Question:
In a group of exponent $2^n$, is the following equality true?

\[ [x^{2^{n-1}}, \underbrace{y^{2^{n-1}}, \ldots, y^{2^{n-1}}}_n] = 1. \]

Here, \([a, b, c] = [[a, b], c]\).

Answer: 1


Question: Determine whether the following statement is true or false: For a prime number $p > 2$ and $n \in \mathbb{N}$, the sum of combinatorial coefficients $$\sum_{i=0}^{\lfloor n/p\rfloor}(-1)^i {n\choose ip}=0$$ if and only if $n=(2k-1)p$ for some $k \in \mathbb{N}$.

Answer: True


Question: Find the maximum area of a triangle formed in the first quadrant by the x-axis, y-axis, and a tangent line to the graph of \( f(x) = (x + 2)^{-2} \).

Answer: \dfrac{1}{4}


Question: Evaluate the limit: \[ \lim_{(x,y)\to (1,2)} \frac{xy^2-4xy-y^2+4x+4y-4}{x^2+y^2-2x-4y+5} \]

Answer: 0


Question: Let \(S\) be a point chosen at random from the interior of the square \(ABCD\), where \(AB\) is a side and \(AC\) is a diagonal. Determine the probability \(P\) that the segments \(AS\), \(SB\), and \(AC\) can form a triangle. Express \(P\) in the form \(\frac{a-\pi\sqrt{b}-\sqrt{c}}{d}\), where \(a\), \(b\), \(c\), and \(d\) are positive integers and \(d\) is minimized. Find the value of \(ab + cd\).

Answer: 160


Question: Let $E$ be the intersection of the cylinders $x^{2}+y^{2} \leq 1$ and $y^{2}+z^{2} \leq 1$. Compute the flux \( \iint_{\partial E} \vec{F} \cdot d\vec{S} \) where \( \vec{F} = (x y^{2} + \cos(y z)) \hat{i} - (x^{2} + \sin(z x)) \hat{j} + (z + \cos(x y)) \hat{k} \) and \( \partial E \) is oriented outward.

Answer: \dfrac{32}{5}


Question: Compute the conditional probability \( P\left(\inf_{t\in [1,2]} W_t < 0 ~ \middle| ~ W_1 >0,~ W_2 >0\right) \) for a Standard Brownian Motion \((W_t)_{t\ge0}\).

Answer: \dfrac{1}{3}


Question: Find the point of contact of the plane \(2x-2y+z+12=0\) with the sphere \(x^2+y^2+z^2-2x-4y+2z=3.\)

Answer: (-1, 4, -2)


Question: Consider a probability space $(\Omega, \mathcal{A}, P)$ and a real Hilbert space $\mathcal{H}$. Let $X: \Omega \to \mathcal{H}$ and $Y: \Omega \to \mathcal{H}$ be two $\mathcal{H}$-valued random variables such that for all $\omega \in \Omega$, $X(\omega)$ and $Y(\omega)$ belong to a ball $C \subset \mathcal{H}$ of radius $\frac{r}{2}$ centered at the origin. Determine whether the inequality $\|X - Y\|^2 \leq r^2$ holds.

Answer: \|X - Y\|^2 \leq r^2


Question: Simplify \( \frac{2^{2017}+1}{3 \cdot 2^{2017}} \) to \( \frac{n}{m} \) where \( n \) and \( m \) are coprime. Find the remainder when \( m+n \) is divided by 1000.

Answer: 763


Question: Suppose \(\phi: \mathbb{Z}_{20} \to \mathbb{Z}_{20}\) is an automorphism and \(\phi(5) = 5\). Determine the number of possible mappings for \(\phi(x)\).

Answer: 4


Question: Given that \( \phi \) is a solution of the integral equation \( \int_{0}^{x} (1-x^2+t^2)\phi(t)dt=\frac{x^2}{2} \), find the value of \( \phi(\sqrt{2}) \).

Answer: \sqrt{2} e^{2}


Question: Evaluate the limit: \[ \lim_{t \to \infty} t\log\left(\dfrac{\dfrac{\log(\alpha + 1)}{t} - \dfrac{\log(t + \alpha)}{t}}{ 1 - \dfrac{1}{t(t + \alpha)}} + 1\right) \] where \( t \in \mathbb{N} \) and \( \alpha > 0 \).

Answer: -\infty


Question:
Consider a short exact sequence of $\mathfrak{g}$-modules:
\[ 0 \rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 \]
where $\mathfrak{g}$ is a finite-dimensional, semisimple Lie algebra over $\mathbb{C}$. If $X$ and $Z$ are in the category $\mathcal{O}$, is $Y$ also in the category $\mathcal{O}$?

Answer: Yes


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

Answer: -\dfrac{4}{3}


Question: Determine the dimension of the vector space \( \mathbb{C}[x,y]/I \) over \( \mathbb{C} \), where \( I = \langle x^2 + 4x + 4, xy+x+2y+2, y^3 + 3y^2 + 3y + 1 \rangle \).

Answer: 4


Question: A coin is tossed repeatedly until either two heads or two tails appear consecutively. The game ended with two tails. What is the probability that the game started with a head? Express your answer as a fraction.

Answer: \dfrac{1}{3}


Question: Determine the degree of extension of the algebraic closure over the field \( \mathbb{Q}_p^{ext} = \mathbb{Z}((X))_{conti}/(X-p) \), where \( p \) is a prime number.

Answer: \infty


Question: Evaluate the integral \( \int_{-2}^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_{-\sqrt{4-x^2-y^2}}^{\sqrt{4-x^2-y^2}} (z^3 \cos xyz - 3) \, dz \, dy \, dx \) without using integration. Provide a brief explanation of your reasoning.

Answer: -32\pi


Question:
Given the recurrence relation with initial conditions $a_0 = a_1 = a_2 = a_3 = a_4 = 0$ and $a_5 = 1$, and the formula:
\[ a_{n+6} = \frac{a_{n+5} + a_{n+4} + a_{n+3} + a_{n+2} + a_{n+1} + a_{n}}{6} \]
Find the limit of $a_n$ as $n$ approaches infinity.

Answer: \dfrac{2}{7}


Question: Determine the values of $p \in (0,1)$ for which the series \( \sum_{n=1}^\infty c_n \cdot \frac{1}{n} \) converges, where \( c_n = \begin{cases} 1 &; \lceil np \rceil - \lceil (n-1)p \rceil = 1 \\ -1 &; \lceil np \rceil - \lceil (n-1)p \rceil = 0 \end{cases} \).

Answer: \dfrac{1}{2}


Question: Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the equation \[ f\left(\frac{x+y}{2}\right) = \frac{f(x) + f(y)}{2} \] for all real numbers \( x \) and \( y \).

Answer: f(x) = ax + b


Question: Are the very large cardinal axioms $I_0$, $I_1$, and $I_2$ consistent with the Continuum Hypothesis (CH) and the Generalized Continuum Hypothesis (GCH)?

Answer: Yes


Question: Determine the number of field homomorphisms from $\mathbb{Q}(\sqrt[4]{2})$ to $\mathbb{R}$.

Answer: 2


Question:
Let A be a 14x14 square matrix with rank 12, and suppose that \( \lambda = 0 \) is an eigenvalue of A with algebraic multiplicity 4. Determine which of the following statements is true:

1. \( \text{rank}(A^2) = 12 \).
2. \( \text{rank}(A^3) \leq 11 \).
3. There is no matrix satisfying the given conditions.

Answer: 2


Question: Solve the system of equations: \[ \begin{cases} x + y^2 = 7 \\ x^2 + y = 11 \end{cases} \]

Answer: (3, 2)


Question: Find the minimum value of \( F(x,y,z) = \frac{1}{x+y} + \frac{1}{x+z} - \frac{1}{x+y+z} \) subject to the constraints \( 0 \leq x+y, y+z, z+x \leq 1 \) and \( 0 \leq x, y, z \leq 1 \).

Answer: 1


Question:
Calculate the line integral \( \oint_C P\,dx + Q\,dy \) over the ellipse \( \frac{x^2}{25} + \frac{y^2}{36} = 1 \), where the vector fields are given by:
\[
P = \frac{-y}{(x-1)^2 + y^2}, \quad Q = \frac{x-1}{(x-1)^2 + y^2}
\]
Determine the value of the integral, considering that the vector field is undefined at the point \((1,0)\) inside the ellipse.

Answer: 2\pi


Question: Is there a calculable function that can transform a single uniformly distributed random value in the range \(0 \leq x < 1\) into a normally distributed value with mean 0 and standard deviation 1? If an exact function does not exist, is there an approximation?

Answer: \Phi^{-1}(x)


Question:
Find the asymptotic expression for the average number of prime factors of a number as the number of digits goes to infinity. More formally, determine the asymptotic behavior as $N \to \infty$ of
\[ \frac{\sum_{1\le k\le N} M(k)}{N} \]
where
\[ M(p_1^{d_1}p_2^{d_2}\cdots p_k^{d_k}) = d_1+d_2+\cdots+d_k \]
For example, $M(24) = M(2^3\cdot3) = 4$. Provide your answer in terms of $N$. \( \boxed{} \)

Answer: \log \log N


Question: Find \( \lim_{r \to \infty} \frac{f(r)}{\pi r^2} \), where \( f(r) \) is the number of integral points inside a circle of radius \( r \) centered at the origin.

Answer: 1


Question: Compute the second homotopy group \( \pi_2(X) \) for the space \( X = D^2 \cup_f S^1 \), where \( f : S^1 \to S^1 \) is a degree \( m \) map.

Answer: \mathbb{Z}


Question: In a triangle $\Delta ABC$ inscribed in a circle $w$ with radius $1$, the angle $\angle BAC$ is $60^\circ$. A circle with center $I$ is inscribed in $\Delta ABC$. The line $AI$ intersects the circle $w$ at point $D$. Determine the length of $ID$. \( \text{Express your answer as a single number.} \)

Answer: 1


Question: Find the value of $y \in [0, 1]$ that maximizes the integral \( \int_{0}^{y} \sqrt{x^4 + (y - y^2)^2} \, dx \).

Answer: 1


Question: Evaluate the integral of the function \( f(x,y) = x^2 \sin(y) \) over the surface defined by \( g(x,y) = 2x - 2y \) on the domain \([0,1] \times [0,\pi]\).

Answer: 2


Question: Calculate \( \lim_{h\rightarrow 0} \dfrac {e^{f(z+h)}-e^{f(z)}}{f(z+h)- f(z)} \) given that \( f \) is a continuous complex function in an open subset \( V \) of \( \mathbb{C} \) and \( z \in V \).

Answer: e^{f(z)}


Question: The curve defined by the differential equation \( \frac{dy}{dx}=\frac{y^3}{e^x +y^2} \) passes through the point \((0,2)\). The line \(x=\ln 5\) intersects the curve at points where \(y=a\) and \(y=b\). Calculate the value of \(\frac{4(a^2+b^2)}{53}\).

Answer: 5


Question: Determine the number of non-isomorphic abelian groups of order $19^5$.

Answer: 7


Question: Let $X_1, X_2, \ldots, X_n$ be a random sample from a normal distribution $N(\mu, \sigma^2)$. Define the random variable $Y = c(\bar{x} - \mu)^2 / S^2$, where $\bar{x}$ is the sample mean and $S^2$ is the sample variance. Find the constant $c$ such that $Y$ follows a named distribution.

Answer: n


Question:
Calculate the variance of a Cox-Ingersoll-Ross process as \( t \rightarrow 0^+ \). The process is defined by:
\[ d X(t) = \alpha (\mu - X(t))dt + \sigma \sqrt{X(t)} dW(t) \]
with the variance given by:
\[ Var(X(t))= X(0)\bigg(\frac{\sigma^2}{\alpha}\bigg)(e^{-\alpha t}-e^{-2\alpha t}) + \mu\bigg(\frac{\sigma^2}{2 \alpha}\bigg)(1-e^{-\alpha t})^2 \]
Find \( \lim_{t \rightarrow 0} Var(X(t)) \).

Answer: 0


Question: Suppose $C, D \subseteq \mathbb{R}$. If $C$ is compact and $D$ is closed, is it true that there exist points $c \in C$ and $d \in D$ such that $d(C, D) = |c - d|$? Justify your answer.

Answer: Yes


Question: Let $\gamma$ be a smooth curve in $M:= \mathbb{R}^2 \setminus \{(0,0)\}$ that goes once around the origin, and assume that the image of $\gamma$, denoted by $N$, is a submanifold of $M$. Endow $N$ with the counterclockwise orientation. Compute $\int_N i^* \alpha$ where $\alpha = \frac{xdy-ydx}{x^2 + y^2}$ and $i: N \to M$ is the inclusion.

Answer: 2\pi


Question:
Calculate the expected value of the random variable \( a_n = \frac{1+i}{2+n} \) where the probability mass function is given by:
\[
P(i) = \binom{n}{i} \frac{(2i-1)!!(2(n-i)-1)!!}{(2n)!!}
\]
Express the expected value \( E(a_n) \) as:
\[
E(a_n) = \sum_{i=0}^{n} \frac{1+i}{2+n} \binom{n}{i} \frac{(2i-1)!!(2(n-i)-1)!!}{(2n)!!}
\]

Answer: \dfrac{1}{2}


Question:
Let $X$ be a separable metric space with the Borel $
obreak\sigma$-algebra $
obreak\mathscr{B}$ generated by its open sets. Does there exist a probability measure $
obreak\mu:
obreak\mathscr{B}
obreak\to [0,1]$ such that $
obreak\mu(B_q(x))>0$ for all $x
obreak\in X$ and $q>0$?

Answer: Yes


Question: Determine whether there exists an unbounded function \( f: \mathbb{R} \to \mathbb{R} \) such that there exists some \( M > 0 \) for which \( f(x) < \log(x) \) for all \( x > M \).

Answer: Yes


Question: Find the arc length of the curve defined by the equation \(24xy = x^4 + 48\) from \(x = 2\) to \(x = 4\).

Answer: \dfrac{17}{6}


Question: Consider a Polish space $X$ and an increasing sequence of compact subsets $(C_m)_{m\in\mathbb{N}}$ of $X$, with $C=\bigcup_{m}C_m$. Let $\{f_n:n\in\mathbb{N}\}$ be a family of functions from $X$ to $[0,1]$, equicontinuous on compact subsets of $X$. By the Arzelà-Ascoli theorem, there exists a subsequence $(f_{k_n})_{n\in\mathbb{N}}$ converging to a function $f$ uniformly on each $C_m$. Is it possible to choose such a subsequence so that the limit function $f$ is continuous on the entire set $C$?

Answer: Yes


Question:
Consider the initial value problem for the classical Burgers equation given by \( u_t + uu_x = 0 \) with the initial condition:
\[
 u(x,0) = \phi(x) = \begin{cases}
 2, & x \leq \pi/2 \\
 \sin x + 1, & \pi/2 < x \leq 3\pi/2 \\
 0, & x > 3\pi/2
\end{cases}
\]
Determine the breaking time \( t_B \) for this problem.

Answer: 1


Question: Given the function $f(x) = a^x + b^x + c^x$ for unknown non-negative integers $a$, $b$, and $c$, and the values $f(1) = 6$, $f(2) = 14$, and $f(3) = 36$, find the value of $f(4)$. Use the given values of $f(x)$ for $x < 4$ to determine $f(4)$ without directly solving for $a$, $b$, and $c$. Provide a method or formula that can be used to find $f(n)$ for $n > 3$ using $f(i)$ for $i < n$.

Answer: 98


Question: Find the point of intersection of the tangents to the parabola \(y^2=4x\) at the points where the circle \((x-3)^2+y^2=9\) meets the parabola, other than the origin.

Answer: (-2, 0)


Question: Determine whether the sequence of independent random variables \(X_n \sim U(1, 1+1/n)\) converges almost surely.

Answer: 1


Question: Is the space $C^1_b(B)$ of continuously differentiable functions, which are bounded and have bounded derivatives, dense in $L^p(B, \mu)$ for every $p \ne \infty$?

Answer: Yes


Question: Evaluate the integral: \(\int_0^{2\pi}\cos(t) \, d\sin(t)\).

Answer: \pi


Question: Determine the number of consecutive composite integers that follow \( k! + 1 \).

Answer: k - 1


Question:
Given the system of equations:
\[\begin{cases}
x + y + z = 1 \\
x^2 + y^2 + z^2 = 14 \\
x^3 + y^3 + z^3 = 36
\end{cases}\]
Find the value of \(x^5 + y^5 + z^5\).

Answer: \dfrac{877}{2}


Question: Find the value of \( \lambda \) such that \( i \) is a root of the quadratic equation \( z^2 + \lambda z - 4 = 0 \).

Answer: -5i


Question: Find \( \lim_{n \to \infty} G_n \), where \( G_n \) is the geometric mean of the set \( \{ \sin(\frac{\pi}{2}\frac{k}{n}): 1 \le k \le n \} \).

Answer: \dfrac{1}{2}


Question:
Verify the computation of the divergence of the gravitational field given by:
\[ \mathbf{g} = \frac{Gm\mathbf{r}}{r^3} \]
Compute \( \nabla\cdot\mathbf{g} \) and show that it equals zero.

Answer: 0


Question: Assuming $P \neq NP$, determine whether the language $L_G = \{H \mid G \text{ contains a copy of } H\}$, for some fixed graph $G$, is in $P$ or $NPC$. Here, $G$ contains a copy of $H = (V_H, E_H)$ if there exists a subset $S \subseteq V_G$ of size $|V_H|$ such that the subgraph induced by $S$ is isomorphic to $H$. Provide a justification for your answer.

Answer: P


Question: Find the length of the curve of intersection between the parabolic cylinder $x^2 = 2y$ and the surface $3z = xy$, from the point $(0, 0, 0)$ to the point $(6, 18, 36)$. Use the parameterization $\langle t, 0.5t^2, \frac{1}{6}t^3 \rangle$ to assist in your calculations.

Answer: 42


Question: You have two coins: a fair coin with a probability of heads as 0.5, and a biased coin with a probability of heads as 0.8. You randomly select one of these coins and flip it repeatedly, obtaining heads each time. Determine the minimum number of consecutive heads required such that the probability of the coin being fair is less than 0.1.

Answer: 5


Question: Find the limit: \[ \lim_{n \rightarrow \infty} \int_0^{1} \min\left( \frac{e^{-nx^{2}}}{\sqrt{x}}, n \right) \, d\lambda \] where \( \lambda \) is the Lebesgue measure on the interval \([0,1]\).

Answer: 0


Question:
Let \( f \) be a differentiable function such that \((x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2)\) and \( f(1)=2 \). Find the area enclosed by the inequality:
\[ \frac{|f(x)-x|^{1/3}}{17} + \frac{|f(y)-y|^{1/3}}{2} \le \frac{1}{4} \]

Answer: \dfrac{17}{4}


Question: Find a closed-form expression for the series \( \sum_{n=0}^\infty\frac{(1/2)_n}{n!}(H_n-H_{n-1/2}) \), where \( H_z \) is the harmonic number.

Answer: 2


Question: Determine the domain of the function \( \ln(x)^{\ln(x)} \).

Answer: (1, \infty)


Question: If a cardinal \( \kappa \) is real-valued measurable, does it necessarily follow that \( 2^\kappa \) is also real-valued measurable?

Answer: No


Question: Is every square traceless matrix unitarily similar to a zero-diagonal matrix? Provide a proof or counterexample to support your answer.

Answer: Yes


Question: Evaluate the double integral: \[ I = \int_{0}^\infty \int_{x}^\infty \frac{1}{y} e^{-\frac{y}{2}} \, dy \, dx \]

Answer: 2


Question: Compute the line integral \( \int _L ye^{xy}dx + xe^{xy}dy \) along the path \( L : x = 2ye^y \) from the point \((0,0)\) to \((2,1)\).

Answer: e^2 - 1


Question:
Determine if the function \( f(x,y) = 2xy + \frac{x}{y} \) is differentiable at the point \( a = (1,1) \). Calculate the partial derivatives and use them to find the linear transformation. Then, evaluate the limit:
\[ \lim \limits_{[h,k] \to [0,0]} \frac{\frac{k^2+hk+2hk^2}{1+k}}{||[h,k]||} = 0 \]
where \(||[h,k]||\) is the Euclidean norm. Show that this limit equals zero to confirm differentiability.

Answer: 0


Question: Determine whether the ring of convergence of the Laurent series for \( f(z) = \frac{1}{e^z - 1} \) centered at 0 is the exterior of the unit disk \(|z| \geq 1\). If so, what is the radius of convergence?

Answer: 2\pi


Question: Consider flipping $n$ independent coins, where the $i$-th coin has a probability of $\frac{1}{i}$ of being heads. As $n$ approaches infinity, prove or disprove that for any given positive integer $k$, the probability of obtaining at least $k$ heads is $1$.

Answer: 1


Question: Compute the dimension of the vector space \( \Bbb C[x,y]/I \) over \( \Bbb C \), where \( I = \langle (x+2)^2, (x+2)(y+1), (y+1)^3 \rangle \).

Answer: 4


Question: Determine the dimension of the splitting field of the cubic polynomial \(x^3 - 5\) over \(\mathbb{Q}\).

Answer: 6


Question: Calculate the integral \( \int_{\Sigma}(2x \, dy\land dz + 2y \, dz\land dx + (x^2+y^2) \, dx\land dy) \), where \( \Sigma \) is the boundary of the region \( D \) defined by \( 0\leq z\leq 1-x^2-y^2 \). The surface \( \Sigma \) is oriented such that the outward normal vectors point away from the region \( D \).

Answer: 2\pi


Question: Let \(\mathcal{F}\) be the family of holomorphic functions \(f\) on the open unit disk such that \(\Re f > 0\) for all \(z\) and \(f(0) = 1\). Compute \(\alpha = \sup\{|f'(0)| : f \in \mathcal{F}\}\). Determine whether or not the supremum \(\alpha\) is attained.

Answer: 2


Question: Is there a first-countable compact space $X$ with an uncountable Cantor-Bendixson index?

Answer: No


Question: Let $x$ be a finite set such that $\bigcup x = x$. Determine whether $x$ must be the empty set.

Answer: \emptyset


Question: Given that \( \frac{\cos x}{\cos y} + \frac{\sin x}{\sin y} = -1 \), find the value of \( 4\left(\frac{\cos^3 y}{\cos x} + \frac{\sin^3 y}{\sin x}\right) \).

Answer: 4


Question:
Determine whether the following inequality about probability is true:
\[ \sum_{i=1}^{\infty}P(T\geq i)\leq \sum_{i=0}^{\infty}cP(T\geq ci) \]
where \(c > 1\) is an integer and \(T\) is a random variable taking non-negative integer values.

Answer: True


Question: Use Cauchy's integral formula to compute the contour integral \( \int \limits_{\Gamma} \frac{e^{-z}}{z-1}dz \), where \( \Gamma \) is the square with sides parallel to the axes, centered at \( i \), and with side length 5, traversed in the anticlockwise direction.

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


Question:
Let $p$ and $q$ be distinct primes. Suppose that $H$ is a proper subset of the integers and $H$ is a group under addition that contains exactly three elements of the set \(\{p, p+q, pq, p^q, q^p\}\). Determine which of the following are the three elements in $H$.

(a) $pq, p^q, q^p$
(b) $p, p+q, q^p$
(c) $p, p^q, q^p$
(d) $p, pq, p^q$

Answer: D


Question: Let $n \geq 2$ and $1 \neq g \in F_n$, where $F_n$ is the free group of rank $n$. Is there an epimorphism $\phi : F_n \to F_2$ such that $\phi(g) \neq 1$?

Answer: Yes


Question: Evaluate the integral \( \int^1_0 \frac{1}{\sqrt{x}} \left( \int^\sqrt{x}_1 e^{-t^2} \, dt \right) \, dx \).

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


Question: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that for every $x \in \mathbb{R}$, there exists an integer $N_x > 0$ with $f^n(x) = c$ for all $n \geq N_x$, where $c$ is a constant. Is it possible for $f^n$ to be non-constant for any $n$?

Answer: Yes


Question: Let \( A \) be a 2x2 square matrix with \(|A| \neq 0\) such that \(|A + |A| \text{adj}(A)| = 0\), where \(|A|\) and \(\text{adj}(A)\) denote the determinant and adjugate of matrix \(A\), respectively. Find \(|A - |A| \text{adj}(A)|\).

Answer: 4


Question: Determine a generator of the multiplicative group \( \mathbb{Z}^{*}_{242} \).

Answer: 7


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

Answer: \ln 2


Question: For iid random variables $X_1, X_2 \sim \mathcal{N}(\mu, \sigma)$, determine the values of $a$ such that $aX_1 + (1-a)X_2$ is independent of $aX_1 - (1-a)X_2$. Provide your answer in terms of $a$.

Answer: \dfrac{1}{2}


Question: Given that $a = 1$, the random variable $m$ follows a uniform distribution $U(0.2, 1)$. If $a = 0$, then $m$ follows a uniform distribution $U(0, 0.5)$. If $m$ is observed to be 0.3, what is the probability that $a$ equals 1?

Answer: \dfrac{5}{13}


Question: Let \( \Omega \) be a bounded domain in \( \mathbb{R}^n \) and let \( y \in H^2(\Omega) \cap H_0^1(\Omega) \) such that the set \( \{x \in \Omega \mid y(x) \neq 0\} \) has non-zero measure and \( \frac{\Delta y}{y} 1_{\{x \in \Omega \mid y(x) \neq 0\}} \in L^\infty(\Omega) \). Determine if there exists a constant \( C > 0 \) such that the inequality \( |\Delta y(x)| \le C |y(x)| \) holds for almost every \( x \in \Omega \).

Answer: Yes


Question: Given two independent standard normal variables $X$ and $Y$, determine the correlation of the samples $(x, y)$ that satisfy the condition $x + y > 0$. Assume $n$ samples are generated. What is the correlation of these samples?

Answer: -\dfrac{1}{\pi - 1}


Question: Does there exist a distance $d$, a point $x_0$ in $\mathbb{R}^2$, and a radius $r \in (0, +\infty)$ such that the ball $B(x_0, r) = \mathbb{R}^2$?

Answer: Yes


Question: Given the function $f: \mathbb{R} \to \mathbb{R}$ satisfying the equation \( \frac{f(x)f(y)-f(xy)}{3} = x+y+2 \) for all real numbers $x$ and $y$, determine all possible values of $f(36)$. Provide your answer as a list of values.

Answer: 39


Question: Evaluate the integral \( \int_{0}^{+\infty }\frac{1}{1+x^{2}} \, dx \) using the Fourier transform of the function \( e^{-\left | x \right |} \), given by \( \mathfrak{F}\left \{ e^{-\left | x \right |} \right \}=\frac{2}{1+\omega ^{2}} \).

Answer: \dfrac{\pi}{2}


Question: In an acute triangle ABC, the altitudes AN, BL, and CM are drawn. Given that \( \angle C = 45^\circ \) and CM = 6, find the distance from vertex C to the line segment NL.

Answer: 3\sqrt{2}


Question: Consider an $n \times n$ nilpotent complex matrix $A$. Determine the minimum integer $d$ such that $A^d = 0$ for any such matrix $A$.

Answer: n


Question: Determine which is larger: $e^\pi$ or $\pi^e$. Provide your reasoning without using a calculator.

Answer: e^\pi


Question: Find the surface area of the solid formed when the curve \(8y^2 = x^2 - x^4\) is revolved about the x-axis, for \(x\) in the interval \([-1, 1]\).

Answer: \dfrac{\pi}{2}


Question: Let $w_1$ and $w_2$ be two permutations of the set $\{1, 2, \ldots, k\}$ such that $w_1(i) \neq w_2(i)$ for all $1 \leq i \leq k$. Given two relatively prime integers $m$ and $n$, determine if there exist two diagonal matrices $D_1, D_2 \in M_k(\mathbb{Z})$ such that $\gcd(\det D_1, n) = 1$, $\gcd(\det D_2, m) = 1$, and $\det(D_1 m (w_1 I) + D_2 n (w_2 I)) = 1$. Here, $\det$ denotes the determinant and $I$ is the $k \times k$ identity matrix, with $w_i I$ being the permutation matrix corresponding to $w_i$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the function \( P(x) \), where \( P(x) \) is a polynomial of degree 101, can be a bijective function from \( \mathbb{R} \) to \( \mathbb{R} \). Answer "True" if it can be bijective, or "False" if it cannot.

Answer: True


Question: Evaluate the flux integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \) where \( \mathbf{F} = \langle 5y, 2z, 3x \rangle \) and \( S \) is the part of the plane \( 6x + 2y + z = 12 \) in the first octant, oriented upward.

Answer: 468


Question:
Consider the following series: \( \sum_{n=1}^{\infty} |a_n| \) is convergent, and \( \sum_{n=1}^{\infty} b_n \) is convergent, but \( \sum_{n=1}^{\infty} |b_n| \) is not convergent. Determine which of the following statements is not always true:

1. You can change the order of the sequence so that \( \sum_{n=1}^{\infty} (a_n + b_n) \) will converge to 0.
2. You can change the order of the sequence so that \( \sum_{n=1}^{\infty} (a_n b_n) \) will converge to 0.
3. Any grouping of terms in \( \sum_{n=1}^{\infty} (a_n b_n) \) will result in a convergent series.
4. Any grouping of terms in \( \sum_{n=1}^{\infty} (a_n + b_n) \) will result in a convergent series.

Identify the statement that is not always true.

Answer: 2


Question: Determine whether every square matrix can be expressed as the product of an invertible matrix and a projection matrix. Specifically, for a given square matrix \( A \), does there always exist an invertible matrix \( B \) and a projection matrix \( P \) such that \( A = BP \)?

Answer: Yes


Question: Consider a 4 × 4 grid of squares, where each square can be colored either black or white. Two colorings are considered identical if one can be transformed into the other by any combination of flipping, rotating, or swapping the two colors (changing all black squares to white and vice versa). Determine the number of non-identical colorings of the grid.

Answer: 4324


Question: Let \( \Gamma_1 \) be a circle with center \( O \) and radius \( R \). Two other circles, \( \Gamma_2 \) and \( \Gamma_3 \), with centers \( O_2 \) and \( O_3 \) respectively, are internally tangent to \( \Gamma_1 \) and intersect each other at points \( A \) and \( B \). Given that \( \angle OAB = 90^{\circ} \), find the sum of the radii of \( \Gamma_2 \) and \( \Gamma_3 \).

Answer: R


Question: Determine the cyclic covering space of the Lens space $L(p,q)$, where $p$ is a composite number $p=ab$, induced by the quotient group homomorphism from $\mathbb{Z}/p$ to $\mathbb{Z}/a$. What is this covering space?

Answer: L(b, q)


Question: For prime numbers \(p\), does the Legendre symbol \(\left(\frac{5}{p}\right)\) depend only on the congruence class of \(p\) modulo 5? Justify your answer.

Answer: Yes


Question: Determine whether the integral \( \int_a^{\infty} \), where \( a \) is a finite number, can be rewritten in the form \( \int_0^1 \).

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{x^{49}}{(1+x)^{51}} \, dx \).

Answer: \dfrac{1}{50}


Question: Evaluate the limit: \[ \lim_{x\rightarrow0} \frac{\tan(x)-x}{x^3} \]

Answer: \dfrac{1}{3}


Question: Given any compact Riemannian manifold \((M,g)\), must the Ricci curvature on \(M\) be bounded? Provide a justification for your answer.

Answer: Yes


Question: Is a function \( f: X \to Y \) uniformly continuous if, for all \( \epsilon > 0 \), there exists \( \delta > 0 \) such that \( d(x,y) < \delta \) implies \( d'(f(x),f(y)) < \epsilon \) for all \( x, y \in X \), where \( X \) and \( Y \) are uniform spaces induced by pseudo metrics \( d \) and \( d' \) respectively?

Answer: Yes


Question: Determine the Galois group of the polynomial \( f(X) = X^4 + 2X^2 + 4 \) over \( \mathbb{Q} \).

Answer: V_4


Question: What is the greatest integer that divides \(p^4 - 1\) for every prime \(p > 5\)?

Answer: 240


Question: Determine the order of the pole of the function \( f(z) = \frac{\sin(3z)}{z^2} \) at \( z = 0 \).

Answer: 1


Question: Evaluate the limit \( \lim_{(x, y)\to (0,0)}\frac{y^3}{x^4+\sin^2y} \).

Answer: 0


Question: Determine whether the following claim is true or false: Let $f(x)$ be $2\pi$-periodic and continuously differentiable on the real line. Then the Fourier series of $f(x)$ converges to $f(x)$ uniformly on $[-\pi, \pi]$. Provide a justification for your answer.

Answer: True


Question: Determine whether the initial object of the category of pointed invertible unary systems is the integers \((\mathbb{Z}, 0, s)\). A pointed invertible unary system is defined as a tuple \((A, q, f)\) where \(f\) is a bijective function.

Answer: (\mathbb{Z}, 0, s)


Question: Evaluate the integral \( \int \int_R (x+y)^2 \sin(x-y) \, dA \), where \( R \) is the region in the plane bounded by the square with vertices \((1,0), (0,1), (1,2), (2,1)\).

Answer: 0


Question: Find the expected value of \( N = \min\{n: U_1 + U_2 + \cdots + U_n > 1\} \), where \( U_1, U_2, \ldots \) are independent uniform (0,1) random variables.

Answer: e


Question: Evaluate the surface integral \(\iint_{S} xz \, dy \, dz\) where \(S = \{(x,y,z) \mid x \geq 0, y \geq 0, z \geq 0, 2x + 2y + z = 2\}\).

Answer: \dfrac{1}{6}


Question: Determine whether the sequence of operators \( A_{n}x(t) = t^{n}(1-t)x(t) \) converges in the operator norm on \( C[0, 1] \).

Answer: Yes


Question: If \( \sum_{n=1}^\infty \frac{\log x^n}{(\log x)^n} = 4, \) find the value of \( x \).

Answer: e^2


Question: Calculate the value of \(i^i\).

Answer: e^{-\pi/2}


Question: Let \( A \in \mathbb{R}^{3\times3} \) be a matrix with eigenvalues \(1\), \(-1\), and \(0\). Calculate \( \det \left(I + A^{50} \right) \).

Answer: 4


Question: Calculate the probability \( P(Z > X + Y) \) where \( X, Y, Z \sim U(0,1) \) are independent random variables.

Answer: \dfrac{1}{6}


Question: Evaluate the integral \( \int_{\gamma} \frac{1}{z^2 + 4} \, dz \) along the positively oriented circle \( |z - i| = 2 \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the integral: \[ \int_0^{\infty}\frac{2+7\cos(x^\pi-e)-7\sin(1+x^8)}{1+x^2} \, \mathrm{d}x \]

Answer: \pi


Question: Calculate the sum of the series \( \sum_{i=1}^n \frac{1}{n}\sqrt{1-\left(\frac{i}{n}\right)^2} \).

Answer: \dfrac{\pi}{4}


Question: Determine the values of $p \in (0, \infty)$ for which the function $\frac{1}{1-z}$ belongs to the Hardy space $H^p(\mathbb{D})$. Here, $H^p(\mathbb{D})$ is defined as the space of analytic functions $f$ on the unit disk $\mathbb{D}$ such that \( \sup_{0 < r < 1} \left( \frac{1}{2 \pi} \int_{0}^{2\pi}|f(re^{i \theta})|^p d \theta \right)^{\frac{1}{p}} < \infty \).

Answer: (0, 1)


Question: A secant line intersects a circle at points $A$ and $C$ and crosses the circle's diameter at point $B$ with a $45^\circ$ angle. Given that $AB = 1$ and $BC = 7$, determine the radius of the circle.

Answer: 5


Question: Given the joint probability density function \( f(x, y) = \begin{cases} 6xy & \text{if } 0 \leq x \leq 1, 0 \leq y \leq 2 - 2x \\ 0 & \text{otherwise} \end{cases} \), find the conditional probability \( \operatorname{Pr}(Y < 0.5 \mid X < 0.5) \).

Answer: \dfrac{3}{22}


Question: Determine the number of solutions for the system of equations: \( x^2 + y^3 = 29 \) and \( \log_3 x \cdot \log_2 y = 1 \), where \( x, y \in \mathbb{R} \).

Answer: 2


Question: Compute the conditional expectation \( \mathbb{E}(\xi | \eta) \) given the random variables \( \xi(x) = x^4 \) and \( \eta(x) = 1 \) for all \( x \in [0,1] \) in the probability space \(([0,1], \mathcal{B}[0,1], \Lambda)\), where \( \Lambda \) is the Lebesgue measure.

Answer: \dfrac{1}{5}


Question: Consider the theorem: If a series \( \sum_{k=0}^{\infty} g_k \) of functions on a set \( S \subset \mathbb{R} \) converges uniformly and each \( g_k \) is continuous on \( S \), then the series represents a continuous function on \( S \). Is this theorem still valid if we replace \( \sum_{k=0}^{\infty} g_k \) with \( \sum_{k \in \mathbb{Z}} g_k \)? Provide a justification for your answer.

Answer: Yes


Question: Find the value of \(x\) such that \(e^{e^{10^{10^{2.8}}}} = 10^{10^{10^{10^{x}}}}\).

Answer: 2.8


Question: Determine whether every prime number can be expressed in the form \( p \cdot a^2 = b^2 \pm 1 \), where \( p \) is a prime number and \( a \) and \( b \) are integers. Provide a justification for your answer.

Answer: Yes


Question: Determine the cardinality of the set \( X = \{ f: \mathbb{N} \rightarrow \mathbb{N} \mid f \text{ is a monotone increasing function} \}. \)

Answer: 2^{\aleph_0}


Question: Solve for $Z$ in the equation: \[ \sum_{t=0}^{N}\left ( \frac{B(t) -C(t)}{(1+Z)^t} \right )=0 \] where $B(t)$ and $C(t)$ are known functions representing benefits and costs at time $t$, respectively.

Answer: Z


Question: If a group $G$ satisfies the maximal condition, is it necessarily finitely generated? Justify your answer.

Answer: Yes


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

Answer: \dfrac{4}{3}


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a positive increasing function such that \( \lim_{x \to \infty} \frac{f(3x)}{f(x)} = 1 \). Determine the value of \( \lim_{x \to \infty} \frac{f(2x)}{f(x)} \).

Answer: 1


Question: Consider a sequence of random variables defined inductively with $X_0=1$ and $X_{n+1}$ selected randomly and uniformly from $[0,X_n]$. Show that $\frac{1}{n}\log X_n$ converges almost surely to a constant. Evaluate this limit.

Answer: -1


Question: Calculate the remainder when \( 63^{63^{63}} \) is divided by 100.

Answer: 67


Question: Find the total number of distinct solutions for \(0 \leq x \leq 1\) such that \(\int_{0}^{x}\frac{t^2}{1+t^4}dt = 2x-1\).

Answer: 1


Question: Determine all subsequential limit points of the sequence \(X_n = \cos(n)\).

Answer: [-1, 1]


Question: Evaluate the integral \[ a_n = \frac{1}{4\pi} \int_0^{2\pi} (\pi - x)^2 \cos(nx) \, dx \] and verify that the result is \( \frac{1}{n^2} \).

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


Question:
Determine the correct general solution for the partial differential equation \( u_{xx} + u_{yy} = 0 \) from the following options:

(a) \( u = f(x+iy) - g(x-iy) \)

(b) \( u = f(x-iy) - g(x-iy) \)

(c) \( u = f(x-iy) + g(x+iy) \)

(d) \( u = f(x+iy) + g(x-iy) \)

Answer: D


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

Answer: \dfrac{1}{4}


Question: Find the number of 1's in the binary representation of the 1717th term in the sequence of positive integers whose binary representation is a palindrome.

Answer: 12


Question: Determine the values of $x \in \mathbb{R}$ for which the series \( \sum\limits_{n=1}^\infty \frac{2\sin^2(nx)}{n} \) converges.

Answer: x = k\pi


Question: Find the volume of the solid bounded above by the plane \( z = y \) and below by the paraboloid \( z = x^2 + y^2 \) using cylindrical coordinates.

Answer: \dfrac{\pi}{32}


Question: Find the value of \( R \in \mathbb{R} \) that maximizes the flux of the vector field \( F(x,y,z) = (xz - x\cos(z), -yz + y\cos(z), -4 - (x^2 + y^2)) \) over the surface \( S_R = \{ x^2 + y^2 + z^2 = R^2 , z \geq 0 \} \).

Answer: 0


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

Answer: \infty


Question: If a measure space \(X\) with measure \(\mu\) is not \(\sigma\)-finite, does this imply that \(X\) has infinite measure? Provide a justification for your answer.

Answer: Yes


Question: In a compact metrizable topological space $X$, if the Boolean algebra of clopen sets is finite, does it imply that the Boolean algebra generated by connected components is also finite? Provide a justification for your answer.

Answer: Yes


Question: Consider a model $V$ of ZFC. If we take an ultrapower $M$ of this model, does $M$ contain non-standard natural numbers?

Answer: Yes


Question: Determine the almost sure limit of \( \frac{X_n}{n} \), where each random variable \( X_n \) is independent and follows a Poisson distribution with parameter \( n \). Specifically, find \( \lim_{n \to \infty} \frac{X_n}{n} \) almost surely.

Answer: 1


Question: Calculate the average value of \( \log |z_0 - z| \) as \( z \) ranges over the unit circle, given that \( |z_0| \le 1 \).

Answer: 0


Question: In a population where mathematical and writing abilities are independently uniformly distributed on the interval [0, 1], students attend college if and only if the sum of their mathematical and writing abilities is greater than one. Calculate the correlation coefficient between writing and math ability among the college students.

Answer: -\dfrac{1}{2}


Question: Show that there exists an ordinal \( \beta \) such that \( \omega_1^\beta = \beta \), where \( \omega_1 \) is the first uncountable ordinal.

Answer: \beta


Question: Find the volume of a cylinder with its base as the disk of unit radius in the $xy$-plane centered at $(1,1,0)$, and its top defined by the surface $z=((x-1)^2+(y-1)^2)^{3/2}$. Use triple integrals to determine the volume.

Answer: \dfrac{2\pi}{5}


Question: Is the condition \( w \left (\dfrac{x+y}{2} \right ) \le \dfrac{1}{2}(w(x) +w(y)) \) for all \( x,y \in \Omega \) sufficient for a continuous function \( w \in C^0(\Omega) \) to be convex?

Answer: Yes


Question: Prove that the automorphism group of \(C_2 \times C_2\) is isomorphic to the dihedral group \(D_6\), which is defined by the relations \(x^3 = 1\), \(y^2 = 1\), and \(xy = yx^2\).

Answer: D_6


Question: For any measurable set $E \subset \mathbb{R}$ with measure 1, determine whether there exists a function $T: [0,1] \to E$ such that $T([0,1]) = E$ and for any Lebesgue integrable function $f$, the following holds: \[ \int_E f = \int_{[0,1]} f \circ T. \]

Answer: Yes


Question: Let $R$ be a reduced commutative non-noetherian ring of dimension $d$, and let $a$ be a non-zero divisor in $R$. Is the Krull dimension of $R/(a)$ at most $d - 1$?

Answer: Yes


Question: Does a set being dense imply that it is not enumerable?

Answer: No


Question: Let $X$ and $Y$ have the joint probability density function $f(x,y) = 8x(1-y)$ for $0 < y < 1$ and $0 < x < 1-y$. Compute the conditional probability $P(Y < X \mid X \leq \frac{1}{4})$. Provide your answer as a single value.

Answer: \dfrac{29}{93}


Question: Is the trace of a matrix \({\bf A}\) less than zero if \({\bf A}\) is a negative definite matrix? Here, Tr\({\bf A}\) denotes the sum of the diagonal elements of the square matrix \({\bf A}\), which is of size \(n\) and comprised of all real elements with real eigenvalues \(\lambda_n < 0\).

Answer: Yes


Question: Determine the number of positive integers \(a\) for which there exists an integer \(b\), with \(0 \leq b \leq 2002\), such that both quadratics \(x^2 + ax + b\) and \(x^2 + ax + b + 1\) have integer roots.

Answer: 44


Question: Determine the number of distinct Jordan normal forms for a matrix \( A \in M_7(\mathbb{C}) \) with the characteristic polynomial \( p(A) = (\lambda+4)^5(\lambda-2)^2 \).

Answer: 14


Question:
Given a function \( \Lambda(t) \) for any \( t > 0 \) with the following properties:

1. \( \Lambda(t) \) is differentiable.
2. \( \Lambda(t) \) is strictly increasing.
3. \( \Lambda(T) = \Lambda(T+S) - \Lambda(S) \) for any \( T, S > 0 \).

Derive the form of the function \( \Lambda(t) \).

Answer: \Lambda(t) = kt


Question: Determine the Galois group of the splitting field of the polynomial \(x^4 - x - 1\) over \(\mathbb{Q}\).

Answer: S_4


Question: Given that there are $n$ people available to complete $m$ tasks, where each person can handle between 0 and $m$ tasks, determine the number of ways to assign the tasks such that all $m$ tasks are completed. For example, if $n=2$ and $m=2$, there are 9 valid ways to assign the tasks. How many valid assignments are there for general $n$ and $m$?

Answer: (2^n - 1)^m


Question: Find the volume of the region \( T = \{(x,y,z) \in \mathbb{R}^3 \mid (x+y+z)^2 + (x-y)^2 \leq 1, 0 \leq z \leq 3 \} \) by evaluating the integral \( \int_{T} \,dx\,dy\,dz \).

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


Question:
Consider the set \( \Xi_A = \{ n \in \mathbb{N} \mid \exists k^2 \in A : k^2 \leq n \} \), where \( A \) is an arbitrary recursively enumerable set. Determine which of the following statements is correct:

I) \( \Xi_A \) is decidable.
II) \( \Xi_A \) is recursively enumerable but not decidable.
III) \( \Xi_A \) is not a recursively enumerable set, but the complement of \( \Xi_A \) is a recursively enumerable set.
IV) \( \Xi_A \) is recursively enumerable only when \( A \) is recursively enumerable-complete.

Select the correct option.

Answer: I


Question: Calculate the conditional probability \( P(X > 0 \mid X+Y > 0) \) where \( X \) and \( Y \) are independent standard normal random variables.

Answer: \dfrac{3}{4}


Question: Suppose you take $n$ independent draws from a uniform distribution on $[0,1]$ and call the set of those draws $N$. Define $X$ as the $K$ largest draws from $N$, and $Y$ as the $N-K$ smallest draws. What is the probability that the mean of $X$ is $k$ times larger than the mean of $Y$?

Answer: 0


Question: Is the language $B = \{a^n b^n c^n \}$ Turing recognizable? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{2\pi} \frac{x(t)x'(t) + y(t)y'(t)}{x^2(t) + y^2(t)}dt \), where \( F(t) = (x(t), y(t)) \) is a closed continuously differentiable curve in \( \mathbb{R}^2 \setminus \{(0,0) \} \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int_0^\frac{1}{n} \frac{e^{-t} \sin{\frac{t}{n}} dt}{(1+\frac{t}{n}^2)\arctan{\frac{t}{n}}} \]

Answer: 0


Question: A man stands at the origin of the real line and plays a game of heads or tails. Each time he gets a head, he moves 1 unit to the right, and each time he gets a tail, he moves 1 unit to the left. What is the expected number of steps he will take to reach either -7 or 10 for the first time?

Answer: 70


Question: Determine whether the following statement is true or false: If $X_n \to X$ in $L^p$ and $Y_n \to Y$ in $L^p$ as $n \to \infty$, then $X_n Y_n \to XY$ in $L^p$ as $n \to \infty$. Justify your answer with a proof or a counterexample.

Answer: False


Question: Determine if there exist matrices, other than the identity matrix or scalar multiples of the identity matrix, that have a condition number equal to 1.

Answer: Yes


Question: Compute the line integral \( \int_\gamma x \, dx \) where \( \gamma \) is parameterized by \( g(t) = (\cos t, \sin t) \) for \( 0 \leq t \leq 2\pi \).

Answer: 0


Question: Consider the subset $S$ of $M(n, \mathbb{R})$, the set of $n \times n$ real matrices, where the absolute value of the eigenvalues of the matrices in $S$ are less than or equal to 2. Is this subset necessarily a connected space?

Answer: Yes


Question: Let $X$ be a connected Hausdorff space such that $H_n(X,\mathbb{Z}) = 0$ for all $n \ge 1$. Does this imply that $X$ is path-connected, i.e., $H_0(X,\mathbb{Z}) = \mathbb{Z}$?

Answer: No


Question: Let $L/F$ be a field extension of infinite degree, where $L$ is algebraically closed. For any integer $d \geq 1$, is there an intermediate field $F \subset K \subset L$ such that $K/F$ is finite and of degree at least $d$?

Answer: Yes


Question: Find the minimum value of the expression \( \sqrt{x^2+1}+ \sqrt {4+(y-z)^2} + \sqrt{1+ (z-x)^2} + \sqrt{9+(10-y)^2} \) where \( x, y, z \in \mathbb{R} \).

Answer: \sqrt{149}


Question: Identify a group $G$ that contains subgroups isomorphic to $\mathbb{Z}_n$ for every positive integer $n$. Provide an example of such a group.

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


Question: Determine whether the following statement is true: For all \(z \in \mathbb{S}^1\) and \(x, y \in \mathbb{S}^7\), the equation \(z \cdot (x \ast y) = (z \cdot x) \ast y\) holds, where \(\cdot\) denotes complex multiplication and \(\ast\) denotes octonionic multiplication.

Answer: False


Question: Let \( f:[a,b] \rightarrow \mathbb{R} \) be a continuous function such that for every \( x \in [a,b] \), there exists a \( y \in [a,b] \) such that \( |f(y)| \leq |f(x)|/2 \). Prove that there exists a \( \xi \in [a,b] \) such that \( f(\xi) = 0 \).

Answer: 0


Question: Calculate the limit: \[ \lim_{n\rightarrow \infty}n\ \int_0^n \sin^2(\pi x)\left(\frac{1}{(x-1)(x-2)}+\frac{1}{(x+1)(x+2)}\right) dx \]

Answer: -1


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

Answer: \dfrac{4}{e}


Question: Compute the limit: \[ \lim_{x\to 0^+}x\cosh^{-1}(x\sinh(\frac{1}{x})) \]

Answer: 1


Question: Find a primitive root modulo $11^2$. Provide your answer as a single integer.

Answer: 2


Question:
A point starts at the origin and can randomly move up, down, left, or right with equal probability. After \( n \) moves, the point's position is described by \( x_1 \) units up, \( x_2 \) units down, \( x_3 \) units left, and \( x_4 \) units right. The squared distance from the origin is given by:
\[ D^2 = (x_4 - x_3)^2 + (x_1 - x_2)^2 \]
Find the expected value \( E(D^2) \) after \( n \) moves.

Answer: n


Question: Evaluate the complex integral \( \int_C z^3\exp{\left(\dfrac{-1}{z^2}\right)} dz \) where \( C \) is the contour \( |z|=5 \). Use the Residue Theorem to find the value of the integral.

Answer: \pi i


Question:
Consider the sequence of functions \(f_n(x)\) defined on the interval \([0,1]\) by:
\[
f_n(x) = \begin{cases}
\sin(n\pi x), & \text{if } x \in [0,1/n], \\
0, & \text{if } x \in (1/n,1].
\end{cases}
\]
Determine the type of convergence of \(f_n(x)\) on \([0,1]\):

- It does not converge pointwise.
- It converges pointwise but the limit is not continuous.
- It converges pointwise but not uniformly.
- It converges uniformly.

Answer: C


Question:
Consider a monotonically decreasing sequence $a_{n}$ such that $a_{n}$ is positive for all $n \in \mathbb{N}^{+}$ and satisfies:
$$\lim_{n\rightarrow \infty} \frac{a_{n+1}}{a_{n}} = 0.$$
Prove that $\lim_{n\rightarrow \infty} a_{n} = 0$. Provide a proof by contradiction.

Answer: 0


Question: Determine whether there exists a transcendental entire function that has no fixed points. Provide a justification for your answer.

Answer: Yes


Question: Does there exist a ring in which a finite degree polynomial has infinitely many roots?

Answer: Yes


Question: Does there exist distinct pseudo-Anosov elements $f$ and $g$ in the mapping class group $Mod(S)$ of a hyperbolic surface $S$ such that their stretch factors are equal, i.e., $\lambda_f = \lambda_g$?

Answer: Yes


Question: Determine whether $2^{1093} - 2$ is divisible by $1093^2$. Provide a justification for your answer.

Answer: Yes


Question: Consider a complex number \( z \) on the unit circle, where \( |z| = 1 \). Determine the number of distinct regions formed by the transformation \( z \to e^z \).

Answer: 2


Question: Evaluate the expression: \[ \frac{100^2}{100!} + \sum_{k=1}^{100} |(k^2 - 3k + 1)S_k| \] where \( S_k \) is the sum of the infinite geometric series with first term \( \frac{k-1}{k!} \) and common ratio \( \frac{1}{k} \).

Answer: 3


Question: Let $G$ be a group of order $pqr$, where $p$, $q$, and $r$ are distinct primes. Suppose $G$ is generated by elements of orders $p$ and $q$. Can $G$ be a simple group?

Answer: No


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

Answer: \dfrac{3}{2}


Question: Determine whether every infinite set has a derangement, where a derangement is defined as a bijection from the set to itself with no fixed points.

Answer: Yes


Question: Determine the number of points on the quadric $x_0^2 + x_1^2 + x_2^2 + x_3^2 = 0$ in the projective space $\mathbb{P}_3$ over the finite field $\mathbb{F}_9$. Provide your answer as a single integer.

Answer: 100


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

Answer: 0


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

Answer: 1


Question: Find a quadratic polynomial that is orthogonal to both \( p(x) = 1 \) and \( q(x) = x \).

Answer: 3x^2 - 1


Question: Consider three random variables uniformly distributed on the interval [0,1]. Let X be the minimum and Y be the maximum of these three variables. Calculate the expected value of the product X*Y.

Answer: \dfrac{1}{5}


Question: Let S and T be disjoint and compact sets in a normed vector space. Define \( f(S,T) = \inf\{||s-t|| : s \in S, t \in T\} \). Determine whether there exist elements \( s \in S \) and \( t \in T \) such that \( f(S,T) = ||s-t|| \).

Answer: Yes


Question: Evaluate the contour integral \( \int_{R} ^{R+i \pi} \frac {1}{\cosh x} \, dx \) and verify the result. The expected result is 0. Identify any mistakes in the calculation if the result differs.

Answer: 0


Question: Let \(\omega\) be a complex number such that \(\omega^7 = 1\) and \(\omega \neq 1\). Define \(\alpha = \omega + \omega^2 + \omega^4\) and \(\beta = \omega^3 + \omega^5 + \omega^6\). If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2 + px + q = 0\), find the ordered pair \((p, q)\).

Answer: (1, 2)


Question: If $X$ is a standard normal random variable and $a > 0$ is a constant, find \( \lim_{x \to 0^+} \mathbb{P}\left(X > x + \frac{a}{x} \mid X > x\right) \).

Answer: 0


Question: Determine whether the following statement is true or false: If a graph has $n$ vertices and $e$ edges, and $e > 100n$, then the graph always contains a route of length 100.

Answer: True


Question: Evaluate the surface integral \( \iint_S z^2 \, dS \) where \( S \) is the surface of the cube defined by \([-1,1] \times [-1,1] \times [-1,1]\).

Answer: \dfrac{40}{3}


Question: Find the residue at \( z=0 \) of the function \( f(z)=\frac{\sinh z}{z^4(1-z^2)} \).

Answer: \dfrac{7}{6}


Question: Is it consistent with ZF set theory that there is no uncountable set of algebraically independent real numbers?

Answer: Yes


Question: Let $(E_1,\tau_1)$ be a locally convex space and $(E_2,\tau_2)$ be a complete locally convex space. Suppose $T:(E_1,\tau_1) \longrightarrow (E_2,\tau_2)$ is a topological isomorphism. Is the space $(E_1,\tau_1)$ necessarily complete?

Answer: Yes


Question: Determine whether the complex function \( f(z) = \cos(|z|^2) \) is differentiable at \( z = 0 \). If it is differentiable, find the derivative \( f'(z) \) at that point.

Answer: 0


Question: Simplify the matrix expression $D - A^{-1}DA$, where $D$ is a diagonal matrix and $A$ is a symmetric matrix. Is there a simpler form of this expression?

Answer: -A^{-1}(DA - AD)


Question: Determine the Galois group of the polynomial $x^4 - a$ over $\mathbb{Q}$, where $a$ is a square-free integer not equal to 0, $\pm 1$. Provide a justification for your answer.

Answer: D_4


Question: Find the limit \( \lim_{n \to \infty} a_n \) for the sequence defined by the recurrence relation \( a_{n+1} = \sin a_n \).

Answer: 0


Question: Evaluate the integral \( \int_{0}^{1}\left\{\frac{1}{x}\left\lfloor\frac{1}{x}\right\rfloor\right\}dx \) in closed form.

Answer: 1 - \gamma


Question: Given two monic polynomials $f(x)$ and $g(x)$ with coefficients in $\mathbb{Z}$ that are coprime over $\mathbb{Q}$, determine if there exist polynomials $u(x)$ and $v(x)$ with coefficients in $\mathbb{Z}$ such that $f(x)u(x) + g(x)v(x) = 1$. Answer "True" or "False".

Answer: False


Question: Let \( X \) be a real-valued stochastic variable with density \( f(x) = 2xe^{-x^2} \) for \( x > 0 \). Let \( Y = X^2 \). Calculate \( E[Y^n] \).

Answer: \Gamma(n + 1)


Question: Given that $y_i \sim N(0,Q)$ are independent for $i = 1, 2, \ldots, N$, and $X = \sum_{i=1}^N y_i^2$, find the variance of $X$, denoted as $\text{Var}(X)$. Express your answer in terms of $N$ and $Q$.

Answer: 2NQ^2


Question: Evaluate the limit without using L'Hopital's rule: \[ \lim_{x\to0}\frac{\sqrt{\cosh{(3x^2)}}\cdot e^{4x^3}-1}{x^2\tan(2x)} \]

Answer: 2


Question: Find the points on the surface defined by the function \( f(x,y) = 4x^2 + 7y^2 + 5xy + 14 \) where the vector \( 6\hat{i} + 69\hat{j} + 3\hat{k} \) is normal to the tangent plane.

Answer: (1, -2, 36)


Question: Determine the maximal time interval $J$ for which the Existence and Uniqueness Theorem guarantees the existence and uniqueness of a solution $y : J \to \mathbb{R}$ for the initial value problem (IVP): \[ y' = y^2 + \frac{1}{t}, \quad y(1) = 1. \]

Answer: (0, 2)


Question: Given an open set $U \subset \mathbb{R}^n$, does there exist a function $f : \mathbb{R}^n \to \mathbb{R}$ such that $f$ is discontinuous at each point of $U$ and continuous on $\mathbb{R}^n \setminus U$? Provide a justification for your answer.

Answer: Yes


Question: Find the minimum value of \( S = \left(a+\frac{1}{b}\right)^2 + \left(b+\frac{1}{c}\right)^2 + \left(c+\frac{1}{a}\right)^2 \) given that \( a, b, c \in \mathbb{R^+} \) and \( a+b+c=12 \).

Answer: \dfrac{867}{16}


Question: Does Parseval's theorem for Fourier series hold when the series are expressed with negative exponents, i.e., when \( A(x) = \sum_{n=-\infty}^{\infty} a_n e^{-inx} \) and \( B(x) = \sum_{n=-\infty}^{\infty} b_n e^{-inx} \)?

Answer: Yes


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

Answer: 0


Question: Given the polynomial \((x^2 + x + 1)^{100} = a_0 + a_1x + a_2x^2 + \ldots + a_{199}x^{199} + a_{200}x^{200}\), find the value of \(\sum_{i=1}^{200} \frac{1}{1+x_i}\), where \(x_i\) are the roots of the polynomial.

Answer: 100


Question: Let $X$ be a real Banach space and $f: X \to \mathbb{R}$ be a continuous linear functional. Determine whether it is true that for any $\varepsilon > 0$, there exists a $\delta > 0$ such that for any $x \in X$, if $|f(x)| < \delta$, then there exists a $z \in \ker f$ such that $\Vert x - z \Vert < \varepsilon$. Provide a proof or a counterexample.

Answer: True


Question:
Given that $X$ and $Y$ are continuous random variables with the joint density function:
\[f_{X,Y}(x,y) = \begin{cases}24xy & \text{if } 0 < x < 1, \ 0 < y < 1, \ 0 < x + y < 1 \\ 0 & \text{otherwise}\end{cases}\]
find the probability density function of $Z = X + Y$. Provide your answer as a function of $z$.

Answer: 4z^3


Question: Let \( a_1 = 1 \) and \( a_n = n(a_{n-1} + 1) \) for \( n = 2, 3, \ldots \). Define \( P_n = \left(1 + \frac{1}{a_1}\right)\left(1 + \frac{1}{a_2}\right)\cdots\left(1 + \frac{1}{a_n}\right) \). Find \( \lim_{n \to \infty} P_n \).

Answer: e


Question: Consider the differential equation \(-u''(x) + u(x) = f(x)\) for \(x \in (-1,1)\), where \(f \in L^2(0,1)\) and \(f(x) = f(-x)\) almost everywhere in \((-1,1)\). The solution \(u(x)\) satisfies the boundary conditions \(u(-1) = u(1) = 0\). Determine whether \(u(x) = u(-x)\) holds for the solution \(u(x)\).

Answer: u(x) = u(-x)


Question: Find the coefficient of $x^{103}$ in the expansion of $$(1+x+x^2+x^3+x^4)^{199}(x-1)^{201}.$$

Answer: 0


Question: Find the limit \( \lim_{x\to e}\frac{x^{e^x}-e^{x^e}}{x-e} \) without using L'Hospital's rule.

Answer: e^{e^e + e - 1}


Question: Evaluate the integral \( \int_{-1}^{1} \text{sgn}(x) \, dx \).

Answer: 0


Question: Given that \(X\) and \(Y\) are jointly continuous random variables with conditional density \(f_{X|Y}(x|y) = I_{(x,x+1)}(y)\) and marginal density \(f_{X}(x) = I_{(0,1)}(x)\), find the probability \(P[X + Y < 1]\).

Answer: \dfrac{1}{4}


Question: Consider a random variable with an expected value \( \mathbb{E}[X] < 0 \) and a parameter \( \theta \neq 0 \). Given that the expectation \( \mathbb{E}[e^{\theta X}] = 1 \), determine whether it is true that \( \theta > 0 \).

Answer: True


Question: Alma and Oscar make a phone call every day. The lengths of Alma's phone calls are distributed as Exponential(1), and the lengths of Oscar's phone calls are distributed as Exponential(2). The lengths of the two phone calls are independent. Let Z denote the length of the longest of the two calls. Determine the mean of Z.

Answer: \dfrac{7}{6}


Question:
Find the closed form of the recursive function defined by:
\[ f(n) = 2f(n-1) + n-1 \text{ for } n > 1, \]
with the initial condition \( f(n) = 0 \) for \( n < 1 \).

Answer: 2^n - n - 1


Question:
Consider an $n \times p$ matrix $A$ and its orthogonally transformed version $B = AQ$, where $Q$ is a $p \times p$ orthogonal matrix such that $QQ^T = I$. Both $A$ and $B$ are centered, meaning each feature has zero mean. Perform PCA on both $A$ and $B$:

\[ \frac{1}{n-1} A^T A = V_A L_A V^T_A \]
\[ \frac{1}{n-1} B^T B = V_B L_B V^T_B \]

Are the projections of $A$ and $B$ on their principal directions equal, i.e., is \( A V_A = B V_B \)? Justify your answer.

Answer: Yes


Question: Determine the maximal possible dimension of an abelian Lie subalgebra of the Heisenberg Lie algebra \( \mathfrak{h}_n \) of dimension \( 2n+1 \). The Heisenberg Lie algebra \( \mathfrak{h}_n \) has a basis \( x_1, \dots, x_n, y_1, \dots, y_n, c \) with the Lie bracket defined by \([x_i, y_j] = \delta_{ij}c, [x_i, x_j] = [y_i, y_j] = [x_i, c] = [y_j, c] = 0\), where \(1 \le i, j \le n\) and \(\delta_{ij}\) is the Kronecker delta.

Answer: n + 1


Question: Calculate \( \lim_{n \to \infty} \mathbb{P}(X + n = Y) \) where \( X \sim \mathrm{Poisson}(n) \) and \( Y \sim \mathrm{Poisson}(2n) \) are independent random variables.

Answer: 0


Question: Use the formula \( 16\tan^{-1}\left(\frac{1}{5}\right) - 4\tan^{-1}\left(\frac{1}{239}\right) = \pi \) and Taylor polynomials to approximate \( \pi \) to 6 decimal places of accuracy. Include error estimates in your solution.

Answer: 3.141593


Question: Find the largest value of \( k \) such that the \( k \)-th absolute moment \( E[|X|^k] \) is finite for a random variable \( X \) with the cumulative distribution function \( F_X(x) = \frac{1}{1+e^{-x}} \) for \( x \geq 0 \).

Answer: \infty


Question: In a game played by two players, Q and A, at each time point i, player Q selects a set $Q_i \subset \mathbb{R}$, and player A responds by choosing $A_i$ such that $A_i = Q_i$ or $A_i = Q_i^c$. Define $C(i) = \bigcap_{k < i} A_k$ and $C(\infty) = \bigcap_{k \in \mathbb{N}} A_k$. Player Q wins if either $C(i)$ has only one element for some finite i, or $C(\infty)$ is empty. Player A wins in all other cases. Determine which player has a winning strategy.

Answer: A


Question:
Determine which of the following functions is a solution to the partial differential equation \( u_{tt} = 2(u_{xx} + u_{yy}) \):

1) \( u(x, y, t) = t \sin(x + y^2) \)
2) \( u(x, y, t) = t \cos(x + y^2) \sin(x + y^2) \)
3) \( u(x, y, t) = \cos x \cos y \cos 2t \)

Answer: 3


Question: Find a basis for the field $F = \mathbb{Z}_5[x]/(x^3-x^2-1)$ over $\mathbb{Z}_5$. Express your answer in terms of the element $u = [x]$.

Answer: \{1, u, u^2\}


Question: Evaluate the integral \( \int\limits_{\mathbb{R}^n} -\Delta \Phi(x) \, dx \), where \( \Phi(x) = \frac{1}{\alpha(n)}((\lvert x\rvert^2 + 1)^{\frac{-n-2}{2}}\lvert x\rvert^2 + (\lvert x\rvert^2 + 1)^{-n/2}) \) and \( x \in \mathbb{R}^n \).

Answer: 0


Question: Calculate \( \lim\limits_{x \to \infty} \left( x - \log(\cosh(x)) \right) \).

Answer: \ln 2


Question: Evaluate the limit: $$\lim_{n\to \infty}\left(\cfrac{1^p+2^p+\cdots +n^p}{n^p}-\cfrac{n}{p+1}\right),$$ where $p\in \mathbb{N}$.

Answer: \dfrac{1}{2}


Question: Find the singular solution of the differential equation \(x+py=(x-y)\sqrt{p^2+1}\), where \(p=\frac{dy}{dx}\), using the substitutions \(x=r\cos{\theta}\) and \(y=r\sin{\theta}\).

Answer: y = 0


Question: Calculate the fundamental group of $S^3 - \gamma$, where $\gamma$ is a circle in $\mathbb{R}^3$ and $S^3 = \mathbb{R}^3 \cup \{\infty\}$ is the one-point compactification of $\mathbb{R}^3$. Provide your answer as a group.

Answer: \mathbb{Z}


Question: If $A$ is a Noetherian domain and $A_p$ is a UFD for some prime ideal $p$, is there an element $f$ not contained in $p$ such that the localization $A_f$ is a UFD?

Answer: Yes


Question: A square is drawn in the Cartesian coordinate plane with vertices at \((2, 2)\), \((-2, 2)\), \((-2, -2)\), \((2, -2)\). A particle starts at \((0,0)\). Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is \(1/8\) that the particle will move from \((x, y)\) to each of \((x, y + 1)\), \((x + 1, y + 1)\), \((x + 1, y)\), \((x + 1, y - 1)\), \((x, y - 1)\), \((x - 1, y - 1)\), \((x - 1, y)\), or \((x - 1, y + 1)\). The particle will eventually hit the square for the first time, either at one of the 4 corners of the square or at one of the 12 lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is \(m/n\), where \(m\) and \(n\) are relatively prime positive integers. What is \(m + n\)?

Answer: 39


Question:
Find the function \( f : \mathbb{R} \to [0, \infty) \) that satisfies the following conditions:
\[ f(x+y) - f(x-y) = f(x) \cdot (f(y) - f(-y)) \]
\[ f'(0) = \log a \]
\[ f(0) = 1 \]
where \( x, y > 0 \) and \( x, y \neq 1 \).

Answer: a^x


Question: Consider the field $K = k(x_{1}, x_{2},...,x_{n})$ where $n \geq 2$ and $k$ is a field. Does there exist a subfield of $K$ that is not a rational function field?

Answer: Yes


Question: Calculate the probability \( P(X < Y+1) \) where \( X \sim \operatorname{Exp}(1) \) and \( Y \sim N(0,1) \), and \( X \) and \( Y \) are independent. Verify that \( P(X < Y+1) = \Phi(1) - \frac{1}{2\sqrt{e}} = 0.5377346701 \).

Answer: \Phi(1) - \frac{1}{2\sqrt{e}}


Question: Determine the values of $x$ for which the series \( \sum_{n=1}^{\infty} \frac{(2n)!\,x^n}{2^{2n}(n!)^2} \) is convergent.

Answer: [-1, 1)


Question:
Let \(a, b, c\) be the roots of the cubic equation \(x^3 - x^2 - 2x + 1 = 0\). Find the value of the expression:
\[ \sqrt[3]{a+b} + \sqrt[3]{b+c} + \sqrt[3]{a+c} \]

Answer: 1


Question: Determine the order of the element \((1, 1) + \langle(2, 2)\rangle\) in the factor group \(\mathbb{Z} \times \mathbb{Z} / \langle (2, 2)\rangle\).

Answer: 2


Question: In an equilateral triangle \( \triangle ABC \), a point \( M \) is located inside the triangle such that \( AM = 3 \) units, \( BM = 5 \) units, and \( CM = 4 \) units. Determine the measure of \( \angle AMC \).

Answer: 150^\circ


Question:
Determine the positive values of \(a\) for which the equation \(a^x = x + 2\) has exactly 2 real solutions. Choose from the following options:

A. \((1, \infty)\)
B. \((0, 1)\)
C. \(\left(\frac{1}{e}, e\right)\)
D. \(\left(\frac{1}{e^e}, e^e\right)\)
E. \((e^{\frac{1}{e}}, \infty)\)

Answer: A


Question:
Consider a quadratic equation \( p(x) = 0 \) with real coefficients that has purely imaginary roots. Determine the nature of the roots of the equation \( p(p(x)) = 0 \).

Options:
(a) Only purely imaginary roots.
(b) All real roots.
(c) Two real and two purely imaginary roots.
(d) Neither real nor purely imaginary roots.

Answer: D


Question: Given a regular hexagon with unit side length, use only a straightedge to construct a line segment of length \(\sqrt{7}\).

Answer: \sqrt{7}


Question: In a contest with \( n \) questions taken by \( m \) contestants, each question is worth a certain positive number of points, and no partial credits are given. After grading, it was observed that by reassigning the scores of the questions, any desired ranking of the contestants could be achieved. What is the largest possible value of \( m \)?

Answer: n


Question: Evaluate the limit: \[ \lim _ {y\rightarrow + \infty } \int _ { 1 } ^ { 2 } \frac { \ln ( x + y ) } { \ln \left(x^{2}+y^{2} \right) } d x \]

Answer: \dfrac{1}{2}


Question: Find the limit of the sequence \(a_n = \sqrt[n]{4^{2n} - 5^{n+2} - \sin(\sqrt{n})}\) as \(n\) approaches infinity.

Answer: 16


Question: Determine the interval of convergence for the series \( \sum_{n=2}^\infty \frac{(-1)^n x^n}{n(n-1)}. \)

Answer: [-1, 1]


Question:
Let $a$, $b$, and $c$ be complex numbers. Let $a_1$ be the image of $a$ when it is rotated $90^\circ$ clockwise around $b$. Let $a_2$ be the image of $a_1$ when it is rotated $90^\circ$ counter-clockwise around $c$.

Then $a_2$ can be written in the form
$$a_2 = pa + qb + rc$$
for some complex numbers $p$, $q$, and $r$. Find $p^2 + q^2 + r^2$.

Answer: 1 - 4i


Question: Let \(a_n\) be a sequence satisfying \[\sum_{n=1}^\infty \left(na_n-\frac{n^2+1}{2n+1} \right)=3.\] Compute \(\lim_{n\rightarrow\infty}(a_n^2+2a_n+2).\)

Answer: \dfrac{13}{4}


Question: Find the function \( f: \mathbb{N} \rightarrow \mathbb{N} \) that satisfies the equation \( f(f(f(n))) + f(f(n)) + n = 3f(n) \) for all \( n \in \mathbb{N} \).

Answer: f(n) = n


Question: Evaluate the integral \( \int_0^i e^z \, \mathrm{d}z \).

Answer: e^i - 1


Question: Given a set \( A = \{a_1, a_2, \ldots, a_{2016}\} \), let \( M \) be the number of subsets of \( A \) where the number of elements is divisible by 4, and \( N \) be the number of subsets where the number of elements is divisible by 2 but not by 4. Calculate \( M - N \).

Answer: 2^{1008}


Question: Let \(A = \{4z^4 : \sin^2z = \sinh^2z\}\). Find \(\sup(A \cap \mathbb{R})\).

Answer: 0


Question: Determine the value of the parameter $m$ such that the equation \(x^8 - mx^4 + m^4 = 0\) has four distinct real roots in arithmetic progression.

Answer: \dfrac{9}{82}


Question: Can the Riemann-Stieltjes integral \( I = \int_{a}^{b} f(x) \, dg(x) \) exist if the integrator \( g \) is not of bounded variation on \([a, b]\)?

Answer: Yes


Question:
Find the closed form expression for the derivative of the determinant with respect to matrix $A$:
\[ \frac{\partial \det(X - XA'(AXA')^{-1}AX)}{\partial A} \] where $X$ is an $n \times n$ full rank matrix and $A$ is an $m \times n$ full rank matrix with $m \le n$. Assume $rank(X) = n$ and $rank(A) = m$. Provide your answer in terms of $X$ and $A$.

Answer: 0


Question: Evaluate the infinite series: $$\sum_{n=0}^\infty \frac{(2n)!}{(n!)^2(2n+1)4^n}$$

Answer: \dfrac{\pi}{2}


Question: Determine the kernel of the ring homomorphism \( \varphi : R[u,v] \to R[x,1/x] \) defined by \( \varphi(p(u,v)) = p(x,1/x) \).

Answer: (uv - 1)


Question: Find the Lipschitz constant for the function \( f(x) = |\sin(x) + \cos(x)| \) on the interval \([-2, 2]\).

Answer: \sqrt{2}


Question: Determine the completion of the metric space consisting of the rational numbers \( \mathbb{Q} \) with the metric \( d(x,y) = |x-y| \).

Answer: \mathbb{R}


Question: Determine the number of roots of the complex equation \( z^4 - 5z + 1 = 0 \) that lie within the annular region defined by \( 1 \leq |z| \leq 2 \).

Answer: 3


Question: Let $G$ be a $g$-dimensional Lie group and $H$ be an $h$-dimensional normal Lie subgroup of $G$. Is the dimension of the quotient group $G / H$ always $g-h$? Justify your answer.

Answer: g - h


Question: In a class trial with 15 participants, each participant votes either "guilty", "not guilty", or "abstain". The voting stops as soon as at least half of the votes are "guilty". Determine the total number of possible voting sequences that can occur.

Answer: 13663745


Question: Let $f$ be a continuous self-map of a topological space $X$ such that $f \circ f = f$. Is $X$ homotopic to its image $f(X)$?

Answer: No


Question: In a non-commutative ring \((R,+,\cdot)\), if \(a \in R\) is invertible and \(x \in R\) is non-invertible, is the product \(ax\) necessarily non-invertible? Provide a justification for your answer.

Answer: Yes


Question: In triangle \(ABC\), points \(D\) and \(E\) are on sides \(BC\) and \(CA\) respectively, and points \(F\) and \(G\) are on side \(AB\) with \(G\) between \(F\) and \(B\). Line \(BE\) intersects \(CF\) at point \(O_1\) and \(BE\) intersects \(DG\) at point \(O_2\). Given that \(FG = 1\), \(AE = AF = DB = DC = 2\), and \(BG = CE = 3\), compute \(\frac{O_1O_2}{BE}\).

Answer: \dfrac{7}{26}


Question: Evaluate the integral \( \int_0^{\pi/2} \sin(2nx) \cot x \, dx \).

Answer: \dfrac{\pi}{2}


Question: Assume $H$ is a separable Hilbert space with an orthonormal basis $\{e_n\}_{n=1}^{\infty}$. Let $p_n$ denote the projection onto the subspace generated by $\{e_1, \ldots, e_n\}$. Let $a$ be a positive trace class operator on $H$. Prove or disprove that \( \|a - p_n a p_n\|_1 = \text{Tr}(|a - p_n a p_n|) \to 0 \) as $n \to \infty$.

Answer: 0


Question: Find all complex numbers $x \in \mathbb{C}$ such that $x^{10} + x^5 + 1 = 0$. Given that $x^{10} + x^5 + 1 = (x^2 + x + 1)(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1) = 0$, determine the roots of the polynomial $x^8 - x^7 + x^5 - x^4 + x^3 - x + 1$.

Answer: e^{\frac{2\pi ik}{15}}


Question: Compute the limit: $$ \lim_{v \rightarrow +\infty} \sum_{n=0}^{+\infty} \frac{\lambda^{n}}{(n !)^v} $$ and verify that it equals $1 + \lambda$ for $\lambda > 0.$

Answer: 1 + \lambda


Question: Find the value of \( x \) such that \( \sin^{-1}\left(x - \frac{x^2}{2} + \frac{x^4}{4} + \cdots \right) + \cos^{-1}\left(x^2 - \frac{x^4}{2} + \frac{x^6}{4} + \cdots \right) = \frac{\pi}{2} \), where \( 0 < |x| < 1.414 \).

Answer: 1


Question: Find the greatest common factor between the polynomials \((x+1)^{4n+3} + x^{2n}\) and \(x^3 - 1\).

Answer: x^2 + x + 1


Question: Let \( C[0,1] \) be the space of continuous functions \( f: [0,1] \to \mathbb{R} \), with the norm \( \|f\| = \max\{|f(x)| : x \in [0,1]\} \). Consider the map \( T: C[0,1] \to \mathbb{R} \) given by \( T(f) = \int_0^1 f(x)\,dx \). Determine the norm \( \|T\| \).

Answer: 1


Question: Find a polynomial $f(x)$ in $\mathbb{Z}_{11}[x]$ such that $(x^2-1)f(x) \equiv (x^3+10x) \pmod{x^6+x^5+x+1}$. Provide your answer as a polynomial in $\mathbb{Z}_{11}[x]$.

Answer: x


Question: Determine all graphs $G$ of order $n \geq 4$ such that the subgraph induced by every three vertices of $G$ is a tree.

Answer: C_4


Question: Evaluate the limit: \[ \lim\limits_{n\to\infty} \int_0^n \left( 1- \frac{x}{n}\right)^n \log \left(2+\cos\left( \frac{x}{n}\right)\right)\, dx \]

Answer: \log 3


Question: Let $F_1$ and $F_2$ be two normal matrices such that $F_1F_2 = F_2F_1$. Consider the set \( W(F_1,F_2) = \{(\langle F_1 y , y \rangle, \langle F_2 y , y \rangle) : y \in F, \|y\| = 1\} \). Is the set $W(F_1,F_2)$ convex?

Answer: Yes


Question: Determine if the expression \(a \cos A + b \cos B + c \cos C\) can be equal to \(4R \sin A \sin B \sin C\) for a triangle with sides \(a, b, c\), angles \(A, B, C\), and circumradius \(R\).

Answer: Yes


Question: Maximize the function \( f(x,y) = \frac{x^2 - y^2}{(x^2 + y^2)^2} \) for \( x, y \in \mathbb{R} \).

Answer: \infty


Question: Two rectangles have their edge lengths chosen uniformly from the interval [0,1]. What is the probability that the second rectangle will fit inside the first one?

Answer: \dfrac{1}{3}


Question: Let \(X_1, X_2, X_3, \ldots\) be a sequence of i.i.d. uniform \((0,1)\) random variables. Calculate the value of \[ \lim_{n \to \infty} P\left(-\ln(1-X_1) - \ln(1-X_2) - \cdots - \ln(1-X_n) \geq n\right). \]

Answer: \dfrac{1}{2}


Question: Find the number of integer solutions to the equation \(a + b + c = 0\) where \(a, b, c\) are integers in the range \([-k, k]\) and \(a \neq 0\). Here, \(k\) is a positive integer.

Answer: 3k^2 + k


Question: Identify the quotient group \(\mathbb{Z}_{ab}/N\) with a familiar group, where \(N\) is the image of the homomorphism \(\mathbb{Z}_a \rightarrow \mathbb{Z}_{ab}\) given by \([k]_a \mapsto [kb]_{ab}\).

Answer: \mathbb{Z}_b


Question: Let \((a_n)_{n \in \mathbb{N}}\) be a sequence in \(\mathbb{R}\). For every \(r \in \mathbb{R}\), there exists an accumulation point \(b \in \mathbb{R} \cup \{-\infty\}\) of \((a_n)_{n \in \mathbb{N}}\) such that \(b < r\). Prove that \(-\infty\) is an accumulation point of \((a_n)_{n \in \mathbb{N}}\).

Answer: -\infty


Question: Evaluate the definite integral \( \int_0^\pi e^{2\cos(\theta)}\cos(2\sin(\theta) - \theta)\,d\theta \).

Answer: 2\pi


Question: Let $R$ be a commutative ring and $p$ be a non-zero prime ideal. If $p^m = p^n \neq 0$ for $m, n \in \mathbb{N}$, does it follow that $m = n$?

Answer: No


Question: Given that \( z_1=2\sqrt{3}\operatorname{cis}\left(\frac{3\pi}{2}\right) \) and \( z_2=2\operatorname{cis}\left(\frac{2\pi}{3}\right) \), find the smallest positive integer \( n \) such that \( \left(\frac{z_1}{z_2}\right)^n \in \mathbb{R}^+ \).

Answer: 12


Question: Does the concept of bounded variation for a function \( g: [a,b] \to X \) in a Hausdorff topological vector space coincide with the standard definition of bounded variation when \( X \) is a Banach space?

Answer: Yes


Question: Evaluate the double integral: $$\int_0^1 \int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \text{d}y \ \text{d}x$$

Answer: \dfrac{\pi}{4}


Question: If A and B are partially-ordered sets, and there exist injective order-preserving maps from A to B and from B to A, does there necessarily exist an order-preserving bijection between A and B?

Answer: No


Question: Find the acute dihedral angle between two faces of a regular tetrahedron, rounded to the nearest tenth of a degree.

Answer: 70.5


Question: Given the digits 0 and 9, form the smallest positive number that is a multiple of a given integer \( n \). For example, if \( n = 7 \), the smallest number is 9009. Determine this number for a given \( n \).

Answer: 9009


Question: Consider the curve given by \( y = x^3 \). A tangent at a point \( C_1 \) on this curve meets the curve again at \( C_2 \). The tangent at \( C_2 \) meets the curve at \( C_3 \), and this process continues such that the x-coordinates of \( C_1, C_2, C_3, \ldots, C_n \) form a geometric progression. Determine the ratio of the area of triangle \( C_1C_2C_3 \) to the area of triangle \( C_2C_3C_4 \).

Answer: \dfrac{1}{16}


Question: Consider an unbounded sequence of positive real numbers \(\{a_n\}\) such that \(\lim_{n\to\infty}(a_{n+1}-a_n)=0\). Is it necessarily true that \(a_n\) can be expressed as a partial sum \(\sum_{k=1}^n b_k\) of some sequence \(\{b_k\}\) where \(\lim_{k\to\infty}b_k=0\) and \(\sum_{k=1}^\infty b_k=\infty\)?

Answer: Yes


Question: Find the degree of the Taylor polynomial centered at \( x = 1 \) for \( \ln(x) \) that approximates \( \ln(1.25) \) with an error less than 0.001.

Answer: 3


Question: Evaluate the limit \( \lim_{n \to \infty} n^3 x^n (1-x)^4 \) for \( x \in [0, 1] \).

Answer: 0


Question: Find the maximum integer value of \( n \) such that \( 2^n \mid 3^{1024} - 1 \).

Answer: 12


Question: Let \( A \) and \( B \) be \( 2 \times 2 \) matrices such that \( A^2 = B^2 = (AB)^2 = 0 \). Does it follow that \( AB = 0 \)?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \rightarrow \infty}\left(\frac{1^3+3^3+5^3+7^3+...+\left(2n-1\right)^3}{\left(1+3+5+7+...+\left(2n-1\right)\right)^2} \right)$$

Answer: 2


Question: Compute the fundamental group \( \pi_1(S^1 \times (0,\infty)) \).

Answer: \mathbb{Z}


Question: Consider a deck of cards consisting only of the jack, queen, king, and ace of each suit (clubs, diamonds, hearts, and spades). You start with the four jacks in your hand. In each round, you randomly select one of the four cards in your hand and replace it with one of the three other cards of the same suit, each with equal probability. What is the expected number of swaps needed to return to having all four jacks in your hand?

Answer: 256


Question: Given the line equation \( y = (2+\sqrt{3}) \cdot x \) and the basis \( B = \left\{ \begin{pmatrix} \frac{\sqrt{3}}{2} \\ \frac{1}{2} \end{pmatrix}, \begin{pmatrix} -\frac{1}{2} \\ \frac{\sqrt{3}}{2} \end{pmatrix} \right\} \), determine the equation of the line \( g \) in the coordinates \( x', y' \) with respect to the basis \( B \).

Answer: y' = x'


Question: Evaluate the limit \( \lim\limits_{n\to \infty}\frac{1\times3+3\times5+\dots+(2n-1)\times(2n+1)}{n^2} \) using the Stolz-Cesaro theorem.

Answer: \infty


Question: At a party, there are 20 girls and 20 boys. How many dance pairs can be formed if boys may dance with boys and girls with girls?

Answer: 780


Question: Evaluate the limit: \[ \lim_{n \to +\infty} \left(0.999 + \frac{1}{n}\right)^{n+3} \]

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\sin^2 x}{x^2} \, dx \) using the given Fourier transform properties. The function \( f(x) = \Theta(x + 1) - \Theta(x - 1) \) has a Fourier transform \( \widetilde{f}(k) = 2\frac{\sin k}{k} \), where \( \Theta(x) \) is the Heaviside step function and the Fourier transform is defined as \( \widetilde{f}(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx \).

Answer: \pi


Question: Can every hyperplane in \( \mathbb{F}^n \) be described as \( \{(x_1, \dots, x_n) \in \mathbb{F}^n \mid a_1x_1 + \dots + a_nx_n = 0\} \) for some coefficients \( a_1, \dots, a_n \) where at least one \( a_i \neq 0 \)? Assume a hyperplane is defined as a subspace with dimension one less than the ambient space.

Answer: Yes


Question: Is there a hyperbolic set \( \Lambda \) such that its set of periodic points is not dense in \( \Lambda \)?

Answer: Yes


Question:
Does there exist a sequence of non-normal trace-class operators \(X_n\) such that in some orthonormal basis \(\{ v_i \}_{i \in \mathbb{N}}\), the following condition holds:
\[
\sum_{i=1}^\infty \vert \langle v_i, X_n v_i \rangle \vert \to 0 \text{ as } n \to \infty
\]
and at the same time \(\| X_n \|_{tc} = 1\) for all \(n \in \mathbb{N}\)?

Answer: Yes


Question: Determine the number of distinct ways to color the sides of a regular octagon using two colors, red and blue, such that each color is used exactly four times. Two colorings are considered the same if one can be obtained from the other by rotating the octagon.

Answer: 10


Question:
Factorize the expression \( \Sigma a^{2}(b^{4}-c^{4}) \). Choose the correct factorization from the options below:

(a) \((a-b)^{2}(b-c)^{2}(c-a)^{2}\)

(b) \((a-b)(a+b)(b-c)(b+c)(c-a)(c+a)\)

(c) \((a+b)^{2}(b+c)^{2}(c+a)^{2}\)

(d) None of these

Answer: B


Question: Let \( \text{Com}(\mathbb{R}^2) \) denote the space of nonempty compact subsets of the plane, equipped with the Hausdorff metric. Consider a continuous path \( S_\bullet:[0,1]\to\text{Com}(\mathbb{R}^2) \). If \( p\in S_0 \), must there exist a path \( \gamma:[0,1]\to\mathbb{R}^2 \) such that \( \gamma(0)=p \) and \( \gamma(t)\in S_t \) for all \( t\in[0,1] \)?

Answer: Yes


Question: Determine the values of \( n \) for which the integral \( \iiint_E \frac{1}{(x^2+y^2+z^2)^{n/2}} \, dV \), where \( E \) is the region bounded by spheres of radii \( r \) and \( R \), has a limit as \( r \to 0^+ \). The integral evaluates to \( 4\pi\left(\frac{R^{3-n}}{3-n}-\frac{r^{3-n}}{3-n}\right) \).

Answer: n < 3


Question: What is the maximum possible value of the Kullback-Leibler (KL) divergence between two probability distributions?

Answer: \infty


Question: Calculate the limit: \( \lim_{x \to 0} \sqrt[x]{1+\sin x} \).

Answer: e


Question: A group D has exactly 100 distinct elements of order 11. Determine the number of distinct subgroups of order 11 in D.

Answer: 10


Question: Three traders have payoffs at the end of the day that are uniformly distributed between $0 and $100. What is the probability that the payoff of trader A is greater than the payoff of trader B, and the payoff of trader B is greater than the payoff of trader C?

Answer: \dfrac{1}{6}


Question: Calculate the flux of the vector field \( \mathbb{v} = (x^2-2xz, -2xy, z^2-x) \) downwards through the paraboloid \( z = 1 - x^2 - y^2 \) for \( x \geq 0 \) and \( z \geq 0 \).

Answer: \dfrac{2}{3}


Question: In a murder investigation, there are three suspects: Adam, Ben, and Charlie. Each suspect claims they did not kill Dave. Adam states that Dave was friends with Ben and that Charlie did not like Dave. Ben claims he did not know Dave and was not in town on the day of the murder. Charlie asserts that he saw both Adam and Ben with Dave on the day of the crime and insists that either Adam or Ben must have killed Dave. Given that two of the three suspects are telling the truth and one of them is guilty, determine who the murderer is.

Answer: Ben


Question: Determine the closure of the set of all polynomials with variable $x \in [0,1]$ in the space of continuous functions $C[0,1]$. Use the Stone-Weierstrass theorem to justify your answer.

Answer: C[0,1]


Question: Calculate \( \lim_{x \to \infty} \left( x - x^2 \ln\left(1+\frac{1}{x}\right) \right) \).

Answer: \dfrac{1}{2}


Question: Convert the double integral \( I = \int_0^1 \int_0^{2-2x} \frac{y+2x}{x^2+y^2} \, dy \, dx \) into polar coordinates and evaluate the integral. Determine the value of \( I \).

Answer: \pi


Question: Solve the equation for \(x \in \mathbb{R}\): \[ 1 - 3^{x-1} + 2^{\frac{x}{2}} - 2^{\frac{x}{2}}3^{\frac{x-1}{2}} = 0. \]

Answer: 1


Question: Determine the order of the splitting field of the polynomial \(x^8 - 1\) over the field with 4 elements.

Answer: 4


Question: Evaluate the contour integral \( \int_{\Gamma} \frac{1}{z^2+1} \, dz \) along the contour \( \Gamma \), which is a closed circle centered around \( i \) and oriented counterclockwise.

Answer: \pi


Question: Find the expected area of the projection of a unit cube onto a plane under an isotropic random direction of projection.

Answer: \dfrac{3}{2}


Question: Evaluate the limit: \[ \lim_{x\rightarrow0^+}(\cos x +|x|+x)^{\cot x} \]

Answer: e^2


Question: In triangle $ABC$, $AB = 3\sqrt{7}$, $AC = 3$, and $\angle ACB = \frac{\pi}{3}$. The angle bisector $CL$ of $\angle ACB$ lies on line $CD$, where $D$ is a point on the circumcircle of triangle $ABC$. Find the length of $CD$.

Answer: 4\sqrt{3}


Question: Let \( \phi : L \to L \) be a linear operator with minimal polynomial \( m_\phi (x) = (x-1)x^8 \). Determine the minimal polynomial of \( \phi^3 \).

Answer: (x - 1)x^3


Question: In \(\Delta ABC\), \(AC = BC\) and \(\angle C = 120^\circ\). Points \(M\) and \(N\) are on sides \(AC\) and \(BC\) respectively, such that \(\angle BAN = 50^\circ\) and \(\angle ABM = 60^\circ\). Find \(\angle NMB\).

Answer: 30^\circ


Question: For which values of \( p > 0 \) does the integral \( \int_0^{\infty} x \sin(x^p) \, dx \) converge?

Answer: p > 2


Question: Find the limit of the sequence \( \sqrt{-n^4+4n^2+4}-in^2 \) as \( n \to \infty \).

Answer: -2i


Question: Let $X$ and $Y$ be stochastic variables such that $\text{Cov}(X,f(Y))=0$ for all bounded measurable functions $f$. Does this imply that $E[X|Y]=E[X]$? Provide a proof or a reference to support your answer.

Answer: E[X|Y] = E[X]


Question: Suppose $A$ is a $4\times4$ matrix with $\det A=2$. Calculate $\det\left(\frac{1}{2} A^T A^7 I A^T A^{-1}\right)$, where $I$ is the $4\times4$ identity matrix.

Answer: 16


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

Answer: 0


Question: Given a non-trivial separative forcing poset, if it is separable (i.e., has a countable dense set), must it have size at most the continuum? Provide a justification for your answer.

Answer: Yes


Question: Calculate $f(f(f(2^{1388})))$, where $f(x)$ is defined as the sum of the digits of $x$. Provide a simple and logical method to find the result.

Answer: 4


Question: Students arrive at a help center according to a Poisson process with rate \( r \). When there are \( n \geq 1 \) students in the center, the first one to leave does so at a random \( \text{Exp}(2r) \) time. Suppose that there are currently no students in the center. What is the expected time until there are two students in the center?

Answer: \dfrac{4}{r}


Question: Evaluate the limit: \[ \lim _{x\to 0^+}\frac{\left(e^x+e^{2x}\right)^2-4}{\sqrt{9+\sin x}-3} \]

Answer: 72


Question: Find the natural number \( x \) such that \( \phi(4x) = \phi(5x) \), where \( \phi(y) \) is the Euler's totient function.

Answer: 2


Question: Given the inequality \(2a^2-b^2\leq C(1+a)\) where \(a > 0\) and \(b > 0\), and assuming \(b \leq C_1\) with \(C_1 = \frac{C+\sqrt{C^2+4C}}{2}\), determine if it follows that \(a \leq C_1\).

Answer: a \leq C_1


Question: Evaluate \( \lim_{x\to\infty} ((x^5+x^4)^{1/6}-(x^5-x^4)^{1/6}) \).

Answer: 0


Question:
Find \( \lim\limits_{n \to \infty} \int_0^1 \sqrt[n]{x^n+(1-x)^n} \,dx \). Choose the correct answer from the following options:

A. \(0\)
B. \(1\)
C. \(\dfrac{3}{4}\)
D. \(\dfrac{1}{2}\)
E. \(\dfrac{1}{4}\)

Answer: C


Question: Calculate the derivative \( \frac{\partial (ABC)}{\partial B} \) where \( A \), \( B \), and \( C \) are matrices.

Answer: C^\top \otimes A


Question:
Evaluate the following trigonometric expression:
\[ \sum_{x=2}^{44} 2\sin x^{\circ} \sin 1^{\circ}[1+\sec (x^{\circ} -1^{\circ})\sec (x^{\circ} +1^{\circ})] \]
This expression is equal to:
\[ \frac{-\sin^2 {\theta_1}^{\circ}}{\cos \theta_1}+\frac{\sin^2 {\theta_2}^{\circ}}{\cos \theta_2}-\frac{\sin^2 {\theta_3}^{\circ}}{\cos \theta_3}+\frac{\sin^2 {\theta_4}^{\circ}}{\cos \theta_4} \]
for distinct values of \(\theta_1,\theta_2,\theta_3,\theta_4 \in \{1,2,3,\ldots,44,45\}\). Find the value of \(\theta_1+\theta_2 +\theta_3 +\theta_4\). Choose the correct answer:
\(a)\ 106\)
\(b)\ 53\)
\(c)\ 92\)
\(d)\ 100\)

Answer: C


Question:
Describe the type of continuous real-valued functions \( f(x_1, \ldots, x_n) \) on the unit disk \( D = \{ (x_1, \ldots, x_n) \in \mathbb{R}^n \mid x_1^2 + \cdots + x_n^2 \leq 1 \} \) that satisfy the following condition:
\[
\int \cdots \int_{D} f(x_1, \ldots, x_n)^2 \, dx_1 \cdots dx_n = 0.
\]

Answer: 0


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

Answer: \ln 2


Question:
Find \(\alpha \in \mathbb{R}\) such that there exists a Lie algebra representation of \(\mathfrak{sl}_2 (\mathbb{R})\) on \(C^\infty(\mathbb{R})\) with image equal to \(\text{span}_\mathbb{C} \{ X , Y , Z_\alpha \}\), where the differential operators are defined as follows:

\[X f(t) = f''(t)\]
\[Y f(t) = t^2 f(t)\]
\[Z_\alpha f(t) = t f'(t) + \alpha f(t)\]

for \(f \in C^\infty(\mathbb{R})\).

Answer: \dfrac{1}{2}


Question: Determine the Jacobson radical of the polynomial ring $S = R[x]$ when $S$ is considered as an $R$-module, i.e., find $J_R(S)$. Provide your answer with justification.

Answer: J(R)[x]


Question: Let \(A\in \mathcal{M}_n(K)\) be a diagonalizable square matrix over a field \(K\). Show that the only nilpotent matrix in the set \(\langle A \rangle = \{p(A) = c_0A^k + \cdots + c_{k-1}A + c_kI \mid c_i \in K\}\) is the zero matrix \(O\). Use the spectral decomposition of \(A\), given by \(A = \sum_{i=1}^n \lambda_i E_i\), where \(\lambda_i\) are the eigenvalues of \(A\) and \(E_i\) are idempotent pairwise orthogonal matrices.

Answer: O


Question: Find the sum of all the digits of the smallest positive integer \( n \) such that the product \( P_2 \times P_3 \times \ldots \times P_n \) exceeds 2010, where \( P_k = 1 + \frac{1}{k} - \frac{1}{k^2} - \frac{1}{k^3} \) for a positive integer \( k \).

Answer: 19


Question:
Find the infinite sum of the alternating series:
\[\sum_{n=0}^{\infty} \frac{(-1)^n \, (5 \pi)^{2n+1}}{6^{2n+1} \, (2n+1)!}\]

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{x \to +\infty}\dfrac{\int_0^{x}|\sin t|{\rm d}t}{x} \]

Answer: \dfrac{2}{\pi}


Question: Given the equation \(x^2 + xy + y^3 = 1\), find the value of \(y'''\) at \(x = 1\).

Answer: 42


Question:
Suppose \(f\) and \(g\) are smooth functions such that \(f(0) = 0\), \(g(0) = 1\), and the following conditions hold:

- \(\{x \in \mathbb{R}: g'(x) > 0\} = \{x \in \mathbb{R}: f(x) > 0\}\)
- \(\{x \in \mathbb{R}: g'(x) < 0\} = \{x \in \mathbb{R}: f(x) < 0\}\)
- When \(x\) is such that \(g(x) = 1\), then \(f'(x) = 0\).

Must it be true that \(g'(x) = 0\) for all \(x \in \mathbb{R}\)?

Answer: No


Question: Is the product \( \phi_{\mathbf{X}}(\mathbf{t})\phi_{\mathbf{Y}}(\mathbf{s}) \) a characteristic function of some \( \mathbb{R}^{k+\ell} \)-valued random vector, given that \( \mathbf{X} \) and \( \mathbf{Y} \) are not independent?

Answer: Yes


Question: Determine whether the function \( f(x) = \begin{cases} \frac{1}{\sqrt{x}}, & x \in \mathbb{I} \cap [0,1] \\ x^3, & x \in \mathbb{Q} \cap [0,1] \end{cases} \) is Lebesgue integrable on the interval \([0,1]\).

Answer: Yes


Question: Find the minimum value of the expression \( S = \vec{a} \cdot \vec{b} + 2 \vec{a} \cdot \vec{c} + \vec{b} \cdot \vec{c} \) given that \(|\vec{a}| = 1\), \(|\vec{b}| = 2\), and \(|\vec{c}| = 3\).

Answer: -10


Question: Given independent random variables $X_i \sim U(0,1)$, determine $P\left(\sum_{i=1}^{\infty} X_i < \infty\right)$. Justify your answer using the Kolmogorov 0-1 law and any other relevant theorems.

Answer: 0


Question: Calculate the Lebesgue integral of the function \( f(x) = x^2 \) over the set \( \bar{C} = [0,1] \setminus C \), where \( C \) is the Cantor set.

Answer: \dfrac{1}{3}


Question: Determine the number of solutions for the matrix equation \(X^n = \begin{pmatrix}3&6\\ 2&4\end{pmatrix}\) when \(n\) is an odd positive integer.

Answer: 1


Question: Determine the number of components in the graph $C_{9}$, where the vertices represent all binary strings of length nine, and two strings are adjacent if and only if they differ in exactly three positions.

Answer: 1


Question: Does there exist a smooth function $u$ on $S^1$ such that the generalized Fourier series $$u = \sum_{n \in \mathbb{N}} \langle u, u_n \rangle u_n$$ does not converge uniformly, where $u_n$ is an orthonormal basis of smooth functions on $S^1$?

Answer: Yes


Question: Let $H$ and $K$ be two subgroups of a group $G$, each of finite order. Can the subgroup generated by $H$ and $K$, denoted as $HK$, be infinite?

Answer: Yes


Question: Find the distance from the ellipsoid \(x^2 + y^2 + 4z^2 = 4\) to the plane \(x + y + z = 6\). Use the method of Lagrange multipliers to solve the problem.

Answer: \sqrt{3}


Question: In triangle ABC, the tangents of angles A, B, and C form a harmonic progression. Given that the length of side BC is 189 and the length of side AB is 459, find the length of side AC.

Answer: 351


Question:
Determine the minimum number of terms required to approximate Euler's constant \( \gamma \) using the series definition, such that the error is less than \( 2 \times 10^{-3} \). \( \gamma \) is defined as the limit:
\[ \gamma := \lim_{n \to \infty} \left( H_n - \log n \right) \]
where \( H_n \) is the \( n \)-th harmonic number. Use the inequality:
\[ \sum_{n=k}^{\infty} \frac{1}{n} - \int_k^{\infty} \frac{1}{t} \, dt < 2 \times 10^{-3} \]
to find the smallest integer \( k \).

Answer: 251


Question: Determine the order of \( \bar{2} \) in the multiplicative group \( \mathbb{Z}_{289}^\times \).

Answer: 136


Question: Compute \( \iint _S (y^2+x^2)\, dA \) by using a change of variables, where \( S \) is the region in the first quadrant bounded by the curves \( xy=1 \), \( xy=3 \), \( x^2-y^2=1 \), and \( x^2-y^2=4 \).

Answer: 3


Question: Solve the equation \(\sqrt{x} + \sqrt{x+1} + \sqrt{x+2} = \sqrt{x+7}\).

Answer: \dfrac{1}{24}


Question: Find the product of the real parts of the roots of the equation \( Z^2 - Z = 5 - 5i \).

Answer: -6


Question: Compute the contour integral \( \int_{C} \log(z+3) \, dz \), where \( C \) is a circle centered at the origin with a radius of 2, oriented counterclockwise.

Answer: 0


Question:
Find the value of the expression:
\[ \sqrt[3]{11+4\sqrt[3]{14+10\sqrt[3]{17+18\sqrt[3]{20+28\sqrt[3]{23+\cdots}}}}} \]

Answer: 3


Question: In triangle ABC, the areas of the shaded regions are denoted as \( S_1 \) and \( S_2 \). Given that \( S_1 \cdot S_2 = 16 \text{ cm}^4 \), calculate the length of segment \( MN \).

Answer: 4


Question: What is the probability \( P(B_1 > 0, B_2 > 0) \) where \( B_t \) is a Brownian motion at time \( t \)?

Answer: \dfrac{3}{8}


Question: Maximize the expression \( \frac{1}{x} + \frac{1}{y} \) given that \( \frac{1}{x^n} + \frac{1}{y^n} < 1 \), where \( x, y, \) and \( n \) are natural numbers.

Answer: 1


Question: Consider a twice continuously differentiable function \( f: \mathbb{R}^n \to \mathbb{R} \) such that the Hessian matrix \( H_f(x) = \textbf{0}_{n \times n} \) for all \( x \in \mathbb{R}^n \). Does this imply that \( f \) is a linear multivariable function?

Answer: Yes


Question: Calculate \( 276^{246} \mod 323 \) without using a calculator. Simplify the expression to find the result.

Answer: 254


Question:
How many lines in a three-dimensional rectangular coordinate system pass through four distinct points of the form \((i, j, k)\) where \(i\), \(j\), and \(k\) are positive integers not exceeding four?
\(\text{(A)} \ 60 \qquad \text{(B)} \ 64 \qquad \text{(C)} \ 72 \qquad \text{(D)} \ 76 \qquad \text{(E)} \ 100\)

Answer: D


Question: Evaluate the limit: $$ \lim _{x\to \infty }\left(\frac{\arctan \left(1-\cos \left(\frac{1}{x}\right)\right)-\sin \:\left(\frac{1}{2x^2}\right)-e^{-x}}{\ln\left(1+\frac{1}{x^2}\right)-\arctan \left(\frac{1}{x^2}\right)}\right) $$

Answer: \dfrac{1}{12}


Question: Calculate the triple integral \( \iiint_{V} \nabla \cdot \vec{v} \, dx \, dy \, dz \), where \( V \) is the region in \( \mathbb{R}^3 \) defined by the inequality \( x^2 + y^2 + z^2 \leq 1 \), \( S \) is the surface bounding \( V \), and \( \vec{v} = r^3\vec{r} \) is the vector field. Here, \( r \) is the radial distance and \( \vec{r} \) is the position vector. Use any method to find the result.

Answer: 4\pi


Question: Evaluate the limit \( \lim_{(x, y, z) \to (0,0,0)} \frac{2x^2 y \cos(z)}{x^2 + y^2} \) if it exists.

Answer: 0


Question: Let \( z = f(x,y) \) be a differentiable function such that \( f(3,3) = 1, \) \( f_x(3,3) = -2, \) \( f_y(3,3) = 11, \) \( f(2,5) = 1, \) \( f_x(2,5) = 7, \) and \( f_y(2,5) = -3. \) Suppose \( w \) is a differentiable function of \( u \) and \( v \) satisfying the equation \( f(w,w) = f(uv, u^2+v^2) \) for all \((u,v)\). Find \( \frac{\partial w}{\partial u} \) at \((u,v,w) = (1,2,3)\).

Answer: \dfrac{8}{9}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int_1^2\left(1+\frac{\ln x}{n}\right)^n \,\mathrm{d}x. \]

Answer: \dfrac{3}{2}


Question: Given the function \( (u,v) = f(x,y) = (x + y, x^2 - y^2) \), compute the second partial derivative \( \frac{\partial^2 x}{\partial v^2} \) at the point \((u,v) = (2,0)\).

Answer: 0


Question: Find the radius of convergence of the power series \( \sum^{\infty}_{n=1}\frac{n!}{n^n}z^n \), where \( z \in \mathbb{C} \).

Answer: e


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = 4x \) if \( x \) is rational and \( f(x) = x + 6 \) if \( x \) is irrational. Define the set \( E = \{ x \in \mathbb{R} \mid f \text{ is continuous at } x \} \). Determine all the limit points of the set \( E \).

Answer: \emptyset


Question: Suppose $F \subset [0,1]$ has positive Lebesgue measure, i.e., $\mu(F) > 0$. If the algebraic difference $F-F = \{f-g \mod 1 \mid f,g \in F\}$ equals $[0,1]$, does it follow that $\mu(F) = 1$?

Answer: No


Question: Compute \( \lim_{\beta\to \infty} \beta \mu(f \geq \beta) \) for the function \( f(x) = \frac{1}{x(1-\log x)} \) on the interval \([0,1]\) in the measure space \((X, \Omega, \mu)\).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{((2x)!)^4}{(4x)! ((x+5)!)^2 ((x-5)!)^2}. \]

Answer: 0


Question:
For a smooth curve \( \alpha: I \to \mathbb{R}^3 \) with \( [a,b] \subset I \), determine if the curve contains a line segment given that the following condition holds:
\[ |
\alpha(b)-\alpha(a)| = \int_a^b |
\alpha'(t)|dt \]
Does this condition imply that the curve contains a line segment between the points \( \alpha(a) \) and \( \alpha(b) \)?

Answer: Yes


Question: Evaluate the integral \( \int_{{\mathbb{R}_+}^2} (x^2 - y) e^{-x-y} \, dy \, dx \).

Answer: 1


Question: Given the joint probability density function \( f_{X,Y}(x,y) = 2e^{-x-y} \) for \( 0 < x < y \) and \( 0 < y \), compute the probability \( P(Y < 3X) \).

Answer: \dfrac{1}{2}


Question: Find the number of ways to arrange the numbers \(\{1,2,...,n\}\) in a row such that for every number \(x\) (except for the very left one), at least one of the numbers \(x\pm 1\) is to the left of \(x\).

Answer: 2^{n-1}


Question: Let $f:[0,\infty)\to(0,1)$ be a continuous function. Define the sequence $a_n$ by $a_1 = 1$ and $a_{n+1}=\int_0^{a_n} f(x)\,\mathrm{d}x$. Find $\lim_{n\to\infty} a_n$. \(\text{Hint: Consider the behavior of the sequence as } n \to \infty.\)

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{(1+3x+2x^2)^{1/x} - (1+3x-2x^2)^{1/x}}{x} \]

Answer: 4e^3


Question: Given a parallelepiped determined by three-dimensional vectors \( \mathbf{x}, \mathbf{y}, \mathbf{z} \) with a volume of 11, find the volume of the parallelepiped determined by the vectors \( \mathbf{x} + 2\mathbf{y}, \mathbf{y} + 2\mathbf{z}, \mathbf{z} + 2\mathbf{x} \).

Answer: 99


Question: Evaluate the limit: \[ \lim_{n \to \infty} n \int_{1}^{\infty} \frac{dx}{1+x^n} \]

Answer: \ln 2


Question: Given that the function \( y \) satisfies the differential equation \( \frac{dy}{dx} = 1 - y^2 \) and is symmetric about the point \((\ln(4), 0)\), find the value of \( y(\ln(2)) \).

Answer: -\dfrac{3}{5}


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

Answer: \infty


Question: Calculate the volume inside the cylinder \(x^2+4y^2=4\) and between the planes \(z=12-3x-4y\) and \(z=1\).

Answer: 22\pi


Question:
Evaluate the limit of the function as \( x \to \infty \):

Given the series for the Mangoldt function \( \Lambda(n) \)
\[ f(x) = \sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(\sqrt{x} \log n + \pi/4) \]

Determine if
\[ \lim_{x \to \infty} \frac{f(x)}{x^{1/4}} = 0 \]

Answer: 0


Question: Evaluate the integral \( \int_{\gamma} \frac{z^5}{z^6-1} \, dz \), where \( \gamma \) is the ellipse defined by \( x^2 + 4y^2 = 16 \).

Answer: 2\pi i


Question: What is the cardinality of \(|\mathbb{R}| + |\mathbb{N}|\)?

Answer: 2^{\aleph_0}


Question: Determine the values of $k \in \mathbb{R}$ for which the fractional Sobolev space $W^{k,2}(\mathbb{R}^n)$ is continuously embedded into the space of Lebesgue-measurable functions $\mathcal{M}$ on $\mathbb{R}^n$. Specifically, for which $k$ does the identity map $C^\infty_c(\mathbb{R}^n) \hookrightarrow \mathcal{M}$ extend by continuity to a map $W^{k,2}(\mathbb{R}^n) \to \mathcal{M}$?

Answer: k \geq 0


Question: Find the maximum value of \( \frac{|(\mathbf{Ax}, \mathbf{x})|}{(\mathbf{x}, \mathbf{x})} \) where \( A = \begin{pmatrix}4 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 2\end{pmatrix} \) and \( \mathbf{x} = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix}^T \in \mathbb{R}^3 \) such that \( \sum_{i=1}^{3}x_i = 0 \).

Answer: 3


Question: Determine whether there exists a function \( \varphi \in C^{\infty}_c(\mathbb{R}) \) such that \( \varphi''(x) \geq 0 \) for all \( x \in \mathbb{R} \). If such a function exists, provide an example.

Answer: 0


Question: Determine whether the difference quotient \( \phi_h(t) = \frac{\phi(t+h) - \phi(t)}{h} \) converges uniformly to \( \phi'(t) \) on \( \mathbb{R} \) as \( h \to 0 \), where \( \phi \in C^\infty_c(\mathbb{R}) \) is a smooth function with compact support.

Answer: Yes


Question: Calculate the value of the integral \( \int_{-\infty}^{\infty} x(t) \, dt \) given that the Fourier transform of \( x(t) \) is \( X(j\omega) = \frac{\cos(2\omega)}{\omega^2 + \omega + 1} \).

Answer: 1


Question: Given that $z_1$ and $z_2$ are complex numbers satisfying $z + \bar{z} = 2|z - 1|$ and $\arg(z_1 - z_2) = \frac{\pi}{4}$, find the value of $\Im(z_1 + z_2)$. \(z_1\) and \(z_2\) both satisfy the equation \(z + \bar{z} = 2|z - 1|\).

Answer: 2