbers is countable.

Answer: B


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{1}{\sqrt[4]{n^4 + n + 2}} + \frac{1}{\sqrt[4]{n^4 + 2n + 1}} + \frac{1}{\sqrt[4]{n^4 + 3n}} + \frac{1}{\sqrt[4]{n^4 + 4n - 1}} + \frac{1}{\sqrt[4]{n^4 + 5n - 1}} \right) \] without using derivatives or integrals.

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^\infty \sin\left(\frac{\pi}{a}x\right)\sin\left(\frac{3\pi}{a} x\right) \, \mathrm{d}x \) by considering the periodicity of the integrand.

Answer: 0


Question: Let $G$ be a group and $f$ be a class function on $G$. If $f(g) = f(h)$ for elements $g, h \in G$, does it necessarily follow that $g$ and $h$ are in the same conjugacy class?

Answer: No


Question: Determine the group of grouplike elements in the Hopf algebra $U(L)$, where $U(L)$ is the universal enveloping algebra of a Lie algebra $L$. The algebra $U(L)$ is constructed as $U(L) = T(L) / I(L)$, with $I(L)$ being the two-sided ideal generated by elements of the form $xy - yx - [x,y]$, and $T(L)$ is the tensor algebra of $L$. A grouplike element in a coalgebra is an element $x \neq 0$ with $\Delta(x) = x \otimes x$. The set of grouplike elements forms a group.

Answer: \{1\}


Question: Let $F$ be a map from $\mathbb{R}^n$ to $\mathbb{R}^n$. Fix $x_0 \in \mathbb{R}^n$. If the spectral radius of the Jacobian of $F$ at $x_0$ is less than 1, is it possible for $F$ to fail to be a local contraction near $x_0$? Provide a justification for your answer.

Answer: Yes


Question: Determine whether there exists a continuous positive function whose integral over \((0, \infty)\) converges, but whose limit as \(x\) approaches infinity is not zero.

Answer: Yes


Question: Is it possible for the function \( x \mapsto \int_{\Xi}f(x,\xi)d\mu(\xi) \) to be discontinuous if we assume that \( f(x,\xi) \) is bounded?

Answer: Yes


Question:
Given the function $f(x,y,z)$ as the determinant of the matrix:

$$\begin{bmatrix}
x^2 & y^2 & z^2 \\
x & y & z \\
1 & 1 & 1
\end{bmatrix},$$

find the expression for $\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$.

Answer: 0


Question: Find the cardinality of the set \( \{ (x,y) \in \mathbb{R}^2 \; | \; y > x > 0, \; x^x = y^y \} \).

Answer: \mathfrak{c}


Question: Determine whether the following statement is true or false: If all groups of order $m$ are solvable, then all groups of order $mp$ are solvable, where $p$ is a prime greater than the square root of $2m+1$. Provide a justification for your answer.

Answer: True


Question: Express the determinant \( f_n(x) = \left|\begin{array}{c c c c c} 1 & x & \cdots & x^{n - 1} & x^n\\ x & 1 & x & \cdots & x^{n - 1} \\ \vdots & x & \ddots & \ddots & \vdots\\ x^{n - 1} & \vdots & \ddots & 1 & x\\ x^n & x^{n - 1} & \cdots & x & 1 \end{array}\right| \) as a polynomial in terms of \( x \).

Answer: (1 - x^2)^n


Question: Evaluate the double integral \( \iint_\Omega \left(\sqrt{\frac xa}+\sqrt{\frac yb}\right)\,dxdy \) where \(\Omega\) is the region enclosed by the curve \(\sqrt{\frac xa}+\sqrt{\frac yb} = 1\) and the coordinate axes.

Answer: \dfrac{2ab}{15}


Question: Determine whether the ideal $I$ generated by $5$ and $3+4i$ in the Gaussian integers $G=\{a+bi:a,b\in\mathbb{Z}\}$ is principal. If it is principal, find a generator for the ideal.

Answer: 2 + i


Question: Given a prime \( p \) and any integer \( n > 1 \), define \[ F(n) = \frac{n^p-1}{n-1} = n^{p-1} + n^{p-2} + \dots + 1. \] Determine whether every prime factor of \( F(n) \) is either \( p \) or congruent to \( 1 \mod p \).

Answer: Yes


Question: Determine whether the Killing form of a semisimple Lie algebra can be degenerate. Answer with true or false.

Answer: False


Question: Calculate the volume of the solid enclosed by the paraboloids $x = y^2 + z^2$ and $x = 16 - y^2 - z^2$ using a triple integral.

Answer: 64\pi


Question:
Let $a > 1$ and consider twice differentiable functions $f, g: [-a, a] \rightarrow \mathbb{R}$ such that for some $c$ with $0 < c < 1 < a$, the following conditions hold:

1. $f(x) = 0$ only at $x = -a, 0, a$.
2. $f'(x) = 0$ only at $x = -1, 0, 1$.
3. $g(x) = 0$ only at $x = -1, 0, 1$.
4. $g'(x) = 0$ only at $x = -c, c$.

Which of the following relations between $f$ and $g$ is possible?

(A) $f = g'$
(B) $f' = g$
(C) $f = -g'$

Answer: B


Question: Determine whether there exists a non-constant holomorphic function $f$ such that $f(x_{n,k})=0$ for all $n,k\in\mathbb{Z}$, where $x_{n,k}=n+ik$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to1}\dfrac{\sqrt{x^3+2}-\sqrt{x+2}}{\sqrt{x^4-x^3}-\sqrt{x^2-x}}$$

Answer: \infty


Question: Determine the number of normal subgroups of order 3 in a non-abelian group of order 21.

Answer: 0


Question: In a topological space, consider two disjoint sets whose union is closed. If one of these sets is open, determine whether the other set can be neither closed nor open.

Answer: No


Question: In an abelian finite group $G$, if $2^n$ divides the order of $G$ for some positive integer $n$, determine the number $N$ of elements in $G$ that have order 2.

Answer: 2^n - 1


Question: Consider a degree 3 polynomial \(P(x)\) with complex coefficients, where the constant term is 2010. Must there necessarily exist a root \(\alpha\) such that \(|\alpha| > 10\)? Provide a justification for your answer.

Answer: No


Question: Calculate the flux \( \iint \vec{F} \cdot d\vec{S} \) of the vector field \( \vec{F}(x, y, z) = x\hat{i} + y\hat{j} + z^4\hat{k} \) through the part of the cone \( z = \sqrt{x^2 + y^2} \) that lies below the plane \( z = 1 \), with the surface oriented downward.

Answer: \dfrac{\pi}{3}


Question: Given the function $y(x) = f(x)(1 - x)$ where $x \in [0, 1]$, with the conditions $y(0) = y(1) = 0$, $f(0) = 0$, $f(1) = c$ where $0 < c < 1$, and $f'(x) > 0$ for all $x$, determine if the unique maximum of $y(x)$ occurs at a single point $x^* \in (0, 1)$. Assume $f(x)$ is continuously differentiable.

Answer: Yes


Question: Determine the number of continuous functions \( f(x) \) defined on \([0,1]\) that satisfy the following conditions: \( I_1 = \int_0^1 f(x) \, dx = 1 \), \( I_2 = \int_0^1 x f(x) \, dx = a \), and \( I_3 = \int_0^1 x^2 f(x) \, dx = a^2 \).

Answer: \infty


Question: Given two functions $f$ and $g$ defined on the same domain $D$, where both $f'$ and $g'$ are non-negative, and the second derivative of $f$ satisfies $f'' \geq g''$ on $D$, determine if we can conclude that $f \geq g$ throughout $D$. Provide a justification for your answer.

Answer: No


Question: Evaluate the line integral \( \int_{\gamma} \frac{1}{z^2-1} \, dz \), where \( \gamma(t) = 1 + e^{it} \) for \( 0 \leq t \leq 2\pi \).

Answer: \pi i


Question: Given a continuous map $\varphi: X \to Y$ between two Hausdorff topological spaces $X$ and $Y$, suppose there exists a point $x \in X$ such that $\varphi(x) = y \in Y$, and $U$ is an open neighborhood of $y$. Is it possible to find an open neighborhood $U'$ of $x$ with the property that $\varphi(U') \subseteq U$? Answer yes or no.

Answer: Yes


Question: Determine whether the limit \( \lim_{(x,y)\rightarrow (0,0)} \frac{x^3-x^2y+xy^2-y^3}{x^2+y^2} \) exists.

Answer: 0


Question:
Let $g :(0,1) \to \mathbb{R}$ be a differentiable function satisfying the differential equation

$$g'(x) = g(x) + \int_0^1 g(x) \,dx$$

for all $x\in (0,1)$. Suppose there are distinct points $c,d\in(0,1)$ such that $g(c) = g(d)$. Determine the value of $g\left(\frac{c + d}{2}\right)$. \(\text{Express your answer as a number or a simple expression.}\)

Answer: 0


Question: Does the convergence of \(\lim\limits_{n\rightarrow\infty}f(a_n) = b\) imply that \(f(a) = b\), given that \(f:\mathbb{R}\rightarrow\mathbb{R}\) is a function and \(a_n\rightarrow a\)? Answer yes or no and justify your answer.

Answer: No


Question: Determine whether $x=0$ is a repeated root of the function \( f(x) = (e^x-1)(\ln(x+1)) \).

Answer: Yes


Question: Find the degree of the field extension $[\mathbb{Q}(\sqrt{2} + \sqrt{5}) : \mathbb{Q}]$. Express your answer as an integer.

Answer: 4


Question:
Is it possible to construct a random vector \( X: [0,1] \rightarrow \mathbb{R}^2 \) such that its distribution is a given probability measure \( \mu \) on \( \mathbb{R}^2 \), and for any Borel subset \( A \) of \( \mathbb{R}^2 \), the following condition holds:
\[ \text{Meas}\{\omega \in [0,1]: X(\omega) \in A\} = \mu(A), \]
where \( \text{Meas} \) denotes the Lebesgue measure on \( [0,1] \)?

Answer: Yes


Question: Determine whether a censored distribution always has lower variance than the original distribution. Answer with true or false.

Answer: False


Question: Find the area of a convex octagon inscribed in a circle, where four of its sides have a length of 2 and the other four sides have a length of 3.

Answer: 13 + 12\sqrt{2}


Question: Evaluate the limit: \[ \lim_{n\to \infty} \frac{1}{n^4}\sum_{i=1}^n \sum_{j=1}^i (i^2+j^2) \]

Answer: \dfrac{1}{3}


Question: Calculate the remainder when \(666^{666}\) is divided by \(1000\).

Answer: 16


Question: Find an approximation for the number of prime factors in the factorization of $N!$ that is less than $4N$.

Answer: \dfrac{N}{\log N}


Question: Calculate the limit \( \lim_{x \to \infty} \frac{\int_0^x|\sin t|\,dt}{x} \). Note that L'Hôpital's rule cannot be applied.

Answer: \dfrac{2}{\pi}


Question: Determine whether the following statement is true or false: A function that is Lipschitz continuous on an interval with a set of zeros of measure zero must have a finite number of zeros.

Answer: B


Question: Find the greatest common divisor (GCD) of expressions of the form \(p^6 - 7p^2 + 6\), where \(p\) is a prime number greater than or equal to 11.

Answer: 672


Question: Determine if the differential equation \( x \cdot \dfrac{du}{dx} + \dfrac{2u^2+2u-2}{1+2u}=0 \) can be solved using the method of integrating factors. If possible, find the integrating factor.

Answer: x^2


Question: A cube has six faces, and we have six distinct colors available. Each face of the cube must be colored with one of these colors, ensuring that no two adjacent faces share the same color. How many distinct ways can the cube be colored under these conditions?

Answer: 30


Question: Find the limit of the sequence \(x_{n}=\frac{1}{n+2^{0}}+\frac{1}{n+2^{1}}+\ldots+\frac{1}{n+2^{n}}\) for \(n \in \mathbb{N}\) and \(n \geq 1\).

Answer: 0


Question: In a group $G$ with exactly 28 elements of order 5, determine the number of distinct subgroups of order 5 that $G$ contains.

Answer: 7


Question: If $x(t)$ and $y(t)$ are functions such that $x(t) \to \infty$ as $t \to \infty$, does it follow that if $x(t) \sim y(t)$ then $\ln x(t) \sim \ln y(t)$? Answer with 'Yes' or 'No' and justify your answer.

Answer: Yes


Question: Evaluate the limit: $$ \lim _{x \rightarrow 0^{+}} \frac{\tan 3 x \sqrt{1-\cos 4 x}}{x^{2}+x^{3}} $$

Answer: 6\sqrt{2}


Question: Determine if there exists a prime number \( p \) such that the function \( f(p) = \frac{1}{(p^2 - 1)} \sum_{q=3}^p \frac{q^2-3}{q}[q\text{ is prime}] \) is greater than 0.2.

Answer: Yes


Question: Find all values of \( k \) for which the matrix \( A = \begin{bmatrix} 2 & -2 & k \\ -1 & 2 & 0 \\ -2 & 0 & 2 \end{bmatrix} \) is not diagonalizable.

Answer: 1


Question: Let $p$ be a prime number and $k$ a positive integer. Let $d$ be the smallest positive integer such that $p^k$ divides $d!$. Determine whether $d$ is necessarily a multiple of $p$. Provide a justification for your answer.

Answer: Yes


Question: Find the derivative of the expression \( ||Mx+n||^2_2d \) with respect to the vector \( x \), where \( M \) is a matrix, and \( n \) and \( d \) are column vectors.

Answer: 2 d (M x + n)^\top M


Question: Let \( G \) be a finite group and consider elements \( g, h \in G \setminus \{1\} \) such that \( g^{-1}hg = h^2 \). Given that \( o(g) = 5 \) and \( o(h) \) is an odd integer, determine the order of \( h \). Use the fact that if \( g^{-1}hg = h^i \) for some natural number \( i \), then \( g^{-n}hg^n = h^{i^n} \) for all \( n \in \mathbb{N} \).

Answer: 31


Question: Calculate the limit: $$\lim_{x \to 0^-}xe^{\frac{1}{x}}$$ without using L'Hôpital's rule, series expansions, or the Squeeze theorem.

Answer: 0


Question: Calculate the sum \( F_0 + F_1 + F_2 + \cdots + F_{2016} \mod 2017 \), where \( \{F_n\} \) is the Fibonacci sequence defined by \( F_0 = 0, F_1 = 1, \) and \( F_{n+2} = F_{n+1} + F_n \).

Answer: 2016


Question: Find the degree of the field extension $\mathbb{Q}(\sqrt[3]{2}, \zeta_3\sqrt[3]{2})$ over $\mathbb{Q}$.

Answer: 6


Question: In a cake-throwing competition with an even number of participants, determine if it is possible for every participant to receive a cake, given that the distances between all pairs of participants are distinct.

Answer: Yes


Question: Suppose $(f_n)$ is a uniformly convergent sequence on the set $A$, and each $f_n$ is bounded on $A$. Does there exist a function $f$ that is bounded on the set $A$?

Answer: Yes


Question: Solve the equation \( \ln(x) = \sin(x) \) for \( x \) using numerical methods. Provide the solution as a decimal approximation.

Answer: 2.219


Question: Let \( A \) be a Dedekind domain, \( K \) its fraction field, \( B \) another Dedekind domain with \( A \subset B \), and \( L \) the fraction field of \( B \). If \( L/K \) is a Galois extension with Galois group \( G \) and \( B^G = \{ b \in B \mid \sigma(b) = b \text{ for all } \sigma \in G \} = A \), determine whether it follows that \( B \) is the integral closure of \( A \) in \( L \).

Answer: Yes


Question:
Compute the probability $P(X<Y<Z)$ for random variables $X$, $Y$, and $Z$ with the joint density function:

$$f(x,y,z) = \begin{cases}e^{-x-y-z}&\text{if }x,y,z>0\\ 0&\text{otherwise}\end{cases}$$

Answer: \dfrac{1}{6}


Question: Find the approximate value of $y$ that satisfies the equation $0.92 = \sqrt{\frac{\tanh(2 \pi y)}{2 \pi y}}$ using the Taylor series expansion up to the fourth degree and the secant method.

Answer: 0.120


Question: Determine the number of solutions to the equation $e^x - 2 = \cos x$. Provide your answer as a single integer.

Answer: 1


Question: Given a standard Brownian motion \( B_t \) starting at zero, and a non-negative stochastic process \( X_t \) that satisfies the stochastic differential equation \( dX_t = \frac{1}{X_t} dt + dB_t \), calculate the expected value \( E[\sigma] \), where \( \sigma = \inf \{ t \geq 0 : X_t = 1 \} \).

Answer: \dfrac{1}{3}


Question: Evaluate the contour integral \( \int_{\gamma} \frac{dz}{z} \) where \( \gamma \) is a closed curve entirely located in the domain \( A = \mathbb{C} \setminus \{z \in \mathbb{C} : \text{Re}(z) \leq 0\} \).

Answer: 0


Question: Find the greatest common divisor (gcd) of the numbers $2^{2^{22}}+1$ and $2^{2^{222}}+1$. Express your answer as a single integer.

Answer: 1


Question: Determine the limit of the infinite product: \[ \prod_{n=0}^{\infty} \left(1 + \frac{1}{5^{2^n}}\right) \]

Answer: \dfrac{5}{4}


Question: When tossing 3 coins 400 times, you score a point for 3 heads, lose a point for 3 tails, and get no points for any other outcome. The total score \( S \) is the difference between the number of 'triple heads' \( X \) and 'triple tails' \( Y \), both following a binomial distribution with parameters \( n=400 \) and \( p=1/8 \). Given that \( E(S) = 0 \), find the variance \( \text{Var}(S) \).

Answer: 100


Question: Evaluate the line integral \( \int_C \sin(y)dx + x\cos(y)dy \) where \( C \) is the ellipse defined by the equation \( x^2 + xy + y^2 = 1 \).

Answer: 0


Question: In the expansion of the determinant of an \( n \times n \) matrix, can a term contain exactly \( n-2 \) diagonal components? Answer with 'Yes' or 'No'.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to \infty}\int_{1000}^{1500}g(nx)\,dx \] where $g(x)$ is a continuous function defined on $[0,+\infty)$ such that $\lim_{x\to \infty}g(x)=2$. Provide your answer as a single value.

Answer: 1000


Question: Determine the number of distinct ways to distribute 17 identical red balls and 10 identical white balls into 4 distinct boxes such that each box contains more red balls than white balls.

Answer: 5720


Question: Evaluate the limit: $$\lim_{x\to +\infty}[(x^2 +1)^{1/2} -2(x^3 + x)^{1/3} +(x^4 + x^2)^{1/4}]$$

Answer: 0


Question: Determine if the function $f: \mathbb{P}^2 \rightarrow \mathbb{P}^2$, defined by $f((1:1:0))=(0:1:1)$, $f((0:1:1))=(1:0:1)$, and $f((1:0:1))=(1:1:0)$, can be represented by a matrix such that the cube of the matrix is the identity matrix.

Answer: Yes


Question: Consider a Hausdorff space $X$ with cardinality greater than $2^{\aleph_0}$. Determine whether there exists a subspace $Y \subseteq X$ such that $|Y| = 2^{\aleph_0}$ and $Y$ is not second-countable.

Answer: Yes


Question: Consider a root system $(V,R)$ and its dual root system $(V^{\vee},R^{\vee})$. A chamber of $R$ is defined as a connected component of the complement in $V$ of the hyperplanes $H_{\alpha} = \{ x \in V : \langle x, \alpha^{\vee} \rangle =0\}$. For a chamber $C$, the corresponding base $B = B(C)$ is the unique vector space basis $\alpha_1, \ldots, \alpha_l$ of roots such that $$C = \{ x \in V : \langle x, \alpha_i^{\vee} \rangle > 0 \text{ for } 1 \leq i \leq l \}.$$ The elements of $B$ are called simple roots. Determine if the set $B^{\vee}=\{\alpha_1^{\vee}, \ldots, \alpha_l^{\vee}\}$, known as the "simple coroots," is always a base of the dual root system $R^{\vee}$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \frac{1}{1+\tan^n(x)} \, dx \) for all natural numbers \( n \).

Answer: \dfrac{\pi}{4}


Question: Determine whether there exists a sequence $\mathbf{a}$ such that the iterated sums $T^p(\mathbf{a})$ are bounded for all $p \geq 0$, where $T$ is the sum operator defined as $T(\mathbf{a}) = \left(\sum_{k=0}^n a_k\right)_{n \geq 0}$. Provide a justification for your answer.

Answer: Yes


Question: Given the transformation $Y = 3X^2 - 2X^3$, where $X$ is a random variable with probability density function $f(x) = 6(x-x^2)$ for $0 \le x \le 1$, find the probability density function of $Y$ without using the inverse function $h^{-1}(y)$. Provide your answer in terms of $y$.

Answer: 1


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{1}{x^2}\left(\frac{1}{2}\right)^{\log \left(\log \left(\frac{1}{x^2}\right)\right)} \] given that \( \log \left(\log \left(\frac{1}{x^2}\right)\right)=\frac{1}{2}\log \left(\log \left(x^2\right)\right)^2 \).

Answer: \infty


Question: Let $A$ and $B$ be two positive semi-definite matrices of size $n \times n$. Suppose $V \in \mathbb{R}^{n \times k}$ is a matrix such that the range of $V$, $\mathcal{R}(V)$, is a subspace of the null space of $A+B$, $\mathcal{N}(A+B)$. If $V^{T}(A+B) V = 0$, does it necessarily follow that $V^{T} A V = 0$ and $V^{T} B V = 0$?

Answer: Yes


Question: Let $L$ be an orthogonal projection on $\mathbb{R}^n$ with rank $1$. Prove that there exists a unit vector $v$ such that $L$ is represented by the matrix $vv^T$.

Answer: vv^T


Question: Evaluate the integral \( \int\limits_{|z| = 2} \frac{z^4dz}{z^4+1} \).

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^\infty f(t-u) \, du \) where \( f(t) = \begin{cases} 0 & \text{if}\ t<0 \\ 1 & \text{if}\ 0\leq t\leq1 \\ 0 & \text{if}\ t>1. \end{cases} \)

Answer: 1


Question: Determine the x-value where the function \( f(x) = \int_{-1}^{x} t(e^t - 1)(t - 1)(t - 2)^3(t - 3)^5 \, dt \) has a local maximum, given that the roots of the derivative \( f'(x) \) are \( x = 0, 1, 2, 3 \). Analyze the behavior of \( f(x) \) around these roots to find the correct x-value.

Answer: 2


Question: Evaluate the integral \( \int_0^{2 \pi} \frac{\sin x}{1+\frac{1}{2}\cos x} \, dx \) using contour integration and the substitution \( z = e^{i\theta} \).

Answer: 0


Question:
Find a formula for the series:

$$ \sum_{i=1}^m {m \choose i} i! S(n,i) $$

where $S(n,m)$ denotes the Stirling numbers of the second kind.

Answer: m^n


Question: Determine the number of equally spaced data points required to uniquely determine a three-harmonic Fourier series representation of a periodic function $f(\theta)$ with a period of $\frac{\pi}{2}$.

Answer: 7


Question: Determine whether there exists an infinitely differentiable function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that is equal to \( |x| \) for all \( x \in \mathbb{Z} \).

Answer: Yes


Question: Given that $X$ follows a uniform distribution over the interval $(0,1)$, and for $X = x$ with $0 < x < 1$, the variable $Y$ is uniformly distributed over $(x,1)$, determine the marginal probability density function $f_Y(y)$ of $Y$. Provide your answer as a function of $y$.

Answer: -\ln(1 - y)


Question: Do the derivatives of functions in the Sobolev space $H_0^1(0,L)$ vanish at the boundary points $x=0$ and $x=L$?

Answer: No


Question: Evaluate the integral \( \int_1^5 \sqrt{(x-1)(5-x)}\ dx \) using an appropriate substitution method.

Answer: 2\pi


Question: Calculate the limit: $$ \lim_{n \to \infty} \frac{1}{n^{2013}} \sum_{k=1}^n k^{2012} $$

Answer: \dfrac{1}{2013}


Question: Find the degree of the algebraic number $\sqrt{3}+\sqrt{7}-\sqrt{5}$ over the field of rational numbers $\mathbb{Q}$.

Answer: 8


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{\arcsin^3x-\sqrt{1-x^2} \arcsin x}{x}$$ without using L'Hopital's Rule.

Answer: -1


Question: Determine whether there exists a continuous function that maps a set of Lebesgue measure zero to a set of positive Lebesgue measure.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{\gamma \rightarrow 0} \frac{1}{\gamma} \int_0^{\gamma}dx \int_0^{\gamma}dy \, f\left( |x-y| \right) \]

Answer: 0


Question: Given positive integers $a, b, m, n$, determine if it is always possible to find relatively prime positive integers $x, y$ such that $ax + m$ and $by + n$ are not relatively prime.

Answer: Yes


Question: Consider the field extension $E$ of $F_2 = \{0, 1\}$, where $E$ is obtained by adjoining a root $\alpha$ of the irreducible polynomial $g = X^4 + X + 1$. Does $E$ have a subfield with $8$ elements?

Answer: No


Question: In triangle $\triangle PQR$, given that $3\sin P + 4\cos Q = 6$ and $4\sin Q + 3\cos P = 1$, determine the measure of angle $R$.

Answer: 30^\circ


Question: Find the minimal polynomial of \(\sqrt{3}\) over the field \(\mathbb{Q}(\sqrt[6]{3})\), given that \(\mathbb{Q}(\sqrt{3}) \subset \mathbb{Q}(\sqrt[6]{3})\) and \((\sqrt[6]{3})^{3} = \sqrt{3}\).

Answer: x - \sqrt{3}


Question: Determine the radius of convergence for the ordinary generating function $f(x) = \tan x$. Express your answer as a numerical value or expression.

Answer: \dfrac{\pi}{2}


Question: Are there uncountably many sub-intervals within the interval $[0,1]$ such that each sub-interval contains uncountably many real numbers?

Answer: Yes


Question: Determine the number of k-tuples of sets \(<S_1, S_2, \ldots, S_k>\) such that each set \(S_i\) is a subset of \(\{1, 2, \ldots, n\}\) and the following conditions hold: \(S_1 \subseteq S_2 \supseteq S_3 \subseteq S_4 \supseteq S_5 \subseteq S_6 \supseteq \ldots\).

Answer: F_{k+2}^n


Question:
Suppose $V$ and $W$ are finite-dimensional vector spaces, and $f:V\rightarrow W$ is a linear map. Given that $\{e_1, \ldots, e_n\} \subset V$ and $\{f(e_1), \ldots, f(e_n)\}$ is a basis of $W$, which of the following statements are true?

I. $\{e_1, \ldots, e_n\}$ is a basis of $V$
II. There exists a linear map $g:W\rightarrow V$ such that $g \circ f = \text{Id}_V$
III. There exists a linear map $g:W\rightarrow V$ such that $f \circ g = \text{Id}_W$

Select the correct option:
(A) I only
(B) II only
(C) III only
(D) I and III
(E) II and III

Answer: C


Question: Determine whether Euler's totient function, \(\phi(n)\), takes on all possible values when \(n\) is an odd integer. Answer with 'True' or 'False'.

Answer: True


Question: Evaluate the infinite series: \[ \sum_{n=0}^{\infty}\frac{1}{2^n(n+1)} \]

Answer: 2 \ln 2


Question: Determine if the piecewise function \(f(x)=\begin{cases} \displaystyle e^{\frac{-1}{x^2}} &\text{if}\, x \ne 0\\ 0 &\text{if}\, x = 0\end{cases}\) has a continuous \(n\)th derivative at \(x=0\) for all positive integers \(n\).

Answer: Yes


Question: Determine the number of ways to arrange $n$ distinct objects in a circle such that no two adjacent objects are the same.

Answer: (n-1)!


Question: Determine whether the sequence \( S_n = \frac{1}{\sum_{i=n}^\infty i^{-2}} \) converges as \( n \to \infty \). If it converges, find the limit.

Answer: \infty


Question: Let $x, y \in \mathbb{R}$ such that \[ \sqrt{\left(\frac{x+y}{2}\right)^3} + \sqrt{\left(\frac{x-y}{2}\right)^3} = 27. \] Find the minimum value of $x$. \( \boxed{} \)

Answer: 9


Question: How many distinct arrangements are there for 61 red cards and 61 blue cards such that no two cards of the same color are adjacent?

Answer: 2 \times (61!)^2


Question: Given two integers $a$ and $b$ such that $$(1+\sqrt{2})^{2020} = a + b\sqrt{2},$$ find the value of $a^2 - 2b^2$. \(\boxed{?}\)

Answer: 1


Question: Let $R$ be an integral domain and $F$ be a subfield of $R$. If $\alpha \in R$ is algebraic over $F$, does it imply that $\frac{1}{\alpha}$ is also algebraic over $F$? Provide a justification for your answer.

Answer: Yes


Question: Find the norm of the differential operator \( \frac{d}{dx} \) on the space of polynomials of degree \( \leq n \) on the interval \([-a,a]\), where \( a > 0 \).

Answer: \dfrac{n^2}{a}


Question: Evaluate the integral \( \int_{0}^{2\pi}\frac{\sin(x)}{\sqrt{5+4\cos(x)}}\,dx \).

Answer: 0


Question: Evaluate the limit: $$\lim\limits_{x \to 1^-} \left[\ln\left(\left(\frac{x-1}{x+1}\right)^2\right) \cdot x - 2 \cdot \ln|x^2 - 1|\right]$$

Answer: -4 \ln 2


Question: Given a set with $n$ elements, determine the number of arrangements in which a specific subset of $k$ elements cannot be placed next to each other, where $k > \frac{n+1}{2}$. Provide your answer as a single integer.

Answer: 0


Question: Determine the singularities and their orders for the function \( \frac{e^{z^3}-1}{z^m} \), where \( m \) is a positive integer.

Answer: m - 3


Question: Calculate the limit of the function \((\sqrt{4+x}-1)^{\frac{1}{e^x-1}}\) as \(x\) approaches 0, using notable limits and without applying l'Hopital's rule.

Answer: \sqrt[4]{e}


Question: Evaluate the improper integral: $$I=\int_{-1}^1 \left(\frac{1}{x}+1-\frac{\sqrt{1-x^2}}{x}\right )\arctan\left(\frac{2}{x^2}\right)dx.$$

Answer: 4 \arctan 2 - \ln 5


Question: Find the value of \( n \) such that \( \lim_{x\rightarrow 0}\frac{x^n\sin^{n}(x)}{x^{n}-(\sin x)^{n}} \) is a finite non-zero number.

Answer: 2


Question: Determine the function $a(t)$ such that the stochastic process $X(t) = e^{Y(t)}$ is a martingale, where $Y(t)$ is given by \[ Y(t) = \int_0^t b(u) \, dW(u) + \int_0^t \left(a(u) - \frac{1}{2}b^2(u)\right) \, dt. \]

Answer: 0


Question: Determine the number of ring homomorphisms from the ring of Gaussian integers \( \mathbb{Z}[i] \) to the ring \( \mathbb{Z}/102\mathbb{Z} \), given that \( 102 = (1-i)(1+i)(2)(3)(17) \).

Answer: 0


Question: Determine the operator norm of the linear operator $A$ defined by \( Ax=(x_1+x_2, x_2+x_3, \ldots) \) for all $x \in \ell_2$. Given that the operator is bounded with \( \sum_{i=0}^\infty|x_i+x_{i+1}|^2 \leq 6\sum_{i=0}^\infty|x_i|^2 \), find the norm of $A$ by evaluating the supremum of $\|Ax\|$ for $\|x\|=1$. \( \| \cdot \| \) denotes the $\ell_2$ norm.

Answer: 2


Question: Calculate the expected value of the ratio \(E\left(\frac{X}{Y}\right)\), where \(X\) is an exponentially distributed continuous random variable with parameter \(\lambda = 1\) and \(Y\) is an independent uniform continuous random variable on the interval \((1, 2)\).

Answer: \ln 2


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

Answer: \dfrac{1}{2}


Question:
Determine the limit of the sequence \( \{x_n\} \) defined recursively by:
\[ x_1 = 2 - \frac{\arctan(1) \cdot 1^2 - 3}{3 + 1^2} \]
and for \( n > 1 \),
\[ x_n = x_{n-1} \cdot \left(2 - \frac{\arctan(n) \cdot n^2 - 3}{3 + n^2}\right) \]
If the limit exists, find its value.

Answer: 0


Question: Determine the number of five-digit sequences \(d_1d_2d_3d_4d_5\), where \(d_i\in\{0,1,\ldots,9\}\), that do not contain any palindromic subsequence of length 3 or greater.

Answer: 52560


Question: Determine whether the inequality \[2 \cdot p_1^{k_1} \cdot p_2^{k_2} \cdot p_3^{k_3} \cdot p_4^{k_4} \cdot p_5^{k_5} \cdot p_6^{k_6} < \frac{(p_1^{(k_1+1)} - 1)}{p_1 - 1} \cdot \frac{(p_2^{(k_2+1)} - 1)}{p_2 - 1} \cdot \frac{(p_3^{(k_3+1)} - 1)}{p_3 - 1} \cdot \frac{(p_4^{(k_4+1)} - 1)}{p_4 - 1} \cdot \frac{(p_5^{(k_5+1)} - 1)}{p_5 - 1} \cdot \frac{(p_6^{(k_6+1)} - 1)}{p_6 - 1}\] holds for any odd primes \(p_1, p_2, p_3, p_4, p_5, p_6\) and their corresponding exponents \(k_1, k_2, k_3, k_4, k_5, k_6\).

Answer: No


Question: What is the probability of selecting $n$ linearly independent vectors from the unit sphere in $\mathbb{R}^n$ using the uniform distribution?

Answer: 1


Question: Find the value of $a$ such that the limit \( \lim_{{x \to \infty}} \left(a^{\frac{1}{x}} + \frac{1}{x}\right)^x = 3 \) holds true.

Answer: \dfrac{3}{e}


Question: Compute the distance from the constant function $g(x) = 0$ to the set $A = \{f \in \mathcal{C}([0,1]) \mid f(0) = 1 \text{ and } \int_0^1 f^2 = 2\}$ in the normed space $(\mathcal{C}([0,1]), \|\cdot\|_\infty)$ with the supremum norm.

Answer: \sqrt{2}


Question: Determine whether it is possible to find algebraic numbers $\alpha$ and $\beta$ such that $[\mathbb{Q}(\alpha, \beta):\mathbb{Q}(\alpha)]$ is not equal to $[\mathbb{Q}(\beta):\mathbb{Q}]$, even though $[\mathbb{Q}(\alpha, \beta):\mathbb{Q}(\alpha)]$ divides $[\mathbb{Q}(\beta):\mathbb{Q}]$. Answer yes or no.

Answer: Yes


Question: Consider a sequence of n-hypercube graphs, where each n-hypercube has \(2^n\) vertices. Assign independent Bernoulli(1/2) random variables to each vertex. Define a random variable on each edge that is 1 if the Bernoulli variables of the two vertices it connects are equal, and 0 otherwise. For each n, sum these random variables over all edges and normalize the sum by subtracting the expectation and dividing by the standard deviation. Does this normalized sum converge to a standard normal distribution as n approaches infinity?

Answer: Yes


Question: Evaluate \( \lim_{k \to \infty} \int_{0}^{\infty} k^{3/2} f(x) e^{-kx} \, dx \) where \( f(x) = \int_{0}^{x} \frac{\sin(t)}{t^{3/2}} \, dt \).

Answer: \sqrt{\pi}


Question: Suppose we have two positive integers $x$ and $y$ such that $$x \bmod p \leqslant y \bmod p$$ for every prime number $p$. Does it follow that $x = y$? Answer yes or no.

Answer: Yes


Question: Let $T$ be an invertible linear operator on a finite-dimensional vector space over a finite field. Determine whether there exists a positive integer $n$ such that $T^n = I$, where $I$ is the identity operator.

Answer: Yes


Question: Determine which function is asymptotically greater as $n$ approaches infinity: $f(n) = 3^{2^n}$ or $g(n) = n! \times n^3$.

Answer: f(n)


Question: Let $A$ be an $n \times n$ matrix over $\mathbb{R}$ with positive entries. Assume $v$ is a generalized eigenvector of $A$ with $(A-\lambda I)^r v=0$, where $r \geq 2$ and $|\lambda|<1$. The unique dominant eigenvalue of $A$ is $\hat{\lambda}=1$. Show that $\lim_{k \to \infty} A^k v=0$. Given that $v'$ is an ordinary eigenvector with $v'=(A-\lambda I)^{r-1} v$, and $A^k v'=\lambda^k v'$, use the fact that $(A^k-\lambda^k I)^r v=0$ to prove the result.

Answer: 0


Question: Determine the values of \( \alpha \) for which the integral \( \int_{1}^{\infty}{\frac{1}{\sqrt{x^\alpha-1}}\, dx} \) converges.

Answer: \alpha > 2


Question: Calculate the limit: $$\lim_{x\rightarrow \infty} \left(\frac{2x+1}{x-1} \right)^x$$

Answer: \infty


Question: Evaluate the integral \( \int_0^{\pi} \frac{dt}{\sin(t)+1} \).

Answer: 2


Question: Let \( F: [0, \infty) \to [0, \infty) \) be a twice continuously differentiable (\( C^2 \)) strictly convex function with \( F''(x) > 0 \) for all \( x \in [0, \infty) \). Given sequences \( \lambda_n \in [0, 1] \), \( a_n \leq c_0 < b_n \in [0, \infty) \) that satisfy \( \lambda_n a_n + (1 - \lambda_n) b_n = c_n \) for some constant \( c_0 \) and \( c_n \to r > c_0 \), and define \( D_n = F(c_n) - \lambda_n F(a_n) - (1 - \lambda_n) F(b_n) \) with \( \lim_{n \to \infty} D_n = 0 \). Determine if \( b_n \) must be bounded.

Answer: Yes


Question: How many subgroups in the Cartesian product $C_9 \times C_9$ are isomorphic to the cyclic group $C_9$?

Answer: 12


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

Answer: 3


Question: Determine whether there exists a matrix A with eigenvalues 1 and -1 such that A ≠ A³.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\sum^{n}_{r=1}\frac{r^2+r}{n^3+r} \]

Answer: \dfrac{1}{3}


Question: Is there a function that is both discontinuous and bijective, and whose inverse function is also both discontinuous and bijective?

Answer: Yes


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

Answer: \dfrac{\pi}{2}


Question: Evaluate the integral \( \int_{|z|=1} \frac{(1-\cos z)}{(e^z-1)\sin z} \, dz \) using the Cauchy residue theorem.

Answer: 0


Question: Given a finite set $S$ and a set $G$ of maps on $S$ with the following properties: (1) each map $g \in G$ has finite order, and (2) there exists a starting element $s_0 \in S$ such that every element $s \in S$ can be reached from $s_0$ through a finite sequence of maps from $G$. Does the composition of these maps form a finite group $H$ that acts on $S$?

Answer: Yes


Question: Evaluate the contour integral \( \frac{1}{2\pi j} \oint_C \Re\{ z\} \, dz \) where \( C \) is the counterclockwise unit circle, by parameterizing the circle as \( z=e^{i\theta} \) and evaluating the integral directly.

Answer: \dfrac{1}{2}


Question: Suppose $\alpha$ is a root of the polynomial $x^3 + x^2 + 1$ in the finite field $\mathbb{F}_2$, and let $K = \mathbb{F}_2(\alpha)$. If $f$ is an irreducible polynomial in $K[x]$ of degree 4, and $\beta$ is a root of $f$, consider the splitting field $L$ of $f$ over $K$. Given that $K$ has 8 elements, determine the number of elements in the field $L$.

Answer: 4096


Question: Find the optimal solution \((x^*, y^*)\) for the following concave maximization problem using the Karush-Kuhn-Tucker conditions: \[\max_{x, y} f(x,y) = 2\ln x - \ln (x+y) - x -2y\] subject to \(x \geq 0, y \geq 0\).

Answer: (1, 0)


Question: Evaluate the limit: $$\lim_{x\rightarrow0} \lim_{y\rightarrow x} \frac{y^2 f(x) - x^2 f(y)}{(1-\cos x)\sin (x-y)}$$ where $f: I \rightarrow \mathbb{R}$ is a differentiable function with a continuous derivative $f'$ on an open interval $I$ containing $0$. Provide the correct solution.

Answer: -2f'(0)


Question: Find the number of automorphisms of the group $\mathbb{Z}_4 \oplus \mathbb{Z}_2$ that map the element $(1, 0)$ to an element of order 4 and the element $(0, 1)$ to an involution.

Answer: 8


Question: What is the number of partitions of 100, denoted by \( p(100) \)?

Answer: 190569292


Question: Determine if the parameter \( b \) can be zero in the confluent hypergeometric function of the second kind \( U(a,b,z) \).

Answer: Yes


Question: Determine the value of \( \alpha \) that results in the fastest convergence of the given iteration formula.

Answer: 1


Question: Determine the number of homomorphisms from the symmetric group $S_n$ to the group $\mathbb{Z}_2$ for $n \geq 5$.

Answer: 2


Question:
Calculate the average value of the function $P(x) = x^2$ on the fat Cantor set $C$, constructed by iteratively removing the middle $1/4$ of the remaining intervals. Use the formula:

$$\lim_{n\to\infty}\sum_{i=1}^{2^n} P(t_{n,i})\left(\frac{1}{2^n}\right)$$

where $\bigcup_{i=1}^{2^n} C_{n,i} = C_n$ and $t_{n,i} \in C_{n,i}$. Determine whether the average value is $\frac{4}{11}$ or $\frac{38}{105}$. Provide the correct value.

Answer: \dfrac{4}{11}


Question: Let \(X_t\) be a Brownian motion and \(a < 0\). Define the stopping time \(T_x = \inf\{t \geq 0: X_t + at \geq x\}\). If \(Y = \sup_{t \geq 0}(X_t + at) = x\), determine whether \(T_x < \infty\).

Answer: T_x < \infty


Question: Determine the number of unique ways to arrange a group of n boys and n girls in a line such that no boy is adjacent to another boy, no girl is adjacent to another girl, substrings "BBB" and "GGG" are not allowed, and the line cannot start or end with both a boy and a girl.

Answer: 0


Question: Determine if the only natural number solutions to the equation \(a^{b^c}=c^{b^a}\), where \(a, b, c \in \mathbb{N}\), occur when \(a = c\) and \(b\) is any natural number.

Answer: a = c


Question: Calculate the volume of the solid enclosed by the surface \((x^2 + y^2 + z^2)^2 = xyz\). Set up and evaluate the appropriate triple integral using spherical coordinates \((r, \theta, \varphi)\), where \(x = r\sin(\theta)\cos(\varphi)\), \(y = r\sin(\theta)\sin(\varphi)\), and \(z = r\cos(\theta)\). Determine the bounds for \(r\), \(\theta\), and \(\varphi\) from the given equation and calculate the volume.

Answer: \dfrac{1}{360}


Question:
Which interpretation correctly represents convergence in distribution for a sequence of real-valued random variables $X_1, X_2, \cdots$ converging to a random variable $X$? Consider the definition: $ \lim _{n\to \infty }F_{n}(x)=F(x) $ for every $x \in \mathbb{R}$ at which $F$ is continuous, where $F_n$ and $F$ are the respective cumulative distribution functions. Choose the correct interpretation:

1) For all $x$ in the set $E$ where $F$ is continuous, there exists an $n$ such that $|F_N(x) - F(x)| < \epsilon$ for all $N \ge n$.
2) For all $\epsilon > 0$, there exists an $n$ such that $|F_N(x) - F(x)| < \epsilon$ for all $x \in E$ and all $N \ge n$.

Select the correct interpretation.

Answer: 1


Question: Determine the critical parameter value \( \mu \) at which a bifurcation occurs in the system given by: \[ \dot{r} = r(r^2 - \mu), \\ \dot{\theta} = r \]

Answer: 0


Question: Evaluate the limit of the integral \( \int_{1/4}^{3/4} G^{n}(x-1/4) \, dF(x) \) as \( n \to \infty \), where \( F \) and \( G \) are cumulative distribution functions (CDFs) with support \([0,1]\), strictly increasing, and with finite probability density functions (pdfs).

Answer: 0


Question: Determine whether the following statement is true or false: Given a prime number $p$, for every $1 \le k, r < p$, there exists a natural number $n$ such that $nk \equiv r \pmod{p}$.

Answer: True


Question:
Given a complex number $z \in \mathbb{C}$ with $|z|=1$, where the imaginary part of $z$ is rational, determine which of the following statements is necessarily true:

(A) For every $n \ge 1$, the imaginary part of $z^{2^n}$ is rational.
(B) For every $n \ge 1$, the real part of $z^{2^n}$ is rational.
(C) There exists an $n_0$ such that for every $n \ge n_0$, both the real and imaginary parts of $z^{2^n}$ are rational.
(D) None of the above.

Answer: B


Question: Given the system of equations: \(x + y = z^2 + 1\), \(y + z = x^2 + 1\), and \(x + z = y^2 + 1\), find the value of \(2x + 3y + 4z\).

Answer: 9


Question: Find the roots of the function \(f(x) = x - 2\ln{(x^2+1)}\) without using a calculator.

Answer: 0


Question: Find the number of elements in a subspace of dimension $k$ of a vector space of dimension $m$ over a field of characteristic $p$.

Answer: p^k


Question: A circle with center at \((15, -3)\) is tangent to the parabola \(3y = x^2\) in the first quadrant. Find the radius of the circle.

Answer: 6\sqrt{5}


Question: Calculate the sum of the series \( \sum\limits_{k \geq 0} \frac{(k+2)^2}{(k+1)!} \).

Answer: 5e - 1


Question: Find the number of ideals in the quotient ring \( k[x, y]/(x^2, y^2) \), where \( k \) is a field.

Answer: 8


Question: Evaluate the limit: $$\lim\limits_{h\to0^+}\dfrac{\sqrt[3]{1+h^6}-\sqrt[3]{1-h^6}}{h^5}$$

Answer: 0


Question: Find the exponential generating function (EGF) for the series \(a_n = (-1)^n \sum_{k=0}^{n}k \binom{n}{k}\).

Answer: -x e^{-2x}


Question: Determine if the recursively defined sequence \((w_n)_{n=1}^\infty\) converges, where \(w_1=i\) and \(w_{n+1}=\frac{w_n+2}{w_n+1}\). If it converges, find its limit.

Answer: \sqrt{2}


Question: Determine whether the integral \( \int_0^\infty \frac{1}{(x^p+k)^q} \,dx \) converges for given parameters \( p, q, k > 0 \).

Answer: pq > 1


Question: Evaluate the limit: $$\lim_{n\to \infty} n \left(\frac{2n \choose n}{4^n}\right)^2$$ and show that it equals $\frac{1}{\pi}$. Provide a detailed explanation of your solution.

Answer: \dfrac{1}{\pi}


Question: Find the Laurent series expansion of the function \( f(z) = \frac{\cos(z^2)}{z^7} \) for \( 0 < |z| < \infty \), and determine the residue of this function at \( z = 0 \).

Answer: 0


Question: Evaluate the integral \( \int_0^n \frac{f(x)}{f(x) + f(n-x)} \, dx \), where \( f(x) \) is a continuous function on the interval \([0, n]\) with \( n > 0 \) and \( f(x) + f(n - x) \neq 0 \) for all \( x \in [0, n] \).

Answer: \dfrac{n}{2}


Question: Let $H$ be a Hilbert space, and let $T:H \to H$ be a bounded self-adjoint operator satisfying the equation $T^3 + T^2 + T + I = 0$. Determine the spectrum $\sigma(T)$.

Answer: \{-1\}


Question: Determine the convergence of the integral \( J_n=\int^\infty_0 e^{-t}\big|\cos(nt+c_n)\big|^n\, dt \) as \( n\rightarrow\infty \), where \( c_n \) is an arbitrary numerical sequence. For which \( p>0 \) does \( n^pJ_n \) converge?

Answer: \frac{1}{2}


Question: Find the limit of the function as both \( \sigma_A \) and \( \sigma_B \) approach infinity: \[ \lim_{{\sigma_A, \sigma_B \to \infty}} e^{-\sigma_A-\sigma_B} \sum_{{n=0}}^{\infty} \frac{(\sigma_A\sigma_B)^n}{(n!)^2} \]

Answer: 0


Question: Find the minimum value of the expression \( \sqrt{2x^2+2x+1} +\sqrt{2x^2-10x+13} \). Determine the value of \( \alpha \) such that the minimum value is \( \sqrt{\alpha} \).

Answer: 20


Question: Let $X = A \cup B$ be an open cover of $X$. Assume that $X$, $A$, and $B$ are simply connected. Determine whether $A \cap B$ must be path-connected.

Answer: Yes


Question: Determine the signature of the quadratic form \( f(x, y, z) = 2xy + 2yz + 2xz + x^2 - z^2 \).

Answer: (1, 2)


Question: Determine whether there exists a ratio of rational numbers to irrational numbers in the open interval (1,2).

Answer: Yes


Question: Determine whether the following statement is true or false: If the matrix $t^{-1}I+A$ is invertible for all $t\neq0$, then the matrix $A$ must be nilpotent.

Answer: True


Question:
Calculate $ab$ given the following expressions for $a$ and $b$:

$$a = \sqrt{7\sqrt{2\sqrt{7\sqrt{2\sqrt{\dots}}}}}$$
$$b = \sqrt{2\sqrt{7\sqrt{2\sqrt{7\sqrt{\dots}}}}}$$

Express $ab$ in a finite form.

Answer: 14


Question: Determine the measure of the set of all numbers in the interval $[0, 1]$ whose decimal expansion does not contain the digit 7.

Answer: 0


Question: Evaluate the double integral of the function \( f(x,y) = x^2 - y^2 \) over the region \( D \) bounded by the curves \( y = x \), \( y = x + 1 \), \( xy = 1 \), and \( xy = 2 \) in the first quadrant.

Answer: -\dfrac{1}{2}


Question: Determine if the sequence defined by \(P(X_n=2^n)= \frac{1}{4^n} ,P(X_n=-2^n)= \frac{1}{4^n}, P(X_n=0)= 1 -\frac{2}{4^n}\) satisfies the strong law of large numbers.

Answer: Yes


Question: Determine the maximum area of a quadrilateral with vertices \((0,0), (x, 0), (0, y), (x,y)\), where the last vertex lies on the line \(\frac{x}{a} + \frac{y}{b} = 1\). Use a modified Lagrange multiplier method to ensure a solution exists.

Answer: \dfrac{ab}{4}


Question: Does there exist a countable separating family for the space of measures \(\mathcal{M}_1\) on a Polish space \((E, \mathcal{A})\)?

Answer: Yes


Question: Transform the integral \( \int_0^4\int_0^{\sqrt{16-y^2}} (x^2+y^2) \, dx \, dy \) into a polar integral and evaluate the result.

Answer: 32\pi


Question: Evaluate the limit: $$\lim_{x \to -\infty}\dfrac {f^{-1}(x)-f^{-1}(-8x)}{x^{1/3}},$$ where $f(x)=8x^3+3x$. Use properties of the inverse function to find the solution.

Answer: \dfrac{3}{2}


Question: Find the equation of the line from the family of lines given by \( a(3x+4y+6)+b(x+y+2)=0 \) that is at the maximum distance from the point \( P(2,3) \).

Answer: 4x + 3y + 8 = 0


Question: Find the value of $x \in [-1.5, 3]$ such that the area of the triangle formed by the points $A=(-1.5, 2.25)$, $B=(3, 9)$, and $C=(x, x^2)$ on the parabola $y=x^2$ is maximized.

Answer: \dfrac{3}{4}


Question: Given a circle with radius 1, an arc subtends an angle of \(x\) radians at the center \(O\). Points \(A\) and \(B\) are the endpoints of the arc, and \(C\) is the intersection point of the tangents at \(A\) and \(B\). Let \(T(x)\) be the area of triangle \(ABC\). Find \(\lim_{x \to 0} \frac{T(x)}{x^3}\).

Answer: \dfrac{1}{8}


Question: Evaluate the line integral of the conservative vector field \(G(x,y,z) = (y^2+z^2)i + (2xy+z)j + (2xz+y)k\) along the parametric curve \(r(t) = (t^2, t^3, t)\) from \(t=0\) to \(t=1\).

Answer: 3


Question: Find the Laurent series representation of the function \(f(z) = \frac{z}{(z-1)(z-3)}\) centered at \(z=3\). Determine the residue of \(f(z)\) at \(z=3\).

Answer: \dfrac{3}{2}


Question: Find the location of the maximum for the function \( f(x) = \frac{x^3}{e^x - 1} \) for \( x > 0 \). Approximate this maximum to two decimal places, using the hint that \( 3e^{-3} \approx 0.15 \).

Answer: 2.82


Question: A teacher has \( n \) students and divides them into groups, assigning one student as president and another as vice president in each group. Determine the exponential generating function for the number of ways \( t_n \) this can be done.

Answer: e^{x^2 e^x}


Question: Find the degree of the splitting field of the polynomial $x^6 + 1$ over the field $\mathbb{Z}_3$. Express your answer as an integer.

Answer: 2


Question: Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that \(f\big(f(x)\big)=2^x-1\) for all \(x\in\mathbb{R}\). Determine the value of \(f(0)+f(1)\).

Answer: 1


Question: Given a nontrivial connected separable topological space $X$, suppose $f: X \rightarrow \mathbb{R}$, $h: X \rightarrow \mathbb{R}$, and $g: \mathbb{R} \rightarrow \mathbb{R}$ are continuous functions such that $f(x) = g(h(x))$ for all $x \in X$. If $h(X) = \mathbb{R}$, determine whether $g$ must be continuous when $X$ is path-connected.

Answer: Yes


Question: Compute the distance $d(S, \mathbb{K}(\mathcal{H}))$, where $S$ is a unilateral shift operator in the algebra of bounded linear operators $\mathbb{B}(\mathcal{H})$ on a separable, infinite-dimensional Hilbert space $\mathcal{H}$ with an orthonormal basis $(e_n)$. The distance is defined as the infimum of $\{\|S-K\|: K\in \mathbb{K}(\mathcal{H})\}$, with $\mathbb{K}(\mathcal{H})$ denoting the set of compact operators on $\mathcal{H}$. Provide your answer as a single numerical value.

Answer: 1


Question:
In a non-homogeneous Poisson process $(N_t)$ with rate function $\lambda(t)$ given by:

$$\lambda(t) = \begin{cases} 1 & t<1 \\ 2 &1\leq t < 2 \\ 3 & t\geq 2 \\ \end{cases} $$

it is known that $N_3 = 10$ events occur over the interval $[0, 3]$. Calculate the conditional probability $P(N_1 = 2 \mid N_3 = 10)$, which is the probability that exactly 2 of these events occurred in the interval $[0, 1]$. Provide your answer as a decimal rounded to four decimal places.

Answer: 0.2907


Question: Calculate the value of the limit: $$\lim_{n\to\infty}\frac{|\cos{1}|+|\cos{2}|+|\cos{3}|+\cdots+|\cos{n}|}{n}.$$

Answer: \dfrac{2}{\pi}


Question: In an extension of algebraic number fields \( L/K \), with \( \mathcal{O}_L \) and \( \mathcal{O}_K \) as the rings of integers, is the prime ideal in \( \mathcal{O}_K \) below a given prime ideal \( \mathfrak{P} \subset \mathcal{O}_L \) unique?

Answer: Yes


Question: Evaluate the integral \( I = \int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} \, dx \).

Answer: 8\pi \ln 2


Question: Find all real values of $x$ that satisfy the equation: $$\arccos\left(\frac{1-x^2}{1+x^2}\right) = -2\arctan(x).$$

Answer: (-\infty, 0]


Question:
Consider a stochastic process \(\{b_t\} \) where \(b_0 = 0\) and \(b_t = \sum_{i=1}^t \varepsilon_i\), with \(\varepsilon_i\) being i.i.d. random variables with zero mean. Given a discount factor \(0 < \beta < 1\), determine if the following limit expectation is true:

\[ \mathbb{E}_{t = 0}\left[ \lim_{t \to \infty} \beta^t b_t \right] = 0 \]

Answer: 0


Question: Find an explicit bijection between the set of equivalence classes of the equivalence relation defined on \( \mathbb{R}[X] \) by \( P_1 \sim P_2 \) if \( P_1 - P_2 \) is divisible by \( X \), and the set of real numbers \( \mathbb{R} \).

Answer: [P] \mapsto P(0)


Question: Let $S$ be a family of functions from $\mathbb{R}$ to $\mathbb{R}$ that are uniformly bounded and belong to $L^1(\mu)$, where $\mu = e^{-2|x|}L$ and $L$ denotes the Lebesgue measure. If $\sup\{v(y): v \in S\} < \infty$ for all $y \in \mathbb{R}$ and $u(x) = \limsup\limits_{y \to x} \sup\{v(y): v \in S\} < \infty$ for all $x \in \mathbb{R}$, determine whether $u$ is also in $L^1(\mu)$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the limit: $$\lim_{x \rightarrow 0^+} \frac{\displaystyle\arctan (\log (1+\sqrt x)) \sin^3(x^\frac34)}{\displaystyle (e^{\tan(x)}-1)(1-\sin^2(x))}$$

Answer: 0


Question: Find the number of elements in the set \(\{a \in \mathbb{Z}_{30} : a^{1005} = [7]\}\). Consider both cases where \(a\) is coprime to 30 and where \(a\) is not coprime to 30.

Answer: 1


Question: Let $f$ be a differentiable function on $\Bbb R$ such that $f\left(\frac{2}{3}\right) = 25$ and $f'\left(\frac{2}{3}\right) = 45$. Define $h(x) = \frac{x}{x^2+1} \cdot f\left( \frac{x}{x+1}\right)$. Find $h'(2)$. \( \boxed{} \)

Answer: -1


Question: Find the norm of the Volterra operator $A$ acting on the space $L_2(0,1)$, where $Ax(t) = \int_0^t x(s) \, ds$. \( \boxed{} \)

Answer: \dfrac{2}{\pi}


Question: Consider a deck of 50 cards, numbered from 1 to 50. Perform the following operation: look at the number on the top card, say it is K, and reverse the order of the first K cards in the deck. For instance, if K=5, reverse the order of the top 5 cards so that the fifth card becomes first, the fourth becomes second, and so on. Determine whether this process will eventually stop.

Answer: Yes


Question: Determine the minimum number of blue squares required on a $7 \times 7$ board such that each square has at least two neighboring blue squares. Two squares are considered neighbors if they share a side.

Answer: 25


Question: Determine the conditions on the parameter $a$ such that the map $h_1$ is a homeomorphism.

Answer: a \neq 0


Question: Determine all prime numbers $p$ for which there exist positive integers $a$ and $b$ such that \( \frac{1-a^p}{1-a} = bp^2 \).

Answer: 2


Question: Determine the degree of the field extension \( \mathbb{Q}(w, \sqrt[3]{5}) \) over \( \mathbb{Q} \), where \( w \) is a primitive cube root of unity and the minimal polynomial of \( \sqrt[3]{5} \) over \( \mathbb{Q} \) is \( x^3 - 5 \).

Answer: 6


Question: Determine the number of steps required for a 4x4 Jordan form matrix with eigenvalues of zero to become nilpotent.

Answer: 4


Question: Determine the number of zeros of the entire analytic function $g(z)$ inside the unit square in the first quadrant, given the conditions $g(z+1) = g(z)$, $g(z+i) = e^{\pi}e^{-2\pi iz}g(z)$, and that $g(z)$ has no zeros on the boundary of the square.

Answer: 1


Question:
Calculate the double integral \( \int_{\mathbb{R}}\left(\int_{\mathbb{R}} f(x,y) \, dx\right)dy \) for the function \( f(x,y) \) defined as follows:

\[ f(x,y) = \begin{cases}
      1 & x\geq 0, \ x\leq y\leq 1+x \\
      -1 & x\geq 0, \ 1+x\leq y\leq 2+x \\
      0 & \text{otherwise}
   \end{cases} \]

Answer: 1


Question: Determine whether the following statement is true or false: A function $f: U \to \mathbb{R}$, where $U \subset \mathbb{R}^n$ is open, is partially derivable if and only if it is derivable in the direction of the standard unit vectors $e_1, \ldots, e_n$. Provide a justification for your answer.

Answer: A


Question: Evaluate the sum \(1^{2017} + 2^{2017} + \cdots + 1000^{2017} \pmod{2016}\).

Answer: 784


Question: Evaluate the limit: $$\lim_{n \to \infty} \left ( 1-\frac{3}{4} \right ) ^{\frac{4}{n}}\cdot \left ( 1-\frac{3}{5} \right ) ^{\frac{5}{n}}\cdot \left ( 1-\frac{3}{6} \right ) ^{\frac{6}{n}}\cdots \left ( 1-\frac{3}{n+3} \right ) ^{\frac{n+3}{n}}$$

Answer: e^{-3}


Question: Evaluate the limit \(\lim_{{(x,y) \to (0,0)}} \frac{x^4 + \sin^2(y^2)}{x^4+y^4}\). Determine whether the limit exists and, if it does, find its value.

Answer: 1


Question: Suppose we have two sequences \((x_n)\) and \((y_n)\). If \((x_n)\) is bounded and \(|x_n - y_n| < \frac{1}{n}\) holds for all \(n\), and a subsequence \((x_{n_i})\) of \((x_n)\) converges to a limit \(a\), does the subsequence \((y_{n_i})\) also converge to \(a\)? Justify your answer.

Answer: Yes


Question: Find the shortest distance from the origin to the surface defined by the equation $xyz^2 = 2$. Express your answer in simplest form.

Answer: 2


Question: Determine whether the function \( g(u) = \int_\Omega (1 - |u|^2)^4\, dx \) is weakly lower semi-continuous on \( H^1(\Omega) \).

Answer: Yes


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

Answer: \dfrac{1}{12}


Question: Determine whether the following statement is true or false: For any subset $S$ of the rational numbers $\mathbb{Q}$, the boundary of $S$ contains both interior and exterior points of $S$. Provide a justification for your answer.

Answer: False


Question: Determine the dimension of the vector space \( \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} \) over the field \( \mathbb{Q} \).

Answer: 1


Question: Determine whether there exist two non-real complex numbers \(a\) and \(b\) such that \(a^2 = b\) and \(b^2 = a\).

Answer: Yes


Question: Find the number of non-isomorphic groups of order 36 that have a normal subgroup of order 4.

Answer: 6


Question: Consider the function \( h(x,y)=\sin(xy+\pi) \). Let \( x(t)=e^t \) and \( y(t) \) be an unknown function. If the rate of change of the height in the direction \( \vec{u}=\vec{i}+\vec{j} \) at the position \( (1,\pi) \) is \( \pi+1 \), find the value of \( y'(0) \).

Answer: 1


Question: Determine whether there exists an injective function from the set of continuous real-valued functions on the interval $[0, 1]$, denoted as $C[0, 1]$, into the set of all infinite sequences of real numbers, $\\mathbb{R}^{\mathbb{N}}$. Provide a justification for your answer.

Answer: Yes


Question: Determine the value of $a$ such that the equation $x^2 + y^2 = axy$ has positive integer solutions for $x$ and $y$, where the greatest common divisor of $x$ and $y$ is 1.

Answer: 2


Question: Compute the limit: $$\lim_{x\to \infty}x\ \log\left(\frac{x+c}{x-c}\right)$$

Answer: 2c


Question: Determine the probability that the total lifetime of the first two type-B bulbs exceeds the lifetime of the first type-A bulb, given the following probability density functions for the lifetimes of the bulbs: \( f_{X}(x) = e^{-x} \) for type-A bulbs and \( f_{X}(x) = 3e^{-3x} \) for type-B bulbs, where \( x \ge 0 \).

Answer: \dfrac{7}{16}


Question: Compute the surface integral \(\int_S\sqrt{x^2+y^2+z^2}\,dS\) over the surface \(S\) defined by \(S=\big\{ \big(x,y,z\big):x^2+y^2+z^2=b^2,z\ge 0, b>0\big\}.\)

Answer: 2\pi b^3


Question: Let $E$ be a set with positive Lebesgue measure in $\Bbb R$. Determine whether there necessarily exists a point $x \in E$ such that $x$ can be expressed as the average of two distinct points $y$ and $z$ in $E$, i.e., $x = \frac{1}{2}(y + z)$. Provide a justification for your answer.

Answer: Yes


Question: Given the sequence $a_{1} = 2020$, $a_{2} = 2021$, and $a_{n+1} = a_{n} + \frac{1}{n^2} a_{n-1}$ for $n \geq 2$, determine which of the following statements about the series $\sum_{n=1}^{\infty} \frac{1}{a_{n}}$ is true: \(\text{A) The series converges to } \frac{1}{2020}.\) \(\text{B) The series is divergent.}\) \(\text{C) The series converges to } \frac{1}{2021}.\) \(\text{D) None of the above.}\)

Answer: B


Question: Evaluate the integral \( \int_{-\pi}^\pi s(x) \, dx \), where the function \( s(x) = \sum_{k=1}^\infty a_k \cos(kx) \) is defined on \( \mathbb{R} \) and the series \( \sum_{k=1}^\infty a_k \) converges absolutely.

Answer: 0


Question: Determine the range of values of \(\beta\) for which the integral \(\int_{e}^{4} \frac{dt}{\ln^\beta(\ln^2(t))}\) converges.

Answer: (-\infty, 1)


Question:
Let $f:X \rightarrow \mathbb{R}$ be a Lipschitz function on a metric space $X$, and let $K < M$ be constants. Define the function $g:X \rightarrow \mathbb{R}$ by:
\[ g(x) = \begin{cases} f(x), & \text{if } K \leq f(x) \leq M, \\ K, & \text{if } f(x) < K, \\ M, & \text{if } f(x) > M. \end{cases} \]
Is the function $g$ Lipschitz?

Answer: Yes


Question: In a regular pentagon $ABCDE$ inscribed in a circle, a point $P$ lies on arc $AB$. Given that $AP + BP = 8$ units, $PD = 12$ units, and $PE = 11$ units, find the length of $PC$. Note: The original problem might have inconsistencies as the given data does not lead to the specified answer of $9$ units for $PC$.

Answer: 9


Question: Is the $p$-adic field $\mathbb{Q}_p$ complete with respect to the $p$-adic norm $|\cdot|_p$?

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty}^{0} \frac{dx}{1+x^2} \) using the residue theorem and a semicircular contour.

Answer: \dfrac{\pi}{2}


Question: Is it possible to construct a closed cover of a non-open set $B \subseteq \mathbb{R}$ that does not have a finite subcover?

Answer: Yes


Question: Consider a triangular array where each element is defined as follows: Let $A_n$ be an independent and identically distributed (iid) sequence of Bernoulli random variables with parameter $\pi$, and $Y_n$ be an iid sequence of normal random variables with mean $\mu$ and variance $\sigma^2$. Define the sequence $\pi_n = \frac{A_1 + A_2 + \ldots + A_n}{n}$. The array consists of terms $\frac{A_iY_i}{\pi_j}$, where $i \leq j$ and $i, j \in [1, n]$. Let $Z_n$ be the average of the $n$th row of this array. Determine if the sequence $Z_n$ converges almost surely, in probability, or in distribution, and to what value.

Answer: \mu


Question:
Determine the ratio of the area of the set \(S_{2}\) to the area of the set \(S_{1}\), where:

\[S_{1} = \{(x, y)| \log_{10}(1 + x^2 + y^2) \leq 1 + \log_{10}(x + y)\}\]
\[S_{2} = \{(x, y)| \log_{10}(2 + x^2 + y^2) \leq 2 + \log_{10}(x + y)\}\]

Answer: 102


Question: Determine the range of values for the parameter \(a\) such that the cubic equation \(2x^3 - 3x^2 - 12x + a = 0\) has all roots real and distinct.

Answer: (-7, 20)


Question:
Let $a, b, c \geq 0$ with $a^2 + b^2 + c^2 = 1.$ Find the minimum value of the expression:
\[ P = \frac{(a + b)^2}{b^2 + 2c^2 + 1} + \frac{(b + c)^2}{c^2 + 2a^2 + 1} + \frac{(c + a)^2}{a^2 + 2b^2 + 1}. \]

Answer: \dfrac{3}{2}


Question: Find the maximum value of the function \( f(t,y) = 1 + \frac{|y|}{t} \) on the rectangle defined by \( |t+1| \le a \) and \( |y-1| \le b \), where \( 0 < a < 1 < b < \infty \).

Answer: 1


Question: Determine the cardinality of the $\sigma$-algebra generated by the sets $X_i := \{(x_1, x_2, ..., x_n): x_i < \frac{1}{2}\}$ for $i = 1, 2, ..., n$ in the space $X = [0, 1]^n$. Is the cardinality of this $\sigma$-algebra $2^n$?

Answer: 2^{2^n}


Question: Determine whether the sequence of random variables \(Y_n = \frac{1}{\sqrt{n}}I_{(n,2n)}(U)\), where \(U\sim \mathrm{Uniform}(0,1)\), converges in \(L^1(\mathbb{P})\) as \(n \to \infty\).

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to0} \frac{\left(1+\sin{x}\right)^{\frac{1}{x}}\left(\frac{e^{\frac{1}{x}\log\left(1+\frac{\sin^2{x}}{1+\sin{x}}\right)}-1}{x}\right)}{1+\sin{x}}$$

Answer: e


Question: Calculate the integral \( \int (10x^4 - 2xy^3) \, dx - 3x^2y^2 \, dy \) along the path \( x^4 - 6xy^3 = 4y^2 \) from the point \( O(0,0) \) to the point \( A(2,1) \).

Answer: 60


Question: Is orientability a topological invariant for surfaces, meaning that an orientable surface cannot be topologically equivalent to a nonorientable one? Provide a yes or no answer.

Answer: Yes


Question: Given a real-valued matrix $A$ such that $AA^{\top} = I$ and $\det(A)<0$, find the value of $\det(A + I)$. \( \boxed{?} \)

Answer: 0


Question: Find a monic integer polynomial of degree $n$ whose Galois group over $\mathbb{Q}$ is $S_n$. Provide the polynomial.

Answer: x^n - x - 1


Question: Consider the lattice $L$ in the complex plane $\mathbb{C}$ defined by $L = \mathbb{Z}w_{1} + \mathbb{Z}w_{2} = \{a_{1}w_{1} + a_{2}w_{2} : a_{1}, a_{2} \in \mathbb{Z}\}$, where $w_{1}$ and $w_{2}$ are linearly independent over $\mathbb{R}$. Determine whether $L$ is uniformly discrete in $\mathbb{C}$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether connected smooth orientable 2-manifolds can have torsion elements in their fundamental group.

Answer: No


Question: Determine the number of permutations of the numbers $1, 2, \ldots, N$ such that each number is either greater than all the numbers to its left or smaller than all the numbers to its left.

Answer: 2^{N-1}


Question: Find the Galois group of the splitting field of the polynomial \( f(x) = x^6 - 1 \) over the finite field \( \mathbb{F}_5 \).

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


Question: In the $3$ hat experiment, let $X$ be the number of people who get their own hat back. If there are $n$ people in the experiment, determine the variance of $X$. Provide your answer in terms of $n$.

Answer: 1


Question: Suppose we have $n$ real numbers $a_1,\ldots,a_n$ where $|a_i| \geq 1$ for all $i$. If we randomly choose $2^n$ vectors $v_1,\ldots,v_{2^n}$ such that $v_i \in \{-1,1\}$, what is the probability that at least one of the vectors satisfies the condition $-1 < \sum_{i=1}^n v_i \cdot a_i < 1$?

Answer: 0


Question: Evaluate \( \lim_{n \to \infty} \int_{\mathbb{R}} \frac{1}{n} \, \mathrm{d}m \), where \( m \) is the Lebesgue measure.

Answer: \infty


Question: Let $A$ be an $n \times n$ matrix with non-negative integer entries. Suppose that the sum of the entries in each row is $d$, and the determinant of $A$ is also $d$. Let $C$ be an $n \times n$ matrix with the same first column as $A$, but with all other entries equal to $0$. Determine whether $\det (A+C)=2d$.

Answer: 2d


Question: Consider the function $f:\mathbb{R}^3\rightarrow \mathbb{R}$, defined by $f(x) = |x|^{-1}$. Its distributional Fourier transform is given by $F(f)(k) = g(k) = 4\pi/|k|^2$. If $\chi$ is the characteristic function of an open, bounded set containing $0$, and $h = F(\chi f)$, determine if $h = O(|k|^{-2})$ as $|k|\rightarrow +\infty$.

Answer: Yes


Question: Suppose we randomly generate a classical Hilbert propositional calculus formula $F$ with $n$ variables, using the following method: $F = x_i$ for each of $i \leq n$ with probability $\frac{1}{n+2}$. $F = \neg F_1$, where $F_1$ is generated independently using the same method. $F = F_1 \to F_2$, where $F_1$ and $F_2$ are generated independently using the same method. It follows from the extinction criterion for the Galton-Watson branching processes, that the process of generation will terminate with probability $1$ and thus our random formula is well defined. What is the probability that $F$ is a tautology?

Answer: 0


Question: Find the residue of the function \( f(z) = \frac{\exp(-z)}{z^2} \) at \( z = 0 \) given that \( |z| = 3 \).

Answer: -1


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{1-e^{\sin(x) \ln(\cos x)}}{x}$$ using the approximations $$\cos{x} \approx 1-\frac{x^2}{2}+o(x^2)$$ and $$\sin{x} \approx x+o(x).$$

Answer: 0


Question: Determine whether there exist subgroups of the set of non-zero complex numbers that are not cyclic.

Answer: Yes


Question: Show that for $Y'>Y>0$ and $N$ being any positive integer, the following holds as $Y$ goes to infinity: $$\int_{Y^2}^{{Y'}^2} z^{-1/2} e^{2\pi i N z}dz = O(Y^{-1}).$$

Answer: O\left(Y^{-1}\right)


Question: Evaluate the integral \( \int_0^{2\pi} \cos(\cos(z))\cosh(\sin(z)) \, dz \) using a corollary of Cauchy's Integral Formula.

Answer: 2\pi


Question:
Let $k$ be a Noetherian ring and $A$ a smooth affine $k$-algebra of dimension $d$. Determine whether the Kahler differentials $
abla^{d+1}_{A|k}$ vanish, i.e., if $
abla^{d+1}_{A|k} \cong 0$.

Answer: \nabla^{d+1}_{A|k} \cong 0


Question: Find the parent function \( f(x, y) \) given the gradient vector field \( \nabla f = \langle 4y\sin(xy), 4x\sin(xy) \rangle \).

Answer: -4\cos(xy)


Question: Determine whether an infinite product of copies of $\mathbb{Z}$ is isomorphic to a direct sum of copies of $\mathbb{Z}$. Provide a true or false answer.

Answer: B


Question: Find the inverse of the polynomial \( g(x) = x^2 + 2x + 1 \) in the field \( \mathbb{Z}_5[x]/\langle x^3+x^2+2\rangle \).

Answer: 4x^2 + 2x + 3


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

Answer: e


Question: Evaluate the limit: $$\lim_{x\to\infty}\frac{(x^{\sqrt 2}+1)^{\sqrt 2}}{x^2+1}$$ without using L'Hopital's rule multiple times.

Answer: 1


Question: Find the limit \( \lim_{{n\to\infty}}n\left(\sqrt[n]{e}-1\right) \) without using Heine's theorem or L'Hôpital's rule.

Answer: 1


Question: Calculate the sum of the infinite series: \[ \sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\sum_{c=0}^{\infty}\frac{1}{(a+b+c)!} \]

Answer: \dfrac{7e}{2}


Question: Determine the asymptotic behavior of the sum \( \sum_{k = 1}^{\frac{n}{2} - \sqrt{n}} n \binom{n - 1}{k - 1} \) as \( n \to \infty \).

Answer: \Theta(n 2^n)


Question: Let \( \mathcal{g} \) be a nilpotent Lie algebra and \( \rho \) be its representation in a vector space \( V \). If all the linear transformations \( \rho(X) \) are nilpotent for \( X \in \mathcal{g} \), determine whether there exists a non-zero vector \( v \in V \) such that \( \rho(X)v = 0 \) for all \( X \in \mathcal{g} \).

Answer: Yes


Question: Is the function \( g: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) defined by \( g(x,y)=(2ye^{2x},xe^y) \) invertible in a neighborhood of the point \((1,0)\)?

Answer: Yes


Question: Derive the expected value of the estimator $\hat{\sigma}^2$ for the multiple linear regression model using the formula $\mathbb{E}(u^T A u) = \mathrm{tr}(AV) + \mu^T A \mu$ and the fact that $\hat{\epsilon}=(I_{n\times n}-P)Y$, where $P$ is the projection matrix.

Answer: \sigma^2


Question: Given the real line $\mathbb R$ with the Lebesgue measure, if $1 \leq p < q < \infty$ and $r \in [p, q]$, it is known that $L^p(\mathbb R) \cap L^q(\mathbb R) \subseteq L^r(\mathbb R)$. Determine if the converse statement holds: If $p, q, r \in [1, \infty)$ with $p < q$ and $L^p(\mathbb R) \cap L^q(\mathbb R) \subseteq L^r(\mathbb R)$, is it necessarily true that $r \in [p, q]$?

Answer: Yes


Question:
Given a function $f(x)$, consider the following limits as $x$ approaches infinity:
\[\lim_{x\to \infty} f(x) - k\ln(x) \leq 0 \quad \text{for all} \quad k \in (0,1)\]
and
\[\lim_{x\to \infty} f(x) - \ln(x) \geq 0.\]
Does this imply that
\[\lim_{x\to \infty} f(x) - \ln(x) = 0?\]
Provide a justification for your answer.

Answer: Yes


Question: Determine the smallest integer \(\alpha\) such that the error of the truncated Taylor series of \(\sqrt{e}\) at \(x=0\) with \(\alpha\) terms is less than 0.001.

Answer: 5


Question: Let \(G = \mathbb{Z} \times \mathbb{Z}\) and \(H = \{(a,b) \in \mathbb{Z} \times \mathbb{Z} : 8 \mid a+b\}\). What is the index \([G:H]\)?

Answer: 8


Question: Determine the global maximum value of the function \( f(x) = \frac{J_2(x)}{x^2} \), where \( J_2(x) \) is a Bessel function of the first kind of order 2.

Answer: \dfrac{1}{8}