Question:
Let $f$ be a non-negative function with a finite third derivative in the open interval $(0,1)$. Determine which of the following conditions ensures that $f'''(c) = 0$ for some $c \in (0,1)$:

(a) $f(x) = 0$ for at least two values of $x \in (0,1)$

(b) $f(x) = 0$ for at least one value of $x \in (0,1)$

(c) $f'(x) = 0$ for at least two values of $x \in (0,1)$

(d) None of these

Choose the correct option.

Answer: A


Question:
Let $X$ be a topological space, and consider two subsets $A$ and $C$ of $X$. Verify if the following equality holds:
$$(C \cap \text{Cl}(A)) \cup (C \cap \text{Cl}(X \setminus A)) = C \cup (C \cap \partial A)$$
Given that $C \cap \text{Cl}(X \setminus A) = C \setminus \text{Int}(A)$, determine if the equality is true.

Answer: True


Question: Let $f(x)$ be a function with a Lipschitz continuous gradient, meaning that $$ \|f'(x)-f'(y)\| \leq \alpha \|x-y\| $$ for all $x, y \in \textbf{dom} f$, where $\|\cdot\|$ denotes the Euclidean norm. Is $f'(x)$ a continuous vector function for all $x, y \in \textbf{dom} f$?

Answer: Yes


Question: In a unital C*-algebra $A$, consider a sequence of positive invertible elements $\{b_{n}\}$. If $\|1_{A} - b_{n}\| \to 0$, does it follow that $\|1_{A} - b_{n}^{-\frac{1}{2}}\| \to 0$?

Answer: Yes


Question: Let $A$ be an $R$-right module, $N$ be a submodule of an $R$-left module $M$, and let $\pi: M \rightarrow M/N$ be the natural epimorphism. Determine $\ker(\pi \otimes 1_A)$. Provide your answer in terms of $M$, $N$, and $A$.

Answer: N \otimes_R A


Question: Determine if there exists a complex number $z$ such that $z\bar{z} = -1$. If so, what is the result of adding such a solution?

Answer: 0


Question: If \(x_1, x_2, \ldots, x_{2019}\) are the roots of the polynomial \(P(x) = x^{2019} - 2019x + 1\), find the value of \(\sum_{i=1}^{2019}\frac{x_i-1}{x_i}.\)

Answer: 0


Question: Let \(P(T) = \sum_{i=0}^n u_iT^i\) be a polynomial whose coefficients \(u_i \in \mathbb{C}[[z]]\) are analytic power series. If \(f \in \mathbb{C}[[z]]\) is a root of \(P\), is \(f\) analytic? Provide a justification for your answer.

Answer: Yes


Question: Calculate \( \lim_{n\to\infty} \frac{\sum_{i=1}^n ia_i}{n^2+n} \) given that \( a_n \to a \).

Answer: \dfrac{a}{2}


Question: Calculate the sum of the infinite series \( \sum_{n=1}^{\infty} \frac{1}{n} \cos \frac{n\pi}{3} \).

Answer: 0


Question: Let $(A, \mathcal{T}_A)$ and $(B, \mathcal{T}_B)$ be two topological spaces such that $B \subset A$ and $\mathcal{T}_B = \{B \cap U \mid U \in \mathcal{T}_A\}$. If $f: A \to X$ is continuous with respect to $\mathcal{T}_A$, is the restriction $f|_B$ continuous with respect to $\mathcal{T}_B$?

Answer: Yes


Question: Determine the number of non-invertible elements in the multiplicative group $(\mathbb{Z}/p^n\Bbb{Z})^\times$, where $p$ is a prime number.

Answer: p^{n-1}


Question: Determine whether the following statement is true or false: For a differentiable function $f\colon\mathbb{R}^n\to\mathbb{R}$, if $\liminf_{x\to x^*}\frac{|f(x)|}{||x-x^*||}=0$ for $n>1$, then $f(x^*)=0$. Provide a justification for your answer.

Answer: True


Question: Evaluate the contour integral \( \oint_{C}\frac{e^z-1}{z}\mathrm{d}z \), where \( C \) is the unit circle traced counter-clockwise.

Answer: 0


Question: Evaluate the limit: $$\lim_{x \to \pi/4} \frac{\ln(\tan x) - \sin(x-\pi/4)}{\cos^2(x)}$$ using L'Hopital's rule or Taylor series expansions.

Answer: 0


Question: If $a$, $b$, and $c$ are 3-dimensional vectors that are linearly dependent, does there always exist a vector that is perpendicular to all three of them? Answer yes or no and justify your answer.

Answer: Yes


Question: In triangle $\triangle ABC$, $AB = AC$ and $\angle BAC = 30^\circ$. Points $A'$, $B'$, and $C'$ are the reflections of points $A$, $B$, and $C$ across lines $BC$, $CA$, and $AB$, respectively. Determine the measure of $\angle A'B'C'$.

Answer: 60^\circ


Question: Determine the number of ordered pairs \((a, b)\) such that \(a^2 + b^2 = 2^3 \cdot 3^4 \cdot 5 \cdot 17^2\).

Answer: 24


Question:
Evaluate the limit for a positive integer $p \geq 2$:

$$ \lim_{n\to\infty}\left(\frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{pn}\right) $$

Determine the value of this limit.

Answer: \ln p


Question: Solve the differential equation: \[ \frac{y \, dx - x \, dy}{x^2 + y^2} = 0. \]

Answer: y = Cx


Question: Evaluate the limit of the function \( f(x,y)=\frac{(x^5 - y^7)}{(x^4+y^4)} \) as \((x,y)\) approaches the origin \((0,0)\).

Answer: 0


Question: Calculate \(2^{32101} \mod 143\).

Answer: 2


Question: Let $a, b,$ and $c$ be three positive real numbers such that $a + b + c = 3$. Determine the minimum value of the expression \[ A = \frac{2-a^3}{a} + \frac{2-b^3}{b} + \frac{2-c^3}{c}. \]

Answer: 3


Question: Find the nth derivative of the function \( g(x) = \int_x^{2x} \sum_{i=0}^{n-1} \frac{(t-x)^i}{i!} \, dt. \)

Answer: 1


Question: Determine whether the following proposition is true or false: If $f(x)$ is absolutely integrable and piecewise-smooth, then $xf'(x)$ is also absolutely integrable.

Answer: False


Question: Determine whether the following statement is generally true: \( \frac{\mathrm{d}}{\mathrm{d}a} \int_{-\infty}^{a-y} f(x)\, \mathrm{d}x = f(a-y) \).

Answer: True


Question: Compute the value of the exterior 2-form \( \theta = \sum_{i=1}^{n-1} x_i \wedge x_{i+1} \) on \( \mathbb{R}^n \) at the vectors \( A = (1, 1, 1, \dots, 1) \) and \( B = (-1, 1, -1, \dots, (-1)^n) \).

Answer: 1 + (-1)^n


Question: If two events $A$ and $B$ are conditionally independent given a third event $C$, does it necessarily imply that they are conditionally independent given the complement of $C$, $C^\complement$? Answer yes or no.

Answer: No


Question:
Determine which of the following second-order differential equations has solutions that oscillate more rapidly:

1. \(x'' + \sqrt{t^6 + 3t^5 + 1} \, x = 0\)
2. \(x'' + 2t^3 \, x = 0\)

Answer: 2


Question: Determine whether a finite nonabelian group can have a proper normal subgroup that is also nonabelian.

Answer: Yes


Question: Find the limit of the sequence \( \{b_n\}_{n=1}^{\infty} \) defined by \( b_n = \frac{n}{\sum_{i=1}^{n}\frac{1}{a_i}} \), where \( \{a_n\}_{n=1}^{\infty} \) is given by \( a_n = (2^n + 3^n)^{\frac{1}{n}} \).

Answer: 3


Question: Find all continuous functions \( g: [c,d] \to \mathbb{R} \) that satisfy the equation \( \int_{y}^d g(u) \, du = \int_{c}^y g(u) \, du \) for all \( y \in (c,d) \).

Answer: 0


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

Answer: 0


Question: Suppose $H$ is a subgroup of order $(n-1)!$ in $S_n$. Determine if there exists an $i \in \{1, 2, \ldots, n\}$ such that $H = \mathbb{Stab}(i)$ for all $n \ge 7$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \( \lim_{x\to 0^{-}}(\cos x-1)e^{\cot x} \).

Answer: 0


Question: Evaluate the limit \(\lim_{x\to 0^+}\frac{\ln(\ln(1+x))}{e^x-1}\) using Taylor series expansion.

Answer: -\infty


Question:
Evaluate the limit as \( n \) approaches infinity:

\[
\lim_{n\to\infty} \frac{\left(\sum_{r=1}^n\sqrt{r}\right)\left(\sum_{r=1}^n\frac{1}{\sqrt{r}}\right)}{\sum_{r=1}^n r}
\]

Answer: \dfrac{8}{3}


Question:
Given the distinct complex roots \(a, b, c\) of the polynomial \(P(x) = x^3 - x - 1\), calculate the sum:

\[a^7 + b^7 + c^7 + \frac{1}{a^7} + \frac{1}{b^7} + \frac{1}{c^7}\]

Use the relations from Vieta's formulas: \(abc = 1\), \(a + b + c = 0\), and \(ab + bc + ac = -1\).

Answer: 6


Question: Find the limit of the sequence \(a_n=\sqrt[n!]{\frac{1}{2^{n!}}-\frac{1}{3^{n!}}}\) as \(n\) approaches infinity.

Answer: \dfrac{1}{2}


Question: Let $\mathcal{D}$ be a collection of subsets of $[m]$ such that if $X, Y \in \mathcal{D},$ then $X \cup Y \neq [m]$. Determine the maximum size of $\mathcal{D}$.

Answer: 2^{m-1}


Question: Let $f$ and $g$ be two real-valued continuously differentiable functions with domain $[a,b]$. If $f(x)g(x)=0$ for all $x$ in $[a,b]$, must either $f$ or $g$ be identically zero on $[a,b]$? Answer "Yes" or "No" and justify your answer.

Answer: No


Question: Compare the numbers: \(9^{8^{8^9}}\) and \(8^{9^{9^8}}\). Determine which is larger.

Answer: 9^{8^{8^9}}


Question: Evaluate the limit \( \lim\limits_{n \rightarrow \infty} \left(\int_0^2 (g(x))^{2n}\,\mathrm dx\right)^\frac{1}{2n} \) for a continuous function \( g:[0,2]\to[0,\infty) \) with a maximum value of \( N \).

Answer: N


Question: Find the equation of the locus of the centers of circles that pass through the points where the line \( L: y = -\frac{1}{5}x + 2 \) intersects the circle \( C: 2x^2 + 2y^2 + 2x + 2y - 13 = 0 \).

Answer: y = 5x + 2


Question: Consider a space $X$ with the discrete topology $2^X$. Let $\mathcal{B}(X)$ be the Borel $\sigma$-algebra on $X$. Are all Borel probability measures on $(X, \mathcal{B}(X))$ supported on finite or countably infinite subsets of $X$? Answer with 'Yes' or 'No'.

Answer: Yes


Question:
Determine whether \(f = \Theta(g)\), \(f = \omega(g)\), or \(f = o(g)\) for the following functions:

\[f(n)=\left( \frac {\ln(n^2+1)}{\ln(n)}\right)^{\ln(n^2)}, \quad g(n)= 2^{\sqrt{\ln(n)}}\]

Answer: f = \omega(g)


Question: Evaluate the integral \(\int\limits_{\Gamma}|z|^2 \, dz\), where \(\Gamma\) is the boundary of a square with vertices at \(0, 1, 1+i, i\), traversed anti-clockwise starting at \(0\).

Answer: -1 + i


Question: Determine if the nonlinear ordinary differential equation \( yy' + f(x)y^2 = yg(x) \) can be transformed into a linear ODE by dividing both sides by \( y \).

Answer: Yes


Question: Given the matrix \( A = \begin{bmatrix}0&0&1&0\\0&0&0&1\\-37&3&-4&0\\3&-37&0&-4\end{bmatrix} \) and the eigenvector \( \begin{bmatrix}-4i\\4i\\24+8i\\-24-8i\end{bmatrix} \), determine the eigenvalue associated with this eigenvector.

Answer: -2 + 6i


Question: Determine whether the following statement is true or false: If \((i^n a_n)_{n=1}^{\infty}\) converges and \(a_n\in\mathbb{R}\) for all \(n\in\mathbb{N}\), then \((a_n)_{n=1}^{\infty}\) converges to 0.

Answer: True


Question: Find the residue of the function \( f(z) = (ze^{2-z}+(2-z)e^z)e^{(z-1)^{2}+(z-1)^{-2}} \) at \( z = 1 \).

Answer: 0


Question: Determine the values of $k$ for which the set of vectors $S = \{1+kx, k + x + x^2 + x^3, kx^2 + x^3\}$ is linearly independent.

Answer: k \neq 1


Question:
Determine the value of \( \lambda \in \mathbb{R} \) for which the improper integral \( I \) exists:
\[ I = \int_0^\infty \left( \frac{x+1}{3x^2+\lambda} - \frac{\lambda}{2x+1} \right) dx \]

Answer: \dfrac{2}{3}


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

Answer: 0


Question: Evaluate the limit of the integral as \( k \) approaches infinity: \[ \lim_{k\to\infty} \int_0^k \left(1-\frac{x}{k}\right)^k e^{\frac{x}{3}} \, dx \]

Answer: \dfrac{3}{2}


Question: Let $k$ be a field and $M$ be an $n \times n$ matrix over $k$. If $k[M] \cong k^m$ for some $m \le n$, is $M$ necessarily diagonalizable? Provide a proof or counterexample.

Answer: Yes


Question: Let $T \sim \mathcal{U}(0,1)$ and define the random variable $Y$ as follows: \[ Y = \begin{cases} -\Phi^{-1}(T+\frac{1}{2}) & \text{if} & 0 \leq T \leq \frac{1}{2} \\ -\Phi^{-1}(T-\frac{1}{2}) & \text{if} & \frac{1}{2} < T \leq 1, \end{cases} \] where $\Phi^{-1}$ is the inverse of the normal distribution function. Calculate the probability $P(Y \leq y)$ for a given $y \in \mathbb{R}$. \( \boxed{} \)

Answer: \Phi(y)


Question: Determine whether there exists a metric space such that for any two distinct points and any radii \( R > r \geq 1 \), the open balls centered at those points with radii \( R \) and \( r \) respectively are equal to the entire space.

Answer: Yes


Question: Is it possible to enumerate the rational numbers \( \mathbb{Q} \) as \( \{q_n\} \) such that only finitely many elements of \( \{q_n\} \) belong to the set \( A_n = \{ q \in \mathbb{Q} : q \in [1/n, 2/n] \} \) for each positive integer \( n \)?

Answer: No


Question: Evaluate the integral \( \int_0^{\infty}(x^3-3x^2)\,e^{-4x}\,dx \) without using numerical methods.

Answer: -\dfrac{9}{128}


Question: Find all natural numbers \( n \) such that the inequality \( \frac{(\ln n)^n}{n^{\ln n}} > \frac{1}{n^3} \) holds.

Answer: n \geq 2


Question: Determine whether it is possible for four vectors in three-dimensional space to have all pairwise dot products negative.

Answer: Yes


Question: What is the last digit of the number \( 2^{3^{4^{\cdots^{2016^{2017}}}}} \)?

Answer: 2


Question: Given that \(\triangle{EBA}\sim\triangle{ABQ}\) and \(PQ\) is parallel to \(DC\), derive an expression for the ratio of \(EB\) to \(DC\) in terms of the golden ratio.

Answer: \phi


Question: Determine the number of continuous and bounded functions on the interval $[0,1]$ that satisfy the equation $f(x) = f(x/2) \frac{1}{\sqrt{2}}$. Provide your answer as a single integer.

Answer: 1


Question: Determine whether the intersection of all ideals of the form \( \langle x_1-c \rangle \) in \( \mathbb{C}[x_1,\ldots,x_n] \), where \( n \geq 2 \), is equal to zero.

Answer: 0


Question: Determine the number of dimensions in which the statement "a projectivity that fixes any three points is the identity" is true.

Answer: 1


Question:
Find the value of \(a+b\) given that
\[\lim\limits_{x\to 0} \frac {x(1+a\cos x)-b\sin x}{x^3}=1.\]

Answer: -4


Question: Determine whether there exists a Borel set \( A \subset \mathbb{R}^2 \) such that its projection onto the first coordinate is not a Borel set in \( \mathbb{R} \).

Answer: Yes


Question: Suppose $p \neq q$ are distinct primes, and $x \in \mathbb{Z}$. Suppose further that $p \mid x$, and there is an integer $k \in \mathbb{Z}$ such that $p^{k} \mid x^{q}$ and $p^{k+1} \nmid x^{q}$. Determine whether it is always true that $k$ is a multiple of $q$. Provide a proof or counterexample.

Answer: Yes


Question: Let $u(x, y)$ be the solution of the first-order partial differential equation $$xu_x+(x^2+y)u_y=u, \ \text{for all}\ x, y\in \mathbb{R}$$ satisfying $u(2, y)=y-4$. Find the value of $u(1, 2)$. \[ \boxed{} \]

Answer: 1


Question: Let $F$ be a ring with characteristic $p$, and let $f(x) = a_0 + a_1x + \cdots + a_nx^n$ be a polynomial in $F[x]$. Define the derivative of $f(x)$ as $f'(x) = a_1 + 2a_2x + \cdots + na_nx^{n-1}$. Consider the homomorphism of abelian groups $D : F[x] \to F[x]$ given by $D(f(x)) = f'(x)$. Determine the kernel of $D$.

Answer: F[x^p]


Question: Evaluate the surface integral \(\int_S z\,dS\) where \(S=\{(x,y,z):x^2+y^2+z^2=a^2, z\geq 0, a>0\}\), a hemisphere of radius \(a\). Verify your result using the Gauss divergence theorem.

Answer: \pi a^3


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

Answer: \dfrac{1}{e}


Question: In a topological space, does every closed set have a point on its boundary? Answer yes or no.

Answer: No


Question: Determine the probability of reaching a state of 100 or more, starting from 20, assuming an optimal strategy is used. Provide your answer as a probability value.

Answer: 0.2


Question: In a series of 100 weighted coin tosses, the probability of obtaining heads, $p$, is either 0.1 or 0.01, but the exact value is unknown. Determine the decision boundary for the number of heads observed that would lead to the maximum likelihood estimate (MLE) of $p$. What is this decision boundary?

Answer: 4


Question: Find the stationary point of the functional \( J[y]=\int_{0}^{1} \left( x^2y'^2+2y^2 \right) dx \) subject to the boundary conditions \(y(0) = 0\) and \(y(1) = 2\).

Answer: 2x


Question: Determine whether a commutative ring can have a noncommutative endomorphism ring for some module. Provide a justification for your answer.

Answer: Yes


Question: Find the minimum value of \( \frac{n}{T(n)} \) for \( n \) in the range \([10000, 99999]\), where \( T(n) \) denotes the number of positive integer divisors of \( n \).

Answer: 140


Question: Determine the cardinality of the set of functions from \( \mathbb{N} \times \mathbb{N} \) to \( \{0, 1\} \).

Answer: 2^{\aleph_0}


Question: Find the Laurent series representation of the function \( f(z) = \frac{1}{z^2 + 4} \) within the annulus \( 0 < |z - 2i| < R \). Determine the maximum value of \( R \) for which the series is valid.

Answer: 4


Question: Determine the number of similarity classes of nilpotent 4x4 matrices over the complex numbers, $\mathbb{C}$. Relate your answer to the possible minimal polynomials.

Answer: 5


Question: Determine whether the intersection of all ideals of a commutative monoid is always a group. Provide a justification for your answer.

Answer: No


Question: Consider a vector space $V$ with a binary operation $*$ defined for all $u, v, w \in V$ such that $u * v \in V$, $u * v = - v * u$, and $u * (v * w) = (u * v) * w$. Determine if this operation can satisfy the Jacobi identity: $u * (v * w) + w * (u * v) + v * (w * u) = 0$ for all $u, v, w \in V$. Provide a justification for your answer.

Answer: Yes


Question:
Verify if the following expression for the conditional probability is true:

$$P(X_n=1|X_1=0,X_2=0,\dots,X_{n-1}=0)=\frac{P(X_1=0,X_2=0,\dots,X_{n-1}=0,X_n=1)}{1-\sum_{i=1}^{n-1} P(X_1=0,X_2=0,\dots,X_{i-1}=0,X_i=1)}$$

where $X_1, X_2, \dots, X_n$ are binary random variables on $\{0,1\}$.

Answer: True


Question: Given that \( \int ^{b}_{a} (f(x)-3x)\; dx=a^2-b^2 \) for all real numbers \(a\) and \(b\) such that \(a\neq b\), find the value of \(f\left(\frac{\pi}{6}\right)\).

Answer: \dfrac{\pi}{6}


Question: Calculate the limit \(\lim_{x\to\infty}\frac{f^{-1}(x)}{\ln(x)}\) given the function \(f(x)=e^x+x^3-x^2+x\) without explicitly calculating the inverse function \(f^{-1}(x)\).

Answer: 1


Question: Suppose $A$ and $B$ are commuting invertible matrices with a common generalized eigenvector $v$ and eigenvalues $a$ and $b$, respectively. Given that there exist positive integers $K$ and $L$ such that $(A-aI)^K v = 0$ and $(B-bI)^L v = 0$, determine if there exists a positive integer $M$ such that $(AB-abI)^M v = 0$. Provide a justification for your answer.

Answer: Yes


Question: Determine the fundamental group of the space obtained by removing four small squares from a large square, with the edges identified as shown in the figure.

Answer: F_5


Question: Given that \(\tan a\) and \(\cot a\) are two real roots of the equation \(x^2 + k^2 - kx - 3 = 0\), and \(3\pi < a < \frac{7\pi}{2}\), find the value of \(\sin a + \cos a\).

Answer: -\sqrt{2}


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{a^{\tan x} - a^{\sin x}}{\tan x - \sin x}$$

Answer: \ln a


Question: Let $A \in \mathcal{M}_{n}(\mathbb{Q})$ be a square matrix such that $A^{2017} = I_n$. For $n \geq 2016$, is there a counterexample where $A \neq I_n$ while still satisfying $A^{2017} = I_n$?

Answer: Yes


Question: If $n$ is not a power of a prime, does there always exist a group with order divisible by $n$ that does not have a subgroup of order $n$?

Answer: Yes


Question: Let $g_n:[0,1] \to \mathbb{R}$ be defined by \( g_n(x) = \frac{nx}{1+n^2 x^q} \) for \( q > 0 \). Determine the values of \( q \) for which the sequence \( g_n \) converges uniformly to its point-wise limit.

Answer: (0, 2)


Question: Determine which of the following functions is differentiably redundant of order 6: (a) $e^{-x} + e^{-x/2} \cos\left(\frac{\sqrt{3x}}{2}\right)$, (b) $e^{-x} + \cos(x)$, or (c) $e^{\frac{x}{2}}\sin\left(\frac{\sqrt{3x}}{2}\right)$. Consider the significance of the complex numbers $1/2 \pm i\sqrt{3} /2$ in your determination.

Answer: C


Question: Determine whether uniform integrability combined with convergence in measure implies convergence in $L^1$. Provide a justification for your answer.

Answer: No


Question: Calculate the limit: \[ \lim_{n\to\infty} \int_{0}^{\pi} \sqrt[n]{\sin(x)}\,dx. \]

Answer: \pi


Question:
Given the following conditions:
1) \( \int_{0}^{\pi} [f(x) + f''(x)]\sin(x)\,dx = 2 \)
2) \( f(\pi) = 1 \)
3) \( f''(x) \) is continuous
Calculate \( f(0) \).

Answer: 1


Question: Given a positive-definite, Hermitian 2x2 matrix $G$ and an invertible 2x2 matrix $W$, does the equation $WGW^\dagger = G$ imply that $W$ is unitary with respect to the standard scalar product? Consider the scalar product defined by $G$: $\langle v, w \rangle_G = v^T G \overline{w}$.

Answer: No


Question: In the projective space $\mathbb{P}^2$, consider the curves defined by the polynomials $f = x^2 - yz$ and $g = (x+z)^2 - yz$. Compute the intersection multiplicity of $V(f)$ and $V(g)$ at the point $p = [-2:1:4]$. Use the fact that the intersection multiplicity at $p$ is equal to the Hilbert polynomial of the $I(p)$-primary component of $(f)+(g)$, where $I(p) = (X+2, Y-1, Z-4)$. Provide your answer as a single integer.

Answer: 1


Question: Evaluate the limit: $$\lim_{n\to\infty}\sum_{k=1}^n\sin^2\left(\frac{\pi}{n+k}\right)$$

Answer: 0


Question: Evaluate the limit \(\lim \limits_{(x, y) \to (0, 0)} \frac{(e^x-1)(e^y-1)}{x+y}\) as \((x, y)\) approaches \((0, 0)\) along the path \(y = -x + mx^2\).

Answer: -\dfrac{1}{m}


Question:
Find the value of $a$ such that the function \( f: \mathbb{R}^2 \to \mathbb{R} \) defined by
\[ f(x,y) = \begin{cases}
      \frac{x\sin^2(x) + axy^2}{x^2 + 2y^2 + 3y^4} & (x,y) \neq (0,0) \\
      0 & (x,y) = (0,0)
   \end{cases} \]
is differentiable at \((0,0)\).

Answer: 2


Question: Use Green's Theorem to evaluate the line integral \( \int_C P\,dx + Q\,dy \), where \( P = xy \) and \( Q = x^2 \). The curve \( C \) is the loop in the first quadrant described by the polar equation \( r = \sin 2\theta \).

Answer: \dfrac{16}{105}


Question: Find an expression for the sum of the infinite series: \[ \sum_{n=1}^{\infty} (-1)^{n}\frac{(2x)^{2n-1}}{(2n-1)!} \]

Answer: -\sin(2x)


Question: Find the limit of the sequence \(\{u_n\}\) defined by \(u_n = \frac{1 + 2^2 + 3^3 + \cdots + n^n}{n^n}\) as \(n\) approaches infinity.

Answer: 1


Question: Using Newton's Method, determine the minimum distance between the origin and the curve \( y = \sin(x) \).

Answer: 0


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

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$\lim _{x \to \infty} \frac {1} {x} \int_1^x \frac {\ln t} {t} \, \mathrm d t$$ using integration by parts.

Answer: 0


Question: Determine the number of solutions to the equation \(e^{x^3-x} - 2 = 0\) within the interval \([0,1]\).

Answer: 0


Question: Let $G$ be a group, $H$ a normal subgroup of index $n$, and $K$ a subgroup of $G$ not contained in $H$. If $n$ is a power of a prime, determine whether $G = HK$.

Answer: No


Question: Evaluate the sum of the series: \[ \sum_{n=1}^{\infty} \frac{(n-1) \bmod 4}{3^{n-1}} \]

Answer: \dfrac{27}{40}


Question: Is the function \( \nu: \mathfrak{N} \rightarrow \mathbb{N}_{>1} \), defined by \( \nu(G) \) as the number of normal subgroups of a non-abelian finite group \( G \) strictly between the trivial subgroup and \( G \), surjective? In other words, does there exist a non-abelian group with exactly \( n \) normal subgroups for any positive integer \( n > 1 \)?

Answer: Yes


Question:
Determine whether \( f'(2) = \lim \limits_{x \to 2}f'(x) \) for the function:

\[f(x) = \begin{cases} (x-2)^3\sin\left(\frac{1}{x-2}\right), & \text{if $x$ $\ne$ 2} \\
0, & \text{if $x$ $=$ 2} \end{cases}\]

Answer: Yes


Question: Given that \( a \) and \( b \) are prime numbers and \( c \) is a natural number, such that \( 2\sqrt{a} + 7\sqrt{b} = c\sqrt{3} \), find the value of \( a^2 + b^2 + c^2 \).

Answer: 99


Question: Compute the covariance \( \text{Cov}[X,Y] \) for the joint mass density function given by: \[ f_{X,Y}(x,y) = \frac{{e}^{-(x + \frac{y}{x})}}{x}, \ y > 0, \ x > 0. \] Assume that \( X \) follows an exponential distribution with parameter 1.

Answer: 1


Question: Find the volume of the solid enclosed by the parabolic cylinder \( z = 1 - y^2 \), the planes \( x = 0 \) and \( z = 0 \), and the plane \( x + z = 1 \).

Answer: \dfrac{4}{5}


Question: Find the extrema of the function \( f(x, y, z) = x^2 + y^2 + z^2 \) subject to the constraint \( x^2 - y^2 + z^2 = 1 \).

Answer: 1


Question: Calculate the integral \( \int_\gamma \frac{dz}{z} \) where \( \gamma = a \cos(t) + i b \sin(t) \) for \( t \in [0, 2\pi] \), using the complex logarithm function.

Answer: 2\pi i


Question: Determine the number of integer solutions \((a, b, c)\) to the equation \(a^2 + b^2 = 10c^2\).

Answer: \infty


Question: Find the ratio \( \overline{PR} : \overline{RQ} \) for the tangent to the curve defined by the equation \( y^2 = x^3 \) at point \( P(1,1) \), where \( Q \) is the point of intersection with the \( x \)-axis and \( R \) is the point of intersection with the \( y \)-axis.

Answer: 3:1


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{n! + 2^n \log(n)}{3n! + n^2}$$

Answer: \dfrac{1}{3}


Question: Calculate the area under one arch of the cycloid given by the parametric equations \(x=6(t-\sin(t))\) and \(y=6(1-\cos(t))\).

Answer: 108\pi


Question: Find the limit of the sequence \( \lim_{n \rightarrow \infty} \frac{(-1)^{n}\sqrt{n}\sin(n^{n})}{n+1} \).

Answer: 0


Question: Find the eigenvalues of the linear operator $M$ on the vector space $W = \mathbb{C}_3[x]$ defined by $Mf = f''' + f'$.

Answer: 0


Question: Find the distance between the two disconnected components of the curve given by the equation \( z^2 + (\overline{z})^2 = 2 \) in the complex plane.

Answer: 2


Question: Determine the number of distinct possible values of $y$ for which there exists a solution $x$ to the congruence $-2131 \times 3^x \equiv y \pmod{(43 \cdot 41)}$. Provide your answer as a single integer.

Answer: 168


Question: Evaluate the integral \( \int_0^{2\pi} e^{\cos\theta}[\sin \theta \cos(\sin \theta) +\cos \theta \sin(\sin \theta)]d\theta \) and determine if it is equal to zero. Provide a proof for your answer.

Answer: 0


Question: Calculate the operator norm \( ||T|| \) of the linear operator \( T: (C[0, \pi], ||.||) \to (C[0, \pi], ||.||) \) defined by \( T(f)(x) = \int_0^x \sin(x+y)f(y)dy \) for \( f \in C[0, \pi] \).

Answer: 2


Question: Given the equations \(x^4 + y^4 = 5\) and \(x^2 + xy + y^2 = 10\), find the value of \((x+y)^4\).

Answer: 195


Question: Determine the Galois group of the polynomial $x^3 - x - 1$ over the field of rational numbers $\mathbb{Q}$. \( \boxed{\text{Galois Group}} \)

Answer: S_3


Question: Let $R$ be an equivalence relation on $P(\Bbb Z)$. Determine the number of elements in the equivalence class of a set $S$ that contains exactly $k$ elements.

Answer: \aleph_0


Question: Let $B(x_0,R)$ be an open ball in $\mathbb{R}^n$ ($n \geq 2$). Let $u \in H^{1}(B(x_0,R))$ be a weak solution of $\Delta u =0$ in $B(x_0,R)$. Let $u_{x_i}, i \in \{ 1,...,n\}$ be a weak derivative of $u$. Determine if the following statement is true: $$ u_{x_i}(x_0) = \frac{1}{|B(x_0,R)|} \int_{B(x_0,R)}u_{x_i} (y) dy ?$$

Answer: True


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

Answer: 1


Question: Determine the maximum number $k$ of linearly independent homogeneous quadratic forms $Q_1, \dots, Q_k$ in five variables over the complex numbers $\mathbb{C}$ such that the intersection $V(Q_1) \cap \dots \cap V(Q_k) \subseteq \mathbb{P}^4$ contains a connected component that is positive-dimensional and does not lie within a hyperplane.

Answer: 6


Question:
Determine whether the following equation is true for a sequence of iid random variables $\{Y_i, \forall i \ge 0\}$:

$$ p(g(Y_{n+1}) = k_{n+1} | f_{n}(Y_0,Y_1,...,Y_n) = k_n, f_{n-1}(Y_0,Y_1,...,Y_{n-1}) = k_{n-1},...,f_0(Y_0) = k_0) = p(g(Y_{n+1}) = k_{n+1}) $$

Given that $g(Y_{n+1})$ is independent of every $f_i(Y_0, ...,Y_i), \forall i \le n$. Is the equation true?

Answer: True


Question: Evaluate the Riemann–Stieltjes integral \( \int_0^1 f(x') \, dc(x') \) where \( f(x) = 1 \) and \( c(x) \) is the Cantor function. What is the value of this integral?

Answer: 1


Question: Let $p(x)$ be a polynomial of degree $n$. If the equation $q(x) = (p(x))^2 + 2p(x) - 2 = 0$ has at least two distinct real solutions, does it follow that the equation $p(x)q(x) = 4$ also has at least two distinct real solutions? Answer yes or no.

Answer: Yes


Question: Determine the number of continuous functions \( f: [0,1] \to [0, \infty) \) that satisfy the inequality \( \int_{0}^{x} f(t) \, dt \geq f(x) \) for all \( x \in [0,1] \).

Answer: 1


Question: Find all quadruples of positive integers \((a, b, c, d)\) such that \(a^3 = b^2\), \(c^3 = d^2\), and \(c - a = 7\).

Answer: (9, 27, 16, 64)


Question: Evaluate the limit: $$\displaystyle\lim_{x\to0}\left(\frac{1}{x^5}\int_0^xe^{-t^2}\,dt-\frac{1}{x^4}+\frac{1}{3x^2}\right)$$

Answer: \dfrac{1}{10}


Question: Determine the integer $a$ such that $83359654581036155008716649031639683153293510843035531^a \equiv 1 \pmod{32003240081}$. Use Euler's totient function to find the solution.

Answer: 32003240080


Question:
Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$. Let $A \subset \mathbb{R}$ be a measurable set such that $\lambda(A) > 0$. Is it always possible to find two measurable sets $K_1, K \subset \mathbb{R}$ such that:
- $K_1 \subset A$
- $K \subset \mathbb{R}$
- $K_1 \subset K$
- $K$ is compact
- $\lambda(K_1) = \lambda(K)$
- $\lambda(K_1) > 0$?

Provide a justification for your answer.

Answer: Yes


Question: Consider a Markov Chain based random number generator with 5 points, starting at the 1st point. Each point has a 25% chance to move forward, a 25% chance to move backward, and a 50% chance to move to an adding point. The adding points add 1 to the final number and return to their connected point. The 1st point cannot move backward and instead returns to itself. If the generator moves forward from the 5th point, it reaches a final absorbing point, ending the process. Calculate the expected number of times the generator visits the adding points before reaching the final absorbing point.

Answer: 30


Question: Evaluate the integral \( \int_{0}^{\pi} \sin(A\sin(x))\sin(x) \,dx \) where \( A \) is a constant.

Answer: \pi J_1(A)


Question: Evaluate the contour integral \(\oint\limits_{|z-3|=4}^{}\frac{z}{\cos z-1}dz\) using the residue theorem.

Answer: -8\pi i


Question: Determine whether the following limit exists: $$\lim_{(x,y) \to (0,0)} \arctan\left(\frac {x^4+y^4}{x^2+y^2}\right)$$

Answer: 0


Question: Given that $z_1$, $z_2$, $z_3$, $z_4$ are the roots of the equation $z^4+z^3+z^2+z+1=0$, find the least value of $\lfloor \text{mod}(z_1 + z_2) \rfloor + 1$, where $\lfloor . \rfloor$ denotes the greatest integer function.

Answer: 1


Question: Evaluate the integral \( I = \frac{1}{2\pi i} \int_{\gamma} \frac{\sin(z^3)}{z^2} \, dz \) where \( \gamma \) is a closed curve containing the origin.

Answer: 0


Question: Find the expected value of the function \( \phi(x) = x + \frac{F(x) - 1}{f(x)} \), where \( x \in [\underline{x}, \bar{x}] \) and \( f(x) \) is a density function.

Answer: \underline{x}


Question: Find the probability distribution function (PDF) for the random variable \( Z = |X - Y| \), where \( X \) and \( Y \) are independent random variables uniformly distributed between 0 and 1.

Answer: 2(1 - z)


Question: Determine whether a group of order 105 must have both a normal Sylow 7-subgroup and a normal Sylow 5-subgroup.

Answer: Yes


Question: Determine whether the following statement is true or false: If $A$ is an $m \times n$ matrix and $B$ is an $n \times m$ matrix such that $AB = I_m$, then $BA = I_n$. Provide a justification for your answer.

Answer: False


Question:
Given a set of independent random variables $X_1, \ldots, X_n$, each with a common density function, mean $\mu$, and variance $\sigma^2$, the sample variance $S^2$ is defined as:
$$S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2$$
where $\bar{X} = \frac{1}{n} \sum^n_{j=1} X_j$. Derive the expected value of the sample variance, $E[S^2]$, using the relation $X_i - \bar{X} = (X_i - \mu) - (\bar{X} - \mu)$. Provide your final expression for $E[S^2]$. \( \boxed{} \)

Answer: \sigma^2


Question: Let $W$ be the vector space of polynomials of degree at most $100$ with complex coefficients. Define a linear map $$S:W\to\mathbb{C}^{500},\quad S(p)=(p(0),p(1),\ldots,p(499)).$$ Find the dimension of the null space of $S$. \( \boxed{\text{Answer}} \)

Answer: 0


Question: Let $g$ be an entire function such that $g(1)=0$ and $|g(z)| \le |z^3-2|$ for all $z \in \mathbb{C}$. Determine the function $g$.

Answer: 0


Question: Find the angle between the two parabolas given by the equations \(y^2=2px-p^2\) and \(y^2=p^2-2px\) at their point of intersection.

Answer: 0


Question: Determine the rank of the linear transformation \( f(X) = AX - XA \) where \( A \) is a given \( n \times n \) matrix in the vector space \( V \) of all \( n \times n \) matrices.

Answer: n^2 - n


Question: Evaluate \( \lim_{x\to 0}\frac{1}{x^2}\int_{0}^{x}\cos(t^2)\,dt. \)

Answer: \infty


Question: Let $g(x)$ be a strictly increasing function that maps positive integers to positive integers. If $g(g(x)) = 4x$, find a possible value of $g(2)$. Provide your answer as a single integer.

Answer: 4


Question: Determine whether the limit \( \lim_{k\to\infty}\prod_{m=2^k}^{m=2^{k+1}} \frac{2m}{2m-1} \) exists, and if it does, find its value.

Answer: \sqrt{2}


Question:
Evaluate the limit of the series as \( n \to \infty \):
\[ x(n) = \sum\limits_{k=1}^n \frac{k^2}{n^3+k^2} \]

Answer: \dfrac{1}{3}


Question: Evaluate the limit \( \lim_{n\rightarrow\infty}\frac{\sqrt{n}}{e^n} \) using the squeeze theorem and Stirling's approximation.

Answer: 0


Question: Consider a function \(g:[a,b]\rightarrow [a,b]\) that satisfies \(|g(x)-g(y)|\leq k|x-y|\) for all \(x,y \in [a,b]\), where \(0<k<1\). If we randomly choose an initial point \(y_0\in[a,b]\) and define a sequence recursively as \(y_{n+1}=g(y_n)\), does the sequence \(\langle y_n\rangle^\infty_{n=1}\) converge to a limit point that is independent of \(y_0\)?

Answer: Yes


Question: Determine whether there exist functions $f(x)$ and $g(x)$, both positive for $x > 0$, such that $f(x) \leq c g(x)$ for some constant $c > 0$, while simultaneously $\log(f(x)) > c \cdot \log(g(x))$ holds.

Answer: Yes


Question: Evaluate the integral \( \displaystyle \int_{\partial \mathcal{D}(0,2)} \frac{8z^2-8z+1}{4z^3-8z^2+z-2}dz \) using the residue theorem.

Answer: 2\pi i


Question: Let $X$ and $Y$ be independent real-valued Borel-measurable random variables. Given an open set $U\subseteq\mathbb{R}$, consider $\Omega_X:= X^{-1}(U)$ and $\Omega_Y:=Y^{-1}(U)$. Are the random variables $X\cdot\mathbb{1}_{\Omega_X}$ and $Y\cdot\mathbb{1}_{\Omega_Y}$ independent?

Answer: Yes


Question: Determine whether the sequence $\{a_n\}_{n=1}^\infty$ defined by \( a_n = \sum_{k=1}^{n}\frac{1}{(n+k)^2} \) converges or diverges. If it converges, find its limit.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{\sqrt{\sin(\tan(x))}}{\sqrt{\tan(\sin(x))}}$$

Answer: 1


Question: Determine the possible values of the greatest common divisor of $n^5 + 6$ and $n^5 + 2$ given that $n^5 + 4$ is a perfect square.

Answer: 2


Question: Evaluate the Riemann-Stieltjes integral \( \int_0^3 x^3 \, \text{d}\left[\frac{x}{2}\right] \) using the properties that \([x/2] = 0\) for \(0 \leq x < 2\) and \([x/2] = 1\) for \(2 \leq x < 4\).

Answer: 8


Question: Determine whether there exists a function that is non-zero only at a countable number of points, yet has a graph that is dense in \( \mathbb{R}^2 \).

Answer: Yes


Question: Given that \(\frac{a+b+c+d}{\sqrt{(1+a^2)(1+b^2)(1+c^2)(1+d^2)}} = \frac{3\sqrt{3}}{4}\) for positive real numbers \(a, b, c, d\), find the value of \(\frac{1}{\sqrt{1+a^2}} + \frac{1}{\sqrt{1+b^2}} + \frac{1}{\sqrt{1+c^2}} + \frac{1}{\sqrt{1+d^2}}\).

Answer: 2\sqrt{3}


Question: Determine the number of roots of the equation \(2\cos^{2} \theta = 1 + 3^{\sin \theta} + 3^{\csc \theta}\) within the interval \([0, 2\pi]\).

Answer: 2


Question: Given non-negative integers $a$ and $b$, and a prime number $p \geq 3$, if $a^2 + b^2$ and $a + b$ are both divisible by $p$, does it necessarily follow that $a$ and $b$ are divisible by $p$? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: Verify the identity: \[ \frac{1}{2}\frac{\Gamma\left(\frac{v}{2}-1\right)}{\Gamma\left(\frac{v}{2}\right)} = \frac{1}{v-2} \] using the property \( \Gamma(z) = (z-1)\Gamma(z-1) \).

Answer: \frac{1}{v-2}


Question: Let $\{\phi_k\}$ be an orthonormal sequence in a Hilbert space $H$. If $\psi_n$ converges weakly to $\psi$ in $H$, does it necessarily follow that $\lim_{n\rightarrow\infty}\langle\psi_n-\psi,\phi_k\rangle _{H}=0$ for each $k$? Provide a justification for your answer.

Answer: Yes


Question: Compute $f(2)$ if $f$ is continuous and satisfies the equation \( \int_0^{x^2(1+x)} f(t) \, dt = x \) for all \( x \geq 0 \).

Answer: \dfrac{1}{5}


Question: Find the value of $k$ such that the fractional part of the number $\frac{3^{500}}{17}$ is $\frac{k}{17}$.

Answer: 13


Question: A cube with side length 12 cm is given. Let point P be the intersection of lines BG and FC, and point Q be the intersection of segment EP with plane DBFH. Calculate the length of segment EQ.

Answer: 4\sqrt{6}


Question:
Let $x, y, z > 0$ such that $x + y + z = 1$. Find the minimum value of the expression:

$$P = \left(\frac{x+1}{x}\right)^3 \cdot \frac{z^2}{z^2+1} + \left(\frac{y+1}{y}\right)^3 \cdot \frac{x^2}{x^2+1} + \left(\frac{z+1}{z}\right)^3 \cdot \frac{y^2}{y^2+1}$$

Answer: \dfrac{96}{5}


Question: Suppose $(x_n)_n$ is a sequence in $\mathbb{C}$ such that for every sequence $(y_n)_n$ in $\mathbb{C}$ where the series $\sum_n y_n$ converges, the series $\sum_n (x_n y_n)$ also converges. Is the sequence $(x_n)_n$ necessarily bounded?

Answer: Yes


Question: Consider a sequence $X_1, X_2, \dots$ of independent exponential random variables, each with a mean of 1. Define $N$ as the minimum value of $n$ such that $\sum_{i=1}^n X_i > k$, where $k$ is a positive real number. Compute the expected value of $N$, $E[N]$, using Wald's equation $E[\sum_{i=1}^N X_i]=E[N]E[X]$. What is $E[N]$?

Answer: k + 1


Question: Find the limit of the sequence \(x_n = \frac{1+2^2+\cdots+n^n}{n^n}\).

Answer: 1


Question: Determine whether the following application of the law of total expectation is valid: \[ \mathbb{E}[X - \mathbb{E}[X \mid Y]] = \mathbb{E}[\mathbb{E}[X - \mathbb{E}[X \mid Y] \mid Y]] = 0. \]

Answer: 0


Question:
Given that the area of a triangle with vertices $(x_1, x_1^2)$, $(x_2, x_2^2)$, and $(x_3, x_3^2)$ is 2 square units, and the area of another triangle with vertices $(y_1, y_1^2)$, $(y_2, y_2^2)$, and $(y_3, y_3^2)$ is 10 square units, calculate the value of the determinant:
$$
\begin{vmatrix}
(1+x_1y_1)^2 & (1+x_2y_1)^2 & (1+x_3y_1)^2 \\
(1+x_1y_2)^2 & (1+x_2y_2)^2 & (1+x_3y_2)^2 \\
(1+x_1y_3)^2 & (1+x_2y_3)^2 & (1+x_3y_3)^2 \\
\end{vmatrix}
$$

Answer: 160


Question: Determine the number of preimages of a general point \((a,b)\) under the mapping \(p\colon S^1\times S^1\rightarrow S^1\times S^1\) defined by \(p(z_1,z_2)=(z_1^2,z_2^3)\).

Answer: 6


Question: Consider the sequence \( \{\alpha_{k} \in \mathbb{N} \mid k \in \mathbb{N}, \sin\alpha_{k} < \sin\alpha_{k+1} \} \). Calculate the limit: \[ \lim_{k \to \infty} \frac{\alpha_{k+1} - \alpha_k}{\alpha_{k+2} - \alpha_{k+1}} \]

Answer: 1


Question: Given a number $a$ such that $RDS_{10}(a) = 7$, determine if it is always true that $RDS_{10}(a^4) = 7$. Provide a justification for your answer.

Answer: Yes


Question:
Consider the sets $S$ and $I$ defined as:

$$S=\left\{\begin{bmatrix} a & b \\ 0 & c \\ \end{bmatrix} \middle| a \in \mathbb{Z}_4, b, c \in \{0, 2\} \right\}$$
$$I=\left\{\begin{bmatrix} 0 & b \\ 0 & c \\ \end{bmatrix} \middle| b, c \in \{0, 2\} \right\}$$

Determine the structure of the quotient set $S/I$ and the size of its multiplication table.

Answer: 16


Question:
Verify whether the operation \(\odot\) defined on the vector space \(\Bbb F_q[X]_{<n}\) by \((f(X), g(X)) \mapsto f(X) \odot g(X) = r(X)\), where \(r(X)\) is the remainder of \(f(X)g(X)\) divided by \(X^n-1\), satisfies the associativity property:
\[
(f(X) \odot g(X)) \odot h(X) = f(X) \odot (g(X) \odot h(X))
\]

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to0} \frac{f\left(x+\sin x\right)-f\left(\sin x\right)}{x} \] where \( f \) is a continuously differentiable function on \( \mathbb{R} \).

Answer: f'(0)


Question: Evaluate the definite integral \( \int_2^4 \frac{\sqrt{\ln(9-x)}}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}} \, dx \).

Answer: 1


Question: Determine the 100th digit to the right of the decimal point in the decimal representation of \((1 + \sqrt{2})^{3000}\).

Answer: 9


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

Answer: \infty


Question: Compute the integral \( \int_{|z| = 2} \frac{dz}{z^2 - 1} \) without using winding numbers.

Answer: 0


Question: Calculate the limit: \[ \lim_{t \to \infty} \ln(t)t^x e^{-t} \] where \( x \in (0,1] \).

Answer: 0


Question: Find the maximum value of \(|2z^2 - 3iz + 1|\) given that \(|z| < \frac{1}{4}\).

Answer: \dfrac{13}{8}


Question: Calculate the limit: $$\lim \limits_{n \to \infty}\left(n-\sum_{k=1}^n\cos{\frac{\sqrt{k}}{n}}\right)$$

Answer: \dfrac{1}{4}


Question: Find the function $f(x)$ such that \( \int_0^1 f(x) - f(x^2)^2 \, dx = \frac{1}{3} \).

Answer: \sqrt{x}


Question: If $f$ is uniformly continuous on $(0,1)$, determine whether $f$ can be extended to $[0,1]$ such that it is right continuous at $0$ and left continuous at $1$. Justify your answer.

Answer: Yes


Question: Consider the set \(A = \{(x, \sin\frac{1}{x}) \mid 0< x \leq 1 \}\), a subset of \(\mathbb{R}^2\). Determine the interior of \(A\).

Answer: \emptyset


Question: Given that \(\gcd(ax+by,cx+dy)=\gcd(x,y)\) where \(ad-bc= \pm 1\), and \(\gcd(p,q)=3\) with \(8p+10q=3\), find \(\gcd(10p+20q, 20p+25q)\).

Answer: 15


Question: Calculate the sum of the alternating series \(\sum_{n=1}^\infty \frac{(-1)^n}{4^n n!}\) correct to four decimal places.

Answer: -0.2212


Question: Determine whether the following claim is true or false: If $f(x)$ is a $2\pi$-periodic function that is continuously differentiable on the real line, then its Fourier series converges uniformly to $f(x)$ on the interval $[-\pi, \pi]$. Provide a justification for your answer.

Answer: True


Question: Evaluate the integral \( \int_0^\infty \int_0^t e^{-t}\frac{\sin(\tau)}{\tau}d\tau dt \) using the Laplace transform.

Answer: \dfrac{\pi}{4}


Question: Let $A$ be the set of nonnegative sequences $(a_n)$ such that $\sum_{n=1}^{\infty}a_n = 1$. Determine the range of the mapping $P: A \to \mathbb{R}$ defined by \[ P((a_n)) = \prod_{n=1}^{\infty}(1+a_n). \]

Answer: [2, e)


Question: Let $T: L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ be a linear operator. If $\langle Tf, g \rangle = 0$ for all compactly supported functions $f$ and $g$, is $T$ necessarily the zero operator?

Answer: No


Question: Solve the recurrence relation \( 2y_n = 11 y_{n-1} - 20y_{n-2} + 12y_{n-3} + 2 \) with the initial conditions \( y_0 = \frac{9}{8} \), \( y_1 = \frac{-5}{4} \), and \( y_2 = \frac{-1}{2} \). Find the next term \( y_3 \).

Answer: \dfrac{35}{2}


Question: Calculate the curl of the radial unit vector $e_r$ in spherical coordinates.

Answer: 0


Question: Evaluate \( \lim_{t \to \infty} \int_{0}^{\infty} e^{-tx} \frac{\sin x}{x} \, dx \).

Answer: 0


Question: Calculate the number $x$ that equals the sum of the prime positive divisors of the number of zeros at the end of the product of the first $784$ integers when expressed in base $15$. Then, find $x - 60$. Choose the correct answer from the following options: \(\begin{array}{ll} 1.&\text{33}\\ 2.&\text{39}\\ 3.&\text{30}\\ 4.&\text{28}\\ \end{array}\)

Answer: B


Question: For a $2\times2$ matrix $A$ defined over the finite field $\mathbb{Z}_p$ with $p>2$, determine the trace of $A$ given that $A^tA=-I$ and $\det(A)=1$. What is the trace of $A$?

Answer: 0


Question: Let $X$ be an infinite set with the cofinite topology. Determine whether the statement "Every infinite, co-infinite subset of $X$ has a non-empty interior" is true or false.

Answer: B


Question: Let $X$, $Y$, and $Z$ be independent Poisson random variables with parameter $\lambda = 1$. Find the expected value of $X + Y$ given that $X + Y + Z = n$.

Answer: \dfrac{2n}{3}


Question: Determine the radius of convergence for the series: $$\sum_{n=1}^{\infty} \left(\frac{(2n-1)!!}{(2n)!!}\right)^3\left(\frac{3z-1}{2}\right)^n$$

Answer: \dfrac{2}{3}


Question: Determine all possible Jordan canonical forms of a $5 \times 5$ complex matrix with a minimal polynomial of $(x-2)^2(x-1)$. How many distinct Jordan forms are there?

Answer: 4


Question: Determine the number of functions \( \phi : \mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\} \) such that \( \phi(ab) = \phi(a) + \phi(b) \) for all \( a, b \in \mathbb{N} \cup \{0\} \).

Answer: 1


Question:
Consider a square matrix $A_n$ with dimensions $n \times n$ where $n \geq 2$, and it satisfies the equation $A^2 + 2A = 0$. Determine which of the following statements is true:

1. $A$ is singular.
2. $A$ is nonsingular.
3. 0 and -2 are eigenvalues of $A$.
4. Either 0 or -2 is not an eigenvalue of $A$.
5. All of the above statements are false.

Using the Cayley-Hamilton theorem, analyze the characteristic polynomial of $A$ to determine the validity of the options.

Answer: E


Question: A train passenger arrives at a busy railway station and chooses a taxi stand. At stand A, taxis arrive according to a Poisson process with a rate of 2 taxis per minute, and passengers choose this stand with a probability of 0.7. At stand B, taxis arrive according to a Poisson process with a rate of 1 taxi per minute, and passengers choose this stand with a probability of 0.3. After waiting for 3 minutes at the front of the queue without a taxi, what is the probability that the passenger is waiting at taxi stand A?

Answer: \dfrac{7}{7 + 3e^3}


Question: Determine if there exist intervals \((a, b)\) such that the number of prime squares \((p^2)\) within the interval exceeds the number of primes \((p)\) within the same interval.

Answer: Yes


Question: Given a matrix $M \in M_{5}(\mathbb{R})$ with the characteristic polynomial $(x-2)^5$, determine the largest possible dimension of the centralizer of $M$ in $M_{5}(\mathbb{R})$. Provide your answer as a single integer.

Answer: 25


Question: Find all natural numbers \( n \) such that \( n = x^y = (x+y)^2 \), where \( x, y \in \mathbb{Z}^+ \) and either \( x \) or \( y \) is a prime number.

Answer: 64


Question:
Calculate the sum \( \sum_{i=1}^{200} f(i) \) for the function \( f(x) \) defined as follows:
\[ f(x)= \begin{cases} \left[\frac{1}{\{\sqrt{x}\}}\right] & \text{if } x \neq k^2, \\ 0 & \text{otherwise}, \end{cases} \]
where \( x \) and \( k \) are natural numbers, \([\cdot]\) denotes the greatest integer function, and \(\{\cdot\}\) represents the fractional part.

Answer: 629


Question: Suppose that X and Y are independent standard normal random variables. Calculate the probability that $X^2 + Y^2 < 1$. Express your answer as a single probability value.

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


Question: Determine the number of elements in the set \( S = \{a \in F \mid \phi^2(a) = a\} \), where \( F \) is a field of \( p^{23} \) elements, \( p \) is a prime number, and \( \phi: F \rightarrow F \) is the field automorphism that sends \( a \) to \( a^p \).

Answer: p


Question: Evaluate the limit: $$\lim_{b \to \infty} \frac{1}{b^3} \int_{2}^{b^3} \left(\frac{3}{x} + \ln\left(\frac{x+2}{x}\right)\right)\, dx$$ using L'Hopital's rule and the Fundamental Theorem of Calculus.

Answer: 0


Question: Let $C \sim \hbox{Bernoulli}(p)$ and $X$ and $Y$ be two independent random variables. Define a new random variable $Z$ by the experiment: $$ Z=\begin{cases} X & \text{if } C=0 \\ Y & \text{if } C=1 \end{cases} $$ Show that $C$ is independent of $Z$ by proving that $E[C|Z] = E[C]$.

Answer: E[C|Z] = E[C]


Question: Evaluate the limit: $$\lim\limits_{x \to 0}\left(\frac{(1+ 4x )^{\frac1{2x}}}{e^2}\right)^{-\frac1{3x}}$$

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


Question: Determine if the Jacobson radical of the ring $S$ obtained by the Dorroh extension theorem, where $S=\mathbb{Z} \times R$ with element-wise addition and the multiplication law $(n,a)(m,b)=(mn, nb+ma+ab)$, is equal to the Jacobson radical of the original ring $R$. Specifically, verify if $J(S) = J(R)$ by applying the rule $J(R_1 \times R_2) = J(R_1) \times J(R_2)$, leading to $J(S) = J(\mathbb{Z}) \times J(R) = 0 \times J(R) = J(R)$. Provide a justification for your conclusion.

Answer: J(S) = J(R)


Question: Let $X$ be a compact and Hausdorff space with a uniformity $\mathcal{U}$. Given that $X$ is a normal space, for every point $x \in X$ and every closed set $A \subseteq X$, there exist open sets $U$ and $V$ such that $x \in U$, $A \subseteq V$, and $U \cap V = \emptyset$. Is there an entourage $U \in \mathcal{U}$ such that $U[x] \cap U[A] = \emptyset$, where $U[x] = \{y : (x, y) \in U\}$?

Answer: Yes


Question: Find the slope of the curve formed by the intersection of the surface given by \( z = x^3 + xy^2 \) and the plane \( 2x - 2y = 1 \) at the point where \( x = 1 \) and \( y = \frac{1}{2} \).

Answer: \dfrac{17}{4}


Question: Determine whether the following statement is true or false: If \(\sum_{n=1}^\infty \frac{1}{|z_n|}\) converges, then \(\sum_{n=1}^\infty \frac{1}{z_n}\) converges absolutely.

Answer: A


Question: In neutral geometry, if opposite angles formed by intersecting lines are equal, does this imply that the lines are parallel? Answer true or false.

Answer: B


Question: Evaluate the limit: $$\lim_{n\to \infty} \frac{n^a}{(\ln n)^b}$$ for all \(a, b > 0\).

Answer: \infty


Question:
Find the total length of the cardioid given by the polar equation \( r = 1 - \cos\theta \). Use the formula for the arc length of a polar curve:
\[ L = \int_{a}^{b}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\, d\theta \]

Answer: 8


Question: Determine the number of ideals in the ring \( \mathbb{R}[x]/\langle x^3+2x^2-x-2 \rangle \), where \( \mathbb{R} \) denotes the field of real numbers.

Answer: 8


Question: Determine whether the following statement is true or false: If $f$ is a measurable function on $[0,\infty)$ with finite $L_p$ norm for $p > 1$, then the limit of $t^{-(1-1/p)} \int_0^t f(x) \, dx$ as $t$ goes to infinity is zero.

Answer: True


Question: Find the value of \(a\) such that \(\lim_{{x \to 0^+}} \ln(x \ln(a))\ln\left(\frac{\ln(ax)}{\ln\left(\frac{x}{a}\right)}\right) = 6\).

Answer: e^3


Question: Determine if the limit \( \lim_{x\rightarrow\infty}\prod_{p\leqslant x}p^{1/(2x)} \) is finite.

Answer: \sqrt{e}


Question:
Given a positive integer $n$ and non-negative numbers $a_1, a_2, \ldots, a_n$ such that $a_1 + a_2 + \cdots + a_n = n$, find the maximum value of the expression:

$$(n^{n-1}-1)\sqrt[n]{a_{1}a_{2}\cdots a_{n}}+\sqrt[n]{\dfrac{a^n_{1}+a^n_{2}+\cdots+a^n_{n}}{n}}$$

Answer: n^{n-1}


Question: Verify the asymptotic approximation \(\int^{1}_{0}t^{-\frac{1}{2}} \cos(t) e^{-xt^{\frac{1}{2}}} \, dt \sim \frac{2}{x}\) as \(x \rightarrow \infty\) using Watson's Lemma or another suitable method.

Answer: \dfrac{2}{x}


Question:
Given a sequence of real numbers $\{a_n\}$, determine which of the following statements is always true:

(a) If $\sum a_n$ converges, then $\sum a_n^4$ converges.

(b) If $\sum |a_n|$ converges, then $\sum a_n^2$ converges.

(c) If $\sum a_n$ diverges, then $\sum a_n^3$ diverges.

(d) If $\sum |a_n|$ diverges, then $\sum a_n^2$ diverges.

Select the correct statement.

Answer: B


Question: Consider a rank deficient periodic function-valued positive semi-definite Hermitian matrix \( \mathbf{A}(t):\mathbb{R}\rightarrow\mathbb{C}^{n\times n} \) with analytic entries \( a_{ij}(t) \). Define \( \mathbf{B}(t)=\lambda(t)\mathbf{x}(t)\mathbf{x}(t)^H \), where \( \lambda(t) \) is the largest eigenvalue of \( \mathbf{A}(t) \) (simple for all \( t\in\mathbb{R} \)) and \( \mathbf{x}(t) \) is its associated normalized eigenvector. Do the Fourier series of the entries \( b_{ij}(t) \) of \( \mathbf{B}(t) \) converge pointwise to \( b_{ij}(t) \)?

Answer: Yes


Question: Find the number of integer solutions \((x, y)\) to the equation \(23107\ | \ 5x^2 + 35y^2 + 2023\) such that \(0 \leq x, y \leq 23106.\)

Answer: 23100


Question: Find the multiplicative inverse of \(n+1\) modulo \(n^2\) for any integer \(n > 1\).

Answer: n^2 - n + 1


Question: Find the remainder when \((177 + 10^{15})^{166}\) is divided by 1003.

Answer: 234


Question: Let $S_n$ be the symmetric group on $n$ letters. Suppose $G$ and $G'$ are transitive subgroups of $S_n$ with order $n$ such that $G \cong G'$. Are $G$ and $G'$ necessarily conjugate in $S_n$?

Answer: Yes


Question: Using Clarke's Theorem, determine the number of labeled trees with 12 vertices where one specified vertex has a degree of 5.

Answer: 372027810


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

Answer: 1


Question: Find the coefficient of \(x^{26}\) in the expansion of \((x^2+x^3+x^4+x^5+x^6)^8\).

Answer: 13140


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{ e^{\frac{1}{x^2}} -1}{\frac{1}{x^2}} \frac{1}{(2\tan^{-1}(x^2)-\pi)x^2}$$

Answer: -\infty


Question: What is the maximum degree of a closed point on a curve of genus $g > 1$ over a non-algebraically closed field, given that the dualizing sheaf has degree $2g-2$ and induces a morphism of degree $2g-2$?

Answer: 2g - 2


Question: Evaluate the improper integral \( \int_{-\infty}^{\infty} \cos(x) \, dx \) using the concept of Schwartz distributions.

Answer: 0


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

Answer: 1


Question: Find the number of 3-Sylow subgroups in the group $SL_2(\mathbb{F}_3)$, using its relationship with the alternating group $A_4$. Provide your answer as a single integer.

Answer: 4


Question: Determine whether any function $f: I \to \mathbb{R}$, where $I$ is an interval of real numbers, can be expressed as $f = f_1 + f_2$, with $f_1, f_2: I \to \mathbb{R}$ both having the intermediate value property.

Answer: Yes


Question:
Evaluate \(x + 2y + 3z\) given that the following limit is a finite non-zero number:

\[\lim_{h \to 0}\frac { \int _{ 2 }^{ h+2 }{ g(t)dt-h(xg(2)+yg(2+h/3)+zg(2+h)) } }{ { h }^{ 4 } }\]

Answer: \dfrac{9}{4}


Question: Find the values of $a$, $b$, and $c$ in the function $f(x)=a+b\sin(x)+c\cos(x)$ that minimize the integral $\int_{-\pi}^{\pi}|f(x)-\cos(2x)|^2dx$. Provide your answer as a set of values $(a, b, c)$. \( \boxed{(a, b, c)} \)

Answer: (0, 0, 0)


Question: Find the set of all complex numbers \( z \) for which the function \( f(z) = |z| \) is differentiable.

Answer: \emptyset


Question: Given a cubic polynomial $f(x) = ax^3 + bx^2 + cx + d$, where $f(3) = 2$, $f(4) = 4$, $f(5) = -3$, and $f(6) = 8$, determine the value of $f(0)$. Express your answer as a single number.

Answer: -328


Question: Suppose $(y_n)_n$ is a sequence in $\mathbb{C}$ such that for every sequence $(x_n)_n$ in $\mathbb{C}$ where the series $\sum_n x_n$ converges absolutely, the series $\sum_n (x_n y_n)$ also converges absolutely. Is the sequence $(y_n)_n$ necessarily bounded?

Answer: Yes


Question: Is it possible to find a non-compact operator $T : l^p \rightarrow l^p$ such that for every complemented subspace $E$ of $l^p$, the restriction $T|_E: E \rightarrow T(E)$ is an isomorphism?

Answer: Yes


Question: Construct a ring homomorphism \( \psi: \mathbb{Z}[X] \rightarrow S_k \) for \( S_k= \left\{\left( {\begin{array}{cc} c & d \\ -kd & c \\ \end{array} } \right)\ \middle| \ c,d\in \mathbb{Z} \right\} , k\in\mathbb{Z} \) such that \( \text{ker}(\psi) = (X^2+k) \).

Answer: \psi(f(X)) = f\left( \begin{pmatrix} 0 & 1 \\ -k & 0 \end{pmatrix} \right)


Question: Given that $a + b + c = 1$ with $a, b, c \geq 0$, determine the maximum value of the expression \[ \frac{1}{a^2 - 4a + 9} + \frac{1}{b^2 - 4b + 9} + \frac{1}{c^2 - 4c + 9}. \]

Answer: \dfrac{7}{18}


Question: Evaluate the double integral \( \int_{0}^{1}\int_{0}^{x}\frac{(x+y)e^{(x+y)}}{{x^{2}}}dydx \) using the change of variables \( u=\frac{y}{x}, \quad v=x+y \). Find the new integral in terms of \( u \) and \( v \) and compute its value.

Answer: e^2 - e - 1


Question: Evaluate the integral \( \int_0^{\pi}\frac{1}{5+3\cos{x}}dx \) using the substitution \( t=\tan{\frac{x}{2}} \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit \( l = \lim_{n \to \infty} \int_{0}^{+\infty} \exp((\cos^n x) -x) \, dx \) using the dominated convergence theorem.

Answer: 1


Question: Evaluate the integral \( \displaystyle\iint_{B} x \,dx\,dy \) where \( B \) is the region defined by the circle \( x^2+y^2 -x \leq 0 \).

Answer: \dfrac{\pi}{8}


Question: Find the sum of the thirteenth powers of the roots of the polynomial equation \(x^{13} + x - 2 = 0\).

Answer: 26


Question:
Given a differentiable function $f$ at the point $a$, with $f'(a) = 2$ and $f(a) = 5$, evaluate the limit:

$$\lim_{n\to\infty}\left(\frac{f(a+\frac{1}{n})}{f(a)}\right)^n$$

Answer: e^{\frac{2}{5}}


Question: Calculate the expected value of the absolute difference between 0.5 and a random variable X, denoted as \(|0.5 - X|\), where X follows a Beta distribution with parameters \(\alpha = 2\) and \(\beta = 2\).

Answer: \dfrac{3}{16}


Question: Suppose $X$ and $Y$ are Banach spaces and $T$ is a continuous operator between them. If for any sequence $(x_n)_{n=1}^{\infty}$ with $\|x_n\|=1$, $\|Tx_n\| \nrightarrow 0$, does it follow that $T$ is bounded below? (Bounded below means there exists a positive constant $C$ such that $\|Tx\| \geq C \|x\|$ for all $x$ in $X$.)

Answer: Yes


Question: Determine whether the following statement is true or false based on the Fundamental Theorem of Calculus: \[ \dfrac{\partial}{\partial y}\int_{a}^{b}dx\ f(x,y)=f(b,y)-f(a,y). \]

Answer: B


Question: Find the minimum possible value of the sum \(\sum_{i=1}^{11}x_i\), where \(x_i\) are positive integers for \(i=1,2,...,11\). The conditions \(x_i+x_{i+1}\geq 100\) and \(|x_i-x_{i+1}|\geq 20\) hold for \(i=1,2,...,10\), and additionally, \(x_{11}+x_{1}\geq 100\) and \(|x_{11}-x_1|\geq 20\).

Answer: 580


Question: Calculate the limit: $$\lim_{n \to \infty} \left[\prod_{k=1}^{n}\left(1+\frac{(k+1)^{n}}{n^{n+1}}\right)\right]$$

Answer: 1


Question: Evaluate the integral \( \iint_{R} xy \, dA \), where \( R \) is the region bounded by the curves \( y = x^2 \), \( y = 2 - x \), \( x = 0 \), and \( x = 1 \).

Answer: \dfrac{3}{8}


Question: Determine the convergence of the Taylor series expansion of \(\frac{1}{(x^2 -1)^2}\) around \(x = -1\), which is given by \(\frac{1}{(x^2 -1)^2} = \frac{1}{4(x+1)^{2}} + \frac{1}{4(x+1)} + \frac{3}{16} + \frac{(x+1)}{8} + \ldots\).

Answer: (-3, 1)


Question: Given that $m$ divides $n$ ($m \mid n$), determine if there exists a subgroup of the dihedral group $D_n$ that is isomorphic to the dihedral group $D_m$.

Answer: Yes


Question: Given a graph with 10 vertices and 5 distinct eigenvalues, determine if all the elements of the matrix \(A + A^2 + A^3 + A^4 + A^5\) are non-zero, where \(A\) is the adjacency matrix of the graph.

Answer: No


Question: Compute the work done by the radial vector field \( F(x,y,z) = xi + yj + zk \) on a particle moving along the curve \( C \) given by \( r(t) = (1+2\sin(t))i + (1+5\sin^2(t))j + (1+4\sin^3(t))k \) for \( 0 \leq t \leq \pi/2 \). The work done is given by the line integral \( \int_C {\bf{F}} \cdot d{\bf{r}} \).

Answer: \dfrac{67}{2}


Question: Determine whether there exist non-zero values of $y$ and a value of $a$ such that the equation \[ \sum_{n=1}^{\infty}\frac{(-1)^n\sin(y\ln n)}{n^{x}}=-\frac{y}{a} \] holds for some $x > 0$. Provide a justification for your answer.

Answer: Yes


Question: Determine the set of all limit points of the set \( C = \left\{ \frac{n}{m} + \frac{m}{n} : n, m \in \mathbb{N} \right\} \).

Answer: [2, \infty)


Question: Let $g$ be a bounded function on $[a,b]$. Determine $\lim_{m\to\infty} J_m$ where $J_m=\int_{a}^{b} \frac{g(t)}{m+e^t}dt$. Provide your answer as a limit expression.

Answer: 0


Question: Let $N$ be a natural number greater than 1 and less than 100. The factors of $N$, arranged in ascending order, are denoted as $F(1), F(2), \dots, F(n)$, with $1 = F(1) < F(2) < F(3) < \dots < F(n) = N$. Define $D$ as the sum of the products of consecutive factors: $D = F(1)F(2) + F(2)F(3) + \dots + F(n-1)F(n)$. If $D$ divides $N^2$, how many possible values of $N$ exist?

Answer: 25


Question: Evaluate the integral \( \int_{-\infty}^\infty x^2 \cdot \frac{1}{2} C e^{-C|x|} \, dx \).

Answer: \dfrac{2}{C^2}


Question: Evaluate the limit: $$\lim _{x \to 0} \left[\frac{3}{x^4}(\cos^{-1}x - \sin^{-1}x )\right]^{\frac{3}{x^3}}$$

Answer: \infty


Question: Evaluate the limit: $$\lim\limits_{h\to0} \frac{\tan(a+2h)-2\tan(a+h)+\tan a}{h}.$$

Answer: 0


Question: Given \( z = \frac{\sqrt{3} - i}{2} \), find the smallest positive integer \( n \) such that \((z^{95} + i^{67})^{94} = z^n\), where \( i = \sqrt{-1} \).

Answer: 10


Question:
Evaluate the following sum and show that it converges to 1:

$$0.5\cdot\sum\limits_{k=0}^\infty \frac{1}{k+1}\binom{2k}{k}\cdot(0.25)^{k}$$

Answer: 1


Question: Determine the optimal value of \(\alpha\) that minimizes the expected value \(E(x - \mu^*)^2\) for the hard thresholding estimator of the mean of a normal distribution, where the true mean \(\mu\) is not necessarily zero.

Answer: 0


Question: In square $ABCD$, points $E$ and $F$ are located on sides $BC$ and $CD$, respectively, such that $\angle EAF = 45^\circ$ and $\angle EAB = 15^\circ$. Diagonal $\overline{BD}$ intersects $\overline{AE}$ at point $P$. If $\overline{PC}$ is drawn, determine the measure of $\angle PFC$. \(\text{Express your answer in degrees.}\)

Answer: 75


Question: Calculate the volume of the solid bounded by the surfaces $z=\sqrt[3]{x^2+y^2}$, $z=2$, and $z=\sqrt{3}$ using a triple integral in cylindrical coordinates.

Answer: \dfrac{7\pi}{4}


Question:
Determine the radius of convergence for the series:
\[ \sum_{n=1}^{\infty} n! x^{n!} \]

Answer: 1


Question: In the symmetric group $S_n$, generated by simple reflections $s_i = (i,i+1)$ for $i=1,2,\ldots,n-1$, a reduced word $w$ is a sequence formed using these simple reflections. Determine if the number of inversions in a reduced word $w$ is equal to its length.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\pi/4}\frac{6 - 6\sin^{2}(x)}{2\cos^2(x)} \mathrm{d}x \).

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


Question: Determine the values of \(\alpha \in \mathbb{R}\) for which the integral \(\int_{\mathbb{R}^2} \frac{\sin(x^2+y^2)}{(x^2+y^2+1)^\alpha} \, dx\,dy\) converges.

Answer: \alpha > 0


Question: Evaluate the integral: $$ \int_0^\infty \frac{x^2 \ln(x)}{(1+x^2)^3} {\rm d}x $$

Answer: 0


Question: Determine whether it is possible for a set in \(\mathbb{R}^n\) to have a finite Lebesgue outer measure, even if the sum of the volumes of any countable covering is greater than some fixed positive number.

Answer: Yes


Question: Is every Borel set in \(\mathbb{R}\) expressible as a countable union of intervals? Answer with "True" or "False."

Answer: B


Question: Let $P$ be a probability distribution on a measurable space $(X, \Sigma)$. Define $Y:=\{x\in X: P(x) > 0\}$. Assume that every singleton set in $X$ is measurable. Is $Y$ necessarily a measurable subset of $X$?

Answer: Yes


Question: Calculate the integral \( \int_{F(S^1)} \omega \) where \( F: S^1 \rightarrow S^1 \times S^1 \times S^1 \) is given by \( F(p) = (p,p,p) \) and \( \omega = dt_1 + dt_2 + dt_3 \) is a 1-form over \( S^1 \). It is given that \( F^{*}\omega = 3\,dt \).

Answer: 6\pi


Question: Determine whether there exist two distinct prime numbers $p$ and $q$ such that the equation $2^a \cdot p^b = q^c - 1$ has no solution for any natural numbers $a$, $b$, and $c$. Provide a justification for your answer.

Answer: Yes


Question: Determine the largest possible area of an ellipse that can be enclosed by the hyperbolas $xy=1$ and $xy=-1$.

Answer: 2\pi


Question: Can a differential 1-form \( \omega \) on \( \mathbb{R}^n \) define a connection on \( M = \mathbb{R}^n \) such that its torsion is given by \( T(X,Y) = \omega(X)Y - \omega(Y)X \)?

Answer: Yes


Question: Given points $P = (1: -2: 3)$, $Q = (2: 2: -1)$, and $R = (3: 0: 2)$ on a line $g$ in $\\mathbb{P}^{2}\mathbb{R}$, find a coordinate for $g$ such that $(P, Q, R)$ has coordinates $(0, 1, ∞)$. Determine the number of possible coordinates for $g$.

Answer: 1


Question: Evaluate the limit of the function \( g(x) = \frac{1-\cos x}{x^2} + \cos x\frac{1-\sqrt{\cos 2x}}{x^2} \) as \( x \) approaches zero.

Answer: \dfrac{3}{2}


Question:
Given non-negative numbers $a, b, c$ such that $a+b+c=3$, determine the maximum value of the expression:
\[ P = a^3b^2 + b^3c^2 + c^3a^2 + 7(ab + bc + ca). \]

Answer: 24


Question: In an abelian group $G$, does the set of elements of exponent $r$ (where $r$ is a positive integer) form a subgroup of $G$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit of the function \( f(x) = \frac{e^{e^x - 1} - \frac{1}{1 - x}}{\ln\left(\frac{1 + x}{1-x}\right) - 2\sin x} \) as \( x \to 0 \).

Answer: -\dfrac{1}{6}


Question: Evaluate the limit: $$\lim_{x\rightarrow 0} \left( \frac{x+1-\sqrt[2022]{2022x+1} }{x^2}\right).$$

Answer: \dfrac{2021}{2}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left[1- p(n) \right]^{n-1} \] where \( p(n) = o \left( \frac{\log n}{n} \right) \).

Answer: 1


Question: Evaluate the limit \(\lim_{x \to 0} \frac{1}{x^2} \int_0^x \frac{ \sqrt{x^2-t^2} }{2+t^{-t}} dt\) using a substitution method.

Answer: \dfrac{\pi}{12}


Question: Evaluate the determinant: $$\Delta = \begin{vmatrix}\sin^2A & \cot A & 1 \\ \sin^2B & \cot B & 1 \\ \sin^2C & \cot C & 1\end{vmatrix}$$ given that $A$, $B$, and $C$ are the angles of a triangle and $B-A=C-A$. Provide your answer as a single numerical value.

Answer: 0


Question: Evaluate the limit: \( \lim_{{x \to 0}}\frac{\int_{0}^{x^2}\sin\sqrt{t}\, dt}{x^3} \). Use L'Hôpital's Rule and the Fundamental Theorem of Calculus to find the value of the limit.

Answer: \dfrac{2}{3}


Question: Let $X$ be a Banach space, and let $T_n, T \in \mathscr{L}(X)$ be bounded linear operators on $X$. Suppose that for any convergent sequence $x_n \to x$ in $X$, we have $T_n(x_n) \to T(x)$. Does this property imply that $T_n \to T$ in norm?

Answer: No


Question: Evaluate the integral \(\int \frac{x + 9}{x^3 + 9x} \, dx\) and express it in the form \(k\arctan(mx) + n\ln (x) + p \ln (x^2 + 9) + c\), where \(m\), \(n\), \(p\), and \(k\) are constants. Determine the value of \(\frac{m+n}{k+p}\).

Answer: -8


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

Answer: \alpha > L


Question: In a gambling game, a player starts with $1 and wagers $1 in each round. The player has a probability $p$ of winning $2 and a probability $1-p$ of winning nothing in each turn. If $p < \frac{1}{2}$, what is the probability that the player will eventually lose all their money?

Answer: 1


Question:
Determine the behavior of the series \( \sum_{n=1}^{+\infty}\dfrac{3\ln n + n^2}{2e^{-n}+n^3} \):

(a) converges to 0
(b) converges to \( S > 0 \)
(c) diverges to \( +\infty \)
(d) none of the others

Answer: C


Question: Given the polynomial equation $x^4 - 2x^3 + 3x^2 - 4x + 2 = 0$ with distinct roots $a$, $b$, $c$, and $d$, find the value of $a^4 + b^4 + c^4 + d^4$. Express your answer as a single number.

Answer: 10


Question: Determine whether the function $X_t = W_t\sin(t)$ is a solution to the stochastic differential equation $$dX_t = \sin(t)dW_t.$$

Answer: No


Question: Evaluate the integral \( \iint_{[-1,1]^2} \sqrt{1-x^2-y^2} \, dx \, dy \) by using an appropriate coordinate transformation.

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


Question: Compute the minimum value of the integral \(\displaystyle\int_{0}^{\infty} \left | x^3-a-bx-cx^2 \right |^2e^{-x}\, dx\) subject to the constraints \(a,b,c \in \mathbb{R}\).

Answer: 36


Question: Determine whether the following statement is true or false: If $P$ is a Sylow $p$-subgroup of a finite group $G$, then for any subgroup $H$ of $G$, $H \cap P$ is a Sylow $p$-subgroup of $H$. Provide a justification for your answer.

Answer: False


Question: Evaluate the limit \( \displaystyle\lim_{n \to \infty} \int_{0}^{\pi} \frac{\sin x}{1+\cos ^2(nx)} \, dx \).

Answer: \sqrt{2}


Question: Determine the values of 's' for which the matrix A is positive semi-definite.

Answer: [-1, 1]


Question: Compute the discriminant of the number field $K=\mathbb{Q}(\sqrt[3]{2})$. \( \boxed{\text{Discriminant value}} \)

Answer: -108


Question: Evaluate the integral \( \int_0^1 \frac{\ln(x)}{\sqrt{1-x}} \, dx \) using a method that does not involve rearranging terms of a series.

Answer: 4 (\ln 2 - 1)


Question: In a game, there are three piles of balls, each initially containing an equal number of balls. On each turn, you select one pile and transfer a ball to another pile. The score for this move is calculated as the number of balls in the source pile minus the number of balls in the target pile (excluding the ball being transferred). If the result is negative, you lose points. What is the highest possible score you can achieve after making moves for 1000 years?

Answer: \infty


Question: Evaluate the limit: $$\lim_{a \to +\infty} \int_a^{a+1} \frac{x}{x+\ln x} \, \text{d}x$$

Answer: 1


Question: Find a formula for the sum of the first n terms of the series given by \( f(n) = 1\binom{n}{1} + 2\binom{n}{2} + 3\binom{n}{3} + \ldots + n\binom{n}{n} \).

Answer: n \cdot 2^{n-1}


Question: Evaluate the limit: $$\lim_{t\rightarrow -\infty} \beta t^\alpha  E_{\alpha ,\alpha  + 1} ( - at^\alpha  ).$$

Answer: \dfrac{\beta}{a}


Question: Let $A$ and $B$ be two $2 \times 2$ matrices with $\det(A) = 1$ and $\det(B) = 2$. Calculate the value of \( \det(A + \alpha B) - \det(\alpha A + B) \) for a real number \( \alpha \).

Answer: \alpha^2 - 1


Question: Evaluate the integral \( A = \int_{0}^{2\pi} |\sin(\theta) + \cos(\theta)| \, d\theta \).

Answer: 4\sqrt{2}


Question: Find the Lebesgue measure of the set \( \{(x_1,\ldots,x_n)\in \mathbb{R}^n: x_1+x_2+\cdots+x_n\leq1, x_i\geq0 \text{ for all } i\} \).

Answer: \dfrac{1}{n!}


Question: Determine the intervals where the function \( f(x) = \left(x+\frac{1}{x}\right)^{x} \) is increasing or decreasing, given that its derivative is \( f'(x) = \left(x+\frac{1}{x}\right)^{x} \left(\ln\left(x+\frac{1}{x}\right) + \frac{x-1}{x+1}\right) \).

Answer: (0, \infty)


Question: Evaluate whether the infinite product \( \lim_{n \to \infty} \frac{\prod_{k=2}^{n}{p_{k}^2}}{\prod_{k=2}^{n}((p_{k}-2)p_{k+1})} \) converges to 0, where \( p_{k} \) is the \( k^{th} \) prime number.

Answer: 0


Question: Determine whether the statement is true: For all finite monoids $M$, $H_i(M^{\rm op};\mathbb Z^l) \cong H_i(M;\mathbb Z^l)$.

Answer: True


Question: Let \( f:[0,+\infty)\rightarrow \mathbb{R} \) be a function such that for every sequence \( \{x_n\} \), where \( x_{n+1}=x_n+1 \), it holds that \( f(x_n)\rightarrow 0 \). Does this imply that \( \lim_{x\rightarrow +\infty}f(x)=0 \)?

Answer: No


Question: Is it possible to construct a non-surjective homomorphism from a group of order 6 to a group of order 3?

Answer: Yes


Question: Let $T: \mathbb{R}^{10} \to \mathbb{R}^{10}$ be a linear transformation. Given that the dimension of $\text{Ker}(T^2) = 5$, $\text{Ker}(T^4) = 9$, and $\text{Ker}(T^5) = 10$, determine the dimension of $\text{Ker}(T^3)$. \( \boxed{?} \)

Answer: 7


Question: Are all compact subgroups of $GL_n(\mathbb{R})$ subconjugates of the orthogonal group $O(n)$? Answer yes or no.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to 0} \frac {e^{-1/x^2}}{x} $$

Answer: 0


Question: Let $T:L^p[0,1]\to L^p[0,1]$ be a linear operator defined by $T(f(x))=x^kf(x)$. Determine the operator norm of $T$. \(1 \leq p < \infty\).

Answer: 1


Question: In triangle ABC, points B' and C' are located on sides AB and AC such that BB' = CC'. Let O and O' be the circumcenters of triangles ABC and AB'C', respectively. The line OO' intersects lines AB' and AC' at points B'' and C'', respectively. Given that AB = \( \frac{1}{2} \) AC, determine the relationship between the lengths of segments AB'' and AC''.

Answer: AB'' = AC''


Question: Evaluate the limit: $$\lim_{n\to\infty} \frac{\left(\sum_{r=1}^n\sqrt{r}\right)\left(\sum_{r=1}^n\frac1{\sqrt{r}}\right)}{\sum_{r=1}^n r}$$

Answer: \dfrac{8}{3}


Question: Let $G$ be a finite group. Determine whether every irreducible representation of $G$ over $\mathbb{C}$ is isomorphic over $\mathbb{C}$ to a representation that has entries in some number field.

Answer: Yes


Question: Find the limit of the expression \( \lim_{n\to\infty} \frac{x_1 y_n + x_2 y_{n-1} + \cdots + x_n y_1}{n} \) where \(x_1, x_2, \ldots, x_n\) and \(y_1, y_2, \ldots, y_n\) are sequences of real numbers such that \(\lim\limits_{n\to\infty} x_n = a\) and \(\lim\limits_{n\to\infty} y_n = b\).

Answer: ab


Question: Calculate the residues of the function \( \frac{1}{1+e^z} \) at its poles.

Answer: -1


Question: Given the joint density function \( f_{X,Y}(x,y) = 6e^{-x}e^{-2y} \) for \( 0 < x < y < \infty \), calculate the expected time of failure for a machine where component \( Y \) becomes operational only when component \( X \) fails, and the machine fails when both components \( X \) and \( Y \) have failed.

Answer: \dfrac{5}{6}


Question: Evaluate the limit: $$\lim_{n\to\infty}{\frac{1^6+2^6+3^6+\ldots+n^6}{(1^2+2^2+3^2+\ldots+n^2)(1^3+2^3+3^3+\ldots+n^3)}}$$

Answer: \dfrac{12}{7}


Question: Evaluate the contour integral \( I = \oint_{\gamma} \frac{dz}{(z^2+4)^2} \), where \( \gamma \) is the circle defined by \( |z-i|=2 \).

Answer: \dfrac{\pi}{16}


Question:
Let $g:\mathbb{R} \to \mathbb{R}$ be a continuous function. Determine which of the following statements implies that $g(0) = 0$:

(A) $\lim_{n \to \infty}\int_{0}^{1}g(nx)^n \, dx=0.$

(B) $\lim_{n \to \infty}\int_{0}^{1}g(x)^n \, dx=0.$

(C) $\lim_{n \to \infty}\int_{0}^{1}g(x/n) \, dx=0.$

(D) None of the above.

Answer: C


Question:
Consider a random walk on the set of natural numbers \(\mathbb{N}\), starting at 0. The transition probabilities are given by:

\[ p(0,1)=0.75, \quad p(n,n-1)=0.5, \quad p(n,n+1)=0.5 \quad \text{for } n>0. \]

Calculate the expected time \(\mathbb{E}[T_{50}]\) for the random walk to reach the value 50.

Answer: \dfrac{7550}{3}


Question: Calculate the volume enclosed by the planes $2x + y = 3$, $z = y$, $y = x$, and $z = 0$. Verify if the volume can be expressed by the integral: $$ \int_{0}^{1}\int_{y}^{\frac{3-y}{2}}(y-0)\,dx\,dy $$

Answer: \dfrac{1}{4}


Question: Find the limit \( \lim _{n \to \infty}(n!)^{\frac{1}{n^2}} \) using mathematical methods other than Stirling's approximation or L'Hôpital's Rule.

Answer: 1


Question: In a street with 65 houses numbered from 0 to 64, both the first and last houses are already occupied. For the remaining 63 unoccupied houses, if a person occupies a house with number \(a\) and then another with number \(b\) consecutively, they must next occupy the house at the midpoint, calculated as \(\left\lfloor\frac{a+b}{2}\right\rfloor\). Determine the total number of distinct ways all houses can be occupied.

Answer: 2^{63}


Question: Show that the composition of the exponential function and the logarithm function, defined by \( E \circ L = \sum_{n=0}^{\infty} \frac{(\int_{1}^{x}\frac{1}{t}dt)^n}{n!} \), is equal to \( x \) for \( x > 0 \).

Answer: x


Question:
Given the system of equations:
\[a - \frac{1}{a} = b\]
\[b - \frac{1}{b} = c\]
\[c - \frac{1}{c} = a\]
Determine the value of:
\[\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ac}\]
Options:
\[a) -3\]
\[b) -1\]
\[c) -6\]
\[d) -9\]

Answer: A


Question: Determine the number of points on the unit circle in the complex plane such that the imaginary parts of $z^{720}$ and $z^{120}$ are equal.

Answer: 1440


Question: Let $A$ and $B$ be two unital $C^*$-algebras, and let $x \in A \otimes B$ be a non-zero element of the algebraic tensor product $*$-algebra. Determine whether there exist states $\omega_x \in A^*_+$ and $\varphi_x \in B^*_+$ such that the product state $\omega_x \times \varphi_x$ satisfies $(\omega_x \times \varphi_x)(x^*x) > 0$. In other words, do the product states separate the points of the tensor product $A \otimes B$?

Answer: Yes


Question: Let $f$ be a continuous function defined on the closed interval $[0,1]$. Determine \( \lim_{n \to \infty} S(n) \), where \( S(n) = n\int_0^1 f(x)x^n\,dx \).

Answer: f(1)


Question: Determine whether the sequence \((x_n)_{n=1}^\infty\), where \(x_n\) is the fractional part of the harmonic series \(1 + \frac{1}{2} + \dots + \frac{1}{n}\), is dense in the interval \((0,1)\). The fractional part of a number \(y\) is denoted by \(y - \lfloor y \rfloor\).

Answer: Yes


Question: Evaluate the improper integral \( \int_0^1 \int_{x_1}^1 x_2^2 \cdot \sin\left(\frac{2\pi x_1}{x_2}\right)\, dx_2\, dx_1 \) without using Fubini's theorem.

Answer: 0


Question:
Given a real-valued harmonic function $u$ on the complex plane $\mathbb{C}$, consider the function $g: \mathbb{R}^2 \to \mathbb{R}$ defined by \[ g(x,y) = \int_0^{2\pi} u(e^{i\theta}(x+iy)) \sin \theta \,d\theta. \] Determine which of the following statements is true about $g$:

(A) $g$ is a harmonic polynomial
(B) $g$ is a polynomial but not harmonic
(C) $g$ is harmonic but not a polynomial
(D) $g$ is neither harmonic nor a polynomial

Select the correct option.

Answer: A


Question: Given functions $F, \varphi : [a, b] \rightarrow \mathbb{R}$, where $F$ is continuous and $\varphi$ is strictly increasing, if $(F + \varphi)(a) > (F + \varphi)(b)$, is $F + \varphi$ necessarily decreasing on some interval within $(a, b)$, even if $\varphi$ has infinitely many points of discontinuity? Answer yes or no.

Answer: Yes


Question: Evaluate the limit \(\lim_{x \to 0} \frac{e^{\tan x} - e^x + \ln(\sec x + \tan x) - x}{\tan x - x}\) using L'Hopital's rule or alternative methods.

Answer: \dfrac{3}{2}


Question: Evaluate the limit \(\lim_{x \to 0}\frac{(1+x)^{1/m}-(1-x)^{1/m}}{(1+x)^{1/n}-(1-x)^{1/n}}\) where \(m\) and \(n\) are positive integers.

Answer: \dfrac{n}{m}


Question: Let $a, b, c \ge 0$ such that $ab + bc + ca = abc + 2$. Find the minimum value of \( P = (a^3 + 1)(b^3 + 1)(c^3 + 1) \).

Answer: 8


Question: Find the limit: \[ \lim_{x\rightarrow\infty}\dfrac{\ln(1+x+x^\frac{1}{3})}{\ln(1+x^\frac{1}{4}+x^\frac{1}{3})} \]

Answer: 3


Question: Determine the value of the limit \( \lim_{{n \to \infty}} \frac{f(n)}{n} \) for a bijective function \( f: \Bbb N^{\star} \to \Bbb N^{\star} \), where \( \Bbb N^{\star} \) denotes the set of positive integers, given that the limit exists.

Answer: 1


Question: Determine the natural density of nonnegative integers whose cubes end with the digits 75.

Answer: \dfrac{1}{20}


Question: Can an analytic function with a constant argument be a constant function? Provide a justification for your answer.

Answer: Yes


Question: Determine whether any fraction \( \frac{a}{b} \), where \( a \) and \( b \) are positive integers, can be expressed as an infinite sum of distinct Egyptian fractions, each having a denominator equal to 1.

Answer: No


Question: Determine the quotient space $V/V$ for a vector space $V$.

Answer: \{0\}


Question:
Determine the interval of convergence for the series:

$$\sum_{n=2}^\infty\frac{(-1)^nx^n}{n(n-1)}.$$

Answer: [-1, 1]


Question: If $A$ and $B$ are rings and $A$ is a field, and $A$ is isomorphic to $B$, does that imply that $B$ is also a field? Answer yes or no and justify your answer.

Answer: Yes


Question: Let $a$ and $b$ be positive real numbers such that \( a^4 + 3ab + b^4 = \frac{1}{ab} \). Evaluate \( {\left(\frac{a}{b}\right)}^{\frac{1}{3}} + {\left(\frac{b}{a}\right)}^{\frac{1}{3}} - {\left(2+\frac{a}{b}\right)}^{\frac{1}{2}} \).

Answer: 2 - \sqrt{3}


Question: Determine whether the following statement is true or false: If the Jacobian of a function \(g:\mathbb{R}^n\to\mathbb{R}^n\) is non-zero at a point \(y\), then \(g\) cannot be constant in a neighborhood of \(y\).

Answer: A


Question: Determine the upper bound for the function \( f(x, y) = \frac{xy^2}{x^2 + y^4} \) for all \((x, y) \in \mathbb{R}^2\) such that \((x, y)^t \neq (0, 0)^t\).

Answer: \dfrac{1}{2}


Question: Let $f_1,..., f_n: \mathbb{R}\rightarrow\mathbb{R}$ be Lebesgue-measurable functions and $F:\mathbb{R}^n\rightarrow\mathbb{R}$ be a continuous function. Is the composition $g(x):=F(f_1(x),f_2(x),...,f_n(x))$ always Lebesgue-measurable?

Answer: Yes


Question:
Find the limit as \( n \to \infty \) of the sequence of integrals:

\[ I_n = \int_{0}^{\frac{\pi}{2}} \left(\tan x\right)^{\frac{1}{n}} \, dx \]

After substituting \( \tan x = z \) and simplifying, the integral becomes:

\[ I_n = \int_{0}^{\infty} \frac{z^{1/n}}{1 + z^2} \, dz \]

Evaluate this integral to find the limit.

Answer: \dfrac{\pi}{2}


Question: In \(\triangle ABC\), \(\angle{ABC} = 90^{\circ}\). Points \(P\) and \(Q\) are on \(BC\) and \(AB\), respectively, such that \(\angle{BAP} = 7^{\circ}\), \(\angle{CAP} = 14^{\circ}\), and \(\angle{BCQ} = 23^{\circ}\). Given that \(AB = AC \cdot \cos (\angle{BAC})\), determine \(\angle{APQ}\).

Answer: 30^\circ


Question: A regular dodecahedron has a volume of 1. Twelve regular dodecahedra are glued to its faces. If the distance from the center of mass of the original dodecahedron to one of its faces is \( \lambda \), what is the volume of the new dodecahedron formed by the outer vertices of the glued dodecahedra?

Answer: 27


Question: In a tetrahedron, let \( \Delta_{1}, \Delta_{2}, \Delta_{3}, \Delta_{4} \) denote the areas of its four triangular faces, and \( h_{1}, h_{2}, h_{3}, h_{4} \) be the corresponding altitudes. Given that the volume of the tetrahedron is 5 cubic units, find the minimum value of the sum \( \sum_{1\leq i<j\leq 4}(\Delta_{i}h_{j}) \).

Answer: 90


Question: Find the unique continuous function \( f(x) \) that satisfies the functional equation \( f(2x) = \frac{f(x) + x}{2} \) for all real numbers \( x \).

Answer: \dfrac{x}{3}


Question: Given a function \( f : \mathbb{R} \setminus \{\frac{1}{2}\} \to \mathbb{R} \) such that \( f\left(\frac{x-2}{2x}\right) = 2x+5 \) for every \( x \in \mathbb{R} \setminus \{0\} \), find \( f^{-1}(3) \).

Answer: \dfrac{3}{2}


Question: For the sum of powers function \(f(n,N) = \frac{\sum^{N}_{r = 1} r^n}{N^n}\), determine the next term in its polynomial approximation for large \(N\).

Answer: \dfrac{n}{12N}


Question: Let $f(x) \in F[x]$ be a polynomial of prime degree that is not irreducible in $F[x]$. Suppose that for any extension $K/F$, if $f(x)$ has a root in $K$, then $f(x)$ splits in $K$. Determine whether $f(x)$ must have a root in $F$. Provide a proof or counterexample.

Answer: Yes


Question: Determine the asymptotic bound for the expression \( \sum_{i=0}^{\log(n)} 2^{i} \sum_{k=0}^{\frac{n}{2^{i}}} (k+2)^{2} + \theta(n) \).

Answer: \Theta(n^3)


Question: Let $\{X(t), t \ge 0\}$ be a process defined by $X(t) = N(t) - \lambda t$, where $N$ is a Poisson process with rate $\lambda > 0$. Compute $\mathbb{E}[X(t_1)X(t_2)]$ for $0 < t_1 \le t_2$. \(\lambda\) is a positive constant.

Answer: \lambda t_1


Question:
Given an $n \times n$ matrix $C$, where $\operatorname{tr}(C^{2m-1}) = 0$ for all $m \in \mathbb{N}$, and all entries in $C$ are natural numbers with their sum being less than $2n$, evaluate the infinite sum:

$$S = \sum_{k=1}^\infty \ln \left| e^{C^k} \right|$$

Simplify the sum using the given trace condition.

Answer: 0


Question: Find the points on the curve defined by \(x=\frac{(y-2)^2}{y}\) where the tangent line is vertical and passes through the origin.

Answer: (0, 2)


Question: Evaluate the integral and express it in exact form: $$\frac{1}{2\pi}\int_{0}^{\infty}\int_{-x}^{\infty}e^{-\frac{(x^2+y^2)}{2}}dydx$$

Answer: \dfrac{3}{8}


Question: Verify Stokes' theorem for the vector field \( \vec{F}(x,y,z)=(y,-x,xz) \) on the surface \( z=9-x^2-y^2 \) with \( z\geq 0 \).

Answer: -18\pi


Question: Let $\overline{X_n}$ denote the mean of a random sample of size $n$ from a Poisson distribution with mean $\lambda = 2$. Use the delta method to find the limiting distribution of $\sqrt{n}\left( (\overline{X_n})^2 -4 \right)$.

Answer: \mathcal{N}(0, 32)


Question: Evaluate the sum \(\sum_{d|n}\tau(d)\mu(d)\).

Answer: (-1)^{\omega(n)}


Question:
Let $f: \Bbb R\rightarrow \Bbb R$ be a differentiable function such that $f '(0)=1$ and $f(x+y)=f(x)f(y)$ for all $x\in\Bbb R$. Determine which of the following statements is true:

(A) Both $f$ and $f '$ are decreasing functions.
(B) Both $f$ and $f '$ are increasing functions.
(C) $f$ is an increasing function but $f '$ is a decreasing function.
(D) $f$ is a decreasing function and $f '$ is a decreasing function.

Answer: B


Question: Let $R=\mathbb{R}[x]$ be the ring of polynomials and $I=(x^2+1)$, $J=(x+1)$, and $K=(x^3+x^2+x+1)$ be ideals of $R$. There is a ring isomorphism between $R/K$ and $R/I \times R/J$. Find the element in $R/K$ whose image in $R/I \times R/J$ is $(2x+1+I,3+J)$.

Answer: 2x^2 + 2x + 3


Question: Determine the radius of convergence of the Taylor series expansion of the function \( h(x) = g\left(\frac{3-x}{5}\right) \) about \( x = 3 \).

Answer: 5


Question: Evaluate the integral \( \int^{\pi/2}_0 \frac{x+\sin x}{1+\cos x}\,\mathrm dx \).

Answer: \dfrac{\pi}{2}


Question: Let $Q(x,y)$ be a bilinear symmetric form on a finite-dimensional vector space $V$, where $Q: V \times V \to \mathbb{R}$. Determine whether there exists a linear self-adjoint map $K$ such that $$(K(x),y) = Q(x,y)$$ for all $x, y \in V$. Provide a justification for your answer.

Answer: Yes


Question: Let $A \in \mathbb{R}^{m \times k}$ be a real matrix, and let $\sigma_{max}$ and $\sigma_{min}$ be its largest and smallest singular values, respectively. If $A$ satisfies the inequality \((1-\delta) \|x\|_2^2 \leq \|Ax\|_2^2 \leq (1+\delta)\|x\|_2^2\) for all $x \in \mathbb{R}^{k}$ and some $\delta \in (0,1)$, determine the rank of $A$. \(\text{Provide your answer as a single number.}\)

Answer: k


Question: Determine whether there exists an uncountable set \( S \) of subsets of the natural numbers such that for any two distinct subsets \( A, B \in S \), either \( A \subset B \) or \( B \subset A \).

Answer: Yes


Question: Determine the values of $q \in \mathbb{N}$ for which the polynomial $20q^4 - 40q^3 + 30q^2 - 10q$ can be expressed as $10t^2$ for some $t \in \mathbb{N}$. Provide all such values of $q$.

Answer: 1


Question: Find the inverse Laplace transform of the function \( F(s) = \frac{e^{-1/s}}{s} \) using the integral property and the unit step function.

Answer: J_0(2\sqrt{t})


Question: Calculate the limit \( \lim_{n\to\infty} \frac{1+xn}{(1+x)^n} \) for \(x\in (0,1]\), using the Lebesgue Dominated Convergence Theorem.

Answer: 0


Question: Let \(f(x)\) and \(g(x)\) be two polynomials such that \(f(0) = -24\), \(g(0) = 30\), and for all \(x \in \mathbb{R}\), \(f(g(x)) = g(f(x))\). Find the ordered pair \((f(3), g(6))\).

Answer: (-21, 36)


Question: Determine the rank of a $9 \times 9$ matrix $A$ over the field $\mathbb{Z}/5$, given that the system $Ax=0$ has between 40 and 150 linearly independent solutions.

Answer: 6


Question: Consider the complex number $\alpha = \cos \frac{\pi }{7} + i\sin \frac{\pi }{7}$. If $T = \sum\limits_{r = 0}^{3n - 1} {{e^{\frac{{i\pi }}{7} \times {2^r}}}}$, find the value of $n$ such that $|T|^2=32$. What is the value of $n$?

Answer: 4


Question: Determine whether a strictly increasing sequence that is not Cauchy must be unbounded. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the sequence of functions \(f_n(x) = \cos\left(\frac{x}{n}\right)\) converges uniformly to 1 on the interval \([0, 2\pi]\].

Answer: Yes


Question: Compute the contour integral \(\int_{|z|=2}\frac{dz}{z^2+1}\) around a positively oriented circle with radius 2 using the Cauchy Integral Formula.

Answer: 0


Question: Is every countable ordinal, when given the order topology, homeomorphic to a subspace of \( \mathbb{R} \)?

Answer: Yes


Question: Evaluate the limit in terms of $a$ and $b$: $$ \lim_{s\rightarrow 0^{+}} \frac{\Gamma(b-as)}{\Gamma(s)\Gamma(1-s)} $$

Answer: 0


Question: Find the limit of the sequence \( x_n \) defined by the recurrence relation \( x_{n+1} = x_n - \frac{4x_n^3}{n+1} \) with the initial condition \( 0 \le x_0 \le \frac{1}{2} \) as \( n \to \infty \).

Answer: 0


Question: In square $ABCD$, points $E$ and $F$ are located on sides $BC$ and $CD$, respectively, such that $\angle EAF = 45^{\circ}$ and $\angle EAB = 15^{\circ}$. Diagonal $BD$ intersects side $AE$ at point $P$. Connect point $P$ to point $C$. Find the measure of $\angle PFC$ in degrees.

Answer: 75


Question:
Let \( x \) and \( y \) be complex numbers such that \( |x| = |y| = 1 \). Find the maximum value of the expression:
\[
|1+x| + |1+xy| + |1+xy^2| + \cdots + |1+xy^{2013}| - 1007|1+y|
\]

Answer: 2014\sqrt{2}


Question: Solve the equation \(x^4 + 6x^3 + 11x^2 + 7x + 1 = (1-x)^{0.5}\) for \(x\) in terms of fractions.

Answer: 0


Question: For an $n \times n$ magic square with $n > 3$, must the sum of the entries be divisible by $n$? Provide a justification for your answer.

Answer: Yes


Question: Let $\triangle ABC$ be a triangle with $BC < AC < AB$ and corresponding angles $\alpha < \beta < \gamma$. Prove that for any point $P$ inside the triangle, the sum of the distances from $P$ to the vertices, $PA + PB + PC$, is less than $\beta + \gamma$.

Answer: PA + PB + PC < \beta + \gamma


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{5^{(1+\tan^2x)} -5}{1-\cos^2x}$$

Answer: 5 \ln 5


Question: Determine the radius of convergence for the power series \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1} (z-1)^{2n+1}}{(2n+1)!} \).

Answer: \infty


Question: Determine the number of ordered triples \((m, n, p)\) of natural numbers such that \(\frac{1}{n} + \frac{1}{m} + \frac{1}{p} = 1\).

Answer: 10


Question: Find the expected value of the integral \( \int_0^t B(s)^2 \, ds \), where \( B(s) \) is a standard Brownian motion.

Answer: \dfrac{t^2}{2}


Question: Find the smallest positive integer $n$ such that there exists a sequence of all rational numbers $a_1, a_2, \dots$ where no subsequence of length $n$ forms an arithmetic progression.

Answer: 4


Question: Does the second countability of a topological space ensure that any open set can be expressed as a countable union of basic open sets from a topological base?

Answer: Yes


Question: Find the natural numbers $x$ and $y$ that satisfy the equation: $5^{2x} - 3 \cdot 2^{2y} + 5^{x} 2^{y-1} - 2^{y-1} - 2 \cdot 5^{x} + 1 = 0$. Provide your answer as a pair $(x, y)$.

Answer: (2, 4)


Question: Among 10 chickens, one is false and weighs either 1kg with probability 1/2 or 3kg with probability 1/2. A machine independently picks the i-th chicken with probability 1/2 for weighing. After each weighing, all chickens are returned to the machine. After 3 weighings, the first set of chickens weighs 11kg, the second set weighs 12kg, and the third set weighs 13kg. What is the probability that the false chicken weighs 3kg?

Answer: \dfrac{3}{5}


Question: In a circular arrangement, determine the number of ways to arrange $n$ men and $m$ women such that no two women are adjacent, given that $n < m$. Provide your answer as a single number.

Answer: 0


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac {1-\frac {1}{2} + \frac {1}{3} -\frac {1}{4}+ ... + \frac {1}{2n-1}-\frac{1}{2n}}{\frac {1}{n+1} + \frac {1}{n+2} + \frac {1}{n+3} + ... + \frac {1}{2n}}$$

Answer: 1


Question:
Given the equations:
\[ ax + by = 7 \]
\[ ax^2 + by^2 = 49 \]
\[ ax^3 + by^3 = 133 \]
\[ ax^4 + by^4 = 406 \]
find the value of \(2014(x+y-xy) - 100(a+b)\).

Answer: 5956


Question: Evaluate the contour integral \( \oint_{\gamma} \frac{\Re(z)}{z - a} \, \mathrm{d}z \), where \( a \) is a complex number with \( 0 < |a| < 2 \), and \( \gamma \) is the unit circle centered at the origin with radius 3, traced counterclockwise.

Answer: \pi i a


Question: Evaluate the limit: \[ \lim_{h \to 0^+} \int_{-1}^1 \frac{h}{h^2+x^2}f(x)\,dx \] where \( f \) is continuous on \([-1,1]\) and \( f(0)=1 \).

Answer: \pi


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

Answer: \dfrac{\pi}{e}


Question: For which integers \(n\) does the set \(H = \{ \alpha \in A_n \mid \alpha^2 = e \}\) form a subgroup of the alternating group \(A_n\)?

Answer: n \leq 4


Question: Calculate the limit: \[ L = \lim_{{n \to \infty}} \left( \frac{n!}{n^n} \right)^{\frac{1}{n}} \]

Answer: \dfrac{1}{e}


Question: Find the limit of the sequence \((a_n)_{n\geq1}\), where \(a_n=\frac{S_n}{n^\alpha}\) and \(S_n=\sum_{k=1}^{n^2} [\sqrt{k}]\), with \(\alpha\in\mathbb{R}\).

Answer: \dfrac{2}{3}


Question: Determine the maximum value of \(|\sin(z)|\) for \(z\) within the closed unit disk \(\{z: |z| \leq 1 \}\).

Answer: \sinh(1)


Question: Find the maximum value of the function \(|\cos{z}|\) where \(z\) is a complex number in the square \([0,2\pi]\times[0,2\pi]\).

Answer: \cosh(2\pi)


Question: Find the value of $m > 0$ such that the area of the set \( A = \{ (x, y) \mid m \leq x \leq 2m, 0 \leq y \leq x + \frac{6}{x^2} \} \) is minimized.

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{2^{1/n} - n^2}{\sum_{k=1}^n \frac{1}{k}} \]

Answer: -\infty


Question:
Assume that $a, b, c$ are nonzero complex numbers satisfying $|a| = |a - b| = |c|$ and $|a + b| = \sqrt{3}|c|$. Determine the value of the logarithm:
$$\log_{|c|}\left|\left(\frac{a}{\overline{b}}\right)^{2017} + \left(\frac{\overline{a}}{b}\right)^{2017}\right|$$

Answer: 0


Question: Find a function \( g: \mathbb{Z}^+ \to \mathbb{Z}^+ \) that satisfies the following conditions: \( \gcd(g(m), g(n)) \geq (\gcd(m, n))^{2014} \) for any positive integers \( m \) and \( n \), and \( n - 2014 \leq g(n) \leq n \) for any positive integer \( n \).

Answer: g(n) = n


Question: Find the value of $m+n$ given that $$\lim\limits_{x\to 0}\frac{x(1+m\cos x)-n\sin x}{x^3}=2.$$

Answer: -10


Question: Let $G$ be a group with a normal subgroup $H$ such that $|H| = 4$. If the coset $gH$ has order 3 in $G/H$, determine the order of the subgroup $K = H \cup gH \cup g^2H$.

Answer: 12


Question: Evaluate the limit: \[ \lim\limits_{x \to 0^+}{\left(2\sqrt{x}+x\right)^\frac{1}{\ln x}} \]

Answer: \sqrt{e}


Question: Given an $8 \times 8$ matrix $A$ with a row space dimension of 5, determine the dimension of the set $S_A = \{ B \in \mathbb{R}^{8 \times 10} \mid AB = 0 \}$. What is the dimension of $S_A$?

Answer: 30


Question: Find the limit of the expression as $x$ approaches 0: $$\lim_{x\to0}\left(\frac{\sin(x)}{x}\right)^{\frac{1}{x^2}}$$

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


Question: Consider a square, symmetric matrix $M$ of size at least $2 \times 2$, with diagonal entries equal to $1$ and off-diagonal entries equal to $\pm 1$. The matrix $M$ is indefinite, meaning there exists a nonzero vector $x$ such that $x^T M x = 0$. Is it always possible to choose such a vector $x$ such that none of its entries are zero, regardless of the specific configuration of $M$?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to\infty}\left(\frac{2^{n+1}+3}{2^{n+1}-5}\right)^{2^{n+1}}$$

Answer: e^8


Question: Find the maximum volume of a rectangular box with a fixed surface area of 150 square units.

Answer: 125


Question: How many distinct garlands can be made using 5 blue, 3 yellow, and 2 pink flowers, such that no two flowers of the same color are adjacent?

Answer: 2


Question: Does the boundedness of a function's derivative on an open interval imply that the function itself is also bounded on that interval? Answer yes or no and justify your answer.

Answer: No


Question: Evaluate the limit \( \lim_{n \to \infty} \int_1^{\infty} \frac{1}{1+x^n} \, dx \) using a dominating function.

Answer: 0


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

Answer: -\dfrac{2}{3}


Question: Find the residue of the function $e^{1/z^2}$ at the point $z=0$.

Answer: 0


Question: Determine whether it is possible to decide if a string belongs to the language $L = \{0^{2^n} \mid n \geq 0\}$ in linear time.

Answer: Yes


Question: Evaluate the limit: $$\lim_{\alpha\rightarrow 0} \int_{0}^1 \frac{(1 - x)^\alpha}{\ln x} dx$$

Answer: -\infty


Question: Let $X$ be a Banach space, $H$ a countably dense subset of $X$, and $\{f_n\}$ a sequence of functions $f_n: X \to \mathbb{R}$ such that \( f_n(x) \to f_x \) for all \( x \in H \), with \(\{f_n\}\) being equicontinuous on $X$. Does it follow that for all $x \in X$, there exists a limit $f_x \in \mathbb{R}$ satisfying \( f_n(x) \to f_x \)?

Answer: Yes


Question: Determine if there is a closed form for the infinite product \(\prod_{k=0}^{\infty} \left( 1-x^{2^k} \right)\) where \(|x|<1\).

Answer: \prod_{k=0}^{\infty} \left( 1 - x^{2^k} \right)


Question: Calculate the limit \(\lim_{t\to \infty} y(t)\) for the differential equation \( y'(t) + ay(t) = de^{-bt} \), where \(a, b, d\) are constants with \(a, b > 0\).

Answer: 0


Question: Determine the number of zeroes of the polynomial \( f(z) = z^{10} + 10z + 9 \) within the annulus \( \{ z : 1 < |z| < 2 \} \).

Answer: 8


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

Lemma: Let $f$ be a one-to-one and continuous function on the interval $[a,b]$. Then, $\lim_{x \rightarrow t} \phi(x) = \lim_{f(x) \rightarrow f(t)} \phi(x)$, where $\lim_{x \rightarrow t} \phi(x)$ exists.

Answer: True


Question: Find the fixed point of the operator \( T(f) = \int_0^x \frac{f(y)}{2} + \frac{yf(y)}{2}\, dy \).

Answer: 0


Question: Determine the value of \(a > 2\) such that \(\int_{0}^{1}\frac{x^8-x^2}{x^{12}+ax^6+1}dx = -\frac{\pi}{24}\).

Answer: 6


Question: Given two independent random variables $X$ and $Y$, both uniformly distributed on the interval $(0,1)$, find the conditional probability $P(X^2+Y^2\leq1|X\leq Y)$. Express your answer as a single number.

Answer: \dfrac{\pi}{4}


Question: Determine the intersection number of the curve given by $f = y - x^3$ with its tangent at the origin.

Answer: 3


Question: If two groups \(G\) and \(H\) have the same order \(|G| = |H| = p^a m\), where \(p\) is a prime and \(\gcd(p, m) = 1\), do we necessarily have that the number of Sylow \(p\)-subgroups \(n_p(G)\) is equal to \(n_p(H)\)? Justify your answer.

Answer: No