Question: Determine the number of Sylow 2-subgroups of the symmetric group $S_5$. Given that $|S_5| = 120 = 2^3 \cdot 3 \cdot 5$, use the Sylow theorems to find the number of Sylow 2-subgroups.

Answer: 15


Question: Find the value of \( n \in \mathbb{N} \) such that the limit \( \lim_{x\to 0}\frac{\cos^2x-\cos x-e^x\cos x+e^x-\frac{x^3}{2}}{x^n} \) is a non-zero finite number.

Answer: 4


Question: Let $f:[0,1]\rightarrow\mathbb{R}$ be a measurable function. Define $F(x,y) = f(x) - f(y)$. Is the function $F$ necessarily measurable on the product space $[0,1] \times [0,1]$?

Answer: Yes


Question: Let \((X_1, X_2)\) be a bivariate normal random vector with mean vector \((0, 0)^T\) and covariance matrix \(\Sigma = \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}\). Calculate \(\mathbb{E}(X_1^4 X_2^2)\).

Answer: 3 + 12\rho^2


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

Answer: 0


Question: Let the curve C be the mirror image of the parabola \(y^2 = 4x\) with respect to the line \(x + y + 4 = 0\). If \(A\) and \(B\) are the points of intersection of \(C\) with the line \(y = -5\), find the distance between \(A\) and \(B\).

Answer: 4


Question: Calculate the flux of the vector field \( \mathbf{F}(x,y,z) = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) through the cylindrical surface \( S = \{ (x,y,z) \mid x^2 + y^2 \le 9, 0 \le z \le 2 \} \).

Answer: 54\pi


Question: Let $V$ be a vector space of even dimension $n$. Does there exist a linear map $T: V \rightarrow V$ such that $\text{Ker}(T) = \text{Im}(T)$?

Answer: Yes


Question: Given a set of points, determine whether they are the vertices of an n-dimensional cube. Answer with "True" if they are, and "False" otherwise.

Answer: True


Question: Evaluate the integral \( \int_\gamma z \ \Im(z^2) \ dz \), where \( \gamma \) is the unit circle traversed once in the anticlockwise direction.

Answer: -\pi


Question: Evaluate the limit of the sequence of improper integrals: \[ \lim_{n \to \infty } \int_0^{\infty} x^{\frac{n}{n+1}} e^{-x^2} \, dx. \]

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{n\to\infty} \left(\sqrt[3]{n^6-6n^4+1} - n^2\right) \]

Answer: -2


Question: Let $K$ be a compact Hausdorff space and $C(K)$ be the space of all scalar-valued continuous functions on $K$. Consider a sequence $(f_{n})_{n}$ in $C(K)$ such that $\sup\limits_{n}\sup\limits_{t\in K}|f_{n}(t)|<\infty$. Define an equivalence relation $R$ on $K$ by $t_{1}Rt_{2}$ if and only if $f_{n}(t_{1})=f_{n}(t_{2})$ for all $n$. Let $K_{1}:=K/R$ be the quotient space with the quotient topology $\tau$. Is the space $(K_{1},\tau)$ metrizable?

Answer: Yes


Question: Does the ring of matrices $M_n(\mathbb{k})$, where $\mathbb{k}$ is a field, have the invariant basis number (IBN) property?

Answer: Yes


Question: Find the greatest common divisor of $3^{3^{333}}+1$ and $3^{3^{334}}+1$.

Answer: 3^{3^{333}} + 1


Question: Determine the limit of the function \( f(x) = \sum_{n=0}^{\infty} \frac{1}{(n!)^x} \) as \( x \to \infty \).

Answer: 2


Question: Find the smallest prime number \( p \) such that 7 is the least quadratic nonresidue modulo \( p \).

Answer: 71


Question: Determine the number of solutions in the real number system for the equation: \[ \sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1. \]

Answer: \infty


Question: Consider two sequences of positive numbers, $(x_n)$ and $(y_n)$. Determine whether the following statement is true: If for every $\epsilon > 0$, the inequality $x_n^{1+\epsilon} > y_n$ holds for all $n$, then it must be that $x_n \geq y_n$ for all $n$. Provide a justification for your answer.

Answer: True


Question: You have 5 red, 5 green, and 5 black rocks, each containing one fake rock, making a total of 3 fake rocks, each of a different color. You can choose any subset of rocks and determine if there is a fake rock in that subset. Devise a strategy to identify all 3 fake rocks using no more than 7 questions.

Answer: 7


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \frac{e^{-1/x}}{x} \cdot \int_x^1 \frac{e^{1/t}}{t} \mathrm{d}t \]

Answer: 1


Question: Find the limit: \[ \lim_{{n}\to {\infty}}\frac{1^p+2^p+\cdots+n^p}{n^{p+1}} \]

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


Question: Let \( f(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \frac{x^5}{5} \) and let \( g(x) = f^{-1}(x) \). Find \( g'''(0) \).

Answer: 1


Question: Evaluate the limit: $$ \lim_{x \rightarrow \infty} x\left(\frac{1}{x^2}+\frac{1}{(x+1)^2}+...+\frac{1}{(2x-1)^2}\right) $$

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim \limits_{x \to 0} \frac{\ln(1+x^5)}{(e^{x^3}-1)\sin(x^2)} \]

Answer: 1


Question: Show that the random variable \( \frac{N_n}{n \ln n} \) converges in probability to 1, where \( N_n \) is the number of cards you need to buy to collect a complete set of \( n \) baseball cards, with each card being randomly chosen one at a time.

Answer: 1


Question: Find the greatest positive real solution for the intersection of the functions \( f(x) = x^{\frac{2}{3}} - \frac{2}{x^{\frac{1}{3}}} + 1 \) and \( g(x) = x - 4 \).

Answer: 8


Question: Given the functional equation \( f(f(x) - 1) = 1 + x^{16} \), find the value of \( f(2) \).

Answer: 17


Question: Evaluate the line integral \( \oint_{C} xy \, ds \) where \( C \) is the curve of intersection of the surfaces \( x^2 + y^2 = 4 \) and \( y + z = 8 \).

Answer: 0


Question: Let $f$ be a complex-valued bounded function on a locally compact group $G$. The function $f$ is called right uniformly continuous if for every $\varepsilon > 0$, there exists an $e$-neighborhood $U$ such that for all $x, x' \in G$, if $x(x')^{-1} \in U$, then $|f(x) - f(x')| < \varepsilon$. Is it always true that the function $L_gf$, defined by $(L_gf)(x) = f(g^{-1}x)$, is right uniformly continuous for any $g \in G$?

Answer: Yes


Question: Given that $y_1$ and $y_2$ are a fundamental set of solutions of the differential equation $(\mathrm{e}^x y')' + y' + y = 0$ for $x > 0$, and the Wronskian $W(y_1, y_2)(0) = 1$, find the value of $W(y_1, y_2)(-1)$. Use Abel's formula to determine your answer.

Answer: e^e


Question: Let \( f(z) \) be a function with a zero of order \( n \) at \( z_0 \). Consider the function \( F(z) = \frac{f'(z)}{f(z)} \). Show that \( F(z) \) has a simple pole at \( z_0 \) and find the residue of \( F \) at \( z_0 \).

Answer: n


Question: Determine the Galois group of the polynomial \(X^{14} - tX^7 + 1\) over the field \(\mathbb{C}(t)\).

Answer: D_7


Question: Evaluate the integral \( \iint_D \lfloor x + y \rfloor \, dx\, dy \) over the domain \( D = [0,2] \times [0,2] \).

Answer: 6


Question:
Determine whether the series expansions for $e^x$, $\sin x$, and $\cos x$ are valid for any real number $x$. The series are given by:

$$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\cdots$$
$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$
$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$$

Are these expansions valid for all real $x$?

Answer: Yes


Question: Consider a sequence of i.i.d. random variables $X_1, X_2, \ldots$ where $P(X_i = 1) = p$ and $P(X_i = -1) = q$ with $p > q$. Define $S_n = a + \sum_{i=1}^{n} X_i$ where $a$ is a positive integer. Let $b$ be a positive integer such that $0 < a < b$, and define the stopping time $T = \inf\{n: S_n = b\}$. Show that $E[T] < \infty$ by finding an upper bound for $P(T=k)$.

Answer: E[T] < \infty


Question: Let $\Omega \subseteq {\mathbb R}^n$ be a possibly nonconvex open set, and suppose $X$ is a compact subset of the interior of $\Omega$, i.e., $X\subseteq {\rm int}\ \Omega$. Moreover, let $f:\Omega \mapsto {\mathbb R}$ be such that for every $x\in\Omega$ there exists an open ball $\cal B$ around $x$ such that $f$ is convex on $\cal B$. Determine if there exists a constant $L > 0$ such that $\|v\| \leq L$ for every $v\in\partial f(x)$ and $x\in X$ (where $\partial f(x)$ is the convex subdifferential of $f$ at $x$). Are the subdifferentials of $f$ uniformly bounded when restricted to a compact subset?

Answer: Yes


Question:
Determine whether the following inequality holds for a function $f(t):[0,t]\rightarrow \mathbb{R}$, where $s\leq t$:
\[\left|\max\limits_{0\leq v\leq t}\left(\int_{0}^{v}f(u)du\right)-\max\limits_{0\leq w\leq s}\left(\int_{0}^{w}f(u)du\right)\right|\leq \max\limits_{s\leq v\leq t}\left|\int_{s}^{v}f(u)du\right|\]

Answer: Yes


Question: Determine whether the series \( S_n = \sum\limits_{k=1}^n \frac{1}{\sqrt{n^2+k}} \) converges as \( n \to \infty \), and if so, find the limit of \( S_n \).

Answer: 1


Question: Can the natural partial order of the power set of real numbers, defined by $A \leq B$ if and only if $A \subseteq B$, be extended to a total order?

Answer: Yes


Question: Given the equations \( x^2 + y^2 = t - \frac{1}{t} \) and \( x^4 + y^4 = t^2 + \frac{1}{t^2} \), find the value of \( \frac{dy}{dx} \) at the point \((1, 1)\).

Answer: -1


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

Answer: 1


Question: A safe has three locks, each with 8 possible settings (1 through 8). The safe opens if any two of the three locks are set correctly. What is the minimum number of combinations needed to guarantee opening the safe?

Answer: 64


Question:
For what real values of \( x \) is the matrix \( A \) positive semidefinite, where \( A = \begin{bmatrix} 1 & (x+1) & 1 \\
(x+1) & 1 & (x+1) \\
1 & (x+1) & 1 \end{bmatrix} \)?

Answer: [-2, 0]


Question: Evaluate the limit of the integral \( \int_0^y \sqrt{\log(1+1/x)} \, dx \) as \( y \to 0 \).

Answer: 0


Question: Find a sufficient statistic for \(\theta\) given a random sample \(X_1, \ldots, X_n\) from a normal distribution with parameters \((\theta, \theta^2)\).

Answer: \left( \sum_{i=1}^n X_i, \sum_{i=1}^n X_i^2 \right)


Question: Determine the magnitude of the vertex angle \( \alpha \) of an isosceles triangle with a given area \( A \) such that the radius \( r \) of the inscribed circle is maximized.

Answer: \dfrac{\pi}{3}


Question: Determine the values of \( p \) such that the series of functions \( f(x) = \sum_{n=1}^\infty \frac{\log(1+ n^2x^2)}{n^p} \) converges and can be differentiated term-by-term for all \( x \in \mathbb{R} \).

Answer: p > 2


Question: Evaluate the integral \( \iiint_V \sqrt{x^2+y^2+z^2}\, dV \), where \( V: x^2 + y^2 + z^2 \leq 2z \).

Answer: \dfrac{8\pi}{5}


Question: Given that \( 0 \le \alpha_1, \alpha_2, \ldots, \alpha_n \le \frac{\pi}{2} \) and \( \cot(\alpha_1)\cot(\alpha_2)\cdots \cot(\alpha_n)=1 \), find the maximum value of \( \cos(\alpha_1)\cos(\alpha_2)\cdots \cos(\alpha_n) \).

Answer: 2^{-n/2}


Question: Find the equation of a curve such that each tangent line to the curve forms a triangle with the coordinate axes having a constant area of $a^2$. What is the equation of the curve?

Answer: xy = \dfrac{a^2}{2}


Question:
Evaluate the following summation for natural numbers $k, m, n$ with $k \leq n$:
$$\sum_{r=0}^{m}\frac{k\binom{m}{r}\binom{n}{k}}{(r+k)\binom{m+n}{r+k}}$$

Answer: 1


Question: Is it true that every finitely generated topologically torsion-free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d \times d$ upper triangular matrices with 1's on the diagonal, for some $d$?

Answer: Yes


Question: Let $G$ be the punctured unit disk, defined as the unit disk missing the point $0$. Let $f: G \to \mathbb{C}$ be an analytic function. Suppose $\gamma$ is a closed curve in $G$ that is homologous to $0$. Determine the value of $\int_{\gamma} f \, dz$. \(\text{Use Cauchy's Theorem to justify your answer.}\)

Answer: 0


Question: Find the focus of the parabola given by the equation: \(x^2 + y^2 + 2xy - 6x - 2y + 3 = 0\).

Answer: (1, 1)


Question: Determine the number of real solutions to the equation: \[ x^2 + e^x = \sin x \]

Answer: 0


Question: Determine the maximum possible number of real roots of the equation \(e^x = mx^2\), where \(m\) is a real parameter.

Answer: 3


Question: Simplify the fraction: \[ \frac{(2^4+2^2+1)(4^4+4^2+1)(6^4+6^2+1)(8^4+8^2+1)(10^4+10^2+1)}{(3^4+3^2+1)(5^4+5^2+1)(7^4+7^2+1)(9^4+9^2+1)(11^4+11^2+1)} \] using algebraic logic.

Answer: \dfrac{3}{133}


Question: Evaluate the limit: \( \lim_{x \to \infty} x^{\sin(1/x)} \) using l'Hôpital's rule.

Answer: 1


Question: Evaluate the infinite product: \[ \prod_{k=1}^\infty \left(1 - \frac{(-1)^k}{(2k-1)}\right) = \sqrt{2}. \]

Answer: \sqrt{2}


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

Answer: 2\pi


Question: Determine the order of the group \( \mathbb{Z}_3^{\times}/(\mathbb{Z}_3^{\times})^3 \).

Answer: 1


Question: Let $G$ be a profinite group, which is a compact and totally disconnected topological group, with the property that all of its normal subgroups of finite index are open sets. Does this imply that all of its subgroups of finite index are open sets?

Answer: Yes


Question: Evaluate the integral \( \int_{1}^5 \frac{\log(7-x)}{\log((7-x)(x+1))}dx \).

Answer: 2


Question: Calculate the line integral of the vector field \( F = \frac{1}{1+x+y^2+z^2} \mathbf{i} + \frac{2y}{1+x+y^2+z^2} \mathbf{j} + \frac{2z}{1+x+y^2+z^2} \mathbf{k} \) from the point \((0,0,0)\) to \((1,1,1)\) along the path \(C\), where \(C\) consists of the line segment from \((0,0,0)\) to \((1,1,0)\) followed by the line segment from \((1,1,0)\) to \((1,1,1)\).

Answer: \ln 4


Question: Calculate the limit of \( \frac{(n-1)^{2n-1-k}}{n^n (n-1-k)^{n-1-k}} \) as \( n \to \infty \).

Answer: e^{k - 1}


Question: Does there exist a finite non-commutative ring with unity that has exactly one invertible (unit) element?

Answer: No


Question: Evaluate the limit \( \lim_{x\rightarrow 0}\frac{\sqrt[m]{1+P(x)}-1}{x} \), where \( P(x) = \sum_{i=1}^{n}a_{i}x^{i} \).

Answer: \dfrac{a_1}{m}


Question: Let $X \sim U(0,1)$ and $Y \sim U(X,1)$. Determine the distribution of the random variable $Y$.

Answer: -\ln(1 - y)


Question:
Consider the linear functional \( f: (c_{00}, \|\cdot\|_1) \to \mathbb{C} \) which is non-zero and continuous. Determine the number of Hahn-Banach extensions of \( f \) to \( (\ell^1, \|\cdot\|_1) \). Choose from the following options:

- One
- Two
- Infinite
- Three

Answer: A


Question: Determine all values of $n$ for which the dihedral group $D_n$ has a unique cyclic subgroup of order $n$. Justify your answer.

Answer: n \neq 2


Question: Evaluate \( \lim\limits_{x \to 0} \left(\frac{x}{\sin x }\right)^{1/x^2} \).

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


Question: Evaluate the limit \( \lim_{x\to+\infty} x \cdot e^{-ax^2} \) for \( a > 0 \).

Answer: 0


Question: Using Rouche’s Theorem, determine the number of solutions to the equation \( e^{z} + 3z = 0 \) within the disk \( D(0,1) \).

Answer: 1


Question:
Determine whether the following equality holds for measurable functions: Let $h$ be a measurable and non-negative function, $f$ another measurable function, and $g$ a step-function. Is it true that:
\[ \sup_g\left\{\int_X(fg)\,d\mu:0\leq g\leq h\right\}=\sup_g\left\{\int_Xg\,d\mu:0\leq g\leq fh\right\} \]

Answer: Yes


Question: Calculate the conditional expectation \( E[X_3 \,|\, \sum_{i=1}^n X_i = t] \) where \( X_1, X_2, \ldots, X_n \) are i.i.d. random variables with a Poisson distribution with parameter \( \lambda \).

Answer: \dfrac{t}{n}


Question: Liz is standing on the real number line at position 0. She rolls a die repeatedly. If the roll is 1 or 2, she takes one step to the right. If the roll is 3, 4, 5, or 6, she takes two steps to the right. Let $X_n$ be Liz's position after $n$ rolls. Determine the limit \( \lim_{n \rightarrow \infty} P(X_n > 1.6n) \).

Answer: 1


Question: Given complex numbers $z_1$ and $z_2$ such that $|z_1| = |z_1 + z_2| = 3$ and $|z_1 - z_2| = 3\sqrt{3}$, determine the value of \( \left\lfloor \log_3 \left| (z_1\bar{z_2})^{2022} + (\bar{z_1}z_2)^{2022} \right| \right\rfloor \).

Answer: 4044


Question: Evaluate the limit: \[ \lim _{(x,y)\rightarrow (0,0)} \ln(\sin(x^2+y^2)) \]

Answer: -\infty


Question: Find all prime numbers $p$ such that the polynomial $x^3 - 120x + p$ is reducible over the rational numbers.

Answer: 11


Question: Let \((G,*)\) be a group and \(H\) and \(K\) be two subgroups of \(G\) with finite index. Is the intersection \(H \cap K\) also a subgroup of finite index in \(G\)?

Answer: Yes


Question: Find the coefficient of $x^{50}$ in the expansion of $\left(\frac{1}{(1+x)(1+x^2)(1+x^4)}\right)^3$. Express your answer as a single number.

Answer: 84


Question: In a two-player game, each player picks an integer between $1$ and $100$ inclusive and reveals it simultaneously. The player who picks the higher number has their number halved. The player with the lower number (after halving) pays the other player an amount equal to the difference. What is the optimal number to choose to maximize your earnings, assuming you have no information about the other player's choice?

Answer: 67


Question: Find the last three digits of \(135797531^{135797531^{320}}\).

Answer: 531


Question: Compute the intersection multiplicity of the curves $V(f)$ and $V(g)$ at the point $p = [-2:1:4]$ in $\mathbb{P}^2$, where $f = x^2 - yz$ and $g = (x+z)^2 - yz$. Use the fact that the intersection multiplicity is the Hilbert polynomial of the $I(p)$-primary component of the ideal $(f) + (g)$. Provide your answer as a single number.

Answer: 1


Question: Let $p(z)$ be a polynomial of degree $n \ge 2$. Determine if there exists a radius $R > 0$ such that \( \int_{|z|=R} \frac{1}{p(z)} \, dz = 0 \).

Answer: Yes


Question: Determine all real numbers \( \alpha \) for which there exists a matrix \( A \in M_{2005}(\mathbb{R}) \) such that \( 5A^2 + (6\alpha - 2)A + (2\alpha^2 - 2\alpha + 1)I = 0 \).

Answer: 2


Question: For which integers \( n > 1 \) does there exist \( n+1 \) distinct points \( \{x_0, x_1, \ldots, x_n\} \) in the interval \([a,b]\) such that for every continuous function \( f: [a,b] \to (0, \infty) \), the unique interpolating polynomial \( p_n(x) \) of \( f \) at the nodes \( \{x_0, x_1, \ldots, x_n\} \) satisfies \( p_n(x) \ge 0 \) for all \( x \in [a,b] \)?

Answer: 2


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

Answer: \dfrac{\pi}{2}


Question: Are the functions of the form \((f+\mathscr Hf)_{|_{(0,1)}}\), where \(f\) is a smooth function on \(\mathbb{R}\) with compact support in \((0,\infty)\), dense in \(L^2((0,1))\)?

Answer: Yes


Question: In triangle $ABC$, the side $BC$ has a length of 293. The side $AB$ is a perfect square, and the side $AC$ is a power of 2 such that $|AC| = 2|AB|$. Find the perimeter of triangle $ABC$.

Answer: 1061


Question: Determine the number of similarity classes of 6x6 matrices over \(\mathbb{C}\) with the minimal polynomial \((x-1)(x-2)^2\) and the characteristic polynomial \((x-1)^2(x-2)^4\).

Answer: 2


Question:
Determine the set of all positive values of \(a\) for which the series \(\displaystyle \sum_{n=1}^{\infty} \left( \frac{1}{n} - \tan^{-1}\left(\frac{1}{n}\right) \right)^{3a}\) converges. Choose from the following options:
(a) \((0,\frac{1}{3}]\)
(b) \((0,\frac{1}{3})\)
(c) \([\frac{1}{3},\infty)\)
(d) \((\frac{1}{3}, \infty)\)

Answer: D


Question:
Given three left exact functors $F, G, H: Mod \to Mod$ such that $R^iF(-) \cong R^iG(-)$ for all $i \in \mathbb{N}$, consider the exact sequence:
\[ \cdots \to R^iF(M) \to R^iG(M) \to R^iH(M) \to R^{i+1}F(M) \to R^{i+1}G(M) \to R^{i+1}H(M) \to \cdots \]
where $M$ is an $R$-module and $R$ is a commutative Noetherian ring. Can $R^iH(M)$ be zero for all $i$?

Answer: Yes


Question:
Express \( \cos^6(x) \) as a linear combination of \( \cos(kx) \) terms in the form:
\[ \cos^6(x) = m\cos(6x) + n\cos(5x) + o\cos(4x) + p\cos(3x) + q\cos(2x) + r\cos(x) + a. \]
Find the value of the constant term \( a \).

Answer: \dfrac{5}{16}


Question:
Consider a homogeneous Markov chain $(X_n)_{n \in \mathbb{N}}$ with the following transition probabilities: the only missing arrow is from state 0 to state 0 with $\mathbb{P}(X_1=0 \mid X_0=0) = 1/5$. Compute $h_0^6$, the probability of starting from state 0 and eventually hitting state 6. Given that $h_1=h_2=h_3=0$ because $\{1,2,3\}$ is a closed class, solve the following system of equations:

$$\begin{aligned}
h_0 &= \frac{1}{5} h_0 + \frac{1}{5} h_4, \\
h_4 &= h_5, \\
h_5 &= h_6, \\
h_6 &= h_4.
\end{aligned}$$

Find the value of $h_0^6$. Provide your answer as a single probability value.

Answer: \dfrac{1}{4}


Question: In 10 Bernoulli trials, calculate the conditional probability that all successes occur consecutively, given that the number of successes is between four and six.

Answer: \dfrac{3}{112}


Question: Evaluate the definite integral: \[ \int_0^{\pi/2} x^2 \csc^2(x) \, dx \]

Answer: \pi \ln 2


Question: Evaluate the integral \( \iiint_B x \, dV \), where \( B \) is the tetrahedron with vertices \((0,0,0), (0,1,0), (0,0,1), (1,0,0)\). Use the order of integration \( dV = dx \, dy \, dz \).

Answer: \dfrac{1}{24}


Question: Evaluate the limit: \( \lim_{x\rightarrow \frac{\pi}{4}} (\tan{x})^{\tan{2x}} \).

Answer: \dfrac{1}{e}


Question: Find the limit of the sequence \( x_n = \frac{1^k + 2^k + 3^k + \ldots + n^k}{n^k} - \frac{n}{k+1} \) as \( n \to \infty \), where \( k \) is a fixed natural constant.

Answer: \dfrac{1}{2}


Question: Let \( f(z) \) be an entire function such that \( |f(z)| > 3 \) for all \( z \). Find \( f'(z) \).

Answer: 0


Question: Given a matrix $A$ with eigenvalues $1, 2, -1$, and $B = I + A - A^{-1} + A^2$, find $\det(B)$. Use the Cayley-Hamilton theorem if necessary.

Answer: 26


Question:
Find the minimum distance between the sets \(A = \{(x,y) \in \mathbb{R}^2 : xy = 0\}\) and \(B = \{(x,y) \in \mathbb{R}^2 : xy = 1\}\) in the plane. Compute:
\[ \inf_{P \in A, Q \in B} d(P, Q). \]

Answer: 0


Question: Calculate the limit: \[ \lim_{x\to1} \left( \frac{40}{1-x^{40}} - \frac{30}{1-x^{30}} \right) \]

Answer: 5


Question: If $a = \frac{1+\sqrt{5}}{2}$, calculate the value of $a^{18} + \frac{323}{a^6}$.

Answer: 5796


Question: Determine the radius of convergence of the series \( \sum_{n=2}^{\infty} \frac{n^{2n}}{4^n(2n+1)!} (3-2x)^n \).

Answer: \dfrac{8}{e^2}


Question: Find the critical points of the function \( f(x,y) = x^2y - 2xy + \arctan y \).

Answer: (1, 0)


Question: Let $C$ be a category and $S \subset C$ be an object. Consider the slice category $C_{/S}$ over $S$ and the forgetful functor $F$ from $C_{/S}$ to $C$. Given morphisms $f:X\rightarrow Z$ and $g:Y\rightarrow Z$ in $C_{/S}$, suppose $F(f)=F(g)\circ h'$ for some morphism $h':F(X)\rightarrow F(Y)$. Does there necessarily exist a morphism $h:X\rightarrow Y$ such that $F(h)=h'$ when $C$ is the category of schemes?

Answer: Yes


Question:
In how many ways can the number 1,000,000 be expressed as the product of three positive integers \(a, b, c\) such that \(a \leq b \leq c\)?

(A) 139
(B) 196
(C) 219
(D) 784
(E) None of the above

Answer: A


Question: Determine which function is larger as $n$ becomes very large: $f(n)=2^{2^{2^n}}$ or $g(n)=256^{256^n}$.

Answer: f(n)


Question: Find the inverse Laplace transform of the function \( e^{-s} \).

Answer: \delta(t - 1)


Question: Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for any \( x, y \in \mathbb{R} \), the equation \( f(x + 2^y) = f(x) + 2^{f(y)} \) holds.

Answer: f(x) = x


Question: Prove that if a matrix $B$ is skew-symmetric and has an eigenvalue $r$, then $-r$ is also an eigenvalue of $B$.

Answer: -r


Question: Given a quartic polynomial with roots \(\alpha, \beta, \gamma, \delta\), where \(\sum \alpha = a\), \(\sum \alpha\beta = b\), \(\sum\alpha \beta \gamma = c\), and \(\sum \alpha\beta\gamma\delta = d\), find the value of \(\sum (\alpha \beta)^2\) in terms of \(a, b, c,\) and \(d\).

Answer: b^2 - 2ac + 2d


Question: Does there exist a complete metric on the open interval $(0,1)$ that induces the usual topology?

Answer: Yes


Question: Let $\alpha$ and $\beta$ be two complex numbers with $\beta$ non-zero. Suppose $f(z)$ is a polynomial function on $\mathbb{C}$ such that $f(z) = \alpha$ whenever $z^5 = \beta$. What is the degree of $f(z)$?

Answer: 5


Question: Find the limit of the expression \( \frac{e^{x}+x-\cos(2x)}{x^2} \) as \( x \to \infty \).

Answer: \infty


Question: Evaluate the limit: \[ \lim_{x\rightarrow0}\frac{(e^x \sin x- (x+1)\tan x)}{(x\log\cos(x))} \]

Answer: 0


Question: Evaluate the line integral \( \int_C \omega \) where \( \omega = xy^2 \, dy - x^2y \, dx \) and \( C \) is the circle defined by \( x^2 + (y-1)^2 = 1 \). Use the parametrization \( \alpha(t) = (\cos(t), \sin(t) + 1) \) for \( t \in [0, 2\pi] \). Verify the result using Green's Theorem and compare the outcomes.

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


Question: Two tangents are drawn from the point \((-2,-1)\) to the parabola \(y^2=4x\). If \(\alpha\) is the angle between these tangents, find the value of \(\tan \alpha\).

Answer: 3


Question: Determine the multiplicity of the zero of the function \( f(z) = \sin z - \tan z \) at \( z_0 = 0 \).

Answer: 3


Question: Prove the identity: \[ \sum_{k=1}^m k(k-1)\binom{m}{k} = m(m-1) 2^{m-2} \]

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


Question: Does the forgetful functor from presentable $\infty$-categories to $\infty$-categories preserve filtered colimits?

Answer: Yes


Question: Given a sequence $(x_n)_n$ of real numbers and a function $f: \mathbb{R} \to \mathbb{R}$ such that the set $\{y \in \mathbb{R} : \lim_{n \to \infty} e^{ix_n y} = f(y)\}$ has positive Lebesgue measure, determine if there exists an interval $I$ such that $\int_{I} f(y) \, dy \neq 0$. Provide a justification for your answer.

Answer: Yes


Question:
Show that the function \( f(z) = \bar{z}^2 \) is differentiable only at \( z = 0 \) using the definition of the derivative:
\[ f'(z) = \lim_{h\rightarrow0} \frac{f(z+h)-f(z)}{h} \]

Answer: 0


Question:
Let $z \in \mathbb{C}$ be a complex number with $|z| < 1$. Determine which of the following curves the set $\left\{ \frac{z-w}{1-z\bar{w}} \mid |w|=1, w \in \mathbb{C} \right\}$ forms:

(A) Straight line.
(B) Hyperbola.
(C) Circle.
(D) Parabola.

Answer: C


Question: Determine whether the matrix \( \begin{bmatrix}t & 1-t \\ 1-t & t \\ \end{bmatrix}^n \) equals the identity matrix in the ring \( \mathbb{Z}_n[t^{\pm 1}]/(t-1)^2 \), where \( n \) is a positive integer.

Answer: Yes


Question: In a group of order 21, there is a conjugacy class of order 3. Determine the order of an element in this conjugacy class.

Answer: 7


Question: Given the logical statements $A \land B \to C$ and $C \to A$, determine if there always exists a $D$ such that $B \to D$ and $A \land D \leftrightarrow C$ are true.

Answer: Yes


Question: Find the smallest non-negative integer value of \( n \) in the range \([0, b-1]\) that satisfies the congruence equation \((a \cdot n + c) \mod (b-n) \equiv 0\). For example, if \( a = 17 \), \( c = -59 \), and \( b = 128 \), the solution is \( n = 55 \).

Answer: 55


Question: Find the numerically greatest term in the expansion of \((3x+5y)^{12}\) when \(x=\frac{1}{2}\) and \(y=\frac{4}{3}\).

Answer: 11


Question: Find the holomorphic function \( \phi(z) = u(x, y) + iv(x, y) \) such that \( \text{Im} \phi(x+iy) = e^x(x \sin y + \sin y + y \cos y) \) and \( \phi(0) = 2 \).

Answer: (z + 1)e^z + 1


Question: Evaluate the integral \( \int_0^1 \frac{f(x)}{f(x) + f(1-x)} \, dx \) where \( f \) is a strictly increasing, continuous function on \([0,1]\). Use the substitution \( y = 1 - x \) to assist in solving the problem.

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_0^\infty \frac{\log(x)}{1+x^2} \, dx \).

Answer: 0


Question: Evaluate the improper integral \( \int_{1}^{\infty}\frac{x^3}{e^x}dx \).

Answer: \dfrac{16}{e}


Question: Evaluate whether the expression \( \sum_{k=0}^n \binom{n}{k}(-1)^k(k+1) \) equals zero. Justify your answer.

Answer: 0


Question: Consider a connected subset $X$ of the circle $S^1$. Is $X$ path connected? Provide a justification for your answer.

Answer: Yes


Question:
Determine whether the following identity holds for all integers \( n \):
\[ \frac{\sum_{k=0}^{\infty}\frac{F_{n+k}}{k!}}{\sum_{k=0}^{\infty}\frac{F_{n-k}}{k!}}=e, \]
where \( F_n \) denotes the Fibonacci sequence.

Answer: e


Question: Determine whether two smooth manifolds without boundaries that have the same universal cover and fundamental group are necessarily diffeomorphic. Provide a justification for your answer.

Answer: No


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

Answer: 0


Question: Calculate the outward flux of the vector field \( F = (x^3, y^3, z^2) \) across the surface of the region enclosed by the circular cylinder \( x^2 + y^2 = 49 \) and the planes \( z = 0 \) and \( z = 2 \).

Answer: 7399\pi


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \[ x\left[f(x+y)-f(x-y)\right]=4yf(x) \] for all real numbers \( x \) and \( y \).

Answer: f(x) = ax^2


Question: Determine the actual growth rate of \( \left|\frac{\zeta'}{\zeta}\left(-\frac{1}{2}+it\right)\right| \) as \( t \to \infty \), where \( \zeta \) is the Riemann zeta-function and \( t > 0 \).

Answer: \log t


Question: Consider the operator on $L^2(\Bbb R)$ defined by $T(f) = f * g$, where $g \geq 0$ is some $L^1$ function. Prove that this operator is a bounded linear operator with operator norm equal to $||g||_1$.

Answer: \|g\|_1


Question: Given $n$ i.i.d Gaussian random variables $x_i$ with mean $0$ and variance $1$, construct two Chi-squared random variables $Y=\sum_{i \in I_1} x_i^2$ and $Z=\sum_{i \in I_2} x_i^2$ with $|I_1|=|I_2|$. Compute $\Pr(Y \le Z)$ when $Y$ and $Z$ are not independent.

Answer: \dfrac{1}{2}


Question: Find the derivative of the function \( F(x) = \int_0^x \log(t) \, dt \) with respect to \( x \).

Answer: \log(x)


Question: Find all bijective functions \( f: [0,1] \to [0,1] \) such that \( f(2x-f(x)) = x \) for all \( x \in [0,1] \).

Answer: f(x) = x


Question: Find the greatest common divisor (GCD) of 24 and 49 in the ring of integers of \(\mathbb{Q}[\sqrt{3}]\), assuming the GCD is defined.

Answer: 1


Question: Determine whether the set of proper fractions \( \frac{p}{q} \), where \( p \) and \( q \) are prime numbers, is dense in the interval \([0,1]\). In other words, does the union of this set with its limit points equal the entire interval \([0,1]\)?

Answer: Yes


Question: Calculate the limit \( \lim_{n \to \infty}  \int_0^1 f(nx) \,dx \), where \( f(x) \) is continuous on \([0, \infty)\) and satisfies \( \lim_{x \to \infty} x^2 f(x) = 1 \).

Answer: 0


Question: Find the volume of the solid obtained by revolving the area under the first arc of the cycloid, given by the parametric equations \(x(t) = a(t - \sin t)\) and \(y(t) = a(1 - \cos t)\), around the \(y\)-axis. Use the formula \( V = \pi \int [f(x)]^2 \, dx \) to find the volume.

Answer: 6\pi^3 a^3


Question: Evaluate the integral \( \int_{0}^{+\infty}\frac{x-\sin{x}}{x^3}dx \) using complex analysis techniques, such as the residue theorem.

Answer: \dfrac{\pi}{4}


Question: Calculate the covariance \( \operatorname{Cov}(X_0, w_0) \) for an ARMA(1,1) process given by \( X_t = a_1X_{t-1} + w_t + b_1w_{t-1} \), where \( w_t \sim \mathcal{N}(0, \sigma^2) \) independently.

Answer: \sigma^2


Question: Classify the group $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$ according to the fundamental theorem of finitely generated abelian groups.

Answer: \mathbb{Z}_4


Question: Verify the inverse Fourier transform of the function \( F(k) = \frac{\sqrt{\pi}}{\sqrt{a}}e^{\frac{k^2}{4a}} \) to confirm that it corresponds to the original function \( f(t) = e^{-at^2} \), where \( a > 0 \).

Answer: e^{-at^2}


Question:
Let $R$ be a valuation ring with maximal ideal $\mathfrak{m}$, residue field $k$, and field of fractions $K$. Assume the valuation on $K$ satisfies $v(K)=\mathbb{Q}$. For each $\lambda \in \mathbb{Q}$, define:
\[ \mathfrak{m}_{\lambda}:=\{ x\in K\:|\:v(x)\geq \lambda \} \]
\[ \mathfrak{m}^+_{\lambda}:=\{ x\in K\:|\:v(x)> \lambda \} \]
Determine the dimension of the $k$-vector space $V_{\lambda}=\mathfrak{m}_{\lambda}/\mathfrak{m}^+_{\lambda}$. Assume $v$ is not discrete.

Answer: 1


Question: Let $W_1, W_2,$ and $W_3$ be three distinct subspaces of $\mathbb{R}^{10}$, each with dimension 9. Find the dimension of the intersection $W = W_1 \cap W_2 \cap W_3$. Provide a single value for $\dim W$. \( \text{(Hint: Use the formula for the dimension of the intersection of subspaces.)} \)

Answer: 7


Question: Find the supremum of the set \( P = \{ x \in \mathbb{R} : x \ge 0, \sum_{n=1}^{\infty} x^{\sqrt{n}} < \infty \} \).

Answer: 1


Question: Let $X$ and $Y$ be Banach spaces, and $U$ a closed subspace of $X$. Suppose $S \in BL(U,Y)$, where $Y = \ell^{\infty}$. Prove that there exists a $T \in BL(X,Y)$ such that $T|_U = S$ and $\|T\| = \|S\|$. Use the Hahn-Banach theorem on the functionals $\ell_n: u \mapsto (Su)(n)$ as a hint.

Answer: T


Question: Determine the maximum number of intersections between the polynomials \( g(x) = a_1x + a_2x^2 + \ldots + a_kx^k \) and \( f(x) = b_jx^j \), where \( a_1, a_2, \ldots, a_k > 0 \), \( j \in \{1, 2, \ldots, k-1\} \), \( b_j > 0 \), and \( x \geq 0 \).

Answer: 3


Question:
In a classroom of 17 students, each student has a unique height, ID, and phone number. Is it possible to find three students A, B, and C such that the following conditions are satisfied?

1. The height of B is between the heights of A and C: \(A_{\text{height}} > B_{\text{height}} > C_{\text{height}}\).
2. The ID of B is between the IDs of A and C: \(C_{\text{id}} > B_{\text{id}} > A_{\text{id}}\).
3. The phone number of B is between the phone numbers of A and C: \(C_{\text{phone}} > B_{\text{phone}} > A_{\text{phone}}\).

Answer: Yes


Question: In a minimal uniquely ergodic system \((X,T)\) with measure \(\mu\), does there exist a point \(p \in X\) such that \(\mu(\partial B(p,t)) = 0\) for all \(t > 0\)?

Answer: Yes


Question: A website offers one of three quizzes, A, B, or C, each day. If the majority of visitors pass the quiz, the next day's quiz is chosen randomly among A, B, and C with equal probability. If the majority fail, quiz A is chosen with certainty. The probabilities of passing each quiz are: \( P(A) = \frac{3}{10} \), \( P(B) = \frac{6}{10} \), \( P(C) = \frac{9}{10} \). Given the stationary distribution \( (G, B) = (\frac{3}{7}, \frac{4}{7}) \), find the limiting probability \( \xi_A \) that quiz A is used.

Answer: \dfrac{5}{7}


Question:
Find the value of the 2-person zero-sum game described by an $n \times n$ square matrix $A$ where:
- $-a_{ii} = \sum_{j \neq i} a_{ij}$
- $a_{ij} \geq 0$ for $i \neq j$
- $[A]_{ij} = a_{ij}$ for $i \neq j$
- $[A]_{ii} = -a_{ii}$

The sum of all diagonal entries is non-positive, the sum of all non-diagonal entries is non-negative, and the sum of the entries in each row is zero, with the diagonal entry equal to the negative of the sum of all other entries in its row. Determine the value of the game.

Answer: 0


Question:
Let \( f(x) \) be a differentiable, continuous function on \([0,1]\) such that:
\[ f(0) + f(1) = 0, \quad \int_0^1 (f(x))^2 \, \mathrm{d}x = \frac{1}{2}, \quad \int_0^1 f'(x) \cdot \cos \pi x \, \mathrm{d}x = \frac{\pi}{2}. \]
Find the value of \( \int_0^1 f(x) \, \mathrm{d}x \).

Answer: \dfrac{2}{\pi}


Question: Let \(A\) be a subset of a finite cyclic group with \(|A| = 5\) and \(|A + A| = 12\). If \(g\) is a group element such that \(g + A \subseteq A + A\), does it necessarily follow that \(g \in A\)? Justify your answer.

Answer: No


Question:
Consider the sum over all permutations \(\sigma \in S_n\) of the rational function:
\[\sum_{\sigma \in S_n} \frac{\sigma_{j1} \sigma_{j2}... \sigma_{jk}}{\prod_{i=1}^{n-1} (\sigma_i - \sigma_{i+1})} \]
where \(k < n-1\) and \(jm \in \{1,2,...,n\}\) for all \(m \in \{1,..,k\}\). Determine if this sum is equal to zero.

Answer: 0


Question: Suppose three distinct points are uniformly and independently distributed on a circle of unit length. Let one of these points be denoted as $B$. Define $Z$ as the random variable representing the distance from point $B$ to its neighbor in the anti-clockwise direction. Find the probability density function (pdf) of $Z$. Assume the circle is bijective with the interval $[0,1)$ and the points are independently and identically distributed as $\text{Unif}(0,1)$. Provide the pdf of $Z$.

Answer: 2(1 - z)


Question: Calculate \( \lim_{n \to \infty} \dfrac{\ln (\log_a (n)) - \ln (\log_n (a))}{\ln (n)} \).

Answer: 0


Question: Find all permutations \( \tau \in S_9 \) such that \( \tau(1,2)(3,4)\tau^{-1}=(5,6)(1,3) \).

Answer: 960


Question: Determine the asymptotic growth of the function defined by the recurrence relation: \[ T(n) = T\left(\frac{2n}{3} + 4\right) + \Theta(n) \]

Answer: \Theta(n)


Question: Evaluate the double integral \( \int_{-1}^{1} \int_{-1}^{1} |x^2 - y| \, dy \, dx \).

Answer: \dfrac{12}{5}


Question: Let $G$ be a finite abelian group and $G^{(p)}$ its unique $p$-Sylow subgroup. Determine if there exists a subgroup $H \subseteq G$ such that $G/H \cong G^{(p)}$. Provide a justification for your answer.

Answer: Yes


Question: Does every analytic function that is $2\pi$ periodic have a finite Fourier series?

Answer: No


Question: Prove that for positive terms \(a_n\), the series \(\sum\frac {a_n}{(1+a_1)\cdots (1+a_n)}=1\) if and only if the series \(\sum a_n\) diverges.

Answer: 1


Question: Calculate the path integral \( \int x^3 \, dx + y^3 \, dy + z^3 \, dz \) over the curve defined by \( z = x^2 + y^2 \) and \( y + z = 3 \). Use the parametrization \( x = r\cos{\alpha}, \, y = r\sin{\alpha} - 0.5, \, z = 3.5 - r\sin{\alpha} \) where \( r = \sqrt{13}/2 \).

Answer: 0


Question: Determine whether the following statement is correct: For a sequence of independent exponential random variables \(X_1, X_2, \ldots, X_n\) each with rate parameter \(\lambda\), the limit \(\lim_{n\to\infty} (X_{1} + \ldots + X_{n})\) is \(\infty\).

Answer: Correct


Question: Let $\psi:\mathbb{R}\to\mathbb{R}$ be a convex function such that $\psi'(0)<0$. Determine whether the point at which $\psi$ attains its minimum is positive.

Answer: Yes


Question:
Evaluate the series representation of \( \sin(i) \) given by:
\[ \sin(i) = i \sum_{j=0}^\infty \frac{1}{(2j+1)!} \]
What does this series converge to?

Answer: i \sinh(1)


Question: Is it possible to find two rotated vectors \( \mid a'\rangle = c_1\mid a\rangle + c_2\mid b\rangle \) and \( \mid b'\rangle = c_3\mid a\rangle + c_4\mid b\rangle \) that are orthogonal, given two abstract orthogonal vectors \( \mid a\rangle \) and \( \mid b\rangle \) with unknown lengths? Provide a justification for your answer.

Answer: Yes


Question: Calculate the limit: \( \lim \limits_{x \to -\infty} \log(\cos\frac{1}{x})(x^3-3x+\sin x) \).

Answer: +\infty


Question: Evaluate the double integral: \[ J = \int\limits_0^1 \int\limits_0^1 \dfrac{x-y}{(x+y)^3} \, dy \, dx. \]

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{x \sin(3x)}{x^2 + 4} \, dx \) using Jordan's Lemma.

Answer: \pi e^{-6}


Question: Determine the radius of convergence of the power series \( \displaystyle \sum_{n=0}^{\infty} \left(\int_{|z|=1} \frac{\cos{\left(\frac{1}{z-10}\right)}}{z^{n+1}} \,dz\right) z^n \).

Answer: 10


Question: Evaluate the integral \( \int_D \ln \sqrt{x^2+y^2} \, dA \), where \( D = \{ (x,y) : x^2 + y^2 < 1 \} \).

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


Question: Let \( A \) be an \( N \times N \) real symmetric matrix. Find the unit vector \( v \in \mathbb{R}^N \) that maximizes the expression \( v^tAv \).

Answer: v


Question: Consider a nontrivial prime ideal $P$ in an arbitrary ring. Can it be the case that $P^2 = P$? Provide an example or a proof to support your answer.

Answer: Yes


Question: Compute the residue of \( \frac{1}{z^2 \sin z} \) at \( z=0 \).

Answer: \dfrac{1}{6}


Question: Suppose $f$ is a function such that $f \\in B[a,b]$ and $V^b_{a+\epsilon}f \leq M$ for all $\epsilon > 0$. Does it follow that $f$ is of bounded variation on $[a,b]$?

Answer: Yes


Question: Find the surface area of the cone defined by \(x^2 + y^2 = z^2\) that lies above the \(xy\)-plane and below the cylinder \(y^2 + z^2 = 16\).

Answer: 16\pi


Question: Consider a symmetric random walk starting at 0. Let $T_a$ and $T_{-b}$ denote the first times the walk visits $a$ and $-b$, respectively, where $a, b > 0$. Find $\mathsf{E}(\min(T_a, T_{-b}))$. Use the fact that $P(T_a < T_{-b})$ can be computed by conditioning on the first step.

Answer: ab


Question: Compute the conditional expectation \( \mathbb{E}_{p}[X_1|X_1+\cdots +X_n=k] \) for \( X_1,\ldots,X_n \sim \operatorname{Ber}(p) \), where \( p \in (0,1) \).

Answer: \dfrac{k}{n}


Question: Find all functions \( f: \mathbb{N} \to \mathbb{N} \) such that \( f(f(m) + f(n)) = m + n \) for all \( m, n \in \mathbb{N} \).

Answer: f(n) = n


Question: Identify the quotient group \( \mathbb{C}^\times / S_1 \), where \( S_1 \) is the unit circle in the multiplicative group of complex numbers without 0. Determine if this quotient group is isomorphic to \( \mathbb{R}_{>0} \).

Answer: \mathbb{R}_{>0}


Question: Determine whether $X_{\theta} \times \mathcal{B}$ is an interpolation space between $X_{0} \times \mathcal{B}$ and $X_{1} \times \mathcal{B}$, given that $X_{\theta}$ is an interpolation space between the Banach spaces $X_0$ and $X_1$, and $\mathcal{B}$ is another Banach space.

Answer: Yes


Question: Let \( A \in \mathrm{GL}_d(\mathbb{Z}) \) be a matrix of finite order \( n \). Suppose \( k \in \mathbb{Z} \) is relatively prime to \( n \). Is \( A^k \) conjugate to \( A \) in \( \mathrm{GL}_d(\mathbb{Z}) \) for \( d \leq 4 \)?

Answer: Yes


Question: Determine the number of 3-Sylow subgroups in a simple group with 168 elements.

Answer: 28


Question:
Determine whether the following inequality is true for any complex numbers \(z_1\) and \(z_2\) such that \(z_1 \neq -z_2\):

If \(\alpha > 0\) and \(\beta > 0\), then
\[|\alpha z_{1}+\beta z_2|\leq \max\{\alpha,\beta\}|z_1+z_2|.\]

Answer: False


Question: Determine the natural numbers \( n \) for which there exists a map \( f: S^{2n} \to \mathbb{C}P^n \) such that \( f \) induces an isomorphism \( H^{2n}(\mathbb{C}P^n) \to H^{2n}(S^{2n}) \) on singular cohomology with coefficients in \( \mathbb{Z} \).

Answer: 1


Question: Evaluate the double integral \( \int_0^1 \left[ \int_0^1 v(u+v^2)^4 \, du \right] dv \).

Answer: \dfrac{31}{30}


Question: Determine the limit of the sequence \( \frac{1}{n} \sqrt[n]{n(n+1)(n+2)\cdots(2n)} \) as \( n \to \infty \).

Answer: \dfrac{4}{e}


Question: In a game slot, the probability of obtaining a number $k$ is given by $p_k=\frac{2^{k-1}}{3^k}$. If the number obtained has a remainder of one when divided by three, you earn 10 dollars. If it is divisible by three, you earn zero dollars. If it has a remainder of two, you lose 10 dollars. Calculate the probability that the total amount of money earned after 1000 attempts is between 50 and 100 dollars.

Answer: 0


Question: Let \( f(x) \) be a continuous function on the interval \([0,1]\). Evaluate the limit: \[ \lim_{n \to \infty} n\int_0^1 x^n f(x)\,dx. \]

Answer: f(1)


Question: Using Wilson's Theorem, compute \((p - 4)! \mod p\) for a prime number \(p\).

Answer: 6^{-1}


Question: Find the minimum value of the function \( f(x,y,z) = \left(x+\frac{1}{y}\right)^2 + \left(y+\frac{1}{z}\right)^2 + \left(z+\frac{1}{x}\right)^2 \) given that \( x > 0 \), \( y > 0 \), \( z > 0 \), and \( x+y+z=6 \).

Answer: \dfrac{75}{4}


Question: Consider a real symmetric matrix \(\mathbf{C} \in \mathbb{R}^{M \times M}\) with non-negative eigenvalues \(\lambda_1, \ldots, \lambda_M\) such that the trace of the matrix is \(M\). The matrix \(\mathbf{C}\) can be decomposed as \(\mathbf{C} = \mathbf{V} \mathbf{\Lambda} \mathbf{V}^T\), where \(\mathbf{\Lambda}\) is the diagonal matrix of eigenvalues and \(\mathbf{V}\) is the matrix of eigenvectors. The diagonal of \(\mathbf{C}\) is all ones, i.e., \(\operatorname{diag}(\mathbf{C}) = [1,\ldots,1]\). Define \(c_\max = \max\limits_{i\neq j}|c_{ij}|\), where \(c_{ij}\) is the element of \(\mathbf{C}\) in the \(i\)-th row and \(j\)-th column. Given that \(\mathbf{V}\) can be chosen freely, what is the minimum value of \(c_\max\)?

Answer: 0


Question: Given matrices $A$ of size $m \times n$, $B$ of size $n \times m$, and $C$ of size $n \times m$, where $m \neq n$, such that $AB = I_{m \times m}$ and $CA = I_{n \times n}$, determine if $B = C$. Provide a justification for your answer.

Answer: B = C


Question: Let $G=(\Sigma,N,R,S)$ be a context-free grammar where $n = |N|$ is the number of non-terminal symbols and $k$ is the maximal length of a production rule. Assume $L(G)$, the language generated by $G$, is not empty. Find a tight upper bound expression $f(n,k)$ for the length of the shortest word in $L(G)$. The expression should satisfy that the shortest word in $L(G)$ is at most $f(n,k)$, and there exists another context-free grammar $F$ such that the shortest word in $L(F)$ is exactly $f(n,k)$. Provide the expression $f(n,k)$. \( \text{Express your answer in terms of } n \text{ and } k. \)

Answer: k^{n-1}


Question: Calculate \( 516489222^{22} \mod 96899 \).

Answer: 4647


Question: Is the set of uniformly bounded non-decreasing functions compact with respect to the metric defined by \( d(f, g) = \left(\int_{a}^b |f(x) - g(x)|^2 \, dx\right)^{1/2} \)? Consider the set \( \Phi = \{ f : [a, b] \to [-M, M] \mid f \text{ is non-decreasing} \} \). Determine if the topology induced by this metric on \( \Phi \) is compact.

Answer: Yes


Question: Determine the Galois group of the polynomial \(x^3 + 3x^2 - 1\) over \(\mathbb{Q}\).

Answer: A_3


Question: Determine the values of \( p \) for which the series \( \sum_{n=3}^{\infty}\frac{1}{n\ln(n)\left(\ln(\ln(n))\right)^p} \) is convergent.

Answer: p > 1


Question: Suppose $\nu$ is a compactly supported signed measure in $\mathbb{R}^{n\geq 3}$. Is the Coulomb energy given by \( \iint \frac{1}{\|x-y\|^{n-2}}d\nu(x)d\nu(y) \geq 0 \)?

Answer: Yes


Question: Calculate \( 42^{17} \mod 3233 \).

Answer: 2557


Question: Evaluate the 2-form \( dxdy + 3dxdz \) on the oriented triangle with vertices \((0,0,0)\), \((1,2,3)\), and \((1,4,0)\) in that order.

Answer: -\dfrac{7}{2}


Question: Determine whether the square of an absolutely continuous random variable is also absolutely continuous. Let $X$ be a real-valued random variable with a distribution that is absolutely continuous with respect to the Lebesgue measure. Is $X^2$ absolutely continuous?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{1 - \sqrt{\cos x}}{x^2} \]

Answer: \dfrac{1}{4}


Question: Evaluate the integral \( \int_{0}^{\infty} e^{-x} x^2 \, dx \).

Answer: 2


Question: Find \( \lim_{n \to \infty} \frac{a_n}{n^2} \), where \( a_n \) is the number of ordered pairs \((a, b)\) of natural numbers such that \( \frac{ab}{a+b} = n \).

Answer: 0


Question: Evaluate the integral \( \int_{0}^{2\pi} \cos(\cos\theta)\cosh(\sin\theta)\,d\theta \) and show that it equals \( 2\pi \).

Answer: 2\pi


Question: Given that \( x, y \in G \) and \( xyx^{-1} = y^{17} \), find the order of \( y \) if \( |x| = 2 \), the order of \( y \) is odd, and \( y \neq e \).

Answer: 9


Question: Calculate the Legendre symbol \( \left(\frac{83}{127}\right) \).

Answer: -1


Question: Given a set of 18 distinct cards, each card is part of four different combinations of three cards, resulting in a total of 24 unique combinations. What is the minimum number of cards that must be drawn to ensure that at least one of these combinations is present in the drawn hand?

Answer: 13


Question: Evaluate the limit \( \lim_{n\to\infty}\frac{1^{k}+2^{k}+\ldots+n^{k}}{n^{k+1}} \) for \( k > 0 \).

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


Question: Evaluate the integral: $$\int_0^{2014} \frac{f(x)}{f(x) + f(2014 - x)}dx$$

Answer: 1007


Question: Find the value of \( n \) such that the limit \( \lim_{x \to 0} \frac{x^n}{\cos(\sin x) -\cos x} \) is a non-zero finite real number. Choose from the following options: (a) 3, (b) 4, (c) 5, (d) 6.

Answer: B


Question: Given that with a probability of \( \frac{1}{3} \), exactly one of eight identical-looking envelopes contains a bill, and with a probability of \( \frac{2}{3} \), all eight envelopes are empty, you open one envelope. Let \( A \) be the event that there is a bill in this envelope, and \( B \) be the event that the other seven envelopes have already been opened and turned out to be empty. Calculate \( P(A \mid B) \).

Answer: \dfrac{1}{17}


Question: Simplify the expression: $$\lim_{x\to0}\left(\prod^{\frac{1}{x}-1}_{i=1}\frac{1}{\sec\frac{xi\pi}{2}}\right)^x$$

Answer: \dfrac{1}{2}


Question: Let S be a circle with center O. A chord AB, not a diameter, divides S into two regions R1 and R2 such that O belongs to R2. Let S1 be a circle with center in R1, touching AB at X and S internally. Let S2 be a circle with center in R2, touching AB at Y, the circle S internally, and passing through the center of S. The point X lies on the diameter passing through the center of S2 and \( \angle YXO = 30^\circ \). If the radius of S2 is 100, find the radius of S1.

Answer: 60


Question: Evaluate the integral \( \int\limits_{B^n}\Delta f(x)dx \), where \( B^n=\{x\in\mathbb{R}^n:|x|<1\} \) and \( f(x)=(|x|^2-1)^{20}\sin(x_1+\cdots+x_n) \).

Answer: 0


Question: Determine the cardinality of the set \(X = \bigcup_{n \in \mathbb{N}} X_n\), where each set \(X_n\) has infinite cardinality \(\alpha\). Is the cardinality of \(X\) equal to \(\aleph_0 \alpha\)?

Answer: \aleph_0 \alpha


Question: Compute the limit: \[ \lim_{x \to 0}{\frac{\int_x^{x+1}f(t)dt-\int_0^{1}f(t)dt}{x}} \] where \( f \) is continuous on \( \mathbb{R} \) and satisfies \( f(0) + 1 = f(1) \).

Answer: 1


Question: Simplify the expression: \( \sqrt[3]{\sqrt{-121} + 2} \) and verify if it simplifies to \( \sqrt{-1} + 2 \).

Answer: \sqrt{-1} + 2


Question:
Given the equations:

\[ ax + by = 7 \]
\[ ax^2 + by^2 = 49 \]
\[ ax^3 + by^3 = 133 \]
\[ ax^4 + by^4 = 406 \]

Find the value of:

\[ 2014(x+y-xy) - 100(a+b) \]

Answer: 5956


Question: In a right triangle \( \triangle ABC \) with \( \angle B = 90^\circ \), the median \( AM \) is drawn. If the measure of angle \( \angle AMB \) is twice the measure of angle \( \angle A \), calculate the ratio \( \frac{AM}{BC} \).

Answer: \dfrac{3}{2}


Question: Find the coefficient of $x^{18}$ in the expansion of the expression $\left(1+x+2x^2+3x^3+\dots +18x^{18}\right)^2$. Provide your answer as a single integer.

Answer: 1005


Question: Let $n \geq 1$ and $A = \mathbb{k}[x]$, where $\mathbb{k}$ is a field. Given $a_1, \dots, a_n \in A$ such that \(Aa_1 + \dots + Aa_n = A\), determine if there exists an invertible matrix $\|r_{ij}\| \in M_n(A)$ such that $r_{1j} = a_j$ for all $j = 1, \dots, n$. Provide a justification for your answer.

Answer: Yes


Question: Let $z > 0$ and $z > x, y > 0$. Prove that if $x \ln \left(\frac{z}{z-y}\right) > y \ln \left(\frac{z}{z-x}\right)$, then $x < y$.

Answer: x < y


Question: Let \( R \) be a domain with finitely many prime ideals such that the localization at each prime, \( R_{\mathfrak{p}} \), is Noetherian. Is \( R \) necessarily Noetherian?

Answer: Yes


Question: Determine the number of positive integers \( n \) such that \( n + 2n^2 + 3n^3 + \cdots + 2005n^{2005} \) is divisible by \( n-1 \).

Answer: 16


Question:
Use the Chinese Remainder Theorem to solve the following system of congruences:
\[\begin{cases}
x \equiv 1 \pmod{4} \\
x \equiv -1 \pmod{25}
\end{cases}\]
Find the value of \( x \) modulo 100.

Answer: 49


Question: Find the distribution of the random variable \( Y = X^2 \), where \( X \sim N(0,1) \).

Answer: \chi^2(1)


Question: Determine whether the function \(|x|^{-r}\) is a tempered distribution. Specifically, find if there exist constants \(C\) and \(\alpha\) such that for all \(\phi \in \mathcal{S}(\mathbb{R})\), the inequality \(|\int_{\mathbb{R}} \frac{1}{|x|^r} \phi(x) \, dx| \leq C \|\phi\|_{(\alpha, \beta)}\) holds for some \(r > 0\).

Answer: r < 1


Question: Determine the radius of convergence of the complex power series \( \sum_{n=0}^{\infty} z^{a^n} \), where \( a > 1 \) is an integer.

Answer: 1


Question: Suppose \( f \in L^1([0,1]) \). Show that \( \lim_{n \to \infty} (f, \chi_n) = 0 \), where \( (f, \chi_n) = \int_{[0,1]} f \bar{\chi_n} \) and \( \bar{\chi_n} = e^{-nx} \).

Answer: 0


Question: Find the greatest common divisor (GCD) of all possible values of the product $a \cdot b \cdot c \cdot d \cdot e \cdot f$, where $a, b, c, d, e, f$ are natural numbers satisfying the equation $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. What is the GCD?

Answer: 24


Question: Evaluate the contour integral \( \oint \frac{z+2}{z^2+9} \, dz \) using the residue at infinity.

Answer: 2\pi i


Question: Find the supremum of the expression \( x - \ln\left(1 + \frac{\sin x - \cos x}{2}\right) \) for \( 0 \leq x \leq \frac{\pi}{4} \).

Answer: \dfrac{\pi}{4}


Question: Express the cardinality of the set \(\mathbb{S}^n\) in terms of \(\lvert \mathbb{S}\rvert\), where \(\mathbb{S}^n = \{(x_1, \cdots, x_n) : x_i \in \mathbb{S}\}\) and \(n \in \mathbb{N}\). Include transfinite cardinals in your consideration.

Answer: |\mathbb{S}|^n


Question: Is the local ring $A_p$ the direct limit of rings corresponding to open subschemes? Suppose $A_p$ is the stalk of a ring $A$ at a prime ideal $p$. Consider the (opposite) system of those open immersions $\operatorname{Spec}(A)\leftarrow \operatorname{Spec}(B)$ such that the scheme map $\operatorname{Spec}(k(p))\to \operatorname{Spec}(A)$ factorizes over these $\operatorname{Spec}(B)\to \operatorname{Spec}(A)$. Here, $k(p)=A_p/m_p$ denotes the residue field of $p$. Is it true that $\underset{\rightarrow}{\lim} B=A_p$?

Answer: A_p


Question: Evaluate the double integral \( \iint_{A} (x^2 + 4y^2) \, dx \, dy \), where the region \( A \) is bounded by the curves \( y = x^2 \) and \( x = y^2 \).

Answer: \dfrac{3}{7}


Question: Evaluate the contour integral \( \int_{C} \frac{dz}{z^2+1} \), where \( C \) is a simple closed contour oriented counterclockwise, containing \( z = i \) as an interior point, and \( C \) lies within the circle \( |z-i| = \frac{1}{2} \). Do not use the Cauchy Integral Formula or the Residue Theorem in your solution.

Answer: \pi


Question: Find the irreducible polynomial over \( \mathbb{Q} \) for \( \beta = \alpha_1\alpha_2 \), where \( \alpha_1, \alpha_2, \alpha_3 \) are the roots of the polynomial \( f(x) = x^3 + x + 1 \).

Answer: x^3 - x^2 - 1


Question: Determine which is greater: $211!$ or $106^{211}$. Provide a logical explanation for your answer.

Answer: 106^{211}


Question: Determine the number of parameter-free definable subsets of the structure $(\mathbb{R}, *)$. Provide your answer as a single number.

Answer: 32


Question: Determine the number of elements $z$ such that $1 < |z| < 2$ and $f(z) = 0$, where $f(z) = z^5 + z^3 + 5z^2 + 2$. Use Rouché's theorem to assist in your solution.

Answer: 3


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty }\sum_{k=n}^{5n}\binom{k-1}{n-1}\left( \frac{1}{5} \right )^n\left( \frac{4}{5} \right )^{k-n} \]

Answer: \dfrac{1}{2}


Question: Find all pairs of natural numbers \((n, m)\) such that \(n^4(n+1) + 1 = 7^m\).

Answer: (2, 2)


Question: A player starts with $X and plays a betting game where they have a 49.8% chance to double their bet and a 50.2% chance to lose it. If the player's total wealth increases to $X + Y, what is the expected total amount of money they must wager to return to $X? For example, if a player starts with $100 and their wealth increases to $150, calculate the expected total amount they must wager to return to $100.

Answer: 250Y


Question: Determine whether the equation \( \lim\limits_{r \to \infty} [ (x - r/\sqrt{2})^2 + (y + r/\sqrt{2})^2 = r^2 ] \) converges to the line equation \( y = x \) as \( r \to \infty \).

Answer: y = x


Question: Let $G$ be a finite group and let $A$ and $B$ be two proper nilpotent subgroups such that $AB = G$. Is it true that $G$ is necessarily solvable?

Answer: Yes


Question: Provide an example of a unital commutative semi-simple Banach algebra that is not amenable.

Answer: A(\mathbb{D})


Question:
Find the minimum value of the function \( f(x) = x + \log_2(2^{x+2} - 5 + 2^{-x+2}) \) from the following options:

A: \( \log_2(1/2) \)

B: \( \log_2(41/16) \)

C: \( 39/16 \)

D: \( \log_2(4.5) \)

E: \( \log_2(39/16) \)

Answer: E


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty} \ln\frac{n-2}{n+3}\arctan\frac{e^{2n}}{n!} \]

Answer: 0


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

Answer: \dfrac{\pi}{4}


Question: Determine the smallest positive integer \( n \) such that the congruence \( x^n \equiv 2 \pmod{13} \) has a solution for \( x \).

Answer: 1


Question: Evaluate the integral: \[ \int_{0}^{\infty} \frac{\ln^{2n}(x)}{1-x^2} \mathrm{d}x \] for \( n \ge 1 \). Determine if the result is zero.

Answer: 0


Question: Maximize the function \( \min\left(\frac{1}{x}, \frac{s+1}{4}\right) - \left(\frac{s}{x} + \frac{1-s}{4}\right) \) subject to the constraints \( 2 < x \leq 4 \) and \( 0 \leq s \leq 1 \).

Answer: \dfrac{1}{16}


Question: Evaluate the integral \( \frac{1}{2 \pi} \int_0^{2 \pi} \frac{1 - r^2}{1 - 2r \cos(\theta) +r^2}d\theta \) for \( 0 < r < 1 \).

Answer: 1


Question: Determine whether the following limit is true: \( \lim\limits_{x\to\infty}{x \cdot P[X>x]}=0 \).

Answer: False


Question: Evaluate the double integral \( I = \int_0^1\int_0^1|k(x,t)|^2dxdt \), where the function \( k(x,t) \) is defined as \( k(x,t) = (1-x)t \) for \( x>t \) and \( k(x,t) = (1-t)x \) for \( x<t \).

Answer: \dfrac{1}{90}


Question:
Let $e_k \in L_2(a,b)$ for $k=1,2,3,...$ be an orthonormal sequence in $L_2(a,b)$. Show that if this sequence is total, then for all $x \in (a,b)$, the following holds:
\[ \sum_{k=1}^{\infty}\left|\int_a^x e_k(t) \, dt\right|^2 = x-a. \]

Answer: x - a


Question:
Determine the limit of the sequence given by:
\[ a_n = \frac{n-1}{n^2+1} + \frac{n-1}{n^2+2} + \cdots + \frac{n-1}{n^2+n} \]

Answer: 1


Question: Determine the number of integer values of \( n \) for which \( n^2+n+1 \) divides \( n^{2019}+20 \).

Answer: 8


Question: Find a number $n$ such that $n$ is a product of three prime numbers and $n$ is a square modulo 389.

Answer: 30


Question: Find the Taylor series of the function \( e^{-1/x^2} \) about \( x = 0 \).

Answer: 0


Question: Evaluate the integral \( \int_0^\pi \frac{d\theta}{5+3\cos\theta} \) using the method of residues.

Answer: \dfrac{\pi}{4}


Question: Given a function \( y = f(x) \) satisfying the equation \( xf(x^2) + 3f(x) = 3x^5 + 9x^2 + x + 3 \) for all real numbers \( x \), evaluate the integral \( \int_{-1}^0 f(x) \, dx \).

Answer: 2


Question: Given that $A$ is a square matrix that satisfies the equation $A^2 - 3A + I = 0$, find the inverse of $A$. Express your answer in terms of $A$.

Answer: 3I - A


Question: Evaluate the contour integral \( \int_{\phi} \frac{dz}{4z^2-1} \) where \( \phi(t) = \exp(2\pi it) \) for \( 0 \leq t \leq 1 \). Justify your answer.

Answer: 0


Question: Evaluate the double integral \( \int \int_{R} (x^2+y^2)^{-2} \, dA \) over the region \( R = \{(x,y) \mid x^2+y^2 \leq 2, x \geq 1\} \) using polar coordinates.

Answer: \dfrac{1}{4}


Question: Determine the range of values of \( k \) for which the function \( f(x) = \frac{x^2 - 1}{(x-2)(x+k)} \) takes all real values, where \( x \) is a real number.

Answer: (-1, 1)


Question: In a checkered plane painted with 10 different colors, each cell is colored with one color, and all colors are present. Any two adjacent cells are colored in different colors. A pair of colors is considered "good" if there are two neighboring cells painted in these colors. What is the least number of good pairs possible?

Answer: 9


Question: Evaluate the integral \( \int_{0}^{1} \frac{x^4+1}{x^6+1} \, dx \).

Answer: \dfrac{\pi}{3}


Question: Find all real eigenvalues of the linear transformation $T: \mathbb{R^3} \rightarrow \mathbb{R^3}$, which represents a rotation around the $x_2$-axis by 90 degrees.

Answer: 1


Question: Let $U \subset \mathbb{C}$ be an open set and $c = n_1 \cdot c_1 + \ldots + n_k \cdot c_k$ be a null-homologous cycle in $U$, where $c_i: [0,1] \to U$ are continuous paths and $n_i \in \mathbb{Z}$. The support $|c| = \cup_i c_i([0,1])$ is compact, so there exists an $R > 0$ such that $|c| \subset B_R(0)$, the open ball of radius $R$ centered at 0. Is the winding number $Ind_c(z)$ zero for all $z \in \mathbb{C} \setminus B_R(0)$?

Answer: Yes


Question: A bag contains three tiles marked $A$, $L$, and $I$. Muhammad Ali wants to pick the letters $A$, $L$, and $I$ in that order. He randomly pulls one tile from the bag. If the letter $A$ is drawn, he keeps it. If the letter pulled is not $A$, he puts it back into the bag. He repeats this process for the next tile, keeping it if it is $L$ and putting it back otherwise. Determine the probability that Muhammad draws from the bag at most 10 times. Express this probability as a fraction $\frac{x_1}{x_2}$, where $x_1$ and $x_2$ are relatively prime numbers. Compute the remainder when $x_1 + x_2$ is divided by 1000.

Answer: 721


Question: Determine whether the following statement is true or false: Let $X$ and $Y$ be metric spaces and let $f : X \to Y$ be a mapping. If $Y$ is complete and $f$ is continuous, then the image of every Cauchy sequence in $X$ is a Cauchy sequence in $Y$. Provide a justification for your answer.

Answer: False


Question: Given a sequence $(x_n)$ of real numbers satisfying the inequality \(|x_n - x_m| \leq \frac{1}{2} |x_{n-1} - x_{m-1}|\), determine whether it follows that $(x_n)$ is a Cauchy sequence.

Answer: Yes


Question:
Consider the series \( S = \sum_{k=1}^{\infty} (-1)^{k-1} \frac{1}{k} \left(\frac{1}{4}\right)^k \) and \( T = \sum_{k=1}^{\infty} \frac{1}{k} \left(\frac{1}{5}\right)^k \). Determine which of the following statements is true:

(A) \( S - T = 0 \)

(B) \( 5S - 4T = 0 \)

(C) \( 4S - 5T = 0 \)

(D) \( 16S - 25T = 0 \)

Answer: A


Question: Let \( \{U_\alpha\} \) be an open cover of the unit circle \( \{(x,y) \in \mathbb{R}^2 : x^2+y^2=1\} \). Does there exist a \( \delta > 0 \) such that \( \{U_\alpha\} \) is also an open cover of the annular region \( \{(x,y) \in \mathbb{R}^2 : 1-\delta < \sqrt{x^2+y^2} < 1+\delta\} \)?

Answer: Yes


Question: Evaluate the improper integral \( \iint_S \frac{dx \, dy}{(1+x^2+y^2)^2} \), where \( S = \{ (x,y) \in \mathbb{R}^2 \mid x = 0 \text{ or } x > 0 \} \).

Answer: \dfrac{\pi}{2}


Question: A regular pentagon has 10 symmetries. Using 4 colors, determine the total number of distinct ways to color the 5 regions of the pentagon, considering two colorings the same if one can be transformed into the other by a symmetry of the pentagon. Use Burnside's Lemma to find your answer.

Answer: 136


Question: What is the minimal dimension \( n \) such that there exists an isometric embedding \( f: SO(3) \to \mathbb{R}^n \)?

Answer: 5


Question: Evaluate the integral \( \int_{0}^{1} (x-f(x))^{2016} \, dx \), given that \( f \) is a differentiable function on \([0,1]\) such that \( f(f(x)) = x \) and \( f(0) = 1 \).

Answer: \dfrac{1}{2017}


Question:
Suppose \(X, Y, Z\) are metric spaces and \(Y\) is compact. Let \(f\) be a function mapping \(X\) into \(Y\). Let \(g\) be a continuous one-to-one map from \(Y\) into \(Z\), and define \(h(x) = g(f(x))\) for \(x\) in \(X\). If \(h(x)\) is uniformly continuous, then which of the following is true about \(f\)?

(a) \(f\) is continuous
(b) \(f\) is uniformly continuous
(c) \(f\) may be discontinuous
(d) \(f\) is strictly increasing

Choose the correct option.

Answer: B


Question: If $f$ is a real-valued function that is integrable over $\mathbb{R}$, does it necessarily follow that $f(x) \to 0$ as $|x| \to \infty$? Provide a justification for your answer.

Answer: No


Question: Let $f$ be an entire function of order $r = 1$. Determine the order of the derivative function $f'$. Is it finite or not?

Answer: 1


Question: Suppose $G$ is an abelian group and $a \in G$. Let $f:\langle a \rangle \to \mathbb{T}$ be a homomorphism, where $\mathbb{T}$ is the circle group. Can $f$ be extended to a homomorphism $g:G \to \mathbb{T}$?

Answer: Yes


Question: Find an equation of the plane that passes through the line of intersection of the planes \(x-y=1\) and \(y+2z=3\) and is perpendicular to the plane \(x+y-2z=1\).

Answer: x - y = 1


Question: Find the value of \( \mathbb{E} [Y + 3X | X] \) given that \( X = \frac{ B_1+ B_3 - B_2}{\sqrt{2}} \) and \( Y = \frac{B_1 - B_3+ B_2}{\sqrt{2}} \), where \( B_t \) is Brownian motion at time \( t \geq 0 \). It is known that \( X \) and \( Y \) are independent, and \( \mathbb{E}[X] = \mathbb{E}[Y] = 0 \).

Answer: 3X


Question:
Determine the values of \(a\) for which the system of differential equations is asymptotically stable:
\[ x' = \begin{bmatrix} a & 2 \\ a & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \]

Answer: a < -1


Question: Let \( A_n = \frac{3}{4} - \left(\frac{3}{4}\right)^2 + \left(\frac{3}{4}\right)^3 - \cdots + (-1)^{n-1}\left(\frac{3}{4}\right)^n \) and \( B_n = 1 - A_n \). Find the smallest integer \( n_0 \) such that \( B_n > A_n \) for all \( n \geq n_0 \).

Answer: 6


Question:
Let $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ be positive real sequences such that \(\lim_{n\to\infty}\frac{a_{n+1}-a_n}{n}=a\in \mathbb{R}_{>0}\) and \(\lim_{n\to\infty}\frac{b_{n+1}}{nb_n}=b\in \mathbb{R}_{>0}\). Compute the limit:
\[\lim_{n\to\infty}\left(\frac{a_{n+1}}{\sqrt[n+1]{b_{n+1}}}-\frac{a_n}{\sqrt[n]{b_n}}\right).\]

Answer: \dfrac{a e}{2 b}


Question:
Consider the partial differential equation \( u_{xx} + u_{yy} + \lambda u = 0 \) for \( 0 < x, y < 1 \) with boundary conditions \( u(x,0) = u(x,1) = 0 \) for \( 0 \leq x \leq 1 \) and \( u(0,y) = u(1,y) = 0 \) for \( 0 \leq y \leq 1 \). Determine which of the following statements is true:

(a) There is a unique solution \( u \) for any \( \lambda \in \mathbb{R} \).
(b) There are infinitely many solutions for some \( \lambda \in \mathbb{R} \).
(c) There is a solution for countably many values of \( \lambda \in \mathbb{R} \).
(d) There are infinitely many solutions for all \( \lambda \in \mathbb{R} \).

Answer: C


Question:
Let $K/F$ be a field extension and $f \in F[x]$ with $\deg{(f)} \geq 1$. For a subset $L$ with $F\subseteq L \subseteq K$, let $p(L)$ denote the property that $f$ can be factored completely into linear factors in $L[x]$. Consider the set
$$\mathcal{L}:=\bigcap\{F\subseteq L \subseteq K \mid p(L) \ \text{holds}\} $$
If property $p(K)$ holds, does it follow that $p(\mathcal{L})$ holds? In other words, if $f$ can be factored completely into linear factors in $K[x]$, does it follow that $f$ can be factored completely into linear factors in $\mathcal{L}[x]$?

Answer: Yes


Question:
Consider the function \( f : [-\pi, \pi) \to \mathbb{R} \) defined by \( f = \chi_{(0, \pi)} - \chi_{[-\pi, 0]} \), and extend \( f \) to \( \mathbb{R} \) such that it is \( 2\pi \)-periodic. The Fourier series \( S f \) of \( f \) is given by:
\[
S f(x) = \frac{4}{\pi} \sum_{k = 1}^{\infty} \frac{\sin (2k - 1)x}{2k - 1}
\]
for all \( x \in \mathbb{R} \). The function \( f \) is differentiable on \( \mathbb{R} \setminus \pi \mathbb{Z} \), where \( \pi \mathbb{Z} = \{\pi k : k \in \mathbb{Z}\} \), and \( S f = f \) on \( \mathbb{R} \setminus \pi \mathbb{Z} \). Determine the behavior of \( S f \) at the points \( \pi k \), where \( k \in \mathbb{Z} \).

Answer: 0


Question: Does there exist a degree 6 irreducible polynomial in \( \mathbb{Q}[x] \) whose Galois group is isomorphic to \( S_3 \), the symmetric group of degree 3?

Answer: Yes


Question: Given that \(m = \sqrt[3]{a + \frac{a+8}{3}\sqrt{\frac{a-1}{3}}} + \sqrt[3]{a - \frac{a+8}{3}\sqrt{\frac{a-1}{3}}}\), find the value of \(m^3\).

Answer: 8


Question: Consider a set $A$ of points on the two-dimensional sphere $S^2$ such that for every great circle $c(\theta, \phi)$, the intersection $A \cap c(\theta, \phi)$ is countable. Can the set $A$ have nonzero Lebesgue measure on the sphere?

Answer: No


Question: Evaluate the definite integral \( \int_{1}^{1+2\pi}\cos(x)e^{(-\sin^2(x))}dx \).

Answer: 0


Question: Compute the limit: \[ \lim_{n\to\infty} n^{-n^2} \prod_{k=1}^n (k+n)^{\frac{1}{n}}. \]

Answer: 0


Question: Determine whether the following statement is true or false: If $Span(A) \cap Span(B \setminus (A \cap B)) = \{\vec 0\}$, then $Span(A) \cap Span(B) = Span(A \cap B)$, given that $A$ and $B$ are two linearly independent sets with $A \cap B = \varnothing$, $A \nsubseteq B$, and $B \nsubseteq A$. Provide a proof or counterexample.

Answer: True


Question: Given that $A, B, C \in \mathbb{R}$ and $\cos(A-B) + \cos(B-C) + \cos(C-A) = -\frac{3}{2}$, evaluate the expression \( \frac{\sum \cos^3(\theta+A)}{\prod\cos(\theta+A)} \), where $\theta \in \mathbb{R}$. Provide your answer in simplest form.

Answer: 3


Question: Evaluate the limit superior: \[ \limsup_{x \to 0} \frac{\int_0^x t \ln(1+t) \, dt}{x^2} \]

Answer: 0


Question: Determine the integer \( n \) such that the function \( f(z) = \begin{cases} \frac{\overline{z}^{n}}{z^{2}} & z \neq 0 \\ 0 & z = 0 \end{cases} \) is continuous but not differentiable at \( z = 0 \).

Answer: 3


Question: Consider a sequence where $a_1$ is the sum of some positive numbers, $a_2$ is the sum of the squares of these numbers, $a_3$ is the sum of the cubes, and so on. Is it possible for this sequence to satisfy $a_1 > a_2 > a_3 > a_4 > a_5$ and then $a_5 < a_6 < a_7 < \ldots$?

Answer: Yes


Question: The base of a solid S is an elliptical region with the boundary curve given by the equation \(9x^2 + 4y^2 = 36\). Cross-sections of the solid perpendicular to the x-axis are isosceles right triangles with their hypotenuses lying along the base. Calculate the volume of the solid S.

Answer: 24


Question: Calculate the value of \( 595^{611} \mod 899 \).

Answer: 119


Question: Evaluate the improper integral: $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(5x^2-6xy+5y^2)}dxdy$$

Answer: \dfrac{\pi}{4}


Question: Evaluate the sum \( \sum_{n=0}^{\infty} \frac{2n}{8^n}{{2n}\choose{n}} \).

Answer: \sqrt{2}


Question: Evaluate the limit: $$\lim_{n\to\infty}\left(\frac{1}{n!} \int_0^e \log^n x \ dx\right)^n$$

Answer: 1


Question: Given that \( \frac{e}{\overline{a}} = \frac{5}{4} + \frac{i}{4} \) and \( ea = 10 + 2i \), compute \( ||a|| \).

Answer: 2\sqrt{2}


Question: Let $A$ be a subset of a sequence of points that converges to a point $a \in \mathbb{R}^n$. Consider a closed subset $B$ of $\mathbb{R}^n$ such that the closure of $A$ and $B$ have no intersection. Is it true that $\inf\{d(x,y) \mid x \in A, y \in B\} > 0$?

Answer: Yes


Question: Let \( \overrightarrow{F} = (x+2y)e^z \hat{i} + (ye^z+x^2) \hat{j} + y^2 z \hat{k} \) and let \( S \) be the surface \( x^2+y^2+z^2 = 1, z \geq 0. \) If \( \hat{n} \) is the unit normal to \( S \) and \( \left | \iint_S \left (\overrightarrow{\nabla} \times \overrightarrow{F} \right ) \cdot \hat{n}\ dS \right | = \alpha \pi, \) find the value of \( \alpha. \)

Answer: 2


Question: Let $f(x)=x^{4}+4x^{3}-16x^{2}+6x-5$. Suppose that the tangent lines to $f$ at $(s,f(s))$ and $(t,f(t))$ are the same, where $s \neq t$. Find $|s-t|$.

Answer: 2\sqrt{11}


Question: Find the order of the quotient ring \( \mathbb{Z}_{10}[x] / \langle 5x^2 + 2x + 1 \rangle \).

Answer: 20


Question: Determine the homotopy type of the space $S^2 - \{p, q\}$, where $p$ and $q$ are two distinct points on the 2-sphere $S^2$. Provide a justification for your answer.

Answer: S^1


Question: Does a normal number contain each sequence of digits infinitely often? Provide a justification for your answer.

Answer: Yes


Question: Prove that if \( \mathbf{A} \) is a positive definite matrix, then all its eigenvalues are positive. Specifically, show that if \( \mathbf{A}\vec{v} = \lambda\vec{v} \) for a non-zero vector \( \vec{v} \), then \( \lambda > 0 \).

Answer: \lambda > 0


Question: Let $D$ be a base-point-free divisor on a normal projective variety $X$, and let $Y$ be the image of the morphism $f_{D}:X\rightarrow Y$ induced by $D$. Assume that $f_D$ is birational. Let $X(D)=\text{Proj}\left(\bigoplus_{k\geq 0}H^{0}(X,kD)\right)$. Is $X(D)$ the normalization of $Y$?

Answer: Yes


Question: Evaluate the limit as the radius $R$ tends to zero of the integral \( \int_{\mathcal C} \frac{e^{iz}-1}{z} \mathrm{d}z \), where \( \mathcal C \) is a semicircle centered at the origin with radius \( R \) and \( y \ge 0 \).

Answer: 0


Question: Calculate \((68! + 68^{86}) \mod 75\).

Answer: 49


Question:
Determine the distance from the origin to the set \( A \) in \( \mathbb{R}^{2d} \) where \( d > 1 \), defined as:
\[
A = \left\{ \begin{pmatrix} x_1 & y_1 & \ldots & x_d & y_d \end{pmatrix}': x_i^2 + y_i^2 > c, \ i = 1, \ldots, d \right\},
\]
with \( c \) being a positive constant. Calculate \( d(0, A) \), the distance from the origin to the set \( A \).

Answer: \sqrt{cd}


Question: Evaluate the integral \( \iint _{\Bbb{R}^2} e^{-(3x+2y)^2 - (4x+y)^2} \, dx \, dy \).

Answer: \dfrac{\pi}{5}


Question: Given that \( \alpha, \beta, \gamma, \delta \) are the roots of the polynomial equation \( x^4 + 4x^3 - 6x^2 + 7x - 9 = 0 \), find the value of \( \prod (1 + \alpha^2) \).

Answer: 13


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{\frac{1}{(n+1)^2}}^{\frac{1}{n^2}} \frac{e^x \sin^2(x)}{x^{\frac{7}{2}}} \, \mathrm{d}x \]

Answer: 2


Question: Find the volume of the region bounded by the surfaces \(x^2 + z = 1\) and \(y^2 + z = 1\) in the first octant.

Answer: \dfrac{1}{2}


Question: Find the value of \(2x+5\) if \(x\) satisfies the equation \(\sqrt{40-9x} - 2\sqrt{7-x} = \sqrt{-x}\).

Answer: -13


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{(4 \cos(n) - 3n^2)(2n^5 - n^3 + 1)}{(6n^3 + 5n \sin(n))(n + 2)^4} \]

Answer: -1


Question: Is there a field extension $L/K$ such that it is an infinite algebraic extension of fields but the separable degree of $L$ over $K$ is finite?

Answer: Yes


Question:
Consider the set \(K=\{x\in \mathbb{R}^4: Ax=a,~x_i\geq 0\}\) with
\[A=\begin{pmatrix}1&-1&-1&2\\ 0&1&0&3\\ 2&0&-2&10\end{pmatrix}\quad a=\begin{pmatrix}2\\ 7\\ 18\end{pmatrix}\]
Determine which of the following vectors is a vertex of the set \(K\):
\[x_1=\begin{pmatrix}10\\ 7\\ 0\\ 0\end{pmatrix},~x_2=\begin{pmatrix}0\\ 1\\ 1\\ 2\end{pmatrix},~x_3=\begin{pmatrix}1\\ 1\\ 2\\ 2\end{pmatrix},~ x_4=\begin{pmatrix}9\\ 7\\ 0\\ 0\end{pmatrix}\]

Answer: x_4


Question: Is it possible for three distinct sets \(S, T, V \subset \mathbb{R}^2\) with non-empty interiors to have coincident boundaries? Provide a justification for your answer.

Answer: Yes


Question: Suppose $f: \mathbb{R}\times\mathbb{R}\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[4f(x,y)=f(x+\delta,y+\delta)+f(x-\delta,y+\delta)+f(x-\delta,y-\delta) + f(x+\delta,y-\delta)\] for all $(x,y)$ in $\mathbb{R}\times\mathbb{R}$ and all $\delta$ in $\mathbb{R}$. Is $f$ necessarily a harmonic function?

Answer: Yes


Question: Let \((S, \leq)\) be a partial order with two minimal elements \(a\) and \(b\), and a maximum element \(c\). Let \(P: S \to \{\text{True}, \text{False}\}\) be a predicate defined on \(S\). Suppose that \(P(a) = \text{True}\), \(P(b) = \text{False}\), and \(P(x) \Rightarrow P(y)\) for all \(x, y \in S\) satisfying \(x \leq y\). Which of the following statements CANNOT be true?\n\n(A) \(P(x) = \text{True}\) for all \(x \in S\) such that \(x \neq b\)\n(B) \(P(x) = \text{False}\) for all \(x \in S\) such that \(x \neq a\) and \(x \neq c\)\n(C) \(P(x) = \text{False}\) for all \(x \in S\) such that \(b \leq x\) and \(x \neq c\)\n(D) \(P(x) = \text{False}\) for all \(x \in S\) such that \(a \leq x\) and \(b \leq x\)

Answer: D


Question: Determine the maximum value of \(|f(z)|\) for \(z\) in the set \(\{z \in \mathbb{C} : |z| \leq 1\}\), where \(f(z) = \frac{z+3}{z-3}\).

Answer: 2


Question: Let $X$ be a Banach space. If $X$ is not reflexive, does there exist a functional $f \in X'$ such that there is no $x \in X$ with $f(x) = \|f\| \cdot \|x\|$?

Answer: Yes


Question: Consider the Diophantine equation \(5x^6 - 4y^3 = N^2\) for integers \(x, y, N\). If there exists another distinct set of integers \((x', y', N')\) such that \(5x'^2 - 4y'^3 = N'^2\), can we conclude that there are no integer solutions to the original equation? Provide a justification for your answer.

Answer: No


Question: Does the existence of a Folner sequence in a discrete group imply that the group is countable?

Answer: Yes


Question: Consider a bounded sequence \( \{a_n\} \) in a general metric space that has no convergent subsequence. Is it true that there exists some \( \epsilon > 0 \) such that \( |a_n - a_m| \geq \epsilon \) for all \( n, m \)?

Answer: No


Question: Evaluate the contour integral \( \int_{\gamma} \frac{dz}{z} \), where \( \gamma \) is the ellipse given by the equation \( x^2 + 2xy + 2y^2 = 1 \) traversed once counterclockwise.

Answer: 2\pi i


Question: Find the minimum distance from the ellipsoid \(x^2+y^2+4z^2=4\) to the plane \(x+y+z=6\).

Answer: \sqrt{3}


Question: Find the volume of the solid formed by rotating the area between the curves \( y = x \) and \( y = \sqrt{x} \) about the line \( y = 1 \).

Answer: \dfrac{\pi}{6}


Question: Compute the triple integral \( \iiint_D (x^2+y^2) \, dxdydz \) over the region \( D = \{(x,y,z); 0 \leq x^2 + y^2 \leq z^2, \, 0 \leq z \leq 1 \} \). Use cylindrical coordinates where \( x = r \cos \theta \), \( y = r \sin \theta \), and the Jacobian of the transformation is \( r \).

Answer: \dfrac{\pi}{10}


Question: Let $M$ be a right module over a ring $R$ with unity, and let $K \leq A \leq M$, where $A$ is a semisimple submodule of $M$ and $K$ is a simple submodule of $M$. Assume that the radical of $M$, $\text{rad } M$, is zero. Is it true that $K$ is a direct summand of $M$?

Answer: Yes


Question:
Prove that for every natural number $n$, there exists a constant $C_n$ such that for any polynomial $f(t) = a_0 + a_1 t + \ldots + a_n t^n$ of degree at most $n$ with real coefficients, the following inequality holds:
\[ |a_0| + |a_1| + \ldots + |a_n| \leq C_n \sup_{t \in [0,1]} |f(t)|. \]

Answer: C_n


Question: Let $A$ be a $3 \times 3$ symmetric matrix over $\mathbb{R}$ such that $A^3 + A^2 - A - E = 0$, where $E$ is the identity matrix. Find the determinant $|A + 2E|$. \(\text{Note: } E \text{ is the identity matrix.}\)

Answer: 3


Question: How many distinct necklaces can be made using 6 beads, each of which can be one of 3 different colors?

Answer: 92


Question: Solve the equation \( x - 8 = x^{\log_{10} 2} \) for \( x \).

Answer: 10


Question: Evaluate the integral \( I = \int_{-\infty}^0 P.V.\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{\partial f}{\partial y} \frac{(x-y)}{(x-y)^2+z^2} \ dy \ dz, \) where \( f \) is a sufficiently well-behaved function of \( y \) alone, and \( P.V. \) denotes the principal value of the integral.

Answer: f(x)


Question: Determine the range of the function \( f(x) = \sqrt{\frac{-x^2 + 2x + 3}{18x - 3x^3}} \) without using limits or derivatives.

Answer: [0, \infty)


Question: Determine the number of elements in the set \( S_A \), where \( S_A = \{ A_{6 \times 6} \mid A^2 = A \} \) and any two matrices \( A \) and \( B \) in \( S_A \) are not similar, i.e., there does not exist an invertible matrix \( P \) such that \( P^{-1}AP = B \).

Answer: 7


Question: Consider an entire function $f$. Determine the value of $\inf_{z\in\mathbb{C}}|f(z)|$.

Answer: 0


Question: Conduct a sequence of 13 independent Bernoulli experiments, each with a probability \(0 < p < 1\) of success. Let \(Y\) denote the number of successes in the sequence, and let \(X\) denote the number of failures before the first success. Find \(\mathbb{E}(X|Y=1)\).

Answer: 6


Question: Determine whether every Hilbert space has an orthonormal basis. Provide a justification for your answer.

Answer: Yes


Question: Let \((X,d)\) be a compact metric space and \(f:X\rightarrow X\) be a continuous function with no fixed points. Prove that there exists an \(\epsilon > 0\) such that \(d(x,f(x)) > \epsilon\) for all \(x \in X\).

Answer: \epsilon


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

Answer: 1


Question: Evaluate the limit: \[ \lim_{n\rightarrow +\infty} n^2 \int_0^{2n} e^{-n|x-n|}\log\left[1+\frac{1}{x+1}\right] dx \]

Answer: 2


Question: Given the projection matrix in least squares, $H=X(X^TX)^{-1}X^T$, and the QR decomposition $X = QR$, express $H$ in terms of $Q$.

Answer: QQ^T


Question: Find the value of \( l = \lim \limits_{k \to \infty} \int_{0}^{k} \left(1-\frac{x}{k}\right)^k \cdot e^{\frac{x}{3}} \, dx \).

Answer: \dfrac{3}{2}


Question: Let $X$ be a normal, projective complex variety with at worst rational singularities, and let $\pi: Y \to X$ be the resolution of singularities obtained by blowing up the singular points. Determine if $R^1 \pi_*\mathbb{Z} = 0$ when $X$ is of dimension 3.

Answer: 0


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

Answer: 1


Question: Given the analytic function \( f(z) = u(x+iy) + iv(x+iy) \) where \( u(x+iy) = x^2 - y^2 + x \), find the constant \( C \) such that the function satisfies the initial condition \( f(i) = -1 + i \). Use the expression \( f(z) = x^2 - y^2 + x + i(2xy + y + C) \) and apply the initial condition to determine \( C \).

Answer: 0


Question: Evaluate the double integral: \[ \int_{0}^{1}\int_{0}^{1-y} \sin\frac{x-y}{x+y}\mathrm{d}x\mathrm{d}y \]

Answer: 0


Question: Evaluate the limit: $$ \lim_{x \to \infty} \frac{\sqrt[5]{x^7+3} - \sqrt[4]{2x^3 - 1}}{\sqrt[8]{x^7 + x^2 + 1} - x} $$

Answer: -\infty


Question: Determine whether the set of normal endomorphisms forms a group. Answer true or false.

Answer: False


Question: Find the area enclosed by the equation: \[ \left\lfloor\frac{\left|3x + 4y\right|}{5}\right\rfloor + \left\lfloor\frac{\left|4x - 3y\right|}{5}\right\rfloor = 3 \] where \(\lfloor\cdot\rfloor\) represents the greatest-integer function.

Answer: 16


Question: Consider the sequence defined by \(a_n=\sum_{k=1}^n\binom{n}{k}\frac{k}{n+k}\). Let \(s(n)\) be the sum of the binary digits of \(n\). Determine if the 2-adic valuation of the denominator of \(a_n\) is equal to \(s(n)\).

Answer: Yes


Question:
Determine the lower bound on the radius of convergence for the power series solution of the differential equation:
\[ e^{-x}\ddot y + \ln(1 + x)\dot y - xy = 0 \]
using the theorem that states if \(x_0\) is an ordinary point of the differential equation \(P(x)\ddot y + Q(x)\dot y + R(x)y = 0\), then the radius of convergence for the series solutions is at least as large as the minimum of the radii of convergence of the series for \(p = Q/P\) and \(q = R/P\).

Answer: 1


Question:
For a random sample \(X_1, X_2, X_3\) from a Bernoulli distribution \(\text{Ber}(p)\), which of the following is a sufficient statistic for \(p\)?

\((A)\ \ X_{1}^{2}+X_{2}^{2}+X_{3}^{2}\)

\((B)\ \ X_1+2X_{2}+X_{3}\)

\((C)\ \ 2X_1-X_{2}-X_{3}\)

\((D)\ \ X_1+X_{2}\)

\((E)\ \ 3X_1+2X_{2}-4X_{3}\)

Use the Neyman-Factorization Theorem to determine the correct answer.

Answer: A


Question: Determine whether there exists an infinite group $G$ such that there is no sequence $(A_n)$ of its subsets satisfying the conditions: \(A_n = A_n^{-1}\) and \(A_{n+1}A_{n+1} \subsetneqq A_n\) for all $n$.

Answer: Yes


Question: Evaluate the integral \( \int_0^\infty \frac{\cos x}{(x^{2} +1)^{2}}dx \) using the residue theorem.

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


Question: Determine the Galois group of the splitting field of the polynomial \( X^5 - 4X + 6 \) over \( \mathbb{Q} \).

Answer: S_5


Question: In a linear space $X$ with $\dim X = 6$, there are three linear subspaces $U$, $V$, and $W$ such that $\dim U = 2$ and $U \cap V = U \cap W = \{0\}$. Determine whether the statement $U \cap (V+W) = \{0\}$ is true or false.

Answer: False


Question:
Determine the values of \(a\) for which the following bilinear form defines an inner product on \(\mathbb{R}^2\):
\[
\left\langle \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}, \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \right\rangle = x_1 y_1 - 3x_1 y_2 - 3x_2 y_1 + ax_2 y_2.
\]

Answer: a > 9


Question: Evaluate the integral \( \int (y + \sin(e^{x^2})) \, dx - 2x \, dy \) over the circle \( x^2 + y^2 = 1 \), traversed anti-clockwise.

Answer: -3\pi


Question: Let $X$ and $Y$ be independent standard normal random variables, i.e., $X, Y \sim N(0,1)$. Define $Z = X \cdot \text{sign}(Y)$ and $W = Y \cdot \text{sign}(X)$. Calculate the expected value $E(ZW)$. Provide your answer in terms of known constants.

Answer: \dfrac{2}{\pi}


Question: If $X_1$ and $X_2$ are independent random variables generated from $\Gamma(\alpha,1)$ and $\Gamma(\alpha+ \frac{1}{2},1)$ respectively, and $Y$ is defined as \(Y=3 \sqrt{X_1X_2}\), determine if \(\mathbb{E}[Y] = 3 \sqrt{\alpha^2+ \alpha\frac{1}{2} }\).

Answer: \mathbb{E}[Y] \neq 3 \sqrt{\alpha^2 + \alpha \frac{1}{2}}


Question: Determine the smallest value of the parameter \( \alpha \) such that the equation \( \sin^2 x \cdot \cos 2x + \alpha (\cos^4 x - \sin^4 x) = -10(2\alpha + 1)^2 \) has at least one real solution.

Answer: -\dfrac{5}{8}


Question: Consider a fixed set of finite discrete symbols $\mathcal{A}$. Equip $\mathcal{A}$ with the discrete topology, denoted by $\theta$, and $\mathcal{A}^{\mathbb{Z}^d}$ with the product topology, denoted by $\tau$. Is $(\mathcal{A}^{\mathbb{Z}^d},\tau)$ a compact metric space?

Answer: Yes


Question:
Consider the matrix $P = \begin{bmatrix}3 & 1\\1 & 3\end{bmatrix}$. Let $S$ be the set of all vectors $\begin{pmatrix}x\\y\end{pmatrix}$ such that $a^2 + b^2 = 1$, where $\begin{pmatrix}a \\ b \end{pmatrix} = P \begin{pmatrix}x \\ y \end{pmatrix}$. Determine the nature of the set $S$ from the following options:

- A circle of radius $\sqrt{10}$
- A circle of radius $\frac{1}{\sqrt{10}}$
- An ellipse with major axis along $\begin{pmatrix}1 \\ 1 \end{pmatrix}$
- An ellipse with minor axis along $\begin{pmatrix}1 \\ 1 \end{pmatrix}$

Provide the correct option.

Answer: D


Question: Evaluate the line integral \( \int_C F \, dr \) using Green's Theorem, where the vector field is given by \( F=\langle e^{\sin{x}}+y^2, x^2+y^2 \rangle \) and the curve \( C \) is oriented as follows: it starts at point \((1,2)\), moves along a line segment to \((1,4)\), then to \((2,4)\), and finally back to \((1,2)\).

Answer: 4


Question: Let \( \{a_n\} \) and \( \{b_n\} \) be two different arithmetic sequences such that \((3n+1)a_n=(2n-1)b_n\) for all positive integers \(n\). Define \(A_n = a_1 + a_2 + \ldots + a_n\) and \(B_n = b_1 + b_2 + \ldots + b_n\). Find the value of \(\frac{A_9}{B_6}\).

Answer: \dfrac{27}{23}


Question: Evaluate the improper integral: $$\int_0^{\infty}\frac{1}{x(1+x^2)}\,dx$$

Answer: \infty


Question: Let \(ABC\) be a triangle with area \(\Delta\). The triangle \(A'B'C'\) is formed by the altitudes \(h_a, h_b, h_c\) of \(ABC\) as its sides, and has an area \(\Delta' = 30\). The triangle \(A''B''C''\) is formed by the altitudes of \(A'B'C'\) and has an area \(\Delta'' = 20\). Find \(\frac{\Delta}{9}\).

Answer: 5


Question: Determine the values of the parameter $p$ for which the improper integral \( \int_{0}^{\pi/2}\tan^{p}(x) \, dx \) is convergent.

Answer: (-1, 1)


Question: Twenty people, consisting of 10 couples, are randomly seated at 5 tables with 4 seats each. What is the expected number of couples that sit together at the same table?

Answer: \dfrac{30}{19}


Question:
Let \( 0 < \lambda < \frac{1}{2} \) be a parameter. Let \( \phi:[0,1] \to [0,\lambda] \) be a smooth strictly increasing bijective function such that \( \phi(0)=0 \) and \( \phi(1)=\lambda \). Determine if the following inequality holds for every such function \( \phi \):
\[ A(\phi):=\int_0^1 (\phi'(r))^2r+\frac{(\phi(r))^2}{r} \, dr \ge \lambda^2. \]
Note that equality holds for the homogeneous scaling \( \phi(r)=\lambda r \).

Answer: A(\phi) \ge \lambda^2


Question: Calculate the surface integral of the vector field \( \vec{F} = k \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \) over the sphere \( S: |\vec{r}| = a \), where \( \vec{r}_0 = \frac{3a}{5}(1,1,-1) \) is a point outside the sphere. Use Gauss' theorem to show that the integral \( \int_S \vec{F} \cdot \mathrm{d}\vec{S} \) is zero.

Answer: 0


Question: Find the value of $R$ such that the maximum value of $x-y+z$ under the constraint $x^2+y^2+z^2=R^2$ is $\sqrt{27}$. Use the method of Lagrange multipliers to solve the problem.

Answer: 3


Question: Let \( X \) be an infinite set and \( f: X \to X \) a function. Determine if the cardinality of the set of functions \( \{g: X \to X : g \circ f = f \circ g\} \) is at least \( |X| \).

Answer: Yes


Question: Given two independent random variables $X$ and $Y$ that are uniformly distributed over the interval $(0,1)$, let $Z = \max(X, Y)$ and $W = \min(X, Y)$. Find the covariance $\mathrm{Cov}(Z, W)$. Justify any assumptions or theorems used in your solution.

Answer: \dfrac{1}{36}


Question: Determine whether the cubic equation \(x^3 + 10x^2 - 100x + 1729 = 0\) has at least one complex root \(\alpha\) such that \(|\alpha| > 12\).

Answer: Yes


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

Answer: e^x


Question: Determine whether the following statement is true or false: A continuous function $f : \mathbb{R} \to \mathbb{R}$ is uniformly continuous if it maps Cauchy sequences into Cauchy sequences.

Answer: B


Question: Solve the equation in \( \mathbb{R} \): \[ 3x + \sqrt[3]{x+1} + \sqrt[3]{(x+1)^2} = -1 \]

Answer: -\dfrac{19}{27}


Question: Find the limit \( \lim_{n \to \infty} a_n \) of the coefficients \( a_n \) in the power series expansion of \((1-x)^{-\frac{1}{2}}\).

Answer: 0


Question: Find a polynomial \( f \in \mathbf{Z}_{21}[x] \) such that \( \deg f = 3 \) and \( f \) has nine distinct roots. Use the Chinese Remainder Theorem to assist in your solution.

Answer: x^3 + 18x^2 + 2x


Question: Find the value of $x \geq 2$ that maximizes the function $f(x) = W(x) - (\ln(x) - \ln{\ln{x}})$, where $W(x)$ is the Lambert-W function.

Answer: e^{e + 1}


Question: Find the coefficient of \(x^{21}\) in the expansion of \((1+x+x^2+x^3+x^4+x^5)^6\).

Answer: 1666


Question: Calculate the value of \( \int \int _ { S } \vec { F } \cdot \hat { n } \, d s \) where \( \vec { F } = x \hat { i } + y \hat { j } + z \hat { k } \) and \( S \) is the sphere defined by \( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4 \). Use the divergence theorem to evaluate the integral.

Answer: 32\pi


Question: Find the value of \( b \) such that the tangent line to the function \( f_a(x) = (x-a)e^{a+2-x} \) at \( x = b \) intersects the y-axis at the point \( A(0, 2012) \).

Answer: -1


Question: Let $V$ be an $n$-dimensional vector space over a field $F$, with basis $\mathcal{B} = \{\mathbf{v_1, \cdots, v_n}\}$. Let $\mathcal{B}^{*} = \{\phi_1, \cdots, \phi_n\}$ be the dual basis for $V^{*}$. Suppose $\psi : V \to V$ is a linear transformation and $A$ is the matrix representing $\psi$ with respect to $\mathcal{B}$. Let $\psi^t : V^{*} \to V^{*}$ be the dual transformation and $B$ be the matrix representing $\psi^t$ with respect to $\mathcal{B}^{*}$. Determine the relationship between the matrices $A$ and $B$.

Answer: B = A^\top


Question: Construct a closed, uncountable, perfect, nowhere dense subset of $[0,1]$ which has Lebesgue measure $\frac{1}{2}$. (Hint: Find the Cantor subset of $[0, 1]$ with Lebesgue measure $\frac{1}{2}$.)

Answer: C


Question: Determine the fundamental group of the complex projective space \( \mathbb{P}^n(\mathbb{C}) \) for \( n \geq 1 \).

Answer: 0


Question:
Given that $f, g \in L_{3}(\mathbb{R})$ with $\|f\|_{L_{3}(\mathbb{R})}=\|g\|_{L_{3}(\mathbb{R})}=2$, find the upper bound for the expression:
$$ \left|\int_{\mathbb{R}} f^{2}(x) g(x) \, dx\right| $$ using Hölder's Inequality.

Answer: 8


Question:
Calculate the positive integer $A$ such that the following equation holds:
\[ 4 + \dfrac{1}{2 + \dfrac{1}{1 + \dfrac{1}{3 + \dfrac{1}{1+\dfrac{1}{2 + \dfrac{1}{8 + \dfrac{1}{\ddots}}}}}}} = \sqrt{A} \]
The pattern of the nested fraction repeats as 2, 1, 3, 1, 2, 8 infinitely, but 4 appears only once at the beginning.

Answer: 19


Question: Evaluate the limit: $$ \lim _{n\rightarrow \infty} \sum _{j=n} ^{4n}  {4n \choose j} \left(\frac{1}{4}\right)^j \left(\frac{3}{4}\right)^{4n-j} $$

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^{n} \left(1 - \frac{x}{n}\right)^{n} e^{x/2} \, dx. \]

Answer: 2


Question: Suppose $f(x + iy) = u(x,y) + iv(x,y)$ is a complex function and $\frac{\partial v(x,y)}{\partial x}$ is not continuous at $(0,0)$. Does this imply that $f(x+iy)$ is not differentiable at $(0,0)$?

Answer: No


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

Answer: 0


Question: Consider a random line in the plane that passes through the origin, where the angle \( \theta \) with respect to the x-axis is uniformly distributed between 0 and \( \pi \). Determine the expected value of the slope of this line.

Answer: 0


Question: Consider a $p$-adic local field $K$, such as $\mathbb{Q}_p$. Let $G$ be the absolute Galois group of $K$, and let $G^v$ (for $v \ge -1$) denote the ramification groups in upper numbering. Determine whether the intersection $\bigcap_{v=0}^\infty G^v$ is trivial, i.e., $\{0\}$. Provide a justification for your answer.

Answer: \{1\}


Question: Determine the minimum number of terms required in the Taylor series expansion of \( \ln(1+x) \) about 1 to approximate \( \ln(4) \) with an error less than \( 2^{-10} \).

Answer: 1024


Question: What is the least possible degree of a polynomial with real coefficients that has the roots \(2\omega, 2+3\omega, 2+3\omega^2, 2-\omega-\omega^2\)?

Answer: 5


Question:
Given the joint probability density function of the continuous random variable $(X,Y)$:
\[
f_{X,Y}(x,y) =
\begin{cases}
ce^{-y}e^{-\frac{(y-1-x)^2}{2}},  & \text{x > 0} \\
0, & {x \le 0}
\end{cases}
\]
Find the probability density function of $X$, denoted as $f_X(x)$. To find $f_X(x)$, use the definition:
\[
f_X(x) = \int_{\mathbb{R}} f_{X,Y}(x,y) \, \mathrm{d}y.
\]
Determine the constant $c$ by ensuring that the double integral of $f_{X,Y}(x,y)$ over all $\mathbb{R}^2$ equals 1.

Answer: e^{-x}


Question: Given the sequence defined by \(a_n = n^{a_{n+1}}\) with \(a_{100} = 100\), find the units digit of \(a_2\).

Answer: 2


Question: Given a complete probability space $(\Omega,\mathcal{A},\operatorname{P})$ with a complete filtration $(\mathcal{F}_t)_{t\ge0}$, and a continuous local $\mathcal{F}$-martingale $(M_t)_{t\ge0}$, consider the process defined by $$N^\sigma := e^{-\frac{\sigma^2}{2}[M] + \sigma M}$$ for all $\sigma \in \mathbb{R}$. Assume that $$\operatorname{E}\left[e^{\lambda[M]_t}\right] < \infty$$ for all $t > 0$ and $\lambda > 0$. Can we conclude that $N^\sigma$ is a $\mathcal{F}$-martingale for all $\sigma \in \mathbb{R}$?

Answer: Yes


Question: Let \( H \) be a Hilbert space with an orthonormal basis \((e_n)_{n \in \mathbb{N}}\) and \( F \) a linear operator such that \( \langle e_k, F e_n \rangle = \phi(n,k) \). Find an estimate for the operator norm \( \lVert F \rVert \) in terms of \( \phi(n,k) \). Apply this estimate to the case where \( \phi(n,k) = \frac{1}{n+k} \).

Answer: \pi


Question: For how many integers \( n \) in the inclusive range \([1, 500]\) is the sum of all positive divisors, \( \sigma(n) \), a prime number?

Answer: 7


Question: Evaluate the limit: \[ \lim_{x\to 0}  \frac{\sqrt{1+4x} -1 -\sin(2x)}{\log(1+x^2)} \]

Answer: -2


Question: Evaluate the surface integral \( \int\int_S z \, dx \, dy \) over the complete surface \( S \) of a circular cylinder with radius \( a \) and axis along the \( OZ \) direction, where the ends of the cylinder are in the planes \( z = 0 \) and \( z = 2a \).

Answer: 2\pi a^3


Question: Evaluate the integral \( \int_{[-1,2]} x \, d\mu(x) \), where \( \mu \) is a measure defined on \([-1,2]\) such that for the set \([-1,2]/\{-1,0,1,2\}\), \( \mu(A) = \int_{A} x^2 \, dx \), and for the set \( B = \{-1,0,1,2\} \), \( \mu(x) = \frac{1}{4} \) for all \( x \in B \).

Answer: \dfrac{17}{4}


Question: For any finite extension of nonarchimedean local fields $K/L$, does there always exist an extension of their residue fields $\kappa/\lambda$?

Answer: Yes


Question: Evaluate the contour integral \( \oint_{|z|=1} \frac{(1 + z)^4}{z^3} \, dz \) using the Cauchy Integral Formula or the Residue Theorem.

Answer: 12\pi i


Question: Provide an example of a Noetherian ring that is not Euclidean.

Answer: \mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]


Question: Find the derivative of \( \text{Tr}(S^T SUU^T) \) with respect to the matrix \( U \).

Answer: 2 S^T S U


Question: Is the set $E_\kappa^+$ always a proper subset of $E_\kappa$ for an uncountable regular cardinal $\kappa$?

Answer: Yes


Question: Evaluate the integral \( \int_{|z|=8}\frac{1+z}{1-\sin z}dz \) using the residue theorem.

Answer: 12\pi i


Question: Given a complete theory $T$ and a collection of formulae $\{\phi_i\}_{i \in I}$ such that for any complete $n$-type $\Sigma$ of $T$, there exists some $i \in I$ with $\Sigma \in A_{\phi_i}$, determine whether there exists a finite subcollection $\{\phi_i\}_{i \in I'}$ that satisfies the same property for all complete $n$-types of $T$. Justify your answer.

Answer: Yes


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

Answer: \infty


Question: Given a Poisson arrival process with an arrival rate of 4 per year, each arrival at time $t$ rewards the holder with $p(t) = \frac{1}{2t}$. Calculate the expected reward gained in a year.

Answer: \infty


Question: Evaluate the limit: $$\lim_{x \rightarrow 0} \frac{x-\overbrace{\sin (\sin (...(\sin x)...))}^{150\ \text{times}\ \sin}}{x^3}$$

Answer: 25


Question: Given that \(A + B + C = \pi\) and \(\sin 2A : \sin 2B : \sin 2C = 5 : 12 : 13\), find the value of \(A\).

Answer: \arctan 5


Question: Let $E$ be an elliptic curve and $d$ be a squarefree integer. If $E'$ and $E$ are isomorphic over $\mathbb{Q}(\sqrt{d})$, must $E'$ be a quadratic twist of $E$? Answer yes or no.

Answer: Yes


Question: Let $f: \mathbb{R}^a \to \mathbb{R}^b$ and $g: \mathbb{R}^c \to \mathbb{R}^d$ be $C^\infty$ functions. Define the Cartesian product of these functions as $(f \times g)(x, y) = (f(x), g(y))$. Is the function $f \times g$ also $C^\infty$?

Answer: Yes


Question: Determine the maximum number of pair-wise non-similar linear transformations on a 3-dimensional vector space, where each transformation has the characteristic polynomial \((\lambda -1)^3\).

Answer: 3