Question: Determine whether there exists a decreasing sequence of positive real numbers \( \{x_n\} \) that converges to 0, such that there exists \( m_0 \in \mathbb{N} \) and a constant \( S(m_0) > 0 \) depending on \( m_0 \), for which \( |x_{j+m_0} - x_{j+m_0+1}| \geq \frac{S(m_0)x_{j+1}}{\log^2 x_{j+1}} \) holds for all \( j \).

Answer: Yes


Question: Let $M$ be a smooth manifold and $p \in M$ be a fixed point. Suppose $v_p, w_p \in T_p M$ are tangent vectors such that $$(f_*)_p(v_p) = (f_*)_p(w_p)$$ for all smooth functions $f: M \to \mathbb{R}$. Prove that $v_p = w_p$. Provide a justification for your answer.

Answer: v_p = w_p


Question: Let $G$ be a compact Hausdorff Lie group. If a topological space $X$ is homeomorphic to $G$, is $X$ necessarily a topological manifold? Provide a justification for your answer.

Answer: Yes


Question: Is it true that for every finite abelian group $H$, there exists a non-abelian group $G$ such that the center of $G$, denoted $Z(G)$, is isomorphic to $H$?

Answer: Yes


Question: Find an integer \( x \) in \( \mathbb{Z}/81\mathbb{Z} \) such that \( x^{31} \equiv 2 \pmod{81} \).

Answer: 47


Question: Suppose $S \subset \mathbb{R}^n$ is a linear affine subspace. If $s \in S$, then $S - s =: U$ is a subspace and we can write $S = s + U$. Consider writing $s = s_{U} + s_{U^{\perp}}$, where the subscripts denote the orthogonal projection. Is the vector $s_{U^{\perp}}$ uniquely defined in this representation?

Answer: Yes


Question: Determine the order of subgroups that appears most frequently in the symmetric group $S_n$. Specifically, find the order $k$ for which the number of subgroups of $S_n$ with order $k$ is maximized.

Answer: 2


Question: Determine the set of values of \(a\) for which the range of the rational function \(y = \frac{x^2 + ax - 2}{x - a}\) is the set of all real numbers.

Answer: (-1, 1)


Question: Find \( \lim_{n \to \infty} f_n(x) \) where \( f_n(x) = \int_{0}^{1} f\left(x + \frac{y}{n}\right) dy \) and \( f: \mathbb{R} \to \mathbb{R} \) is a continuous function.

Answer: f(x)


Question: Given two independent random variables, where \(X\) follows a uniform distribution \(U(0,\pi)\) and \(Y\) follows a uniform distribution \(U(0,2\pi)\), calculate the probability \(\Pr(Y \in [\pi, X+\pi] \mid X)\).

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


Question: Evaluate the limit: \[ \lim \limits_{x \to \infty} \dfrac{\lfloor P(x) \rfloor}{P(\lfloor x \rfloor)} \] where \( P(x) \) is a polynomial with positive real coefficients.

Answer: 1


Question: Determine whether there exists a Borel measurable function \( f: [0, 1] \times \mathbb{R} \to [0, \infty) \) such that \( \inf_{g \in \mathcal{L}} \int_{0}^{1} f(t, g(t)) \, dt < \inf_{h \in C^2([0, 1])} \int_{0}^{1} f(t, h(t)) \, dt \), where \( \mathcal{L} \) is the set of continuously differentiable real-valued functions on \([0, 1]\) with Lipschitz continuous derivative.

Answer: Yes


Question:
Given the final amount $A$, the principal amounts $P_1, P_2, \ldots, P_n$ invested at the beginning of each year, and the number of years $n$, determine the average annual rate of return $x\%$ that satisfies the equation:

\[ A = P_1(1+x\%)^1 + P_2(1+x\%)^2 + \cdots + P_n(1+x\%)^n \]

Solve for $x$.

Answer: x


Question: Determine the number of subgroups of the group \( \mathbb{Z}_p \times \mathbb{Z}_p \), where \( p \) is a prime number.

Answer: p + 3


Question: What is the smallest integer $k$ such that $PSU_3$ is a subgroup of $GL_k(\mathbb{C})$?

Answer: 8


Question:
Find the equation of the circle whose center is at \(a + i\) (where \(a\) is a real number) and that intersects the circles \(|z| = 1\) and \(|z - 1| = 4\) orthogonally. Choose the correct option:

(a) \(|z-7+i| = 7\)
(b) \(|z-2+i| = 7\)
(c) \(|z+7-i| = 7\)
(d) \(|z+2+i| = 7\)

Answer: C


Question: Let \(K\) be a non-empty compact subset of \(\mathbb{R}^3\). Define \(G = \{f \in \mathrm{Isom}~\mathbb{R}^3 \mid f(K) = K\}\), and let \(G'\) be the identity connected component of \(G\). Is \(G'\) isomorphic as a Lie group to \(SO(k)\) for some \(k \in \{1, 2, 3\}\)?

Answer: Yes


Question: Evaluate the improper integral: \[ \int_{0}^{\infty}\frac{\ln\left(1+x^2\right)}{1+x^2}dx \]

Answer: \pi \ln 2


Question: Let $f$ be a Lipschitz function with compact support, satisfying $|f(x)-f(y)|\leq L|x-y|$ for some constant $L$. If $p > 1$, is the function $f^p$ also Lipschitz?

Answer: Yes


Question: Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth function and $M$ be a smooth manifold of $\mathbb{R}^n$. Assume that $Df(x)v \neq 0$ for all $v$ being tangent to $M$ at $x$ and for all $x$ in $M$. Is $f$ locally injective on $M$?

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to0}\frac{\ln(\sin x)}{\ln(\cos x)}$$

Answer: \infty


Question: Determine all local extreme points of the function \( f(x,y) = x^2y^2 + x^2y + 2y^2 - 4y \).

Answer: (0, 1)


Question: Determine the number of elements in the set \( S = \{ M \in \mathbb{F}_3^{2 \times 2} : M \text{ is orthogonal} \} \), where \( \mathbb{F}_3 = \{ 0, 1, 2 \} \).

Answer: 8


Question: Find the integer part of the sum \( \sum_{k=2}^{9999} \frac{1}{\sqrt{k}} \).

Answer: 197


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{1}{\sqrt[4]{n^4+n+2}} + \frac{1}{\sqrt[4]{n^4+n+3}} + \cdots + \frac{1}{\sqrt[4]{n^4+5n-1}} \right) \]

Answer: 4


Question: Determine all values of \( \alpha \) such that all solutions of the differential equation \( x^2y'' + \alpha xy' + 4y = 0 \) approach zero as \( x \to \infty \).

Answer: \alpha > 1


Question: For a positive integer $n \ge 3$, let $A_n = \{\ln 3, \ln 4, \dots, \ln n\}$. Determine whether there exists an integer $N$ such that for all $n > N$, the set $A_n$ can be partitioned into two subsets whose sums differ by no more than 1.

Answer: Yes


Question: Let \( f : \mathbb{R}^n \to \mathbb{R}^n \) be a function such that \( f(ax) = af(x) \) for every \( a \in \mathbb{R} \) and \( x \in \mathbb{R}^n \). Does it follow that \( f \) is a linear function? Justify your answer.

Answer: No


Question: Suppose \(x_n\) is a sequence in \(\mathbb{R}\) and \(x\) is a real number such that for every bounded continuous function \(f: \mathbb{R} \to \mathbb{R}\), the sequence \(f(x_n)\) converges to \(f(x)\) as \(n \to \infty\). Does it follow that \(x_n\) converges to \(x\) as \(n \to \infty\)?

Answer: Yes


Question: Determine whether the following statement is true or false: Given a measurable function \( g(x) \) such that \(|g(x)| \leq M\) for some \( M > 0 \), and \( \int_a^b g(x)\,dx = 0 \) for every interval \([a,b]\subset\mathbb{R}\), is it true that \( \forall f\in L^1, \int_{\mathbb{R}} f(x)g(x)\,dx = 0 \)?

Answer: True


Question: Evaluate the limit: \[ \lim_{x\to 0} \sum^{2013}_{k=1} \frac{\left\{\frac{x}{\tan x}+2k\right\}}{2013} \] where \( \{x\} \) denotes the fractional part of \( x \), satisfying \( 0 \leq \{x\} < 1 \).

Answer: 1


Question: Consider the plane $X = \mathbb{R}^2$ with the equivalence relation $x_0 \times y_0 \sim x_1 \times y_1$ if $x_0 + y_0^2 = x_1 + y_1^2$. Let $X^*$ be the corresponding quotient space. Identify the familiar space to which $X^*$ is homeomorphic.

Answer: \mathbb{R}


Question: Find the modular inverse of $x^4$ in $GF(2^5)$ modulo the polynomial $x^5 + x^2 + 1$. Provide your answer as a polynomial in $GF(2^5)$. What is the inverse?

Answer: x^3 + x + 1


Question: Determine the Galois group of the polynomial \(x^3 - x - 1\) over \(\mathbb{Q}\).

Answer: S_3


Question: Calculate \( \lim_{x\rightarrow 0^+} \int_0^1 \ln(1+\sin(tx))\,dt \).

Answer: 0


Question:
Determine which of the following graphs cannot have its edges partitioned into triangles:

(a) K7
(b) K12
(c) K3,3,3
(d) K5,5,5

Select the correct option.

Answer: B


Question: Determine whether the vector field \( F(x,y) = y^2e^{xy}\vec{i} + (1+xy)e^{xy}\vec{j} \) is conservative. If it is, find a function \( f \) such that \( F = \nabla f \).

Answer: f(x, y) = y e^{xy}


Question: Determine whether the sequence of functions \((f_n)_{n\geq1}\), where \(f_n = \sin(x+n)\) and \(x \in \mathbb{R}\), has a converging subsequence.

Answer: Yes


Question: Find the remainder when \(24^{1202}\) is divided by \(1446\).

Answer: 576


Question: Find the exponential generating function for the sequence \( \{0, 1, 2a, 3a^2, 4a^3, \ldots\} \).

Answer: x e^{a x}


Question:
Given the joint density function of random variables X and Y:
\[ f(x,y) = \begin{cases}
2e^{-x}e^{-2y} & 0<x<\infty, 0<y<\infty \\
0 &\text{otherwise}
\end{cases} \]
Find the probability \( P\{ X < Y \} \).

Answer: \dfrac{1}{3}


Question:
If the line \(z\overline\alpha + \overline z\alpha + i\beta = 0\) makes an angle of \(45^\circ\) with the real axis, find the value of \((1+i)(-\frac {2\alpha} {\overline\alpha})\). Choose from the following options:

(a) \(2\sqrt{2}\)
(b) \(2\sqrt{2} i\)
(c) \(2(1-i)\)
(d) \(-2(1+i)\)

Answer: C


Question: Can Tarjan's strongly connected components algorithm be used to determine if a directed graph has a cycle?

Answer: Yes


Question:
Evaluate the integral \( I = \int_{0}^{\pi/2} \frac{\sin\frac{7x}{2}}{\sin\frac{x}{2}} \, dx \). Determine the value of \( I \) from the following options:

\( (A) \pi - 1 \hspace{1cm} (B) \frac{\pi}{2} + \frac{1}{3} \hspace{1cm} (C) \frac{\pi}{2} + \frac{4}{3} \hspace{1cm} (D) \pi - \frac{4}{3} \)

Answer: C


Question: Find the inverse Laplace transform of \( \frac{4s}{(s^2+4)^2} \).

Answer: t \sin(2t)


Question: Is the number of topologies that can be defined on a finite set with \( n \) elements finite?

Answer: Yes


Question: Given a sequence of random variables \(x_1, x_2, x_3, \ldots\) drawn from a normal distribution with mean \(\mu\) and variance \(\text{var}\), construct a sequence of cumulative products: \((x_1), (x_1 \cdot x_2), (x_1 \cdot x_2 \cdot x_3), \ldots\). What is the formula for the expected value of the 15th value in this sequence, in terms of \(\mu\) and \(\text{var}\)?

Answer: \mu^{15}


Question: Determine the second homotopy group of the space obtained by removing a line and a point (where the line does not intersect the point) from \( \mathbb{R}^3 \).

Answer: \mathbb{Z}


Question:
Determine the value of \( \alpha \in \mathbb{R} \) such that the second derivative of the function \( f(x) \) exists at \( x = 0 \), where:

\[
f(x) =
\begin{cases}
e^{-rac{1}{x}},  & \text{if } x > 0 \\
\sin(x) + \alpha x^2 - \log(1+x), & \text{if } x \leq 0
\end{cases}
\]

Answer: -\dfrac{1}{2}


Question: Compute the limit of the integral \( I = \lim_{n \to +\infty} \int_0^n \frac{x^n \text{e}^{-x}}{n!} \text{d}x \).

Answer: \dfrac{1}{2}


Question: Using Rouché's Theorem, determine how many roots of the polynomial \( z^5 + 4z^2 + 1 \) lie in the annulus defined by \( 1 < |z| < 2 \).

Answer: 3


Question: Consider two independent simple random walks on the integers, \(X_n\) and \(Y_n\), starting at positions \(-1\) and \(1\) respectively. At each time step \(n\), each walk moves left or right with equal probability. Determine the expected time at which the two walks collide.

Answer: \infty


Question: Determine if the expression \( \frac{f(r+ct) - f(ct-r)}{2r} \) tends to the derivative of \( f \) at some point, possibly multiplied by a constant, as \( r \to 0 \).

Answer: f'(ct)


Question: Determine the transcendence degree of the field \( \mathbf{Q}(x,y,\exp(x),\exp(y)) \) using the Lindemann-Weierstrass Theorem.

Answer: 4


Question: Find the whole number \( x \) such that \( x < 33 \), \( x^7 \equiv 29 \pmod{33} \), and \( x^{13} \equiv 8 \pmod{33} \).

Answer: 2


Question: Let $f$ be an entire function such that $|f(z)| \le k|z|$ for all $z \in \mathbb{C}$, and $f(1) = i$. Determine the value of $f(i)$. Assume $\frac{f(z)}{z}$ is constant for $z \in \mathbb{C}$. What is $f(i)$?

Answer: -1


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables, each having a $Uniform[0, 0.5]$ distribution. Define $Y_n = \frac{\sum_{i=1}^{n} X_i(X_i+1)}{n}$. Find a number $c$ such that $Y_n$ converges in probability to $c$ as $n \to \infty$. What is the value of $c$?

Answer: \dfrac{1}{3}


Question: Let $G$ be an infinite, finitely generated group with one end. Consider a word-metric $d$ on the Cayley graph of $G$ with respect to some finite symmetric generating set $S$. Does there exist a good bi-infinite path $\{g_i\}_{i\in\mathbb{Z}}$ in $G$, where $d(g_i, g_{i+1})=1$ for all $i$ and $\lim_{|i-j|\to\infty}d(g_{i}, g_j)=\infty$? Assume $G$ is an infinite torsion group.

Answer: Yes


Question: Given the function \( f(x) \) defined as \( f(x) = 0 \) for \( x \leq 0 \) and \( f(x) = e^{-x} \) for \( x > 0 \), the Fourier transform \( F(\omega) \) is calculated to be \( F(\omega) = \frac{1}{\sqrt{2\pi}(1+i\omega)} \). Use this result to evaluate the integral \( \int_{0}^{\infty} \frac{\cos(2\omega) + \omega\sin(2\omega)}{1+\omega^2} \, d\omega \).

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


Question: Evaluate the integral \( \int_{|z| = 1}\left(\frac{z}{z - a}\right)^n dz \) for \(|a| < 1\) and \(n > 1\).

Answer: 2\pi i n a


Question: Given that \( \sin^3 x \sin 3x = \sum^n_{m=0}C_m \cos mx \), where \( C_n \neq 0 \), find the value of \( n \).

Answer: 6


Question: How many 4-permutations can be formed from the letters in the word "aaabbccdef"?

Answer: 758


Question: Identify and sketch the locus of points in the complex plane where \( \operatorname{Re}(z^3) = 1 \).

Answer: \operatorname{Re}(z^3) = 1


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( 1 + \sin \left( \frac{3}{x} \right) \right)^x \]

Answer: 1


Question: Let \( \phi : \operatorname{End}(E) \to \mathbb{K} \) be a linear form where \( E \) is a finite-dimensional vector space. Suppose there exists \( x \in E \) such that \( \operatorname{Span}\{f_1(x), \ldots, f_p(x)\} = E \), where \( f_i \) are a basis of \( \ker \phi \). Does it necessarily follow that \( \phi = 0 \)?

Answer: No


Question: Calculate the limit: \[ \lim_{x \to 0} \left(1+\frac {1-\cos x} {x} \right)^\frac{1}{x} \]

Answer: \sqrt{e}


Question:
Consider a sequence \(\{ f_k \}_{k=1}^{\infty}\) of locally-bounded functions \(f_k: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}\). Assume that for any sequence \(\{X_k\}_{k=1}^{\infty}\) of compact sets \(X_k \subset \mathbb{R}^n\) such that \(X_k \subseteq X_{k+1}\) and \(X_k \rightarrow \mathbb{R}^n\), there exists a uniform \(M \in \mathbb{R}_{>0}\) such that
\[ \sup_{x \in X_k} f_k(x) \leq M \]
Determine if there exists \(K \in \mathbb{Z}_{\geq 1}\) such that
\[ \sup_{x \in \mathbb{R}^n} f_K(x) < \infty \]

Answer: Yes


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

Answer: 0


Question: Is the ring $A[x, a/x]$ always integrally closed if $A$ is a unique factorization domain and $a$ is an element of $A$?

Answer: Yes


Question: Is it true that every smooth curve \( \gamma:[a,b]\to \mathbb{R}^n \), where \( \gamma \in C^1 \), has a length given by \( L(\gamma)=\int_{a}^b\|\gamma'(t)\|\,dt \)?

Answer: Yes


Question: Determine the set of values of \( t \in [0, \infty) \) for which the series of functions \( \sum_{n=1}^\infty \frac{7^n + 3t^n}{3 + 7^n t^n} \) converges pointwise.

Answer: (1, \infty)


Question:
Given the recursive system:
\[\begin{cases}
& a_{n+1}=-2a_n -4b_n\\
& b_{n+1}=4a_n +6b_n\\
& a_0=1, b_0=0
\end{cases}\]
Calculate the value of \( \frac{a_{20}}{a_{20}+b_{20}} \).

Answer: -39


Question: Let \(a, b, c, d\) be real numbers such that \(a+b+c+d=0\) and \(abcd=1\). Find the minimum value of \(a^4+b^4+c^4+d^4+a^2+b^2+c^2+d^2\).

Answer: 8


Question: Evaluate the integral \( \int_0^1{\frac{(2x^3-3x^2)f'(x)}{f(x)}}\,dx \) given that the function \( f(x) \) is differentiable on \([0, 1]\) and satisfies the conditions \( f(1) = 1 \) and \( f(x)f(1-x) = e^{x^2 - x} \).

Answer: -\dfrac{1}{10}


Question: Consider the set \( S = \{3^n 4^m : n, m \in \mathbb{N}\} \). Let \( x_1, x_2, x_3, \dots \) be the elements in \( S \) arranged in increasing order such that \( x_1 < x_2 < x_3 < \dots \). Prove that as \( k \to \infty \), the ratio \( \frac{x_{k+1}}{x_k} \to 1 \).

Answer: 1


Question: Given two power series \(\sum_{n=0}^{\infty} a_nx^n\) and \(\sum_{n=0}^{\infty} b_nx^n\) with radii of convergence \(R_{1}\) and \(R_{2}\) respectively, where \(R_{1} < R_{2}\), determine the radius of convergence of the series \(\sum_{n=0}^{\infty} (a_n+b_n)x^n\).

Answer: R_1


Question: Let $X$ be a separable, metric, compact space, such as an interval in $\mathbb{R}$ like $[0,10]$. Consider $M(X)$, the set of all finite signed measures over $X$ with the weak-* topology. Define the set $A = \{ \mu \in M(X) : |\mu|(X) \leq a \}$ for some $a > 0$, where $|\mu|(X)$ denotes the total variation norm. Is the set $A$ compact in $M(X)$?

Answer: Yes


Question: Find the sum of the infinite series \( \sum\limits_{n=1}^{\infty} (2n-1)^2 \left(\frac{1}{2}\right)^n \).

Answer: 17


Question: Find the residue of the function \( f(z) = \frac{z}{1 - \cos(z)} \) at \( z = 2k\pi \), where \( k \in \mathbb{Z} \).

Answer: 2


Question: Solve for \(x\) in the equation: \[ x = 1 + \ln(1 + \ln(1 + \ln(x))) \]

Answer: 1


Question: The population \( P(t) \) of a species satisfies the logistic differential equation \( \frac{dP}{dt} = P\left(224 - \frac{P^2}{56}\right) \), where the initial population \( P(0) = 30 \) and \( t \) is the time in years. What is the limit of \( P(t) \) as \( t \) approaches infinity? (Calculator allowed)

Answer: 112


Question: Determine the number of decimal digits in the number \(2^{3021377}\).

Answer: 909526


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

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit: \[ \lim _{x\to 0}\frac{\exp(x^2)-\cos x-x\sin x}{\sinh^2x} \]

Answer: \dfrac{1}{2}


Question: Determine the period of the function \( f(x) \) if it satisfies the equation \( f(x+1) + f(x-1) = \sqrt{3}f(x) \).

Answer: 12


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

Answer: Yes


Question: A particle moves along the integers from 1 to 10. It starts at position 1 and moves to an adjacent integer at each step. When at position 1, it can only move to position 2. From any other position n (2 ≤ n ≤ 9), it moves to n-1 or n+1 with equal probability. What is the expected number of steps for the particle to reach position 10 for the first time?

Answer: 81


Question: Consider a field extension $F \subset E$. Suppose there exists an integer $n$ such that for each element $\alpha \in E$, the degree $|F[\alpha]: F| \leq n$. Can we conclude that the extension $|E:F|$ is finite?

Answer: Yes


Question: Let $f$ and $g$ be polynomials with real coefficients in several variables $x_1, \ldots, x_n \in \mathbb{R}$. Assume that $f$ and $g$ are equal on an open ball $X = B_\delta(a)$ around some point $a \in \mathbb{R}^n$. Does this imply that $f \equiv g$? Provide a justification for your answer.

Answer: f \equiv g


Question: Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ and $g: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be two smooth functions with $m, k < n$. Given that $f^{-1}[\{0\}] = g^{-1}[\{c\}]$ and $c$ is a regular value of $g$, determine whether $0$ is a regular value of $f$.

Answer: No


Question: Calculate the limit of the sequence \( f_n = \frac{e^x \sin(x) \sin(2x) \cdots \sin(nx)}{\sqrt{n}} \) as \( n \to \infty \).

Answer: 0


Question:
Verify the identity \((1-x^2)\frac{\partial^2 \Phi}{\partial x^2}-2x\frac{\partial\Phi}{\partial x}+h\frac{\partial^2}{\partial h^2}(h\Phi)=0\) using the generating function for Legendre Polynomials:
\[\Phi(x,h)=(1-2xh+h^2)^{-1/2}\quad\text{for}\quad |h|<1\]

Answer: 0


Question: Find all functions \( f(x) \) such that for any \( x > 0 \), the equation \( 2f\left(\frac{1}{x}+1\right) + f(2x) = 1 \) holds.

Answer: \dfrac{1}{3}


Question:
Given a matrix $A \in \mathbb{R}^{4\times4}$ with the following properties:

- The spectrum of $A$ has 3 distinct eigenvalues.
- The trace of $A$, $Tr(A) = 92$.
- The trace of $A^2$, $Tr(A^2) = 4060$.
- The rank of $(A - I)$ is 2.

Determine the determinant of the matrix $A$. Express your answer as a single number.

Answer: 2021


Question: Calculate \(\lim _{n\to \infty }a_n\int _0^1 x^{2n}\sin \frac{\pi x}{2}dx\), where \(a_n = \sum _{k=1}^n\sin \frac{k\pi }{2n}\). It is given that \(\lim _{n\to \infty} \frac{a_n}{n} = \frac{2}{\pi} \).

Answer: \dfrac{1}{\pi}


Question: Derive the sufficient statistic for \(\lambda\) based on the data \(X_1\) and \(X_2\), where \(X_1 \sim \operatorname{Poisson}(\lambda p)\) and \(X_2 \sim \operatorname{Poisson}(\lambda(1-p))\), with \(p\) being a known number in \((0,1)\).

Answer: X_1 + X_2


Question: Let \( \mathbf{A} \) be a matrix, and let \( \mathbf{x} \) and \( \mathbf{y} \) be linearly independent vectors such that \( \mathbf{A}\mathbf{x} = \mathbf{y} \) and \( \mathbf{A}\mathbf{y} = \mathbf{x} + 2\mathbf{y} \). Find the ordered pair \((a, b)\) such that \( \mathbf{A}^{5} \mathbf{x} = a \mathbf{x} + b\mathbf{y} \).

Answer: (12, 29)


Question: Compute \( \lim_{n \to \infty}\int_{\mathbb{R}}g(x)\sin^2(nx)\,dx \) given that \( \int g(x)\,dx = 1 \) and \( g \in L^1(\mathbb{R}) \).

Answer: \dfrac{1}{2}


Question: Let $X$ and $Y$ be random variables with a joint uniform distribution over the triangle with vertices $(0,0)$, $(1,0)$, and $(1,1)$. Compute the conditional expectation $E[Y|X]$. Provide your answer as a function of $X$.

Answer: \dfrac{X}{2}


Question: Find the value of \(a \in \mathbb{R}\) such that the solution set of the linear system \[ \begin{cases} -x+2y+az=3 \\ y-z=a \\ x+y-z=-9 \end{cases} \] represents a line perpendicular to the plane \(K\) given by \[ \begin{cases} x=t+1 \\ y=-t+2s-1 \\ z=s-1 \end{cases} \].

Answer: -2


Question: A person has 10 distinct footballs and 3 distinct bins. Each bin must contain at least 2 footballs and no more than 5 footballs. How many ways can the footballs be distributed among the bins under these conditions?

Answer: 37170


Question: Find the value of \(a_2 + 11a_3 + 70a_4\) for which the function \(f(x) = x^3 + 3(a-7)x^2 + 3(a^2-9)x - 1\) has a positive point of maximum in the interval \((a_1, a_2) \cup (a_3, a_4)\).

Answer: 320


Question: Given a set of positive distinct integers \(a_1, a_2, \dots, a_n\), does there exist a positive integer \(N\) such that the set \(S_N = \{Na_1, Na_2, \dots, Na_n\}\) has the property that the divisor function \(\tau(Na_i)\) is distinct for each \(i\)? Here, \(\tau\) denotes the number of divisors function. For example, if we start with \(\{2, 3, 5, 8\}\), can we find such an \(N\)?

Answer: Yes


Question: Determine the dimension of the complex vector space generated by the set \( E_n = \{ e^{2 \pi i k x} \mid -n \leq k \leq n \} \).

Answer: 2n + 1


Question: Find the image of the right half-plane \( \text{Re}(z) > 0 \) under the linear transformation \( w = f(z) = \frac{i(1-z)}{1+z} \).

Answer: |w| < 1


Question: In a triangle with an acute angle at $ABC$, the Euler line intersects the sides of the triangle to form an inscribed quadrilateral. Determine the measure of the angle between the Euler line and the circumradius that passes through point B.

Answer: 45^\circ


Question: For a given integer \( n \), find the value of \( x \) such that the equation \( \pi(xn) = \frac{xn}{\frac{n}{\pi(n)} + \ln(x)} \) holds, where \( \pi \) is the prime counting function.

Answer: 1


Question: Is the formula \( s^2 = \frac{1}{n-1} \cdot \sum_{i=1}^n (X_i-\bar{X})^2 \) an unbiased estimator for the population variance \( \sigma^2 \) in simple random sampling with replacement?

Answer: Yes


Question: Let $A_1, A_2, \dots, A_{11}$ be the vertices of a regular 11-gon inscribed in a circle of radius 2. A point $P$ is located such that its distance from the center of the circle is 3. Calculate the sum $PA_1^2 + PA_2^2 + \dots + PA_{11}^2$. What is this sum?

Answer: 143


Question: Determine the number of rotation matrices that can rotate line A to line B in 3D space, given that both lines are of the same length, centered at the origin, and are not identical.

Answer: \infty


Question: Find the limit of the sequence \( P_{n} = \frac{\log \binom{n}{0} + \log \binom{n}{1} + \dots +\log \binom{n}{n}}{n^{2}} \) as \( n \to \infty \). Here, \( \log \) denotes the natural logarithm.

Answer: \dfrac{1}{2}


Question: Evaluate the double integral \( \iint_D \frac{\sin x}{x} \,\mathrm{d}x\mathrm{d}y \), where \( D = \{(x,y): 0 \le x \le 1 , 0 \le y \le x\} \).

Answer: 1 - \cos 1


Question: Determine the values of \( p \) for which the integral \( \int_0^{\infty} \frac{e^{-x}}{x^p} \, dx \) converges.

Answer: p < 1


Question: Determine whether the set \( \{(x_1, x_2) \in \mathbb{R}^2 \mid x_1 \neq 0 \text{ and } x_2 = 1/x_1 \} \) is complete in the metric space \( (\mathbb{R}^2, \| \cdot \|) \) with the norm \( \| (x_1, x_2) \| = \sqrt{x_1^2 + x_2^2} \).

Answer: Yes


Question: Evaluate the line integral \( \oint\limits_C \sqrt{x^2 + y^2}\,\mathrm ds \) where \( C \) is the circle defined by the equation \( x^2 + y^2 = ax \).

Answer: 2a^2


Question: Let \( f: \text{M}_n(\mathbb{C}) \to \text{M}_n(\mathbb{C}) \) be a \( \mathbb{C} \)-linear map. Determine whether there exist matrices \( A_1, \ldots, A_d \in \text{M}_n(\mathbb{C}) \) and \( B_1, \ldots, B_d \in \text{M}_n(\mathbb{C}) \) such that \[ f(X) = \sum_{j = 1}^d A_j X B_j \quad \forall X \in \text{M}_n(\mathbb{C}). \]

Answer: Yes


Question: Let $R$ be a commutative ring and $E$ and $F$ be two $R$-modules such that $E \cong_{R} F$ and $E \subset F$. Can we conclude that $E = F$?

Answer: No


Question:
In a rectangular Cartesian coordinate system \(\mathcal{C}\), the coordinates are \((x, y)\). In another coordinate system \(\mathcal{C'}\), the coordinates are defined by the transformations:
\[\begin{align}x' &= 2x + 3y\\ y' &= -3x + 4y\end{align}\]
What is the area element in the \(\mathcal{C'}\) coordinate system?

(A) \(\frac{1}{17}\ dx'dy'\)
(B) \(12\ dx'dy'\)
(C) \(dx'dy'\)
(D) \(x'\ dx'dy'\)

Answer: A


Question: Determine whether any ring of order 15 with identity is commutative.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left(\sqrt[9]{x^9+x^8} - \sqrt[9]{x^9-x^8}\right) \]

Answer: \dfrac{2}{9}


Question: Given that \( 3f(x) = 3x^4 + x^3 + 3x^2 \), evaluate the limit \( \lim_{a \to \infty} \int_{2a}^{8a} \frac{1}{(f^{-1}(x))^2 + (f^{-1}(x))^4} \, dx = \ln(n) \). Find the value of \( n \).

Answer: 4


Question: Consider the function \(f(x)=\int_x^{2x}\frac{\sin t}{t}dt\). Determine the value of \(x\) on the interval \((0,\frac{3\pi}{2})\) at which this function attains a local maximum.

Answer: \dfrac{\pi}{3}


Question: Evaluate the integral \( \int_{D} (x^2+y^2) \, \mathrm{dx} \wedge \mathrm{dy} \) over the region \( D \) defined in polar coordinates by \( 0 < r < 1 \) and \( 0 < \theta < 2\pi \).

Answer: \dfrac{\pi}{2}


Question: Let \( G = \mathbb{Z} \times \mathbb{Z} \) and \( H = 35 \mathbb{Z} \times 97 \mathbb{Z} \). Determine the order of the quotient group \( |G/H| \).

Answer: 3395


Question: Find all incongruent roots modulo 13 of the polynomial \(x^4 + x^2 + x + 1\).

Answer: 6


Question: Find all integer values of \( n \) such that \( 3(1^2 + 2^2 + \ldots + n^2) \) is a perfect square.

Answer: 0


Question: Consider a linear regression model given by \( y = \alpha + \beta_1 x_1 + \beta_2 x_2 \). According to the Gauss-Markov theorem, the ordinary least squares (OLS) estimates \( \hat{\beta}_1 \) and \( \hat{\beta}_2 \) have the smallest variance among all unbiased linear estimators. Determine whether the estimate of the linear combination \( \hat{\beta}_2 - \hat{\beta}_1 \) also has the smallest variance among unbiased linear estimators.

Answer: Yes


Question: Given the recursive sequence defined by \( x_n = x_{n-1}^2 \cdot K \) with \( K < 1 \) and \( x_0 = 3.089 \), find the sum of the sequence \( \sum_{n=0}^{\infty} x_n \). Assume \( K \approx 0.01685 \).

Answer: 3.250


Question: Evaluate the integral: \[ \int_{-1/2}^{+1/2} \sum_{n = -\infty}^{+\infty} \delta(\nu+n) \cdot e^{i\cdot2\pi\nu\cdot k} \,d\nu \] where \( n, k \in \mathbb{Z} \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to \infty} x^4 \left( \arctan \frac{2x^2+5}{x^2+1} - \arctan \frac{2x^2+7}{x^2+2} \right) \]

Answer: \dfrac{3}{5}


Question: Determine whether there exists a group \( G \) such that the center \( Z(G) \) has order \( p^2 \), where \( p \) is a prime number.

Answer: Yes


Question: Find the greatest common divisor (GCD) of the expressions \(61^{610} + 1\) and \(61^{671} - 1\).

Answer: 2


Question: Consider the group \(G=\langle a,b\mid a^6=1,b^2=a^3,bab^{-1}=a^{-1} \rangle\). The derived subgroup is \(G'=\{e,a^2,a^4\}\). Determine whether the factor group \(G/G'\) is isomorphic to \(C_4\) or \(C_2 \times C_2\).

Answer: C_4


Question: Evaluate the line integral \( \int_C (3x - 4x^2y) \, dx + (4x^2y + 2y) \, dy \) where \( C \) is the circle centered at the origin with radius 2. Use Green's theorem to find the value of the integral.

Answer: 16\pi


Question: Let $f:[0,1]\times [0,2\pi]\to \mathbb{R}$ be a function such that $f$ is in $L_1[0,1]$ with respect to the first parameter and is continuous with respect to the second parameter. Is $f$ in $L_1([0,1]\times [0,2\pi])$?

Answer: Yes


Question: Find the last two digits of \(9^{9^9}\).

Answer: 89


Question:
Which of the following is the correct expression for the differential operator in a Sturm-Liouville problem?

1. \( L(y) = \frac{d}{dx}\left(P(x) \frac{d}{dx}\right) + q(x) \)

2. \( L(y) = \frac{-d}{dx}\left[P(x) \frac{dy}{dx}\right] + q(x)y(x) \)

Answer: 2


Question: Find the remainder when $49!$ is divided by $53$. Use Wilson's Theorem to assist in your calculation.

Answer: 9


Question: A raindrop increases in mass as it falls, with its mass at time \( t \) given by \( m(t) \). The rate of growth of the mass is \( km(t) \), where \( k \) is a positive constant. Applying Newton's Law of Motion to the raindrop gives \( (mv)' = gm \), where \( v \) is the velocity of the raindrop and \( g \) is the acceleration due to gravity. Find an expression for the terminal velocity of the raindrop, \( \lim_{t \to \infty} v \), in terms of \( k \) and \( g \).

Answer: \dfrac{g}{k}


Question: Find the sum of the infinite series: \[ \frac{1^2}{1!} + \frac{2^2}{2!} + \frac{3^2}{3!} + \frac{4^2}{4!} + \cdots \]

Answer: 2e


Question: Consider the double series \( \sum\limits_{n = 1}^{\infty}\sum\limits_{m = 1}^{\infty} a_{m, n} \) where \( a_{m, n} \geq 0 \) for all \( m \) and \( n \). Can the order of summation be interchanged? Justify your answer.

Answer: Yes


Question:
Let $G \subset GL(n)$ be a Lie subgroup and denote \( M:=G x_0 = \{ Ax_0\ :\ A\in G\}\subset \mathbb R^n,\) where $x_0\ne 0$ is a fixed vector in $\mathbb R^n$. Then $M$ is a smooth submanifold of $\mathbb R^n$. Determine whether the following statement is true for all $A \in G$:
\[ A(T_{x_0} M) = T_{Ax_0} M. \]

Answer: True


Question: In triangle ABC, angle ABE is 50 degrees, angle DAC is 20 degrees, angle BAD is 60 degrees, and angle ACB is 20 degrees. It is also given that side CD is equal to side DA. Find the measure of angle ADE in degrees.

Answer: 30


Question:
Consider the manifold \(M=\{(\cos(\theta),\sin(\theta),z):\theta\in[0,2\pi],z\in[0,1]\}\). A bending of \(M\) is defined as a smooth map \(\Gamma:M\times [0,1]\rightarrow \mathbb{R}^3\) such that:
1) \(\Gamma[M\times\{t\}]\) is a submanifold with boundary of \(\mathbb{R}^3\) (denoted as \(M_t\)).
2) For every \(m\in M\), \(\Gamma(m,0)=m\).
3) For every \(t\in[0,1]\), the map \(m\rightarrow \Gamma(m,t)\) is an isometry between \(M\) and \(M_t\).

Must any bending of \(M\) leave the bases planar?

Answer: Yes


Question: Let $X$ be a connected oriented triangulation (polyhedron) space, homeomorphic to a geometric realization of an oriented simplicial complex $S$ with dimension $n$, where the boundary $\partial S$ of $S$ is 0. Is the homology group $H_n(X, \mathbb{Z})$ isomorphic to $\mathbb{Z}$?

Answer: \mathbb{Z}


Question: Determine the largest interval in which the initial value problem (IVP) has a differentiable solution, given that the solution is unique in the interval $(-2,2)$. You do not need to solve the equation.

Answer: (-2, 2)


Question:
Let $x$, $y$, and $z$ be positive integers satisfying the equations:
\[x^2 + y^2 + z^2 = 2(xy + 1)\]
and
\[x + y + z = 2022.\]
If $x_1$ and $x_2$ are two distinct solutions for $x$, find the value of $x_1 + x_2$. \(\text{(Express your answer as a single integer.)}\)

Answer: 2021


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{e^{\tan^3x}-e^{x^3}}{2\ln (1+x^3\sin^2x)} \]

Answer: \dfrac{1}{2}


Question:
Determine the minimum distance of the ternary linear code with the following parity check matrix:
\[
H=\begin{pmatrix}
2 & 0 & 1 & 1 & 1 \\
0 & 2 & 2 & 0 & 0 \\
0 & 0 & 2 & 2 & 1 \\
1 & 1 & 2 & 2 & 0
\end{pmatrix}.
\]

Answer: 4


Question: Let \( f(x) = \frac{\sin(nx)}{x} \varphi(x) \), where \( \varphi \in C^\infty_c(\mathbb{R}) \). Suppose that \( \text{Supp}(\varphi) \subset [-a,a] \) with \( a > 0 \). Is there a function \( g \in L^1(\mathbb{R}) \) such that \( |f(x)| \leq g(x) \) for all \( x \in \mathbb{R} \)? Justify your answer.

Answer: Yes


Question: Calculate \(442^{260} \mod{616}\) using Euler's theorem and the Chinese Remainder Theorem.

Answer: 232


Question: If $x$ is an even integer and $f(x) = x^2 + 1$ is composite, are all prime divisors of $f(x)$ congruent to $1 \pmod{4}$?

Answer: Yes


Question: Find the largest integer $k$ such that the function \( x \mapsto \sum_{n=1}^\infty \frac{\cos nx}{2^n} \) is in the differentiability class $C^k$ with respect to $x$.

Answer: \infty


Question: Let $G$ be a finite group with a normal subgroup $H$ such that $H \cong A_5$ and $G/H \cong A_5$. Is $G \cong A_5 \times A_5$?

Answer: A_5 \times A_5


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a function that is Riemann integrable on any bounded interval and satisfies \( \lim_{x \to \infty} f(x) = 0 \). Define \( g(x) = \int_{x}^{x+1} f(t) \, dt \). Prove that \( \lim_{x \to \infty} g(x) = 0 \).

Answer: 0


Question: Let \( \omega \) be a primitive third root of unity. Determine the number of distinct possible images of \( \omega \) under all field homomorphisms from \( \mathbb{Q}(\omega) \) to \( \mathbb{C} \).

Answer: 2


Question: Evaluate the limit: $$\lim_{x \to 0}\frac{\cos(\sin x)-\cos x}{x^4}$$

Answer: \dfrac{1}{6}


Question: Let \( Q \in \mathbb{Z}[x] \) be a polynomial that defines an injective function from \( \mathbb{Z} \) to \( \mathbb{Z} \). Determine whether there exists a prime \( p \) such that \( Q \) defines an injective function from \( \mathbb{Z}/p\mathbb{Z} \) to \( \mathbb{Z}/p\mathbb{Z} \).

Answer: Yes


Question: Let $M$ be an $n \times n$ matrix consisting entirely of 1's. Compute the operator norm $\|M\|_{op}$ and show that $\|M\|_{op} = n$.

Answer: n


Question:
Given the conditions:
\[ |a| = 1, \quad |b-2| = 3, \quad |c-5| = 6 \]
where \(a, b, c\) are complex numbers, find the maximum value of the expression:
\[ |2a - 3b - 4c| \]

Answer: 61


Question: Calculate the mass of a half disc of radius $R$ centered at the origin $(0,0)$, with $y>0$, given that the density is \( \rho = \frac{\rho_oy}{\sqrt{x^2+y^2}} \) for \( y>0 \), where \( \rho_o \) is a constant with units of \( \text{kg.m}^{-3} \).

Answer: \rho_o R^2


Question: If \(e^{xA}\) and \(e^{yB}\) commute for all \(x, y \in \mathbb{R}\), do the matrices \(A\) and \(B\) commute? Provide a proof or counterexample.

Answer: Yes


Question: Calculate the circulation of the vector field \( F = (5x - 5y)\hat{i} + (x + 2y)\hat{j} \) around the circle \( C \) of radius 7 centered at the origin in the \( yz \)-plane, oriented clockwise as viewed from the positive \( x \)-axis, using Stokes' Theorem.

Answer: 0


Question: Is the converse of Lewy's theorem true? Lewy's theorem states that if \( u = (u_1, u_2) : \subseteq \mathbb{R}^2 \to \mathbb{R}^2 \) is one-to-one and harmonic in a neighborhood \( U \) of the origin \((0,0)\in \mathbb{R}^2\), then the Jacobian \( J(x,y) = \left[\frac{\partial(u_1,u_2)}{\partial(x,y)}\right] \) does not vanish at the origin. Is it true that if the Jacobian of a harmonic function \( u \) does not vanish at a point, such as the origin, then \( u \) is always one-to-one in a neighborhood of the origin?

Answer: Yes


Question:
Find the equilibrium point of the system of differential equations given by:
\[ \dot{x}=1+x^{2}y-(1+A)x \]
\[ \dot{y}=Ax-yx^{2} \]
in terms of the parameter $A$. Express your answer as a point $(x, y)$.

Answer: (1, A)


Question:
Does there exist a function \( f: \mathbb{R} \to \mathbb{R} \) that satisfies the following properties?
1) \( f(\mathbb{R}) = \mathbb{R} \)
2) \( f \) is everywhere discontinuous
3) \( \mathbb{Q} \subseteq f(\mathbb{I}) \)
4) \( f(\mathbb{Q}) \subset \mathbb{I} \)

Answer: Yes


Question: Find the minimum value of \(|z_1 - z_2|\), where \(z_1\) and \(z_2\) are the solutions of the equation \(e^{ia}z^2 - (i + 3e^{3ia})z + 3ie^{2ia} = 0\), with \(a\) being a real parameter.

Answer: 2


Question: Evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \) where \( \mathbf{F}(x, y, z) = \langle z, x-z, y \rangle \) and \( S \) is the triangle with vertices \((1,0,0), (0,1,0), (0,0,1)\), oriented downwards.

Answer: -\dfrac{1}{3}


Question: Determine the threshold value \( c > 0 \) such that for the function \( f_a(x) = e^{-x^2/2}\cosh(ax) \), the maximum occurs at \( x = 0 \) for \( a \in (0,c) \) and at nonzero values \( \pm M_a \) for \( a \in (c,\infty) \).

Answer: 1


Question: Is there a vector space $X$ with two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ such that $(X,\|\cdot\|_1)$ is separable and $(X,\|\cdot\|_2)$ is not separable, considering the topology induced by each norm?

Answer: Yes


Question: Let $T$ be a full binary tree with depth $k$, with levels $L_0, \ldots, L_k$. The number of leaves is $n=2^k$. Let $A$ be the full automorphism group of $T$, and let $f$ be its faithful action on the leaves of the tree, i.e., on $L_k$. The group $f(A)$ is an iterated wreath product of $\mathbb{Z}_2$ with itself and has order $2^{n-1}$. It is the Sylow 2-subgroup of $S_n$. Determine the subgroups of $f(A)$ of index 2.

Answer: 2^k - 1


Question: Calculate the integral \( \int_1^\infty x^2 e^{1-x} \, dx \) using integration by parts. Verify that the correct value of the integral is 5.

Answer: 5


Question: Calculate the work done by the force field \( F(x,y) = -xi + 6yj \) along the path \( C: y = x^3 \) from the point \((0,0)\) to \((6,216)\).

Answer: 139950


Question: Evaluate the integral \( \int_{C}{e^{\frac{1}{z}} \, dz} \) over the contour \( C: |z|=1 \) using the Residue Theorem.

Answer: 2\pi i


Question: Evaluate the limit: \[ \lim \limits_{x \to \infty} \frac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}} \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{n\to \infty}{\frac{n}{(n_p)!}} \] where \((n_p)!\) is the product of the first \(n\) prime numbers.

Answer: 0


Question: Calculate the area bounded by the curve \((x^2+y^2)^2=a^2(x^2-y^2)\), where \(a\) is a real constant, using double integrals.

Answer: a^2


Question: Find the number of integer values of \(c\) such that the equation \(||20|x|-x^2|-c|=21\) has 12 distinct real solutions.

Answer: 57


Question: For a 3x3 matrix $A$ with $\text{Rank}(A) = 2$, determine the angle of rotation from $A$ to its transpose $A^T$. Assume that the elements of $A^T$ are the same as the elements of another matrix $B$. What is the angle of rotation from $A$ to $B$?

Answer: 90^\circ


Question: Compute the minimal number of generators, \( \mu(mA_m) \), of the module \( mA_m \), where \( m \) is the maximal ideal \( (\bar{x} - 1, \bar{y} - 1) \) of \( A = \mathbb{C}[x,y]/(x^3-y^2) \).

Answer: 1


Question: Let $E$ and $F$ be two disjoint closed sets in a metric space, with $E$ being compact. Prove that the distance $d(E, F) > 0$. Use the extreme value theorem in your proof.

Answer: d(E, F) > 0


Question:
Determine the critical exponent \( \lambda \) of the function \( f(x) = Ax^{1/2} + Bx^{1/4} + Cx \), where \( A, B, C \in \mathbb{R} \). The critical exponent is defined as:
\[
\lambda := \lim_{x \to 0} \frac{ \ln(|f(x)|) } { \ln(|x|) }
\]

Answer: \dfrac{1}{4}


Question:
Find the value of \( n \) that is the closest solution to the equation:
\[
\frac{(4.554 \times 10^{9})!(4.6 \times 10^9 - n)!}{(4.554 \times 10^{9} -n)!(4.6 \times 10^9)!} \approx 0.997
\]

Answer: 0


Question: Determine the radius of convergence for the power series \( \sum_{n = 0} ^ {\infty} \frac {1}{\pi (n)} x^n \), where \( \pi(n) \) is the number of prime numbers less than or equal to \( n \).

Answer: 1


Question: Calculate the flux of the vector field \( \mathbf{F}(x, y, z) = \langle e^{z^2}, 2y + \sin(x^2z), 4z + \sqrt{x^2 + 9y^2} \rangle \) through the surface \( S \) defined by \( x^2 + y^2 \leq z \leq 8 - x^2 - y^2 \).

Answer: 96\pi


Question: Given integers \(0 \le a_1 < a_2 < \dots < a_c\), determine a bound on the least modulus \(m\) such that for all \(i \neq j\), \(a_i \not\equiv a_j \pmod{m}\).

Answer: a_c - a_1 + 1


Question: Is the relation "$p$ determines $\tau$" equivalent to "$(p,p)\Vdash l(\tau)=r(\tau)$"?

Answer: Yes


Question: Given a matrix $A \in M_3(\mathbb{R})$ with eigenvalues $\{-1, 0, 1\}$, calculate $\det(I + A^{100})$. Provide your answer as a single number.

Answer: 4


Question: Solve for \(x\) in the equation: \[ \frac{72_8!}{18_2!} = 4^x \] where \(n_a!\) is defined as \(n(n-a)(n-2a)(n-3a)\ldots(n-ka)\) for \(n > ka\).

Answer: 9


Question: In a 4-dimensional space divided into 16 regions about the origin, what is the maximal number of these regions that a plane can intersect? For reference, in a 3-dimensional space divided into 8 regions, a plane can intersect up to 7 of these regions.

Answer: 11


Question:
Determine the largest open interval $I$ containing $x = 0$ on which there exists a unique solution $y(x)$ to the initial value problem:
\[ (\sin(x)-1)y''' + (x^2-x)y'' + \frac{1}{x-1}y' + x^5y = e^{x^3} \]
with initial conditions $y(0) = 1$, $y'(0) = 5$, and $y''(0) = 2$. What is the largest open interval $I$ containing $x = 0$ on which there exists a unique solution $y(x)$ to this problem?

Answer: \left( -\frac{3\pi}{2}, 1 \right)


Question: Evaluate the integral \( \int_{0}^{1} 4x^3 \cdot \left\{ \frac{d^2}{dx^2}(1-x^2)^5 \right\} \, dx \).

Answer: 2


Question: Calculate the expected value \( E[\exp(XY)] \) where \( X = Z + \epsilon_1 \) and \( Y = Z + \epsilon_2 \). Here, \( Z, \epsilon_1, \epsilon_2 \sim N(0,1) \) are independent and identically distributed normal random variables.

Answer: \infty


Question: Is the set of states $S(A)$ weak${}^*$ compact if $A$ is a non-zero, non-unital $C^*$-algebra?

Answer: No


Question: Let $\mathscr{H}$ be a Hilbert space and $f: \mathscr{H} \to \mathbb{R}$ be a convex function, meaning for all $x, y \in \mathscr{H}$ and $\alpha \in [0,1]$, \[ f(\alpha x +(1-\alpha)y) \leq \alpha f(x) + (1-\alpha)f(y). \] Define the set $C = \{x \in \mathscr{H} : f(x) \leq 1\}$. Is the set $C$ convex? In other words, is it true that $\alpha x + (1-\alpha)y \in C$ for all $x, y \in C$ and $0 \leq \alpha \leq 1$?

Answer: Yes


Question: Find the value of $c$ that minimizes the sum of the areas of two regions bounded by the curve $f(x) = x^3 - x^2$ and the line $y = c + x$. Assume $c$ is such that exactly two of these regions have finite area.

Answer: -\dfrac{11}{27}


Question: Determine whether the following statement is true or false: If \( \lim \limits_{n\to \infty}\ (a_{2n}-a_n)=0 \), then \( \lim \limits_{n\to \infty}\ a_n \) exists.

Answer: False


Question: A miner is trapped in a mine with 3 doors. The first door leads to a tunnel that takes him to safety after 3 hours. The second door leads to a tunnel that returns him to the mine after 5 hours. The third door leads to a tunnel that returns him to the mine after 7 hours. Assuming the miner is equally likely to choose any door at any time, calculate the expected length of time until he reaches safety.

Answer: 15


Question: Suppose $f(x)$ is a polynomial of degree 5 with leading coefficient 1. Given that $f(1)=1$, $f(2)=3$, $f(3)=5$, $f(4)=7$, and $f(5)=9$, find the value of $f(6)$. \( \boxed{} \)

Answer: 131


Question: Given an ordered multiset, such as \(\{1,2,2,3,3,3,4,4,4,4\}\), determine the expected number of matching elements under a random permutation. Specifically, calculate the expected number of times an element in the permutation matches the original element in the same position. Assume the multiset contains \(n\) unique values with multiplicities \(m_i\), where \(i \in \{1,2,\ldots,n\}\) and \(\sum_i m_i = N\).

Answer: \dfrac{\sum_{i=1}^{n} m_i^2}{N}


Question:
Consider the sequence of functions \((f_n)\) defined by:
\[
f_{0}(x)=\frac{1}{(1+x)^3}, \quad f_{n}(x)=\int^{x}_{0}f_{n-1}(y)\,dy, \quad n\ge1.
\]
Find \(\lim_{n\to \infty}f_{n}(1)\).

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{\pi\sin{x}-\sin{\pi x}}{x(\cos{x}-\cos(\pi x))}$$

Answer: \dfrac{\pi}{3}


Question: Evaluate the product \( \prod_{k=0}^{2^{1999}}\left(4\sin^2\left(\frac{k\pi}{2^{2000}}\right)-3\right) \).

Answer: 3


Question: Consider a holomorphic function $f: \mathbb{C}^n \to \mathbb{C}$ that vanishes on a set $E \subset \mathbb{R}^n \subset \mathbb{C}^n$ with positive Lebesgue measure in $\mathbb{R}^n$. Is it true that $f$ must be identically zero?

Answer: Yes


Question: Find the area of the intersection between the sphere \(x^2 + y^2 + z^2 = 1\) and the cylinder \(x^2 + y^2 - y = 0\).

Answer: 2\pi - 4


Question: Evaluate the integral \( \int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} \, dx \) and verify that it equals \( 8\pi \ln 2 \).

Answer: 8\pi \ln 2


Question: Evaluate the integral of the function \( f(z) = \tan z \) around a rectangular contour in the complex plane. The rectangle extends from \( 0 \) to \( 2\pi \) in the real direction and from \( -0.5i \) to \( 0.5i \) in the imaginary direction. Determine the value of this integral.

Answer: -4\pi i


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

Answer: 0


Question: Find the equation of the plane tangent to the surface given by \( z = \sqrt{2x^2+y^4+1} \) that is parallel to the plane \( x-y-z=0 \).

Answer: x - y - z = 0


Question: Determine the number of solutions in the third quadrant of the complex plane for the equation \(z^{100} = 1+i\).

Answer: 25


Question: In the ring of integers $\mathcal{O}_K = \mathbb{Z}[\sqrt{15}]$ of the number field $K = \mathbb{Q}(\sqrt{15})$, does there exist an ideal $\mathfrak{a}$ such that the norm $N(\mathfrak{a}) = 13$?

Answer: No


Question: Find all functions \( f:(0,\infty)\rightarrow(0,\infty) \) that satisfy the following conditions: \( f(f(f(x))) + 2x = f(3x) \) for all \( x > 0 \) and \( \lim_{x\to\infty}(f(x)-x)=0 \).

Answer: f(x) = x


Question: Determine whether the estimator \( Y = 2\overline{X} \) is strongly consistent for the parameter \( \theta \) when \( X_1, \ldots, X_n \) are independent and identically distributed random variables from a uniform distribution on \([0, \theta]\).

Answer: Yes


Question: Evaluate the line integral \( \oint_C x^4 \, ds \), where \( C \) is the curve of intersection of the sphere \( x^2 + y^2 + z^2 = 1 \) and the plane \( x + y + z = 0 \).

Answer: \dfrac{\pi}{3}


Question: Evaluate the limit using polar coordinates: \[ \lim_{(x,y)\to(1,0)} \frac{y^2\log(x)}{(x-1)^2+y^2}=0. \]

Answer: 0


Question: Let \(x\) be a real number such that \(\frac{(1+x)^3}{1+x^3} = \frac{9}{13}\). If \(\frac{(1+x)^5}{1+x^5} = \frac{a}{b}\) where \(a\) and \(b\) are positive integers, find the minimal value of \(a+b\).

Answer: 2078


Question: Find the values of \( \lambda \) for which the system of equations \( \sin x \cos y = 2\lambda - 4 \) and \( \sin y \cos x = 1- \lambda \) has solutions.

Answer: 2


Question:
Evaluate the integral \( \int_{|z| = 2} \frac{e^{2z}}{(z-2)^4} \, dz \). Choose the correct answer from the following options:

A. \( \frac{8\pi i e^4}{3} \)

B. \( \frac{\pi i e^4}{3} \)

C. \( \frac{\pi i e^4}{3!} \)

D. \( \frac{8\pi i e^4}{3!} \)

Answer: A


Question: Let \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL(2, \mathbb{R}) \) with \( \text{trace}(A) = 0 \). Determine if \( A \) is conjugate to \( \begin{pmatrix} 0 & t \\ -t^{-1} & 0 \end{pmatrix} \).

Answer: Yes


Question: Determine the cardinality of the set of continuous functions \( f : [0,1] \rightarrow \mathbb{R}^2 \) such that \( f(0) = a \) and \( f(1) = b \), where \( a \) and \( b \) are distinct points in \( \mathbb{R}^2 \). Express your answer in terms of beth numbers.

Answer: \beth_1


Question: Evaluate the line integral \( \int_C x \, ds \) where the curve \( C \) is parametrized by \( x = \frac{3}{4} \sin(2t) \), \( y = \cos^3 t \), and \( z = \sin^3 t \).

Answer: 0


Question: Evaluate the integral: $$\int_0^{\infty } {\frac{(x^7)(1-x^{12})}{(1+x)^{28}}}dx$$

Answer: 0


Question: Determine the number of non-isomorphic Abelian groups of order \(2^7 \times 3^4 \times 5^2\).

Answer: 150


Question: Let $A = [0, 1] \times [0, 1]$. Let $h$ be a continuous function on $\mathbb{R}$, and let $\phi$ be defined by \( \phi(x, y) = (x + h(x+y), y - h(x+y)) \). Determine the area of $\phi(A)$.

Answer: 1


Question: Find the maximum and minimum values of the expression \(a^2\sin^2\theta + b^2\csc^2\theta\) for real numbers \(a\) and \(b\).

Answer: 2|ab|


Question: Consider a matrix $A$ and two left inverses $L$ and $L'$ of $A$. If $LB = L'B$ for some matrix $B$, does it imply that $X = LB$ is a solution to the equation $AX = B$? Justify your answer.

Answer: Yes


Question: Evaluate the double integral \( \iint dydx \) over the domain defined by the conditions \( 0 \leq r \leq 1 \) and \( \frac{\pi}{3} \leq \theta \leq \frac{2\pi}{3} \) without using polar coordinates.

Answer: \dfrac{\pi}{6}


Question: Evaluate the integral: $$\int_0^1 \sqrt{\frac{1-x^2}{x+2x^3+x^5}}\,dx$$ using Euler's integral.

Answer: \dfrac{\pi}{2}


Question: Find the derivative of the trace of the product of a matrix and its transpose, \( \frac{d}{dX} \text{Trace} (XX^T) \).

Answer: 2X


Question: Compute the line integral of the function \( f(x,y) = xy \) over the ellipse \( \frac{x^2}{4} + y^2 = 1 \) oriented counterclockwise.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{1}{x^5}\int_0^{x} \frac{t^3\ln(1-t)}{t^4 + 4}\,dt$$

Answer: -\dfrac{1}{20}


Question: Calculate the surface integral \( \int\int_{S}\frac{1}{\sqrt{1+4z}}dS \), where \( S \) is the part of the paraboloid \( z=x^{2}+y^{2} \) that lies below the plane \( z=3-2y \).

Answer: 4\pi


Question: Evaluate the limit: \[ \lim_{x\to 0}{\frac{(\sin x)^{13} - \ln(1 + (\sin x) ^{13} )} {(\tan x) ^{26} }} \]

Answer: \dfrac{1}{2}


Question: Find the limit: \[ \lim_{x \to 0} \left( \frac{\sin(x)}{8} \right)^{\frac{1}{x^2}} \]

Answer: 0


Question: Evaluate the integral \( \iiint_E 1 \, dx \, dy \, dz \) where \( E = \{(x,y,z) \mid |x| + |y| + |z| \leq 1\} \).

Answer: \dfrac{4}{3}


Question: Determine whether the following statement is true or false: If $a$ is algebraic over a field $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$. Provide a proof or a counterexample.

Answer: True


Question: If $R$ is a field and $f(x) \in R[X]$ is an irreducible polynomial, is $(f(x))$ a principal ideal in $R[X]$?

Answer: Yes


Question: Compute the line integral of the vector field \( \vec{F}(\vec{x}) = \begin{pmatrix} x_1^2 + 2x_3 \\ x_1x_2 \\ x_3^2 - 2x_1 \end{pmatrix} \) along the straight line path \( C \) from the origin to the point \( P(1, 2, 3) \).

Answer: \dfrac{32}{3}


Question: Find the closed form of the sum \( S = \sum_{n=1}^{99} \frac{5^{100}}{25^n + 5^{100}} \).

Answer: \dfrac{99}{2}


Question: Evaluate the limit: \[ \lim_{n\to\infty} n^2 \int_{0}^{1} \frac{x\sin{x}}{1+(nx)^3} \, \mathrm{d}x \]

Answer: 0


Question: In how many ways can 10 people be seated on 24 seats placed in a row, such that in each pair of seats equidistant from the beginning and end, at least one seat is empty?

Answer: 245248819200


Question: Find the coefficient of $x^6$ in the expansion of the expression $(1+x)(1+x^2)^2\cdots(1+x^n)^n$.

Answer: 28


Question: Find all constants \( K > 0 \) such that for any positive measure space \((X, \Sigma, \mu)\) and any \(\mu\)-integrable function \(f: X \to \mathbb{R}\) satisfying \(\left|\int_E f\,d\mu\right| < K\) for all \(E \in \Sigma\), it follows that \(\|f\|_1 < 1\).

Answer: \dfrac{1}{2}


Question: Use Lagrange multipliers to find the maximum value of the function \( f(x,y,z) = 13x^4 + 13y^4 + 13z^4 \) subject to the constraint \( 13x^2 + 13y^2 + 13z^2 = 3 \).

Answer: \dfrac{9}{13}


Question: A bag contains 100 uniquely labeled marbles. You draw 9 marbles at random from the bag and then return them. This process is repeated 50 times. What is the probability that you have selected less than 70% of all the marbles at least once during these 50 draws?

Answer: 0


Question: Can every closed and bounded set $S \subseteq [0,1]$ be expressed as a countable union of disjoint intervals, given that $S$ has the property that if $x_1, x_2, \dots$ all belong to $S$ and converge to $x$, then $x$ also belongs to $S$?

Answer: No


Question: Find all functions \(f(x)\) such that \(D^*f = Df\), where \(D^*f(x) = \lim_{h \to 0} \frac{f^2(x + h) - f^2(x)}{h}\).

Answer: f(x) = C


Question: If $A$ is a Cohen Macaulay local ring and $B$ is a quotient ring of $A$ that is also Cohen Macaulay, is $B$ always a quotient by a regular sequence of $A$? Answer yes or no.

Answer: No


Question: In triangle $ABC$, it is given that $\angle A = 2\angle C$ and $IG \parallel AC$, where $I$ is the incenter and $G$ is the centroid. If $IG = x$, calculate the distance $ID$, given that $BD$ is an interior angle bisector. Express your answer in terms of $x$.

Answer: 3x\sqrt{2}


Question: Determine whether it is true that any fraction \( \frac{a}{b} \) can be expressed in infinitely many ways as a sum of distinct Egyptian fractions.

Answer: True


Question: For what integer values of $d$ does the twisted tangent bundle $E = T\mathbb{P}^n \otimes \mathcal{O}_X(d-1)$ have a globally nonvanishing section, where $X = \mathbb{P}^n$ is the projective space over an algebraically closed field?

Answer: -1


Question: If $G$ is a group and $H$ is a proper subgroup of $G$, is $H$ a proper subgroup of the quotient group $G/H$?

Answer: No


Question:
Given a sphere with the parameterization \(X(\theta,\phi)=(r \sin\phi \cos\theta,r \sin\phi \sin\theta, r \cos \phi)\), find the Christoffel symbols for the metric:

\[
(g_{ij})=\begin{pmatrix} r^2\sin^2\phi& 0\\ 0& r^2 \end{pmatrix}
\]

Calculate the Christoffel symbol \(\Gamma^1_{22}\).

Answer: 0


Question: Given that \(x^4 + x^2 = \frac{11}{5}\), find the value of \(\sqrt[3]{\frac{x+1}{x-1}} + \sqrt[3]{\frac{x-1}{x+1}}\).

Answer: 2 + \sqrt{5}


Question: Given the equations \(2a^2 + 2007a + 3 = 0\) and \(3b^2 + 2007b + 2 = 0\), where \(ab \neq 1\), find the value of \(\frac{a}{b}\) without solving for the roots of the equations.

Answer: \dfrac{3}{2}


Question: Estimate the improper integral \( \int_{\gamma}\dfrac{e^{i\,t\,z}}{z^2+1} \, dz \), where \( \gamma \) is a semicircle in the complex plane with the parametrization \( \gamma(t) = Re^{i\varphi} \) for \( \varphi \in [0,\pi] \), and \( R \to \infty \). Show that the integral tends to zero as \( R \to \infty \).

Answer: 0


Question: Given the affine variety $X = V(x^{2}+y^{2}-1, x-1)$, determine the ideal $I(V)$. Verify if $I(V) = (x-1, y^2)$ is correct.

Answer: (x - 1, y)


Question: Does there exist an uncountable group such that every proper subgroup is countable?

Answer: Yes


Question: Suppose $\{x_n\}_{n=1}^{\infty}$ and $\{y_n\}_{n=1}^{\infty}$ are two distinct sequences of positive numbers, each of which sums to 1. Determine whether it is necessarily true that $\sum_{n=1}^{\infty}\sqrt{x_ny_n}<1$. Provide a justification for your answer.

Answer: Yes


Question: Consider a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a reverse filtration $(\mathcal{F}_1, \mathcal{F}_2, \ldots)$ where $\mathcal{F}_{n+1} \subseteq \mathcal{F}_n$ for each $n$. For each $n$, let $X_n : (\Omega, \mathcal{F}_n) \rightarrow (\overline{\mathbb{R}}, \mathcal{B}(\overline{\mathbb{R}}))$ be an extended random variable, and define $X_{\infty} = \lim_{n \rightarrow \infty} X_n$. Define $\mathcal{F}_{\infty} = \bigcap_{n=1}^{\infty} \mathcal{F}_n$. Is $X_{\infty}$ measurable with respect to $\mathcal{F}_{\infty}$, i.e., is $X_{\infty} : (\Omega, \mathcal{F}_{\infty}) \rightarrow (\overline{\mathbb{R}}, \mathcal{B}(\overline{\mathbb{R}}))$?

Answer: Yes


Question: Let $(X_n)_{n\ge 1}$ be a sequence of dependent nonnegative random variables, where each $X_n$ has a density with respect to the Lebesgue measure on $[0,n]$ and satisfies $\mathbb{E}(X_n) < \infty$. Suppose $X_n$ converges weakly to a random variable $X$, which has a density with respect to the Lebesgue measure on $[0,\infty]$ and satisfies $\mathbb{E}(X) < \infty$. Determine whether $X_n/\log(n)$ converges almost surely to 0.

Answer: No


Question: Find the correct value of the integral: $$\int_0^{\pi/2}\frac{\cosh[a \cos x]\cos[a\sin x]}{\cosh[2a\cos x]+\cos[a\sin x]}dx.$$

Answer: \dfrac{\pi}{4}


Question: Solve for $x$ in the equation: $$x^{x^x} = 2^{-rac{1}{	ext{sqrt}(2)}}$$

Answer: \dfrac{1}{2}


Question: Solve for the time \( t \) in the equation \( 318.45 = 288.45e^{-t/4.5} + 44.1t \) without using graphing methods.

Answer: 5.13


Question: If $f \circ g$ is injective and $g$ is surjective, is $f$ necessarily injective? Justify your answer.

Answer: Yes


Question: Given that \( \lim_{n\to\infty}(p_{n+1}/p_{n})=1 \), where \( p_k \) denotes the sequence of prime numbers, determine if it follows that \( \lim_{n\to\infty}\log(p_{n+1})/\log(p_{n})=1 \), where \( \log \) is the natural logarithm.

Answer: 1


Question: Let \( f: \mathbb{R}^+ \rightarrow \mathbb{R} \) be a differentiable function satisfying the equation \( f(x) = e + (1-x)(\ln(x)-1) + \int_{1}^{x} f(t) \, dt \) for all \( x \in \mathbb{R}^+ \). Find the value of \( e - \int_{0}^{1} f(t) \, dt \).

Answer: 2


Question: Given a random variable $X$ that follows a Poisson distribution with parameter $\lambda$, calculate the expectation $\mathbb{E}[X(X-1)(X-2)(X-3)]$. Provide your answer in terms of $\lambda$. \(X \sim Poi(\lambda)\)

Answer: \lambda^4


Question: Is it true that if \((R, m)\) is an infinite local ring that is not a division ring, then \(R\) and \(m\) have the same cardinality? Exclude the cases when \(R\) is finite or when \(R\) is a division ring.

Answer: Yes


Question: Find a function \( \beta \) such that \( \beta \circ f \), where \( f \) is the Dirichlet function, equals the characteristic function of the rationals, \( x_\mathbb{Q} \).

Answer: \beta(y) = y


Question: Given the function \( f(x) = 2x^{3} - 9ax^{2} + 12a^{2}x + 1 \), find the value of the parameter \( a \) such that if \( x_1 \) is the x-coordinate of the maximum point and \( x_2 \) is the x-coordinate of the minimum point, then \( x_1^{2} = x_2 \).

Answer: 2


Question: Consider the function \( f(x) = 2x + \ln{x} \) for \( x > 0 \), and let \( g = f^{-1} \). Find the slope of the tangent line to the graph of \( g \) at the point \( (2, g(2)) \).

Answer: \dfrac{1}{3}


Question: Let $R$ be a commutative ring with identity, and let $P$ be a prime principal ideal of $R$. Suppose there exists a proper principal ideal $I$ of $R$ such that $R \supsetneq I \supsetneq P$. Determine if $P^2 = P$.

Answer: P^2 = P


Question: Let $E$ be a normed $\mathbb{R}$-vector space and $\lambda$ be a measure on $\mathcal{B}(E)$. Given that \( \int \min(1,\|x\|_E)\,\lambda(\mathrm{d}x) < \infty \), does it follow that \( \int \left| e^{\mathrm{i}\langle x, x'\rangle} - 1 - \mathrm{i}\langle x, x'\rangle 1_{\overline{B}_1(0)}(x) \right| \, \lambda(\mathrm{d}x) < \infty \) for all $x' \in E'$?

Answer: Yes


Question:
Find the value of the infinite sum \( \sum_{n=1}^{\infty} a_n \), where the sequence \( a_n \) is defined as follows:

\[ a_1 = 1, \quad a_2 = 2^{-1}, \quad a_n = n^{-\sum_{k=1}^{n-1}a_k} \text{ for } n \geq 3. \]

Answer: 2


Question: Verify the correctness of the equation: \( \sum_{k=0}^{\infty}\sum_{m=0}^{\infty}S(m,k)\frac{1}{k!}=e^{e-1} \), where \( S(m,k) \) is the number of partitions of a \( k \) element set into \( m \) nonempty parts.

Answer: e^{e-1}


Question: Determine the values of the parameter \( p \) for which the series \( \sum_{n=1}^\infty \sqrt{n} \ln^{p} \left(1+ \frac{1}{\sqrt{n}}\right) \) converges.

Answer: p > 3


Question:
Find the derivative of the function \( f(x) = \frac{x^2}{2} \cdot (I[x \ge 0] - I[x < 0]) \) using the limit definition:
\[ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

Answer: |x|


Question: An abelian group $G$ is generated by elements $x$ and $y$ such that $O(x) = 16$, $O(y) = 24$, and $x^2 = y^3$. Determine the order of the group $G$. \( \boxed{} \)

Answer: 48


Question:
Find the largest non-negative value of \(a\) such that the following limit evaluates to \(\frac{1}{4}\):
\[
\lim\limits_{x \to 1} \left(\frac{-ax + \sin(x-1)+a}{x+\sin(x-1)-1}\right)^\frac{1-x}{1-\sqrt{x}}.
\]

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{x \sqrt{1 + \sin x} - \ln{(\sqrt{1 + x^2} - x)}}{\tan^3{x}} \] using the Maclaurin series.

Answer: \infty


Question: Use Fermat's Little Theorem to find the remainder of the expression \( x \uparrow \uparrow k \mod m \), where \( x \) is a prime number, \( k \) is the number of times \( x \) is repeated in the power tower, and \( m \) is any number. For example, if \( x=5 \), \( k=3 \), and \( m=3 \), find the remainder of \( 5^{5^5} \mod 3 \).

Answer: 2


Question: Real numbers \(x, y\) satisfy \(x^2 + y^2 = 1\). If the minimum and maximum values of the expression \(z = \frac{4-y}{7-x}\) are \(m\) and \(M\) respectively, find \(2M + 6m\).

Answer: 4


Question:
Consider the functions $f_1(x) = \sum_{n=1}^{\infty} \frac{x \sin(n^2 x)}{n^2}$ and $f_2(x) = \sum_{n=1}^{\infty} x^2 (1-x^2)^{n-1}$ defined on the interval $[0,1]$. Determine which of the following statements is true:

(a) $f_1$ is continuous but $f_2$ is not continuous.
(b) $f_2$ is continuous but $f_1$ is not continuous.
(c) Both $f_1$ and $f_2$ are continuous.
(d) Neither $f_1$ nor $f_2$ is continuous.

Answer: A


Question: Evaluate the line integral \( \int_C F \cdot ds \) where \( F = \left( \frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2} \right) \) and \( C \) is the closed curve formed by the parabolas \( x = y^2 + 1 \) and \( y = (x-3)^2 \), along with the x-axis, joining the points \((1,0)\) and \((3,0)\).

Answer: 0


Question: Find the modular multiplicative inverse of 13 modulo 27.

Answer: 25


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

Answer: \sqrt{e}


Question: Let $V$ be a normed space with two equivalent norms, $\|\cdot\|_1$ and $\|\cdot\|_2$. If $A$ is a closed subspace of $V$ with respect to $\|\cdot\|_1$, is $A$ also a closed subspace of $V$ with respect to $\|\cdot\|_2$? Provide a proof or counterexample.

Answer: Yes


Question: Evaluate the double integral of $x$ over the region $S$, where $S$ is the area bounded by $x^2 + y^2 \leq 2$ and $x \geq 1$. Use the substitution $x = r\cos\theta$, $y = r\sin\theta$. Express your answer as a single numerical value.

Answer: \dfrac{2}{3}


Question: Find the equation of the image of the line $x+y=1$ under the Möbius transformation $w=\frac{z+1}{z-1}$.

Answer: u - v = 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1 + \sqrt{2} + \sqrt{3} + \sqrt{4} + \ldots + \sqrt{n}}{n^{3/2}}. \]

Answer: \dfrac{2}{3}


Question: Find the limit of the sequence \( p_n = \frac{1}{2^n} \sum_{j=0}^{\frac{a\sqrt{n}-1}{2}}\binom{n}{j} \), where \( a \) is a natural number. Show that \( \lim_{n \to \infty} p_n = 0 \).

Answer: 0


Question:
Find the sum of all solutions to the equation:
\[
(\log_2 x)(\log_3 x)(\log_4 x)(\log_5 x) = (\log_2 x)(\log_3 x)(\log_4 x) + (\log_2 x)(\log_3 x)(\log_5 x) + (\log_2 x)(\log_4 x)(\log_5 x) + (\log_3 x)(\log_4 x)(\log_5 x).
\]

Answer: 121


Question: Approximate \( \arctan\frac{8}{9} \) using a first-order Taylor expansion around \( \arctan 1 \). How many digits of accuracy can you guarantee in your approximation?

Answer: 2


Question:
Does there exist a real continuous function $f(x)$ defined on $\mathbb{R}$ such that:
\[\int_{-\infty}^{\infty} \lvert f(x)\lvert \, dx < \infty\]
and
\[\int_{-\infty}^{\infty} \lvert f(x)\lvert^2 \, dx = \infty\]?

Answer: Yes


Question: Find the maximum value of the function \( f(x) = \int_0^x \sqrt{(x^2-x)^2+y^4} \, dy \) for \( x \in [0,1] \).

Answer: \dfrac{1}{3}


Question: Calculate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \), where \( S \) is the boundary of the solid \( E \) enclosed by the paraboloids \( z = x^2 + y^2 \) and \( z = 12 - 2x^2 - 2y^2 \), and \( \mathbf{F}(x, y, z) = \langle x^3 + y^2, 2yz + e^z, y^2 - z^2 \rangle \). Use the Divergence Theorem to find the value of the integral.

Answer: 48\pi


Question: Let \( P(x) = (x-1)(x-2) \cdots (x-50) \) and \( Q(x) = (x+1)(x+2) \cdots (x+50) \). If \( P(x)Q(x) = a_{100}x^{100} + a_{99}x^{99} + \cdots + a_{1}x^{1} + a_0 \), compute \( a_{100} - a_{99} - a_{98} - a_{97} \).

Answer: 42926


Question: Let $U$ be a unipotent upper triangular group over a local field $K$ of characteristic zero. Is there a right translation invariant metric on $U$ such that any ball of finite radius is relatively compact?

Answer: Yes


Question: Determine the number of roots of the polynomial \( p(z) = z^8 + 3z^7 + 6z^2 + 1 \) that lie inside the annulus \( 1 < |z| < 2 \).

Answer: 5


Question: For what positive integer $A$ is the sequence $(a_k)_{k=0}^\infty$, defined by $a_0 = A$ and $a_{k+1} = a_k + S(a_k)$ for $k \ge 0$, eventually constant? Here, $S(n) = n - m^2$, where $m$ is the greatest integer such that $m^2 \le n$. Determine the positive integer $A$ for which the sequence is eventually constant.

Answer: 1


Question: Determine the limit \( \lim_{n \to \infty} n\sin(2^{-n}) \).

Answer: 0


Question: Let $A$ be a finite Artinian ring. For an element $x$ in $A$, is it true that the cardinality of the right annihilator $\text{Ann}_r(x)$ equals the cardinality of the left annihilator $\text{Ann}_l(x)$?

Answer: No


Question: Find the maximum distance from the origin of a point on the curve defined by the parametric equations \(x = a\sin t - b\sin\left(\frac{at}{b}\right)\) and \(y = a\cos t - b\cos\left(\frac{at}{b}\right)\), where \(a, b > 0\).

Answer: a + b


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

Answer: e^2


Question: Does the Baumslag Solitar group $B(2,3)$ contain a non-trivial element with arbitrary roots? The group $B(2,3)$ is defined by the presentation $\langle a,b \mid b a^3 b^{-1} = a^2 \rangle$. An element $g$ in a group $G$ has arbitrary roots if for every $n \geq 1$, there exists an element $x$ in $G$ such that $g = x^n$. Provide a justification for your answer.

Answer: No


Question: Determine the number of solutions to the congruence equation \(x^2 \equiv 121 \pmod{1800}\).

Answer: 16


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

Answer: 0


Question: Calculate \( \ln(i^i) \).

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


Question:
Determine the number of ways to distribute 80 candies to 3 kids such that:
- The first kid receives any number of candies (including 0).
- The second kid receives an even positive number of candies.
- The third kid receives 0, 2, or 5 candies.
- All candies are distributed.

Answer: 116


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0}\int_{\sin x}^{x}\dfrac{1}{t^3(1+t^{1/2})}\,dt. \]

Answer: \dfrac{1}{6}


Question: Evaluate the integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}{\frac{\sin^{2014}x}{\sin^{2014}x + \cos^{2014}x}}dx \).

Answer: \dfrac{\pi}{2}


Question: Consider the c.e. set $W_e$, which is the domain of the partial computable function $\varphi_e$. Define the equivalence relation $\sim$ such that $x \sim y$ if and only if $\varphi_x = \varphi_y$. Does the infiniteness of $W_e$ imply that $W_e/\sim$ is infinite?

Answer: No


Question: Determine whether the following statement is true for any finite field extension $L/K$: \[ \min_{\substack{\{\alpha_1, \ldots, \alpha_n\} \\ L = K(\alpha_1, \ldots, \alpha_n)}} \left(\prod_{i=1}^n [K(\alpha_i): K] \right) = [L:K]. \]

Answer: True


Question: A coin is tossed until two consecutive heads appear. The probability of the coin landing heads, denoted by \( p \), follows a beta distribution with parameters \( p \sim \beta(6, 8) \). What is the expected number of coin tosses required to achieve two consecutive heads?

Answer: \dfrac{52}{5}


Question: Solve the differential equation \( \frac{dy}{dx} - \frac{1}{x}y = \frac{1}{x^3}y^2 \) with the initial condition \( y(1) = 1 \). Find the correct expression for \( y \).

Answer: x^2


Question:
Let $S: \mathbb{R}^3 \to \mathbb{R}^4$ and $T: \mathbb{R}^4 \to \mathbb{R}^3$ be linear transformations such that $T \circ S$ is the identity map on $\mathbb{R}^3$. Determine which of the following statements is correct about the composition $S \circ T$:

1) $S \circ T$ is the identity map on $\mathbb{R}^4$.
2) $S \circ T$ is one-to-one, but not onto.
3) $S \circ T$ is onto, but not one-to-one.
4) $S \circ T$ is neither one-to-one nor onto.

Answer: D


Question: Let \( F \) be a linear functional over \( \mathbb{R}^3 \) such that \( F(1,0,0) = 2 \), \( F(0,1,0) = 1 \), and \( F(0,0,1) = -1 \). Find an orthogonal set to \( \ker(F) \) and use it to calculate the Riesz representative of \( F \).

Answer: (2, 1, -1)


Question: Let $T: H \to H$ be a right shift operator defined by $Te_n = e_{n+1}$ for $n = 1, 2, \ldots$, where $(e_n)$ is a complete orthonormal sequence in a separable Hilbert space $H$. Find the Hilbert adjoint operator $T^*$ of $T$. Specifically, determine the action of $T^*$ on the basis elements $e_n$, including $T^*e_1$. What is $T^*e_1$?

Answer: 0


Question: Let \( f \) be an analytic function on the domain \( \{ z \in \mathbb{C} : |z| < 2 \} \) such that \( f(1) = 0 \) and \( f'(1) \neq 0 \). Define \( u = \text{Re}(f) \) and \( v = \text{Im}(f) \). Assume that in a neighborhood of 1, \( u(z) = 0 \) defines a smooth path \( \gamma_0 \) and \( v(z) = 0 \) defines a smooth path \( \gamma_1 \). Calculate the angle between the paths \( \gamma_0 \) and \( \gamma_1 \) at the point 1.

Answer: \dfrac{\pi}{2}


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

Answer: \dfrac{1}{\pi}


Question:
Determine the value to which the Fourier series of the function \( F(x) \) converges at the point \( x = \pi \), where \( F(x) \) is defined as:
\[
F(x)=\left\{
\begin{array}{cl}
-1 & \text{for}~-\pi\leq x\leq 0\\
1 & \text{for}~0\leq x\leq \pi
\end{array}\right..
\]

Answer: 0


Question:
Given the differential equation:
\[ y'(t) = a \cdot x'(t) + \frac{1}{b} \cdot x(t) + c \cdot x''(t) \]
with initial conditions \( x(f) = g \) and \( x(h) = k \), find the value of:
\[ P = a \lim_{n \to \infty} \frac{1}{n} \int_0^n x^2(t) \, \text{d}t \]
for the solution:
\[ x(t) = \exp(-\alpha t) \cdot \{K_1 + K_2 \cdot \exp(\beta t)\} \]
where:
\[ \alpha = \frac{a + \frac{\sqrt{ba^2 - 4c}}{\sqrt{b}}}{2c} \]
\[ \beta = \frac{\sqrt{ba^2 - 4c}}{c \cdot \sqrt{b}} \]

Answer: 0


Question: Find integer solutions to the equation $x^6 + 5y^2 = z^2$ where $x \neq 1$. Provide an example or prove whether such solutions exist.

Answer: (2, 4, 12)


Question: In an acute triangle $\triangle ABC$, the segment $CD$ is an altitude and $H$ is the orthocenter. The circumcenter of the triangle lies on the line containing the bisector of $\angle DHB$. Determine all possible values of $\angle CAB$. \(\text{(Express your answer as a set of angles)}\)

Answer: 60^\circ


Question: Calculate the conditional mean $E(X \mid X < Y)$ given the joint probability density function $f(x,y) = 1$ for $0 < x < 1$ and $0 < y < 1$. Assume $X$ and $Y$ are continuous random variables.

Answer: \dfrac{1}{3}


Question: Students p, q, r, s, and t are members of three committees A, B, and C. Committee A includes members r and s, committee B includes members p, r, and t, and committee C includes members p, q, and t. Determine if it is possible to select a distinct student representative for each committee using the Max-flow-Min-cut theorem.

Answer: Yes


Question: Find the real number \( x \) represented by the continued fraction \([12;2,2,12,2,2,12,2,2,12\dots]\).

Answer: 6 + \sqrt{41}


Question: Evaluate the limit \( \lim_{n \to \infty} \left(1 - \frac{c \ln(n)}{n} \right)^n \) where \( c \neq 1 \) is a positive real number.

Answer: 0


Question: Evaluate the contour integral \(\oint_{|z| = 3} \frac{z^3e^{1/z}}{z^3+1}dz\) using the Residue Theorem.

Answer: 2\pi i


Question: Given that $i$ is an annual effective interest rate, $1$ accumulates to $3$ in $r$ years, $2$ accumulates to $7$ in $s$ years, and $3$ accumulates to $16$ in $t$ years. Determine the value of $10(1+i)^{(2r+s-t)}$. Provide your answer as a numerical value.

Answer: \dfrac{945}{16}


Question: Determine the value of the real number \(a\) for which the integral \(\int_0^{\infty} \frac{1-\cos x}{x^a} \, dx\) converges.

Answer: (1, 3)


Question: The surface defined by the equation \( z = y^2 - x^2 \) is intersected by the plane \( y = 3x \). Determine the slope of the resulting curve at the point \((1, 3, 8)\).

Answer: 16


Question: Evaluate the contour integral \( \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \) using the Residue Theorem, where \(C_3(0)\) is the circle of radius 3 centered at the origin, oriented counter-clockwise.

Answer: 0


Question: What is the smallest non-trivial $E_8$-module?

Answer: 248


Question: Find the derivative of the function \( f(v) = \frac{v+4}{|v+4|} \) using the definition of a derivative.

Answer: 0


Question: Determine whether $e^{\pm 2\pi i/3} = 1$ in the splitting field of the polynomial $x^3 - t$ over the field $\mathbb{F}_3(t)$, where $t$ is a variable. Consider $\omega$ as an abstract 3rd root of unity. Does this imply that $\omega = 1$ in the splitting field of $x^3 - t$?

Answer: Yes


Question: Compute the line integral \( \oint_C (y^2 \, dx - xy \, dy) \) where \( C \) is the boundary of the triangle with vertices \((1,0)\), \((4,0)\), and \((1,10)\), oriented counterclockwise. Use Green's Theorem to evaluate the integral.

Answer: -150


Question: Determine the number of zeroes of the function \( f(z) = e^z - 3z - 2 \) inside the unit circle \( |z| = 1 \).

Answer: 1


Question: Determine the value of $f(0)$ that makes the function \( f(x) = \frac{\log(1+ax) - \log(1-bx)}{x} \) continuous at \( x=0 \).

Answer: a + b


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_0^\pi \sin^n x \, dx \]

Answer: 0


Question: Can a subset of the 2D plane have a Hausdorff dimension greater than 2?

Answer: No


Question: Evaluate the integral \( I = \int\limits_{-\infty}^{+\infty} \frac{\sin x}{x} \cdot \frac{\sin\frac{x}{3}}{\frac{x}{3}} \cdot \frac{\sin\frac{x}{5}}{\frac{x}{5}} \, dx \).

Answer: \pi


Question: Suppose \(X\) and \(Y\) are independent random variables where \(X \sim N(1,4)\) and \(Y \sim N(1,9)\). Find the value of \(a\) such that \(P(2X + Y \le a) = P(4X - 2Y \ge 4a)\).

Answer: \dfrac{4}{3}


Question: Show that the order of the automorphism group of \(Z_3 \times Z_3\) is 48.

Answer: 48


Question: Find the next number after 365 that can be expressed as both a sum of two consecutive squares and a sum of three consecutive squares. Provide the last three digits of this number.

Answer: 645


Question: Is it correct to conclude that the multiplicity $m$ of irreducible $\mathbb{C}S_n$-modules in the decomposition of $M$ is equal to $\dim \operatorname{span}\{e_{T_{\lambda}}f: f \in M\}$?

Answer: Yes


Question: Consider the function \( f(x) = 2 + x^{-2} \) on the interval \([2, \infty)\). Using the contraction mapping principle, determine the number of iterations required for the fixed-point iteration starting at \( x_0 = 2 \) to approximate the fixed point to three decimal places. The contraction constant is \( \frac{1}{4} \). Provide a bound for the error at the \( n \)-th stage in terms of the initial point of the iteration, and find the smallest \( n \) such that the error is less than \( 0.0005 \).

Answer: 5


Question: Compute the limit: $$\lim_{n\to\infty} \frac{n!\cdot(3n)!}{(4n)!}$$

Answer: 0


Question: Is the score function, which is the gradient of the likelihood function, a random variable? Consider that the score has a mean of zero and variance equal to the Fisher information.

Answer: Yes


Question:
Let $a_1 = 2$ and for all natural numbers $n$, define $a_{n+1} = a_n(a_n + 1)$. As $n \rightarrow \infty$, what happens to the number of prime factors of $a_n$? Does it:
1. Go to infinity.
2. Go to a finite limit.
3. Oscillate boundedly.
4. Oscillate unboundedly.

Answer: 1


Question: Let $M$ and $N$ be $d$-dimensional Riemannian manifolds, and let $f: M \to N$ be a function that is continuous and differentiable almost everywhere. Suppose that the differential $df$ is an orientation-preserving isometry almost everywhere. Is it true that there exists a ball $B_{\epsilon}(p) \subseteq M$ such that the restriction $f|_{B_{\epsilon}(p)}$ is injective?

Answer: Yes


Question: Let $[a,b] \subset \mathbb{R}$ be a bounded interval, and let $f: [a,b] \to \mathbb{R}$ be a function of bounded variation. Does there exist a sequence of piecewise constant functions $f_k$ such that $[f_k - f]_{BV} \to 0$ as $k \to \infty$?

Answer: No


Question: Consider a representable functor \( \mathcal{F}:\mathrm{Sch}/S \to \{\mathrm{Sets}\} \) represented by a scheme \( X \). Let \( \{Z_i\}_{i \in I} \) be an inverse system of affine schemes with limit \( Z:=\varprojlim\limits_i Z_i \). For a sequence of objects \((a_i)_{i \in I}\) with \(a_i \in \mathcal{F}(Z_i)\) and corresponding \(f_i \in \mathrm{Hom}_S(Z_i,X)\), suppose \((f_i)\) and \((a_i)\) converge to \(f \in \mathrm{Hom}_S(\varprojlim\limits_i Z_i,X)\) and \(a \in \mathcal{F}(\varprojlim\limits_i Z_i)\) respectively. Does \(a\) map to \(f\) under the natural transformation \(\mathcal{F}(-) \to \mathrm{Hom}_S(-,X)\)?

Answer: Yes


Question: A group of 10 people went to a bookstore. Each person bought exactly 3 books, and for every pair of people, there is at least 1 book that both of them bought. What is the minimum number of people that could have bought the book purchased by the greatest number of people?

Answer: 5


Question: Does there exist a function defined on the interval $[0,1] \rightarrow \mathbb{R}$ that contains only jump discontinuities but is not piecewise continuous?

Answer: Yes


Question: Evaluate the triple integral \( \iiint_W \sqrt{x^2 + y^2 + z^2} \, dx \, dy \, dz \) over the region \( W = \{ (x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq x \} \) using an appropriate change of variables.

Answer: \dfrac{\pi}{10}


Question: Let \( g \) be a continuous function on \([0,1]\) such that \( \left|\int_0^1 g(t) \, dt \right| = 0 \). Can we conclude that \( g(t) = 0 \) for all \( t \in [0,1] \)?

Answer: No


Question: Consider a function $F \in L^2(X \times Y)$, where $X = \mathbb{R}/\mathbb{Z}$ and $Y$ is an arbitrary probability space. For almost every $y$, $F(\cdot, y) \in L^2(X)$, and thus can be expressed as a Fourier series: $F(x, y) = \sum_{k=-\infty}^\infty c_k(y)e^{2i\pi kx}$. Determine whether the coefficients $c_k(y)$ belong to $L^2(Y)$ for each $k$. Provide a justification for your answer.

Answer: Yes


Question: Let \(P(x) = ax^{2014} - bx^{2015} + 1\) and \(Q(x) = x^2 - 2x + 1\) be polynomials where \(a\) and \(b\) are real numbers. If \(Q(x)\) divides \(P(x)\), find the value of \(a + b\).

Answer: -4029


Question: What is \(\lim_{n \to \infty} m((n, \infty))\)?

Answer: \infty


Question:
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x \) and \( y \), the following equation holds:
\[ f(x+f(x+y))=f(x-y)+f(x)^2. \]

Answer: 0


Question: Show that the sequence of functions \(f_n(x) = n^p x e^{-n^q x}\) tends to 0 pointwise on the interval \([0,1]\) as \(n\) approaches infinity, where \(p\) and \(q\) are positive constants.

Answer: 0


Question: Compute the sixth derivative of the function \( F(x) = \int_1^x \sin^3(1-t)\, dt \) at \( x_0 = 1 \).

Answer: 60


Question: Let $(E, | \cdot |)$ be a normed space and $E'$ its topological dual. Consider the weak topology $\sigma(E,E')$ on $E$. Suppose $(x_d)_{d\in D}$ is a net in $E$ that converges weakly to $x \in E$ and satisfies $|x_d| \le r$ for all $d \in D$. Is it true that $|x| \le r$?

Answer: Yes


Question: Find the function \( f:(0,\infty)\to(0,\infty) \) that satisfies the functional equation \[ f\left(\frac {x} {f(y)}\right)=\frac {x} {f(x\sqrt y)} \] for all \( x,y\in(0,\infty) \).

Answer: \sqrt{x}


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

Answer: \dfrac{1}{2}


Question: Given unit vectors \(\vec{a}, \vec{b}, \vec{c}\) such that \(\vec{a} \cdot \vec{b} = 0\) and \(\vec{a} \cdot \vec{c} = 0\), and the angle between \(\vec{b}\) and \(\vec{c}\) is \(\pi/3\), find the value of \(|\vec{a} \times \vec{b} - \vec{a} \times \vec{c}|\).

Answer: 1


Question: Let \( f_n : \mathbb{R} \to [0,\infty) \) be a sequence of smooth functions such that the series \( \sum_{n=1}^\infty \int_{\mathbb{R}} f_n(x) \, dx \) is convergent. Additionally, assume that \( \sum_{n=1}^\infty f_n''(x) \) converges for every \( x \in \mathbb{R} \). Does this imply that \( \sum_{n=1}^\infty f_n(x) \) converges for all \( x \in \mathbb{R} \)?

Answer: Yes


Question: Find the operator norm \( \|T\|_{\ell^\infty \rightarrow \ell^2} \) for the linear operator \( T: \ell^\infty \rightarrow \ell^2 \) defined by \( T(a_1, a_2, \ldots) = (v_1a_1, v_2a_2, \ldots) \), where \((v_n)_{n \geq 1} \in \ell^2\) is a fixed bounded sequence of real numbers. Use the fact that \( \|Tx\|_2 \leq \|(v_n)\|_2 \cdot \|x\|_\infty \) for all \( x \in \ell^\infty \).

Answer: \|(v_n)\|_2


Question: In a linear regression model given by \( Y = XB + U \), where \( U \) is the vector of error terms normally distributed with mean 0 and variance \( \sigma^2 \), and all classical assumptions are satisfied, determine the variance of \( Y \).

Answer: \sigma^2 I


Question: Evaluate the limit: \[ \lim_{x \to 1} \left( \frac{2017}{1-x^{2017}} - \frac{2018}{1-x^{2018}} \right) \]

Answer: -\dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{\sum_{k=1}^n \frac{1}{k}}{\sum_{k=0}^n \frac{1}{2k+1}} \]

Answer: 2


Question: Determine the dimension of the quotient ring \( \mathbb{Z}[x,y,z]/\langle xy+2, z+4 \rangle \), where \( \mathbb{Z} \) is the ring of integers.

Answer: 2


Question: Determine whether there exists a $G_\delta$ set in $\mathbb{R}^n$ with positive Lebesgue measure and empty interior.

Answer: Yes


Question: In triangle \(ABC\), \(\angle C = 30^\circ\) and \(\angle B = 40^\circ\). A cevian line \(AD\) is drawn such that \(AB = CD\). Find the exact value of \(\angle CAD\).

Answer: 50^\circ


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

Answer: -1


Question: Determine the interval of convergence of the series: \[ \sum_{k=1}^\infty \sin\left(\frac{\pi}{k}\right)(x-1)^k \]

Answer: [0, 2)


Question: Evaluate the limit: \[ \lim_{x\to+\infty}\frac{\sin x\tan x }{x^3+x^2} \]

Answer: 0


Question: Find the volume of the solid region enclosed laterally by the elliptical cylinder \(\frac{(x-2)^2}{4}+\frac{(y-1)^2}{9}=1\) and bounded by the planes \(z+x=5\) and \(z+x=6\).

Answer: 6\pi


Question: Calculate \(x_{2016}\) for the sequence defined recursively by \(x_{n+1} = \frac{n+1}{n+3} \left(x_n + \frac{1}{2}\right)\) with the initial condition \(x_1 = \frac{1}{6}\).

Answer: 336


Question: Find the minimum value of $PA + PB$ where $P(x, y, z)$ lies on the line given by $\frac{x+2}{2} = \frac{y+7}{2} = \frac{z-2}{1}$, and $A(5, 3, 4)$ and $B(1, -1, 2)$ are fixed points.

Answer: 6\sqrt{2}


Question: Determine whether the statement "$a^b + b^c + c^a$ is a multiple of 8 if and only if $a + b + c$ is a multiple of 8" is true for odd natural numbers $a$, $b$, and $c$.

Answer: True


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

Answer: \dfrac{1}{2}


Question: Let \( v \) be a random unit vector in \( \mathbb{R}^3 \). Calculate the expected length of the projection of \( v \) onto a given plane.

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit \( \lim _ { n \rightarrow \infty } \int _ { 0 } ^ { \pi / 2 } e ^ { - n \sin x } \, dx \).

Answer: 0


Question: What is the expected number of coin tosses required to obtain the sequence 'Head-Tail-Head' for the first time?

Answer: 10


Question: Evaluate the limit of the sequence \(a_n = \left(\left( 1 + \frac{1}{n} \right) \left( 1 + \frac{2}{n} \right)\cdots\left( 1 + \frac{n}{n} \right) \right)^{1/n}\) as \(n\) approaches infinity.

Answer: \dfrac{4}{e}


Question: Given the input signal $x(t) = \cos(t)$ and the impulse response $h(t) = u(t)$, find the output $y(t)$ using the Fourier Transform.

Answer: \sin(t)


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

Answer: 8


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

Answer: \dfrac{1}{2}


Question: Consider a random permutation of n elements, where \( n \geq 2 \). Calculate the probability that the elements 1 and 2 are in the same cycle of this permutation.

Answer: \dfrac{1}{2}


Question: Find the value of \( a \) such that \[ \lim_{n\to\infty}\frac{1}{{2n+1}}\sqrt{(25-a^2)n^2+2n+3}=2. \]

Answer: \pm 3


Question: Evaluate the double integral \( \int_{0}^{a} \int_{0}^{\sqrt{a^2-y^2}} xy(x^2+y^2)^\frac{3}{2} \, dx \, dy \).

Answer: \dfrac{a^7}{14}


Question: Find the volume of the region below the surface \( z = r \), above the \( x \)-\( y \) plane, and inside the region defined by \( r = 1 + \cos \theta \).

Answer: \dfrac{5\pi}{3}


Question: If a normed vector space $V$ has the property that any finite dimensional linear subspace of it has the unicity property, does it follow that $V$ is strictly convex?

Answer: Yes


Question: Given the vector field \( E = (3x+2y)I_x + (2-ax+2z)I_y + (2y-z)I_z \), where \( I_x, I_y, I_z \) are the unit vectors in the x, y, and z directions respectively, determine the value of \( a \) such that \( E \) is a gradient vector field.

Answer: -2


Question: Let $X$ be a compact Hausdorff space such that for every $x \in X$, there exists a nested sequence of open sets $\{U_n\}$ such that $\{x\}=\bigcap_{n=1}^\infty U_n$. Is it true that $X$ is first-countable?

Answer: Yes


Question: Calculate the binomial coefficient \( C\left(-\frac{1}{2}, 1\right) \).

Answer: -\dfrac{1}{2}


Question: Compute the limit: \[ \lim_{x \to \infty}\left(\frac{1}{n}\sum_{k=1}^{n} k^{1/x}\right)^{nx} \] where \( n \in \mathbb{N} \).

Answer: n!


Question: Evaluate the double integral \( \int\int_Q \lfloor x+y \rfloor \, dx \, dy \) over the region \( Q = [0,2] \times [0,2] \), where \( \lfloor . \rfloor \) denotes the greatest integer function.

Answer: 6


Question: Let $X$ be an infinite set, and let $\mathscr{F}$ be the filter on $X$ generated by the filter base consisting of all complements of finite sets. Determine the set of points to which $\mathscr{F}$ converges when $X$ is given the cofinite topology.

Answer: X


Question:
Determine the limit function of the sequence of functions \( f_n(x) \) defined by:
\[
f_n(x) = \begin{cases}
 n & \text{if } -n \leq x \leq n, \\
 0 & \text{if } |x| > n
\end{cases}
\]

Answer: \infty


Question: Determine the value of the series \( \sum_{x=0}^{\infty}\frac{\cos(\pi x)}{x!} \). Is it equal to \( \frac{1}{e} \)? If not, find the closed form of the series.

Answer: \dfrac{1}{e}


Question: Does a non-empty locally closed subset of a $k$-scheme of finite type always contain a closed point? Let $k$ be a field, $X$ be a scheme of finite type over $k$, and $Y$ be a locally closed subset of $X$. Is $Y \cap X_0$ non-empty whenever $Y$ is non-empty, where $X_0$ denotes the set of closed points of $X$?

Answer: Yes


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \frac{d\theta}{1 + \sin^2 \theta} \) using the method of residues.

Answer: \pi \sqrt{2}


Question: If a function $f$ has an anti-derivative on the interval $[a,b]$, does it imply that $f$ is Riemann integrable on $[a,b]$? Provide a justification for your answer.

Answer: No


Question: If $u \in \mathcal{S}'(\mathbb{R}^n)$, determine whether there exists an integer $m \ge 0$ and a constant $C > 0$ such that for all $\phi \in \mathcal{S}(\mathbb{R}^n)$, the inequality $$|u(\phi)| \le C\|\phi\|_m$$ holds, where $$\|\phi\|_m = \sum_{|\alpha| + |\beta| \le m} \sup_{x \in \mathbb{R}^n} |x^\alpha \partial_x^\beta \phi|.$$

Answer: Yes


Question: Find the value of \(x > 0\) such that the matrix \(\begin{pmatrix} 100\sqrt{2} & x & 0 \\ -x & 0 & -x \\ 0 & x & 100\sqrt{2} \end{pmatrix}\) has two equal eigenvalues.

Answer: 50


Question:
Let \( \mathcal{N} \) be a Vitali set and define \( A = \{(x, a_0) \in \mathbb{R}^2 \mid x \in \mathcal{N} \} \), where \( a_0 \) is a fixed real number. Determine which of the following statements is true about \( A \):

1. \( A \) is a measurable set in \( \mathbb{R}^2 \).
2. \( A \) is a closed set in \( \mathbb{R}^2 \) for the Euclidean distance.
3. \( A \) is a measurable set but not closed.
4. \( A \) is a measurable set with infinite measure.

Answer: 3


Question: Consider $N$ independent and identically distributed random variables $X_1, X_2, \ldots, X_N$ drawn from a Uniform(0, 1) distribution. Define a good neighbor for $X_i$ as the point that is closest to $X_i$ in absolute value. A pair $(X_i, X_j)$ is called a good pair if $X_i$ is $X_j$'s good neighbor and $X_j$ is $X_i$'s good neighbor. What is the expected number of good pairs?

Answer: \dfrac{N}{3}


Question: Determine the number of invertible elements in the quotient ring \(\mathbb{F}_5 [x]/\langle x^4 + 2x^2 \rangle\).

Answer: 480


Question:
Find the coefficient $a_n$ in the generating function given by:
\[ G(z) = \frac{1}{1-z-z^2-z^3-z^4} = \sum_{k \geq 0} a_k z^k \]
where the sequence $\{a_n\}$ satisfies the recurrence relation:
\[ a_n = a_{n-1} + a_{n-2} + a_{n-3} + a_{n-4} \]
with initial conditions $a_0 = 1$, $a_1 = 1$, $a_2 = 2$, and $a_3 = 4$. Determine $a_n$.

Answer: a_n = a_{n-1} + a_{n-2} + a_{n-3} + a_{n-4}


Question: Determine whether there exists an even permutation $a \in A_{10}$ such that $a\tau a^{-1} = \sigma$, where $\sigma = (1\ 7)(5\ 6\ 9)(2\ 3\ 4\ 8\ 10)$ and $\tau = (1\ 2)(3\ 4\ 5)(6\ 7\ 8\ 9\ 10)$. Provide a justification for your answer.

Answer: Yes


Question: Is it possible to choose a countable sequence $(A_n)_{n\in\mathbb{N}}$ of subsets of a metric space $X$ such that $X=\bigcup_{n\in\mathbb{N}} A_n$ and a bounded continuous function $f:X\to\mathbb{R}$ is uniformly continuous on each $A_n$? Provide a justification for your answer.

Answer: No


Question:
Given that 2cos(x), |cos(x)|, and (3sin^2(x) - 2) form a geometric sequence, find one possible value of |x_1 - x_2|, where x_1 and x_2 are two positive values of x. Choose from the following options:

(a) \( \frac{2\pi}{3} \)
(b) \( \frac{\pi}{3} \)
(c) \( 2\cos^{-1}\left(\frac{2}{3}\right) \)
(d) \( \cos^{-1}\left(\frac{2}{3}\right) \)

Answer: C


Question:
Let \( S \) be the open unit disk in \( \mathbb{C} \) and \( f: S \to \mathbb{C} \) be a real-valued analytic function with \( f(0) = 1 \). Determine the nature of the set \( \{z \in S : f(z) \neq 1\} \):

(a) empty,
(b) non-empty finite,
(c) countably infinite,
(d) uncountable.

Answer: a


Question: Let $X_1, \ldots , X_n$ be independent random variables with a common density, mean $\mu$, and variance $\sigma^2$. Define $\bar{X}=\frac{1}{n}\sum_{k=1}^n X_k$. Calculate $\operatorname{Cov}(\bar{X}, X_k-\bar{X})$. Provide your answer in terms of $\mu$, $\sigma^2$, and $n$.

Answer: 0


Question: Consider a random variable $Y$ with finite support $\mathcal{Y}$ and a continuous random variable $X$ with cumulative distribution function $G$ and probability density function $g$. All random variables are defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. For some $y \in \mathcal{Y}$, let $\mathbb{P}(Y=y| X) \equiv h_y(X)$, where $h_y: \mathcal{X} \rightarrow [0,1]$. Determine if the function \((y,x) \in \mathcal{Y} \times \mathcal{X} \mapsto h_y(x) \cdot g(x) \in \mathbb{R}\) is the joint probability density function of $(Y,X)$. ($\cdot$ denotes scalar multiplication.)

Answer: Yes


Question: A person draws cards one at a time from a standard deck of 52 cards without replacement, continuing until they draw the 3 of Hearts. What is the expected maximum value of the cards drawn? Provide your answer as a numerical value.

Answer: 12.35


Question: Calculate the expected value \( E\left(\frac{X_1}{X_2+1}\right) \) where \( X_1 \) and \( X_2 \) are independent random variables with the density function \( P[X_1=k]=P[X_2=k]=pq^k \) for \( k=0,1,2,\ldots \).

Answer: -\ln p


Question: Given that \(\lim_{x\rightarrow \infty}(f(x)+x^2)=l\) for a real number \(l\), calculate \(\lim_{x\rightarrow \infty}\left(\frac{f(x)}{1+x-x^2}\right)\).

Answer: 1


Question:
Solve the system of congruences using the Chinese Remainder Theorem:

\[ 560x \equiv 1 \pmod{3} \]
\[ 560x \equiv 1 \pmod{11} \]
\[ 560x \equiv 1 \pmod{13} \]

Find the solution for \(x\) modulo 429.

Answer: 131


Question: Evaluate the infinite sum: \[ \sum_{n=0}^\infty \left[ 1-p(1-q)^{n-1} \right] \] where \(0 < p < 1\) and \(0 < q < 1\).

Answer: \infty


Question: Evaluate the limit: \[ \lim _{x\to 0}\frac{\cos x-1+\frac{x}{2}\cdot \sin x}{\ln ^4(x+1)} \]

Answer: -\dfrac{1}{24}


Question: Calculate the limit \( \lim_{x\to0^+}\frac{\sqrt{\cos(2x)}-\sqrt{1+x\sin(x)}}{\tan^2\frac x2} \).

Answer: -6


Question: Determine whether there exists a periodic function \( x(t) \) such that the integral of \( |x(t)| \) over its period is finite.

Answer: Yes


Question: Given that the function $f(x,y)$ satisfies Laplace's equation \( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 \), determine whether the function \( f\left(\frac{x}{x^2+y^2}, \frac{y}{x^2+y^2}\right) \) also satisfies Laplace's equation.

Answer: Yes


Question: Evaluate the definite integral: \[ \int_1^e (\ln x + 1)x^x \, dx \]

Answer: e^e - 1


Question: Prove that the ring \( k[x,y]/(y-x^2) \) is isomorphic to \( k[x] \), where \( k \) is a field.

Answer: k[x]


Question: Is there a sequence of square matrices \( M_n \) such that \( \lim_{n\to\infty} \det(M_n)=0 \) but with \( \kappa_2(M_n)=1 \) for all \( n \)?

Answer: Yes


Question: Find the Fourier transform of the function \( xe^{-x^2/2} \).

Answer: -i \sqrt{2\pi} \xi e^{-\xi^2/2}


Question: Evaluate the limit using a Riemann sum: \[ \lim_{n \to \infty} \frac{1+\sqrt[n]{e}+\sqrt[n]{e^2}+\cdots+\sqrt[n]{e^{n-1}}}{n} \]

Answer: e - 1


Question: Solve the equation: \[ 8x^2 + 2\ln x - 6 = 0 \]

Answer: 0.8837


Question: Is the normaliser \( N_L(K) = \{ x \in L : [x, y] \in K \ \ \forall y \in K \} \) of a Lie subalgebra \( K \subseteq L \) a subalgebra of \( L \)?

Answer: Yes


Question: If $J$ is a homogeneous ideal of height 2 in $\mathbb{C}[X,Y]$ such that $J \subseteq (X,Y)$, does there necessarily exist an integer $n \ge 1$ such that $X^n, Y^n \in J$? Answer yes or no.

Answer: Yes


Question: Is the relation $\subseteq_{ec}$ a well-quasi-ordering on the set of finite sequences of finite trees? In other words, does every infinite set of finite sequences of finite trees contain two elements $X$ and $Y$ such that $X\subseteq_{ec}Y$?

Answer: Yes


Question: Evaluate the triple integral \( \iiint_B xyz \, \mathrm{d}V \), where \( B \) is the portion of the unit ball in the first octant (i.e., all coordinates are positive).

Answer: \dfrac{1}{48}


Question: Is it possible to find 2014 distinct positive integers such that their sum is divisible by each of them?

Answer: Yes


Question: Given that $Y$ is uniformly distributed over $(0,2)$ and $X$ is exponentially distributed with mean $Y$, find the expected value of the process output $XY^3$. Assume $X$ and $Y$ are dependent as described.

Answer: \dfrac{16}{5}


Question: Find the arc length of the function \( y = (x-x^2)^{1/2} + \sin^{-1}(x^{1/2}) \) over the interval \([0, 1]\).

Answer: 2


Question:
Find a function \( f(x) \) for \( x > 0 \) that satisfies the equation:
\[ x[f(x)]^2 = \int_0^x f(t) \, dt \]

Answer: 1


Question: Calculate the conditional expectation \( E[W|X>Y] \) where \( W = X+Y \) and \( X, Y \sim \exp(2) \) are independent exponential random variables.

Answer: 1


Question: Determine the value of the parameter $a$ such that the equation \( \text{arccos}(\sin(x)+a)=\text{arcsin}(\cos(x)+a) \) has exactly one solution for $x$ in the interval $(0, 2\pi)$.

Answer: \sqrt{2}


Question: Given that $(g'(x))^2 = g(x)$ for all real $x$, with $g(0) = 0$ and $g(4) = 4$, find the value of $g(1)$.

Answer: \dfrac{1}{4}


Question: Consider an $n \times n$ array where each site is either open or blocked. A fluid falls from the top, and a site becomes "full" if it has fluid. Fluid can only move downwards and stops at blocked sites. The array percolates if fluid reaches the bottom row in at least one column. If each site is open with probability $p$, what is the probability that the $n \times n$ array percolates?

Answer: 1 - (1 - p^n)^n


Question: Is there an ellipse with nonzero rational minor and major axis lengths $a$ and $b$ such that the circumference of the ellipse is also rational?

Answer: No


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

Answer: 2


Question: Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Consider a cosemisimple pointed Hopf $\mathbb{K}$-algebra $A$. If $A$ is cocommutative, does it necessarily imply that $A$ is the group Hopf algebra $\mathbb{K}G$ for some group $G$?

Answer: Yes


Question: Let \( f(x) = 0 \) when \( x \) is irrational and \( f\left(\frac{p}{q}\right) = \frac{1}{q} \) if \( \frac{p}{q} \) is in lowest terms. Determine the function defined by \( g(x) = \lim_{y \to x}{f(y)} \).

Answer: 0


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

Answer: 0


Question: Find the area of the region bounded by the curves $xy=4$, $xy=8$, $xy^3=5$, and $xy^3=15$. Express your answer in terms of natural logarithms.

Answer: 2 \ln 3


Question: Find the area of the small loop of the graph given by the polar equation \( r = 2 + 2 \sqrt{2} \sin \theta. \)

Answer: 2\pi - 6