Question: Determine whether there exists a $C^{\infty}$ function $f$ such that the radius of convergence of the Taylor series \( \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n \) is zero.

Answer: Yes


Question: Find all pairs of natural numbers \((x, y)\) such that \(5^x + 3^y\) is a perfect square.

Answer: \emptyset


Question: Prove that for any given $r > 0$, there exists an integer $n_0$ such that for all integers $n \ge n_0$, the polynomial $1 + z + \frac{z^2}{2!} + \cdots + \frac{z^n}{n!}$ has all its zeros outside the disk $|z| \le r$.

Answer: n_0


Question: Find the area between the curves given by the parametric equations \( x = \sqrt{2} \cos{t} \) and \( y = 4\sqrt{2} \sin{t} \), and the line \( y = 4 \) for \( y \ge 4 \).

Answer: 2\pi - 4


Question: Determine the cofinality of the ordinal $\omega+1$.

Answer: 1


Question: Let $X$ be a compact connected complex manifold. Consider a surjective holomorphic map $f: X \to X$. Is it true that $f$ is a finite map, meaning every point in $X$ has finitely many preimages under $f$?

Answer: Yes


Question: Let \(Z\subset \mathbb{C}\) be the splitting field of the polynomial \(f(X)=X^7-3\). Also, let \(\alpha = 3^{\frac{1}{7}}\) and \(\zeta=e^{\frac{2\pi i}{7}}\). Calculate the degree of the field extension \([Z : \mathbb{Q}]\].

Answer: 42


Question: Let $p_1, \dotsc, p_k: \mathbb{N} \to \mathbb{Z}$ be rationally independent polynomials with zero constant term. If $t_1, \dotsc, t_k \in [0,1)$ are not all rational, determine whether the polynomial \( p(n) = \sum_{j=1}^k t_j \cdot p_j(n) \) necessarily has at least one irrational coefficient.

Answer: Yes


Question: Calculate the limit \( \lim_{n \to \infty} S_n \), where \( S_n = \sum_{k=1}^{n} \frac{kn}{k+n^3} \).

Answer: \dfrac{1}{2}


Question:
Find \( \mathbb{E}[XY] \) given the joint probability density function:
\[
f_{X,Y}(x,y)=
\begin{cases}
12y^2 & 0 \leq y\leq x\leq 1 \\
0 & {\rm otherwise}
\end{cases}
\]

Answer: \dfrac{1}{2}


Question: Given an affine scheme that is of finite type over \(\mathbb{Z}\), it can be covered by \(\text{Spec } B_i\) for \(1 \leq i \leq n\), where each \(B_i\) is a finitely generated \(\mathbb{Z}\) algebra. If \(\text{Spec } A\) is quasi-compact and can be covered by a finite number of these affine open sets, does it follow that \(A\) is also a finitely generated \(\mathbb{Z}\) algebra?

Answer: Yes


Question: Calculate the closed surface integral \( \iint\limits _S \vec F \cdot d \vec S \), where \( \vec F = x \cos^2(\pi z) \hat{i} + 2y \cos(\pi x) \hat{j} + xy^2 \hat{k} \) and \( S \) is the surface of the box defined by \( 0 \le x, y, z \le 2 \). You may use one of the integral theorems if you wish.

Answer: 4


Question: Evaluate the improper integral: \[ \int\limits_{-\infty}^{+\infty} \frac{2}{(x-1)\cdot(x^2-6x+10)}\,\mathrm{d}\,x \]

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


Question: Compute the Legendre symbol \( \left(\frac{3^{24671}}{105953}\right) \).

Answer: -1


Question: Determine the order of the smallest non-commutative ring without unity.

Answer: 4


Question: Evaluate the sum \( \sum_{r=1}^n r \binom{n}{r} x^r (1-x)^{n-r} \).

Answer: nx


Question: Find a parametrization of the curve \(x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1\) and use it to compute the area of the interior.

Answer: \dfrac{3\pi}{8}


Question: In a non-commutative ring $A$, do the ideals $I$ and $J$ satisfy $IJ = JI$? Here, $IJ$ denotes the product of the ideals $I$ and $J$.

Answer: No


Question: Calculate the total flux of the vector field \( F = (3x, xy, 1) \) across the boundary of the box \( D = \{|x| \leq 1 , |y| \leq 2, |z|\leq 3\} \).

Answer: 144


Question: If $A$ is a local domain where each non-trivial ideal factors uniquely into prime ideals, must $A$ be a principal ideal domain?

Answer: Yes


Question: Using the central limit theorem, determine the minimum number of 100W bulbs needed to ensure that an advertising board is lit for at least 20,000 hours with a probability of at least 0.9. Assume that the life of each bulb follows an exponential distribution with an average life of 600 hours.

Answer: 42


Question: Find the remainder when the number formed by writing the digits '1234' a total of 1234 times is divided by 13.

Answer: 12


Question: Solve the modular equation: \( 7x + 5 \equiv 2^{11^{2017}} \pmod{31} \). Find the integer value of \( x \) that satisfies this congruence.

Answer: 4


Question: Evaluate the limit: \[ \lim_{x \to 1^+} \frac{x-\sqrt{\arctan(x)-\frac{\pi}{4}}-1}{x-1} \]

Answer: -\infty


Question: Evaluate the surface integral \( \int_M (\nabla \times F) \cdot N \, dS \) where \( F = (e^x + z^2, \sin(yz), x^3) \) and \( M = \{(x,y,z) : y = \frac{x^2}{2} + z^2 - 4, y \leq 0\} \). The vector \( N \) is the outward-pointing unit normal to the surface \( M \).

Answer: 24\sqrt{2}\pi


Question: Three fair twenty-sided dice are rolled and arranged in decreasing order. The expected value of the largest die can be expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers. Calculate \( p+q \).

Answer: 1319


Question: Find an irreducible polynomial of degree 4 in \(\mathbb{Z}_2[x]\).

Answer: x^4 + x + 1


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

Answer: 1


Question: Given the sequence defined by the recurrence relation \(x_{n+1}=x_n + \frac{2}{x_n}\) with \(x_0 = 1\), find \(\lim\limits_{n \to \infty} \frac{x_n}{\sqrt{n}}\).

Answer: 2


Question: In trapezoid $ABCD$, the sides are $AB = 92$, $BC = 50$, $CD = 19$, and $AD = 70$, with $AB \parallel CD$. Let $P$ be a point on $AB$ such that the perpendicular distances from $P$ to $AD$ and $BC$ are equal. Determine the length of $AP$.

Answer: \dfrac{161}{3}


Question: Determine whether the following statement is true or false: If $P(|a|>k)\leq p^k$ for some $p \in (0,1)$, then $$\lim_{k\to \infty}\int_{\{|a|>k\}}a^2d\mathbb{P} = 0.$$

Answer: True


Question: Evaluate the limit: \[ \lim_{x\to\infty}4 x \left(1 + \frac{\log\frac{1 + e^3}{1 + e^x}}{x}\right) \]

Answer: 4 \ln\left(1 + e^3\right)


Question: Let $A$ be a commutative ring with an identity element, and let $f: A \to A$ be an isomorphism. If for any prime ideal $\mathfrak{p}$ of $A$, $f^{-1}(\mathfrak{p}) = \mathfrak{p}$, is $f$ necessarily the identity map?

Answer: No


Question: Calculate the integral \(\iint_{\mathbb{R}^2} e^{-(|2x-y| + |2y-x|)} \, dx \, dy\).

Answer: \dfrac{4}{3}


Question: Given that \( a = \cos \left(\frac{2 \pi}{7}\right) + i \sin \left(\frac{2 \pi}{7}\right) \), find a quadratic equation whose roots are \( \alpha = a + a^2 + a^4 \) and \( \beta = a^3 + a^5 + a^6 \). Use the fact that the sum of the 7th roots of unity is 0 to assist in your solution.

Answer: x^2 + x + 2 = 0


Question: Suppose \( k \) is a field that is finitely generated as a \( \mathbb{Z} \)-algebra. Does it follow that \( k \) is finite?

Answer: Yes


Question: Determine which of the following expressions decays faster as \( t \to \infty \): \( \frac{c^t e^t}{t^{t+1/2}} \) or \( e^{-kt^2} \), where \( c \) and \( k \) are constants.

Answer: e^{-kt^2}


Question: Calculate the integral \( S = \int_0^{5p} |\sin x + \cos x| \, dx \), where \( p \) is the period of the function \( f(x) = |\sin x + \cos x| \). Find the value of \( S \).

Answer: 10\sqrt{2}


Question: Let $S$ be a piecewise smooth, orientable, closed surface enclosing a region in $\mathbb{R}^3$. Calculate the surface integral \( \iint\limits_S \mathbf{a} \cdot \mathbf{n} \, \text{d}S \), where \( \mathbf{n} \) is the outward unit normal vector to \( S \), and \( \mathbf{a} \) is a constant vector field in \( \mathbb{R}^3 \).

Answer: 0


Question: Evaluate the integral: \[ \int_0^2 \frac{\sqrt{4-2y}}{\sqrt{2y-y^2}}\,dy. \]

Answer: 4


Question: Given \( \phi = \frac{1}{|\underline{r}|} \) and \( E = -\operatorname{grad} \phi \), show that \( \operatorname{div} E = 0 \).

Answer: 0


Question: In a group $(G, \cdot)$, there exists an element $a \in G$ such that for all $x \in G$, $ax = x^4a$. Solve the equation $x^7 = e$ in this group.

Answer: e


Question: Let $f$ be a function from the set $\{1,2,3, \dots ,10 \}$ to $\mathbb{R}$ such that \[ \left( \sum_{i=1}^{10}\frac{|f(i)|}{2^i}\right)^2 = \left( \sum_{i=1}^{10} |f(i)|^2 \right) \left(\sum_{i=1}^{10} \frac{1}{4^i} \right). \] Let $S$ be the set of all functions that satisfy this equality. Determine the cardinality of the set $S$. \(\text{Express your answer as a number.}\)

Answer: 1024


Question: Calculate $3^{500001} \bmod{1000003}$, where $1000003$ is a prime number.

Answer: 1000002


Question: Is it true that for any integer $k \geq 3$, there exist countably infinite, pairwise non-isomorphic $k$-regular graphs? Answer with 'True' or 'False'.

Answer: True


Question:
Determine the type of singularity at the point $z=0$ for the function $f(z)= \sin\bigg(\frac{1}{\cos(\frac{1}{z})}\bigg)$. Choose from the following options:
1) Removable singularity
2) Pole
3) Essential singularity
4) Non-isolated singularity

Answer: D


Question: Evaluate the integral: \[ \int_0^\pi \cos(a \sin(x)) e^{a \cos(x)} \, dx \] where \( a \neq 0 \) and \( a \in \mathbb{R} \).

Answer: \pi


Question: Determine the testing procedure that uses exactly 43 servants or rats to identify two poisoned bottles out of 1000.

Answer: 43


Question: Determine the number of subspaces of the vector space $\mathbb{Z}_p^3$, where $\mathbb{Z}_p$ is the finite field with $p$ elements.

Answer: 2p^2 + 2p + 4


Question: In triangle ABC, let H, G, and O be the orthocenter, centroid, and circumcenter, respectively. Given that the Euler line is parallel to AC and \( \angle HBC = 2 \angle OCA \), calculate the distance GO if AH = a. Express your answer in terms of a.

Answer: \dfrac{a}{3}


Question:
Let $v_1, \dots, v_n \in \mathbb{R}^n$ be vectors that form a basis. Consider the kernel of the matrix formed by these vectors as follows:
\[ \mathrm{Ker}\begin{pmatrix}v_1-v_n\\\vdots \\ v_{n-1}-v_n\end{pmatrix}. \]
Does there exist a non-zero vector $N$ in this kernel such that the following equation holds for all $v \in \mathbb{R}^n$?
\[ \sum_{i=1}^{n}\det\begin{pmatrix}v_1 \\ \vdots \\ v_{i-1} \\ v \\ v_{i+1} \\ \vdots \\ v_n\end{pmatrix} =\langle N,v\rangle. \]

Answer: Yes


Question: Let $(X,d_X)$ and $(Y,d_Y)$ be compact metric spaces, each with at least two points. Consider a Lipschitz function $f:X\times Y\to\mathbb{R}$, meaning there exists a constant $C>0$ such that for all $x,x'\in X$ and $y,y'\in Y$, \[|f(x,y)-f(x',y')|\leq C(d_X(x,x')+d_Y(y,y')).\] Define the zero marginal superlevel set of $f$ for each $x\in X$ as \[F(x) = \{y\in Y \mid f(x,y)\geq 0\},\] which is a compact subset of $Y$. The space $\mathcal{K}(Y)$ consists of non-empty compact subsets of $Y$, equipped with the Hausdorff distance \[d_H(K,K') = \sup_{y\in Y} |d_Y(y,K) - d_Y(y,K')|,\] where \[d_Y(y,Z) = \inf_{y'\in Z} d_Y(y,y').\] Consider the set-valued map $F:X\to\mathcal{K}(Y)$ defined by $F(x)$. Is $F$ Lipschitz as a map from $(X,d_X)$ to $(\mathcal{K}(Y),d_H)$? That is, does there exist a constant $C'>0$ such that for all $x,x'\in X$, \[d_H(F(x),F(x'))\leq C'd_X(x,x')?\]

Answer: Yes


Question: Let \( P \) be a non-zero projector. Determine whether the statement \( ||P||_F \geq 1 \) holds true, with equality if and only if \( P \) is an orthogonal projector.

Answer: ||P||_F \geq 1


Question: Given a complex square matrix with 1s on the main diagonal and arbitrary values elsewhere, does its column space necessarily contain a vector in which each component is nonzero?

Answer: Yes


Question: Do there exist functions $f, g: [1, 2] \to [1, 2]$ such that $f \circ g = g \circ f$ and for no $x \in [1, 2]$ do we have $f(x) = g(x) = x$?

Answer: Yes


Question: Let \( f: [0,1] \rightarrow \mathbb{R} \) be a continuous function. Does there always exist a sequence of continuous functions \( \{f_n: [0,1] \rightarrow \mathbb{R}\}_{n \in \mathbb{N}} \) such that \( f_n \rightarrow f \) pointwise, but not uniformly on \([0,1]\)?

Answer: Yes


Question:
Given two sequences of real numbers, the target sequence \( a_{1t} = 10, a_{2t} = 20, a_{3t} = 30 \) and the current sequence \( a_{11} = 1, a_{21} = 1, a_{31} = 5 \), you can change the current sequence using the following rules:
\[
a_{12} = a_{11} + \eta \cdot \mu_1 \\
a_{22} = a_{21} + \eta \cdot \mu_2 \\
a_{32} = a_{31} + \eta \cdot \mu_3
\]
where \( \mu_1 = 1, \mu_2 = 2, \mu_3 = 3 \) are fixed factors. Find the value of \( \eta \) that minimizes the mean square error between the modified sequence \( a_{12}, a_{22}, a_{32} \) and the target sequence \( a_{1t}, a_{2t}, a_{3t} \).

Answer: \dfrac{61}{7}


Question: Find the sum of the infinite series \( \sum_{n=2}^\infty \frac{7n(n-1)}{3^{n-2}} \).

Answer: \dfrac{189}{4}


Question: Find the minimum value of \( \alpha \) such that the solution \( y \) of the initial value problem \( \frac{dy}{dx} = y(2-y) \) with \( y(0) = \alpha \) exists on \([0, \infty)\).

Answer: 0


Question: Calculate the 25th derivative of the function \( f(x) = \cos{x^3} \) at \( x = 0 \), denoted as \( f^{(25)}(0) \).

Answer: 0


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

Answer: 1


Question: If $a, b, c, d$ are non-negative real numbers such that $a + b + c + d = 1$, find the maximum value of $ab + bc + cd$ in the form $\frac{p}{q}$. What is the value of $p + q$?

Answer: 5


Question: How many ways are there to roll 10 different dice such that the sum of the results is 25?

Answer: 831204


Question: Determine the value of \( n \) for which the iterative method \( x_{k+1} = \frac{1}{2} \left[x_k + \frac{a}{x_k^{n-1}}\right] \) fails to converge when finding the \( n \)-th root of a positive number \( a \).

Answer: 4


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

Answer: 0


Question: Suppose $A:(\mathbb{C}^n,+,\diamond) \to (\mathbb{C}^n,+,\cdot)$ is a unital homomorphism. Is $A$ necessarily invertible?

Answer: No


Question: Calculate the volume of the region bounded between the surfaces \( x = z^2 + 2y^2 \) and \( 2z + 4y + x = 1 \).

Answer: 4\sqrt{2}\pi


Question: Find a non-zero function in the kernel of the transpose of the linear transformation \( T: C^0(\mathbb{R}, \mathbb{R}) \rightarrow C^0(\mathbb{R}, \mathbb{R}) \) defined by \( T(f)(x) = \int_0^x f(t) \, dt \).

Answer: \delta(x)


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

Answer: \dfrac{3}{2}


Question: Given that $Z = 1$, $Y$ is a Rademacher random variable, $W \sim N(0, 1)$, and $X = WY$, determine if $X$ is conditionally independent of $Y$ given $Z$. Specifically, does $f(x, y|z) = f(x|z)f(y|z)$ hold when $Z = 1$? Here, $f$ denotes the probability density function.

Answer: Yes


Question: Let $f, g: \mathbb{R} \to \mathbb{R}$ be increasing functions such that $f(r) = g(r)$ for every $r \in \mathbb{Q}$. Must it be true that $f(x) = g(x)$ for every $x \in \mathbb{R}$?

Answer: No


Question: Let $X_1, \dots, X_n$ be independent and identically distributed standard normal random variables. Define the vector $X = (X_1, \dots, X_n)$ and the vector $Y = \frac{1}{\|X\|}(X_1, \dots, X_k)$ for $k < n$. Calculate $\mathbb{E}[\|Y\|^2] = \mathbb{E}\left[\frac{\sum_{i=1}^k X_i^2}{\sum_{i=1}^n X_i^2} \right]$. Provide your answer as a single value.

Answer: \dfrac{k}{n}


Question: Find the coefficient \( a_{2^n} \) in the power series expansion of \( \left(\frac{x}{1-x^2}+\frac{3x^3}{1-x^6}+\frac{5x^5}{1-x^{10}}+\frac{7x^7}{1-x^{14}}+\cdots\right)^2 = \sum_{i=0}^{\infty}a_{i}x^i \).

Answer: 8^{n-1}


Question: Given a probability space \((\Omega, \mathcal{F}, \mathbb{P})\), where \(\mathbb{P}(A) \geq \frac{2}{3}\), \(\mathbb{P}(B) \geq \frac{2}{3}\), \(\mathbb{P}(C) \geq \frac{2}{3}\), and \(\mathbb{P}(A \cap B \cap C) = 0\), determine the value of \(\mathbb{P}(A)\).

Answer: \dfrac{2}{3}


Question: How many k-member teams can be created from a group of n people? Express your answer in terms of combinations.

Answer: \dbinom{n}{k}


Question: Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function that is not bounded above or below. Prove that the range of \( f \) is the entire set \( \mathbb{R} \).

Answer: \mathbb{R}


Question: Is the affine function $f:\mathbb{R}^n \to \mathbb{R}^n$ defined by $f(x) = T(x) + a$, where $T$ is an invertible linear map and $a \in \mathbb{R}^n$, a diffeomorphism?

Answer: Yes


Question: Evaluate the integral \( \int_C \left( \frac{1}{z} + \frac{1}{z-1} \right) \, dz \) where \( C \) is the rectangle with corners at \( \pm 2 \pm 3i \) oriented in the anticlockwise direction.

Answer: 4\pi i


Question: Let $P(x)$ and $Q(x)$ be polynomials with integer coefficients. Let $R(P,Q)$ be the resultant of $P(x)$ and $Q(x)$. For a prime number $p$, if there exists an integer $n$ such that $p$ divides both $P(n)$ and $Q(n)$, is it true that $p$ divides $R(P,Q)$?

Answer: Yes


Question:
Let $n > 15$ and $A = A_n$ be the alternating group of degree $n$ on the set $\{1, \dots, n\}$. For any subset $X$ of $\{1, \dots, n\}$, let $\text{Stab}_A(X)$ denote the stabilizer of $X$ in $A$. Determine if the following set has a size greater than $n!/4$:
\[ \bigcup \text{Stab}_A(X) \times \text{Stab}_A(Y), \]
where the pair $(X, Y)$ runs over all pairs of subsets $X$ and $Y$ of $\{1, \dots, n\}$ such that $X \cap Y = \varnothing$, $X \cup Y = \{1, \dots, n\}$, $|X| \leq n-4$, and $|Y| \leq n-4$. Provide a justification for your answer.

Answer: Yes


Question: Find the particular solution of the differential equation given that the homogeneous solution is $28e^{-2x} - 18e^{-3x}$ and the differential equation is equal to 8.

Answer: \dfrac{4}{3}


Question: Is finiteness a quasi-isometric invariant property? Specifically, if $G$ and $H$ are two groups such that $G$ is finite and $G \sim_{QI} H$, is $H$ also finite?

Answer: Yes


Question: Verify Stokes' Theorem for the hemisphere defined by \(D: x^2 + y^2 + z^2 = 9, z \geq 0\), its bounding circle \(C: x^2 + y^2 = 9, z=0\), and the vector field \(\overrightarrow{A} = y\overrightarrow{i} - x\overrightarrow{j}\).

Answer: -18\pi


Question: How many strict total orderings are there for a set of \( N \) objects? Express your answer as a function of \( N \).

Answer: N!


Question: Consider two independent geometric random variables \(X \sim \text{Geometric}(1/2)\) and \(Y \sim \text{Geometric}(3/4)\). Find the probability \(\mathbb{P}(X-Y=2)\).

Answer: \dfrac{3}{28}


Question: Count the number of solutions \((x, y)\) in \(\mathbb{F}_q^2\) to the equation \(x^2 - ay^2 = 1\), where \(a\) is not a square in \(\mathbb{F}_q^*\). Assume \(q = p^r\) for a prime \(p\) and \(r > 0\).

Answer: q + 1


Question: Given the distance $d$ between two random points on a segment of unknown length $L$, find the maximum likelihood estimate (MLE) of $L$. Assume the points are uniformly distributed over the segment.

Answer: 2d


Question: In the universal enveloping algebra \(U(\mathfrak{g})\) of a semisimple Lie algebra \(\mathfrak{g}\), which is a Hopf algebra, what is the coproduct \(\Delta(1)\) of the identity element \(1\)? Is it \(1 \otimes 1\) or \(1 \otimes 1 + 1 \otimes 1\)?

Answer: 1 \otimes 1


Question: Let \( X \) and \( Y \) be compact, connected, smooth manifolds of the same dimension, and let \( f : X \rightarrow Y \) be a surjective smooth map such that \( |f^{-1}(p)| \leq k \) for all \( p \in Y \). Given an open dense subset \( U \subset X \), determine whether there exists a point \( p \in Y \) such that \( f^{-1}(p) \subset U \).

Answer: Yes


Question: Calculate the limit: \[ \lim_{n\rightarrow +\infty }\left ( 2\log n-\frac{n}{\log n} \right ) \]

Answer: -\infty


Question: If symmetric matrices $A$ and $B$ have identical eigenvalues, is there always an orthogonal matrix $O$ such that $B = OAO'$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \frac{3\pi}{4}}\frac{1+(\tan x)^{1/3}}{1-2\cos^2x} \]

Answer: -\dfrac{1}{3}


Question: If \((1-2x)(1-2y)(1-2z) = -27\), evaluate \(x+y+z\) where \(x = \frac{b}{a+b}\), \(y = \frac{c}{b+c}\), and \(z = \frac{a}{c+a}\).

Answer: -12


Question: Find the last two digits of $5312^{442}$. In other words, determine $5312^{442} \mod 100$. Provide your answer as a number between 0 and 99.

Answer: 44


Question: Let \( G \) be a finite, planar, disconnected graph with two components. Is it always possible to add an edge to \( G \) to make it connected while still maintaining its planarity?

Answer: Yes


Question: Evaluate the triple integral \( \int \int \int_{D} y \, dV \), where \( D \) is the region below the plane \( z = x + 1 \), above the \( xy \) plane, and between the cylinders \( x^2 + y^2 = 1 \) and \( x^2 + y^2 = 9 \). Use cylindrical coordinates to find the bounds and evaluate the integral.

Answer: 0


Question: Let $A$ and $B$ be $3 \times 3$ matrices such that $A^3 + 5AB = I$ and $A^3 - 5BA = 2I$. Find $\det(A)$.

Answer: \dfrac{3}{2}


Question: Calculate the average IQ of Mensa members, given that Mensa is a group of people with an IQ of at least 130. Assume the IQ is normally distributed with a mean (\(\mu\)) of 100 and a standard deviation (\(\sigma\)) of 15. Use the formula for the mean of a function over an interval to find the average IQ for this group.

Answer: 135.6


Question: Given the polynomial \( X^3 + pX + q = 0 \) with roots \( x_1, x_2, x_3 \), express \( x_1^8 + x_2^8 + x_3^8 \) in terms of \( p \) and \( q \).

Answer: 2p^4 - 8pq^2


Question: Does there exist a real sequence \((a_n)_{n \in \mathbb{N}} \subset [0,1)\) such that for each \(n \in \mathbb{N}\), there is a bijection \(f: [n] \to [n]\) satisfying \(a_{f(k)} \in \left[ \frac{k-1}{n}, \frac{k}{n} \right)\) for all \(k \in [n]\)? In other words, can we construct a sequence \((a_n)_{n \in \mathbb{N}} \subset [0,1)\) such that for each \(n \in \mathbb{N}\), there do not exist distinct \(i, j \in [n]\) with \(a_i, a_j \in \left[ \frac{k-1}{n}, \frac{k}{n} \right)\) for any \(k \in [n]\)?

Answer: Yes


Question: Does the set-theoretic bijection \( \Bbb{Z}_p \cong \lim_{\leftarrow n} \Bbb{Z}_p/p^n \Bbb{Z}_p \) imply that \( \Bbb{Z}_p \) is complete as a metric space? Answer 'Yes' or 'No' and provide a brief justification.

Answer: Yes


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables from a continuous distribution with a density symmetric about 0. Suppose $\mathbb{E}(|X_1|) = 2$. Define $Y = \sum_{i=1}^{n} X_i$ and $Z = \sum_{i=1}^{n} I(X_i > 0)$. Calculate the covariance between $Y$ and $Z$. \( \text{Cov}(Y, Z) = \boxed{n} \)

Answer: n


Question: Find the derivative of the quadratic form \( z^t M z \) with respect to the vector \( z \), where \( M \) is an \( n \times n \) matrix and \( z \) is an \( n \times 1 \) vector. Express your answer in terms of \( M \), \( M^t \), and \( z \).

Answer: (M + M^t) z


Question: Determine the minimum value of \( \frac{\int_0^1{x^2\left( f'(x) \right)^2 \, dx}}{\int_0^1{x^2\left( f(x) \right)^2 \, dx}} \) for all non-zero continuously differentiable functions \( f: [0,1] \to \mathbb{R} \) with \( f(1) = 0 \).

Answer: \pi^2


Question: Consider the sequence of random variables \(X_1, X_2, \ldots\) given by \(X_n = \exp(-n^2(Z - \frac{1}{n}))\), where \(Z\) is a uniformly distributed random variable on the interval \([0,1]\). Does this sequence converge almost surely? If so, to what random variable does it converge?

Answer: 0


Question: Find the infimum of the integral \[ \iint_{x^2+y^2\leqslant1}\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2\,\mathrm{d}x\,\mathrm{d}y \] for \( C^\infty \)-functions \( u \) that vanish at the origin and are equal to 1 on the boundary \( x^2+y^2=1 \).

Answer: 0


Question: Find the limit of the integral \( \int_{0}^{1}u^k\cot{\frac{\pi(1-u)}{m}}\sin{\frac{2\pi n(1-u)}{m}}\,du \) as \( n \) approaches infinity, where \( k \) is a non-negative integer and \( m \) is a real number greater than or equal to 1.

Answer: \dfrac{m}{2}


Question: Evaluate the limit \( \lim_{n \to \infty} n^{A\lg n} \cdot 2^{n-n^A} \) for \( A > 1 \).

Answer: 0


Question: Is it possible for the winding number of a closed curve around a point in the complex plane to be infinite? Specifically, can the integral \( n(\gamma,z) = \frac{1}{2\pi i}\int_\gamma \frac{dw}{w-z} \) become unbounded for some closed curve \( \gamma \) and point \( z \)?

Answer: No


Question:
Consider three random variables $v_1, v_2, v_3$ drawn independently from the uniform distribution on the interval $(0,1)$. Determine which of the following equalities correctly applies the law of total expectation:

1. \[
\begin{align}
& E_2[E_3[\max\{v_2,v_3\}\mid v_1<\max\{v_2,v_3\}]] \\
= {} &\Pr(v_2=\max\{v_2,v_3\})E_2[v_2\mid v_1 < \max\{ v_2, v_3\}, v_2=\max\{v_2,v_3\}] \\
& {} +\Pr(v_3=\max\{v_2,v_3\})E_3[v_3\mid v_1< \max\{v_2,v_3\}, v_3=\max\{v_2,v_3\}]
\end{align}
\]

2. \[
\begin{align}
& E_2[E_3[\max\{v_2,v_3\}\mid v_1<\max\{v_2,v_3\}]] \\
= {} & \Pr(v_2=\max\{v_2,v_3\}\mid v_1<\max\{v_2,v_3\}) E_2[v_2\mid v_1 < \max\{v_2,v_3\},v_2 =\max\{v_2,v_3\}] \\
& {} + \Pr(v_3 = \max\{v_2,v_3\}\mid v_1 < \max\{v_2,v_3\}) E_3[v_3\mid v_1 < \max\{v_2,v_3\}, v_3=\max\{v_2,v_3\}]
\end{align}
\]

Answer: 2


Question: Consider a supermarket with three cash registers and a single queue for customers. Four people, A, B, C, and D, enter the supermarket in that order. A, B, and C immediately go to the available cash registers, while D waits for the first available one. The service time at each cash register is exponentially distributed with a mean of 5 minutes. What is the probability that D is the last person to leave the supermarket?

Answer: \dfrac{1}{3}


Question: Find all integer solutions \((m, n)\) to the equation: \[ \frac{1}{m} + \frac{1}{n} - \frac{1}{mn^2} = \frac{3}{4} \]

Answer: (3, 2)


Question: Evaluate the line integral \( \int_C{F \cdot \hat n \, ds} \) where \( F = (2xy, -y^2) \) and \( \hat n \) is the unit outward normal to the curve \( C \) in the xy-plane. The curve \( C \) is the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) traversed in the anticlockwise direction.

Answer: 0


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

Answer: \dfrac{1}{5}


Question: Calculate the Lebesgue integral of the function $g(x) = x^2$ over the entire real line with respect to the Lebesgue measure $\lambda$. Express your answer as a limit if necessary.

Answer: \infty


Question: Evaluate the limit \( \lim\limits_{x \to 0} \frac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x^{3}} \) without using L'Hospital's rule or the conjugate method.

Answer: \dfrac{1}{4}


Question: In the context of \( \mathbb{R} \), is every point in a perfect set necessarily a condensation point? Provide a justification for your answer.

Answer: Yes


Question: Find the positive integer \(x\) such that \(x^{13} = 21982145917308330487013369\).

Answer: 89


Question: Determine whether the identity matrix \(I\) is an extreme point of the set \(S = \{A \in M_{2\times2} : \|A\|_1 \leq 1\}\).

Answer: Yes


Question: Let $A \subseteq \mathbb{R}$ be an uncountable set. Is the set of limit points $A'$ necessarily uncountable? Justify your answer.

Answer: Yes


Question: Find \( \lim_{k \to \infty} \sup_{x \in [0,1]} |f_k(x) - f(x)| \), where \( f_k(x) = x^k \) and \( f(x) = \begin{cases} 0, & \text{if } 0 \leq x < 1, \\ 1, & \text{if } x = 1. \end{cases} \)

Answer: 1


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{x\sqrt{y^2-(y-x)^2}}{(\sqrt{8xy-4x^2}-\sqrt{8xy})^3} \]

Answer: -\infty


Question: Find the minimum value of the function \( f(x,y,z) = xy + 2xz + 3yz \) subject to the constraint \( xyz = 6 \) and \( x \ge 0 \), \( y \ge 0 \), \( z \ge 0 \). Use the method of Lagrange multipliers.

Answer: 18


Question: Determine whether the lower density of the set of positive integers that are not consecutively summable is zero. That is, if $N$ is the set of positive integers that cannot be expressed as a sum of consecutive positive integers, is $\text{ld}(N) = 0$?

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to\infty}\frac{x - \sqrt{x^2+5x+2}}{x-\sqrt{x^2+0.5x + 1}}. \]

Answer: 10


Question: Determine whether the following statement is true or false: If the radius of convergence of the power series \(\sum_{n=0}^\infty a_n z^n\) is 2, then the series \(\sum_{n=0}^\infty |a_n|\) converges.

Answer: True


Question: Evaluate the limit: $$\lim_{x\to1}{\frac {\cos(\frac {\pi x} {2})} {1-\sqrt x}}$$ without using L'Hôpital's rule.

Answer: \pi


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_1^\infty \frac{2n}{e^{2x}+n^2x} \, dx \]

Answer: 0


Question: Prove the trigonometric identity: \( \tan(A) + \tan(60^\circ+A) - \tan(60^\circ-A) = 3\tan(3A) \)

Answer: 3\tan(3A)


Question: Determine the number of solutions to the equation $x^{3^{n+1}+1} = 1$ in a field of order $3^{2n+1}$. Let $F$ be a field such that $|F|=3^{2n+1}$. Find the number of elements $x \in F$ that satisfy this equation.

Answer: 2


Question: Find the minimum value of \(D(x,y,z) = \frac{(x+y+z)^3}{xy+yz+zx}\) given that \(x, y, z \geq 0\) and \(x^2 + y^2 + z^2 \geq 3\).

Answer: 9


Question: Determine the number of solutions for the equation \( \sin^{-1}[x^2+\frac{1}{3}]+\cos^{-1}[x^2-\frac{2}{3}]=x^2 \) within the interval \( x \in [-1,1] \), where \([x]\) denotes the greatest integer less than or equal to \(x\).

Answer: 0


Question: Determine whether the sequence \((x_n)\) defined by \(x_0 = a\), \(x_1 = b\), and \(x_{n+2} = -\frac{1}{2}(x_{n+1} - x_n^2)^2 + x_n^4\) for all \(n \in \mathbb{N}\) converges. If it converges, compute \(\lim_{n \to \infty} x_n\). Assume \(|x_n| \leq \frac{3}{4}\) for all \(n \in \mathbb{N}\).

Answer: 0


Question: Determine the total number of maximal ideals in the quotient ring \( \frac{\mathbb{Q}[x]}{x^4-1} \).

Answer: 3


Question: Given sequences of measurable sets $A_1, A_2, \ldots$ and $B_1, B_2, \ldots$ in a sigma-algebra $Q$, suppose $P(A_k \text{ infinitely often }) = 1$ and $P(B_k^c \text{ infinitely often }) = 0$. What is the probability that infinitely many of the joint events $A_k \cap B_k$ will occur?

Answer: 1


Question: Evaluate the integral \( \int_\gamma \frac{1}{z\sin z}dz \) where \( \gamma \) is the circle \( |z| = 5 \). Use the residue theorem to determine the value of the integral.

Answer: 0


Question: Find all non-negative integer solutions \((x, y, z)\) to the equation \(x^3 - y^3 = z! - 18\).

Answer: (9, 3, 6)


Question: Given the moment generating function \( F(t) = \frac{1}{(1-2500t)^4} \), calculate the standard deviation.

Answer: 5000


Question: Find a function \( a_n \) such that \( a_n \sim \sum_{k=1}^n (\ln k)^{1/3} \) as \( n \to \infty \).

Answer: n (\ln n)^{1/3}


Question: Evaluate the limit \( \lim_{n \to \infty} f(n) \), where \( f(n) = \lim_{x \to 0}\left(\prod_{k=1}^{n} \left(1+\sin\frac{x}{2^k}\right)\right)^{1/x} \).

Answer: e


Question: Determine the value of \( a \) such that the graphs of the functions \( y = \ln x \) and \( y = ax \) intersect at exactly two points.

Answer: \dfrac{1}{e}


Question: Calculate the line integral \( E = \int_\gamma \frac{-y \cdot dx + x \cdot dy}{x^2 + y^2} \) for the curve \( \gamma \), which is the parabola \( y = 2x^2 - 1 \) from \((1, 1)\) to \(\left(\frac{1}{2}, -\frac{1}{2}\right)\).

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


Question:
For all functions \( f: \mathbb{R} \setminus \{0\} \to \mathbb{R} \) that satisfy the equation:
\[
f\left(x+\frac{1}{x}\right)f\left(x^3+\frac{1}{x^3}\right) - f\left(x^2+\frac{1}{x^2}\right)^2 = \left(x-\frac{1}{x}\right)^2,
\]
find the sum of all distinct values of \( f(2014) \).

Answer: 0


Question: Find the limit superior of the sequence \((n+2^n)^{1/n}\).

Answer: 2


Question: Calculate the line integral of the vector field \( F(x,y) = (3x^2 \cos y + 2 \cos x, -x^3 \sin y) \) along the curve \( \gamma(t) = (t, (\pi-t)^2) \) for \( t \) in the interval \([0, \pi]\).

Answer: \pi^3


Question: Determine the asymptotic behavior of $R_n$ as $n \to +\infty$, where $R_n$ is defined as follows: Let $X_n$ be the set of transpositions in $n$ elements, with cardinality $N = \frac{n(n-1)}{2}$. For a bijection $f: {\mathbb Z}/N{\mathbb Z} \to X_n$, define \[ r(f) := \min\{|\ell-m|; \ell \ne m \quad \text{and} \quad f(\ell) \cap f(m) \ne \emptyset\}. \] Then, \[ R_n := \max\{r(f); \text{bijections} \quad f: {\mathbb Z}/N{\mathbb Z} \to X_n\}. \] Is it true that $R_n \sim cn$ for some $c \in (0, \frac{1}{2})$, or do we have $R_n = o(n)$?

Answer: R_n \sim \frac{n}{2}


Question: Determine whether the following statement is true or false: Given a variety $X=V(f)$ where $f$ is a polynomial, is it true that $I(X)=\sqrt{(f)}$?

Answer: True


Question: Find \( \lim\limits_{|X| \rightarrow \infty} f(X) \), if it exists, where \( f(X) = \frac {\log(x^2+2y^2+4z^2)}{x^2+y^2+z^2} \).

Answer: 0


Question: Find the coefficient of $x^{80}$ in the power series $$(1+x+x^{2}+x^{3}+x^{4}+	ext{...})(x^{2}+x^{4}+x^{6}+x^{8}+	ext{...})(1+x^{3}+x^{5}).$$

Answer: 115


Question:
Let \(( X_i )_{i \in \mathbb{N}}\) be a series of independent, identically distributed random variables, with \(\mathbb{E}[X_i] = 1\) and \(\mathbb{V}[X_i] = 1\). Determine the limit:
\[
\lim_{n \to \infty} \mathbb{P}\left(\frac{1}{\sqrt{n}} \sum_{i=1}^n X_i \leq \sqrt{n}\right)
\]

Answer: \dfrac{1}{2}


Question: Calculate the minimum value of the integral \( \int_{-1}^{1} |x^3 - a - bx - cx^2|^2 \, dx \) with respect to the parameters \(a, b,\) and \(c\).

Answer: \dfrac{8}{175}


Question: Find all real numbers \(a\) such that there exists a bijection \(f : [0,1] \rightarrow [0,1]\) satisfying \(f(f(x) + ax) = x\) for all \(x \in [0,1]\).

Answer: 0


Question: Assume $X_1, X_2, \ldots$ are i.i.d. positive random variables with $\mathbb{E}[X_i^4] < \infty$. Does $\sup\limits_{i=1, \ldots, n} \frac{X_i}{\sqrt{n}} \rightarrow 0$ hold in probability?

Answer: Yes


Question: Calculate the average value per second of a process where you earn $20 per second, but with a starting probability of 5% to earn $50 instead. Every second that passes without earning $50, the probability increases by 3%. Once $50 is earned, the probability resets to 5%. What is the average value per second of this process?

Answer: 24.72


Question: Let $A$ be a real $n \times n$ matrix such that $AB = BA$ for any real skew-symmetric $n \times n$ matrix $B$. Prove that $A = kI_n$ for some $k \in \mathbb{R}$.

Answer: A = kI_n


Question: Three cousins multiplied their ages together today and obtained 2016. Last year on the same day, the product of their ages was 1377. What was the product of their ages two years ago on the same day?

Answer: 832


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

Answer: 1


Question: Find the minimal number of elements in a set such that when all possible quotients of each pair of elements are taken, the remainders cover the set \( \{0, 1, \ldots, 100\} \).

Answer: 101


Question: Let $R$ be a finitely generated $\mathbb{Z}$-algebra with a linear algebraic action of $G(\mathbb{Z})$, where $G$ is a split simply-connected semisimple group. For any prime $p$, consider the map $R^{G(\mathbb{Z})} \otimes \mathbb{F}_p \rightarrow (R \otimes \mathbb{F}_p)^{G(\mathbb{F}_p)}$. Is this map necessarily surjective for sufficiently large $p$?

Answer: Yes


Question: Determine the number of ways to form 5 unique committees of 3 members each from a group of 6 mathematicians, given that no two committees can have the same members and a person can be in multiple committees.

Answer: 15504


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{(x^2+y^2+b^2)^{3/2}}\,dxdy \).

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


Question: Find the sum of the series: \[ \sum_{n=1}^{\infty}\left(\frac{\frac{3}{2}}{2n+3}-\frac{\frac{3}{2}}{2n-1}\right) \]

Answer: -2


Question: Let \( p = \operatorname{exp} : \mathbb{C} \to \mathbb{C}^* \) be a covering map and \( (U,z) \) a chart of \( \mathbb{C}^* \) with \( z = x + iy \). Given the one-form \( \omega = \frac{dz}{z} \) on \( U \), find the pullback \( p^*\omega \).

Answer: dw


Question: Find the slope of the tangent line to the curve \( \sin(\pi x) + \ln(x^2)y + \sec(\pi x) = xy \) at \( x = 1 \). Choose the correct answer from the following options: \( A: -2\pi \), \( B: -\pi \), \( C: -1 \), \( D: -(\pi+1) \).

Answer: D


Question: Evaluate the limit \( \lim_{Z \to Zi} (Z^2 - \bar{Z}) \).

Answer: -1 + i


Question: Find the number of ways to collect a total of $20 from 4 different children and 3 different adults, where each child can contribute up to $6 and each adult can give up to $10. Use generating functions to determine the coefficient of $x^{20}$ in the expansion of the generating function \((1 + x + x^2 + \ldots + x^6)^4 \cdot (1 + x + x^2 + \ldots + x^{10})^3\).

Answer: 112567


Question: Find the maximum value of \( M \) such that for all real numbers \( a, b, c \), there exists a point \( x_0 \) in the interval \([0, 4]\) where \(|x_0^3 + ax_0^2 + bx_0 + c| \geq M\).

Answer: 2


Question: Find the limit: \[ \lim_{n \to \infty} \frac{ \sum_{i=1}^n \lfloor i^3 x \rfloor}{n^4} \]

Answer: \dfrac{x}{4}


Question: Find \( \lim_{x \to 0} \frac{1-\cos(ax)}{1-\sqrt{1+x^2}} \) in terms of \( a \) without using L'Hospital's Rule.

Answer: -a^2


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

Answer: 2\pi


Question: Compute the limit: \[ \lim_{n \to \infty} \sqrt[n]{ \int_{0}^{1} (1+x^n)^n \, dx} \]

Answer: 2


Question: Determine the Galois group of the polynomial \(X^5 - 1\) over the finite field \(\mathbb{F}_7\).

Answer: \mathbb{Z}/4\mathbb{Z}


Question: Is it always possible to solve a system of $n$ linear equations with $n$ unknowns, where all coefficients and right-hand sides are nonnegative integers and the unique solution consists of nonnegative rational numbers, using only restricted subtraction-moves, unrestricted addition, and multiplication by positive rational numbers? The restricted subtraction-move allows subtracting one equation from another only if the coefficients and right-hand side of the latter dominate those of the former.

Answer: Yes


Question: Determine the norm of the integral operator \( A: (C^1[0,1];|||\cdot|||)\rightarrow(C^1[0,1];||\cdot||_{\infty}) \) defined by \( Af(x)=\int_0^x f(t)dt \), where \( |||f|||= ||f||_\infty+||Af||_\infty \). What is \( ||A||? \)

Answer: \dfrac{1}{2}


Question: Solve the equation \( 6^{\log_{5}{x}} \log_{3}(x^{5}) - 5^{\log_{6}{6x}} \log_{3}{\frac{x}{3}} = 6^{\log_{5}{5x}} - 5^{\log_{6}{x}} \). The sum of the solutions can be expressed as \( a^{b/c} + d \). Find the value of \( abc + d \).

Answer: 91


Question: Let $X$ be a Hausdorff, locally compact but non-compact topological space. If the one-point compactification of $X$ is connected, can $X$ have compact connected components? Answer yes or no and justify your answer.

Answer: No


Question: Calculate the value of the following sum: \[ \sum_{u=0}^{22} u(u-1) \binom{22}{u} \]

Answer: 484442112


Question: Evaluate the integral \( \int_0^1 \frac{x^{(m-1)} + x^{(n-1)}}{(1+x)^{(m+n)}} \, dx \) in terms of the Beta function.

Answer: B(m, n)


Question: Determine whether the sequence \( x_n = (\sin(n), \cos(n), 1 + (-1)^n) \) in \( \mathbb{R}^3 \) has a convergent subsequence.

Answer: Yes


Question: Find the volume of the solid bounded by the surfaces \( z = x^2 + y^2 + 1 \) and \( z = 2 - x^2 - y^2 \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the integral: \[ \int_{0}^{\infty}\text{sech}^2(x+\tan(x))\,dx \]

Answer: 1


Question: Find all functions \( f: \mathbb{Q} \to \mathbb{Q} \) such that \( f(x+f(x)+2y)=2x+2f(f(y)) \) for all \( x,y \in \mathbb{Q} \).

Answer: f(x) = x


Question: Consider a vector field $X$ on the plane. Define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ by $D(f)=(\Delta\circ L_X-L_X\circ \Delta)(f)$, where $\Delta$ is the standard Laplacian. Is there a vector field $X$ with two nested closed orbits $\gamma_1 \subset \gamma_2$ such that there exists a smooth function $f$ for which $D(f)$ does not vanish on the closure of the annular region surrounded by $\gamma_1$ and $\gamma_2$?

Answer: Yes


Question: Evaluate the integral: $$ \int_0^4 z \left(\sqrt{16 - z^2} - \sqrt{25 - z^2} + 3 \right) dz $$

Answer: \dfrac{38}{3}


Question: Find the area of the part of the surface defined by $w = z^2$ in the complex space $\mathbb{C}^2$, where $|z| \le 1$. Express your answer in terms of real numbers.

Answer: 3\pi


Question: Let $A$ and $B$ be real symmetric $n \times n$ matrices with the same rank, where $B$ differs from $A$ only by two sufficiently small nondiagonal entries. Does $B$ have the same signature as $A$?

Answer: Yes


Question: Determine the speed of convergence of Newton's method for the function \( f(x) = x^3 - 5x^2 + 8x - 4 \) with the initial guess \( x_0 \neq 0 \) lying in the convergence ball of the solution \( \alpha = 2 \).

Answer: \frac{1}{2}


Question: Evaluate the limit: $$\lim_{h\to 0}\frac{1}{h}\int_{h}^{2h}f(t)\,dt$$ where $f(t)$ is a continuous function.

Answer: f(0)


Question: Determine the values of \( \alpha \in \mathbb{R} \) for which the function \( f(x) = \mathbb{1}_{(0,1)}(x)x^\alpha \) belongs to the space \( L^p(\mathbb{R}) \), where \( p \in [1, \infty) \) and \( \lambda \) is the Lebesgue measure on \( \mathbb{R} \).

Answer: \left( -\frac{1}{p}, \infty \right)


Question: Consider a normed space $\mathbb{X}$ that is complete (a Banach space) and another normed space $\mathbb{Y}$ that is not complete. Can a bounded linear map $A: \mathbb{X} \to \mathbb{Y}$ be bijective?

Answer: Yes


Question: Determine whether there exists a Hausdorff non-discrete topology on the set of integers \( \mathbb{Z} \) that makes it a topological group with respect to the usual addition operation.

Answer: Yes


Question: A company produces devices, each of which is defective with probability \( p \), independently of others. The number of devices produced in a day, \( X \), follows a Poisson distribution with parameter \( \mu \). Calculate the probability that no defective devices are produced on a given day.

Answer: e^{-\mu p}


Question: In a category, if there exists a unique morphism \( f: A \to B \) and a unique morphism \( g: B \to A \), are \( f \) and \( g \) necessarily isomorphisms?

Answer: No


Question: Determine the degree of the splitting field of the polynomial \( f(t) = t^3 - 1 \) over \( \mathbb{Q} \).

Answer: 2


Question:
A hotel has 100 rooms and charges guests for their rooms in advance. The probability that any booked guest shows up by 10pm is \( p = \frac{19}{20} \), and guests show up independently. Let \( X \) denote the number of booked guests who show up before 10pm.

The hotel earns $50 in profit for every guest and no-show, but must pay $60 in compensation to each guest who is unable to stay due to overbooking. Assuming unlimited demand for pre-booked rooms, how many rooms should the hotel sell to maximize expected profit?

Answer: 108


Question: Evaluate the integral \( \int_0^1 \frac{x-1}{\ln(x)} \, \mathrm{d}x \).

Answer: \ln 2


Question: Calculate the variance of the random variable with probability density function \( f(x) = 2(1+x)^{-3} \) for \( x > 0 \).

Answer: \infty


Question:
Evaluate the improper integral to find the surface area of a cylinder:
\[ 4 \int_{0}^{r} \int_{0}^{h} \frac{r}{\sqrt{r^2-x^2}} \, dy \, dx \]

Answer: 2\pi rh


Question: Inside the square $MNPK$, a point $O$ is chosen such that the distances from $O$ to the vertices $M$, $N$, and $P$ are in the ratio $MO:ON:OP = 1:2:3$. Find the measure of the angle $\angle MON$.

Answer: 135^\circ


Question: Consider a sequence of i.i.d. random variables $(X_i)_{i \geq 1}$ with mean 0 and variance 1. It is known from Donsker's theorem that $$\left(\frac{1}{\sqrt{n}}\sum\limits_{i=1}^{nt}{X_i}\right)_t \rightarrow (W_t)_t$$ where $(W_t)_t$ is a standard Brownian motion. Determine whether the following statement is true: $$\left(\frac{1}{\sqrt{n}}\sum\limits_{i=1}^{n^a t}{X_i}\right)_t \rightarrow 0$$ for $0<a<1/2$. Justify your answer.

Answer: True


Question: In a first-price auction with two bidders, each bidder may have either a low valuation $V_L$ or a high valuation $V_H$. Bidders do not know the valuations of others. Determine the minimum bid (i.e., the lowest bid in the support $[\underline{b_i},\overline{b_i}]$) for a bidder with a low valuation $V_L$. Assume that strategies involve a probability distribution over the domain $[\underline{b_i},\overline{b_i}]$. What is $\underline{b_i}$ for a $V_L$ type bidder?

Answer: 0


Question: Determine the value of $k$ such that the image of the circle $|z - 1| = k$ under the complex function $f(z) = \frac{z-3}{1-2z}$ is a line.

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_C \frac{e^z}{z^3} \, dz \), where \( C \) is the boundary of the square with corners at \( \pm 4 \pm 4i \).

Answer: \pi i


Question: Evaluate the contour integral \( \int_C \frac{1}{z^3(z^2+1)} \, dz \), where \( C \) is the circle \( |z-1| = \frac{3}{2} \).

Answer: 0


Question: Evaluate the integral \( \iint_R \frac{y^2}{x} \, dy \, dx \), where \( R \) is the region in the \( xy \)-plane bounded by the curves \( y = x^2 \), \( y = 4x^2 \), \( xy = 1 \), and \( xy = 5 \).

Answer: 12


Question:
A light can emit 4 different colors: blue, red, green, and yellow. Initially, the light emits blue. Each second, the color changes according to these probabilities:

- From blue, it turns red the next second.
- From red, there is a 1/3 probability of turning blue and a 2/3 probability of turning green.
- From green, there is a 1/2 probability of turning red and a 1/2 probability of turning yellow.
- From yellow, there is a 2/3 probability of turning green and a 1/3 probability of turning off.

Calculate the expected number of seconds it takes for the light to turn off.

Answer: 18


Question: Determine the dimension of the vector space $V$ of functions that are holomorphic on the extended complex plane except possibly at the points $0$ and $i$, where they have poles of order at most two. Provide an explicit basis for this vector space.

Answer: 5


Question: Evaluate the integral: \[ \int_{0}^{2\pi} \frac{\cos(3\theta)}{5-4\cos(\theta)}d\theta \] without using the residue theorem.

Answer: \dfrac{\pi}{12}


Question: Evaluate the contour integral \( \oint_{|z|=1}\log\left|\frac{1}{1+z^p}\right|\frac{dz}{z} \) where \(0<p<\frac{1}{2}\).

Answer: 0


Question: Consider the \( \mathbb{R} \)-linear space \( \mathbb{R}^n \) and subspaces \( U, W \le \mathbb{R}^n \) with \( \dim(U)=\dim(W)=n-1 \). Determine whether there exists a subspace \( V \le \mathbb{R}^n \) such that \( U \oplus V = W \oplus V = \mathbb{R}^n \).

Answer: Yes


Question: Find \(x \in \mathbb{Z}_{250}\) such that \(x \equiv 248^{156454638} \pmod{250}\).

Answer: 194


Question:
Let \( f(z) = u(r,\theta) + iv(r,\theta) \) be an entire function in \( \mathbb{C} \). Compute the integral:
\[
\int_{0}^{2 \pi} [u(r,\theta)\cos(\theta) - v(r,\theta)\sin(\theta)] \, d\theta
\]
Use the hint: write \( z = e^{i \theta} \) and consider the contour integral on the curve \( |z| = 1 \).

Answer: 0


Question:
Given the cumulative distribution function:
\[ F(x) = \begin{cases} 0 & \text{if } x < 0 \\ \frac{1}{4} + \frac{1}{6}(4x - x^2) & \text{if } 0 \leq x < 1 \\ 1 & \text{if } x \geq 1 \end{cases} \]
Calculate the probability \( P(X = 0 \mid 0 \leq x < 1) \).

Answer: \dfrac{1}{3}


Question: Let \( f \in \mathbb{C}[x] \) be a monic polynomial of prime degree \( p \) such that \( f \) is not co-prime with any of its derivatives \( f^{(k)} \) for \( 1 \leq k < p \). Prove or disprove that there exists \( a \in \mathbb{C} \) such that \( f(x) = (x-a)^p \).

Answer: (x - a)^p


Question: Consider a compact connected Riemannian 2-manifold $M$ with positive curvature and diameter 1. Determine the supremum of the average distance between two randomly chosen points on $M$, distributed according to the volume form on $M$. Assume $M$ is diffeomorphic to a 2-sphere. What is this supremum?

Answer: \dfrac{1}{2}


Question: Let \((f_n)_n\) be a sequence in \(C^2([0,1])\) converging pointwise to \(g \in L^1([0,1])\). Assume that for all \(n \in \mathbb{N}\), \(f_n'' < h\), where \(h \in L^1([0,1])\). Is it true that \(\lim \int_0^1 f_n = \int_0^1 g\)?

Answer: Yes


Question: Evaluate the integral \( I(k) = \int_{-\pi/2}^{\pi/2} e^{i k \tan(\phi)} \, d\phi \) as a function of \( k \), where \( k \in \mathbb{R} \).

Answer: \pi e^{-|k|}


Question: Determine all linear transformations \( T: V \to V \) for a real vector space \( V \) of dimension \( n > 1 \) such that the matrix representation of \( T \) is independent of the basis. This means that for any invertible matrix \( P \), the condition \( TP = PT \) holds. Describe the form of such transformations.

Answer: T = \lambda I


Question: Let $f, g: [0, 1] \to [0, M]$ be measurable functions for some $M < \infty$. Is the $L^1$ norm submultiplicative for these functions, i.e., does the inequality \( \|fg\|_1 \leq \|f\|_1 \|g\|_1 \) hold?

Answer: No


Question: Let $A$ and $B$ be C*-algebras, where $A$ is nonunital and $B$ is unital. Suppose $\phi: A \rightarrow B$ is a contractive completely positive map. Consider the double adjoint map $\phi^{**}: A^{**} \rightarrow B^{**}$. Identify the double duals with their enveloping von Neumann algebras. Does $\phi^{**}$ map positive operators to positive operators?

Answer: Yes


Question: Let $G$ be the group of all nonzero complex numbers under multiplication, and let $\bar{G}$ be the group of all real $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ -b & a \end{pmatrix}$, where not both $a$ and $b$ are $0$, under matrix multiplication. Prove that $G$ and $\bar{G}$ are isomorphic by constructing an explicit isomorphism from $G$ to $\bar{G}$. Provide the mapping and verify that it is a homomorphism, bijective, and preserves the group operation.

Answer: \phi(a + bi) = \begin{pmatrix} a & b \\ -b & a \end{pmatrix}


Question: If the length of a cycloid is 4 times the diameter of the rotating circle, determine how many times the area under the arch of the cycloid is compared to the area of the rotating circle.

Answer: 3


Question: Evaluate the limit: \[ \lim_{n \to \infty }\frac{n!}{(n+1)!}\frac{\sin((n+1)x)}{\sin(nx)} \]

Answer: 0


Question: Evaluate the integral \( \int_{0}^{\pi} e^{\cos\theta} \cos(\sin\theta) \, d\theta \).

Answer: \pi


Question: Given that $x, y, z > 0$ are distinct and $x + y + z = 1$, find the minimum value of the expression \[ \frac{(1+x)(1+y)(1+z)}{(1-x)(1-y)(1-z)}. \]

Answer: 8


Question: Compute the limit: $$ \lim_{x \to 0}{\frac{1-\cos^3(x)}{x\sin(2x)}}$$

Answer: \dfrac{3}{4}


Question: A coin is flipped 100 times. Let $X$ be the number of heads in the first 70 flips and $Y$ be the number of heads in the last 50 flips. Compute the correlation of $X$ and $Y$. Provide your answer as a decimal rounded to two decimal places.

Answer: 0.34


Question: Given a matrix $A$ with a spectral radius less than 1, and assuming the matrix is non-negatively valued, does the geometric series of each entry of the matrix converge?

Answer: Yes


Question: Find the derivative of \( \tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right) \) with respect to \( \tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right) \) at \( x=0 \).

Answer: \dfrac{1}{4}


Question: Determine whether there exists a real $4 \times 4$ matrix $A$ such that $A^2 + I = 0$, where $I$ is the $4 \times 4$ identity matrix.

Answer: Yes


Question: Let $A$ be a positive definite matrix. Compute the derivative of $\log \det A$ with respect to $A$. Express your answer in terms of $A^{-1}$. \( \frac{d}{dA}\log (\det (A)) = A^{-1} \).

Answer: A^{-1}


Question: Let \( f: \mathbb{R}^2 \to \{0,1\} \) be a function. Determine whether there exists an equilateral triangle in the plane with vertices \( x, y, z \) such that \( f(x) = f(y) = f(z) \).

Answer: Yes


Question: Solve the equation \( z^{\sqrt{5}} = 1 \) for \( z \) and determine the number of unique solutions.

Answer: \infty


Question:
Given the transition matrix of a Markov chain:
\[
\begin{pmatrix} \frac{7}{10} &  \frac{3}{10} & 0 & 0 \\
0 & \frac{5}{10} & \frac{4}{10} & \frac{1}{10} \\
0 & 0 & \frac{3}{5} & \frac{2}{5} \\
0 & 0 & 0 & 1\end{pmatrix}
\]
calculate the expected number of steps required for a process starting in the first state to reach the absorbing state.

Answer: \dfrac{22}{3}


Question:
Find the limit of the function as \( n \to \infty \):
\[ \lim_{n\rightarrow\infty}n\left[1-\left(1+\frac{1}{n}\right)^{e}\right] \]

Answer: -e


Question: If \( f(z) \) is defined in the unit disk such that both \( f^2(z) \) and \( f^3(z) \) are analytic in the disk, is \( f(z) \) necessarily analytic in the disk?

Answer: Yes


Question: Determine the smallest integer $k$ such that the semigroup $C_n$ is equal to the union of the $k$-th powers of the set $A$, where $A = \{\begin{bmatrix}2\\1  \end{bmatrix},\begin{bmatrix}3\\2  \end{bmatrix},\cdots, \begin{bmatrix}n\\n-1  \end{bmatrix}\}$ and $C_n$ is the semigroup of all order-preserving and decreasing transformations in the full transformations semigroup $T_n$. \[ C_n = \bigcup\limits_{i=1}^{k} A^k \]

Answer: n-1


Question: Determine the radius of convergence of the power series \( \operatorname{f}(x) = \sum_{n \ge 0} a_n x^n \) given that \( a_i \ge 0 \) and \( a_{n+1} + a_{n+2} + \cdots < 1/n! \).

Answer: \infty


Question: Let the sequence \(\{a_n\}\) be defined by \(a_1=1\) and \(a_{n+1}=a_n+[\sqrt{a_n}]\) for \(n \geq 1\), where \([x]\) denotes the integer part of \(x\). Find the limit \(\lim\limits_{n\to\infty}\frac{a_n}{n^2}\).

Answer: \dfrac{1}{4}


Question: Determine whether the function \( h(x) = \begin{cases} \exp\left(-\frac{1}{x^2}\right) & x \neq 0 \\ 0 & x = 0 \end{cases} \) is differentiable at \( x = 0 \).

Answer: 0


Question: Determine the field to which the quotient \( \frac{\mathbb{Z}[x]}{\langle 2x-1, 5 \rangle} \) is isomorphic.

Answer: \mathbb{F}_5


Question: Calculate the volume of the set \( B = \{(x,y,z) \in \mathbb{R}^3 : 0 \leq y \leq x\sqrt{3}, 0 \leq z \leq 2, 1 \leq x^2 + y^2 \leq 4\} \) using cylindrical coordinates. Express your answer as an integral and evaluate it.

Answer: \pi


Question: Let $X, Y, Z$ be independent standard uniform random variables. Find the probability $P\left(\frac{X}{Y} < Z\right)$. Provide your answer as a single probability value.

Answer: \dfrac{1}{4}


Question: In the given figure, $AB = BC$ and $AC = BD$. Find the angle $x$.

Answer: 30^\circ


Question: Isaac is planning a nine-day holiday. Each day, he will either go surfing, go water skiing, or rest. On any given day, he does only one of these three activities. He never does different water sports on consecutive days. How many possible schedules are there for his holiday?

Answer: 3363


Question: Determine whether the following statement is true or false: If $f$ is a real-valued injective differentiable function on the interval $(a,b)$, then $f'(x) \neq 0$ for all $x \in (a,b)$. Justify your answer.

Answer: B


Question: Compute the limit: \[ \lim_{x\rightarrow 0}\frac{\cos{x}-1}{\ln{(1+\sin^2{x})}}. \]

Answer: -\dfrac{1}{2}


Question: Evaluate the limit: \( \lim_{n\to\infty} \frac{(\log n)^n}{n^{\log n}} \)

Answer: \infty


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{m(m - 1)\cdots(m - n + 1)}{(n - 1)!}x^n \] where \( x \in (-1, 1) \).

Answer: 0


Question: Express the integral \( \int_1^e \frac{(\ln(\ln(x)))^3}{x}\,dx \) in terms of the Gamma function.

Answer: -\Gamma(4)


Question: Solve the differential equation \( y'' + (y')^2 \cdot e^x = 0 \) with the initial conditions \( y(0) = 1 \) and \( y'(0) = 1 \). Find \( y(x) \).

Answer: 2 - e^{-x}


Question: In a commutative monoid \((M,+,e)\) with unit \(e\), an element \(a\in M\) is called a cancellative element if for any \(b,c \in M\), the equation \(a+b=a+c\) implies \(b=c\). Consider the commutative monoid of natural numbers \((\mathbf{N},+,0)\). Suppose there are two morphisms of monoids \(f:(\mathbf{N},+,0)\rightarrow (M,+,e)\) and \(g:(M,+,e)\rightarrow (\mathbf{N},+,0)\) such that \(g\circ f= \text{id}\). Additionally, the monoid \((M,+,e)\) is torsion-free, meaning there does not exist a natural number \(n>0\) and some element \(x\in M-\{e\}\) such that \(n x=e\). Is the element \(a=f(1)\) automatically a cancellative element in \((M,+,e)\)?

Answer: Yes


Question: Three points are located at the vertices of an equilateral triangle with side length $a$. Each point moves with a constant speed $v$, with the first point always heading towards the second, the second towards the third, and the third towards the first. Determine the time it takes for the points to converge.

Answer: \dfrac{2a}{3v}


Question:
Let $a, b, c$ be non-zero real numbers such that the following condition holds:
\[ \int_{0}^{1} (1+\cos^8 x)(ax^2+bx+c) \, dx = \int_{0}^{2} (1+\cos^8 x)(ax^2+bx+c) \, dx. \]
Determine the nature of the roots of the equation $ax^2 + bx + c = 0$. Choose the correct option:
1. No root in $(0,2)$
2. At least one root in $(0,2)$
3. Both roots in $(0,2)$

Answer: B


Question: Suppose $\lambda_1, \lambda_2, \dots, \lambda_k$ are complex numbers such that for all natural numbers $n$, the equation $\lambda_1^n + \lambda_2^n + \dots + \lambda_k^n = 0$ holds. Can we conclude that $\lambda_1 = \lambda_2 = \dots = \lambda_k = 0$?

Answer: Yes


Question: Given the differential equation \( y'' + ay' + by = 0 \) and a solution \( y(x) = e^x \cos(x) \), find the value of \( a + b \).

Answer: 0


Question: Determine the number of \( n \times n \) matrices with entries of either 0 or 1 such that the sum of the entries in each row and each column is even.

Answer: 2^{(n-1)^2}


Question: Determine whether the following statement is true or false: For any convex, twice-differentiable vector-valued function \( f: \mathbb{R}^n \to \mathbb{R}^m \), its \( n \times n \) Hessian matrix is positive semi-definite.

Answer: B


Question:
Determine the radius of convergence of the power series \( f(z) = \sum_{n=0}^{\infty} p(n)z^n \), where \( p(x) \) is a polynomial of degree \( k \geq 1 \) and \( z \) is a complex variable. Choose the correct answer from the following options:

- \( 0 \)
- \( 1 \)
- \( k \)
- \( \infty \)

Answer: B


Question: Let $X$ be a completely regular space, and let $C = C(X,[0,1])$ be the collection of continuous functions from $X$ to $[0,1]$. Consider the map $\phi: X \to [0,1]^C$ defined by $\phi(x) = (f(x))_{f \in C}$. By Tychonoff's theorem, $[0,1]^C$ is compact in the product topology, and $\overline{\phi(X)}$ is compact in $[0,1]^C$. Define $\beta X = \overline{\phi(X)}$, the Stone-Cech compactification of $X$. If $V$ is an open subset of $\beta X$, is $U = V \cap X$ open in $\beta X$?

Answer: No


Question: Find the number of positive integral solutions to the equation \(2a + 3b + 4c + 5d = 25\).

Answer: 7


Question: Find the limit of \( S_n \) as \( n \to \infty \), where \( S_n = \int_{0}^{1} \frac{n x^{n-1}}{1+x} \, dx \) for \( n \ge 1 \).

Answer: \dfrac{1}{2}


Question:
For given integers \( n, m \in \mathbb{N} \) such that \( n^9 = 19m + r \), determine the possible values of \( r \) modulo 19. Choose the correct option:

(a) Only 0
(b) Only 0, 1, -1
(c) Only 1, -1
(d) None of the above

Answer: B


Question: Find all prime numbers \( p \) such that \( 2017^{p-1} + p^3 \) is a perfect square.

Answer: 2


Question:
Evaluate the limit as \( n \to \infty \) for the sequence defined by \( a_1 = 1 \) and \( a_n = n(1 + a_{n-1}) \) for all \( n \geq 2 \):
\[
\lim_{n\to \infty} \left(1+\frac{1}{a_1}\right)+\left(1+\frac{1}{a_2}\right)+\cdots+\left(1+\frac{1}{a_n}\right)
\]

Answer: \infty


Question: Is it possible to construct an algorithm that exactly simulates an exponential random variable of rate $p$, using the ability to simulate a Bernoulli random variable with success probability $p > 0$ and a uniform random variable on $[0, 1]$, given that $p$ is unknown? The algorithm can have a random run-time but must have an almost-surely finite run-time.

Answer: Yes


Question: Evaluate the limit: \( \lim_{x\rightarrow -3} f(f(x)) \) given that \( f(x) = 2 \) for \( x \) near \( -3 \) and \( f(2) = -1 \).

Answer: -1


Question:
Use integration by parts to prove the Riemann–Lebesgue lemma for the integral \( \int_a^b f(x) \sin(\lambda x) \, dx \), where \( f' \) is continuous on \([a,b]\). Show that:
\[ \lim_{\lambda \to \infty}\int_a^b f(x)\sin(\lambda x) \, dx =0 \]

Answer: 0


Question: Consider the equation $y + xy^2 - e^{xy} = 0$. Determine if this equation can be solved for $y$ as a function of $x$ in a neighborhood of $(x_0, y_0)$ with $x_0 = 0$ and a suitable $y_0$. If it can, find the value of $f'(0)$, where $y = f(x)$. Provide your answer as $f'(0) = \boxed{\text{value}}$.

Answer: 0


Question: Determine the number of real roots of the equation \((x-1)x(x+1)(x+2) - 1 = 0\).

Answer: 2


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\frac{\cos x+x \sin x}{x^2+\cos^2x} \, dx \).

Answer: \pi


Question: Find all complex numbers $z$ that satisfy the equation $z^4 = (z-4)^4$. Provide your answer in the form of a set of complex numbers.

Answer: \{2, 2 + 2i, 2 - 2i\}


Question: Find the value of the limit \( \lim_{a\searrow 0}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^{a}} \).

Answer: \dfrac{1}{2}


Question: Solve the differential equation using Laplace transforms: \[ Y''(t) + (t+1)Y'(t) + tY(t) = 0 \] given the initial conditions \( Y(0) = 1 \) and \( Y'(0) = -1 \).

Answer: e^{-t}


Question: Find the slant asymptote of the function \( y = x \operatorname{arccot}(x) \).

Answer: y = \pi x + 1


Question: Evaluate the limit: \[ \lim_{x \to \frac{\pi}{2}} \biggl|\frac{1}{(1+\sin x)\log(\sin x)}+\frac{1}{\cos^2x} \biggl| \]

Answer: \dfrac{1}{4}


Question: Find the second degree Taylor polynomial of the function \( g(x,y) = e^{f(x,y) - 5} \sin(y) \) at the point \((1,0)\), where \( f: \mathbb{R}^2 \to \mathbb{R} \) is a \( C^2 \) function with the second degree Taylor polynomial centered at \((1,0)\) given by \( P(x,y) = 4 + x + xy + \frac{y^2}{2} \).

Answer: xy + y^2


Question: Find all positive integer solutions \((x, y)\) to the equation \[ y^3 = x^3 + 8x^2 - 6x + 8. \]

Answer: (9, 11)


Question:
Find all real-valued functions \( f \) that are continuous on \([0,1]\) and satisfy the condition:
\[ \int_0^1 f(x) x^n \, dx = 0, \quad n = 0, 1, 2, \ldots \]

Answer: 0


Question: Evaluate the triple integral \( \int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz \, dx \, dy \).

Answer: \dfrac{26}{105}


Question: Determine whether there exists a real-valued, non-negative, bandlimited function. Provide a justification for your answer.

Answer: Yes


Question: Solve the equation \( e^{-x} - x = 0 \) using the Lambert W function.

Answer: W(1)


Question: Evaluate the integral: \[ \int_0^{+\infty} \frac{e^{-3t} - e^{-6t}}{t} \, dt \] and verify that its value is \( \ln(2) \).

Answer: \ln(2)


Question: In a setup with 5 blue nails in one column and 3 red nails in another column, you can attach a string between any red nail and any blue nail. Determine the number of ways to attach strings such that every nail has at least one string attached to it.

Answer: 16081


Question: Determine the Galois group of the splitting field of the polynomial \(x^3 - 2\) over \(\mathbb{Q}\). Is it isomorphic to \(\mathbb{Z}_6\) or \(\mathbb{Z}_2 \times \mathbb{Z}_3\)?

Answer: S_3


Question: In \(\triangle ABC\), \(AB=13\), \(BC=14\), \(CA=15\). Points \(D, E, F\) lie on \(BC, CA, DE\) respectively such that \(AD, DE, AF\) are perpendicular to \(BC, AC, BF\). If segment \(DF = \frac{M}{N}\) where \(M\) and \(N\) are coprime, find \(M+N\).

Answer: 21


Question: Determine whether the function \( f(t) = \sin t + \log(\tan(\frac{t}{2})) \) has derivatives of all orders.

Answer: Yes


Question: Evaluate the integral: $$\int_{-\infty}^\infty \frac{e^{-x} \, dx}{1-e^{-2x}}$$

Answer: 0


Question: If the projections of a space curve \( \gamma:[0,1] \to \mathbb{R}^3 \) onto the planes perpendicular to the coordinate axes are rectifiable, does it imply that the space curve \( \gamma \) itself is rectifiable?

Answer: Yes


Question: Find the minimum distance between the point \((1,1,0)\) and the points on the sphere defined by the equation \(x^2+y^2+z^2-2x-4y=4\) using the method of Lagrange multipliers.

Answer: 2


Question: Given that \(a, b, c > 0\) and \(a^2 + b^2 + c^2 = 4\), and \(4(a^2 + 2) = (a^2 + b + c)^2\), find the maximum possible value of \(a + b + c\).

Answer: 2 + \sqrt{2}


Question: Determine the necessary and sufficient condition for the point \((X_0, Y_0)\) to be an interior point of the parabola given by the equation \(y^2 = 2px\). A point \(M\) is considered an interior point if any line passing through \(M\) (not parallel to the parabola's axis) intersects the parabola at two distinct points.

Answer: Y_0^2 < 2pX_0


Question: Find the value of the limit: \[ \lim_{n\to\infty} \frac{a_n}{b_n} \] where \( a_n = \sum_{m=1}^{\lfloor n/2 \rfloor} \frac {n!}{2^m(n-2m)!} \) is the number of elements of order 2 in the symmetric group \( S_n \), and \( b_n = \sum_{m=1}^{\lfloor n/4 \rfloor} \frac {n!}{2^{2m}(n-4m)!} \) is the number of elements of order 2 in the alternating group \( A_n \).

Answer: 2


Question: Determine whether the space $X = [0, \omega_1]$, where $\omega_1$ is the first uncountable ordinal, satisfies the property $S_1(\Omega, \Omega)$. Specifically, given a sequence $\langle \mathcal{U}_n: n \in \mathbb{N} \rangle$ of open $\omega$-covers of $X$, can we always find a sequence $\langle F_n: n \in \mathbb{N} \rangle$ with each $F_n \in \mathcal{U}_n$ such that $\cup F_n$ is an $\omega$-cover of $X$?

Answer: No


Question: Given that \( \lim_{(x, y) \to (0,0)} \frac{f(x, y) - \tan{(x)}\sin{(y)}}{x^2 + y^2} = 0 \) and \( f \in C^2(\mathbb{R^2}) \), find \( \frac{\partial^2 f}{\partial x \partial y}(0, 0) \).

Answer: 1


Question: Determine the limit: \[ \lim_{x \to \infty}\left(\tan{\frac{\pi x}{2x+1}}\right)^\frac{1}{x} \]

Answer: 1


Question: If $f(x)$ is $O(g(x))$, does it follow that $2^{f(x)}$ is $O(2^{g(x)})$? Answer yes or no.

Answer: No


Question:
Consider the Sturm-Liouville problem given by the differential equation:

\[-y'' + (1+x)y = \lambda y, \quad x \in (0,1)\]

with boundary conditions \(y(0) = y(1) = 0\). Determine under which condition the problem has a non-zero solution:

1. For all \(\lambda < 0\)
2. For all \(\lambda \in [0,1]\)
3. For some \(\lambda \in (2, \infty)\)
4. For a countable number of \(\lambda\)'s

Select the correct option.

Answer: 4


Question: Evaluate the integral \( \int_{0}^{\infty} \left( \frac{f(t)}{e^{2t}} \right) dt \) using the Laplace transform properties, given that \( F(s) = \frac{s-2}{s^{2}-1} \) is the Laplace transform of \( f(t) \) and \( s > 1 \). Use the fact that \( L\left[ e^{-2t}f(t) \right] = F(s-2) = \frac{s}{(s+2)^{2}-1} \) and \( L\left[ \int_{0}^{\infty} \left( \frac{f(t)}{e^{2t}} \right) \right] = \frac{F(s)}{s} = \frac{1}{(s+2)^{2}-1} \).

Answer: 0


Question:
Find the largest possible value of the expression:
\[
\sin(a_1)\cos(a_2) + \sin(a_2)\cos(a_3) + \cdots + \sin(a_{2014})\cos(a_1)
\]

Answer: 1007


Question:
Consider the projective space \(\mathbb{P}^n\) over the field \(\mathbb{C}\). It corresponds to a principal \(\mathbb{C}^*\)-bundle:
\[ \mathbb{C}^* \to \mathbb{C}^{n+1}-0 \to \mathbb{P}^n, \]
where \(\mathbb{C}^*\) acts on \(\mathbb{C}^{n+1}-0\) by multiplication:
\[ t\cdot(z_0,\dots,z_n) = (tz_0,\dots,tz_n).\]

Fix an integer \(d\). Consider the action of \(\mathbb{C}^*\) on \(\mathbb{C}\) given by:
\[ t \cdot z = t^d z.\]

This gives a line bundle \(\left(\mathbb{C}^{n+1}-0\right) \times_{\mathbb{C}^*} \mathbb{C}\) over \(\mathbb{P}^n\), where \(\mathbb{C}^*\) acts on \(\left(\mathbb{C}^{n+1}-0\right) \times \mathbb{C}\) by:
\[ t \cdot ((z_0, \dots, z_n),z) = ((tz_0, \dots, tz_n), t^d z). \]

Show that the line bundle \(\left(\mathbb{C}^{n+1}-0\right) \times_{\mathbb{C}^*} \mathbb{C} \cong \mathcal{O}(d)\).

Answer: \mathcal{O}(d)


Question: Let $x^n - a \in F[x]$ be an irreducible polynomial over $F$, and let $b \in K$ be its root, where $K$ is an extension field of $F$. If $m$ is a positive integer such that $m \mid n$, find the degree of the minimal polynomial of $b^m$ over $F$.

Answer: \dfrac{n}{m}


Question: Consider a three-dimensional space with origin $O$. You have a finite number of points $P_1, P_2, \cdots, P_n$, each assigned a nonzero integer charge $q_i$. For any other point $R$ in the space, define the vector function $$\vec{F(R)} = \sum_{i = 1}^{n} \frac{q_i}{D(P_i, R)^2} \vec{r_i},$$ where $D(P_i, R)$ is the Euclidean distance between $P_i$ and $R$, and $\vec{r_i}$ is a unit vector directed from $P_i$ to $R$. Now, choose a ray $\vec{\ell}$ originating from $O$ in any direction. Is it true that for any configuration of points and charges, there exists a rational number $\alpha$ such that $$\lim_{x \rightarrow \infty} \| F(R_x) \| x^{\alpha}$$ converges to a nonzero constant, where $R_x \in \ell$ with $D(O, R_x) = x$ and $\| F(R_x) \|$ is the magnitude of the function at $R_x$?

Answer: Yes


Question: Given the polynomial \( f(x) = \frac{x^3}{\alpha} + \frac{x^2}{\beta} + \frac{x}{\gamma} + \frac{1}{\delta} \) where \( \alpha, \beta > 0 \), and the condition \( |f(x)| \leq 1 \) for \( |x| \leq 1 \), determine the maximum value of \( \frac{1}{\alpha} + \frac{1}{\beta} + \left| \frac{1}{\gamma} \right| + \left| \frac{1}{\delta} \right| \).

Answer: 7


Question: Evaluate the complex line integral: \[ I = \int_{0}^{4+2i} z^* \,\mathrm{d}z \] where \( z^* = x - iy \).

Answer: 10


Question: Let $A \in \mathbb{R}^{n \times n}$ be a symmetric positive definite matrix with all non-diagonal elements non-positive. Determine whether all the non-diagonal elements of $A^{-1}$ are non-negative.

Answer: Yes


Question: Evaluate the double integral \( \iint_D xy^2 \, dx \, dy \), where the region \( D \) is the area between the circles defined by \( x^2 + (y-2)^2 = 4 \) and \( x^2 + (y-1)^2 = 1 \).

Answer: 0


Question: Let \( \{N(t), t \geq 0\} \) be a Poisson process with rate \( \lambda \), independent of the sequence \( X_1, X_2, \ldots \) of independent and identically distributed random variables with mean \( \mu \) and variance \( \sigma^2 \). Find the covariance \( \operatorname{Cov} \left(N(t), \sum_{i=1}^{N(t)} X_i \right) \).

Answer: \lambda \mu t


Question:
Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{\sqrt{n}} \left( \frac{1}{\sqrt{1} + \sqrt{3}} + \cdots + \frac{1}{\sqrt{2n-1} + \sqrt{2n+1}} \right) \] Choose the correct answer from the following options:

A) \( \sqrt{2} \)

B) \( \frac{1}{\sqrt{2}} \)

C) \( \sqrt{2} + 1 \)

D) \( \frac{1}{\sqrt{2} + 1} \)

Answer: B


Question: Evaluate the sum \( \sum_{k=m}^{n} (-1)^k \binom{n}{k} \binom{k}{m} \) for all positive integers \( n \) and \( m \) such that \( n > m \).

Answer: 0


Question: Given permutations $f$ and $g$ in the permutation group $S_E$, where the sign of $f \circ g \circ f^{-1}$ is $-1$, determine the sign of the permutation $f \circ g \circ f \circ g \circ f \circ g \circ f$. Provide your answer as either $+1$ or $-1$.

Answer: -1


Question: Find all positive integer solutions $(n, m)$ such that $n! = m^2$.

Answer: (1, 1)


Question: Given the power curve defined by the expression \( c \cdot x^b \), where \( x \) ranges from 1 to 52, the sum of the curve is initially 10. If this sum is increased by 10% to become 11, determine the new value of the exponent \( b \) that achieves this result, assuming \( c \) remains constant.

Answer: 1.03


Question: Determine whether the following statement is true: For all natural numbers \( k \), there exists a natural number \( m \) such that \( k!\pi + \frac{\pi}{6} \le m! \le k!\pi + \frac{5\pi}{6} \).

Answer: False


Question: Let \( V \) be an \( n \)-dimensional real inner product space. Define the operator \( M: L(V) \to L(V) \) by \( M(S) = S + S^* \), where \( S^* \) is the adjoint of \( S \). Determine the trace of the operator \( M \).

Answer: n(n + 1)


Question: Determine the range of the function \( f(x) = \int_x^{2x} \left(\frac{e^{-t^2x}}{t}\right) \, dt \) for \( x > 0 \).

Answer: (0, \ln 2)


Question: Evaluate the limit \( l = \lim_{n\rightarrow \infty}\sum^{n}_{r=1}\frac{(\cos (2x))^{2r-1}}{2r-1} \) for \( x\in(\cot^{-1}(2),\cot^{-1}(1)) \). Find \( \lfloor l \rfloor \).

Answer: 0


Question:
Determine which of the following 1-D representations of the incompressible Navier-Stokes equations is correct. Consider the equations:

A. \( \frac{\partial (\rho v)}{\partial t} + \frac{\partial (\rho v^2)}{\partial z} = S \)

B. \( \frac{\partial v}{\partial t} + v\frac{\partial v}{\partial z} = \frac{S}{\rho} \)

Assume \( \rho = 1 \) and simplify both equations. Which version is correct?

Answer: B


Question: Determine whether there exists a nowhere dense set in $\mathbb{R}$ that is not an $F_\sigma$ set.

Answer: Yes


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function defined by the equation \( xf(x) = e^x - 1 \). Determine the value of \( \lim_{n \to \infty} n f^{(n)}(x) \).

Answer: e^x


Question: Determine the radius of convergence for the series \( \sum_{n=1}^{\infty} \frac{(3+(-1)^n)^n}{n}x^n \).

Answer: \dfrac{1}{4}


Question: How many ways are there to partition the set \([8]\) into 3 disjoint pairs? For example, one such partition is \((12)(45)(67)\).

Answer: 420


Question: Let \( h: X \rightarrow Y \) be a homeomorphism. If \( h(x) \) and \( h(y) \) lie in different connected components of \( h(X) \), is it true that \( x \) and \( y \) lie in different connected components of \( X \)? Justify your answer.

Answer: Yes


Question: Let $L$ be the linear subspace of $C[0,1]$ defined by $L = \{x(t) \in C[0,1] : x(t) = \text{constant}\}$. Consider the functional $f_0 \in L^*$ given by $f_0(x) = x(0)$. Define $L_1 = \text{span}(L, t)$. Find a functional $f \in L_1^*$ such that $f|_L = f_0$ and $\|f\| = \|f_0\|$. Provide the functional $f$.

Answer: f(x) = x(0)


Question: Evaluate the limit \( \lim_{r\to 0}\frac{\text{Area}(D')}{\text{Area}(D)} \), where \( D \) is the disc in the complex plane centered at \( \frac{\pi}{4} \) with radius \( r \), and \( D' \) is the image of this disc under the map \( z \to \sin z \).

Answer: \dfrac{1}{2}


Question:
Given the integral \( \int \frac{ ( \cos x + \sin 2x ) \ \mathrm{d}x}{( 2 - \cos^2 x)(\sin x)} \), it is expressed as:
\[ \int \frac{A \ \mathrm{d}x}{(\sin x)} + B \int\frac{\sin x \ \mathrm{d}x}{ 1 + \sin^2 x} + C \int \frac{\mathrm{d}x}{1 + \sin^2 x} \]
Determine which of the following equations is correct:

- \(A + B + C = 4\)
- \(A + B + C = 2\)
- \(A + BC = 1\)
- \(A + B + C = 5\)

Answer: B


Question: Let $X = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Determine the maximum number of 3-element subsets $\mathcal{C}$ of $X$ such that for any two subsets $A, B \in \mathcal{C}$, the intersection $A \cap B$ has at most one element. What is the maximum cardinality of $\mathcal{C}$?

Answer: 12


Question: Given the Fourier transform pair $h(t) \leftrightarrow H(\omega)$, determine the time-domain function whose Fourier transform is $H(-\omega)$. What is the time-domain function corresponding to $H(-\omega)$?

Answer: h(-t)


Question:
Evaluate the limit of the integral:
\[ \lim_{n\to\infty}\int_A{(|x|(1-|x|))^{(1/n)}\arctan(ny)\over{1+x^2+y^2}} \,dx\,dy, \]
where
\[ A = \left\lbrace w\in \mathbb{R}^2: |w|\le \sin(3\cdot \angle(w,e_1)) \right\rbrace, \]
and $e_1=(0,1)\in \mathbb{R}^2$. Use Lebesgue's theorem to find the limit.

Answer: 0


Question: Solve the congruence \(x^3 + x^2 - 5 \equiv 0 \pmod{7^3}\) using Hensel's lemma.

Answer: 23


Question: Determine whether the power series \( \sum\limits_{k=1}^{\infty} \frac{x^k}{k} \) converges uniformly on the interval \([-s,s]\) for \(s \in [0,1)\). Justify your answer.

Answer: Yes


Question: Given that $f$ is entire, use a power series representation of $f$ about $0$ to solve the differential equation $f''(z)-3f'(z)+2f(z)=0$ with initial conditions $a_0 = 1$ and $a_1 = 2$. Determine the function $f(z)$. \( \text{(Express your answer in terms of known functions if possible.)} \)

Answer: e^{2z}


Question: Evaluate the limit \( \lim_{x \to 0^+} \frac{x^x - 1}{\ln(x) + x - 1} \) using L'Hôpital's rule.

Answer: 0


Question: Find the multiplicative inverse of the polynomial $4x+3$ in the field $\frac{\Bbb F_{11}[x]}{\langle x^2+1 \rangle}$. Express the inverse as a polynomial $px+q$ where $p, q \in \Bbb F_{11}$. Verify that $(4x+3)(px+q) \equiv 1 \pmod{x^2+1}$.

Answer: 6x + 1


Question: Determine if there are any other semiprime numbers of the form obtained by concatenating the first even numbers, such as $2468101214$, up to a certain limit.

Answer: 2468101214


Question: Let $L$ be a field of characteristic $p$, and $E$ be a field extension of $L$. Suppose $a$ is a $p$th root of an element of $L$ such that $a$ is not in $E$. Consider the polynomial $p(x) = x^p - a^p$. If $g(x)$ is a polynomial in $E[x]$ and for some integer $n$, $p(x)$ divides $g(x)^n$, does it follow that $p(x)$ divides $g(x)$?

Answer: Yes


Question: Find the value of \( \lim_{x \to \infty} \left( \frac{x+\pi}{x+e} \right)^x \).

Answer: e^{\pi - e}


Question: Let \(X\sim N(1,1)\). Find \(\operatorname{Var}(X^2)\).

Answer: 6


Question: Find the greatest lower bound of the expression $xy + yz + zx$ given the constraints $x+y+z=k$ for some $k > 0$, and $0 < x,y,z < \frac{k}{2}$.

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


Question: Evaluate the limit: \[ \lim_{(x,y) \to (1,0)} \frac{(x-1)\sin y}{y \ln x} \]

Answer: 1


Question: Determine the multiplicity of the root $a$ of the polynomial \( \frac{x-a}{2}(f'(x) + f'(a)) - f(x) + f(a) \), where the polynomial is defined over some field.

Answer: 3


Question: Evaluate the limit without using L'Hôpital's rule: \[ \lim_{x \to 0} \left( \frac{\sqrt{\cos(x)} - \sqrt[3]{\cos(x)}}{\sin^2(x)} \right) \]

Answer: -\dfrac{1}{12}


Question: Consider the equation \( \ln(x^2+7x-6) - \ln(2x) = 0 \). Solve for \(x\) and determine which solutions are valid given the domain restrictions of the logarithmic function.

Answer: 1


Question: Determine whether the following statement is true: $O(M^3 + NM^2) = O(M^3 + N)$, where $M$ and $N$ are variables of the function.

Answer: False


Question: Determine the smallest value of \(k\) such that \(\ln(x) = O(x^k)\).

Answer: 0


Question:
Find a linearly independent solution to the Bessel's equation given that $J_0(x)$ is a solution. The Bessel's equation is given by:
\[ x^2y'' + xy' + x^2y = 0 \]

Answer: Y_0(x)


Question: Evaluate the integral \( \int_C \frac{z^2}{(z^2+4)(z^2+9)} \, dz \), where \( C \) is the contour \( |z| = 4 \) traversed in the counterclockwise direction, using the Residue Theorem.

Answer: 0


Question: Suppose \( \kappa \) is a regular cardinal. Determine whether there necessarily exists a poset \( \mathbb{P} \) that collapses \( \kappa^+ \) while preserving all other cardinals.

Answer: Yes


Question: Find the Jordan measure of the set \(A = \{(x,y,z) \mid z > 0, 1-x^2 = z, y^2+z^2 = z\}\).

Answer: 0


Question: In an equilateral triangle ABC, three rods of lengths 3, 4, and 5 units are placed such that they intersect at a common point O, with the other ends at vertices A, B, and C, respectively. Given that BO = 3 units and CO = 4 units, find the angle ∠BOC.

Answer: 150^\circ


Question: Find the sum of all the residues of the function \( \frac{a(z)}{b(z)} \), where \( a(z) \) and \( b(z) \) are polynomials such that \( \deg(b) \ge \deg(a) + 2 \).

Answer: 0


Question: Calculate the expected position \(\mathbb{E}X_2\) of the second ace in a shuffled deck of 52 cards, given that the expected number of cards between any two consecutive aces is the same. Use the fact that \(4 + \sum_{i=1}^5 \mathbb{E}\Delta_i = 52\) to find your answer.

Answer: \dfrac{106}{5}


Question: Calculate the value of the following summation: \[ \sum_{k=1}^{6} \left( \sin\frac{2\pi k}{7} - i\cos\frac{2\pi k}{7} \right) \]

Answer: i


Question: Determine whether the improper integral \( \int_{-\infty}^\infty x^3e^{-x^4}dx \) is convergent or divergent. If it is convergent, evaluate the integral.

Answer: 0


Question: If $G$ is a group and $N$ is a nontrivial normal subgroup, is it possible for $G/N$ to be isomorphic to $G$?

Answer: Yes


Question: Calculate the discriminant of the number field \(\mathbb{Q}(\sqrt{-3})\) using the basis \(\{1, \sqrt{-3}\}\) and the complex embeddings \(\{\sigma_1, \sigma_2\}\), where \(\sigma_1\) is the identity map and \(\sigma_2\) is the conjugate map. Verify your result.

Answer: -12


Question: Let \(a_i\) be the n-th roots of unity in \(\mathbb{C}\). Prove that \((1-a_2)\cdot(1-a_3)\cdot...\cdot(1-a_n) = n\).

Answer: n


Question: Find all polynomial functions \( f(x, y) \) such that \( f(x+y, y-x) = f(x, y) \) for all \( x, y \).

Answer: f(x, y) = c


Question: Find the shortest path that traverses all edges in a complete graph with 14 vertices, where all edges have equal weight.

Answer: 98


Question: Evaluate the limit: $$\lim_{x \to 0} \dfrac{\displaystyle \int_0^x \sin \left(\frac{\pi t^2}{2}\right) dt}{x^3}$$

Answer: \dfrac{\pi}{6}


Question: Determine the shortest sequence of letters using the alphabet A-E that guarantees the inclusion of any possible code of length up to 4. The code panel will only remember the last number of buttons pressed equal to the length of the code. For example, if the code is 2 letters long and you entered ABECD, the panel would only remember CD.

Answer: 628


Question: Evaluate the integral \( \int_0^{2\pi}\frac{1}{5-4\cos x}\ dx \).

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


Question: Given a target number in a $2^{64}$ space, how many random tries are needed to be the closest to this target number if there are $X$ other random numbers present?

Answer: X + 1


Question: Determine the inner measure of the set of irrational numbers within the interval \([0,1]\) using Lebesgue measure.

Answer: 1


Question: Let $(M,d_M)$ be a separable metric space, meaning there exists a countable subset $S \subset M$ such that the closure of $S$ in $M$ is $M$ itself, i.e., $\text{cl}_M(S) = M$. Suppose there is a homeomorphism $f: M \to N$. Must the space $N$ be separable? Justify your answer.

Answer: Yes


Question: Determine whether the function \( f: [0,1] \rightarrow \mathbb{R} \) defined by \( f(x) = \frac{1}{n} \) for \( x = \frac{1}{n} \) (where \( n \in \mathbb{N} \)) and \( f(x) = 1 \) otherwise, is Riemann integrable over the interval \([0,1]\).

Answer: Yes


Question:
Let $f$ be a function defined on $(a,b)$, integrable on any $[c,d]$ with $a<c<d<b$, but not integrable in the standard Riemann sense on the entire $[a,b]$. Define the function $F$ as an improper integral:
\[ F(u) = \int_{a}^{u} f(x) \, dx = \lim_{c \to a} \int_{c}^{u} f(x) \, dx. \]
Assuming $F(u)$ is well-defined for any $u \in (a,b)$, determine if \( \lim_{u \to a} F(u) = 0 \). Consider whether the assumption that $f$ is non-decreasing affects this result.

Answer: 0


Question: Find the first homology group $H_1(K)$ of the Koszul complex $K : 0 \rightarrow R \rightarrow R^3 \rightarrow R^3 \rightarrow R \rightarrow 0$, where $R = \mathbb{Q}[x,y,z]$ and $I = \langle x,y,z\rangle$. Express $H_1(K)$ as $\ker (\phi_1)/ \operatorname{im} (\phi_2)$. Determine if $H_1(K)$ is zero, given that the sequence is regular.

Answer: 0


Question: Suppose there is an exact triangle $A \to B \to C \to A[1]$ in $D^b(\mathcal A)$, where $A, B, C$ are concentrated in degree $0$. Determine if it follows that there is a short exact sequence $0 \to A \to B \to C \to 0$ in $\mathcal A$. Provide a justification for your answer.

Answer: Yes


Question: Determine the maximal number of vectors in \(\mathbb{R}^n\) such that each pair of vectors has a negative scalar product. Let this maximal number be denoted as \(f(n)\). What is \(f(n)\)?

Answer: n + 1


Question: Let $f$ be a holomorphic function on the open unit disk $\mathbb{D}$ and continuous on the closed disk $\mathbb{\overline{D}}$. Given that $f\left(\frac{z}{2}\right) = \frac{1}{2}f(z)$ for all $z \in \mathbb{\overline{D}}$ and $f(1) = 1$, prove that $f(z) = z$ for all $z \in \mathbb{\overline{D}}$.

Answer: f(z) = z


Question: Show that if $X \sim N(0, \sigma^2)$, then the expected value of $\exp(X)$ is $\exp(\sigma^2/2)$. Provide a detailed calculation to demonstrate this result.

Answer: \exp(\sigma^2/2)


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

Answer: 8


Question: Prove or disprove the following identity for all positive real numbers \(a\): \[ a^{i\frac{\pi}{\log(a)}} = -1 \]

Answer: -1


Question: Find the sum of the series \( \sum_{n=1}^{\infty} \frac{(n^2+n) \cdot 3^{n-1}}{4^n} \).

Answer: 32


Question: Calculate \( \operatorname{Cov}(X + 3, XZ) \) given that \( f_X(x) = \begin{cases} 4x^3 &, \text{if} \ 0 < x < 1 \\\ 0 &, \text{otherwise} \end{cases} \), where \( Z \sim N(1,2) \) and \( X \) and \( Z \) are independent.

Answer: \dfrac{2}{75}


Question: Let $f: X \to Y$ be a continuous surjective function such that the preimage of every point is compact, and $Y$ is compact. Does it follow that $X$ is compact?

Answer: No


Question: Given an epimorphism $f: B_4 \longrightarrow S_4$ from the braid group on 4 strands to the symmetric group on 4 elements, determine if the kernel of this epimorphism is isomorphic to $P_4$, the pure braid group on 4 strands.

Answer: Yes


Question: Solve the differential equation \((y)^{y'} = (y')^{y+c}\) for the case when \(c = 0\), assuming that \(y\) and \(y'\) are strictly positive functions.

Answer: y = Ce^x


Question: Given the integral equation \( \int {yy''dx} = 3xy \), determine if it is possible to find the derivative \( y' \).

Answer: y' = 9x


Question: Calculate $g(5000)$, where $g(n) = \sum_{d|n} f(d)$ and $f(n)$ is the greatest power of 2 that divides $n$. For example, $f(20) = 4$, $f(32) = 32$, and $f(72) = 8$. Given that $5000 = 2^3 \cdot 5^4$, find the value of $g(5000)$. \( \text{Note: } f(n) \text{ is multiplicative.} \)

Answer: 75


Question: Find the radius of convergence of the series \( \sum a(n)x^n \), where \( a(n) = n^{-\sqrt{n}} \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \rightarrow 0^{+}} \frac{x^x - 1}{x} \]

Answer: -\infty


Question: Evaluate the definite integral: \[ \int_{0}^{2} x^3 \sqrt{2x - x^2} \, dx \]

Answer: \dfrac{7\pi}{8}


Question: Determine whether it is possible to express $x$ in terms of $y$ for the equation $x \log x = y$. Provide a justification for your answer.

Answer: Yes


Question: Given the surface parameterization \( g(s, t) = (s^2 + 2t, s + t, e^{st}) \), find \( \frac{\partial y}{\partial z} \) at the point \((3, e)\) near \( g(1, 1) = (3, 2, e) \).

Answer: \dfrac{1}{3e}


Question: Evaluate the limit: $$\lim_{x\to 0}\left(\frac{e}{(1+x)^{1/x}}\right)^{1/x}$$

Answer: \sqrt{e}


Question: Determine whether the series \( \sum_{k=0}^\infty \frac{x^k}{2^k(k+1)!} \) is defined at \( x=0 \).

Answer: 1


Question: Determine the number of homomorphisms from the group \( \mathbb{Z}_3 \times \mathbb{Z}_4 \times \mathbb{Z}_9 \) to the group \( \mathbb{Z}_{18} \).

Answer: 54


Question: Find the limit of the sequence \(a_n = \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{n+n}\). Show that the sequence \(a_n\) converges to \(\log 2\).

Answer: \log 2


Question: Determine whether there exists a Lebesgue measurable set $D \supset [0,1]$ such that $D \neq [0,1]$ and $\lambda^{*}(D) = 1$. Provide a justification for your answer.

Answer: Yes


Question: Let $C \subset \mathbb{R}^3$ be a regular connected curve. Does there exist a parametrized regular curve $\gamma: I \rightarrow \mathbb{R}^3$ such that $\gamma(I)=C$? Answer yes or no and justify your answer.

Answer: Yes


Question: Determine if the second singular cohomology group of the product of projective space over an algebraically closed field $k$ and a discrete valuation ring $R$ (which is also a $k$-algebra) is isomorphic to the integers: \( H^2(\mathbb{P}^n_k \times_k R, \mathbb{Z}) \cong \mathbb{Z} \).

Answer: \mathbb{Z}


Question: Given a flow network represented as a directed graph with a single source vertex \(s\) and a single sink vertex \(t\), and a total of \(|V|\) vertices, determine the number of possible cuts in the network.

Answer: 2^{|V| - 2}


Question: Calculate the probability that \( \min(X_1, X_2, X_3) < \max(Y_1, Y_2) \), where \( X_1 \sim \text{Exp}(1) \), \( X_2 \sim \text{Exp}(2) \), \( X_3 \sim \text{Exp}(3) \), \( Y_1 \sim \text{Exp}(4) \), and \( Y_2 \sim \text{Exp}(4) \).

Answer: \dfrac{27}{35}


Question: Calculate the residue of the function \( g(z) = \frac{1}{\sin(z)} \) at the simple pole \( z = 0 \).

Answer: 1


Question: Determine the value of the limit \( \lim_{n\rightarrow\infty}\frac{\sum_{k=n}^{\infty}\frac{\lambda^{k}}{k!}e^{-\lambda}}{\sum_{k=n-1}^{\infty}\frac{\lambda^{k}}{k!}e^{-\lambda}} \).

Answer: 0


Question: Evaluate the integral \( \sqrt{\frac{2}{\pi}}\int_{-\infty}^{\infty} e^{-\frac{z^2}{2}}\,dz \).

Answer: 2


Question: Evaluate \(12(c - \alpha)\) given that \(\lim_{n\to{\infty}}\frac{e(1-\frac{1}{n})^n-1}{n^{\alpha}}=c\), where \(c\) is a non-zero constant.

Answer: 6


Question: Find the residue of the function \( xe^{\frac{1}{x}} \) at \( x = 0 \).

Answer: \dfrac{1}{2}


Question: Given a group element \(a\) such that \(|a^{28}| = 10\) and \(|a^{22}| = 20\), determine the order \(|a|\) of the element \(a\).

Answer: 40


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{n^{1/3}}{2} \arccos \left(\frac{1}{\sqrt{1+\frac{4}{(k(n)-1)^2}}\sqrt{1+\frac{8}{(k(n)-1)^2}}} \right) \right), \] where \( k(n) = \frac{1}{12} (108n+12 \sqrt{768+81n^2})^{1/3}-\frac{4}{ (108n+12 \sqrt{768+81n^2})^{1/3}} + 1. \)

Answer: 2\sqrt{3}


Question: Given the equation \((x^2 + 2xy + 3y^2) \, f(x,y) = (4x^2 + 5xy + 6y^2) \, g(x,y)\) for any real numbers $x$ and $y$ satisfying $x^2y + 6y = xy^3 +5x^2 +2x$, and knowing that $g(0,0) = 6$, find the value of $f(0,0)$. Provide your answer as a single number.

Answer: 19


Question: Determine whether there exists a measurable function \( f: [0,1] \to \mathbb{R} \) such that every modification of \( f \) on null sets is nowhere continuous. Justify your answer.

Answer: Yes


Question: Determine the minimum value of \(c\) such that for all \(n \in \mathbb{N}\), the number \(c^n + 2014\) has all its digits less than 5. Prove that this value of \(c\) satisfies the condition.

Answer: 10


Question: Calculate \(\theta = \sum_{0 \le i, j \le 5 \ (i, j) \neq (0, 0)}{A_{(i, j)}}\), where \(A_{(i, j)}\) is the measure of angle \(\angle V_{(i, j)}V_{(0, 0)}V_{(1, 0)}\) in the Cartesian coordinate system. Find \(\theta \pmod{2\pi}\).

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


Question: Verify the limit \( \lim \limits_{n\to\infty} n \left( p_n(x) - f(x)\right) = x(1-x) \), where \( p_n(x) \) is the Bernstein polynomial approximating \( f(x) = x^2 \).

Answer: x(1 - x)


Question:
Determine the set of values of the parameter \(a\) for which the function \(f(x) = 8ax - a \sin 6x - 7x - \sin 5x\) is increasing and has no critical points in \(\mathbb{R}\). Choose from the following options:

(A) \([-1, 1]\)
(B) \((-\infty, -6)\)
(C) \((6, +\infty)\)
(D) \([6, +\infty)\)

Answer: C


Question: Evaluate the infinite sum: $$\sum_{n=0}^\infty \binom{2n+2}{n}\left(\frac{2}{3}\right)^n\left(\frac{1}{3}\right)^{n+2}$$

Answer: \dfrac{3}{4}


Question: Let $H$ be a Banach space and $C \subset H$ be a convex cone. Consider $H' \subset H$ as a subspace and a functional $\phi: H' \to \mathbb{R}$ that is continuous in the norm of $H$ and nonnegative on $H' \cap C$. Does there exist an extension $\tilde{\phi}: H \to \mathbb{R}$ that is both bounded on $H$ and nonnegative on $C$?

Answer: Yes


Question: Given $n$ balls, where $\frac{n}{3}$ are blue and $\frac{2n}{3}$ are red, each ball is thrown randomly into one of $k$ bins. Define $M_k$ as 1 if a bin has more red balls than blue balls, and 0 otherwise. Let $A = \sum_{i=0}^k M_i n_i$, where $n_i$ is the number of balls in bin $i$. What is the upper bound of $A$?

Answer: n


Question: Identify which element of the group $\Bbb Z_2 \otimes \Bbb Z_2$ corresponds to the composite cycle $(14)(23)$. The elements of the group are $I = (1,1)$, $A = (-1, 1)$, $B = (1,-1)$, and $C = (-1,-1)$. Choose from $I, A, B,$ or $C$.

Answer: C


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{1 - \cos x \cdot (\cos 2x)^{\frac{1}{2}} \cdot (\cos 3x)^{\frac{1}{3}}}{x^2} \] without using L'Hospital's rule.

Answer: 3


Question: Evaluate \( \lim_{n \to \infty} \int_{0}^{\infty} \frac{n \sin (x/n)}{x(1+x^2)} \, dx \) using the Dominated Convergence Theorem.

Answer: \dfrac{\pi}{2}


Question: Determine the number of elements in the quotient ring \(\Bbb{Z}[i]/\langle3+i\rangle\).

Answer: 10


Question: Let $X$ be a compact Riemann surface with genus $g(X) = 1$, and let $p \in X$. Consider the divisor $D = n[p]$, where $n$ is a natural number. Determine the dimension of the space $L(D)$. Provide your answer.

Answer: n


Question: Calculate the sum \( S = \sum_{n=0}^{\infty} \frac{1}{3^n} \sum_{k=0}^{n} \frac{k}{2^k} \) using the Z-transform. Given that if a signal \( y(n) = \sum_{k=0}^{n} x(k) \), its Z-transform is \( \frac{z}{z-1} X(z) \), where \( X(z) \) is the Z-transform of signal \( x \). Let \( x(n) = \sum_{k=0}^{n} \frac{k}{2^k} \). Find the value of \( S \).

Answer: \dfrac{9}{25}


Question: Find the area bounded by the curve \( y = f(x) \) from \( x = 1 \) to \( x = 3 \), given that \( f(x) \) satisfies the equation \( \int_0^1 (x - f(x)) f(x) \, dx = \frac{1}{12} \).

Answer: 2


Question:
Find the n-dimensional Lebesgue measure of the set \( A \subset \mathbb{R}^{n} \) defined as:
\[
A = \left\{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^{n}: \left( \sum_{i=1}^n |x_{i}| \right) \leq 1 \right\}
\]

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


Question: Given that \(a, b, c\) are real numbers satisfying the equation \(a^2 + b^2 + c^2 = ab + bc + ca + 6\), find the minimum value of the expression \(P = (a - b)(b - c)(c - a)\).

Answer: -4\sqrt{2}


Question: If $F$ is a finite field such that every element is a square, prove that the characteristic of $F$ is 2.

Answer: 2


Question: Let $N$ be a normal subgroup of a group $G$ such that $G/N$ is abelian. Suppose for any $n \in N$, $n = g_1^{\alpha_1}\ldots g_k^{\alpha_k}$. Also, assume $G/N = Nh_1^{\beta_1} \times \ldots \times Nh_r^{\beta_e}$. Is it true that for all $g \in G$, there exists $n' \in N$ such that \[ g = n' h_1^{\gamma_1}h_2^{\gamma_2} \ldots h_e^{\gamma_e} \] where each $\gamma_i$ is less than or equal to $\beta_i$?

Answer: Yes


Question: Compute the integral \( \int_{C}\cos\left(\frac{z}{2}\right)\,dz \) where the contour \( C \) is given by \( \gamma(t) = t + i\sqrt{\pi^2 - t^2} \) for \( -\pi \leq t \leq \pi \).

Answer: 4


Question: Find the sum of the series \( \sum_{n=0}^{\infty} \frac{n^2}{2^n} \).

Answer: 6


Question: Consider the ring map \( f: \mathbb{C}[y] \rightarrow \mathbb{C}[x] \) given by \( y \mapsto x^2 \). This induces a map on the spectra: \( \operatorname{Spec} \mathbb{C}[x] \rightarrow \operatorname{Spec} \mathbb{C}[y] \) sending the ideal \( (x-a) \) (where \( a \in \mathbb{C} \)) to \( f^{-1}((x-a)) \). Determine the ideal \( f^{-1}((x-a)) \) concretely.

Answer: (y - a^2)


Question: Let \( f :\mathbb{R}\to \mathbb{R} \) be a continuous function with period 1. Given that \[ \lim_{n\to\infty}\int_0^1\sin^2(\pi x)f(nx)dx= \frac{1}{k}\int_0^1f(x)dx, \] find the value of \( k \).

Answer: 2


Question: If \( f(x) \) is a continuous, one-to-one function with \( f(3)=5 \), \( f(10)=1 \), and \( \int_{3}^{10} f(x) \, dx = 20 \), find \( \int_{1}^{5} f^{-1}(x) \, dx \).

Answer: 25


Question: Let \( X = \{ x = (x_1, x_2, \ldots) : x_i \in \mathbb{R}, x_i = 0 \text{ for almost all } i \in \mathbb{N} \} \) with the norm \( \|x\| = \sup_{i \in \mathbb{N}} |x_i| \). Let \( S: X \to X \) be a linear operator defined by \( Sx = (x_1 + x_2 + x_3 + \ldots, x_2 + x_3 + x_4 + \ldots, x_3 + x_4 + \ldots, \ldots) \). Compute the operator norm \( \|S\| \).

Answer: \infty


Question: Consider the Prokhorov metric $d_P$ on the space of probability measures on $\mathbb{R}$, defined as \[d_P (\mu , \nu ) := \inf \{ \varepsilon > 0 : \mu (B) \leq \nu (B^\varepsilon) + \varepsilon \text{ for all Borel sets } B \},\] where $B^\varepsilon := \{ x\in \mathbb{R} : \inf_{y\in B} \vert x - y \vert \leq \varepsilon \}$. Let $\mathcal{N}(0,1)$ denote the normal distribution with mean 0 and variance 1, and $*$ denote the convolution of measures. Does the inequality \[d_P ( \mathcal{N} (0,1) * \mu , \mathcal{N} (0,1) * \nu ) \leq d_P (\mu , \nu ) \] hold for all probability measures $\mu$ and $\nu$ on $\mathbb{R}$?

Answer: Yes


Question: Calculate the expected value of \( \max(R_1, R_2) \) where \( R_1 \) and \( R_2 \) are independent random variables uniformly distributed over the interval \([-1, 1]\).

Answer: \dfrac{1}{3}


Question: Let \( \alpha: [0,1] \rightarrow \mathbb{R} \) be a strictly monotonically increasing function with \( \alpha(0) = 3 \) and \( \alpha(1) = 4 \). Find all continuous functions \( f: [0,1] \rightarrow \mathbb{R} \) such that \( \int_0^1 f \, d\alpha = \int_0^1 f^2 \, d\alpha = 1 \).

Answer: 1


Question: For any given positive integer \( k \), does there exist a sequence of positive numbers \( \left(x_{n}\right)_{n\geq1} \) such that \[ \limsup_{n\rightarrow\infty}\left(\frac{x_{1}+x_{n+k}}{x_{n}}\right)^{n}=e^{k}? \]

Answer: Yes


Question: Determine the rank of the first homology group $H_1$ for a compact, connected, orientable 2-manifold $M$ of genus $g$ with $b$ boundary components. Express your answer in terms of $g$ and $b$. Assume $M$ is connected and use the Euler characteristic formula: $$\chi = 2 - 2g - b$$ and $$\chi = H_0 - H_1 + H_2$$ where $H_0 = 1$ and $H_2 = 0$ for a manifold with boundary.

Answer: 2g + b - 1


Question:
Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Consider the polynomial \( P(x) = (x - a_1)(x-a_2) \cdots (x-a_n) \). For any \( 0 \le i \le n-1 \), define the \( i \)th mean \( m_i \) as the number such that \((x - m_i)^n\) and \(P(x)\) have the same coefficient of \(x^i\). For example, with three variables \(x, y, z\), we have:
\[
m_0 = \sqrt[3]{xyz}, \\
m_1 = \sqrt{\frac{xy + yz + zx}{3}}, \\
m_2 = \frac{x + y + z}{3}
\]
Determine whether it must be true that \(m_0 \le m_1 \le m_2 \le \cdots \le m_{n-1}\).

Answer: Yes


Question: For every nonnegative integer \( k \), does there exist a \( k \)-universal integer quadratic form?

Answer: Yes


Question: Calculate the flux of the vector field \( F(x,y,z) = (2x, 2y, 2z) \) through the surface \( S \) defined by the cylinder \( x^2 + y^2 = 9 \) with \( 0 \leq z \leq 5 \).

Answer: 180\pi


Question: Evaluate the integral: \[ \int_{0}^{1} \frac{x^{a - 1}}{(1 - x)^{a + 1}} e^{\frac{x}{x - 1}} \, dx \] where \( a \in \mathbb{R} \) is a constant.

Answer: \Gamma(a)