Question: What is the probability that any two distinct order statistics from a set of independent and identically distributed continuous random variables are equal?

Answer: 0


Question: Determine the number of zeros of the function \( f(z) = z^5 + iz^3 - 4z + i \) within the annular region \( \{1 < |z| < 2\} \).

Answer: 4


Question: For a given positive integer $k$, find the values of $\alpha$ for which the series \[ \sum_{a\in \mathbb{N}^n} \left( \sum_{i=1}^n a_i^k\right)^{-\alpha} \] converges.

Answer: \alpha > \dfrac{n}{k}


Question: Determine whether the inequality \( b \geq a \) holds for all positive integers \( n \), where \( 0 \leq p_i \leq 1 \) and \( t > 0 \). Here, \( a = \prod_{i=1}^n (p_i e^t + (1-p_i)) \) and \( b = (\bar{p} e^t + (1-\bar{p}))^n \), with \( \bar{p} = \frac{1}{n} \sum_{i=1}^n p_i \).

Answer: b \geq a


Question: Evaluate the double integral: $$ \int_0^\infty\int_0^\infty xye^{-(x^2+y^2)}\,dx\,dy $$

Answer: \dfrac{1}{4}


Question: Find all positive integer pairs \((x, y)\) such that \(3^x - 26 = y^5\).

Answer: (3, 1)


Question: Evaluate the limit: $$ \lim_{{(x,y)\to (0,0)}}\frac{\sin(x^2+9y^2+|x|+|3y|)}{|x|+|3y|} $$

Answer: 1


Question: Given i.i.d random variables with mean $\mu$ and variance 4, let $\bar{X}_n = \frac{1}{n}(X_1+\cdots+X_n)$. Use the Central Limit Theorem to approximate the probability $P(|\bar{X}_n - \mu| \ge \frac{1}{\sqrt{n}})$.

Answer: 0.617


Question: Given a graph with n vertices labeled from 1 to n and m distinct pairs of these labels, determine the minimum number of vertices that can have an in-degree of 0 after optimally placing directed edges between the pairs. An optimal placement maximizes the number of vertices with an in-degree greater than 0.

Answer: \max(n - m, 0)


Question: Suppose $f_n$ is a sequence of bounded measurable functions on a probability space, and $f_n$ does not converge to $0$ almost everywhere. Determine whether there exists a set of positive measure $E$ such that $\limsup\limits_{n\to\infty}|f_n(x)|>2\epsilon$ for all $x\in E$. Answer "yes" or "no".

Answer: yes


Question:
Given the system of equations for $x$ and $y$ in terms of $a, b, c \in \mathbb{R}$, where $a \neq 0$ and $b^2 - 4ac = 0$:

1. $ax^2 + (b+3)x + c = 3y$
2. $ay^2 + (b+3)y + c = 3x$

Determine the value of $\frac{x}{y}$.

Answer: 1


Question: Determine whether it is true that for any subgroup $H$ of $S_n$ that acts transitively on $(1,2,...,n)$, any non-trivial normal subgroup $M$ of $H$ must also act transitively on the same set, given that $H$ is not primitive.

Answer: False


Question: Consider a monic polynomial $f(x)$ of degree $5$, where the graphs of $f(|x|)$ and $|f(x)|$ are identical. If $4$ is a root of $f(x)$, find the value of $f(1)$. Provide reasoning based on the graph and properties of odd and even functions.

Answer: 225


Question: Evaluate the limit: $$\lim_{x\to\left(\frac{\pi}{2}\right)}\left(1+\cos x\right)^{\tan x}.$$

Answer: e


Question: Determine the radius of convergence of the Taylor series expansion of the function \( f(x) = \frac{1}{1+x^2} \) about a point \( a \in \mathbb{R} \).

Answer: \sqrt{a^2 + 1}


Question: Can a CW complex have a 2-dimensional cell without any 1-dimensional cells, such that the 1-skeleton is equal to the 0-skeleton?

Answer: Yes


Question: Find all sets of three positive integers \((a, b, c)\) such that the sum of the factorials of two of the integers equals the factorial of the third integer.

Answer: (1, 1, 2)


Question: Determine the integer value of $n$ for which the expression $\frac{500^n}{n!}$ reaches its maximum value.

Answer: 500


Question: In the expansion of \((x^3 + x^2 + x)^{36}\), how many coefficients of the resulting polynomial are not divisible by 2?

Answer: 9


Question: Evaluate the limit \( \lim_{n \rightarrow \infty} \int_{0}^{\frac{\pi}{2}} \frac{\sin^n x}{\sqrt{x}} \, dx \) using Lebesgue's Dominated Convergence Theorem.

Answer: 0


Question: Given the functional equation $2 + f(x)f(y) = f(x) + f(y) + f(xy)$ and the condition $f(2) = 5$, find the value of $f(5)$. Provide your answer as a single number.

Answer: 26


Question: Let $f_n(x) = \sin(2^n x)$ for $n \in \mathbb{N}$. Show that the distance function \( d(f_n, f_m) = \sup_{x \in [0, 2\pi]} \{|f_n(x) - f_m(x)|\} \) satisfies \( d(f_n, f_m) \geq 1 \) when \( n \neq m \).

Answer: d(f_n, f_m) \geq 1


Question: Given that $x, f(x), a, b$ are positive integers and $f$ is a strictly increasing function such that $f(f(x)) = x^2 + 2$, determine the value of $f(3)$.

Answer: 6


Question:
Determine whether the following limit exists:
\[\lim_{(x,y)\rightarrow(0,0)} \frac{7 \sin(2 x) x^2 y}{2 x^3 + 2 x y^2}\]

Answer: 0


Question: Let \(G\) be a positive-definite, Hermitian \(2 \times 2\) matrix, and \(W\) be an invertible \(2 \times 2\) matrix such that \(WGW^\dagger = G\). Does this imply that \(W\) is unitary with respect to the standard scalar product?

Answer: No


Question: Determine if there exists a continuous surjective map \( g : T \rightarrow T \), where \( T = \mathbb{S}^1 \times \mathbb{S}^1 \) is a torus, such that the induced homomorphism \( g^* : H_1(T,x) \rightarrow H_1(T,x) \) multiplies every element by 2.

Answer: Yes


Question:
Let $X_1, \ldots, X_n$ be independent, identically distributed, and nonnegative random variables, and let $k \leq n$. Compute the expected value:

$$E\left[\frac{\sum_{i=1}^k X_i}{\sum_{i=1}^n X_i}\right].$$

Answer: \dfrac{k}{n}


Question: Let \( f \) be a real-valued continuous function defined on the interval \( [0,3] \) such that \( f(x) \) assumes only rational values. If \( f(1) = 1 \), what is the value of \( f(2) \)? Choose from the following options: \(1)\ 2, \(2)\ 4, \(3)\ 8, \(4)\ \text{None of the above}.

Answer: D


Question: True or false: If E and F are dense uncountable sets in the real numbers \(\mathbb{R}\), then their intersection \(E \cap F\) is also dense in \(\mathbb{R}\).

Answer: B


Question: Determine the value of \(\alpha\) such that the operator \(F\) defined by \(Fu(x) = y_0 + \int_{x_0}^x f(s, u(s)) \, ds\) becomes a contraction mapping on the space \(C^0(I)\) with respect to the norm \(||u||_\alpha = \max_{x \in I} |u(x)e^{-\alpha x}|\). Explain how this value of \(\alpha\) is derived.

Answer: \alpha > L


Question: Determine whether the function \( f(x,y) = \frac{x^2y^2}{x^2+xy+y^2} \) with \( f(0,0) = 0 \) is continuous at the point \((0,0)\).

Answer: Yes


Question: Determine whether $\frac{1}{4}$ is an element of the Cantor set.

Answer: Yes


Question: Simplify the expression: $$\sqrt{\dfrac{\sqrt[4]{x^3}-8}{\sqrt[4]{x}-2}+2\sqrt[4]{x}}\left(\dfrac{\sqrt[4]{x^3}+8}{\sqrt[4]{x}+2}-\sqrt{x}\right)$$

Answer: 8 - 2\sqrt{x}


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

Answer: 8


Question: Find the function $f(x)$ that satisfies the equation: \[ f(x) + xf(-x) = x + 1. \]

Answer: 1


Question: Determine the number of divisors of the number $2^{2} \cdot 3^{3} \cdot 5^{3} \cdot 7^{5}$ that are of the form $4n+1$, where $n$ is a natural number.

Answer: 48


Question: Let $R$ be a Noetherian ring and $M$ a finitely generated $R$-module. If $P$ is minimal over the annihilator of $M/M'$, where $M' = \ker (M \to M_P)$, is $M'$ necessarily $P$-primary? Provide a justification for your answer.

Answer: Yes


Question: Let $(X,f)$ be an expansive map with constant $c(f)$. Suppose there exists an infinite set $\{n_k\}$ such that $d(f^{n_k}(x), f^{n_k}(y)) < c(f)$ for all $k$. If $x$ and $y$ are not periodic points, does it necessarily follow that $x = y$?

Answer: No


Question: Let $A \subset \mathbb{R}$ be a Lebesgue measurable set such that $0 < \lambda(A) < \infty$. Consider the function $f: \mathbb{R} \to [0, \infty)$ defined by $f(x) = \lambda(A \cap (-\infty, x))$. Compute $\lim\limits_{x \to \infty} f(x)$. \( \boxed{} \)

Answer: \lambda(A)


Question: Is it possible for a $2 \times 2$ matrix $A$ to have real eigenvalues such that $A - rI$ is invertible for all real numbers $r$?

Answer: No


Question: Let $A$ and $B$ be two $n \times n$ matrices with real entries. Suppose that $A$ and $B$ have the same first $k$ rows and the same first $k$ columns, where $k < n$. If $\det A$ and $\det B$ have the same sign, does it follow that $A$ and $B$ have the same signature? Answer yes or no and justify your answer.

Answer: No


Question: Evaluate the limit: \( \lim_{x\to 0^+} \left\lfloor \dfrac{x^2-2x}{\ln (1-x)}\right\rfloor \)

Answer: 1


Question: Find the intersection point of the functions $x^{0.999}$ and $\frac{x}{\log(x)}$ for $0 < p < 1$. Assume $p = 0.999$. Determine the value of $x$ where these two functions are equal.

Answer: 2.721


Question: Find the number of elements in the center of the group of all $3 \times 3$ orthogonal matrices over $\mathbb{R}$. Provide your answer as a single integer.

Answer: 2


Question: Let $a$ and $b$ be two positive integers that satisfy the equation $a^2 + 84a + 2008 = b^2$. Determine the value of $a + b$.

Answer: 80


Question: Determine the number of elements \( \alpha \) in the finite field \( \mathbb{F}_{2^4} \) such that the field extension \( \mathbb{F}_2(\alpha) \) is equal to \( \mathbb{F}_{2^4} \).

Answer: 12


Question: Determine the values of $n$ for which the sequence $0, 1, \cdots, n-1$ can be the degree sequence of a graph.

Answer: 1


Question: Determine whether there exist integer functions $f(n)$ and $g(n)$ such that \( \lim_{n \to \infty} f(n)\zeta(2) + g(n) = 0 \) and \( f(n)\zeta(2) + g(n) \neq 0 \) for all positive integers $n$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of Sylow 5-subgroups in the alternating group $A_5$.

Answer: 6


Question: Evaluate the limit \( \lim_{x\to 0}\frac{f(2+\sin x)-f(2+x\cos x)}{x-\sin x} \) given that the function \( f(x) \) is differentiable and its derivative \( f'(x) \) is continuous on \( \left(-\infty,\infty \right) \) with \( f'(2)=14 \).

Answer: 28


Question: Given a monic polynomial with non-zero integer coefficients, \(P(x) = x^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0\), and Vieta's Formulas, which state that the constant term \(a_0 = (-1)^nr_1r_2\dots r_n\) represents the product of the roots, where repetition is allowed and the roots are denoted by \(r_i\) (\(i = 1, 2, \dots, n\)). If \(P(x)\) has a rational root \(r_1 \in \mathbb{Z}\) and \(\text{GCD}(r_1, r_2r_3\dots r_n) = 1\), does it follow that for any other rational root \(r_2 \in \mathbb{Z}\), we have \(\text{GCD}(r_1, r_2) = 1\)? Answer with 'yes' or 'no'.

Answer: yes


Question: Find the determinant of a weighted orthogonal projection matrix based on the weighted pseudo-inverse.

Answer: 0


Question: Find the condition that must be satisfied by the positive real numbers \(a, b, c\) such that the iterative process of constructing a new tuple \((S - 2a, S - 2b, S - 2c)\), where \(S = a + b + c\), results in a sequence of tuples with all positive elements.

Answer: a = b = c


Question: Calculate the value of \(-\int_{\partial\Omega}\langle b,v\rangle d\mu_{\partial\Omega}\) for the vector field \(b(x,y,z)=(0,0,qy)\) with \(q>0\), where \(\Omega \subset \mathbb{R}^3\) is an open bounded set with a smooth boundary \(\partial \Omega\) and the unit normal \(v\).

Answer: 0


Question: Determine if there are any other solutions to the differential equation \( y = 2\sqrt{x} - \frac{1}{2} \frac{(y')^2}{y''} \) besides \( y = 4\sqrt{x} \).

Answer: y = 4\sqrt{x}


Question: Determine whether there exist unique factorization domains (UFDs) that have only finitely many prime elements up to associativity, but are not fields.

Answer: Yes


Question: Determine whether the closure of the set \( N = \{ f(x) \in C[0,1] \mid f(1) = 0 \} \) under the usual norm of \( L^2[0,1] \) is equal to \( L^2[0,1] \).

Answer: L^2[0,1]


Question: Calculate the sum of the interior angles of a star formed by connecting every other point on a convex curve labeled $1, 2, 3, \ldots, n$ in a clockwise direction, where $n$ is an odd number.

Answer: 180(n-4)


Question: Given a positive integer \( n \), let \( \mathrm{rad}\; n \) be the product of \( n \)'s distinct prime factors and \( \mathrm{spf}\; n \) be the sum of \( n \)'s distinct prime factors. For any \( k \ge 1 \), let \( a_{n,k} \) be defined as \[ a_{n,k} = \begin{cases} 1 & \frac{\mathrm{rad}(n)}{\mathrm{rad}(n+k)} > 1 \\ 0 &\text{ otherwise} \end{cases} \] Find the limiting value of \[ \lim_{x \to \infty}\dfrac{1}{x}\sum_{n \le x}a_{n,k} \]

Answer: \dfrac{1}{2}


Question: Determine the limiting behavior of the sequence \( \left\langle \frac{f(n)}{n^2} \right\rangle \), where \( f(n) \) is the number of ordered pairs \((a, b)\) such that the least common multiple (lcm) of \(a\) and \(b\) is \(n\), with \(a, b \in \{1, 2, \ldots, n\}\).

Answer: 0


Question: Calculate the expected value of $B^4(t)$, where $B_t$ is a Brownian motion with $B_t \sim N(0, t)$. Express your answer in terms of $t$.

Answer: 3t^2


Question: Determine whether the statement is true or false: If $G/C(G)$ is cyclic, then $G$ is abelian.

Answer: A


Question:
Evaluate the limit:
\[
\lim_{n \to \infty} \frac{7^n + 23^n - 3(11)^n + 5^n}{n^2 + 3n^7 + 6n^{13} + 1}
\]

Answer: \infty


Question: Calculate the limit: $$ \lim _{n\rightarrow \infty} \left[ \left( (4^n+3)^{\frac{1}{n}} - (3^n+4)^{\frac{1}{n}} \right)^{n \cdot 3^n} \right] $$

Answer: e^{-12}


Question: Find the limit of the function \[ f(x) = \left\{1 + 6\left(\frac{\sin x}{x^{2}}\right)^{x}\frac{\log(1 + 10^{x})}{x}\right\} \] as \(x \to 0^{+}\).

Answer: \infty


Question: Determine if the sequence defined by the iterative method \(x_{n+1} = \frac{x_n(x_n^2 + 3a)}{3x_n^2 + a}\) converges to \(\sqrt{a}\) as \(n\) approaches infinity.

Answer: \sqrt{a}


Question: Is a random variable $X$ independent from the $\sigma$-algebra generated by a collection of $\sigma$-algebras $F_t$, if $X$ is independent from each $F_t$ individually?

Answer: No


Question: Find the radius of convergence for the infinite series \( \sum_{n=0}^{\infty} (n+2^n)z^n \) using the formula \( R = \lim_{n \rightarrow \infty} \left| \frac{a_n}{a_{n+1}} \right| \), where \( a_n = n+2^n \).

Answer: \dfrac{1}{2}


Question: Let \( f:\mathbb{R}\to \mathbb{R} \) be a continuous and convex function such that \( f(0) \leq 0 \). Define the set \( \mathcal{D} = \{x \mid f(x) \leq 0\} \cap \{x \mid x \geq 0\} \). If \( \mathcal{D} \) is bounded, determine the set \( \mathcal{D} \).

Answer: [0, M]


Question: Consider a sequence of complex numbers lying in a sector with an angular width less than \( \pi \). If the sum of the real parts of the sequence converges for any two non-collinear directions in the sector, does the sum of the moduli of the sequence also converge? Answer 'yes' or 'no'.

Answer: yes


Question: Determine the number of distinct solutions to the congruence $x^{50} \equiv 1 \pmod{181}$. Use the fact that $\varphi(181) = 180$ and that $181$ is a prime number to assist in your solution.

Answer: 10


Question: Evaluate the integral \( \int_{|z|=2} ze^{\frac{3}{z}} \, dz \) using the residue theorem.

Answer: 9\pi i


Question: Determine the probability density function for the y-coordinates of a random point uniformly sampled on the triangle with vertices A = (1,0,0), B = (0,1,0), and C = (0,0,1).

Answer: 2(1 - y)


Question: Given a problem where the sets $A_i$ are not sorted, determine if the problem can still be solved in polynomial time.

Answer: Yes


Question: Determine if a concave polygon can be divided into two convex polygons of equal area by a straight line moving parallel to a given direction. Answer "Yes" or "No."

Answer: No


Question: Calculate the limit: $$\lim_{n\rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n}$$

Answer: 1


Question: Determine the number of critical points of the function \( g(x) = 2\cos\left(\frac{x}{2} \left(\sqrt{2} +1\right)\right)\cos\left(\frac{x}{2} \left(1-\sqrt{2} \right)\right) \).

Answer: \infty


Question: Evaluate the contour integral \( \oint_{ |z| = 1} \frac{ 30z^{2}-23z+5 }{ (2z-1)^{2}(3z-1) } \ dz \) using the residue theorem and partial fraction decomposition.

Answer: 5\pi i


Question: Calculate the surface integral \(\iint_S(\nabla \times \vec f) \cdot \vec n\,\mathrm dS\) where \(\vec f(\vec r) = (y,-x, zxy)\) and \(S\) is the surface defined by \(x^2+y^2+3z^2=1\) for \(z\leq0\), with the unit normal vector \(\vec n\) pointing in the positive \(z\)-direction. Use Stokes' theorem if applicable.

Answer: -2\pi


Question: Does there exist an infinite set of sets \( A \) such that for every positive integer \( n \), there are exactly \( n \) pairwise disjoint sets in \( A \), but no infinite sequence of pairwise disjoint sets in \( A \)?

Answer: Yes


Question:
Evaluate $b+c+d$ given the integral:
$$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\left(\frac{\pi}{2}-x\right)\cos x}{3+\sin^2 x}\mathrm{d}x= \frac{\pi^a}{b\sqrt c+d}$$
where $a$, $c$, and $d$ are positive integers.

Answer: 9


Question: Express $E(YX)$ as a function solely of the first two moments of the random variable $X$, given that $E(Y|X) = aX + b$, where $a$ and $b$ are constants.

Answer: aE(X^2) + bE(X)


Question: Determine the values of $n$ for which the expression \( \frac{(n^2)!}{(n!)^{n+1}} \cdot \frac{n!}{n^2} \) is an integer, given that \( \frac{n!}{n^2} \) is not an integer.

Answer: \emptyset


Question: How many ordered pairs of permutations \((\pi, \sigma)\) in the symmetric group \(S_n\) satisfy \(\pi \circ \sigma = \sigma \circ \pi\)?

Answer: n! \cdot p(n)


Question: Given the polynomial $x^4 - 8x^3 + 24x^2 - 32x + 15$, let its roots be $a, b, c, d$. Calculate the value of $(a^2 + b^2)(a^2 + c^2)(a^2 + d^2)$. Express your answer as a single number.

Answer: 320


Question: Evaluate the triple integral \( \iiint y^2z \,dx \,dy \,dz \) over the ellipsoid defined by \( \frac{x^2}{4}+\frac{y^2}{1}+\frac{z^2}{9}=1.\)

Answer: 0


Question: Suppose that $X$ and $Y$ are jointly continuous random variables such that $X\sim U(0,1)$ and $f_{Y\mid X}(y\mid x) = 1$ where $ x < y < x+ 1$. Find the covariance $\operatorname{Cov}(X,Y)$. \( \boxed{?} \)

Answer: \dfrac{1}{12}


Question: Determine whether the following statement is true or false: Every closed subspace of a Banach space is also a Banach space.

Answer: A


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{(2n)!}{4^n(n!)^2} \]

Answer: 0


Question: Find the 6-digit number such that when it is multiplied by 2, 3, 4, 5, and 6, the resulting numbers are permutations of the original number's digits.

Answer: 142857


Question: Find the value of $a \in \mathbb{R}$ that minimizes the maximum value of $|x^2 - ax|$ for $x \in [0, 1]$. What is this value of $a$?

Answer: 2\sqrt{2} - 2


Question: Determine whether the function \( f(x)=\cosh^2(\cosh (x)) - \sinh^2 (\sinh (x)) \) has any local minima other than at \( x=0 \).

Answer: No


Question: Let \( f:\mathbb{R}\to\mathbb{R} \) be a continuous non-negative function such that \( \int_{-\infty}^{+\infty} f(x)dx = 1 \). Define \[ I_n(r) = {\idotsint}_{x_1^2+\dots x_n^2 \le r^2} \prod_{k = 1}^{n} f(x_k)dx_1\cdots dx_n \] Find the limit: \( \lim_{n\to\infty} I_n(r) \) for a fixed \( r \).

Answer: 0


Question: In a shop, two employees are each serving a customer. The service time for each customer is an independent random variable following an exponential distribution with parameter \( \lambda \). After finishing with their current customer, each employee will serve you. What is the probability that you complete your transaction and leave the shop before both of the initial customers finish their transactions?

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(1+\dfrac{1}{n+1}\right)\left(1+\dfrac{1}{2(n+1)}\right)\left(1+\dfrac{1}{4(n+1)}\right)\ldots\left(1+\dfrac{1}{2^{n-1}(n+1)}\right). \]

Answer: 1


Question: Determine whether the statement \( \frac{f(x)}{g(x)} = \frac{a+bx}{c+dx} + O(x^2) \) is true, given that \( f(x) = a + bx + O(x^2) \) and \( g(x) = c + dx + O(x^2) \).

Answer: True


Question: Consider a ring homomorphism \( \phi: K[x] \rightarrow L \), where \( K \) is a field, \( L \) is an extension of \( K \), and \( \alpha \in L \). If \( \phi(a) = a \) for all \( a \in K \) and \( \phi(x) = \alpha \), determine if it is possible for \( \alpha \) to be an element of \( K \).

Answer: Yes


Question: Find the infimum of the integral \( \int_{\mathbb{R}^5} u^2 - \ln(1+u^2) \, \text{dx} \) subject to the constraint \( \int_{\mathbb{R}^5} u^{14} = 1 \).

Answer: 0


Question: The coordinates of the vertices of a triangle \(\triangle ABC\) are \(A(6,0)\), \(B(0,8)\), and \(C(x,y)\) such that \(x^2-16x+y^2-12y+91=0\). Find the largest possible value of the area of the triangle \(\triangle ABC\).

Answer: 41


Question: Evaluate the integral \( \iiint_V xe^{-(x^2+y^2+z^2)^2}\,\mathrm{d} V \) using spherical coordinates, where \( V \) is the solid that lies between \( y=0 \) and the hemisphere \( x^2+y^2+z^2=4 \) in the right half-space \( (y>0) \).

Answer: 0


Question: Find the integer $a$ in the range $0 \leq a < 113$ such that $a^{37} \equiv 99 \pmod{113}$, given that $102^{70} + 1 \equiv 99 \pmod{113}$. Determine the value of $a$. \( \boxed{\ } \)

Answer: 60


Question: Let $X$ be a compact space and $Y$ be a metric space. Consider continuous mappings $f: X \to Y$ and $g: X \to Y$. Is the set $A = \{(f(x), g(x)) \mid x \in X\}$ necessarily compact in $Y \times Y$?

Answer: Yes


Question: Given that $\vec{u}$, $\vec{v}$, and $\vec{w}$ are vectors satisfying $\vec{v}\times\vec{w}=\vec{u}+\vec{v}\times\vec{u}$ and $\vec{u}$, $\vec{v}$ are unit vectors, determine the value of $\vec{u}\cdot(\vec{v}\times\vec{w})$. \(\boxed{?}\)

Answer: 1


Question: Suppose \(H\) and \(H^*\) are two non-isomorphic graphs with the same number of vertices. Let \(G\) and \(G^*\) be constructed from \(H\) and \(H^*\), respectively. Is it possible for the spectral radii \(\rho(G)\) and \(\rho(G^*)\) to be equal?

Answer: Yes


Question: Determine whether the limit of the sequence \( u_n = A^n u_0 \) is the zero vector as \( n \to \infty \), where \( A = \begin{pmatrix} \frac{1}{2} & 0 & 0 \\ 0 & 1 & -\frac{\sqrt{5}}{2} \\ 0 & \frac{\sqrt{5}}{2} & -1 \end{pmatrix} \) and \( u_0 \) is an arbitrary 3-dimensional vector. Given that all eigenvalues of \( A \) have an absolute value of \( \frac{1}{2} \), does this imply that the limit of \( u_n \) is the zero vector for any initial vector \( u_0 \)?

Answer: Yes


Question: If the expression \(z^5-32\) can be factorized into linear and quadratic factors over real coefficients as \((z^5-32)=(z-2)(z^2-pz+4)(z^2-qz+4)\), find the value of \(p^2+2p\).

Answer: 4


Question:
Find the vector \( \mathbf{s} = (s_1, s_2, s_3, s_4)^T \), where each \( s_i \in \{-1, 1\} \) and the number of 1's and -1's in \( \mathbf{s} \) is equal, that minimizes the maximum of the following correlation values:

\[ c_1 = \mathbf{s} \cdot (1, -1, 1, -1)^T \]
\[ c_2 = \mathbf{s} \cdot (-1, 1, -1, s_1) \]
\[ c_3 = \mathbf{s} \cdot (1, -1, s_1, s_2) \]
\[ c_4 = \mathbf{s} \cdot (-1, s_1, s_2, s_3) \]

Determine the vector \( \mathbf{s} \) that achieves this minimization.

Answer: (1, -1, -1, 1)


Question: Consider two sequences, $\{a_n\}$ and $\{b_n\}$, where $\lim_{n \to \infty} b_n = b$. If there exist constants $M > 0$ and $N \in \mathbb{N}$ such that $|a_n - a| \geq M \cdot |b_n - b|$ for all $n \geq N$, does it follow that $\lim_{n \to \infty} a_n = a$? Answer with 'Yes' or 'No'.

Answer: No


Question: Determine the set of real values of \(\alpha\) for which the improper integral \(\int_0^\infty x^\alpha \cos x\, dx\) converges.

Answer: (-1, 0)


Question: Evaluate the limit \( \lim_{x\rightarrow 0}\frac{\int_{\cos x}^{1}{}\arctan t ^2\ \mathrm dt}{\tan^2 x} \) using L'Hopital's rule.

Answer: \dfrac{\pi}{8}


Question:
Given a sequence of nonnegative real numbers $(p_n)$, consider a positive sequence $\\{x_n\\}$ satisfying the recurrence relation:

$$x_{n+1}=x_n+\frac{p_n}{x_1+x_2+\cdots+x_n}.$$

Let $F_n:=\sqrt{\frac{p_1}{1}+\frac{p_2}{2}+\cdots+\frac{p_{n-1}}{n-1}}$, and assume that:

1. $\lim_{n \to \infty} F_n = \infty$.
2. $\lim_{n \to \infty} \frac{nF_n}{F_1+F_2+\cdots+F_n}=c \in (1,\infty)$.

Find the limit:

$$\lim_{n \to \infty} \frac{x_n}{F_n}.$$

Answer: \sqrt{2c}


Question: Suppose that n observations are chosen at random from a continuous and uniform probability density function fY(y). What is the probability that the second to last observation recorded is the smallest number in the entire sample?

Answer: \dfrac{1}{n}


Question: Determine whether there exists a prime number $p$ such that $p \neq (n+1) \bmod (n+3)$ for all positive integers $n \leq 500000$. Provide a justification for your answer.

Answer: Yes


Question: Let $n$ be a positive integer and $s > 0$ be a real number. Determine whether there must exist positive real numbers $x_1, x_2, \dots, x_n < s$ such that no nontrivial linear combination of $x_1, x_2, \dots, x_n$ (with rational coefficients) is an integer.

Answer: Yes


Question:
Simplify the following limit expression into a computable form:

$$\lim_{n\to\infty}\left(\frac{n^4-3\cdot n^3-n^2+2\cdot n-1}{n^5+n^4-n^3-3\cdot n^2-3\cdot n+1}\right)^{\frac{6\cdot n^5-2\cdot n^4-2\cdot n^3+n^2-2\cdot n}{9\cdot n^4-2\cdot n^3+n^2+3\cdot n}}$$

Answer: 0


Question: Evaluate the series \( \sum_{n=1}^\infty\frac{n}{(n+1)!} \).

Answer: 1


Question: Find the value of $n$ between 1 and 1024 that minimizes the difference between $\pi^n$ and its nearest integer.

Answer: 3


Question: Evaluate the limit: $$\lim_{x \to 0}\frac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x^3}$$ You may use L'Hôpital's rule in your solution.

Answer: \dfrac{1}{4}


Question: Find the limit of \( \frac{\sqrt[n]{n!}}{n} \) as \( n \to \infty \).

Answer: \dfrac{1}{e}


Question: Determine whether a bijective function can be constructed between the set of prime numbers and the set of natural numbers.

Answer: Yes


Question: For a fixed natural number $n$, are there infinitely many pairs of prime numbers whose difference is a multiple of $n$? In other words, does the set of differences $\{n, 2n, 3n, 4n, \ldots\}$ contain infinitely many differences between prime numbers?

Answer: Yes


Question: Let $G$ be a finitely generated abelian group, and let $a$ be a nontrivial element belonging to all nontrivial subgroups of $G$. Is $G$ necessarily cyclic? Provide a justification for your answer.

Answer: Yes


Question: Calculate the expected length of runs in a sequence of 52 binary digits (0s and 1s) that contains an equal number of 0s and 1s.

Answer: \dfrac{52}{27}


Question: Consider a non-atomic measure space and a non-negative function in L^1. Is it possible to partition the space into a finite number of disjoint measurable subsets such that the integral of the function over each subset is equal?

Answer: Yes


Question: Find the Jacobian matrix of the expression $x^T A x$, where $A$ is a symmetric matrix in $\mathbb{R}^{n \times n}$ and $x$ is a vector in $\mathbb{R}^n$. Provide your answer in terms of $A$ and $x$.

Answer: 2x^T A


Question: Suppose $f : \mathbb{R}^n \to \mathbb{R}$ is a $C^\infty$ function with a non-degenerate critical point at $0 \in \mathbb{R}^n$. Given that $\triangledown f (0) = 0$ and the Hessian $\left((\partial^2 f/ \partial x_i \partial x_j ) (0) \right)$ is invertible, determine if the critical point at $0$ is an isolated critical point.

Answer: Yes


Question: Determine the number of distinct real solutions for the equation $f(f(x)) = 0$, where $f(x) = x^3 - 3x + 1$. Provide your answer as a single integer.

Answer: 7


Question: Determine whether there exists a real number $x$ such that its irrationality measure $\mu(x)$ equals $\lambda$ for any given $\lambda \in [2, +\infty)$. Provide a justification for your answer.

Answer: Yes


Question:
Calculate the limit:

$$\lim_{x\rightarrow 0}\frac{d}{dx}\left(\frac{F(x)-F(-x)}{x-(-x)}\right)$$

where $F(x)$ is an even function, differentiable, and $F'(x)$ is continuous at $x=0$. Provide your answer as a single value.

Answer: 0


Question: Let $X_n$ be a sequence of random variables defined as $X_n=\sum_{i=1}^{n}U_i$, where $U_i$ are independent random variables such that $P(U_i=1)=P(U_i=-1)=\frac{1}{2}$. For $m\geq n$, find the variance of $E(U_n|X_m)$. \( \text{Express your answer as a single number.} \)

Answer: \dfrac{1}{m}


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

Answer: x = k\pi


Question: Determine whether the statement is true or false: For an odd prime $p$, $x^2 \equiv 1 \pmod {p^k}$ if and only if $x \equiv \pm 1 \pmod {p^k}$. Provide a proof or counterexample.

Answer: True


Question: Let $Y \sim \text{Exp}(3)$. The random variable $X$ is defined such that, given $Y = y$, $X$ is uniformly distributed over the interval $[y, 3y]$. Compute the covariance of $X$ and $Y$.

Answer: \dfrac{2}{9}


Question: Calculate the sum of the series: $$\sum\limits_{a=0}^{\infty}\sum\limits_{b=0}^{\infty}\sum\limits_{c=0}^{\infty}\frac{1}{(a+b+c)!}$$

Answer: \dfrac{7e}{2}


Question: Evaluate the limit \( \lim_{n\rightarrow \infty} \int_1^{n^2} e^{-x^n} \, dx \).

Answer: 0


Question: Let \( f(x) = 2x^3 + 7x - 1 \), and let \( g(x) \) be the inverse of \( f(x) \). Find \( g'(191/4) \).

Answer: \dfrac{2}{89}


Question: If $U$ is a Lie group with a closed subgroup $K$ such that both $U$ and $U/K$ are simply connected, determine whether $K$ must be connected.

Answer: Yes


Question: Find all real numbers \(x\) such that \(x^{x^5} = 5\).

Answer: \sqrt[5]{5}


Question:
Evaluate the limit using the Taylor approximation for \((1+x)^p\):

\[\lim_{n\to\infty}n\left(1-\left(\frac{3n-1}{3n+1}\right)^p\right)\]

Answer: \dfrac{2p}{3}


Question: Solve the equation \(2f(x) + 3f^{-1}(x) = 10\), where \(f(x) = x^3 + x - 8\) for all \(x \in \mathbb{R}\).

Answer: 2


Question: Calculate the volume of the region bounded by the surfaces $z = y^2$, $x = 0$, $y = 0$, and $z = 9 - x$.

Answer: \dfrac{324}{5}


Question: Find the maximum value of \( \max_{-1\le x\le 1}\Big\vert\frac{1-(-x^2)^{n+1}}{1+x^2}\Big\vert \).

Answer: 1


Question: Find the value of $m$ that minimizes the expression $\binom{m}{k} + \binom{n-m}{k}$, given that $n$ and $k$ are fixed integers and $m = n/2$ if $n$ is even.

Answer: \left\lfloor \dfrac{n}{2} \right\rfloor


Question: Let $f(t)$ be a Riemann-integrable function with $\int_{-\infty}^{\infty} f(t) \, dt = 1$. Determine whether $\lim\limits_{x \to \infty} F(x) = 0$, where $F(x) = \int_{x}^{x^2} f(t) \, dt$. Justify your answer.

Answer: 0


Question: Verify that \( t = \frac{1}{2} \) is a root of the equation \( \tan 4\theta = \frac{4t-4t^3}{1-6t^2+t^4} = \frac{-24}{7} \).

Answer: t = \frac{1}{2}


Question: Evaluate the limit \( \lim_{x\to \infty} \frac{1}{x} \cdot \int_{x}^{2x}(1+\cos(t))^\frac{1}{t}dt \).

Answer: 1


Question: Let \(n\equiv1\pmod{4}\) be a squarefree number and \(p\equiv1\pmod{4n}\) be a prime number. Determine whether there exist natural numbers \(x\) and \(y\) such that \(p = x^2 + ny^2\).

Answer: Yes


Question: Determine whether the limit \( \lim_{n \to \infty} \cos\left(\pi\sqrt{n^4+n^2}\right) \) exists, where \( n \) is a positive integer.

Answer: 0


Question: Evaluate the limit \(\lim_{(x,y)\rightarrow\infty, y\geq x}\left(\frac{3y-1}{y-1}-\frac{y}{x}\right)\). Determine the possible limit values.

Answer: (-\infty, 2]


Question: Let $A$ be an $n \times n$ real matrix such that all its conjugates (matrices similar to $A$ over $\mathbb{R}$) have zero diagonal entries. Determine whether it is necessarily true that $A=0$. Provide a justification for your answer.

Answer: A = 0


Question:
If \( A \) is a \( 3 \times 3 \) matrix with \( \det(A) = 2 \), find the value of \( \det(\text{adj}(\text{adj}(\text{adj}(A^{-1})))) \). Choose the correct answer from the following options:

1. \( \dfrac{1}{512} \)
2. \( \dfrac{1}{1024} \)
3. \( \dfrac{1}{128} \)
4. \( \dfrac{1}{256} \)

Answer: D


Question: Determine whether the following equation is true: \[ \lambda_1\sum\limits_{j=1}^{\infty} f_1(E_j)+ \lambda_2\sum\limits_{j=1}^{\infty}f_2(E_j)+...+\lambda_n\sum\limits_{j=1}^{\infty}f_n(E_j)=\sum\limits_{j=1}^{\infty}\sum\limits_{i=1}^{n}\lambda_if_i(E_j) \]

Answer: True


Question: Suppose $g(x)$ satisfies the functional equation $g\left(\frac{x+8y}{9}\right)=\frac{g(x)+8g(y)}{9}$ for all real $x$ and $y$. Given that $g'(0) = 3$ and $g(0) = 1$, find $g(5)$. \( \boxed{} \)

Answer: 16


Question: Evaluate the limit \( \lim_{r\rightarrow \infty}r\cdot \bigg(\frac{2}{\pi}\bigg)^{r+1}\cdot f(r) \), where \( f(r) = \int^{\frac{\pi}{2}}_{0}x^r\cdot \sin x \, dx \). Use integration by parts to find \( f(r) \) and then determine the limit.

Answer: 1


Question: Determine the end limit of a probability density function defined on the set of non-negative integers, \(\mathbb{N}_0\).

Answer: 0


Question: Find the limit: \[ \lim_{x\to 2^+}\frac{\ln(x^2-3)}{\sinh(x-2)} \]

Answer: 4


Question: Determine the values of $a$ for which the infinite series \( \sum_{n=1}^{\infty} \frac{a^n (\ln n)^{1/n}}{n^{0.5}} \) converges.

Answer: [-1, 1)


Question: Let $V$ be a complex vector space, possibly infinite dimensional. Consider the completion $V[[t]] \cong V\hat{\otimes}_{\mathbb{C}} \mathbb{C}[[t]]$ with respect to the $(t)$-adic topology on $\mathbb{C}[[t]]$. Let $K := \mathbb{C}((t))$ denote the field of formal Laurent series. Determine the completion of the tensor product $V[[t]] \otimes_{\mathbb{C}[[t]]} K$ with respect to the $(t)$-adic topology on $V[[t]]$ and the topology on $K$ generated by the subspaces $t^i \mathbb{C}[[t]]$. Provide your answer as a completed vector space.

Answer: V((t))


Question:
Does Taylor's theorem with the Lagrange remainder hold when the roles of \( a \) and \( b \) are reversed? Specifically, if \( f \) is \( \mathcal{C}^n([a,b]) \) and \( n+1 \) times differentiable, can we find a \( c \in (a,b) \) such that:
\[ f(a) = f(b) + f'(b)(a-b) + \dots + \frac{f^{(n)}(b)}{n!}(a-b)^n + \frac{f^{(n+1)}(c)}{(n+1)!}(a-b)^{n+1} \]

Answer: Yes


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

Answer: \sqrt{51}


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

Answer: 1


Question: Determine the values of $k$ for which the function \( f(r) = \left(\frac{r}{1+r}\right)^{k} \) has an upper bound for \( r > 0 \).

Answer: k \geq 0


Question: Find the interval of convergence for the Maclaurin series of the function \( f(x) = \ln(1+x) \) using the formula for the remainder term in Taylor's Theorem.

Answer: (-1, 1]


Question: Evaluate the limit: \( \lim_{x \to 0^+} x \log\{\sin(x) \} \).

Answer: 0


Question: Evaluate the limit: $$\lim_{p\to\infty}\frac{\int_{0}^{p}x^{m+1}e^{x}dx}{\int_{0}^{p}x^{m}e^{x}dx},$$ where \(m\in \mathbb{N}\).

Answer: \infty


Question: Determine the number of binomial coefficients in the expansion of \((1+x)^{2018}\) that are divisible by 13.

Answer: 1395


Question:
Determine if the following equation is correct by expressing \( \pi \) in terms of \( \Delta \) and \( \Omega \), where \( \Omega=\frac{1}{(\pi-3)^{2}}\approx49.8790939 \):
\[ \pi=3+\cfrac{\log\left(\frac{1}{\Delta}\right)}{\log\left(\frac{1}{\Delta^{\sqrt{\Omega}}}\right)} \]

Answer: \pi = 3 + \cfrac{\log\left(\frac{1}{\Delta}\right)}{\log\left(\frac{1}{\Delta^{\sqrt{\Omega}}}\right)}


Question: Determine the number of terms required in the alternating series representation of the integral \( \int_{0}^{\frac{1}{2}} \frac{1}{1+x^6} \, dx \) to ensure the error is less than \( 10^{-5} \).

Answer: 2


Question: Consider a quarter circle defined by the equation \(x^2 + y^2 = 16\) in the first quadrant. Points \(B\) and \(C\) are chosen on the arc \(AD\), with \(C\) lying between \(A\) and \(D\). Determine the maximum possible area of the pentagon \(OABCD\), where \(O\) is the origin.

Answer: 12


Question:
Calculate the limit of the sequence as \( n \) approaches infinity:

\[
\lim_{{n \to \infty}} \frac{\left(2\,\sqrt[n]{\, n\,}\, - \,\sqrt[n]{\, 2\,}\right)^{n}}{n^2}
\]

Answer: \dfrac{1}{2}


Question: Suppose $M$ and $N$ are manifolds, and $\mathbb{R}^n \times M$ is diffeomorphic to $\mathbb{R}^n \times N$ for some natural number $n$. Determine if it is possible for $M$ and $N$ to not be diffeomorphic.

Answer: Yes


Question: Determine the values of $x \in \mathbb{R}$ for which the matrix $M_x = \begin{pmatrix} 1 & x+1 & 0 & 0 \\ 0 & x & x-1 & 0 \\ 0 & x-1 & x & 0 \\ 1 & 0 & x+1 & x \end{pmatrix}$ is diagonalizable.

Answer: -1


Question: Determine whether there exists a noncommutative ring with prime characteristic.

Answer: Yes


Question:
Given the set $X = (0,1]$ and two metrics: $d_1$, the standard metric, and $d_2$, the discrete metric, determine which of the following statements is true:

1. $(X,d_1)$ is compact, but $(X,d_2)$ is not.
2. $(X,d_2)$ is compact, but $(X,d_1)$ is not.
3. Both $(X,d_1)$ and $(X,d_2)$ are compact.
4. Neither $(X,d_1)$ nor $(X,d_2)$ is compact.

Answer: 4


Question:
Consider a strictly decreasing function $f:(0, \infty) \rightarrow (0, \infty)$. Define the function $h(x)$ as follows:

$$h(x)=\frac{f\left(\frac{x}{1+x}\right)}{1+f\left(\frac{x}{1+x}\right)}$$

Which of the following statements is correct?

A) $h(x)$ is strictly decreasing.
B) $h(x)$ is strictly increasing.
C) $h(x)$ is strictly decreasing initially and then strictly increasing.
D) $h(x)$ is strictly increasing initially and then strictly decreasing.

Determine the correct behavior of $h(x)$ based on the properties of $f$. \(\frac{x}{1+x}\) is strictly increasing, and $f\left(\frac{x}{1+x}\right)$ is an increasing function.

Answer: A


Question: Determine the range of values of \( p \) for which the equation \( \left(\frac{1}{\pi}\right)^{|x|} = x^2 - p \) has real solutions.

Answer: [-1, \infty)


Question: Determine whether for every positive integer $m$, there exists a second-degree real polynomial $P(x)$ with real coefficients and a natural number $k(m)$, such that the sequence $\{P(k(m)), P(k(m)+1), \ldots, P(k(m)+m-1)\}$ consists entirely of prime numbers.

Answer: Yes


Question: Provide an example of a non-abelian group $G$ such that for some positive integer $n$, the subset $H = \{g \in G \mid g = x^n \text{ for some } x \in G\}$ is not a subgroup of $G$. Justify why $H$ is not a subgroup.

Answer: S_3


Question: Suppose $(f_n)_n$ are analytic functions with power series representation $$ f_n(x) = \sum_{k=0}^\infty a_{3k}^n (x-x_0)^{3k}, \qquad x \in [a,b]. $$ If the $(f_n)_n$ converge uniformly on $[a,b]$ to a function $f$, does $f$ have a power series representation where all coefficients of powers that are not multiples of 3 are zero?

Answer: Yes


Question: Determine the cardinality of a set \(X\) that is the union of an infinite number of sets \(X_n\), each having a cardinality of \(\alpha\), where \(\alpha\) is an infinite cardinal number. Is the cardinality of \(X\) equal to \(\aleph_0\alpha\)?

Answer: \aleph_0 \alpha


Question: Is it possible to construct an uncountable set of non-overlapping rectangles in the plane, even if degenerate rectangles are allowed? Provide a justification for your answer.

Answer: Yes


Question: Consider a tower of fields $K \subset L \subset M$ where both extensions $L/K$ and $M/K$ are Galois. Let $f(x) \in K[x]$ be an irreducible polynomial over $K$ that splits completely in $M$. Suppose $f(x)$ splits into factors $f_1f_2\dots f_k$ over $L$, with each $f_i$ irreducible over $L$. Let $G = \operatorname{Gal}(M/K)$ and $H = \operatorname{Gal}(M/L)$. Is the action of $H$ on the set of roots of $f_i$ the same as its action on the set of roots of $f_j$ for any $i$ and $j$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{(x,y)\to (0,0)} \frac {\sin(x^2-xy)}{\vert x\vert} \]

Answer: 0


Question: Determine whether there exists a function \( y = f(x) \) such that \( \frac{df}{dx}\big|_{x=a} = |a| \) for all \( a \in \mathbb{R} \).

Answer: Yes


Question:
Given a function $f(x)$ that satisfies the following properties:
1) $f(x) + f(-x) = 2$
2) $f(1-x) = f(1+x)$
Evaluate the integral $\int_0^{2016} f(x) \, dx$. Use the given properties to determine the value of the integral.

Answer: 2016


Question: Given that the operation time (X) and repair time (Y) of a machine are independent and each follows an exponential distribution with density function \( f(x) = e^{-x} \) for \( x > 0 \) and \( f(y) = e^{-y} \) for \( y > 0 \), find the probability density function (PDF) for the proportion of time, denoted by \( U \), that the machine operates without failure during one operation-repair cycle. The transformation from \( (X, Y) \) to \( (U, V) \) is given by \( U = \frac{X}{X+Y} \) and \( V = X+Y \). Calculate the Jacobian determinant to determine the PDF of \( U \).

Answer: 1


Question: In the Kepler equation, if the radius $r$ and the gravitational constant $G$ are both divided by a factor $q$, by what factor must the mass $M$ be multiplied to keep the orbital period $T$ unchanged?

Answer: \dfrac{1}{q^2}


Question: Determine the number of solutions to the equation \(x^2 = y^3\) in the finite field \(\mathbb{Z}_{57}\).

Answer: 57


Question: Find the volume of a generalized tetrahedron in \(\mathbb{R}^n\) that is bounded by the coordinate hyperplanes and the hyperplane \(x_1 + x_2 + \ldots + x_n = 1\).

Answer: \dfrac{1}{n!}


Question: Suppose $Y(t)$ is the price of SomeCo stock at time $t$ years from now. Assume that $Y(t)$ follows a geometric Brownian motion with a drift rate of 0.05 per year and a volatility of $\sigma = 0.3$ per year. If the current price of SomeCo stock is 10.00 USD, calculate the probability that the price will be at least 11.00 USD one year from now.

Answer: 0.3817


Question: A fair coin is flipped indefinitely. What is the probability that a run of three consecutive tails occurs before a run of four consecutive heads?

Answer: \dfrac{15}{22}


Question: Determine whether every field extension can be considered as a vector space over the base field, regardless of its properties (e.g., finite, algebraic).

Answer: Yes


Question: Find the maximum value of $x^2 + y^2 + z^2$ subject to the constraints: $x \ge y \ge z \ge 0$, $x + y \le a$, and $y + z \le b$, where $a, b \ge 0$.

Answer: a^2


Question: Find the natural numbers $k$ and $p$ that satisfy the equation $(k+p)^{k+p} = (p(k+1))^p p^k$. Provide your answer as a pair $(k, p)$.

Answer: (3, 3)


Question: Determine the Krull dimension of the formal power series ring $R[[X]]$, where $R$ is a Noetherian ring.

Answer: \dim R + 1


Question: Evaluate the integral \(\int_0^{2\pi} \frac{2+\sin\theta}{5+4\sin\theta}\,\mathrm{d}\theta\).

Answer: \pi


Question: Given a bounded sequence \((x_n)\), find a subsequence that converges to \(\limsup x_n\).

Answer: \limsup x_n


Question: Determine whether there exists an affine scheme $X$ with exactly two points and dimension 1.

Answer: Yes


Question: In triangle ABC, points E and F divide sides AC and AB respectively such that \( \frac{AE}{EC} = 4 \) and \( \frac{AF}{FB} = 1 \). Let D be a point on side BC, and G be the intersection of lines EF and AD. If \( \frac{AG}{GD} = \frac{3}{2} \), determine the ratio \( \frac{BD}{DC} \).

Answer: \dfrac{4}{5}


Question: Determine whether the union of distinct cosets of a group $G$ spanned by elements $a$ and $b$, where $a \neq b$, always equals the entire group $G$. Provide a justification for your answer.

Answer: No


Question: Determine whether the function \( f(z) = \frac{\log(z)}{z} \), where \( \log(z) \) is the principal branch of the logarithm defined on \((-\pi, \pi]\), is analytic on the domain \( \Sigma = \mathbb{C} - ((-\infty, 0]) \).

Answer: Yes


Question: Compute the conditional expectation $E\left(W_6 | W_2 , W_4\right)$, where $W$ is a standard Brownian motion.

Answer: W_4


Question: Suppose that $h$ is the inverse function of a differentiable function $f$ and $H(x) = \frac{2}{h^3(x)}$. Given that $f(-2) = 1$ and $f'(-2) = \frac{1}{2}$, find $H'(1)$. \(\boxed{\text{Answer}}\)

Answer: -\dfrac{3}{4}


Question: Compute the winding number, \(i_p(\alpha)\), of the curve \(\alpha(t)=((2\cos t - 1)\cos t, (2\cos t - 1)\sin t)\) with respect to the point \(p=\left(\frac{1}{2},0\right)\), where \(t \in [0,2\pi]\). Express the integral for the winding number as a complex integral, and evaluate it using Cauchy's integral formula. The curve is parameterized by \(z=e^{it}\).

Answer: 2


Question: Given that \( \int_a^b |\sin x| \, dx = 8 \) and \( \int_0^{a+b} |\cos x| \, dx = 9 \), find the value of \( \int_a^b x \sin x \, dx \).

Answer: -2\pi\sqrt{2}


Question: Determine whether it is possible to construct an invertible element in $\tilde{A}$ from an element $a$ in $A$ that satisfies $a^d = 0$ for some $d > 0$. Provide a justification for your answer.

Answer: Yes


Question: In a commutative ring with unity, let \( q \) be a \( p \)-primary ideal such that \((q:x)\) is \( p \)-primary for \( x \notin q \). If \((q:x) \neq q\), does this imply that \( x \in p \)? Answer yes or no.

Answer: Yes


Question: In a circle of 20 people, determine the number of ways to select a committee of 7 people such that no two selected members are adjacent.

Answer: 2640


Question: Determine the points on the paraboloid $z = x^2 + 2y^2$ where the tangent plane is parallel to the plane $2x + 4y - z = 5$. Provide the coordinates of these points.

Answer: (1, 1, 3)


Question:
Let $G: \mathbb{R}^3 \to \mathbb{R}$ be a function that satisfies the following properties:

* $G(tx,ty,tz) = t^2G(x,y,z)$ for all $(x,y,z) \in \mathbb{R}^3$ and $t > 0$
* $G$ is increasing in each variable, i.e., if $x_1 \ge x_2$, $y_1 \ge y_2$, and $z_1 \ge z_2$, then $G(x_1,y_1,z_1) \ge G(x_2, y_2, z_2)$

Is $G$ necessarily continuous?

Answer: Yes


Question: Suppose we have an orthonormal system $\{ f_n \}$ in $L^2(0,1)$ where $|f_n(x)| \leq M$ almost everywhere for some $M > 0$. If the series $\sum_{n = 1}^{\infty} c_n f_n$ converges almost everywhere, does it follow that the sequence $\{ c_n \}$ must converge to zero? Answer yes or no.

Answer: Yes


Question: Let $X$ be a locally compact Hausdorff space and $f:X\to [0,\infty]$ a Borel measurable function. Suppose that for every compact subset $K$ of $X$, the inequality \( \int_K f\,d\mu \leq a \) holds, where $\mu$ is a Radon measure on $X$. Can we conclude that \( \int_X f\,\mathrm{d}\mu \leq a \)?

Answer: Yes


Question: Given the joint probability density function (PDF) of two random variables X and Y as \(f_{XY}(x,y)=1\) for \(0<x<1\) and \(0<y<1\), and the conditional joint PDF as \(f_{XY}(x,y | X>Y)=2\) for \(0<x<y<1\), find the conditional expectation \(E(X|X>Y)\). The conditional marginal PDF is given by \(f_{X}(x | X>Y)=2x\) for \(0<x<1\).

Answer: \dfrac{2}{3}


Question: Evaluate the contour integral \(\oint_{C} \frac{e^z}{z-\pi i}\ \mathrm{d}z\), where \(C\) is the contour enclosing the region bounded by the ellipse \(|z - 2| + |z+2| = 6\) and does not contain the point \(z = \pi i\).

Answer: 0


Question: Determine the number of abelian groups of order 540 and express each group in the form \(C_{n_1}\times C_{n_2}\times \cdots \times C_{n_r}\).

Answer: 6


Question: Calculate the limit of the function \( f(x) = \int_{x}^{+\infty} \frac{e^{-t}}{t} \, dt \) as \( x \) approaches \(+\infty\).

Answer: 0


Question: Determine whether the sequence \(\{(n+1)^{\alpha}-n^{\alpha}\}_{n=1}^{+\infty}\), where \(0 < \alpha < 1\), converges or diverges. If it converges, find its limit.

Answer: 0


Question: Determine if the function \( f: \mathbb{R}^2 \to \mathbb{R} \) defined by \[ f(x, y) = \begin{cases} \frac{x^3+y^3}{\sqrt{x^2+y^2}} & \text{if } (x,y) \neq (0,0) \\ 0 & \text{if } (x,y) = (0,0) \end{cases} \] has a directional derivative in all directions at the origin.

Answer: Yes


Question: For \(x > 0\), let \(f(x) = \int_{1}^{x} \left( \sqrt{\log t} - \frac{1}{2} \log \sqrt{t} \right) dt\). Determine the number of tangents to the curve \(y = f(x)\) that are parallel to the line \(x + y = 0\).

Answer: 1


Question: Determine the radius of convergence for the power series \( \sum_{n=0}^{\infty} z^n \sin(n) \).

Answer: 1


Question: Let $f:[0,1]\to [0,1]$ be a continuous function with a unique fixed point $x_{0}$. Assume that for every $x\in [0,1]$, there exists a natural number $n$ such that $f^{n}(x)=x_{0}$. Does this imply that there exists a uniform natural number $N$ such that for every $x \in [0,1]$, $f^N(x) = x_0$?

Answer: Yes


Question: Is the set \(V_\epsilon = \left( \bigcup_{j \in \mathbb{N}} (x_n - \frac{\epsilon}{2^{n+1}} , x_n + \frac{\epsilon}{2^{n+1}}) \right) \ \bigcap \ (0,1)\), where \(x_n\) is an enumeration of the rational numbers in \([0,1]\), dense in \((0,1)\) for \(\epsilon > 0\)?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to \infty} \frac{1}{n^3}\sum_{k=1}^n \frac{k^2}{a_k} \] where \( a_n \) is a sequence of real numbers defined by \( a_1=1 \) and \( a_{n+1}=\dfrac{a_n}{1+na_n} \).

Answer: \infty


Question: Let $f_1, g_1, f_2, g_2 \in \mathbb{C}[x]$ such that $\gcd(f_1, g_1) = 1$ and $\gcd(f_2, g_2) = 1$. It is also given that $\deg(g_i) \ge \deg(f_i)$ for $i = 1, 2$, and $\gcd(g_1, g_2) = 1$. Determine if $\gcd(f_1g_2 + f_2g_1, g_1g_2) = 1$. Provide a proof or counterexample.

Answer: 1


Question: Consider the function \( F = \exp(\text{trace}(A) + \frac{1}{2}\text{trace}(A^2) - \lambda) \) where \( A \) is a non-negative matrix. Determine if minimizing the function \( G = \text{trace}(A) + \frac{1}{2}\text{trace}(A^2) - \lambda \) will yield the same optimal solution for \( A \) as minimizing \( F \).

Answer: Yes


Question: Let $V$ be a vector space of dimension at least 3 over an infinite field $k$. Let $W_1, W_2, \ldots, W_n$ be any $n$ 1-dimensional subspaces of $V$. Determine whether there must exist a two-dimensional subspace $W$ of $V$ such that $W \cap \left(\bigcup_{i=1}^n W_i\right) = \{0\}$.

Answer: Yes


Question: Evaluate the integral \( \int_{[0,1]} f(x) \, du \), where \( f(x) = 0 \) on the Cantor set \( C \) and \( f(x) = x \) on \( [0,1] \setminus C \).

Answer: \dfrac{1}{2}


Question: Let $\{u_{k}\}_{k=1}^{\infty}$ be a sequence of complex numbers. If \(\sum_{k=1}^{\infty}\lambda^{k}u_{k}=0\) for every \(\lambda\) in the open disc \(\mathbb{D}(0, 1/3)\), can we conclude that \(u_{n}=0\) for all \(n=1, 2, 3, \ldots\)?

Answer: Yes


Question: Determine if the function \( f(z) = e^{\frac{1}{2}\log(z\sin z)} \) can be defined as a single-valued analytic function in the region \( 0 < |z| < \pi \). If it can, find the radius of convergence of its power series expansion around \( z=0 \).

Answer: \pi


Question: Evaluate the limit: \[ L = \lim_{x \to 1^+} \ln(x) \ln(\ln(x)) \] using methods other than L'Hôpital's rule.

Answer: 0


Question: Determine the radius of convergence for the infinite series: $$\sum_{n=0}^\infty\frac{2^n}{2^n+1}x^{n^2}$$

Answer: 1


Question: Determine whether a group $G$ of order $1000$ contains a proper normal subgroup $H$. Provide a justification for your answer.

Answer: Yes


Question: Let $n$ be an integer greater than $1$ and $x$ be an integer between $1$ and $10^{12}$. Calculate the probability that $x^{2n+1}$ ends with the digit sequence $11$. Express your answer as a fraction.

Answer: \dfrac{1}{100}


Question: Find the smallest positive integer that completely divides the expression \(12! + 6! + 12! \times 6! + 1!\). Use Wilson's Theorem in your solution.

Answer: 7


Question: Determine the number of homomorphisms from the ring of Gaussian integers \( \mathbb{Z}[i] \) onto the ring \( \mathbb{Z}_{13}(i) \).

Answer: 2


Question: Find the closed-form expression for the limit of the sequence \((a_n)\), where \(a_n = \prod_{k=1}^n \left(1 - \frac{c}{k}\right)\) with \(c > 0\) and \(c \neq 1\).

Answer: 0


Question: Evaluate the integral: $$\int_{-2}^{2}\frac{\sin^2x}{[\frac{x}{\pi}]+\frac{1}{2}}dx$$ where [•] denotes the Greatest Integer Function.

Answer: 0


Question: Determine whether the statement is true or false: For a sequence of functions $g_n(x)$, if $\sup_{x\in[a,b]}|g_n(x)-g(x)|$ converges to $0$ for some function $g(x)$, then $g_n(x)$ converges uniformly to $g(x)$ on $[a,b]$. Provide a justification for your answer.

Answer: A


Question: Determine if there exist positive rational numbers $x$, $y$, $z$, and $w$ such that $x^3 y - z^2 = w^4$ and $x^3 y > z^2$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$ \lim_{x\rightarrow \infty} \log [e^{x^2} - e^{-x^2}] - x^2 $$

Answer: 0


Question: Let \(p\) be a prime number and \((G,\cdot)\) be a group with \(p^3\) elements. We denote by \(\operatorname{C}(x)\) the centralizer of \(x\in G\). If \(|\operatorname{Z}(G)|=p\), determine the number of distinct centralizers \(\operatorname{C}(x)\) for \(x\in G\).

Answer: p + 2


Question: Determine the set of values of $x$ for which the infinite series $\sum_{k=0}^{\infty} E_k(x)$, where $E_0(x) = x$ and $E_{n+1}(x) = e^{E_n(x)}$, converges.

Answer: \emptyset


Question: Let $X$ be a random vector in $\mathbb{R}^n$ with a spherically symmetric distribution. Given two points $x_1$ and $x_2$ in $\mathbb{R}^n$, determine the probability that $X$ is closer to $x_1$ than to $x_2$. Express your answer as a single probability value.

Answer: \dfrac{1}{2}


Question: Determine the annihilator of the tensor $e_1 \wedge e_2 + e_3 \wedge e_4$ in the vector space $V = \langle e_1, e_2, e_3, e_4 \rangle$.

Answer: \{0\}


Question: Determine whether the statement is true or false: If $m$ and $n$ are odd positive integers, then $n^2 + m^2$ is not a perfect square.

Answer: A


Question: Evaluate the integral \( \int_{0}^{2022} (x^2 - \lfloor x \rfloor \lceil x \rceil) \, dx \).

Answer: 674


Question: Determine the cardinality of the ultrapower $M^{\langle U\rangle}$ when $U$ is a principal ultrafilter and $M$ is an infinite structure.

Answer: |M|


Question: Evaluate the improper integral \( \int_0^\infty \left( \frac{x^{10}}{1+x^{14}} \right)^{2} \, dx \).

Answer: \dfrac{\pi}{28}


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

Answer: 2^{2^n}


Question: Evaluate the integral \( \int_{0}^{1} \left(e^{-\frac{x}{2}}-2\left(1-e^{-\frac{1}{2}}\right)\right)^2 dx \). The expected result is 0.0128.

Answer: 0.0128


Question: Determine the least upper bound of the set \(A=\left\{\frac{1}{n}-\frac{1}{m}:n,m\in\Bbb{N}\right\}.\)

Answer: 1


Question: Find the remainder when \(3^{2015} + 11^{2014}\) is divided by 61, given that \(3^5 \equiv -1 \pmod{61}\) and \(11^2 \equiv -1 \pmod{61}\).

Answer: 59


Question: Evaluate the limit \( \lim\limits_{x\to 0}\left(\frac{\sin(x)}{x}\right)^{\frac{\sin(x)}{x-\sin(x)}} \).

Answer: \dfrac{1}{e}


Question: Determine whether the statement is true or false: For any prime number \( p \), the reciprocal of \( p \) in base 10 has a repeating set of digits with a period of \( p-1 \).

Answer: B


Question: Calculate the area of the region enclosed by the curves $x \ge 0$ and $(x^2 + y^2)^2 = x^2 - y^2$ in the first quadrant using a double integral.

Answer: \dfrac{1}{4}


Question: If a function $g(x)$ is analytic at $x=a$ and its Taylor series at $x=a$ has a radius of convergence $R$, is $g(x)$ equal to its Taylor series for all $x$ within a distance of $R$ from $a$?

Answer: Yes


Question: In a room illuminated by two light bulbs, each bulb has an exponential lifetime with a parameter of \( \lambda = 1 \). The bulbs are replaced only when both have burned out. What fraction of the time is the room half-lit, meaning only one bulb is functioning?

Answer: \dfrac{2}{3}


Question: Given a short exact sequence $$1 \rightarrow A \xrightarrow{\phi} B \xrightarrow{\psi} C \rightarrow 1,$$ where $|A|=10$ and $|C|=5$, determine the possible values for the order of the group $B$. \(\boxed{\text{Answer}}\)

Answer: 50


Question: Let $H$ be a Hilbert space and $T$ be a linear operator on $H$. Suppose that \( \sum_{j=1}^\infty\langle T\xi_j,\eta_j\rangle \) converges for any choice of orthonormal bases \(\{\xi_j\}\), \(\{\eta_j\}\). Does this imply that $T$ is a Hilbert-Schmidt operator?

Answer: Yes


Question: Calculate \( \lim_{a \rightarrow 0^+} \int_{D_a} \frac{|x|}{y(1+y^2)} \, dx \, dy \) where \( D_a=\{(x,y) \in \mathbb{R}^2: \frac{x^2}{2} \le y \le 2x^2 ; a \le \sqrt{x^2+y^2} \le \frac{1}{a} \} \) without using polar coordinates.

Answer: \dfrac{3\pi}{4}


Question:
Determine the number of connected components in the set \(\{t\in \Bbb{R}: f \text{ is differentiable at } t\}\), where the function \(f: \Bbb{R} \to \Bbb{R}\) is defined as follows:

\[f(t) = \begin{cases}
\frac{|t|}{2}, & t <-2,\\
\frac{|t+3|}{2} + \frac{1}{2}, & -2\le t<-1,\\
|t^3|, & -1\le t< 1,\\
\left(\frac{|t-3|}{2}\right)^2 + \frac{3}{4}, & 1\le t <2,\\
\frac{|t|}{2}, & t\ge 2.
\end{cases}\]

Answer: 5


Question: In triangle $\Delta ABC$, point $A$ is located at $(1,2)$. The angle bisector of $\angle B$ is given by the line $y = 2x + 10$, and the perpendicular bisector of side $AC$ is given by the line $y - x = 0$. Determine the equation of side $BC$. Additionally, justify whether the reflection of point $A$ across the angle bisector of $\angle B$ lying on $BC$ can be supported by a known theorem or property.

Answer: 5x + 9y = 19


Question: Consider an open and connected subset of the plane that intersects the unit circle. Does this subset necessarily contain points within a small distance of the circle's center?

Answer: No


Question: Given $n$ values $U_1, U_2, \ldots, U_n$ chosen randomly and independently from the interval $[0,1]$, where the probability that $U_i$ is smaller than $x$ is given by $\Pr(U_i < x) = x^2$, find the expected value of the $K$'th smallest element after sorting the values. Specifically, determine the expected value of $U_K$ if $U_1 < U_2 < \ldots < U_n$. Assume $n=2$ and $K=1$. Provide your answer as a numerical value.

Answer: \dfrac{8}{15}


Question: Determine an upper bound on the number of iterations required for the recurrence relation \( R(0) = n \) and \( R(t) = R(t-1) - \max\left(1, \left\lfloor\sqrt{R(t-1)}\right\rfloor\right) \) to reach zero, where \( n \) is a positive integer. Experimentally, it seems to end at \( t \leq 2 \sqrt{n} \). Prove or disprove this bound.

Answer: 2\sqrt{n}


Question: Determine whether the following statement is true or false: Let $f$ be a positive measurable function defined on a measurable space $(X, S)$, where $S$ is a $\sigma$-algebra. If the integral of $f$ with respect to a measure $\mu$ is finite, i.e., $\int_{X} f \, d\mu < +\infty$, then for all $E \in S$ such that $\mu(E) = +\infty$, it follows that $\int_{E} f \, d\mu = 0$. Provide a justification for your answer.

Answer: False


Question: In an Abelian category \(\mathscr{A}\), for an object \(A\) in \(\mathscr{A}\), what category does the Hom functor \(\mathrm{Hom}_\mathscr{A}(A,-)\) take values in? Is it always the category of abelian groups \(\mathbf{Ab}\), or can it be the category \(\mathscr{A}\) itself, particularly when \(\mathscr{A}=R\hbox{-}\mathbf{Mod}\)?

Answer: \mathbf{Ab}


Question: Find the values of $x \in (0, 1)$ for which the series \( \sum_{k=1}^{\infty} \frac{\cos(2\pi kx)}{k^2} \) converges.

Answer: (0, 1)


Question: Determine the range of \( \lambda > 0 \) for which the integral \( \int _0 ^\infty \frac{e^{\sin x}\sin 2x}{x^\lambda} \, dx \) converges.

Answer: (0, 2)


Question:
In the category \(\mathsf{C}_{\alpha, \beta}\), consider a morphism \(\sigma: (Z_1, f_1, g_1) \to (Z_2, f_2, g_2)\). Determine which of the following conditions must hold true for \(\sigma\):

1. \(f_1 = f_2\sigma\) and \(g_1 = g_2\sigma\)
2. \(\alpha f_1 = \alpha f_2\sigma\) and \(\beta g_1 = \beta g_2\sigma\)
3. \(\alpha f_1 = \beta g_2\sigma\) and \(\beta g_1 = \alpha f_2\sigma\)

Select the correct condition.

Answer: 1


Question: The function $f$ is differentiable and satisfies the equation \( \int_0^x (6f(t) + 4t) \, dt = \sin(x) \). Find the value of \( f'\left(\frac{\pi}{6}\right) \).

Answer: -\dfrac{3}{4}


Question: Calculate the limit of the sum \( \lim_{n\to\infty}\sum_{r=1}^{n}\frac{r}{n^2+n+r} \) using the integral method.

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{0}^{1}\frac{x}{x^2+(1-x)^2}\,dx \) using the substitution \(x = \frac{1}{2}(1+\tan \theta)\).

Answer: \dfrac{\pi}{4}


Question: Find the length of $BP$, in terms of the width $w$ of a sheet of paper, such that the area of triangle $B'PQ$ is minimized when the bottom right corner is folded over crease $\overline{PQ}$, assuming the sheet extends vertically indefinitely.

Answer: \dfrac{2}{3}w


Question: In a measurable space $(X, \mathcal{A})$ with a signed measure $\mu$, if $E, F \in \mathcal{A}$ with $F \subset E$, is it necessarily true that $\mu(F) = 0$ whenever $\mu(E) = 0$? Answer true or false and justify your answer.

Answer: False


Question: Evaluate the integral \( \int_0^{\pi/2} \frac{\sin((2n+1)t)}{\sin t} \, \mathrm{d}t \).

Answer: \dfrac{\pi}{2}


Question: Given the power series representation \( \sum_{i=0}^{+\infty}a_{i}x^i=\frac{1}{1-x-x^2-x^3}, \) find all positive integers \( n \) such that \( a_{n-1}=n^2. \) It is known that \( n=1 \) satisfies the condition. Determine the remaining values of \( n \).

Answer: 9


Question: Consider a compact 1-dimensional $C^1$-submanifold $\Gamma$ of $\mathbb{R}^3$, defined by a regular smooth injective map $\gamma: S^1 \to \mathbb{R}^3$ with $\gamma(S^1) = \Gamma$. Determine if there exists a fixed $\epsilon > 0$ such that for every point $p \in \Gamma$, the intersection $B_\epsilon(p) \cap \Gamma$ is a connected subset of $\Gamma$, where $B_\epsilon(p)$ denotes an open ball of radius $\epsilon$ in $\mathbb{R}^3$. Provide a justification for your answer.

Answer: Yes


Question: Find the limit of the sequence \( \lim _{n\rightarrow \infty }\frac{(n!)^{2}}{(2n)!} \).

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 1^-}\sum\limits_{r=1} ^\infty (x)^{\frac{1}{r}}$$

Answer: \infty


Question: Let \( B \in M_{15}(\Bbb R) \), the vector space of \( 15 \times 15 \) matrices with entries in \( \Bbb R \). Let \( V_B \) be the subspace of \( M_{15}(\Bbb R) \) spanned by \( \{B^n : n \geq 0\} \). What is the maximum possible dimension of \( V_B \)?

Answer: 15


Question: Determine whether a Reidemeister III move can be performed on a knot with minimal crossings.

Answer: Yes


Question: Let $A$ be a compact self-adjoint operator on an infinite-dimensional Hilbert space $H$. Consider the sequence $(\lambda_n)$ of distinct eigenvalues of $A$ and the corresponding sequence $(x_n)$ of unit eigenvectors. Determine whether the set \( S = \left\{ \sum_{i=1}^{\infty} \alpha_n x_n \mid \alpha_n \in \mathbb{C}, \sum_{i=1}^{\infty} |\alpha_n|^2 < \infty \right\} \) is a closed subset of $H$. Provide a justification for your answer.

Answer: Yes


Question: Let $M$ be a smooth manifold on which a Lie group $G$ acts properly and freely. Suppose there exists a set $D_M \subset M$ which is a smooth submanifold and contains exactly one representative from each orbit of $M$. Is it possible to construct a chart on $D_M$ from a given chart on $M$ that is adapted to the group action?

Answer: Yes


Question: Compute \( \lim_{n\to\infty} \int_{0}^{\pi/4}\tan^n x \, dx \).

Answer: 0


Question: Determine whether the following statement is true or false: If $f$ is continuous at $x_0$ and $\lim\limits_{x \to x_0}f'(x)$ does not exist, then $f$ is not differentiable at $x_0$. Provide a justification for your answer.

Answer: B


Question: Find the volume of the set \(S(a)=\{x\in \mathbb{R}^n \mid 0 \le x_i \text{ for all } i \text{ and } x_1+...+x_n\le a\}\).

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


Question: Evaluate the integral \( I = \frac{1}{2\pi i} \int_{\gamma} z^7 \cos \frac{1}{z^2} \, dz \) where \( \gamma \) is a positively oriented circle centered at the origin with radius \( R \).

Answer: \dfrac{1}{24}


Question: Find the orthogonal projection of the polynomial \( p(x) = x^3 \) onto the subspace \( P_2 \) using the inner product \( \langle f, g \rangle = \int_{-1}^1 f(t)g(t) \, dt \).

Answer: \dfrac{3}{5}x


Question: Determine whether the following statement is true or false: If $X \times Y$ is path connected, then both $X$ and $Y$ are path connected.

Answer: A


Question: Calculate the line integral of the gradient of the function $\varphi(x,y) = x^3y + xy^3$ along the curve $C$ defined by $\varphi(x,y) = 5$. What is the value of this integral?

Answer: 0


Question: Determine all natural numbers \(n\) such that the polynomial \(n^7+n^6+n^5+1\) has exactly 3 divisors.

Answer: 1


Question: Find the localization $T^{-1}R$ of the ring $R=\mathbb{Z}/m\mathbb{Z}$ at the set $T$ of all non-units of $R$.

Answer: 0


Question: Determine the order of the pole of the function \( g(z) = \frac{\sin(\pi z)}{z^3} \) at \( z=0 \).

Answer: 2


Question: In a one-dimensional random walk, at position $n > 0$, the probability of moving to position $(n-1)$ is $\frac{n+2}{2n+2}$, and the probability of moving to position $(n+1)$ is $\frac{n}{2n+2}$. Calculate the average time to reach position $0$ starting from position $n$. Provide your answer as a function of $n$.

Answer: n(n + 2)


Question: Find a polynomial with rational coefficients that has $i - \sqrt{2}$ as a root, given that $\mathbb{Q}(i, \sqrt{2}) = \mathbb{Q}(i + \sqrt{2})$. Provide the polynomial.

Answer: x^4 - 2x^2 + 9


Question: Determine the values of \(\alpha\) for which the series \(\displaystyle \sum_{n\geq 1}n^\alpha x^n(1-x^2)\) converges uniformly on the interval \([0, 0.5]\).

Answer: \alpha \in \mathbb{R}


Question: Determine the range of $p$ for which the inequality $$|\xi|^2 = 1 - 4p^2(1-p^2)s^4$$ ensures stability, given that $s = \sin\left(\frac{\omega}{2}\right)$ and stability is defined by the condition $|\xi| \leq 1$. Find the range of $p$.

Answer: [-1, 1]


Question: Determine whether the function \( f(x,y) = e^{-(x^2+y^2)^{-1}} \) with \( f(0,0) = 0 \) is differentiable at the origin.

Answer: Yes


Question: Calculate the value of $248^{156,454,638}$ modulo 250.

Answer: 194


Question: Determine all real numbers $x$ such that the series \( S = \sum_{n=1}^{\infty} \frac{1}{n^2+n} \cdot \tan^2\left(\frac{1}{\sqrt{n}}\right) \cdot n^x \) converges.

Answer: (-\infty, 2)


Question: Find the limit of the sequence \( b_n = \frac{\Gamma \left( n + \frac{3}{2} \right)}{\Gamma \left( n + 2 \right)\Gamma \left( \frac{1}{2} \right)} \) as \( n \to \infty \).

Answer: 0


Question: Determine the limit of the sequence \( \{a_n\} \) defined by \( a_{n+1} = \frac{a_n}{1 + \frac{1}{n} + a_n + a_n^3} \) for all positive integers \( n \), given that \( a_1 > 0 \).

Answer: 0


Question: Calculate the integral \( \oint_{L}{\frac{e^{\frac{1}{z-a}}}{z}}dz \) using the Residue Theorem, where \( L=\{z\in\mathbb{C}:|z|=r\} \) for some \( r>|a| \).

Answer: 2\pi i


Question: Determine the Galois group \( Gal \left(\frac{\mathbb{Q}(\sqrt[3]{3},\sqrt{-3})}{\mathbb{Q}}\right) \).

Answer: S_3


Question: Given that $a, b, c$ are the side lengths of a triangle with an area of 1, find the minimum value of the product $ab$.

Answer: 2


Question: Find the equation of the tangent plane to the surface patch \(\sigma(r, \theta) = (r\cosh\theta, r\sinh\theta, r^2)\) at the point \((1, 0, 1)\).

Answer: 2x - z = 1


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

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


Question: Determine whether there exists an integer $n$ such that for all integers $m$, the inequality \[ \left|\left|\sin(m)\right| - \left|\sin\left(n+m+\frac{1}{2}\right)\right|\right| \leq \epsilon \] holds for a given arbitrary $\epsilon > 0$. Provide a justification for your answer.

Answer: Yes


Question: You have 1000 bottles of wine, and 1 bottle is poisoned. You have 10 lab mice and 20 hours before a party to identify the poisoned bottle. The poison will kill any mouse that drinks it within 18 hours. Can you determine which bottle is poisoned using the 10 mice before the party?

Answer: Yes


Question: For the sequence $x_n = \frac{x_{n-1}}{2} + \frac{3}{x_{n-1}}$ where $n \geq 1$ with $x_0 > 0$, determine the values of $x_0$ for which the sequence converges. If the sequence converges to a limit $L > 0$, find the value of $L$.

Answer: \sqrt{6}


Question: Given that $ab + bc + ca = 0$, find the value of $\frac{1}{a^2 - bc} + \frac{1}{b^2 - ac} + \frac{1}{c^2 - ab}$.

Answer: 0


Question: Determine if the cardinality of the sequence \( \{ I_r \mid r < \varkappa \} \) is equal to \( \varkappa \), where \( \varkappa \) is the least ordinal with infinite cardinality and the sequence is constructed as follows: Initialize with \( I_0 = \varnothing \). For limit ordinals \( r \), \( I_r \) is the union of all preceding subsets \( I_s \). If \( r = s + 1 \), choose \( I_r \) as the smallest initial segment of \( \varkappa \) containing \( I_s \) and a nonempty finite subset \( U_s \) of \( \varkappa \), with the conditions that \( I_s \cap U_s = \varnothing \) and \( s \in U_s \).

Answer: \varkappa


Question: Evaluate the limit \( \lim _{x\to 0}\:\:\frac{x^m-\cos^n(x)}{x^{n+2}} \) for different values of \( n \in \mathbb{N} \).

Answer: -\infty


Question: Evaluate the limit \( \lim\limits_{n\to +\infty}\int\limits_{[0,n]}\left(1-\frac{x}{n}\right)e^{\frac{x}{2}}\,dx \) using the Dominated Convergence Theorem or the Bounded Convergence Theorem. Establish the measurability and boundedness of the function \( \left(1-\frac{x}{n}\right)e^{\frac{x}{2}}\chi_{[0,n]}(x) \), where \( \chi_{[0,n]}(x) \) is the characteristic function of the interval \([0,n]\).

Answer: \infty


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

Answer: 1


Question: Determine the real parameter values \( n \) for which the equation \( \frac{4n+3}{6} - \sin(4x)\cos(4x) - (n + \frac{2}{3})\sin(4x - \frac{\pi}{4}) = 0 \) has exactly three solutions for \( x \) in the interval \( \left[\frac{\pi}{16}, \frac{5\pi}{16}\right] \).

Answer: 1


Question: Consider the group of orthogonal matrices of order n over the field \( \mathbb{R} \) of reals, equipped with the topology induced by the Euclidean norm of matrices. Let \( g \) be one of such matrices and denote by \( X \) the topological closure of the cyclic group generated by \( g \). Is there a matrix \( g \) such that if the identity matrix \( I \) is the accumulation point in \( X \) of some sequence of powers of \( g \), then the sequence is definitely trivial, meaning all the elements (except finitely many) equal \( I \)?

Answer: Yes


Question: Calculate the surface integral of the vector field \( \mathbf{G} = \begin{pmatrix} y \\ z \\ x \end{pmatrix} \) over a sphere \( S \) using the divergence theorem.

Answer: 0


Question:
Given $A, B, C$ are $n \times n$ matrices, where $B$ and $C$ are nonsingular, and $b$ is an $n$-vector, solve for the vector $x$ using LU factorization, where:
\[ x = C^{-1}(3A-I)(B^{-1}-A)b \]

Answer: C^{-1}(3A - I)(B^{-1} - A)b


Question: Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot d\mathbf{S}\) using the Divergence Theorem, where \(\mathbf{F}(x, y, z) = y\mathbf{j} - z\mathbf{k}\) and \(S\) is the surface composed of the paraboloid \(y = x^2 + z^2\) for \(0 \leq y \leq 1\) and the disk \(x^2 + z^2 \leq 1\) at \(y = 1\), both oriented outward. Determine the flux through the paraboloid alone.

Answer: -\pi


Question: Determine the complete range of possible face counts for a convex polyhedron formed by folding and pasting two convex quadrilateral sheets of paper, $S$ and $T$, with equal perimeter lengths. The quadrilaterals must be joined at their edges to form the polyhedron, and folding is allowed along a line.

Answer: 4


Question: In triangle $ABC$, $AB = AC = 2AD$. Let $H$ be the intersection of $BD$ and $AE$, where $AE$ is perpendicular to $BD$. Given that the ratio of the areas of $\triangle ABH$ to $\triangle ADH$ is $8:1$, determine the ratio $BH:HD$. \( \text{Express your answer as a simplified ratio.} \)

Answer: 8:1


Question: Calculate the limit: $$\lim_{x \rightarrow 0} \frac{x-\sin (\sin (...(\sin x)...))}{x^3}$$ where the sine function is nested 150 times.

Answer: 25


Question: Calculate the fourth moment of Brownian motion at time \( t \).

Answer: 3t^2


Question: Consider the one-point compactification of the real numbers, denoted as $\mathbb{R}^\ast$. Let $A$ be an infinite set. In the product topology $(\mathbb{R}^\ast)^A$, suppose $U$ is an open set such that $U \supset \mathbb{R}^A$. Is it true that $U$ must contain $p_{a_1}^{-1}(\mathbb{R}) \cap \ldots \cap p_{a_n}^{-1}(\mathbb{R})$ for some $a_1, \ldots, a_n \in A$, where $p_{a_i}$ is the $a_i$-th coordinate map from $(\mathbb{R}^\ast)^A$ to $\mathbb{R}^\ast$?

Answer: Yes


Question: Let $\{A_n\}$ be a sequence of subsets of a set $X$. Suppose there is a set $B$ such that for all measurable $C \subset X$, the limit of $|A_n \cap C|$ as $n \to \infty$ is equal to $|B \cap C|$. Does it follow that the symmetric difference between $A_n$ and $B$ tends to the empty set as $n \to \infty$? Answer yes or no.

Answer: No


Question: Calculate the integral \( \int _0 ^ 1 \int_0^ {x} (x ^ 2 + y ^ 2) \, dy \, dx \) using the change of variables \(x = u + v, y = u-v\). Given that \(|Jg(x)|=2\) and \(f(g)=2(u^2+v^2)\), determine the new limits of integration and evaluate the integral.

Answer: \dfrac{1}{3}


Question: Is the smallest complete metric space containing a given metric space unique up to isometry?

Answer: Yes


Question: Calculate the value of \(1234^{1234} \mod 5379\).

Answer: 4603


Question: Consider the sets $C_b(X)$ and $C_b(Y)$, which consist of all bounded, real-valued, continuous functions on topological spaces $X$ and $Y$, respectively. These sets form rings under pointwise addition and multiplication and are metric spaces with the supremum metric. Is any ring homomorphism from $C_b(X)$ to $C_b(Y)$ continuous?

Answer: Yes


Question: Calculate the limit: $$\lim_{n\to\infty}\left(\alpha^{\frac{1}{n \cdot \ln n}}-1\right)^{\frac{1}{n}}$$ where $2<\alpha<3$ is a real number.

Answer: 1


Question: Find all real values of $x$ such that \((x+1)^7 - x^7 - 1 = 7x(x+1)(x^2+x+1)^2\).

Answer: \mathbb{R}


Question: Let $f$ be a function that is bounded over $[a, \\infty)$, and suppose $f'(x)$ exists for all $x \geq a$. If $\lim_{x \to \infty} f'(x) = b$, prove that $b = 0$.

Answer: 0


Question: Given that an IQ score is normally distributed with a mean of 100 and a standard deviation of 15, compute the variance of the square of the IQ score, X, for a person randomly selected from the population.

Answer: 9101250


Question: Given a positive definite matrix $A$, if the inverse is expressed as $A^{-1} = \frac{1}{\text{det}(A)} \cdot B$, determine whether matrix $B$ is also positive definite.

Answer: Yes


Question: In a Banach space $X$, if a sequence $\{x_n\} \subset X$ satisfies $f(x_n) \to 0$ for all continuous linear functionals $f \in X^*$, does it follow that $x_n \to 0$ in the norm topology? Answer yes or no.

Answer: No


Question: Find the minimum value of \( I(f) = \int_0^e f(x) \, dx \), where \( M \) is the set of continuous functions from \( \mathbb{R} \) to \( \mathbb{R} \) such that \( e^{f(x)} + f(x) \geq x + 1 \) for all \( x \in \mathbb{R} \) and for all \( f \in M \).

Answer: \dfrac{3}{2}


Question: Determine the type of singularity and the residue of the function \( g(z) = \frac{\tan(z)}{(z-\frac{\pi}{4})^3} \) at \( z = \frac{\pi}{4} \).

Answer: 2


Question: Find \( \lim\limits_{x \rightarrow \infty} \frac{\sin x}{|x|^\alpha} \) for \( \alpha > 0 \), using the definition: \( \lim\limits_{x \rightarrow \infty} f(x)=L \) if \( f \) is defined on an interval \((a, \infty)\) and for each \( \epsilon >0 \) there is a number \( \beta \) such that: \( | f(x) - L| < \epsilon \) if \( x> \beta. \)

Answer: 0


Question: Determine the distribution of the random variable $X = W(s) + W(s + t)$, where $W(t)$ is a Wiener process with $W(0) = 0$. Assume that for $t \le s$, $W(s) - W(t)$ follows a normal distribution with mean $0$ and variance $(s - t)$. Provide the distribution of $X$.

Answer: \mathcal{N}(0, 4s + t)


Question: Find the limit of the sequence \( \lim\limits_{n \to \infty}{\frac{1^1+2^2+3^3+\cdots+n^n}{n^n}} \).

Answer: 1


Question: Find the remainder when $50!$ is divided by $10^{12}$, expressed as $N \equiv a \pmod{10}$, where $a$ is the remainder.

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^{+\infty} \text{sinc}(8t) \text{sinc}^2(t) \cos(8\pi t) \, dt \) where the sinc function is defined as \( \text{sinc}(t) = \begin{cases} \frac{\sin(\pi t)}{\pi t} \quad &\text{for } t \neq 0, \\ 1 \quad &\text{for } t = 0. \end{cases} \)

Answer: \dfrac{1}{16}


Question: For a given natural number \( n \), determine the smallest natural number \( m \) such that the expression \( 3^{3^{3^{3^{\cdot^{3}}}}} \) (with \( m \) layers of exponents) is greater than \( 9^{9^{9^{9^{\cdot^{9}}}}} \) (with \( n \) layers of exponents).

Answer: n + 1


Question: Calculate the fractional change in the wavelength of the Lyman-alpha line (transition from n = 2 to n = 1) for a hydrogen atom when considering the reduced mass of the electron.

Answer: 5.45 \times 10^{-4}


Question:
Find the equilibrium points of the following system of differential equations:

\[ \frac{du(t)}{dt}=au(t)v(t)-bu(t) \]
\[ \frac{dv(t)}{dt}=-au(t)v(t) \]
\[ \frac{dw(t)}{dt}=u(t) \]
\[ u(t)+v(t)+w(t)=1 \]

Answer: (0, v, 1 - v)


Question: Determine the minimum value of \( n \) for which the function \( y(x) = x\sin x \) is a solution to an \( n^{th} \) order linear non-homogeneous differential equation with constant coefficients.

Answer: 2


Question: Find the 2018th value of $x$ such that $15^x \equiv x \pmod{1000}$. What is this value of $x$?

Answer: 2017375


Question: Find the remainder when \(50!\) is divided by \(1904\), given that \(1904 = 4 \times 8 \times 59\). Use the Chinese Remainder Theorem.

Answer: 0


Question: Find the number of roots of the polynomial \( q(z) = z^3 - 2z + 5 \) within the annulus \( \{1 < |z| < 3\} \).

Answer: 3


Question: In the context of a finite group $H$, what does the notation $2^H$ represent?

Answer: 2^H


Question: Find the radius of convergence of the power series \( \sum\limits_{n=1}^{\infty}a_nZ^n \), where \( a_n \) is the number of divisors of \( n^{50} \).

Answer: 1


Question: Determine the values of \( \rho \in \mathbb{R} \) for which the function \( g(x) = e^{\rho x} + \sin(\rho |x|) \) is convex on \( \mathbb{R} \).

Answer: 0


Question: Determine the amortized time complexity of performing a series of union and find operations on disjoint sets, assuming that all union operations occur after all find operations.

Answer: O(\alpha(n))


Question: Determine the points at which the function \( f(x) = \begin{cases} \cos x & x \in \mathbb{Q} \\ x & x \notin \mathbb{Q} \end{cases} \) is differentiable.

Answer: \emptyset


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

Answer: 1


Question: Let $f(x) \in k[x]$ be an irreducible, separable polynomial, and let $E$ be the splitting field of $f(x)$ over the field $k$. If the Galois group $\text{Gal}(E/k)$ is abelian, does it follow that $E = k(a)$ for every root $a$ of $f(x)$? Provide a justification for your answer.

Answer: Yes


Question: Suppose that $F(z)$ is an analytic function with $f(z)$ as its derivative. Consider the complex line integral \( \int_\gamma f(z)\ dz \). Is this integral equal to zero if $\gamma$ is a closed curve? Justify your answer.

Answer: 0


Question: Determine whether the expression \[ e=\left(1+\pi-(5^{\pm})\arccos\left(\frac{\varphi}{2}\right)\right)^{\frac{1}{\pi-(5^{\pm})\arccos\left(\frac{\varphi}{2}\right)}} \] is true, where $5^{\pm}$ represents an infinitesimal quantity approaching zero ($5 \pm \varepsilon$, with $\varepsilon$ a small positive real number).

Answer: e


Question: Determine whether differentiation and summation commute for power series within their interval of convergence. Provide a justification for your answer.

Answer: Yes


Question: Determine the power of the group of languages A, where A consists of languages L such that L = L* and L is defined over the alphabet Σ = {1}.

Answer: 2^{\aleph_0}


Question: Determine whether a free group of a given rank can be a subgroup of a free group of a different rank.

Answer: Yes


Question: Determine the number of ways to distribute 12 distinct objects into 3 indistinguishable boxes such that each box contains at least one object.

Answer: 86526


Question: Let $G$ be a filter finer than the filter $F$. Is it necessary that the net based on the filter $G$ is a subnet of the net based on $F$? Answer yes or no.

Answer: No


Question: Determine whether the integral of a discontinuous Riemann integrable function is always continuous. Provide a justification for your answer.

Answer: Yes


Question: Find the series solution to the differential equation \(x^2y' = y\) assuming the solution is of the form \(y = \sum a_n x^n\).

Answer: 0


Question: Evaluate the limit: $$\lim \limits_{x \to \infty} \frac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$

Answer: 1


Question: If two topologies, \(\tau_1\) and \(\tau_2\), on a set \(X\) have the same ultrafilter convergence, are these topologies equivalent, meaning \(\tau_1 = \tau_2\)? Provide a justification for your answer.

Answer: Yes


Question: Given a continuous function $f$ such that $f(f(x)) = 1 + x$, find the value of $f'(1)$. Provide a formal approach to solving this problem.

Answer: 1


Question: Determine whether it is possible to construct angles that are not multiples of 15° using only a compass and a straightedge.

Answer: Yes


Question: Determine the number of integer solutions \((x, y)\) to the equation \(x^3 - y^3 = 271\), where both \(x\) and \(y\) are integers.

Answer: 2


Question: Determine the limiting distribution of the sequence of random variables \( Y_n = \frac{X_n}{n} \), where \( X_n \sim \text{Poisson}(np_n) \) and \( p_n \to 0 \) as \( n \to \infty \).

Answer: 0


Question: Determine whether a square root of a normal operator on a complex inner product space can also be normal.

Answer: Yes


Question: Determine the domain of the function: \[ f(x)= \sqrt\frac{(e^x-2)^x + \arccos(e^x -2)}{x- \sqrt {x^2 -1}} \]

Answer: [1, \ln 3]


Question: Find the maximum value of $y$ given the equation $2(x^3+y)^4+(x^3+y)^2=2x^3+y$.

Answer: \dfrac{5}{8}


Question: In the secretary problem with 500 numbers, determine the optimal number of numbers to observe before selecting the next largest number.

Answer: 184


Question: In a square $ABCD$ with side length 1, equilateral triangles $AYB$ and $CXD$ are drawn inside the square such that points $X$ and $Y$ lie inside the square. Find the length of the segment $XY$. Express your answer in simplest form.

Answer: \sqrt{3} - 1


Question: Find the conditional extrema of the function \( f(x,y) = \sin^2 x + \sin^2 y \) subject to the constraint \( x + y = \frac{\pi}{2} \).

Answer: 1


Question: Evaluate the integral \( \int^1_0 \frac{1}{\sqrt{x}} \int^{\sqrt{x}}_1 e^{-t^2} \, dt \, dx \).

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


Question: Calculate the limit: $$\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$$

Answer: -1


Question: Determine the number of critical points for the function \( x(1-y) \) on the unit circle \( x^2+y^2=1 \) using the method of Lagrange multipliers.

Answer: 3


Question: Consider two Hausdorff topological spaces $(X,\tau)$ and $(X,\tau')$ with the same underlying set. If a sequence $(x_n)_{n\in \Bbb{N}}$ converges to $x\in X$ in the topology $\tau$, does it necessarily converge to $x$ in the topology $\tau'$?

Answer: No


Question: Determine whether the integral \( \int_{-\infty}^\infty \frac{2\cos(w)\sin(w)}{w} \, dw \) converges. If it converges, find its value.

Answer: \pi


Question: Let $X$ be a compact topological space and $f: X \to Y$ a continuous function into a metric space $(Y, d)$. If $f(X)$ is closed in $Y$, is there a compact subset $K \subseteq X$ such that $f(K) = f(X)$?

Answer: Yes


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

Answer: 2


Question: Determine the values of $x$ for which the series \( \sum_{n=1}^\infty \frac{x^{n^2}}{5^n} \) converges.

Answer: [-1, 1]


Question: Evaluate the limit: $$\lim_{n\to \infty} \frac{\sqrt[n]{n!}}{n^{1+\frac{1}{2n}}}$$

Answer: \dfrac{1}{e}


Question: Calculate the sum \( S = \sum_{k=1}^{2015} (-1)^{\frac{k(k+1)}{2}} k \) by considering the residue of \( k \) modulo 4.

Answer: 0


Question: Determine the maximum value of $| \cos(z) |$ where $z$ is a complex number with real and imaginary parts restricted to the interval $[0, 2\pi]$. Provide the maximum value.

Answer: \cosh(2\pi)


Question: Evaluate \( \lim_{x\to 1} \int_{x}^{x^2}\frac{1}{\ln {t}}\mathrm dt \).

Answer: \ln 2


Question: Find the coefficient of \((z-1)^{-3}\) in the Laurent series expansion of \(R(z)=\frac{1}{(z^3-1)(z^5-1)}\) around the point \(z=1\).

Answer: 0


Question: Given the joint probability density function \( f(x_1, x_2) = 8x_1x_2 \) for \( 0 < x_1 < x_2 < 1 \) and \( f(x_1, x_2) = 0 \) elsewhere, determine the expected value \( E(X_1^2 X_2) \).

Answer: \dfrac{2}{7}


Question: Determine whether a Lie group can be a manifold with boundary. Provide a yes or no answer.

Answer: No


Question: Let \( f(x,y,z,u)=\frac{24}{(1+x+y+z+u)^5}, x>0,y>0,z>0,u>0 \) be a density function. Calculate the probability \( P(X+Y+Z+U \ge 1) \).

Answer: \dfrac{15}{16}


Question: Let $f(x)$ be a polynomial with complex coefficients such that the greatest common divisor of $f(x)$ and its derivative $f'(x)$ is 1. Does this imply that $f(x)$ has no repeated irreducible factors in $\mathbb{C}[x]$? Answer with 'yes' or 'no' and justify your answer.

Answer: yes


Question: Evaluate the limit \( \lim_{n \to \infty} \sum_{k=1}^{n} \frac{(n+1)^k}{n^{k+1}} \) using Riemann sums.

Answer: e - 1


Question: Find the maximum value of \( F(x,y) = x^2 + 4y^2 \) subject to the constraints \( x, y \ge 0 \) and \( x^3 + y^3 - x + y = 0 \). Do not use Lagrange multipliers.

Answer: 1


Question: Given sets X, Y, and Z with cardinalities l, m, and n respectively, where m < l < n, determine the number of possible one-to-one functions of the form g∘f, where f: X → Y and g: Y → Z.

Answer: 0


Question: Determine the big-O bound for the expression: $$\sum^{ N}_{k=0}{2^k \log_2\left(\frac{2^N}{2^k}\right)}$$ where $N = \log M$. Provide your answer in big-O notation.

Answer: O(M)


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

Answer: -x e^{-2x}


Question: Consider the heat equation \(u_t = u_{xx}\) for \(0 < x < \pi\) and \(t > 0\), with boundary conditions \(u(0, t) = 0\) and \(u(\pi, t) = 0\), and the initial condition \(u(x, 0) = \sin x\). What is the value of \(u\left(\frac{\pi}{2}, 1\right)\)?

Answer: \dfrac{1}{e}


Question: Calculate the probability that a list of length \(b^{n/2}\) contains only unique numbers when selecting from a set of \(b^n\) distinct numbers, with replacement, until the list is filled.

Answer: e^{-1/2}


Question: In a field \((A, +, *)\), is there a non-zero element \(x \in A\) such that for all \(y \in A\), \(x * y = 0\)? Provide a justification for your answer.

Answer: No


Question:
Given a function $f: A \rightarrow A$, where $A \in (-1,\infty)$, and the functional equation $f(x + f(y) + xf(y)) = y + f(x) + yf(x)$ holds for all $x,y \in A$, and $\frac{f(x)}{x}$ is strictly increasing on $(-1,0) \cup (0,\infty)$. Determine $f(x)$ from the following choices:

(A) $f(x) = xe^{1+x}$
(B) $f(x) = x\ln(1+x)$
(C) $f(x) = \frac{x}{1+x}$
(D) $f(x)$ is undefined or has no specific form

Which is the correct function $f(x)$?

Answer: D


Question: A crew of 8 members needs to secure a safe such that any group of 5 or more members can open it, but any group of 4 members cannot. Each member receives an equal number of keys. Determine the minimum number of locks required to achieve this security condition.

Answer: 70


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty}\sqrt[n]{\sum^{n}_{k=1}\left(k^{999}+\frac{1}{\sqrt{k}}\right)}$$

Answer: 1


Question: In an isosceles triangle $ABC$ with $AB = AC$, the angle bisector from $B$ intersects $AC$ at $D$ such that $AD + BD = BC$. Find the measure of angle $A$.

Answer: 100^\circ


Question: Calculate the volume of the solid generated by revolving the region bounded by the curves \(x^2-y^2 = 7\) and \(x = 4\) about the \(y\)-axis using the shell method.

Answer: 36\pi


Question:
Evaluate the limit for a continuous function $f: \mathbb{R} \to \mathbb{R}$:

$$ \lim_{x \to 0} \frac{1}{x^2} \int_{0}^{x} f(t)t \, dt $$

where $F$ is an antiderivative of $f$ such that:

$$\int_{a}^{b} f(x) \, dx = F(b) - F(a) \quad \text{for} \quad a, b \in \mathbb{R} $$
and:

$$ \lim_{x \to x_{0}} \frac{F(x) - F(x_{0})}{x - x_{0}} = f(x) $$

Answer: \dfrac{f(0)}{2}


Question: Find a non-zero function $f(x)$ such that $\frac{d}{dx}f(x) = f(x-1)$. Provide the function $f(x)$.

Answer: e^{W(1) x}


Question: Determine whether there exist three distinct points P, Q, and R on an elliptic curve such that P + Q + R = O, where O is the point at infinity.

Answer: Yes


Question: Determine whether there exists an elliptic operator of order $2m$ with real variable coefficients that is not strongly elliptic.

Answer: Yes


Question: Calculate the sum of the infinite series: \[ \sum_{n=0}^{\infty} \frac{(2n+1)^2}{4^n}. \]

Answer: \dfrac{164}{27}


Question: Calculate the flux of the vector field $\mathbb{v} = \operatorname{curl}\mathbb{u}$, where $\mathbb{u} = (xy, xz^2, x^2y)$, downwards through the paraboloid $z = 1 - x^2 - y^2$ for $x \geq 0$ and $z \geq 0$. What is the value of this flux?

Answer: \dfrac{2}{3}


Question: Determine whether the following statement is true or false: If a linear operator \( T \) from a Banach space to itself maps every weakly convergent sequence to a weakly convergent sequence, then \( T \) must be continuous.

Answer: True


Question: Find the minimum value of \( f(x,y,z) = x^2 + y^2 + z^2 \) subject to the constraint \( (x+y)^2 = xy \).

Answer: 0


Question: Let $A \subset \mathbb{R}$ be a Lebesgue measurable set with $0 < \lambda(A) < \infty$. Consider the function $f: \mathbb{R} \to [0, \infty)$ defined as $f(x) = \lambda(A \cap (-\infty, x))$. Determine the value of $\lim\limits_{x \to \infty} f(x)$ and explain how this limit relates to the Lebesgue measure of $A$. \( \text{Provide your answer in the form } \boxed{\text{answer}}. \)

Answer: \lambda(A)


Question: Evaluate the limit: $$\lim_{(x,y)\to(0,0)}\frac{x^3}{x^2+y^2}$$

Answer: 0


Question: Calculate the contour integral \( \int_C \bar{z}^2 \, dz \) where \( C \) is the unit circle defined by \( |z| = 1 \).

Answer: 0


Question: Let $p = 10007$ be a prime number. Determine the number of $2 \times 2$ matrices $X$ with elements from $\mathbb{Z}_p$ such that $X^2 \equiv I \pmod{p}$, where $I$ is the identity matrix.

Answer: 100150058


Question: Let $g: W_m \to W_m$ be an endomorphism such that $g^2 = 0$ and the rank of $g$ is $k$. Calculate the dimension of the intersection of the kernel and image of $g$. Express your answer as a single integer.

Answer: k


Question: Find the angle between the curves $(C1) : x = y+2$ and $(C2): x=4-y$ on the surface $(S):$ $$\textbf{r}(x,y) = x\ \textbf{i} + y\ \textbf{j} +(x+y)\ \textbf{k}$$.

Answer: \dfrac{\pi}{2}


Question: In a complete and separable metric space $(X, \mathrm{d})$, given an open set $U$ and a closed set $K \subset U$, determine if there exists a Lipschitz function $f$ such that $f|_K = 1$ and $f|_{X \setminus U} = 0$. Provide a justification for your answer.

Answer: No


Question:
Let $x, y, z$ be real numbers such that $xy + yz + zx = 3$, $x + y + z = 5$, and $xyz = 2$. Calculate the value of:

$$\frac{1}{xy + z - 2} + \frac{1}{yz + x - 2} + \frac{1}{zx + y - 2}$$

Answer: \dfrac{2}{13}


Question: Determine the genus of the complete 4-partite graph \( K_{4,2,2,2} \).

Answer: 2


Question: Evaluate the limit \( \displaystyle \lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}\left(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n}}\right) \) without using a Riemann sum.

Answer: 2


Question: Consider a vector $x \in \mathbb{R}^n$ and a real $n \times n$ matrix $A$. Assume that $n > 2$, $x \neq 0$, $A$ is not a scalar multiple of the identity matrix $I_n$, and the column space of $aI_n - A$ is not spanned by $x$ for any real scalar $a$. Does the set of $y \in \mathbb{R}^n$ such that $x, y, Ay$ are linearly dependent have zero $n$-dimensional Lebesgue measure?

Answer: Yes


Question: Evaluate the limit: \[ \lim _ { n \rightarrow \infty } \frac { 2 ^ { - n ^ { 2 } } } { \sum _ { k = n + 1 } ^ { \infty } 2 ^ { - k ^ { 2 } } }. \]

Answer: \infty


Question: Evaluate the definite integral \( \int_0^1 \frac{f(x)}{f(x) + f(1 - x)} \, dx \) where \( f \) is a positive and continuous function.

Answer: \dfrac{1}{2}


Question: Given real numbers \( A \), \( B \), and \( C \) such that \( A^2 + B^2 + C^2 = 1 \) and \( A + B + C = 0 \), find the maximum value of \((ABC)^2\).

Answer: \dfrac{1}{54}


Question: Consider a partially ordered set \((P, \leq)\) with cardinality \(|P| > 2^\lambda\). Must \(P\) necessarily contain a chain or an antichain of cardinality \(\lambda^+\)?

Answer: Yes


Question: Determine the generator of the ideal \( I = \{a+bi \in \mathbb{Z}[i] \mid a \equiv b \pmod{2}\} \) in the ring of Gaussian integers \( \mathbb{Z}[i] = \{a+bi \mid a,b \in \mathbb{Z}\} \).

Answer: 1 + i


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

Answer: (-\infty, 1)


Question: Evaluate the integral \( \int_0^{\pi/2} \sin^{10}x\cdot \cos(12x)\,\text{d}x \).

Answer: 0


Question: Evaluate \( \lim_{x\to1^{+}}\dfrac { g(2-x){ (x-1) }^{ 3 } }{ \left\{ \frac { \sin(x-1) }{ [g(x)] } \right\} } \), where \(g(x)\) is a real-valued function that is continuous everywhere except at \(x=1\), \(g'(x)>0\) for \(x<1\), \(g'(x)<0\) for \(x>1\), \(\lim_{x\to1^{+}} g(x)=5\), \(\lim_{x\to1^{-}} g(x)=7\), and \(g(1)=3\). Here, \([\cdot]\) and \(\{\cdot\}\) denote the greatest integer and fractional part functions, respectively.

Answer: 0


Question: Find the order of the element \((2, 3, 1)\) in the group \((\mathbf{Z}/105\mathbf{Z})^\times\) using the Chinese Remainder Theorem.

Answer: 4


Question: Given that $u$ is a twice continuously differentiable function with linear growth, such that \(\lim_{x\to\infty} \left(u'(x) - \frac{1}{g(x)}u(x)\right) = 0\) where \(g(x)\) is a Lipschitz continuous function with Lipschitz constant \(L < 1\). For the first order linear homogeneous differential equation \(y'(x) - \frac{1}{g(x)}y(x) = 0\), its general solution is \(y(x) = c\exp\left(\int\frac{1}{g(x)}dx\right)\) with a constant \(c \in \mathbb{R}\). If any solution with linear growth satisfies \(\lim_{x\to\infty}y(x) = 0\), can we infer that \(\lim_{x\to\infty}u(x) = 0\)?

Answer: 0


Question: Calculate the work done by the radial vector field \( F(x, y, z) = xi + yj + zk \) on a particle moving along the curve \( C \) defined by \( r(t) = (1 + 2\sin(t))i + (1 + 5\sin^2(t))j + (1 + 4\sin^3(t))k \) for \( 0 \leq t \leq \frac{\pi}{2} \).

Answer: \dfrac{67}{2}


Question: Given non-square matrices $A$ and $B$ such that $AB = 0$, determine whether it necessarily follows that $BZA = 0$ for any matrix $Z$ such that the product $BZA$ is defined.

Answer: No


Question: Find \( \alpha \in \mathbb{Z}[i] \) such that the ideal \((3+5i,1+3i)\) is equal to the ideal \((\alpha)\).

Answer: 1 - i


Question: Determine the classification of the discontinuity for the function \(f(x)=\frac{\sin^2(x)}{x|x(\pi-x)|}\) at \(x=\pi\). Investigate whether the limit \(\lim _{x \to \pi} f(x)\) exists and find its value if it does.

Answer: 0


Question: Calculate the probability that the sum of $n$ randomly chosen elements from the set $\mathbb{Z}/p\mathbb{Z}$, where $p$ is a prime number, is congruent to $0 \pmod{p}$. Assume the selection is done with a uniform distribution.

Answer: \dfrac{1}{p}


Question: Let $(B_t)_{t \ge 0}$ be a standard one-dimensional Brownian motion, independent of $W_1$. Calculate the probability that $W_1 + B_t > 0$ for all $t \in (0, 1)$. Provide your answer as a probability value.

Answer: \dfrac{1}{4}


Question:
Given a nonempty set $X$ and a neighborhood function $\mathscr{N}:X\to2^{2^X}$ satisfying the following properties:
1. For every $x\in X$, $\mathscr{N}(x)\neq\{\emptyset\}$.
2. If $U\in\mathscr{N}(x)$, then $x\in U$.
3. If $U,V\in\mathscr{N}(x)$, then $U\cap V\in\mathscr{N}(x)$.
4. If $U\in\mathscr{N}(x)$ and $V\subseteq X$ is such that $U\subseteq V$, then $V\in\mathscr{N}(x)$.

The topology induced by $\mathscr{N}$, denoted by $\tau_\mathscr{N}$, is defined as:
$$\tau_\mathscr{N}:=\{\mathscr{O}\subseteq X:\mathscr{O}\in\mathscr{N}(x)\mbox{ for every }x\in\mathscr{O}\}.$$

Determine if, for a given $x\in X$ and $N\in\mathscr{N}(x)$, $N$ is a neighborhood of $x$ in the topology $\tau_\mathscr{N}$. Provide a justification for your answer.

Answer: Yes


Question: Determine the sum function of the infinite series: \[ \sum_{n=1}^{\infty} \frac{n-2}{(n-1)!} z^{n+1} \] The series has been transformed to \[ \sum_{m=0}^{\infty} \frac{m-1}{m!} z^{m+2} \] which resembles the series for $e^x$. Use the properties of the exponential function to find the sum function of the series.

Answer: z^2 (z - 1) e^z


Question: Determine the value of the quadratic variation \(<r_t dt, r_t dt>\) in the context of stochastic calculus, where \(r_t\) is a stochastic process. Is it equal to zero, considering that \(<dt, dt> = 0\) regardless of \(r_t\) being stochastic?

Answer: 0


Question: Determine the dimension of the vector space $\mathfrak{m}/\mathfrak{m}^2$ for the curve $\mathcal{C}=\{ (x,y): x^3+y^3-1=0 \} \subset \mathbb{C}^2$, where $\mathfrak{m} = \langle x-a,y-b\rangle/\langle x^3 + y^3 -1\rangle$ and $p=(a,b) \in \mathcal{C}$. Provide your answer as a single integer.

Answer: 1


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

Answer: (-2)^n n!


Question: Find the sum of the infinite series \( \displaystyle \sum\limits_{n=1}^{\infty} \dfrac{n^2}{2^n} \).

Answer: 6


Question: Evaluate the limit \( \lim_{x \to \frac{\pi}{4}} \frac{\ln (\cot x)}{1-\tan x} \) without using L'Hopital's rule or Taylor series.

Answer: 1


Question: Compute the definite integral of $e^{-x^2 - y^2}$ over the entire $xy$-plane.

Answer: \pi


Question: Find the limit of the function \( f(x,y) = \frac{e^{xy^2}-1}{x^2+y^2} \) as \((x,y) \to (0,0)\).

Answer: 0