Question: Let $M$ be a finite-dimensional subspace of $c_{0}$, and let $\\varepsilon > 0$. Does there exist a finite rank projection from $c_{0}$, with norm $\leq 1+\\varepsilon$, onto a subspace $N$ of $c_{0}$ such that $M \subseteq N$ and the Banach-Mazur distance $\textrm{d}(N,l_{\infty}^{n}) \leq 1+\\varepsilon$, where $n=\textrm{dim} N$?

Answer: Yes


Question: Show that the Lebesgue measure of the set \( A \) of real numbers \( r \) such that there exists a constant \( C \) and a sequence \( \frac{p_n}{q_n} \) of rational numbers (where \( p_n \) and \( q_n \) are integers) with \( q_n \rightarrow \infty \) and \( \left\vert r - \frac{p_n}{q_n} \right\vert < \frac{C}{q_n^3} \) is zero.

Answer: 0


Question: Evaluate the limit: \[ \lim_{V\rightarrow {\bf{0}}^{n\times n}}\frac{|\text{tr}(V^2)|}{\|V\|_{op}} \] where $\text{tr}(V^2)$ is the trace of the matrix $V^2$ and $\|V\|_{op}$ is the operator norm of $V$. Determine if this limit is equal to 0.

Answer: 0


Question: Evaluate the line integral \( \int_C \vec{F} \cdot \vec{dS} \), where \( C \) is the circle of radius 3 in the \( xz \)-plane oriented counter-clockwise when viewed from the point \((0, 1, 0)\) into the plane. The vector field is given by \( \vec{F}(x, y, z) = \langle 2x^2z + x^5, \cos(e^y), -2xz^2 + \sin(\sin(z)) \rangle \). Use Stokes' Theorem to find the value of the integral.

Answer: 81\pi


Question: Calculate the Lebesgue integral of the function \( f(x) = x \) over the Smith-Volterra-Cantor Set.

Answer: \dfrac{1}{4}


Question: Find the greatest common divisor of the numbers \(2^{71} - 2, 3^{71} - 3, \ldots, 100^{71} - 100\).

Answer: 4686


Question: Consider the space of tempered distributions. If $f \in \mathcal{S}'(\mathbb{R}^3,\mathbb{R}^3)$ and $\textbf{curl} f = 0$, does there exist a $u \in \mathcal{S}'(\mathbb{R}^3)$ such that $\nabla u = f$?

Answer: Yes


Question: Find the maximum value of $ad-bc$ given that $a^2+b^2+c^2+d^2=1$ and $a, b, c, d \in \mathbb{R}$. Provide your answer as a single number.

Answer: \dfrac{1}{2}


Question: Find the value of \( \lim_{x \to 0}\left (\frac {\sin x}{x}\right)^{\frac {1}{x^2}} \).

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


Question: Consider a one-way street with $n$ parking lots. One-by-one, $n$ cars numbered 1, 2, 3, ..., n enter the street. Each driver $i$ heads to their favorite parking lot $a_i$. If it is free, they occupy it. Otherwise, they continue to the next free lot and occupy it. If all succeeding lots are occupied, they leave for good. Determine the number of sequences $(a_1, a_2, ..., a_n)$ such that every driver can park.

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


Question: Determine whether an integer polynomial with constant term \( \pm 1 \) must have at least one zero within the unit disc \(|z| \leq 1\).

Answer: Yes


Question: Calculate \(3^{187} \mod 1000\).

Answer: 387


Question: Suppose \( \{a_n\} \) is a sequence in \([0,1]\) that converges to \( L \), and \( \{f_n\} \) is a sequence of continuous functions from \([0,1]\) to \( \mathbb{R} \) that converges pointwise to a function \( f \). Does it follow that \( f_n(a_n) \to f(L) \)? Justify your answer.

Answer: No


Question: Consider a Markov chain with state space $\mathbb{R}$ and transition function $P(x,A) = \lambda([x - 1/2, x + 1/2] \cap A)$, where $\lambda$ is the Lebesgue measure. If the initial distribution is concentrated at the origin, find the probability $P(|\omega_{2}| \leq 1/4)$. Provide your answer as a single probability value.

Answer: \dfrac{7}{16}


Question: Evaluate the triple integral \( \iiint\limits_G (x^2 - z^2) \, dx \, dy \, dz \), where the region \( G \) is bounded by the planes \( y = -x \), \( z = x \), \( z = y \), and \( z = 1 \).

Answer: -\dfrac{4}{15}


Question: Find the covariance function \( C_X(s, t) \) for the Gaussian process defined by \( X(t) = \varepsilon_1 \cos(t) + \varepsilon_2 \sin(t) \), where \( \varepsilon_1 \) and \( \varepsilon_2 \) are independent standard normal random variables.

Answer: \cos(s - t)


Question:
Let \( f(x) \) be a continuous function satisfying the integral equation:
\[ \int_{0}^x t f(x-t) \, dt = 1 - \cos x. \]
Compute \( \int_0^{\pi /2} f(x) \, dx. \)

Answer: 1


Question: Given a quartic monic polynomial \(P(x) = x^4 + ax^3 + bx^2 + cx + d\), where \(a, b, c,\) and \(d\) are constants, and the conditions \(P(1) = 10\), \(P(2) = 20\), and \(P(3) = 30\), find the value of \(\frac{P(12) + P(-8)}{10}\).

Answer: 1984


Question: Compute the limit: \[ \lim_{n\to\infty}\sqrt[n]{\log\left|1+\left(\frac{1}{n\cdot\log n}\right)^k\right|}. \]

Answer: 1


Question: Calculate the mass of the sphere defined by \(x^2 + y^2 + z^2 = 1\) with a point density given by \(e^x + e^y + e^z\). Use Cartesian coordinates to set up and evaluate the triple integral.

Answer: \dfrac{12\pi}{e}


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

Answer: a(1 + 2a)


Question: Given a Brownian motion $X_t$ with $X_0 = 0$ and $X_1 > 0$, find the probability that $X_2 < 0$. Express your answer as a single value.

Answer: \dfrac{1}{4}


Question: Let \( \{X(t), t \geq 0 \} \) be a pure birth process with \( X(0)=1 \) and birth rates \( \lambda_k = k \) for \( k=1,2, \dots \). Let \( T=\inf \{t \geq 0: X(t)=4\} \). Find \( E\left(\int_{0}^{T}X(t)\,dt\right) \).

Answer: 3


Question: Find the smallest positive integer \( n \) such that \( 5000 n \log(n) \leq 2^{n/2} \).

Answer: 39


Question: Let $k$ be a field that is not algebraically closed, and let $n > 1$ be an integer. Does there exist a polynomial $f \in k[X_1, \ldots, X_n]$ such that the zero set $Z(f) = \{(a_1, \ldots, a_n) \in k^n : f(a_1, \ldots, a_n) = 0\} = \{0\}$?

Answer: Yes


Question: A random walk moves from position k to k+1 with probability 1/2 and to k-1 with probability 1/2, except when k=n, where it remains at n with probability 1/2 and moves to n-1 with probability 1/2. Suppose the random walk starts at position n. Let T be the first time the path reaches position 0. What is the expected value of T?

Answer: n(n+1)


Question: Let $G$ be a finite group and $F$ be a field. Consider a linear action $h: G \times F^n \to F^n$ defined by a group homomorphism $f: G \to GL(n, F)$ such that $h(g, v) = f(g)v$ for all $g \in G$ and $v \in F^n$. Is it true that there exists a positive integer $n$, a linear action $h$, and a non-zero vector $v \in F^n$ such that $h(g, v) = v$ for all $g \in G$? Provide a justification for your answer.

Answer: Yes


Question: Find the minimum value of the expression \( I = a_1^2 + a_2^2 + \cdots + a_n^2 - 2a_1a_2 - 2a_2a_3 - \cdots - 2a_{n-1}a_n - 2a_na_1 \), where \( a_1, a_2, \ldots, a_n \geq 0 \) and \( a_1 + a_2 + \cdots + a_n = 1 \). Assume \( a_{n+1} = a_1 \).

Answer: -\dfrac{1}{n}


Question: Find the unique positive integer \( n \) such that \( S_n \) is an integer, where \( S_n \) is the minimum value of the sum \[ \sum_{k=1}^n \sqrt{(2k-1)^2+(a_k)^2} \] and \( a_1, a_2, \ldots, a_n \) are positive real numbers whose sum is 17.

Answer: 12


Question: Find the greatest common divisor (GCD) of the polynomials \(f(x) = (x^2+1)(x-2)\) and \(g(x) = (x^3+7)(x-2)\) over the finite field GF(2).

Answer: x^2 + x


Question: Given the function $L(x) = x - \frac{x^2}{4}$, define $a_n$ as the result of applying $L$ iteratively $n$ times to $\frac{17}{n}$, i.e., $a_n = L(L(\cdots L(\frac{17}{n}) \cdots ))$ with $n$ iterations. Determine the limit of $n \cdot a_n$ as $n$ approaches infinity.

Answer: \dfrac{68}{21}


Question: Let $G$ be a finite simple group such that there is no element of order $p^2$ for any prime $p$ dividing $|G|$. Suppose $H$ is a finite group with the same number of nontrivial proper subgroups as $G$, and there exists a bijection $\phi$ from the set of nontrivial subgroups of $G$ to the set of nontrivial subgroups of $H$ that preserves subgroup inclusion. Determine if $G \cong H$. Provide a justification for your answer.

Answer: G \cong H


Question: Calculate the intersection number of the curves given by \((x^2+y^2)^3-4x^2y^2=0\) and \(x=0\) at the point \((0,0)\). The intersection number is defined as \(\dim_k(O_p/(F,G))\), where \(O_p\) is the local ring at the point \(P\).

Answer: 6


Question: Determine the range of Kendall's tau that can be used in the Frank copula.

Answer: (-1, 1)


Question:
Let $f:[0,1]\to \mathbb{R}$ be a differentiable function such that $f(0)=0$ and $|f'(x)| \le k|f(x)|$ for all $x \in [0,1]$, where $k>0$. Determine which of the following statements is always true:

(A) $f(x)=0$ for all $x \in \mathbb{R}$

(B) $f(x)=0$ for all $x \in [0,1]$

(C) $f(x) \ne 0$ for all $x \in [0,1]$

(D) $f(1) = k$

Answer: B


Question: Find the pointwise limit of the sequence \(f_n(x) = \frac{x^n}{n}\) on the interval \([0,1]\).

Answer: 0


Question: Evaluate the double integral: \[ \int_{1}^{\infty} \int_{1}^{\infty} \frac{1}{x^4y^2\ln (xy)} \, dx \, dy \]

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


Question: Let $q_k$ be the product of the first $k$ primes. Is the density of square-free numbers in the sequence $p_n + q_k$ strictly increasing with $k$ and higher than that of any other sequence $p_n + d$ where $0 \le d < q_k$?

Answer: Yes


Question: Find the positive integer value of \(n\) such that \(\pi^n\) is as close to an integer as possible.

Answer: 3


Question: Determine the number of directed graphs \( G = (V, E) \) with \( |V| = N \) nodes, where each node has an out-degree of 2, that contain a given directed cycle \( C = (V, D) \) of length \( L \) as a subgraph. The cycle \( C \) visits each node at least once, and \( |D| = L \). Find the number of such graphs \( G \) that contain exactly \( C \) as a subgraph, denoted as \( n \).

Answer: 0


Question: Let $A$ be a square matrix of dimension $n$, and let $\lambda \in \mathbb{C}$ be an eigenvalue of $A$. Denote $X_\lambda$ as the eigenspace of $A$ associated with $\lambda$. Under what condition do we have $AX_\lambda = \{0\}$? Provide the value of $\lambda$ that satisfies this condition.

Answer: 0


Question:
Consider the function \( f(x) = e^{2-x}x^2 \). Which of the following assertions is TRUE?

A: \( f \) has no relative extremes;
B: \( f \) restricted to \( ]-\infty, 0[ \) is increasing;
C: \( f \) has two points of inflection;
D: \( f \) has no asymptotes;
E: None of the previous answers is true.

Answer: C


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

Answer: 3


Question: Evaluate the limit: $$\lim_{N\to \infty}  \sum_{r=1}^N \frac{2^r}{(1+5^{2^r})}.$$

Answer: \dfrac{1}{12}


Question: Find the volume of the solid in \( \mathbb{R}^3 \) bounded by the surfaces \( y = x^2 \), \( x = y^2 \), \( z = x + y + 21 \), and \( z = 0 \).

Answer: \dfrac{73}{10}


Question: Let $G$ be an abelian group and $H$ a subgroup of $G$ such that $n(G/H)=0$ for some integer $n$. Does this imply that $H$ has finite index in $G$? Justify your answer.

Answer: No


Question: Calculate \( \lim_{x\to\frac{\pi}{2}^-} (1-\cos x)^{\tan x} \).

Answer: \dfrac{1}{e}


Question: Calculate the limit: $$\lim_{a\to \infty}\,\frac{\lfloor{\frac{{1+\sqrt{a}}}{2}}\rfloor+\lfloor{\frac{a}{2}}\rfloor}{a},$$ if it exists.

Answer: \dfrac{1}{2}


Question: Evaluate the limit \( \lim_{x\to\infty}\frac{x^{\log_2 x}}{(\log_2 x)^x} \).

Answer: 0


Question: Evaluate the integral \( \int_0^1 r(x) \, dx \), where \( r(x) \) is defined as follows: \[ r(x) = \begin{cases} 1 & \text{if } x \text{ is rational}, \\ 0 & \text{if } x \text{ is irrational}. \end{cases} \]

Answer: 0


Question: In triangle \(ABC\), \(\angle BAC = 18^\circ\) and \(\angle BCA = 24^\circ\). Point \(D\) is on \(AC\) such that \(\angle BDC = 60^\circ\). The bisector of \(\angle ADB\) meets \(AB\) at \(E\). Find \(\angle BEC\).

Answer: 30^\circ


Question: Find the smallest integer \(x\) such that \(2^x \equiv 166 \pmod{330}\).

Answer: 20


Question: Find the value of \( \frac{d^7y}{dx^7}\bigg|_{x=0} \) for the function \( y = \frac{\sin(x^2) - x^2}{x^3} \).

Answer: 42


Question: Consider a diffusion process defined by the stochastic differential equation \(dX_t = g(X_t,t) \, dt + \sigma \, dW_t\), which induces a path measure \(Q\) over the interval \([0,T]\). Let \(f\) be a bounded function. Determine whether the expectation \[ \left\langle \int^T_0 \frac{f(X_t)-g(X_t,t)}{\sigma^2} \, dW_t  \right\rangle_Q \] is constant or zero.

Answer: 0


Question: Does there exist a sequence \((x_n)\) belonging to \(\ell_1 \cap \ell_2\) which converges in one space but not the other?

Answer: Yes


Question:
Consider a genus two projective smooth complex curve $Y_0$ and an étale cover of degree 2, $Y_1$, with involution $
abla$ over $Y_0$. Let $J_1$ be the Jacobian of $Y_1$, and define the Prym variety $P$ as the image $(1-
abla)(J_1)$. This Prym variety $P$ is an elliptic curve. Examine the map $f$ induced by $1-
abla$ from $J_1[2]$ to $P[2]$. Can the map $f$ be surjective?

Answer: Yes


Question: Let $\alpha$ be a root of $x^3+3x-1$ and $\beta$ be a root of $x^3-x+2$. Determine the degree of the field extension $\mathbb{Q}(\alpha^2+\beta)$ over $\mathbb{Q}$. Provide your answer as a single integer.

Answer: 9


Question: Determine whether the function \( u(t) \) defined by \( u(t) = 0 \) for \( t \neq 0 \) and \( u(t) = \infty \) for \( t = 0 \) is integrable with respect to the Lebesgue measure.

Answer: Yes


Question: Determine the radius of convergence for the Taylor series of the function \( f(z) = \frac{1}{1+e^z} \) centered at \( z = 0 \).

Answer: \pi


Question: Given that \(a+b+c=10\), \(abc=2015\), and \(ab+bc+ca=1\), find the value of the expression \((bc-a^2)(ac-b^2)+(ac-b^2)(ba-c^2)+(ba-c^2)(bc-a^2)\).

Answer: -97


Question: What is the smallest dimension of a Lie algebra that is neither solvable nor reductive? Provide an example of such a Lie algebra.

Answer: 5


Question: Given that $a, b, c$ form a geometric sequence and $\\log_c a, \\log_b c, \\log_a b$ form an arithmetic sequence, the common difference of the arithmetic sequence can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 5


Question: Let \( G \) be a finite group and \( g, h \in G - \{1\} \) such that \( g^{-1}hg = h^2 \). Given that the order of \( g \) is 5 and the order of \( h \) is an odd integer, determine the order of \( h \).

Answer: 31


Question: Let $R$ be an integral domain and $P, Q$ be proper prime ideals of $R$. Let $R_P$ and $R_Q$ be the localizations of $R$ at $P$ and $Q$, respectively. If $R_P \subseteq R_Q$, does it follow that $Q \subseteq P$?

Answer: Yes


Question: Compute \( \lim_{x \to \frac{\pi}{2}} \{1^{\sec^2 x} + 2^{\sec^2 x} + \cdots + n^{\sec^2 x}\}^{\cos^2 x} \).

Answer: n


Question: Let $n \in \mathbb{N}^*$ and $k \in \{0, 1, 2, \ldots, n - 1\}$. Given the complex number \( z = \left( \cot \frac{(2k + 1)\pi}{2n} + i \right)^n \), find the real part of \( z \).

Answer: 0


Question: Evaluate the integral \( \oint_{\gamma} \frac{z}{\cos z} \, dz \), where \( \gamma \) is the circle \( |z| = 5 \) oriented in the counterclockwise direction.

Answer: 4\pi^2 i


Question: Find the correct bounds for the double integral of \( \sin(x) \) by reversing the order of integration for the region bounded by \( x \leq y \leq 2x \) and \( 1 \leq x \leq 5 \). The correct value of the double integral is \(-2.67\).

Answer: -2.67


Question: Determine if there exists an ellipse given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) that is congruent to its image under the shear transformation \( T(c) = \begin{pmatrix} 1 & 0 \\ c & 1 \end{pmatrix} \). If such an ellipse exists, specify the conditions on \( a \), \( b \), and \( c \).

Answer: c = 0


Question: Evaluate the integral \( \iint_{D} \frac{\mathrm{d}x\,\mathrm{d}y}{\sqrt{1 - x^{2} - y^{2}}} \), where \( D = \{(x,y) \mid x^{2} + 4y^{2} \leq 1\} \).

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


Question: Is the isometry group of a simply-connected Ricci-flat closed manifold finite?

Answer: Yes


Question: Determine the genus of the complete 4-partite graph $K_{4,2,2,2}$, where the vertex set is partitioned into 4 disjoint parts of sizes 4, 2, 2, and 2, and any two vertices are adjacent if and only if they belong to different parts.

Answer: 2


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

Answer: 1


Question:
Determine which of the following sets S makes the statement false: "There exists a continuous function f : S → S such that f(x) ≠ x for all x ∈ S." Choose the correct option:

(A) [2, 3]
(B) (2, 3]
(C) [−3, −2] ∪ [2, 3]
(D) (−∞, ∞)

Answer: A


Question: Let X be a Bernoulli random variable with success parameter p, where p is uniformly distributed over the interval (0,1). Suppose p is chosen, then two independent observations of X (call them X_1 and X_2) are made. Calculate the unconditional correlation coefficient between X_1 and X_2.

Answer: \dfrac{1}{3}


Question: Determine whether there exists an unbounded function that is uniformly continuous. Provide a justification for your answer.

Answer: Yes


Question: Determine whether every compact metric space can be continuously embedded into a Hilbert space, possibly with large distortion.

Answer: Yes


Question: In triangle $\triangle ABC$, $\angle C = 120^\circ$. Point $M$ is located on side $AB$ such that $\angle CMB = 60^\circ$ and the ratio $BM:AB = 1:3$. Determine the measure of $\angle B$.

Answer: 45^\circ


Question: Evaluate the integral \( \int_0^{2 \pi } \frac{ R^2 - r^2 }{R^2-2rR \cos( \phi - \theta ) + r^2 } d\theta \) and show that it equals \( 2 \pi \) for any \( R > r \) and \( \phi \).

Answer: 2\pi


Question: Verify the calculation of the Alexander polynomial for the unknot using the determinant of the sum of a Seifert matrix and its transpose, as described in Lickorish's book "Introduction to Knot Theory." Is the determinant of a (0x0)-matrix equal to 1, and does this correctly yield \( \Delta_{\text{unknot}} = 1 \)?

Answer: 1


Question: Find the exact length of the polar curve given by \( r = 1 - \cos(\theta) \) for \( 0 \leq \theta \leq 2\pi \).

Answer: 8


Question: Evaluate the limit: \[ \lim_{x\rightarrow \frac{\pi}{4}}\left(\frac{\sin x}{\cos x}\right)^\left(\frac{\sin 2x}{\cos 2x}\right) \]

Answer: \dfrac{1}{e}


Question: Find the coefficient of \(x^{17}\) in the expansion of \((1+x^5+x^7)^{20}\). Choose the correct answer from the following options: (A) 3400 (B) 3410 (C) 3420 (D) 3430 (E) 3440.

Answer: C


Question: Evaluate the definite integral: $$\int_0^1 x\arcsin\left(\sin\left(\pi x\right)\right) dx$$

Answer: \dfrac{\pi}{8}


Question:
Let $(X,d)$ be a separable compact metric space and fix $c > 1$. Does there exist a Borel probability measure $\nu$ on $(X,d)$ such that:

1. $\sup_{x \in X, r > 0} \frac{\nu(\mathrm{Ball}(x, cr))}{\nu(\mathrm{Ball}(x, r))} < \infty$,
2. $\nu(\mathrm{Ball}(x, r)) > 0$ for every $x \in X$ and $r > 0$?

Provide a justification for your answer.

Answer: Yes


Question: Determine whether the matrix $(I \circ A - I \circ B)$ is positive semi-definite, given that $A$ is a positive definite matrix, $B$ is a positive semi-definite matrix, and $A - B$ is positive semi-definite. Justify your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty} \frac{1}{n}\left[\log\left( \frac{\sqrt{2\pi n} (x-p) - p(1-p) e^{-\frac{n}{2}(\frac{x-p}{p(1-p)})^2}}{\sqrt{2\pi n} (x-p)}  \right)\right] \]

Answer: 0


Question: Does there exist a function \( f: [0,1] \rightarrow \mathbb{R} \) that is almost everywhere zero but has a range equal to \( \mathbb{R} \)?

Answer: Yes


Question:
Let $f$ be a real-valued function on $\mathbb{R}^3$ satisfying $f(rx) = r^\alpha f(x)$ for any $r > 0$ and $x \in \mathbb{R}^3$, where $\alpha \in \mathbb{R}$ is fixed. Determine which of the following statements is true:

A. If $f(x) = f(y)$ whenever $\|x\| = \|y\| = \beta$ for some $\beta > 0$, then $f(x) = \beta \|x\|^\alpha$.
B. If $f(x) = f(y)$ whenever $\|x\| = \|y\| = 1$, then $f(x) = \|x\|^\alpha$.
C. If $f(x) = f(y)$ whenever $\|x\| = \|y\| = 1$, then $f(x) = c \|x\|^\alpha$ for some constant $c$.
D. If $f(x) = f(y)$ whenever $\|x\| = \|y\| = 1$, then $f$ must be a constant function.

Answer: C


Question: Compute the value of \( \int_{S} \vec{F} \cdot \hat{n} \, dS \) where \( S = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + 2z = 2, z \geq 0\} \) and \( \vec{F} = \langle y, xz, x^2 + y^2 \rangle \). Here, \( \hat{n} \) is the outward unit normal to the surface \( S \).

Answer: 2\pi


Question: If \(c_1\) and \(c_2\) are two different piecewise differentiable simple closed curves in \(\mathbb{R}^2\), is the number of points in the intersection of these curves necessarily a finite set?

Answer: No


Question: Find the least positive real number $k$ such that \( 7\sqrt{a} + 17\sqrt{b} + k\sqrt{c} \ge \sqrt{2019} \) for all positive real numbers \( a, b, c \) with \( a + b + c = 1 \).

Answer: 41


Question:
Compute the covariance \( \text{Cov}(X, Y) \) for the random variables \( X \) and \( Y \) with the joint probability density function given by:
\[
f(x,y) = \begin{cases}
\left(\frac{2e^{-2x}}{x}\right) & \text{, $0 < y < x <  ∞$} \\
0 & \text{, elsewhere}
\end{cases}
\]

Answer: \dfrac{1}{8}


Question: Given the set of numbers \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \), how many ways can you choose 5 numbers such that their sum is 22?

Answer: 11


Question:
Consider \(\mathbb{R}^2\) with the \(||.||_1\) norm and the subspace \(M=\{(x,0):x\in \mathbb{R} \}\). Define the linear functional \(g:M \to \mathbb{R}\) by \(g(x,0)=x\). Which of the following is the Hahn-Banach extension \(f\) of \(g\) to \(\mathbb{R}^2\)?

- \(f(x,y)=2x\)
- \(f(x,y)=x-2y\)
- \(f(x,y)=x+2y\)
- \(f(x,y)=x+y\)

Answer: D


Question: Let $X$ be a Banach space over $\mathbb{C}$, and let $A$ and $B$ be closed subspaces of $X$ such that $X = A \oplus B$. Assume that $\|a + b\| = \|a\| + \|b\|$ for each $(a, b) \in A \times B$. Does there exist a unique projection $T$ on $X$ such that $\text{im}(T) = A$?

Answer: Yes


Question: Is it true that the distributive number \( \mathfrak{h} \) is the least cardinal \( \kappa \) such that there exist \( \kappa \) open dense subsets in \( ([\omega]^\omega,\subset^*) \) whose intersection is not dense?

Answer: Yes


Question: Evaluate the line integral \( \int_D x\ln(y) \, dx \) over the region \( D \) defined by \( 1 \leq x \leq 2 \) and \( e^x \leq y \leq e^{x^2} \) using Green's Theorem. The answer is given as \( \frac{-17}{12} \). Verify this result.

Answer: -\dfrac{17}{12}


Question: Find the approximate variance of the angle \(\theta\) from the origin to the point \((X, Y)\), where \(X - \mu_X\) and \(Y - \mu_Y\) are independent and identically distributed as \(N(0, \sigma^2)\). Assume that after transforming \((X, Y)\) to \((X', Y')\) by rotating onto the positive x-axis and scaling such that \(\sigma' = 1\), we have \(X' \sim N(r, 1)\) and \(Y' \sim N(0, 1)\), where \(r = \frac{\sqrt{\mu_X^2 + \mu_Y^2}}{\sigma}\). Consider the variance of the angle's distribution from \(-\pi\) to \(\pi\) for \(r > 2\).

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


Question:
Find the smallest natural number \( n \) such that there exists a 4-digit number \( \overline{abcd} \) satisfying the equation:
\[
\overline{abcd} + \frac{\overline{abcd}}{6} + \frac{\overline{abcd}}{6^2} + \cdots + \frac{\overline{abcd}}{6^n} = \frac{6^{n+1} - 1}{5}
\]

Answer: 4


Question: If $p,q$ are coprime polynomials in $\Bbb C[X]$ and $a\in\Bbb C$, does the following hold: If the equation \( \frac{p(x)}{q(x)}=a \) has a multiple solution, can we arbitrarily move $a$ a little to make the solution simple?

Answer: Yes


Question: Let $f : \mathbb{R}\to\mathbb{R}$ be a function that is continuous at $0$ with $f(0) = 1$. The function satisfies the equation \(f(x) - f\left(\frac{x}{2}\right) = \frac{3x^2}{4} + x\) for all $x$. Determine the value of $f(3)$. \(\boxed{\text{Your answer here}}\)

Answer: 16


Question: Suppose we have a $C^2(\mathbb{R}^2, \mathbb{R})$ function $f$ with globally bounded first and second derivatives. Additionally, $f(x, 0) = f(0, y) = 0$ for any $x, y \in \mathbb{R}$. Determine if there exists a constant $C > 0$ such that for any $(x, y) \in \mathbb{R}^2$, the inequality $|f(x, y)| \leq C|xy|$ holds.

Answer: Yes


Question: Find the coefficient of $x^{21}$ in the expansion of \( \left(\frac{1-x^{10}}{1-x}\right)^3 \).

Answer: 28


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\delta(y-x)\delta(y-z)\,dy \).

Answer: \delta(x - z)


Question: If $V$ is a free module and also a finite dimensional module, and $V$ is simple, must $V$ have dimension 1? Justify your answer.

Answer: Yes


Question:
Calculate the product of the following cycle permutations and verify the result:

$$(x_d,x_a,x_c)^{-1}(x_c,x_e,x_b)^{-1}(x_d,x_a,x_c)(x_c,x_e,x_b).$$

The expected result is $$(x_a,x_b,x_c).$$

Answer: (x_a, x_b, x_c)


Question: Calculate the volume of the region bounded by the surfaces $z + x^2 = 1$, $z + y^2 = 1$, and the coordinate planes $x = 0$, $y = 0$, $z = 0$. Express your answer as a single value.

Answer: \dfrac{1}{2}


Question: Determine the largest natural number \( m \) such that the polynomial \( P(x) = x^5 - 3x^4 + 5x^3 - 7x^2 + 6x - 2 \) is divisible by \((x-1)^m\).

Answer: 3


Question: Let $X$ and $Y$ be two independent random variables with densities. Let $c \in \mathbb{R}$ be a constant. Determine whether the conditional distribution of $X+Y$ given $X=c$ is the same as the distribution of $c+Y$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the infinite series: \[ \sum_{k=1}^{\infty} (-1)^{k+1}\frac{\pi^{6k}}{(6k)!} \]

Answer: \dfrac{4}{3}


Question: Find \( \lim_{n\to\infty}\int_0^{\frac{1}{2^n}}(1+\cos(x))^n\,dx \).

Answer: 1


Question: Find the limit superior of the sequence \( \{a_n\} \), where \( \{a_n\} \) is an enumeration of the rational numbers in the interval \([0,1]\).

Answer: 1


Question: Determine the Galois group of the polynomial $x^{4}-4x^{2}+5$ over the field $\mathbb{Q}$. Provide a detailed explanation of your reasoning.

Answer: D_4


Question: Evaluate the following limit: $$\lim_{n\to\infty}\int_0^2f(n-x)dx$$ where $f$ is continuous on $[0, \infty)$ and $\lim_{x \to\infty}f(x)=L$. Can the limit be moved inside the integral?

Answer: 2L


Question: Let $f: X \to Y$ be an open immersion of schemes, where $X$ is Noetherian. Is $Y$ the scheme theoretic image of $f$?

Answer: No


Question: Evaluate the limit \( \lim_{x \to 0} \frac{(\sin(x-x^2))^2 - x^2 + 2x^3}{\sin^{4}(2x)} \) using Taylor series.

Answer: \dfrac{1}{24}


Question: Calculate the projection of the function \( f(x) = \frac{1}{x} \) onto the subspace \( W = \text{span}\{x\} \) in the vector space \( C^0[1,3] \) with the usual scalar product defined by the integral.

Answer: \dfrac{3}{13}x


Question: Let $A$ be an $n \times n$ real matrix. Suppose that each $e_i$ ($1 \leq i \leq n-1$) is in the direct sum of all eigenspaces of $A$ corresponding to eigenvalues with positive real part, where $e_i$ denotes the vector with $i$th component $1$, and any other component $0$. If a sufficiently large positive number is added to the $(n,n)$th entry $a_{nn}$ of $A$, can the resulting matrix be assured to have only eigenvalues with positive real parts?

Answer: Yes


Question: Let $Q \in M_{10\times 10}(\mathbb{R})$ be a matrix of rank 5. Define the linear transformation $T: M_{10\times 10}(\mathbb{R}) \to M_{10\times 10}(\mathbb{R})$ by $T(P) = QP$. What is the rank of $T$?

Answer: 50


Question: Let \( f :[0,1]\to \mathbb{R} \) be a function such that \( f'(x) = \sqrt{1+f^2(x)} \) for all \( x \in [0,1] \). Given that \( f(0) + f(1) = 0 \), calculate the integral \[ I=\int_{0}^{1}f(x)\,dx. \]

Answer: 0


Question: Let $X$ be a random variable with cumulative distribution function $F$ and finite expectation, $\mathbb{E}(X) < \infty$. Prove that \( \int_M^\infty (1-F(x))\,dx \rightarrow 0 \) as \( M\rightarrow\infty \).

Answer: 0


Question: A bag contains two coins: Coin A, which lands heads with a probability of 90%, and Coin B, which lands tails with a probability of 90%. One coin is randomly selected from the bag and tossed 10 times, resulting in 10 heads. Based on this outcome, if the same coin is tossed 10 more times, what is the expected number of heads?

Answer: 9


Question: Calculate the sum of the infinite series: \[ \sum_{n=0}^{\infty} \frac{n^2 - 2}{n!} \]

Answer: 0


Question: Determine the region of convergence for the power series \( \sum_{n=1}^{\infty} a_{n}(x-1)^{n} \), given that the sequence \( \{a_{n}\} \) decreases monotonically, \( \lim_{n \rightarrow \infty} a_{n}=0 \), and the series \( S_{n}=\sum_{k=1}^{n} a_{k} \) diverges.

Answer: [0, 2)


Question: Does every field \( \mathbb{K} \) have a cubic extension? Provide a justification for your answer.

Answer: No


Question: Determine the familiar space that is homeomorphic to the result of gluing a Möbius band to the boundary of a disk.

Answer: \mathbb{RP}^2


Question: A number $P$ is randomly chosen from the uniform distribution on the interval [0,1]. A coin with probability $P$ of landing heads is flipped $n$ times. Let $X$ be the number of heads obtained. Compute the probability $P(X=k)$ for a given integer $k$.

Answer: \dfrac{1}{n+1}


Question:
Let \( A \in M_{3\times3} \) and \( x, y, z \in \mathbb{R}^3 \) be linearly independent vectors. Given that:
\[ Ax = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \quad Ay = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad Az = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \]
Find the determinant of the matrix \( A \).

Answer: 0


Question: Given the survival function \( S(x) = 1 - \frac{x}{120} \) for \( x \in [0, 120] \), determine the premium intensity \( H \) using the equivalence principle. Thomas, currently 30 years old, will have his family receive $1,000,000 with probability 0.2 if he dies. At the beginning of each year, Thomas pays a premium \( H \) with probability 0.8. The discount factor is \( v = \frac{1}{1.06} \). Use the formula \( _kp_x = \frac{S(k+30)}{S(30)} \) to calculate the premium intensity \( H \). The answer should be \( H = 3195 \).

Answer: 3195


Question: Given the function \(f(x) = a(x-2)(x-b)\), where \(a, b \in \mathbb{R}\) and \(a \neq 0\), and the condition that \(f(f(x)) = a^3\left(x^2-(2+b)x+2b-\frac{2}{a}\right)\left(x^2-(2+b)x+2b-\frac{b}{a}\right)\) has exactly one real root at \(x = 5\), find the minimum and maximum values of \(f(x)\).

Answer: 2


Question: Find the minimum value of the function $xe^x$ without using calculus. Assume the limit or series definition of $e$. What is the minimum value?

Answer: -\dfrac{1}{e}


Question: Expand the function \( \frac{1}{\log\left( \frac{z+1}{z-1} \right)} \) at infinity and find the first term of the expansion.

Answer: \dfrac{z}{2}


Question: Find the value of \( \lim_{n \to \infty} (\arctan (n+1) - \arctan (n)) \).

Answer: 0


Question:
Solve the system of equations for real numbers $x$ and $y$:
\[ \begin{align*} x^3 - y^3 &= (y + 1)x^2 - (x - 1)y^2, \\ x^2 + 4\sqrt{y + 4} &= 2x + y + 7. \end{align*} \]

Answer: (1, 0)


Question: Determine whether the following inequality holds: \[ \inf_{\begin{array}{c} v_{1},v_{2},v_{3},v_{4}\in\mathbf{D},\\ v_{0}\in\mbox{convexhull}\left(v_{1},v_{2},v_{3},v_{4}\right) \end{array}}\max_{0\le i,j,k\le4}\frac{\mbox{perimeter}\left(\triangle v_{i}v_{j}v_{k}\right)}{\mbox{area}\left(\triangle v_{i}v_{j}v_{k}\right)} \] is no less than \[ \min_{v_{0},v_{1},v_{2},v_{3},v_{4}\in\partial\mathbf{D}}\max_{0\le i,j,k\le4}\frac{\mbox{perimeter}\left(\triangle v_{i}v_{j}v_{k}\right)}{\mbox{area}\left(\triangle v_{i}v_{j}v_{k}\right)}? \]

Answer: Yes


Question: Determine the order of the poles for the function \( f(z) = \frac{3z - 1}{z^2 + 2z + 5} \).

Answer: 1


Question: Calculate the line integral \( \int_\gamma f \, dx \) for the vector field \( f(x) = (x_1^2 + 5x_2 + 3x_2x_3, 5x_1 + 3x_1x_3 - 2, 3x_1x_2 - 4x_3) \), where \( \gamma \) is the helix with radius 1 and pitch 1 from \((0, 1, 0)\) to \((0, 1, 2\pi)\). Use the parameterization \( \gamma(t) = (\cos{t}, \sin{t}, t) \) for \( t \in [0, 2\pi] \).

Answer: -8\pi^2


Question: Find the coefficient of \(x^{18}\) in the expansion of \((x + x^2 + x^3 + x^4 + x^5)(x^2 + x^3 + x^4 + \cdots)^5\).

Answer: 771


Question: Find a generator of the multiplicative group of integers modulo 27, denoted as \( Z_{27}^* \).

Answer: 2


Question: Determine the range of the function \( f(x) = \arccos \frac{2x}{1+ x^2} \) without using the continuity of the \( \arccos x \) function.

Answer: [0, \pi]


Question: In how many ways can 24 bullets be distributed among four burglars such that each burglar receives at least three bullets but no more than eight bullets?

Answer: 125


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k = 0}^{n} \frac{\binom{n}{k}}{n2^n+k} \]

Answer: 0


Question: Find the linear function \(g(x) = mx + b\) that is tangent to the graph of \(f(x) = x^4 - 6x^3 + 13x^2 - 10x + 7\) at two distinct points.

Answer: g(x) = 2x + 3


Question: Evaluate the contour integral \(\int_{|z-i|=\frac{1}{9}} \frac{1}{z^4+16} \, dz\).

Answer: 0


Question: Let $A$ be a 3x3 real symmetric matrix, and let $v_1$ and $v_2$ be orthogonal eigenvectors of $A$. If $v_3$ is orthogonal to both $v_1$ and $v_2$, is $v_3$ necessarily an eigenvector of $A$? Provide a justification for your answer.

Answer: Yes


Question: Consider two non-negative sequences $f(n)$ and $g(n)$. Suppose there exists a constant $C > 0$ such that for every $\varepsilon > 0$, there exists an integer $N$ satisfying \( \forall n \geq N, \ f(n) < C \cdot g(n) + n^{\varepsilon}\varepsilon. \) Determine if this implies that \( f(n) = O(g(n)) \) as \( n \to \infty. \)

Answer: No


Question: Evaluate the integral \( \int_{|z|=5} \frac{z^2}{(z-3i)^2} \, dz \).

Answer: -12\pi


Question:
Find the set of all limit points of the set \( S = \left\{ n + \frac{1}{3m^2} ;\, n, m \in \mathbb{N} \right\} \). Choose the correct option:
(a) \( \mathbb{Q} \)
(b) \( \mathbb{Z} \)
(c) \( \mathbb{R} \)
(d) \( \mathbb{N} \)

Answer: D


Question: Determine the number of real solutions to the equation \( 2017^x = x^{2017} \).

Answer: 2


Question: Let $E$ be the splitting field of $x^4+x+1$ over $\mathbb{Z}_2$. Determine the structure of the Galois group $\mathrm{Gal}(E/\mathbb{Z}_2)$. Provide your answer as a description of the group.

Answer: \mathbb{Z}_4


Question: Find all positive integer pairs \((n, k)\) such that \(\binom{n-1}{k+1} = \binom{n+1}{k-1}\).

Answer: (3, 1)


Question: Determine the number of solutions to the equation \(6\ln(x^2+1)-e^x = 0\).

Answer: 3


Question: Find the locus of the point $P$ such that the sum of the angles $Q_1$ and $Q_2$, made by the tangents from $P$ to the parabola $y^2 = 4x$ with the axis, is $45^{\circ}$. Provide your answer as a geometric locus.

Answer: y = x - 1


Question: Let $\mathcal{H}$ be a Hilbert space. Given a unitary operator $U$ and a projection operator $A$ on $\mathcal{H}$, where $U$ satisfies $UU^{*}=U^{*}U = I$ and $A$ satisfies $A^{*}=A=A^{2}$, determine whether $UA=AU$ holds.

Answer: UA \neq AU


Question: Determine whether there exists a function $f(x)$ such that for every positive integer $n$, the $n$-th derivative of $f$ at $x=0$ is $f^{(n)}(0) = \frac{1}{n}$. Provide a justification for your answer.

Answer: Yes


Question: Given the equation \( \log_{4n} 40 \sqrt{3} = \log_{3n} 45 \), find the value of \( n^3 \).

Answer: 75


Question: Compute the limit: \[ \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{\left\lfloor \frac{n}{2} \right\rfloor} \cos \left(\frac{k \pi}{n}\right) \]

Answer: \dfrac{1}{\pi}


Question: Evaluate the integral \( \int_{-\infty }^{\infty }|x| \delta \left(x^2\right) \, dx \).

Answer: \dfrac{1}{2}


Question: Let \( f(x, y) \) be a real-valued function such that \( f(1, 1) = 1 \) and its gradient at this point is \( \nabla f(1, 1) = (-4, 5) \). Define a function \( g(t) \) by \( g(t) = f(t, f(t^2, t^3)) \). Find the value of \( g'(1) \).

Answer: 31


Question: Let $\mathbb{F}$ be a finite field and $X$, $Y$ be two algebraic varieties over $\mathbb{F}$. Let $R_1$ and $R_2$ be the coordinate rings of $X$ and $Y$, respectively. If $R_1$ and $R_2$ are not isomorphic, can we conclude that $X$ and $Y$ do not have the same number of points?

Answer: No


Question: Evaluate the multiple integral \( \idotsint_{[0,1]^n} \max \{u_1,u_2,...,u_n\} \,du_1 \dots du_n \).

Answer: \dfrac{n}{n+1}


Question: Determine whether the expression \( \lim_{n \to \infty} \frac{2}{n^2} \sum_{i=1}^{n}{\sqrt{n^2 - i^2}} \) converges or diverges.

Answer: \dfrac{\pi}{2}


Question:
Find the values of $x$ that satisfy the equation:
\[ \sin(x) = x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5 \]

Answer: 0


Question:
Given a $5 \times 5$ matrix $A \in M_5(\mathbb{R})$ that satisfies the equation $A^2 - 4A - I = 0$, where $I$ is the identity matrix, let $a_1, a_2, a_3, a_4, a_5$ be the eigenvalues of $A$. Calculate the value of the expression:
\[ \left(a_1 - \frac{1}{a_1}\right) + \left(a_2 - \frac{1}{a_2}\right) + \cdots + \left(a_5 - \frac{1}{a_5}\right) \]

Answer: 20


Question:
Find an efficient method to derive the coefficients $K_p$ in the power series expansion of the function:
$$\frac{1-z^2}{1+z^2-2z\cos(\theta)} = \sum_{p=0}^{\infty}K_pz^p$$
where $\theta$ is a constant. Verify that $K_p = 2\cos(p\theta)$ for $p \geq 1$.

Answer: K_p = 2\cos(p\theta)


Question: Given $X \sim \text{Binomial}(n, p)$ and $Y \sim \text{Bernoulli}\left(\frac{X}{n}\right)$, find the conditional expectation $E[X \mid Y]$. Express your answer in terms of $n$, $p$, and $Y$. \( \text{Note: } Y \text{ is a Bernoulli random variable whose parameter depends on } X. \)

Answer: Y + (n - 1)p


Question: Determine the number of index 3 normal subgroups of the group given by the presentation \( \langle a, b, c \mid aba^{-1}b^{-1}c^2 = 1 \rangle \).

Answer: 4


Question: Determine all functions \( f \) such that \( f(x+1) = 2f(x) \) for all real numbers \( x \).

Answer: f(x) = 2^x \cdot g(x)


Question:
Find the coefficient of the term $x_1^2x_2^2x_3^2\cdots x_{12}^2$ in the expansion of the polynomial:
\[(x_1-x_2)(x_1-x_3)(x_1-x_4)(x_1-x_{10})(x_2-x_3)(x_2-x_5)(x_2-x_{11})(x_3-x_6)(x_3-x_{12})(x_4-x_5)(x_4-x_6)(x_4-x_7)(x_5-x_6)(x_5-x_8)(x_6-x_9)(x_7-x_8)(x_7-x_9)(x_7-x_{10})(x_8-x_9)(x_8-x_{11})(x_9-x_{12})(x_{10}-x_{11})(x_{10}-x_{12})(x_{11}-x_{12})\]

Answer: 0


Question:
Evaluate the limit:
\[ L = \lim_{n \to 0} \left( \frac{2}{1}\left(\frac{3}{2}\right)^2\left(\frac{4}{3}\right)^3\cdots\left(\frac{n+1}{n}\right)^n \right)^{1/n} \]
Choose the correct answer from the following options: \( e \), \( \pi \), \( \frac{1}{e} \), \( \frac{1}{\pi} \).

Answer: e


Question: Let $X$ be the number of tosses of a fair coin required to get the first head. If $Y | X = n$ is distributed as Binomial$(n, \frac{1}{2})$, what is $P(Y = 1)$?\n(A) \frac{4}{9} \n(B) \frac{1}{4} \n(C) \frac{1}{3} \n(D) \frac{5}{9}

Answer: A


Question: Find the smallest natural number \( n > 1 \) such that for any set of \( n \) natural numbers, there exist two numbers \( a \) and \( b \) for which \( a^2 - b^2 \) is divisible by 595.

Answer: 109


Question:
Consider the function \( f: \mathbb{R}^2 \to \mathbb{R} \) defined by
\[
f(x,y) =
\begin{cases}
\dfrac{x^2y}{x^4+y^2},  & \text{if \((x,y)\neq(0,0)\)}, \\
0, & \text{if \((x,y)=(0,0)\)}
\end{cases}
\]
Determine which of the following statements is true regarding the continuity and existence of partial derivatives of \( f \) at \((0,0)\):

(a) Both partial derivatives of \( f \) exist at \((0,0)\) and \( f \) is continuous at \((0,0)\).

(b) Both partial derivatives of \( f \) exist at \((0,0)\) and \( f \) is NOT continuous at \((0,0)\).

(c) One partial derivative of \( f \) does not exist at \((0,0)\) and \( f \) is continuous at \((0,0)\).

(d) One partial derivative of \( f \) does not exist at \((0,0)\) and \( f \) is NOT continuous at \((0,0)\).

Answer: B


Question: Compute the integral \( \int_{0}^{\infty} \frac{x}{x^{4}+1} \, dx \) using complex analysis.

Answer: \dfrac{\pi}{4}


Question: Given a continuous random vector with joint density \( f_{X,Y}(x,y) = 2(x + y) \) for \( 0 < x < y < 1 \), find \( \mathbb{E}(XY) \).

Answer: \dfrac{1}{3}


Question: Find the smallest natural number $n$ such that the group $\mathbb{Z}_{6} \times \mathbb{Z}_{20} \times \mathbb{Z}_{45}$ is isomorphic to the direct product of $n$ cyclic groups.

Answer: 2


Question: In a finite-dimensional complex Hilbert space \( \mathcal{H} \), is every isometric map also a symmetry transformation? Justify your answer.

Answer: Yes


Question:
Determine which of the following properties can be satisfied by a map \( f: \mathbb{Z} \rightarrow \mathbb{Q} \):

1. \( f \) is bijective and increasing.
2. \( f \) is onto and decreasing.
3. \( f \) is bijective and satisfies \( f(n) \geq 0 \) if \( n \leq 0 \).
4. \( f \) has an uncountable image.

Answer: 3


Question: Determine the number of relations on a set $A$ with $|A| = n$, where the relations are reflexive, symmetric, transitive, and anti-symmetric.

Answer: 1


Question: Find the expected distance between two randomly chosen points on a circle of radius 1, where the points are selected independently from the interval $[0, 2\pi]$. Treat these points as forming a chord on the circle.

Answer: \dfrac{4}{\pi}


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

Answer: 0


Question: Approximate the definite integral \( \int_{0}^{1} x^{2} \sin(x^{4}) \, dx \) using a power series such that the error is less than 0.001.

Answer: 0.132


Question:
Let \( f:[0,1] \to [0,\infty) \) be a continuous function such that \( \int\limits_0^x f(t) \, dt \ge f(x) \) for all \( x \in [0,1] \). Determine which of the following statements is true:

A. No such function exists
B. There are infinitely many such functions
C. There is only one such function
D. There are exactly two such functions

Choose the correct option.

Answer: C


Question: Calculate the value of the following sum: \[ \sum^{10}_{k=1} \left( \sin\left(\frac{2k\pi}{11}\right) + i\cos\left(\frac{2k\pi}{11}\right) \right) \]

Answer: -i


Question: Evaluate the limit: $$\lim_{x\to \infty}\left(\dfrac{x+15}{x+3}\right)^{\large{\ln\left(\sin{\frac{1}{x}}\right)}}$$

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} n \sin(2\pi n!e) \]

Answer: 2\pi


Question: Find the y-coordinate of the center of gravity of the arc of the astroid $x^{2/3} + y^{2/3} = a^{2/3}$ in the first quadrant. Use the formula for the center of mass of an arc, $y_{cm} = \frac{\int y \, dl}{\int dl}$, where $dl$ is the differential arc length. Verify if the y-coordinate of the center of gravity is $\frac{2}{5}a$. Provide your answer in the form of a single value or expression.

Answer: \dfrac{2}{5}a


Question: Let $\mathcal{F}$ be the family of all analytic functions $f: D_1(0) \to D_1(0)$ such that $f(\frac{i}{3}) = 0$. Determine the supremum $M = \sup\{\operatorname{Im} f(0) : f \in \mathcal{F}\}$.

Answer: \dfrac{1}{3}


Question: Find all prime numbers \( p \) such that \( 2p^4 - p^2 + 16 \) is a perfect square.

Answer: 3


Question: Find all entire functions \( f \) of finite order such that \( f(\log(n)) = n \) for all positive integers \( n \).

Answer: e^z


Question: Determine the number of real roots of the equation: \[ x^{13} + x - \frac{1}{e^x} - \sin(x) = 0 \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{n^3}{(3n)!^{\frac{1}{n}}} \]

Answer: \dfrac{e^3}{27}


Question: Determine the order of the group of automorphisms of \( \mathbb{Z}_{15} \times \mathbb{Z}_3 \).

Answer: 192


Question: Prove that in any clique with 17 vertices, where the edges are colored with 3 different colors, there must exist a monochromatic triangle.

Answer: 17


Question: Evaluate the integral \( \int_0^\infty \frac{\sin s \cos s}{s} \, ds \) given that \( \int_0^\infty \frac{\sin s}{s} \, ds = \frac{\pi}{2} \).

Answer: \dfrac{\pi}{4}


Question: Find all real-valued functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the equation \( f(2x+f(y))+f(f(y))=4x+8y \) for all real numbers \( x \) and \( y \).

Answer: f(x) = 2x


Question: Evaluate the limit of the integral: $$\lim_{n\to \infty} \int_0^{\infty} \frac{\cos(nx)}{1+x^2} \, dx$$ without using complex variable theorems (i.e., residues).

Answer: 0


Question: Evaluate the limit \( \lim_{h \to 0} \int \vert f(x+h) - f(x) \vert^p \, dx \) for a function \( f \in L^p \), where \( p \in [1, \infty) \).

Answer: 0


Question:
Evaluate the determinant:
\[\begin{vmatrix}
a_1 & a_{51} & a_{101} \\
a_2 & a_{52} & a_{102} \\
a_3 & a_{53} & a_{103} \\
\end{vmatrix}\]
where \(a_n = \int^{\frac{\pi}{2}}_0 \frac{\sin^2(nx)}{\sin^2(x)} \, dx\).

Answer: 0


Question: Determine the first cohomology group \( H^{1}(\mathbb{C}^{n}\setminus\{0\},\pi^{*}\mathcal{T}_{\mathbb{CP}^{n-1}}) \), where \( \pi:\mathbb{C}^{n}\setminus\{0\}\rightarrow\mathbb{CP}^{n-1} \) is the projection from affine space without the origin to the projective space, and \( n \geq 3 \).

Answer: 0


Question: Compute the 2-adic expansion of \( \frac{2}{3} \).

Answer: \overline{10}110


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

Answer: 0


Question: Does there exist a real-valued function \( f \) that is smooth (infinitely differentiable) but not analytic at any point \( x \in \mathbb{R} \)?

Answer: Yes


Question: Compute the genus of the curve given by the intersection \( \{X^4+Y^4+Z^4=0\} \cap \{X^3+Y^3+(Z-tW)^3=0\} \subset \mathbb{P}^3 \). Use the adjunction formula and the fact that \( K_{\mathbb{P}^3}=O(-4) \).

Answer: 19


Question: Find one integer solution to the Diophantine equation \(18x + 14y + 63z = 5\).

Answer: (3, 1, -1)


Question: Find the minimum value of \(P = \left(3+\frac{1}{a}+\frac{1}{b}\right)\left(3+\frac{1}{b}+\frac{1}{c}\right)\left(3+\frac{1}{c}+\frac{1}{a}\right)\) given that \(a, b, c\) are positive numbers satisfying \(a+b+c \leq \frac{3}{2}\).

Answer: 343


Question: Can every vector space, regardless of dimension or underlying field, be equipped with a norm to become a normed vector space?

Answer: Yes


Question: Find the conditional expectation \(E(2\xi_1 \mid \xi_1, \xi_1+\xi_2)\), where \(\xi_1\) and \(\xi_2\) are independent identically distributed random variables with \(P(\xi=1)=P(\xi=-1)=1/2\).

Answer: 2\xi_1


Question: Determine the degree of the splitting field of the polynomial \( f(x) = x^4 - x^2 + 4 \) over \( \mathbb{Q} \).

Answer: 4


Question:
Calculate the expected number of steps to transition from state 3 to state 1 in a Markov chain with the following transition matrix:
\[
\begin{pmatrix}
1 & 0 & 0 \\
0.5 & 0.25 & 0.25 \\
0.5 & 0.25 & 0.25
\end{pmatrix}
\]

Answer: 2


Question:
Consider the sequence \( x_n = \left(1 - \frac{1}{n}\right) \sin \left(\frac{n\pi}{3}\right) \) for \( n \ge 1 \). Determine the limit inferior \( l \) and limit superior \( s \) of this sequence. Choose the correct option:

- \(-\frac{\sqrt{3}}{2} \le l < s \le \frac{\sqrt{3}}{2}\)
- \(\frac{1}{2} \le l < s \le \frac{1}{2}\)
- \(l = -1, s = 1\)
- \(l = 0 = s = 0\)

Answer: -\frac{\sqrt{3}}{2} \le l < s \le \frac{\sqrt{3}}{2}


Question:
Let $S \subset \mathbb{R}^{2}$ be an infinite collection of non-collinear points with the following properties:

1. For any $a \in \mathbb{R}$, $|S \cap \{ (x,a) : x \in \mathbb{R} \}| = |S \cap \{ (a,x) : x \in \mathbb{R}\}| = 1$.
2. For $(x_{0}, y_{0})$, $(x_{1} , y_{1}) \in S$ such that $x_{0} < x_{1}$, we have that $y_{0} < y_{1}$.

Must there exist an infinite subset $T \subset S$ such that for all $x \in T$, $x$ is not a convex combination of points in $T \setminus \{ x\}$?

Answer: Yes


Question: Determine whether two different polynomials of the same degree $d$ can appear in the factorization of $x^{p^n} - x$ over the field $\Bbb F_p$, given that $x^{p^n} - x$ is the product of all distinct irreducible polynomials in $\Bbb F_p[x]$ of degree $d$, where $d$ runs through all divisors of $n$. Provide a justification for your answer.

Answer: Yes


Question:
Find the Bayes estimator of \(\theta\) given the following information:

- Loss function: \(l(a,\theta)=\frac{(a-\theta)^2}{\theta(1-\theta)}\)
- Prior distribution: \(\pi(\theta)\) is uniformly distributed from 0 to 1.
- Likelihood function: Bernoulli, where \(X\) is the total number of ones in a sequence of \(n\) independent Bernoulli trials, each with sample space \(\{0,1\}\), so \(f(x|\theta) = \theta^x(1-\theta)^{1-x}\), \(x=0,1\).

Calculate the Bayes estimator of \(\theta\).

Answer: \dfrac{X}{n}


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

Answer: 1


Question: A listener receives messages with a wait time between consecutive messages that is exponentially distributed with a mean of 1 time unit. After any given message, there is a 1/2 probability that no further messages will be received. What is the variance of the time until the final message is received?

Answer: 4


Question: Consider the reductive group $\operatorname{GSp}_{4}$. Let $\pi$ be a smooth admissible cuspidal representation of $\operatorname{GSp}_{4}(\mathbb{A}^{(\infty)})$ of dominant weight, satisfying a multiplicity one hypothesis. Fix an odd prime $p$. To $\pi$ is attached a $p$-adic representation $\rho$ of the absolute Galois group of $\mathbb{Q}$, unramified outside a finite set of places, such that the characteristic polynomial of the Frobenius morphisms $Fr\ell$ for $\ell$ outside this set coincides with the Euler factor at $\ell$ of the degree 4 $L$-function of $\pi$. This Galois representation occurs in the degree 3 cohomology of the étale cohomology of a Siegel-Shimura variety. The image of complex conjugation under $\rho$ is semi-simple and can be chosen to be diagonal with eigenvalues 1 and -1. Determine the number of $-1$ eigenvalues.

Answer: 2


Question: Calculate the line integral \( \int_{\kappa} \arctan\left(\frac{y}{x}\right) dy - dx \), where \( \kappa \) is the part of the parabola \( y = x^2 \) from the point \((1,1)\) to \((0,0)\).

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


Question: Find the smallest positive integer \( k \) such that there exist distinct integers \( m_1, m_2, m_3, m_4, m_5 \) for which the polynomial \( p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5) \) has exactly \( k \) nonzero coefficients. Provide a set of integers \( m_1, m_2, m_3, m_4, m_5 \) for which this minimum \( k \) is achieved, along with a proof.

Answer: 3


Question: Let $S$ be the disk $|z|<3$ in the complex plane, and let $f: S \to \mathbb{C}$ be an analytic function such that $f\left(1+\frac{\sqrt{2}}{n}i\right) = -\frac{2}{n^2}$ for all natural numbers $n$. Determine the value of $f(\sqrt{2})$. \( \boxed{} \)

Answer: 3 - 2\sqrt{2}


Question: Let $f: \mathbb{N}_0 \to \mathbb{N}_0$ be a function such that $f(n+1) > f(n)$ for all $n \in \mathbb{N}_0$, and $f(n+f(m)) = f(n) + m + 1$ for all $n, m \in \mathbb{N}_0$. Find $f(2017)$. \( \text{Provide your answer as a single integer.} \)

Answer: 2018


Question: Given the permutations \( \theta = (13)(14)(567) \), \( \tau = (137)(2456) \), and \( \sigma = (13456)(27) \) in the symmetric group \( S_7 \), find the intersection of the left cosets \( \sigma \langle \theta \rangle \) and \( \tau \langle \theta \rangle \).

Answer: \emptyset


Question: Evaluate the double limit or determine if it diverges: $$\lim_{(x, y)\to(0,2)}\left(\frac{y\sin(xe^y)}{x}\right)$$

Answer: 2e^2


Question: How many values of \(x\) in the interval \([0,1]\) satisfy the equation \(f_7(x) = x\), where the function \(f(x)\) is defined as \(f(x) = 4x(1-x)\)?

Answer: 128


Question: Evaluate the complex integral \( \int_{|z|=2} \frac{1}{z^2+z+1} \, dz \).

Answer: 0


Question: Calculate \( \left\lfloor \underbrace{\sqrt{2017+\sqrt{2017+\ldots+\sqrt{2017}}}}_{2017 \text{ roots}} \right\rfloor \).

Answer: 45


Question: Determine the radius of convergence for the series \( \sum_{n=1}^\infty c_n x^n \), where \( c_n = \frac{1}{\sqrt{n^2+1}} + \frac{1}{\sqrt{n^2+2}} + \cdots + \frac{1}{\sqrt{n^2+n}} \).

Answer: 1


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

Answer: 2


Question:
Evaluate the limit: \[ \lim_{n\rightarrow \infty}n\int_{0}^{1}\left(\cos x-\sin x\right)^n\text{ d}x \] Choose the correct answer from the following options:
A) $ \infty$
B) $ 0$
C) $ 1$
D) $ \frac{1}{2}$
E) $\cos 1$

Answer: C


Question: Calculate the contour integral \( \int_\gamma \frac{1}{(z-z_0)^2}dz \) where \( \gamma(t)=Re^{it} \) for \( 0\le t \le 2\pi \) and \( R>0 \). Show that \( \int_\gamma \frac{1}{(z-z_0)^2}dz = 0 \).

Answer: 0


Question: Determine whether the measure \( \mu(T(C) \cap C) = 0 \), where \( T \) is the shift modulo 1 of the Cantor set by an irrational number \( \alpha \in (0,1) \), and \( \mu \) is the measure on the interval \([0,1]\) generated by the Cantor staircase.

Answer: 0


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

Answer: \dfrac{1}{2}


Question: Let \(G\) be a group and \(a, b \in G\) such that \(ab^{3}a^{-1} = b^2\) and \(b^{-1}a^2b = a^3\). Determine if \(ab = e\), where \(e\) is the identity element of the group.

Answer: ab = e


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

Answer: e^2


Question:
Minimize the function \( f(x, y, z) = x^2 + y^2 + z^2 \) subject to the constraints:
\[ x^2 + y^2 + z^2 - 5 = 0 \]
\[ x^2 + y^2 + z^2 - 2x - 3 = 0 \]

Answer: 5


Question: Determine the number of times a biased coin must be flipped to estimate the probability of it landing heads with an accuracy of 0.01 and a confidence level of 99%.

Answer: 16590


Question: Does there exist a constant $A$ such that the absolute value of the regulator is bounded above by $A$ for infinitely many totally real number fields?

Answer: No


Question: Consider a monic quartic polynomial with four distinct real roots. The turning points of this polynomial form an equilateral triangle. Find the area of this triangle.

Answer: 3^{\frac{5}{6}}


Question: Determine the constant $b$ such that the functions $\sin(3x)$ and $1 - e^{bx}$ overlap on the interval $(-1,1)$ when graphed.

Answer: -3


Question: Evaluate the line integral \( \int_{0}^{1+i} (y - x - 3x^2 i)\, dz \), where \( z = x + yi \) and the path is a straight line from \( 0 \) to \( 1+i \).

Answer: 1 - i


Question: In the town of Discretion, duels are rarely fatal. Each dueler arrives at a random time between 5 a.m. and 6 a.m. and leaves exactly 5 minutes later, unless their opponent arrives during this interval, in which case they fight. What fraction of duels lead to violence?

Answer: \dfrac{23}{144}


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \sqrt{x} (\ln(x))^{2015} \]

Answer: 0


Question: Compute the conditional expectation $E[t^Y|X=k]$ for $X \sim \text{Bin}(n,p)$ and $Y|X=k \sim \text{Bin}(k,p)$. Verify that the result is $(1-p+pt)^k$. Provide the steps leading to this expression.

Answer: (1 - p + pt)^k


Question: Evaluate the integral \( \int_x^\infty \frac{1}{1+e^t} \, dt \).

Answer: \ln\left(1 + e^{-x}\right)


Question: For which integers $n$ is the complete bipartite graph $K_{4,n}$ Hamiltonian?

Answer: 4


Question:
Consider the stochastic differential equations (SDEs):
\[dX_t = \mu_X (t, \omega) \, dt + \sigma_X (t, \omega) \, dW_t\]
\[dY_t = \mu_Y (t, \omega) \, dt + \sigma_Y (t, \omega) \, dW_t\]
with initial conditions \(X_0 = x_0\) and \(Y_0 = y_0\) almost surely, where \(\mu_X, \mu_Y, \sigma_X, \sigma_Y \geq 0\) are progressively measurable with respect to the natural filtration of a standard one-dimensional Brownian motion \(W\), and \(x_0, y_0\) are constants. Assume solutions exist up to a deterministic time \(T\). If \(\sigma_X \neq \sigma_Y\) on a subset of \(\Omega \times [0, T]\) of positive measure, is it true that \(\mathbb{P}(Y_T > X_T) > 0\)?

Answer: Yes


Question:
Find whole numbers \(X\), \(Y\), and \(Z\) such that:
\[ X(20A + 88C) + Y(32B + 72C) + Z(40A + 40B) \ge 616A + 890B + 982C \]
and the sum \(X + Y + Z\) is minimized. Provide one possible solution.

Answer: (3, 10, 15)


Question: Given vectors $u, v, w \in \mathbb{R}^n$ with $\|u\| = \|w\| = 7$, $\|v\| = 1$, and $\|u - v + w\| = \|u + v + w\|$, and the angle between $u$ and $v$ is $\frac{\pi}{3}$, find the angle between $v$ and $w$. Express your answer in radians.

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


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

Answer: 0


Question: Determine the value of $p$ that maximizes the expected amount of money gained when flipping a biased coin with probability $p$ of landing heads and $1-p$ of landing tails, until 4 consecutive tails are obtained. For each tail, you earn $2, and for each head, you lose $1.

Answer: 0.522


Question: Calculate the limit: \( \lim_{x \to 1^-} (1-x)^{\sin(\pi x)} \)

Answer: 1


Question: Determine whether the following statement is true: If \([e^X, Y] = 0\), does it imply that \([X, Y] = 0\)? Justify your answer.

Answer: False


Question: Evaluate the integral \( \iint\limits_D \max\{\sin x, \sin y\} \, dx \, dy \), where \( D = \{0 \le x \le \pi, \ 0 \le y \le \pi\} \).

Answer: 8


Question: Suppose \( f(z) = \prod_{n=1}^{17}(z-\frac{\pi}{n}) \) for \( z \in \mathbb{C} \), and let \( \gamma(t) = e^{2it} \) for \( t \in [0, 2\pi] \). If \( \int_{\gamma} \frac{f'(z)}{f(z)}\, dz = \alpha \pi i \), find the value of \( \alpha \).

Answer: 56


Question: Let \(\mathcal{F}\) denote either \(\mathbb{R}\) or \(\mathbb{C}\) and let \(V\) denote an inner product space over \(\mathcal{F}\). Let \(T\) be a normal operator on \(V\). Suppose \(\mathbf{v}\) and \(\mathbf{w}\) are elements of \(V\) such that \(||\mathbf{v}|| = 3\), \(||\mathbf{w}|| = 3\), \(T\mathbf{v} = 4\mathbf{v}\), and \(T\mathbf{w} = 2\mathbf{w}\). Find the value of \(||T(\mathbf{v} - \mathbf{w})||^2\).

Answer: 180


Question:
Let \(X\subseteq\mathbb{R}^n\) be a convex set. Determine whether there always exists a sequence \((X_n)_{n\geq 0}\) of sets such that:

1. The sequence \((X_n)_{n\geq 0}\) is increasing, i.e., \(X_0\subseteq X_1\subseteq X_2\subseteq\cdots\).
2. Each \(X_n\) is convex and compact.
3. The union \(\bigcup_{n\geq 0}X_n\) equals \(X\).

Answer: Yes


Question:
Find the function \( u \in C^1[0,1] \) that minimizes the functional \( J(y) \), where
\[ J(y) = \frac{1}{2}\int_0^1 (y'^2+y^2) \, \mathrm{d}x + \frac{1}{2}[y^2(0)+y^2(1)] - 2y(0). \]

Answer: e^{-x}


Question: Determine the minimal number \( M \) of telephone connections needed to ensure that no more than 1% of calls from city A to city B are blocked. Assume city A has 2000 phones, each user needs a connection for 2 minutes on average during 10 working hours, and calls are random. Use a Gaussian distribution with a mean of \( 6.6667 \) connections. Ignore impossible cases of less than 0 or more than 2000 calls.

Answer: 13


Question: Determine whether the following inequality is true: \[ \int_0^{100}\frac{e^{-x}}{x+100}dx > 0.005 \]

Answer: True


Question: Suppose the sphere $S^2 = A \cup B$, where $A$ and $B$ are closed subsets of $S^2$. Is it true that either $A$ or $B$ must contain a closed connected set $X$ such that $X = X^*$, the set of antipoles of points of $X$?

Answer: Yes


Question: Given a set $A$ with $|A| = n$, and two random subsets $B, C \subseteq A$ with $|B| = |C| = k$, find the expected value of $|B \cap C|$. Provide your answer in terms of $n$ and $k$.

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


Question: Evaluate the integral \( \int_{C} \frac{f(z)}{z^{3}} \, dz \), where \( f(z) = \sum_{k=0}^{\infty} k^{4} \left( \frac{z}{4} \right)^{k} \) and \( C \) is the circle \( |z| = \pi \), positively oriented.

Answer: 2\pi i


Question: Determine whether there exists an integer $n$ with exactly 2000 factors such that $n$ divides $2^n + 1$. In other words, does $n \mid 2^n + 1$?

Answer: Yes


Question: Consider a function f that is continuous on the interval [0,1] and takes no rational values. Can f(x) = 1 for some x in [0,1]?

Answer: No


Question: Find the maximum value of the integral \( \int_{0}^{1} t^2 x^2(t) \, dt \) subject to the constraint \( \int_{0}^{1} x^2(t) \, dt = 1 \). Determine the function \( x(t) \) that achieves this maximum value.

Answer: 1


Question: Given the matrix \( P = \begin{pmatrix} 2 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{pmatrix} \), define the matrix \( Q = P^{50} - I \). Calculate the value of \( \frac{q_{32} + q_{31}}{q_{21}} \).

Answer: 8


Question: Compute \( \mathbb{E}[\tilde{X}_t] \), where \( \tilde{X}_t = (1-t)\int_0^t \frac{1}{1-s} dW_s \) for \( 0 \le t < 1 \) and \( \tilde{X}_t = 0 \) for \( t = 1 \).

Answer: 0


Question:
Let \( a_{n+1} = a_n^2 + 1 \) with \( a_0 = 0 \). Define \( b_n(h) \) as follows:

- \( b_n(h) = a_n \) for all \( n \) such that \( a_n < 2^h \).
- \( b_n(h) = 2^{2\lceil\log_2 b_{n-1}(h)\rceil} \) for all \( n \) such that \( a_n \ge 2^h \).

Let \( m(h) \) be the maximum \( n \) such that \( \lceil\log_2 b_n\rceil < 2^h \). Is the ratio \( \rho(h) = \frac{a_{m(h)}}{b_{m(h)}(h)} \) upper bounded by an absolute constant for all \( h \in \mathbb{N} \)?

Answer: Yes


Question: Find the arc length of the cardioid given by the polar equation \( r = 3 - 3\cos \theta \).

Answer: 24


Question: Find the smallest positive integer \( b \) such that there exists an integer \( a \) for which \( \frac{43}{197} < \frac{a}{b} < \frac{17}{77} \).

Answer: 32


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

Answer: 0


Question: Let $F$ be a real closed field and $X$ be a positive-dimensional algebraic variety over $F$. If $X$ has finitely many $F$-points, are all these points singular?

Answer: Yes


Question: Determine the number of isomorphism classes of associative rings with identity that have 35 elements.

Answer: 1


Question: Determine whether the symmetric group $S_n$ can be generated by the transposition $(12)$ and the set of 3-cycles $\{(123),(124),\dots ,(12n)\}$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the surface integral \( \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{\hat{N}} \, dS \), where \( S \) is the hemisphere defined by \( x^2 + y^2 + z^2 = a^2, z \geq 0 \) with outward normal, and \( \mathbf{F} = 3y \hat{i} - 2xz \hat{j} + (x^2 - y^2) \hat{k} \).

Answer: -3\pi a^2


Question: Find the probability that the sum of the squares of two independent standard normal random variables is less than 1, i.e., calculate \( P(X^2 + Y^2 < 1) \) where \( X \) and \( Y \) are independent \( N(0,1) \) random variables.

Answer: 1 - e^{-1/2}


Question: Compute the value of \( s = (3(16)^2 + 9) \cdot (2 \cdot 5)^{-1} \bmod{23} \).

Answer: 11


Question: Let $m$ be an odd prime number, and define the set $A = \{0, 1, 2, \ldots, 2m-1\}$ as the set of all remainders modulo $2m$. Determine the number of elements $x$ in $A$ such that $x^2 \equiv 1 \pmod{2m}$.

Answer: 2


Question: Let \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) be a linear map. If \( K \) is a right-inverse of \( T \), does \( K \) have to be a linear map? Provide a justification for your answer.

Answer: No


Question: Find the limit of the sequence \((b_n)_{n=1}^\infty\), where \(b_n = \int_{-2\pi n}^{\pi + 2\pi n} x \cos(x) \, dx\).

Answer: -2


Question: Is a $p$-Sylow subgroup of a profinite group of order $lp^n$, where $l$ and $p$ are primes with $l<p$, necessarily normal?

Answer: Yes


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a function that is three times differentiable and satisfies the equation \( f(x^2)f''(x) = f'(x)f'(x^2) \) for all \( x \in \mathbb{R} \). Given that \( f(1) = 1 \) and \( f'''(1) = 8 \), find the value of \( f'(1) + f''(1) \).

Answer: 6


Question: Let \(A\) be an \(n \times n\) matrix with full rank. Let \(B\) be another \(n \times n\) matrix. Can we always find a constant \(c > 0\) sufficiently small such that the matrix \(A + cB\) also has full rank?

Answer: Yes


Question: Let \( f: (0,+\infty) \rightarrow \mathbb{R} \) be a continuous function such that \( f(x + y) = f(x) + f(y) \) for all \( x, y > 0 \). Prove or disprove that \( f \) is a linear function.

Answer: f(x) = kx


Question: Consider a real vector space $V$ of dimension $2m$, where $m$ is a positive integer, and a non-singular quadratic form $Q$ on $V$. If $Q$ vanishes on a subspace of dimension $m$, determine the signature of $Q$.

Answer: (m, m)


Question: Let $X_1, X_2, \ldots, X_{2n}$ be random variables such that $\operatorname{Var}(X_i) = 4$ for $i = 1, 2, \ldots, 2n$ and $\operatorname{Cov}(X_i, X_j) = 3$ for $1 \leq i \neq j \leq 2n$. Calculate the variance of the expression $X_1 - X_2 + X_3 - X_4 + \cdots + X_{2n-1} - X_{2n}$. \[ V(X_1 - X_2 + X_3 - X_4 + \cdots + X_{2n-1} - X_{2n}) = ? \]

Answer: 2n


Question: Evaluate the integral: \[ \int_{-\infty}^\infty \exp(-\sqrt{1+x^2})\,dx. \]

Answer: 2 K_1(1)


Question: Determine the norm of the operator \( T: L^1[0,1] \to L^1[0,1] \) defined by \( T(f)(t) = \int_0^t f(s) \, ds \).

Answer: 1


Question:
Calculate the value of the determinant \(\Delta\) given by:
\[\Delta=\begin{vmatrix}
\dfrac{\pi}{2} & u_2 & u_3 \\
u_4 & u_5 & u_6 \\
u_7 & u_8 & u_9
\end{vmatrix}\]
where \(u_n=\displaystyle\int^{\frac{\pi}{2}}_{0}{\dfrac{1-\cos{2nx}}{1-\cos{2x}}dx}\).

Answer: 0


Question: Let $M$ be a compact oriented smooth $n$-manifold with $H_{dR}^1(M) = 0$. Determine the integers $k$ for which there exists a smooth map $f : M \to T^n$ of degree $k$. Provide a justification for your answer.

Answer: 0


Question: Let $L/K$ be a finite extension of fields with characteristic 0, and let $G$ be a finite group. Suppose $V_1$ and $V_2$ are irreducible representations of $G$ over $K$. If $V_1 \otimes_K L$ and $V_2 \otimes_K L$ are isomorphic as representations over $L$, does it follow that $V_1 \cong V_2$ as representations over $K$?

Answer: Yes


Question: Consider the ideal $I = (f, g, h)$ in the polynomial ring $k[x, y, z]$, where $LT(f) > LT(g) > LT(h)$ in the lexicographic order, and $I$ is reduced such that $LT(g) \nmid LT(f)$, $LT(h) \nmid LT(g)$, and $LT(h) \nmid LT(f)$. Is it necessary that the reduced Groebner basis $G$ of $I$ contains at least 3 elements? Assume the case is nontrivial.

Answer: Yes


Question: Find the inverse Z-transform of the function \( \frac{1}{z-a} \).

Answer: a^{n-1} u[n-1]


Question: Find the value of $k$ for which the integral \( \int_{0}^{1} \frac{\ln(x)}{x^k} \, dx \) has a finite value.

Answer: k < 1


Question: Solve the equation \( x^{\log(x)-1 } = \frac{1}{\sqrt[4]{10}} \) for \( x \).

Answer: \sqrt{10}


Question:
Compute the limit of the sequence \(a_n\) as \(n\) approaches infinity, where:
\[a_n=\int_n^{2n} \frac{1}{x}e^\frac{1}{x} \,dx\]

Answer: \ln 2


Question:
For what value of \( x \) are all the eigenvalues of the matrix \( A \) real?
\[
A=\begin{pmatrix}
10 & 5+i & 4 \\
x & 20 & 2 \\
4 & 2 & -10
\end{pmatrix}
\]

Answer: 5 - i


Question: Let $(X, \mathcal F, \mu)$ be a finite measure space and let $f \in L^\infty(X, \mu)$. Define $\alpha_n=\int_X |f|^n\ d\mu$. Prove that $$\lim_{n\to \infty}\frac{\alpha_{n+1}}{\alpha_n}=\|f\|_\infty.$$

Answer: \|f\|_\infty


Question: Find the derived set \( A' \) of the set \( A = \{1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n}, \ldots\} \) in the metric space \((\mathbb{R}, |\cdot|)\).

Answer: \{0\}


Question:
Let \(X\) and \(Y\) be random variables with the joint probability density function given by:
\[
f_{X,Y}(x,y) = \begin{cases}
0.25ye^{-y} & \text{if } 0 \leq |x| \leq y \\
0 & \text{otherwise}
\end{cases}
\]
Find the conditional density \(f_{X|Y = y}(x)\) and identify the conditional distribution \(X|_{Y = y}\). Then, use the law of total expectation to find \(E[X^2]\).

Answer: 4


Question: Find all polynomials \( P(x) \) such that \( P(x^2+1) = P(x)^2 + 1 \) and \( P(0) = 0 \).

Answer: x


Question:
Calculate the Jacobian determinant of the transformation given by the inverse functions:
\[ x_1 = y_1 y_n, \]
\[ x_2 = y_2 y_n - y_1 y_n, \]
\[ \vdots \]
\[ x_{n-1} = y_{n-1} y_n - y_{n-2} y_n, \]
\[ x_n = y_n (1 - y_{n-1}). \]
Verify that the Jacobian determinant is \( y_n^{n-1} \).

Answer: y_n^{n-1}


Question: In the Erdős–Rényi model $G(n, p)$, calculate the expected number of different spanning trees of the graph $G$. Provide your answer as a function of $n$ and $p$.

Answer: n^{n-2} p^{n-1}


Question: Determine whether the series \( \sum_{n=1}^{\infty} (-1)^n \zeta(n + 1) \) converges to \(-1\).

Answer: -1


Question: Find the integer value of $x$ such that \( \sqrt{x+\frac{1}{2}\sqrt{2011}}-\sqrt{x-\frac{1}{2}\sqrt{2011}} \) is an integer.

Answer: 503


Question: Given a function $f:[0,\infty)\to\mathbb{R}$, where $f$ is an increasing function, and its primitive $F$ satisfies $F(0)=0$ and $F(x+y)\leq F(x)+F(y)$ for all $x,y \geq 0$, determine the function $f$.

Answer: f(x) = c


Question: Compute the radius of convergence for the function \( f(z)=\frac{z^{5}e^{z}}{(2-z)(3i-z)} \) expanded at the point \( z=4+4i \).

Answer: \sqrt{17}


Question: Determine whether the following limit exists: \[ \lim \limits_{(x,y) \to (0,0)} \left(\frac{(x^4+y^4) \sin\left(\frac{1}{x}\right)}{x^2+y^2}\right) \]

Answer: 0


Question: Provide an example of a set that contains infinitely many irrational numbers and is compact.

Answer: \left\{\sqrt{2} + \frac{1}{n} \mid n \in \mathbb{N}\right\} \cup \{\sqrt{2}\}


Question: Compute \( \max_{x \in [0,1]} \min_{y \in [0,1]} \max \{x(1-2y), y(1-2x)\} \).

Answer: \dfrac{1}{8}


Question: Given a triangle $ABC$ with sides $a \ge b \ge c$, if \( \frac{a^3+b^3+c^3}{\sin^3(A)+\sin^3(B)+\sin^3(C)}=7 \), find the maximum value of $a$.

Answer: \sqrt[3]{7}


Question: Determine the values of $d$ for which the equation $x - d\tanh(x) = 0$ has a positive solution.

Answer: d > 1


Question: Determine the basis or dimension of the vector space $A_k$, consisting of polynomials $f(x_1,x_2,\ldots,x_n)$ in $\mathbb{Q}[x_1,x_2,\ldots,x_n]$ that are symmetric in the first $k$ variables, i.e., $$f(x_{\sigma(1)},\ldots, x_{\sigma(k)},x_{k+1}, \ldots,x_n)=f(x_1,x_2,\ldots,x_n),\,\, \sigma \in S_k.$$

Answer: \infty


Question: Determine whether the matrix \( \begin{bmatrix} A & A \\ 0 & A \end{bmatrix} \) is diagonalizable if and only if \( A = 0 \), where \( A \) is an \( n \times n \) complex matrix.

Answer: A = 0


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{\log x!}{x \log x} \]

Answer: 1


Question: Determine the number of elements in the multiplicative group of the ring \( \mathbb{Z}_2[x] / (x^3 + x^2 + 1) \).

Answer: 7


Question: Consider the space of positive bounded sequences and the functional \((x_i) \rightarrow \sum_{i=1}^{\infty}\beta(i) x_i\), where \(\beta: \mathbb{N} \rightarrow (0,1]\) is a strictly decreasing function such that \(\sum_{i=1}^\infty \beta(i)=B<+\infty\). Is the functional \((x_i) \rightarrow \sum_{i=1}^{\infty}\beta(i) x_i\) continuous with respect to the product topology on the set of positive bounded sequences?

Answer: No


Question: Find the value of \( k \) for which the solution to the differential equation \( y'' + 4y' + ky = 0 \) vanishes at \( t = 2 \) and \( t = 3 \), but not in between.

Answer: 4 + \pi^2


Question: If the probability of an event is \( \frac{1}{\infty} \) and \( \infty \) trials are conducted, how many times will the event occur: 0, 1, or \( \infty \)?

Answer: 0


Question: Evaluate the limit \( \lim_{R \to \infty} \int \int _{S} e^{-x^2-y^2} \, dx \, dy \) where \( S = \{ (x,y) : x \in [-R,R], y \in [-R,R] \} \).

Answer: \pi


Question: A creature called "Charza" lives in blocks of an infinite square grid. Initially, there is one Charza in a block. After one hour, each Charza divides into 4 Charzas, and each one moves into one of the adjacent blocks. After 6 hours, how many Charzas will be in a block that shares only one common vertex with the initial block?

Answer: 300


Question: Evaluate the expression: \[ \sum _ {k=1}^{n}{\frac{k}{n}\binom{n}{k}t^k(1-t)^{n-k}} \] where \( \binom{n}{k} \) is the binomial coefficient representing the number of ways to choose \( k \) objects from \( n \) objects.

Answer: t


Question: Determine the number of solutions to the congruence \(x^2 \equiv 3 \pmod{11^2 \cdot 23^2}\) without actually finding the solutions.

Answer: 4


Question: Determine whether the following logical equivalence is true: \[ a(x) \Rightarrow \forall{y} \varphi(x,y) \equiv \forall{y} \left( a(x) \Rightarrow \varphi(x,y) \right), \] where \( a(x) \) is a quantifier-free formula with only one free variable \( x \) and \( \varphi(x,y) \) is a quantifier-free formula with free variables \( x,y \).

Answer: True


Question: Given $n$ real numbers $a_1, a_2, \ldots, a_n$ arranged on a circle such that their total sum is zero, and a fixed integer $k < n$, let $S_i$ be the sum of $k$ adjacent numbers starting at $a_i$ and counting clockwise. Determine the largest possible number of strictly-positive $S_i$ as a function of $k$. Provide your answer in terms of $k$ and $n$.

Answer: n - k


Question: In a semicircle with diameter $AB$ and center $S$, points $C$ and $D$ are located such that point $C$ is on arc $AD$. Given that angle $CSD$ is 120°, let $E$ be the intersection of lines $AC$ and $BD$, and $F$ be the intersection of lines $AD$ and $BC$. Find the ratio $\frac{EF}{AB}$.

Answer: \sqrt{3}


Question: A machine produces a weighted coin that lands on heads with an unknown probability $p$, where the distribution of $p$ is given by $P(p \le x) = x^4$. You flip the coin 5 times, and it lands on heads each time. What is the probability that the next flip will also land on heads?

Answer: \dfrac{9}{10}


Question: Consider a function \( f: \mathbb{R}^n \to \mathbb{R}^n \) that is differentiable only at a single point \( p \in \mathbb{R}^n \) and not differentiable at any other point in \( \mathbb{R}^n \). According to the inverse function theorem, if the Jacobian of \( f \) is non-singular at \( p \), then \( f \) would be differentiable on some open set containing \( p \). Does this imply that the Jacobian of \( f \) at \( p \) must be singular?

Answer: Yes


Question: Suppose $X$ and $Y$ are independent and identically distributed random variables from a distribution function $F$, which is continuous. Is it always true that $P(X = Y) = 0$? Provide a justification for your answer.

Answer: 0


Question: A thirty-year annuity X has annual payments of $1,000 at the beginning of each year for twelve years, followed by annual payments of $2,000 at the beginning of each year for eighteen years. A perpetuity Y has payments of Q at the end of each year for twenty years, then payments of 3Q at the end of each year thereafter. The present values of X and Y are equal when calculated using an annual effective discount rate of 10%. Find the value of Q.

Answer: 1070.44


Question: Find the value of \( b \) such that the process \( M_t = e^{5B_t - bt} \) is a martingale, where \( B_t \) is a Brownian motion at time \( t \).

Answer: \dfrac{25}{2}


Question: Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that for $|x| > M$, \( \frac{{C_1|\sin(\lambda x)|^a}}{{|x|^b}} \leq |f(x)| \leq \frac{{C_2|\sin(\lambda x)|^a}}{{|x|^b}} \) and $f(x) \neq 0$ for $|x| \leq M$, where $0<C_1<C_2$, $\lambda>0$, $a\geq 1$, $b>0$, $M>0$. For each $\varepsilon>0$, define $B_\varepsilon = \{x \in \mathbb{R} : |f(x)| < \varepsilon \}$. Does the Lebesgue measure $m(B_\varepsilon)$ tend to $0$ as $\varepsilon \to 0$?

Answer: No


Question: Determine whether the statement is true or false: The modules \( \mathbb{C}[X]/(x-c) \) and \( \mathbb{C}[X]/(x-d) \) are isomorphic as \( \mathbb{C}[X] \)-modules if and only if \( c = d \).

Answer: True


Question: Compute the limit: $$\lim_{a \to 0^+} \left(a \int_{1}^{\infty} e^{-ax}\cos \left(\frac{2\pi}{1+x^{2}} \right)\,\mathrm dx\right)$$

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{n!}{k! \times n^{k+1}} \] where \( k \) is a constant.

Answer: \infty


Question: Let $\{f_n\}$ be a sequence of continuous functions on the real line that converges pointwise. Is it true that for every $c \in \mathbb{R}$, the set $\{x \in \mathbb{R} : \lim_{n \to \infty} f_n(x) \ge c\}$ is a $G_{\delta}$ set in $\mathbb{R}$?

Answer: Yes


Question:
Determine the units of the ring of dual numbers \(\mathbb{C}[\varepsilon]\). Choose the correct set from the following options:

(a) \(\{a+b\varepsilon : a,b \in \mathbb{C}, a \neq 0\}\)

(b) \(\{1\}\)

(c) \(\{b\varepsilon : b \in \mathbb{C} \setminus \{0\}\}\)

Answer: a


Question:
In a Boolean algebra \( A = (A, \vee, \wedge) \), which of the following conditions is sufficient to ensure that \( \overline{x} = y \) for all \( x, y \in A \)?

1. \( x + y = 1_A \).
2. \( x \cdot y = 0_A \; \wedge \; x \neq 0_A \).
3. \( x \cdot (x + y) = \overline{y} \cdot x + \overline{(y + x)} \).
4. None of the above.

Answer: D


Question: Determine whether the following proposition is true or false: If $a_n \neq 0$ for all natural numbers $n$, $\sum_{n=1}^{+\infty} a_n$ converges, and $\lim_{n\to+\infty} \frac{a_n}{b_n} = 1$, then $\sum_{n=1}^{+\infty} b_n$ converges.

Answer: False


Question: Is it necessary for a finite field with order \(p^n\), where \(p\) is a prime number, to be a field extension of a subfield isomorphic to \(\Bbb{Z_p}\) and consequently have characteristic \(p\)?

Answer: Yes


Question: Does there exist a group $G$ such that its automorphism group $Aut(G)$ is perfect?

Answer: Yes


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

Answer: 1


Question: Find integers \((a, b)\) such that \(\gcd(a, b) = 13\) and they satisfy the equation \(49a - 125b = 169\).

Answer: (1131, 442)


Question: Find the closure of the set \((0,1]\) in the topological space \((\mathbb{R}, T_1)\), where \(T_1\) is the minimal topology such that the functions \(f_1(x) = x^2\) and \(f_2(x) = -x^2\) are continuous. The topology \(T\) on \(\mathbb{R}\) has a basis consisting of sets of the form \((a,b]\) for \(a, b \in \mathbb{R}\) with \(a < b\).

Answer: [0,1]


Question: Evaluate the limit: $$\lim_{x\rightarrow +\infty} \sum_{k=0}^\infty \frac{(-1)^k x^k}{k!} \frac{(4N+2k-1)!(k+N-1)!}{k!(k+2N-1)!^2}$$ where $N> 1$.

Answer: 0


Question: In a competition with 37 contestants, each contestant has a non-negative integer score. The sum of the scores of any 17 contestants is at least 45, and the total score of all 37 contestants is 150. What is the maximum score that a single contestant can achieve?

Answer: 48


Question:
Compute the cross-product term \(d(x_1)d(x_2)\) for the following independent Wiener processes:
\[ dx_1 = a_1 dt + b_1 dw_1(t) \]
\[ dx_2 = a_2 dt + b_2 dw_2(t) \]
where \(w_1\) and \(w_2\) are independent Wiener processes.

Answer: 0


Question: Given a 5th-degree polynomial \( f(x) \) with a leading coefficient of 2009, and the conditions \( f(1) = 1 \), \( f(2) = 3 \), \( f(3) = 5 \), \( f(4) = 7 \), and \( f(5) = 9 \), find the value of \( f(8) \).

Answer: 5062695


Question: Evaluate the integral \( \int_{|z|=1} \frac{|dz|}{z} \).

Answer: 0


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

Answer: -\dfrac{1}{6}


Question: Evaluate the expression \( \lim\limits_{t\rightarrow 0}\int_0^t f(s)\,\delta(t-s)\,ds \), where \( f(s) \) is a continuous function.

Answer: f(0)


Question: Calculate the volume of the region defined by the inequality \(|x| + |y| + |z| \leq 1\) using integration.

Answer: \dfrac{4}{3}


Question: Find the total mass of the solid defined by the inequalities \(x^2 + y^2 + z^2 \ge 1, \hspace{.1cm} x \ge 0,\hspace{.1cm} y \ge 0\) with mass density \(z^2\) using cylindrical coordinates.

Answer: \dfrac{\pi}{30}


Question: Determine the number of epimorphisms from the free group with 2 generators, $F_2$, to the cyclic group $\mathbb{Z}_5$. Provide your answer as a single integer.

Answer: 24


Question: Determine the values of \( \alpha \in \mathbb{R} \) for which the integral \( \int_0^1 \frac{\sin(\frac{1}{x})}{(\sqrt{x} - x)^\alpha} \, dx \) converges.

Answer: (-\infty, 1)


Question: In a sequence of vehicles, two-thirds of all trucks have a car behind them, and one-fifth of all cars have a truck behind them. What proportion of the vehicles are cars?

Answer: \dfrac{10}{13}


Question: Evaluate the Cauchy principal value of the improper integral: \[ \int_{-\infty}^{\infty} \frac{1}{(x^2+4)^2} \, dx \]

Answer: \dfrac{\pi}{16}


Question: Find the limit of the series \( \sum^{\infty}_{n=0} \frac {(-1)^n 2^{-n} x^{2n+1}} {n!} \) where \( x \in \mathbb{R} \).

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


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

Answer: 1


Question: Determine the shortest sequence of digits such that every 3-digit number from 000 to 999 appears as a substring. What is the length of this sequence?

Answer: 1002


Question: Find the measure of the set \(\{x \in [0,1] : -x^2 \cos\left(\frac{1}{x}\right) \ge 1\}\).

Answer: 0


Question: Let $H=(V,E)$ be a Hausdorff hypergraph, where $V$ is an infinite set. A subset $D \subseteq V$ is called dense if $D \cap e \neq \emptyset$ for all $e \in E \setminus \{\emptyset\}$. Determine if it is possible for $2^{|D|} < |E|$ to hold.

Answer: Yes


Question: Determine the number of pairwise non-isomorphic 2-connected graphs with 20 vertices that do not contain a $K_4^-$ minor.

Answer: 1


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

Answer: \int_{0}^{1} x^{-x} \, dx


Question: Determine whether the set \( A = \{\sqrt{2}\sin(\sqrt{n^2+a} \pi x)\}_{n=1}^\infty \), where \( a \) is a positive real number, is a Riesz basis for \( L^2(0,1) \).

Answer: Yes


Question: Determine the cardinality of the set $F$ of all entire functions $f$ such that $f(x) = |x^k|$ for all $x \in (-1, 1)$, where $k$ is an odd integer $k \geq 1$. Provide your answer as a single number.

Answer: 0


Question:
Find the units digit of the sum:
\[2^{2^2} + 4^{4^4} + 6^{6^6} + \cdots + 2016^{2016^{2016}} + 2018^{2018^{2018}}.\]

Answer: 8


Question: Evaluate the infinite series: \[ \sum_{n=1}^{\infty} \frac{n \cdot 3^n}{(n+3)!} \]

Answer: \dfrac{1}{2}


Question: A sphere is inscribed in a right circular cone with a volume of \(10\, \text{cm}^3\). Determine the maximum volume of the sphere.

Answer: 5


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

Answer: \pi \ln 2


Question: Does there exist a non-separable topological space $(X, \tau)$ such that the sigma algebra generated by $\tau$ is equal to the sigma algebra generated by a minimal topological base $\tau_0$ of $\tau$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \frac{\pi}{2}} \left( \tan^2{x} \left( \sqrt{2\sin^2{x}+3\sin{x}+4} - \sqrt{\sin^2{x}+6\sin{x}+2} \right) \right) \]

Answer: \dfrac{1}{12}


Question: Let $A$ be a convex subset of a real normed linear space $V$, and let $D$ be a dense subset of $V$. Is it true that for every non-empty open subset $U$ of $V$, the intersection $U \cap D \cap A$ is non-empty?

Answer: No


Question: Consider the sequence of polynomials defined by $P_0(x) = 0$ and for $n > 0$, $P_n(x) = \frac{x + P_{n-1}^2(x)}{2}$. Let $Q_n(x) = P_n(x) - P_{n-1}(x)$. Determine whether all the coefficients of the polynomials $Q_n(x)$ are nonnegative.

Answer: Yes


Question: Find the multiplicative inverse of $2x + 3 + I$ in the ring $\mathbb{Z}_5[x]/I$, where $I = \langle x^2 + x + 2 \rangle$. Express your answer in the form $ax + b + I$, where $a, b \in \mathbb{Z}_5$. \(\text{The answer is } 3x + 1 + I.\)

Answer: 3x + 1 + I


Question: Consider the differential equation \( \frac{dy}{dx} + 10y = f(x) \) for \( x > 0 \), where \( f(x) \) is a continuous function such that \( \lim_{x \to \infty} f(x) = 1 \). Find the value of \( \lim_{x \to \infty} y(x) \).

Answer: \dfrac{1}{10}


Question: Evaluate the limit: \( \lim_{x \to 0^+} \sqrt{\tan x}^{\sqrt{x}} \)

Answer: 1


Question: In triangle $\Delta ABC$, $AB = 12$ and $AC = 16$. Let $M$ be the midpoint of side $BC$. Points $E$ and $F$ are on sides $AC$ and $AB$, respectively. The lines $EF$ and $AM$ intersect at $G$. Given that $AE = 2AF$, find the ratio $\frac{EG}{GF}$.

Answer: \dfrac{3}{2}


Question: Let $U_\infty$ be a compact space, and let $U_r$ be an increasing family of compact subspaces whose closure is all of $U_\infty$. For $r \in [1,\infty]$, let $Y_r = C(U_r,\mathbb R)$ be the Banach space of real-valued continuous functions over $U_r$ with the supremum norm. For $r \le r'$, let $\phi_{r,r'} : Y_{r'} \to Y_r$ be the restriction maps, so that $Y_\infty$ is the inverse limit of the spaces $Y_r$. Write $\phi_r : Y_\infty \to Y_r$ for the restriction map $\phi_{r,\infty}$. Suppose there exists a family of continuous linear operators $m_r : Y_r \to Y_\infty$ such that $\|m_r\| \le M$ for all $r$, and $\phi_r \circ m_r$ is the identity map on $Y_r$. If $\Gamma \subseteq Y_\infty$ is compact, does $m_r \circ \phi_r$ converge strongly to the identity operator on $\Gamma$? That is, for all $\epsilon > 0$, does there exist $R > 0$ such that if $r \ge R$, then $$\sup_{y \in \Gamma} \left\| (m_r \circ \phi_r)(y) - y \right\|_{Y_\infty} < \epsilon?$$

Answer: Yes


Question: Let \( A \) be a square matrix of order \( n \), and \( \phi \) a linear form on the set of square matrices of order \( n \) with real entries. Define the function \( f: M \mapsto M - \phi(M)A \). Determine a necessary and sufficient condition on \( \phi(A) \) for \( f \) to be a bijective function.

Answer: \phi(A) \neq 1


Question: Calculate the cardinality of the set \(\left(\{1,2,3\}^{\mathbb{N}} - \{1,2\}^{\mathbb{N}}\right)\cap\mathcal{P}(\mathbb{N}\times\mathbb{N})\), where \(A^B\) denotes the set of all functions from \(B\) to \(A\).

Answer: 2^{\aleph_0}


Question: Determine the condition on the prime number $p$ such that the equation $x^2 + 3 \equiv 0 \pmod{4p^2}$ has solutions. Additionally, find the number of solutions to this equation.

Answer: 4


Question: Consider the irreducible polynomial $g = X^4 + X + 1$ over $F_2$ and let $E$ be the extension of $F_2 = \{0, 1\}$ with a root $\alpha$ of $g$. Could $E$ have a subfield of order 8?

Answer: No


Question: Let $C$ be a $4 \times 4$ matrix with a nonzero determinant. Given that $8CC^T=2I_4$, find $C^{-1}$. Express your answer in terms of $C^T$.

Answer: 4C^T


Question:
Determine the correct differential equation to solve the following problem: A line of initial length 1 is reduced in a process where, in each step, two points $x_1$ and $x_2$ are selected on the remaining line, and the segment from the left end to the maximum of $x_1$ and $x_2$ is removed. What is the expected number of steps $E(x)$ needed for a fraction $x$ of the initial line to disappear? Choose the correct differential equation:

1. $$E(x) = 1-x^2 + \int_0^x 2y+2yE(\frac{x-y}{1-y}) \, dy$$
2. $$E(x) = 1-x^2 + \int_0^x 1+2yE(\frac{x-y}{1-y}) \, dy$$

Decide whether the first term of the integrand should be $1$ or $2y$. Consider the following information:
- $1-x^2$ is the probability that one step is enough to remove more than $x$.
- $E(\frac{x-y}{1-y})$ is the expected number of steps needed for a fraction $x$ to disappear, starting from length $y < x$.
- $2y$ is the probability distribution function for the maximum of $x_1$ and $x_2$ with $0 \leq y \leq 1$. Choose the correct differential equation.

Answer: 1


Question: Given a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that every derivative $\frac{d^n}{dx^n}(f(x)), \,n > 0$ is continuous, determine if there exists a function $g: \mathbb{R} \rightarrow \mathbb{R}$, for which every derivative $\frac{d^n}{dx^n}(g(x)), \,n > 0$ is also continuous, such that $g(x) = f(x)$ for all $x \in [a, b]$ and there exists some $x \notin [a, b]$ where $f(x) \neq g(x)$, with $a \neq b$. Provide a justification for your answer.

Answer: Yes


Question: In how many different ways can you select 2005 balls from a collection of 10000 identical red balls, 10000 identical yellow balls, and 10000 identical green balls, such that the number of red balls is even and the number of yellow balls is odd?

Answer: 503506


Question: Given an $n \times n$ square matrix $A$ with rank $r$, and matrices $B$ and $C$ such that $AB = AC$, determine the maximal rank of the matrix $B - C$.

Answer: n - r


Question:
Determine the correct form of the derivative of the degree matrix with respect to the adjacency matrix for a symmetric adjacency matrix $A \in \{0, 1\}^{n \times n}$. The degree matrix is defined as $D_{ii} = \sum_j A_{ij}$. Given the following options for $n=3$, identify the correct derivative $\frac{\partial D_{00}}{\partial A}$:

1. \( \begin{pmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \)
2. \( \begin{pmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} \)
3. \( \begin{pmatrix} 1 & 1/2 & 1/2 \\ 1/2 & 0 & 0 \\ 1/2 & 0 & 0 \end{pmatrix} \)
4. \( \begin{pmatrix} 0 & 1/2 & 1/2 \\ 1/2 & 0 & 0 \\ 1/2 & 0 & 0 \end{pmatrix} \)

Consider that the correct derivative should ideally be symmetric, and the element on the diagonal should be 1 or 0.

Answer: 3


Question: Use Green's Theorem to evaluate the line integral \( \int_C (y + e^\sqrt{x})\,dx + (2x + \cos y^2)\,dy \), where \( C \) is the boundary of the region enclosed by the parabolas \( y = x^2 \) and \( x = y^2 \). Determine the appropriate limits of integration for the double integral.

Answer: \dfrac{1}{3}


Question: Compute the surface integral \( \int\!\!\!\int_S \vec{F}\cdot \vec{n}\,dA \) for the vector field \( \vec{F}=(-a^2y, b^2x, z^2) \) over the surface \( S \) of the ellipsoid \( \left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2=1 \), with \( \vec{n} \) oriented outwards. Use the Divergence Theorem to evaluate the integral.

Answer: 0


Question: Find the value of \( y \) such that \( \tan(y^\circ) = (4\cos^2 9^\circ - 3)(4\cos^2 27^\circ - 3) \).

Answer: 9


Question:
Consider the sequence \(x_n=\frac{(\alpha+1)^n+(\alpha -1)^n}{{(2\alpha)}^n}\) for \(\alpha >0\) and \(n=1,2,3,\ldots\). Determine \(\lim_{n\rightarrow\infty}x_n\). Choose the correct option:

(A) 0 for any \(\alpha>0\)
(B) 1 for any \(\alpha>0\)
(C) 0 or 1 depending on what \(\alpha>0\) is
(D) 0, 1 or \(\infty\) depending on what \(\alpha>0\) is

Answer: D


Question: Determine whether the sequence \( \frac{n^n}{n!} \) converges, and if so, find its limit.

Answer: \infty


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

Answer: \dfrac{1}{6}


Question: Mike makes $40 for every apple he sells but incurs a net loss of $10 for every apple he doesn't sell. The demand for apples, $X$, is uniformly distributed on $[0,5]$. How many apples should Mike produce to maximize his expected profit?

Answer: 4


Question:
Let \(\omega\) be a complex number such that \(\omega^5 = 1\) and \(\omega \neq 1\). Compute the value of the expression:
\[
\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \omega^3}.
\]

Answer: 2


Question: Evaluate the limit: \[ \lim_{n \to \infty } \frac{\sum_{k=1}^{n} \frac{1}{k^{1/k}}}{n} \]

Answer: 1


Question: Find the sum of all possible real solutions of the equation \((x^2+4x+5)^{(x^2+4x+5)^{(x^2+4x+5)}}=2018\).

Answer: -4


Question: Let $X\subseteq \mathbf{P}^n(\mathbb{C})$ be a non-singular projective variety defined by the vanishing of homogeneous polynomials $f_1, \dots, f_r$. Determine if it is true that for sufficiently small variations of the coefficients of the polynomials $f_1, \dots, f_r$, the resulting variety $X'$ is isomorphic as a topological manifold to $X$. Provide a justification for your answer.

Answer: True


Question: A manufacturer of cameras offers a two-year warranty for each camera, which consists of 8 components. Each component has a 3% chance of breaking during the warranty period. If a component breaks, it costs $50 to repair or replace it. The company plans to sell 800 cameras at launch. Determine the price at which each camera should be sold so that there is a 98% probability of making a profit of at least $120,000. Use the normal approximation to solve this problem.

Answer: 163.75


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

Answer: 0


Question: Evaluate \( \lim_{n \to \infty} \int_{[0,2]} t^{1/n} \, d\mu(t) \), where \( \mu = \delta_0 + \delta_1 + \delta_2 \) is the Dirac measure.

Answer: 2


Question: If \( \int^{1}_{0} \frac{\tan^{-1}x}{x} \, dx = k \int^{\pi/2}_{0} \frac {x}{\sin x} \, dx \), find the value of \( k \).

Answer: \dfrac{1}{2}


Question: Find the area of the part of the cylinder \(x^2+y^2=2ay\) that lies inside the sphere \(x^2+y^2+z^2=4a^2\).

Answer: 16a^2


Question: Express the function \( f(z) = (3x^2 + 2x - 3y^2 - 1) + i(6xy + 2y) \) in terms of the complex variable \( z \).

Answer: 3z^2 + 2z - 1


Question: Determine whether the set of non-diagonalizable matrices over \( \mathbb{C} \) has Lebesgue measure zero in \( \mathbb{R}^{n \times n} \) or \( \mathbb{C}^{n \times n} \).

Answer: Yes


Question: Given a monic polynomial \( \prod_{j=1}^n (z-z_j) \) with integer coefficients, determine if \( \prod_{j=1}^n (z-z_j^k) \) also has integer coefficients for \( k = 1, 2, 3, \ldots \).

Answer: Yes


Question: Determine the largest number of edges that can be removed from the hyper-cube graph $Q_{10}$ such that the graph still contains a Hamiltonian cycle.

Answer: 4096


Question: Solve the inverse trigonometric equation \(\sin\left(\operatorname{cot^{-1}}(x + 1)\right) = \cos\left(\tan^{-1}x\right)\). Use the identity \(\cos\left(\frac{\pi}{2} - \theta\right) = \sin\theta\) to rewrite \(\cos\left(\tan^{-1}x\right)\) as \(\sin\left(\frac{\pi}{2} - \tan^{-1}x\right)\), and then find the value of \(x\) by equating both sides.

Answer: -\dfrac{1}{2}


Question:
In an obtuse-angled triangle, the obtuse angle measures 105º. Determine the value of \( n \) such that the acute angles of the triangle are the roots of the equation:
\[
3\sec(x)+n\left(\frac{1}{\sec(x)}-\frac{1}{\csc(x)}\right)=3\left(\frac{1}{\sec(x)}+\frac{1}{\csc(x)}\right)
\]

Answer: \sqrt{3}


Question: Find the largest subset $U$ of the set $S = \{(x,y) \in \mathbb{C}^2 : y^2 = x\}$ such that for every point $s \in U$, there exists a neighborhood $N_s$ of $s$ which is homeomorphic to an open disk in $\mathbb{C}$. Provide a justification for your answer.

Answer: S


Question: Determine the number of zeros of the function \( f(z) = e^z - 4z^n + 1 \) inside the unit disk \( |z| < 1 \) using Rouche's Theorem, where \( n = 1, 2, 3, \ldots \).

Answer: n


Question: If two finite groups are isomorphic, do they have the same character table?

Answer: Yes


Question: Find the value of the real number \( a \) that minimizes the area bounded between the curves \( y = \frac{x^2}{4} \) and \( y = ax + 9 \).

Answer: 0


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{\cos(xe^x) - \cos(xe^{-x})}{\arcsin^3 x} \] You may use asymptotic simplifications such as big and small o notations and Taylor series, but L'Hopital's rule is not allowed.

Answer: -2


Question:
Given \( I = \int_{-1}^{1} \frac{e^x}{\sqrt{1-x^2}} \, dx \), determine the value of \( S = \frac{1}{2^2} + \frac{1}{2^2 4^2} + \frac{1}{2^2 4^2 6^2} + \ldots \). Choose the correct option:

A) \( \frac{I-\pi}{\pi} \)

B) \( \frac{I-1}{\pi} \)

C) \( I-\pi \)

D) \( \frac{I-\pi}{2} \)

Answer: A


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

Answer: 3


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

For all vectors \(a, b \in \mathbb{R}^d\), \((\text{sgn}(a)^T\text{sgn}(b))^2 \le (a^Tb)^2\), where \(\text{sgn}(x) = \{1 \text{ if } x \ge 0, -1 \text{ if } x < 0 \}.\)

Answer: False


Question: If $u \in L^p(\Omega)$ is continuous and its distributional derivative $u' \in L^p(\Omega)$ is also continuous, does it follow that $u$ is $C^1(\Omega)$? Justify your answer.

Answer: Yes


Question: If a sequence of functions \(f_n\) does not converge uniformly to a function \(f\), does there exist an \(\varepsilon_0 > 0\), a point \(x_0\), and a sequence of integers \(n_k \to \infty\) such that \(|f_{n_k}(x_0) - f(x_0)| \geq \varepsilon_0\)?

Answer: No


Question: Let \(A\) be a non-symmetric matrix with real entries. Does there exist a natural number \(k\) such that \(A^k = A^t\), where \(A^t\) is the transpose of \(A\)?

Answer: No


Question: Given $n > 8$, let $a_n = \sqrt{n}$ and $b_n = \sqrt{n+1}$. Determine which is greater: $a_n^{b_n}$ or $b_n^{a_n}$?

Answer: a_n^{b_n}


Question: Find the limit: \[ \lim_{x \to 0} \frac{(\sinh(x))^n - x^n}{(\sin(x))^n - x^n} \] using Taylor's theorem.

Answer: -1


Question: Find the residue of the function \( \operatorname{sech}^2(z) \) at the second order pole \( z = i\pi/2 \).

Answer: 0


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

Answer: 16


Question: Evaluate the limit: \[ \lim_{x \to 0}\left(x+e^{\frac{x}{3}}\right)^{\frac{3}{x}} \]

Answer: e^4


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

Answer: e^2


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

Answer: \infty


Question: Find the last three digits of the number \(2022^{{2023}^{2024}}\).

Answer: 872


Question: Determine the value of \( \alpha \in \mathbb{R} \) such that the integral \( \int_{\mathbb{S}^{n}} \frac{d\theta}{|\theta-\theta_{0}|^{\alpha}} \) converges, where \( \theta_{0} \in \mathbb{S}^{n} \) is fixed.

Answer: \alpha < n


Question: Evaluate the limit: \[ \lim_{x \to 2}\frac{\ln(3-x)+3\ln(x-1)}{\ln(2x-3)+3\ln(5-2x)} \] without using L'Hôpital's rule.

Answer: -\dfrac{1}{2}


Question: Determine the value of \(a\) for which the line \(y = \frac{3}{2}x - 2\) is tangent to the curve \(y^2x + a = x^2 + y^2\).

Answer: 3


Question:
Given the joint probability density function of \(X = (X_1, X_2)\):
\[
f(x_1, x_2) = \begin{cases} \frac{e^{-\frac{x_2^2}{2}}}{x_2\sqrt{2\pi}}, & \text{if } 0 < |x_1| \le x_2 < \infty, \\ 0, & \text{otherwise} \end{cases}
\]
Find the variance of \(X_1\).

Answer: \dfrac{1}{3}


Question: Evaluate the limit: \[ \lim_{x\to 5} (6-x)^{1/(x-5)} \]

Answer: \dfrac{1}{e}


Question: Let $a \in \mathbb{R}$ and $v > 0$. Consider a sequence $(X_n)_n$ of real-valued random variables such that $\sqrt{n} (X_n - a) \xrightarrow{d} N(0, v)$ as $n \to \infty$. Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable at $a$ with $f'(a)^2 \cdot v = 1$. Prove that $\sqrt{n} (f(X_n) - f(a)) \xrightarrow{d} N(0, 1)$ as $n \to \infty$.

Answer: N(0, 1)


Question: Solve the congruence equation \(x^{15} \equiv 2 \pmod{47}\).

Answer: 6


Question: Calculate the length of the closed curve defined by the equation \(x^{2/3} + y^{2/3} = 4\).

Answer: 48


Question: Determine whether the following asymptotic equation is true: \( \log_2((n!)^n) = O(n \sin(n \frac{\pi}{2}) + \log_2{n}) \).

Answer: False


Question: Show that the integral \( \int_0^\infty \frac{\cos^2 x}{x^2} \, dx \) diverges to infinity.

Answer: \infty