Question: Solve the differential equation using Fourier transforms: $$u(x) - u''(x) = x^2$$ You may use the Fourier transform identity: $$\mathcal{F} \{ e^{-a|x|} \} = \frac{2a}{a^2 + s^2}$$

Answer: x^2 + 2


Question: Determine the degree of the field extension \( F_{p}(X^p,Y^p) \subset F_{p}(X,Y) \). Prove that the degree \([F_{p}(X,Y) : F_{p}(X^p,Y^p)] = p^2\).

Answer: p^2


Question: Let $J_c$ be the Julia set for the quadratic polynomial $f_c(z) = z^2 + c$, and let the Mandelbrot set be $M = \{ c \in \mathbb{C} : J_c \text{ is connected} \}$. Consider the closed disc of radius $2$ centered at the origin, $D = \{ c \in \mathbb{C} : |c| \le 2 \}$. Given that $M \subseteq D$, does $c \in M$ imply $J_c \subseteq D$?

Answer: Yes


Question: Calculate the double limit \( \lim_{n\to\infty}(\lim_{k\to\infty}\int_{0}^{1}\left ( \frac{[kx]}{k} \right )^{n}dx)) \).

Answer: 0


Question: Identify the quotient group \( \frac{\mathbb{C}^*}{P} \), where \( \mathbb{C}^* \) is the multiplicative group of non-zero complex numbers and \( P \) is the subgroup of positive real numbers.

Answer: S^1


Question: Suppose $M$ and $N$ are monoids such that the categories $C(M)$ and $C(N)$ are isomorphic. Is it true that $M$ is isomorphic to $N$?

Answer: Yes


Question: Determine the number of distinct terms in the expansion of \((x+\frac{1}{x}+x^2+\frac{1}{x^2})^n\).

Answer: 4n + 1


Question: In a bicycle race between two competitors, let $Y(t)$ denote the amount of time (in seconds) by which the racer that started in the inside position is ahead when $100t$ percent of the race has been completed, where $0 \leq t \leq 1$. The process $Y(t, 0 \leq t \leq 1)$ can be modeled as a Brownian motion with variance parameter $\sigma^2$. If the inside racer is leading by $\sigma$ seconds at the midpoint of the race, what is the probability that she is the winner?

Answer: \Phi(\sqrt{2})


Question: Given a Poisson process $N(t), t \geq 0$ with rate $\lambda$, calculate the covariance of $N(2)$ and $N(3)$. Express your answer in terms of $\lambda$.

Answer: 2\lambda


Question: Evaluate the definite integral \( \int_1^3 \frac{2x-3}{\sqrt{4x-x^2}} \, dx \) by completing the square and using an appropriate substitution.

Answer: \dfrac{\pi}{3}


Question:
Find the maximum value of \( \int_0^1 (f(x))^3 \, dx \) given the following conditions on the function \( f(x) \):

- \( \int_0^1 f(x) \, dx = 0 \)
- \( -1 \leq f(x) \leq 1 \) for all \( x \in [0, 1] \)
- \( f(x) \) is a real-valued function.

Answer: \dfrac{1}{4}


Question: Find the radius of convergence of the series \( \sum_{n=1}^{\infty} (-1)^n x^n \ln\left(1+\frac{1}{n}\right) \).

Answer: 1


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

Answer: 0


Question: The diagonals of a rectangle are both 10 units long and intersect at the point (0,0). Calculate the area of this rectangle, given that all of its vertices lie on the curve \( y = \frac{12}{x} \).

Answer: 14


Question:
Determine whether the following equality holds for the double integral of a function \( f(x, y) \):
\[ \int_{y_1}^{y_2}\int_{x_1}^{x_2}f(x,y)\,dx\,dy = \int_{x_1}^{x_2}\int_{y_1}^{y_2}f(x,y)\,dy\,dx \]

Answer: Yes


Question: Find the mass of the solid cone $C$ with boundary $x^2+y^2=(1-z)^2, 0\leq z \leq 1$, given that the density at any point $(x,y,z)$ is $z$. The mass is given by the integral \( \text{Mass}=\iiint_{C}z\,dV=\int_{0}^{2\pi}\int_{0}^{1}\int_{0}^{1-r}z\,\,dz\,r\,dr\,d\theta. \)

Answer: \dfrac{\pi}{12}


Question: Let $G$ be a group with a subgroup $H$, and let $M$ be a $G$-module over a field $k$. Consider the module $kG \otimes_{kH} kG \otimes_{kG} M$. Is this module isomorphic to $M$ as a $G$-module? Specifically, is $kG \otimes_{kH} kG \simeq kG$ as a $(G,G)$-bimodule, and does this imply $kG \otimes_{kG} M \simeq M$?

Answer: No


Question: Consider a functor $F: \mathcal{A} \to \mathcal{B}$ between additive categories. If $F$ either preserves binary coproducts, meaning $F(A) \oplus F(B) \cong F(A \oplus B)$ for all $A, B \in \mathcal{A}$, or preserves binary products, meaning $F(A \oplus B) \cong F(A) \oplus F(B)$ for all $A, B \in \mathcal{A}$, does it necessarily follow that $F$ preserves the zero object, i.e., $F(0) = 0$?

Answer: Yes


Question: Let $X$ be a $3$-fold, and $f:Y\rightarrow X$ a birational $\mathbb{Q}$-factorial divisorial terminal contraction (of relative Picard number one) contracting a divisor $E\subset Y$ to a point $p\in X$. Is the exceptional divisor $E$ necessarily irreducible?

Answer: Yes


Question:
Find the supremum of \( \operatorname{tr}(Q^T A Q) \) where \( Q \in M_{4\times 2} (\mathbb{R}) \) and \( Q^T Q = I_2 \). The matrix \( A \) is given by:
\[
A = \begin{bmatrix}
3 & 1 & 0 & 0 \\
1 & 3 & 0 & 0 \\
0 & 0 & 6 & 2 \\
0 & 0 & 2 & 6
\end{bmatrix}
\]

Answer: 12


Question: Consider a sample of size \(n=8\) from the Uniform \((\theta, \theta+4)\) distribution where \(\theta>0.\) Two estimators of \(\theta\) are given: \(T_1=\overline{X}\) and \(T_2=5\overline{X}\), where \(\overline{X}\) denotes the sample mean. By comparing the Mean Squared Errors (MSEs) of these estimators, determine whether \(T_1\) is a better estimator than \(T_2\) for \(\theta.\)

Answer: T_1


Question: An unbiased coin is tossed repeatedly. What is the expected number of tosses needed to obtain the sequence HT (one head followed by one tail) for the first time?

Answer: 4


Question:
Does there exist a closed Riemannian manifold $M$ with distinct closed geodesics $\gamma$ and $\alpha$ that satisfy the following properties?
1. $\gamma$ and $\alpha$ have the same length $l$.
2. $\gamma$ is isolated in the space of loops of length $l$.
3. For any $\varepsilon > 0$, there exists a path of loops, each of length between $l$ and $l+\varepsilon$, starting at $\gamma$ and ending at $\alpha$.

Consider the loop space $C^0(S^1,M)$ with the compact-open topology, where $\gamma$ has a neighborhood in which the only loops of length $l$ are reparameterizations of $\gamma$, and the path in (3) is a path of rectifiable loops.

Answer: Yes


Question: Find the Lebesgue measure of the set \( \{ (x,y,z) \in \mathbb{R}^3 : x \in \mathbb{R}, 0 \leq y \leq 10, z \in \mathbb{Z} \} \).

Answer: 0


Question:
Given the probability density function (pdf) of a random variable $X$:

$$ f(x) = \begin{cases} 0 &\mbox{ } \text{elsewhere} \\
\frac{4x}{\theta^2} & \mbox{for } 0 < x \le \frac{\theta}{2} \\
\frac{-4x}{\theta^2}+\frac{4}{\theta} & \mbox{for } \frac{\theta}{2}<x\le\theta  \end{cases} $$

where $0<\theta\le2$, find the method of moments estimator for $\theta$. Assume that the sample mean is $\bar{X}$. Express your answer in terms of $\bar{X}$. \( \theta_{MOM} = 2\bar{X} \)

Answer: 2\bar{X}


Question: Determine the number of permutations of the set \( \{1, 2, \ldots, 9\} \) in which exactly one odd integer is in its natural position.

Answer: 120120


Question:
Determine which of the following rings is the ring of integers of the number field $K = \mathbb{Q}(\alpha)$, where $\alpha$ is the unique real root of the polynomial $f(T) = T^3 - T^2 + 2T + 8$:

1. $\mathbb{Z}\left[\alpha, \frac{\alpha^2}{2}\right]$
2. $\mathbb{Z}\left[\alpha, \frac{\alpha + \alpha^2}{2}\right]$

Answer: B


Question: Let $a, b, c, p$ be rational numbers, with $p$ not being a perfect cube. If $a + b p^{1/3} + c p^{2/3} = 0$, prove that $a = b = c = 0$.

Answer: a = b = c = 0


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac {1}{n^k}\binom{n}{k} \] where \( k \in \mathbb{Z}_+^* \).

Answer: \dfrac{1}{k!}


Question:
Find the smallest constant \( c \) such that for any function \( f \in C^4([0,1]) \) and any \( x, y \in [0,1] \), the following inequality holds:
\[ \left\vert\frac{f(x)-f(y)}{x-y}-\frac{f'(x)+f'(y)}{2}\right\vert\le c\vert x-y\vert ^2\max_{\xi\in[0,1]}\vert f'''(\xi)\vert \]

Answer: \dfrac{1}{12}


Question: Determine the number of permutations of the multiset \( \{1,1,2,2,3,3,4,4\} \) such that no two consecutive numbers are the same.

Answer: 864


Question: Two planes \(\pi_1\) and \(\pi_2\) intersect in a line \(l\). The angle between planes \(\pi_1\) and \(\pi_2\) is \(45^\circ\). Let \(A\) be a point on \(l\), and let \(m\) be the line in plane \(\pi_1\) passing through \(A\) that is perpendicular to \(l\). Let \(B\) be a point in plane \(\pi_1\) such that the angle between line \(m\) and line \(AB\) is \(45^\circ\). Let \(P\) be the projection of \(B\) onto plane \(\pi_2\). Find the angle between lines \(AB\) and \(AP\).

Answer: 30^\circ


Question: Let \( D = \{(x,y) \in \mathbb{R}^2: x^2 + y^2 \leq 1 \} \) be a closed disk. Suppose \( A \subset \mathrm{int}D \) is connected and compact, and \( D \setminus A \) is connected. Let \( f: A \to A \) be a continuous function, and \( g: D \to D \) be a continuous function such that \( g_{|A} = f \). Does there exist a point \( x \in \mathrm{int}D \) such that \( g(x) = x \)?

Answer: Yes


Question: Find the value of \(\frac{a+b}{ab}\) such that \(\int_{-1/2}^{1/2} \cos x \ln\frac{1+ax}{1+bx} \, dx = 0\), where \(a\) and \(b\) are real numbers and not equal.

Answer: 0


Question: Solve the equation \(x^{2^{\sqrt{2}}} = {\sqrt{2}}^{2^x}\) for \(x \in \mathbb{R}^{+}\).

Answer: \sqrt{2}


Question: Consider the function \( F(x) = \sum_{j=1}^k \frac{1}{|x-z_j|^2} \), where \( z_j \) are \( k \) points on the complex plane, none of which lie on the real line. Determine whether it is always true that \( F(x) \) has at most \( k \) local maxima on the real line.

Answer: Yes


Question:
Suppose $V$ is a finite-dimensional complex Hilbert space, and let $A_1, \dots, A_r: V \rightarrow V$ be linear operators. Define the $L_{2,d}$-spectral radius of $A_1, \dots, A_r$ as:
\[ \rho_{2,d}(A_1, \dots, A_r) = \sup \left\{ \frac{\rho(A_1 \otimes X_1 + \dots + A_r \otimes X_r)}{\rho(\Phi(X_1, \dots, X_r))^{1/2}} \mid \rho(\Phi(X_1, \dots, X_r)) \neq 0, X_1, \dots, X_r \in M_d(\mathbb{C}) \right\}. \]
Given that $1 \leq d < \dim(V)$, is there a set of linear operators $A_1, \dots, A_r: V \rightarrow V$ such that $\rho_{2,d}(A_1, \dots, A_r) < \rho_{2,d+1}(A_1, \dots, A_r)$?

Answer: Yes


Question: Can a non-amenable connected Lie group have a faithful finite-dimensional unitary representation?

Answer: No


Question: Evaluate the limit: \[ \lim_{x\to0^+}(\cot x-\frac{1}{x})(\cot x+\frac{1}{x}) \]

Answer: -\dfrac{2}{3}


Question:
Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that for all $x, y \in \mathbb{R}$, the following conditions hold:
\[ f(xy) = xf(y) + yf(x) \]
\[ f(x+y) = f(x^{2021}) + f(y^{2021}) \]
Calculate $f(\sqrt{2020})$. \( \)

Answer: 0


Question: Compute \( \lim_{s \to \infty} \int_0^1 \left|\cos(sx)\right| \, dx \).

Answer: \dfrac{2}{\pi}


Question: Prove that the equation \(x^n + ax + b = 0\) has at most two real solutions for any real numbers \(a\) and \(b\), where \(n\) is an even natural number.

Answer: 2


Question: Determine whether every natural number can be extended to form a prime number by appending digits. Specifically, for any natural number \( n \), does there exist a prime number \( p \) such that \( n \) is a truncation of \( p \)?

Answer: Yes


Question: Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth function such that for every $x \in \mathbb{R}^n$, the derivative $A = Df(x)$ satisfies the condition $a_{ii} = 1$ and $a_{ij} = k(x)$ for all $i \neq j$, where $k(x)$ depends only on $x$. Does this imply that $f$ is a linear map?

Answer: No


Question: Find the volume of the solid bounded by the surfaces $x^2 + y^2 - 2y = 0$, $z = x^2 + y^2$, and $z = 0$. Use polar coordinates to set up and evaluate the appropriate integral.

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


Question: Determine the real dimension of the cone of $2 \times 2$ Hermitian matrices with at least one eigenvalue equal to $0$. Provide your answer as a single integer.

Answer: 3


Question: If $X$ is a Tychonoff space and $\beta X$ is its Stone-Cech compactification, is $X$ a dense $G_\delta$ subset of $\beta X$?

Answer: No


Question: Find the largest integer \( n \) such that \( 64n\log_2 n < 8n^2 \).

Answer: 43


Question: Given that \(x > 1\), \(y > 1\), and \((\log_e x)^2 + (\log_e y)^2 = \log(x^2) + \log(y^2)\), find the maximum value of \(x^{\log_e y}\).

Answer: e^4


Question: Find the greatest real number \( C \) such that the inequality \( |\tan x - \tan y| \geq C|x-y| \) holds for all \( x, y \in \left(\frac{-\pi}{2}, \frac{\pi}{2}\right) \).

Answer: 1


Question:
Let $f: \mathbb{R} \to \mathbb{R}$ be a bounded function and let $a < b$. Determine whether the following equality holds:
\[ \sup_{x \in (a,b)} f(x) - \inf_{y \in (a,b)} f(y) = \sup_{x,y \in (a,b)} | f(x)-f(y)|. \]

Answer: Yes


Question: Let $f(x)$ and $g(x)$ be two polynomials in $\mathbb{C}[x]$. If $f^{-1}(0) = g^{-1}(0)$ and $f^{-1}(1) = g^{-1}(1)$, prove that $f(x) = g(x)$. Here, $f^{-1}(a) = \{z \in \mathbb{C} \mid f(z) = a\}$.

Answer: f(x) = g(x)


Question: Determine the number of distinct integral solutions to the equation \(a \times b \times c \times d = 600\).

Answer: 6400


Question: Evaluate the line integral \( \int z \sin z \, dz \) over the path \( \gamma(t) = \pi t + (t^2-t)i \) for \( 0 \leq t \leq 1 \).

Answer: \pi


Question: Find the operator norm \( ||\phi|| \) of the linear functional \( \phi : \mathcal{L}^1(m) \to \mathbb{R} \) defined by \( \phi(f) = \int (x - \frac{1}{2}) f(x) \, dm(x) \), where \( (X, \mathcal{A}) = ([-1, 1], \mathcal{B}_{[-1, 1]}}) \) is a measurable space equipped with the Lebesgue measure \( m \).

Answer: \dfrac{3}{2}


Question:
Consider the function \(x : \mathbb{R} \to \mathbb{R}\) given by
\[x(t) = \frac{1}{30000} \frac{1}{\mathrm{e}^t}+ \frac{2}{30000} \mathrm{e}^{\frac{t}{2}} \cos \left(\frac{\sqrt{3}}{2}t\right), \quad t\in \mathbb{R}.\]
Arrange the roots of this function \(x\) in increasing order. Let \(t_n\) denote the \(n\)th root of the function. The sequence \(\{t_n\}_{n = 1}^\infty\) begins with (rounded to one decimal place)
\[1.8, 5.4, 9.1, 12.7, 16.3, \ldots\]
Does there exist a positive integer \(N\) such that, for each positive integer \(n \ge N\),
\[t_{n + 1} \le t_n + 5?\]

Answer: Yes


Question: Let the function \( g:(-\infty,\infty)\rightarrow(-\frac{\pi}{2},\frac{\pi}{2}) \) be given by \( g(u) = 2\tan^{-1}(e^u) - \frac{\pi}{2} \). Determine whether \( g \) is (A) even and strictly increasing in \((0,\infty)\), (B) odd and strictly decreasing in \((-\infty,\infty)\), (C) odd and strictly increasing in \((-\infty,\infty)\), or (D) neither even nor odd and is strictly increasing in \((-\infty,\infty)\).

Answer: C


Question: Find the radius of convergence of the power series \( \sum a_{n}x^{n} \), where \( a_{0}=0 \) and \( a_{n}=\frac{\sin(n!)}{n!} \) for \( n\geq 1 \).

Answer: \infty


Question:
Determine the number of zeros in the right half-plane \( \text{Re} \, z > 0 \) for the polynomial:
\[
f(z) = z^3 - z + 1
\]

Answer: 2


Question: Evaluate the integral \( \int_c z^2 dz \) where \( c \) is the circle \( |z| = 2 \).

Answer: 0


Question: Is there a ring $R$ with infinite global dimension that does not have any finitely generated module of infinite projective dimension?

Answer: Yes


Question: Does the poset of free factors of a free group form a lattice? Specifically, for a free group $F_n$ of rank $n$, is the set of its free factors, partially ordered by inclusion, a lattice?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\rightarrow0}\frac{\int_{\cos(\sqrt{x})}^{1}\frac{\cos(t)}{1-t}dt}{\sqrt{x}} \]

Answer: \infty


Question: Is it possible to find an uncountable chain of subsets of the natural numbers \( \mathbb{N} \) such that for every pair of elements in the chain, one is a subset of the other?

Answer: Yes


Question: Let $S$ be a smooth projective algebraic surface over $\mathbb{C}$ and $C$ be a smooth curve on $S$. Determine whether $\dim_{\mathbb{C}} H^0(S, \mathcal{O}_{S}(-C)) = 0$. In particular, verify if this holds when $S$ is a K3-surface.

Answer: 0


Question: Find the minimum value of \((\tan C - \sin A)^2 + (\cot C - \cos B)^2\) given that \((\sin A, \cos B)\) lies on a unit circle centered at the origin, and \(\tan C\) and \(\cot C\) are defined.

Answer: 3 - 2\sqrt{2}


Question: Find the inverse Laplace transform of \( \frac{3}{(s-1)^2} \).

Answer: 3te^t


Question: Determine whether it is possible to find 99 consecutive natural numbers such that the first number is divisible by 100, the second by 99, the third by 98, and so on, with the last number divisible by 2.

Answer: Yes


Question: Determine whether the following statement is true: If dilators $D_0$ and $D_1$ are bi-embeddable, meaning there exist embeddings $T_0: D_0 \Rightarrow D_1$ and $T_1: D_1 \Rightarrow D_0$, then $D_0 = D_1$. Provide a justification for your answer.

Answer: True


Question: Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(x) = f(x^2 + 1) \) for all \( x \in \mathbb{R} \).

Answer: f(x) = c


Question: Evaluate the integral \( \int_{0}^{1}\sqrt{\frac{x}{1-x}}\,\mathrm{d}x \).

Answer: \dfrac{\pi}{2}


Question: Find the pair of positive integers \((a, b)\) such that \(3^a + 7^b\) is a perfect square.

Answer: (2, 1)


Question: Let \( \alpha > 0 \). Consider the sequence of independent random variables \( \{ X_n^{(\alpha)} \}_{n \geq 1} \) such that \( P(X_n^{(\alpha)} = 1) = \frac{1}{n^{2\alpha}} = 1 - P(X_n^{(\alpha)} = 0) \). Define the set \( S = \{ \alpha > 0 : X_n^{(\alpha)} \overset{a.s}{\rightarrow} 0, n \rightarrow \infty \} \). Find \( \inf S \).

Answer: \dfrac{1}{2}


Question:
Find the limit of \( y(x) \) as \( x \to \infty \), where \( y(x) \) is defined by the integral equation:
\[
y(x) = 10 + \int_0^x \frac{22(y(t))^2}{1 + t^2}\,dt
\]

Answer: \infty


Question: If $|z_1 - z_2| = |z_1 + z_2|$, prove that $\arg(z_1) - \arg(z_2) = \frac{\pi}{4}$.

Answer: \frac{\pi}{2}


Question: Find a continuous and absolutely integrable function \( f(x) \) that satisfies the integral equation \( \int_{-\infty}^{\infty} e^{-|t-x|}f(x) \, dx = e^{-|t|} + |t|e^{-|t|} \) using the Convolution Theorem.

Answer: e^{-|x|}


Question: Evaluate the limit: $$\lim_{n\to\infty} \int_{0}^{\pi} e^x\cos(nx)\space dx$$

Answer: 0


Question: Determine the set of values for the parameter $a \in \mathbb{R}$ such that the function $f(x) = x^2 - a |3x - 1| + 4$ is convex on the interval $[0, 1]$. Provide your answer as an interval.

Answer: (-\infty, 0]


Question: Calculate the derivative of \( \text{Tr}(X^TP^TPX) \) with respect to the matrix \( P \), where \( X \) and \( P \) are real matrices.

Answer: 2 P X X^T


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables following a Bernoulli distribution with parameter $p$. Let $\overline{X}$ be the sample mean. Show that the expression \( \frac{\overline{X}(1-\overline{X})}{\sqrt{n} - \frac{1}{\sqrt{n}}} \) converges in probability to zero as $n$ approaches infinity.

Answer: 0


Question: Find $H'(2)$ given that \[ H(x) = \int^{x^3-4}_{2x} \frac{x}{1+\sqrt{t}}\,dt. \]

Answer: \dfrac{20}{3}


Question: Compute the standard part of the expression \(\frac{3-\sqrt{c+2}}{c-7}\) given that the standard part of \(c\) is 7 and \(c \neq 7\).

Answer: -\dfrac{1}{6}


Question: Find the spectrum of the operator \( T \) on the Banach space \( \ell^1 \), where \( \|x_n\|=\sum_{n=1}^\infty |x_n| \) and \( T\{x_n\}=\{x_2,x_1, x_4, x_3, x_6, x_5, x_8,x_7, \ldots \} \).

Answer: \{-1, 1\}


Question:
Evaluate the integral \( \int\limits_{-1/2}^{1/2}(\sin^{-1}(3x-4x^3)-\cos^{-1}(4x^3-3x))dx \). Choose the correct answer from the following options:

\(A)\ 0\hspace{1cm}(B)\ \frac{-\pi}{2}\hspace{1cm}(C)\ \frac{\pi}{2}\hspace{1cm}(D)\ \frac{7\pi}{2}\)

Answer: B


Question: Let $R$ be an integral domain, and let $a, b \in R$ with $a \neq 0$ and $b \neq 0$. Given that $15a = 0$ and $3b \neq 0$, determine the characteristic of $R$.

Answer: 5


Question: Let $\{B_t: 0 \leq t < \infty\}$ be a standard Brownian motion and $T_n$ be an increasing sequence of finite stopping times converging to infinity almost surely. Determine whether the following property holds: $$\lim_{n \to \infty}\frac{B_{T_n}}{T_n} = 0$$ almost surely.

Answer: 0


Question: Evaluate the sum \( \sum_{k=0}^{2013} 2^k a_k \) for the sequence \( a_0, a_1, a_2, \ldots, a_n \) defined by \( a_0 = 2013 \) and \( a_n = -\frac{2013}{n} \sum_{k=0}^{n-1} a_k \) for \( n \geq 1 \).

Answer: -2013


Question: Calculate the expected value of \( \frac{1}{1+X} \) where \( X \sim \text{Beta}(2,3) \).

Answer: 34 - 48 \ln 2


Question: Determine the maximum variance of a distribution with support in the interval $[0, 1]$ and a fixed mean $\mu \in [0, 1]$. Is there an upper bound on this variance?

Answer: \mu(1 - \mu)


Question: Let \( a, b, c \) be the roots of the polynomial \( x^3 - (k+1)x^2 + kx + 12 = 0 \), where \( k \) is a real number. Given that \( (a-2)^3 + (b-2)^3 + (c-2)^3 = -18 \), find the value of \( k \).

Answer: 5


Question: Lotta is a consulting mathematician with 100 clients ranked 1 through 100 in order of importance. Each day, she visits the most important client who feels mistreated. A client feels mistreated if Lotta has never visited them, or if she has visited someone less important since the last time she visited them. On the first day, she visits client 1; on the second day, she visits client 2; on the third day, she goes back to client 1, and so on. Determine the number of days it will take for Lotta to retire when none of her clients feel mistreated.

Answer: 2^{100} - 1


Question: Given that $x^2 + y^2 + z^2 = 1$ where $x, y, z$ are real numbers, find the maximum value of the expression $x^3 + y^3 + z^3 - 3xyz$. Provide your answer as a single number.

Answer: 1


Question: Express \( \sum_{i,j\in\{0,1,2\}\atop i\neq j} x_ix_j^2 \) using the elementary symmetric functions \( e_1 = x_0 + x_1 + x_2 \), \( e_2 = x_0x_1 + x_1x_2 + x_0x_2 \), \( e_3 = x_0x_1x_2 \), and the power sums \( p_2 = x_0^2 + x_1^2 + x_2^2 \), \( p_3 = x_0^3 + x_1^3 + x_2^3 \).

Answer: e_1 p_2 - p_3


Question:
Consider the pair of first-order ordinary differential equations: \( \frac{dx}{dt} = Ax + By, \frac{dy}{dt} = x, \) where \( B < -1 < A < 0. \) Let \((x(t), y(t))\) be the solution of the above that satisfies \((x(0), y(0)) = (0, 1).\) Determine which of the following statements is correct:
1. \((x(t), y(t)) = (0, 1)\) for all \( t \in \mathbb{R} \)
2. \(x(t)\) is bounded on \( \mathbb{R} \)
3. \(y(t)\) is bounded on \( \mathbb{R} \)
4. \(x(t)\) is bounded on \([0, \infty) \).

Answer: 4


Question: Let $q$ be a prime and $n \ge 2$ an integer. Define $f_q(n)$ as the number of $q$-th residues modulo $n$. Is it true that for any positive constant $K$, there exist infinitely many integers $n$ such that $f_q(n) \le Kn$?

Answer: Yes


Question: A cylindrical container is to be made from a solid material with the following constraints: it has a fixed inner volume \( V \) mm\(^3\), a 2 mm thick solid wall, and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and has a radius equal to the outer radius of the container. If the volume of the material used to make the container is minimized when the inner radius of the container is 10 mm, find the value of \( \frac{V}{250\pi} \).

Answer: 4


Question: Given that \(0 \le a, b \le 3\), find the value of \(a+b\) such that the equation \(x^2 + 4 + 3\cos(ax+b) = 2x\) has at least one real solution.

Answer: \pi


Question: On a train route with 18 stations, in how many ways can a train stop at 4 stations such that there are at least 2 and at most 6 stations between any two stops?

Answer: 390


Question: Consider \(n\) linearly independent unit vectors \(\nu_1, \nu_2, \ldots, \nu_n \in \mathbb{R}^n\). For each unit vector \(\nu_i\), define an affine hyperplane \(A_{\nu_i}\) orthogonal to \(\nu_i\). Is the intersection of the hyperplanes \(A_{\nu_1}, A_{\nu_2}, \ldots, A_{\nu_n}\) necessarily a single point in \(\mathbb{R}^n\)?

Answer: Yes


Question: A weight is to be suspended 10 feet below a horizontal line AB by a wire in the shape of a Y. If the points A and B are 8 feet apart, determine the shortest total length of wire that can be used.

Answer: 10 + 4\sqrt{3}


Question: Find the minimum value of \[ \frac{(\sum_{i=1}^n x_i)^2}{\sum_{i=1}^n \sqrt{x_i^2 - i^2}} \] given that \( x_i > i \) for all \( 1 \le i \le n \).

Answer: n(n+1)


Question: Evaluate the integral \( \int_{0}^{2\pi} \exp(\sin \theta) \cos(\cos \theta) \mathrm{d}\theta \).

Answer: 2\pi


Question: Find all pairs \((p, n)\) of positive integers where \(p\) is a prime number and satisfies the equation \(p^3 - p = n^7 - n^3\).

Answer: (5, 2)


Question: Evaluate the limit: $$ \lim_{x\to\infty} \left(\frac {e^x-x^2 }{e^x-5x}\right)^{\log x} $$

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \cos^2(\pi \sqrt[3]{n^3+n^2+2n}), \quad n \in \mathbb{N} \]

Answer: \dfrac{1}{4}


Question: Evaluate the integral \( \int_0^{\frac{\pi}{2}} \log\left(\frac{1+a\cos(x)}{1-a\cos(x)}\right)\frac{1}{\cos(x)}dx \) for \( |a| \le 1 \).

Answer: \pi \arcsin a


Question: Calculate \( \int_0^1 \ln^{2020}(x) \, dx \).

Answer: 2020!


Question: Determine whether the outer measure of the real numbers, \( m^*(\mathbb{R}) \), is infinite. Recall the definition of outer measure: \[ m^*(E) = \inf\left\{\sum_{n=1}^{\infty}|I_n| : E \subseteq \bigcup_{n=1}^{\infty} I_n\right\} \] where \( I_n \) are intervals covering \( E \).

Answer: \infty


Question: Can every odd prime number, except 11, be the smallest prime factor of a Carmichael number with exactly three prime factors?

Answer: Yes


Question: Is the support of functions in \(\ell^1(I)\) always countable for an arbitrary set \(I\)? Here, the support refers to the set of elements \(i \in I\) such that \(f(i) \neq 0\) for a function \(f \in \ell^1(I)\).

Answer: Yes


Question: Determine whether there exists a 47-factor of the complete graph on 100 vertices, denoted as \( K_{100} \).

Answer: Yes


Question: Evaluate the limit \( \lim_{R \rightarrow \infty} \int_{\vert z \vert = R} \frac{e^{\frac{1}{z}}}{z^k}dz = 0 \) for all \( k \geq 2 \).

Answer: 0


Question: Determine whether the integral \( \int_0^\infty \cos x^2 \, dx \) converges. Answer "Yes" or "No".

Answer: Yes


Question: Given that $X$, conditioned on $Y = y$, follows a geometric distribution with parameter $y$, and $Y$ follows a uniform distribution on the interval $(0,1)$, determine the expected value $E(X)$. Use the probability mass function (PMF) for the geometric distribution and the probability density function (PDF) for the uniform distribution in your calculations.

Answer: \infty


Question:
Determine the order of growth for the recurrence relation given by:

\[ T(n) = n + \sum\limits_{k=1}^{n-1} [T(n-k) + T(k)] \]

with the initial condition \( T(1) = 1 \). What is the order of \( T(n) \) in big O notation?

Answer: O(3^n)


Question: Let $\mathcal{H}$ be a Hilbert space, $B(\mathcal{H})$ the bounded linear operators on $\mathcal{H}$, and $M_n(B(\mathcal{H}))$ the $n \times n$ matrices with operator-valued entries. Consider a matrix $A = [a_{ij}]$ in $M_n(B(\mathcal{H}))$. If $\sum_{i,j=1}^n u_i^* a_{ij} u_j$ is a positive semi-definite operator in $B(\mathcal{H})$ for all $n$-tuples $(u_1,...,u_n)$ of unitary elements in $B(\mathcal{H})$, does this imply that $A$ is positive semi-definite in $M_n(B(\mathcal{H}))$?

Answer: Yes


Question: Let $z_0 \in \mathbb{C}$ and $R>0$. Suppose that $f$ is analytic on $B(z_0,R)-\{z_0\}$ and has a pole of order $m \geq 1$ at $z_0$. Prove that $\lim_{z \rightarrow z_0} |f(z)| = \infty$. Justify your answer using the properties of the Laurent series expansion.

Answer: \infty


Question: Evaluate the limit: \[ \lim_{h \to 0} \frac{f(2-h^2) - f(2)}{h^2} \] where \( f(x) = (x^3 + 2x) \left\lfloor \frac{x}{2} \right\rfloor \).

Answer: -\infty


Question: If $H$ is a Hilbert space with a subspace $A$ such that the orthogonal complement $A^\perp$ of $A$ is trivial, i.e., $A^\perp = \{ 0 \}$, is $A$ necessarily dense in $H$? Provide a justification for your answer.

Answer: Yes


Question: Consider a sequence of conformal injective mappings \(f_n\) of the unit disk \(D\) onto the nested smooth Jordan domains \(D_n\subset D\), such that \(\cup_{n=1}^\infty D_n=D\) and \(D_n\) are images of \(n/(n+1) D\) under a fixed diffeomorphic quasi-conformal mapping of the unit disk onto itself. Assume that \(f_n\to id\) uniformly on \(D\). Determine whether the sequence \[ I_n=\int_D\frac{|f_n'(z)|^2}{|1-z|} \, dx \, dy \] is bounded.

Answer: Yes


Question: Let \( A, B \in M_2 \) be 2x2 matrices, and define \( C = AB - BA \). Prove that \( C^2 = \lambda I \) for some scalar \( \lambda \).

Answer: C^2 = \lambda I


Question: Let $G$ be a group where $a$ and $b$ are non-identity elements such that $ab = b^2a$. If the subgroup of $G$ generated by $a$ has order 3, determine the order of the subgroup of $G$ generated by $b$.

Answer: 7


Question: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that for $x_0 \in \mathbb{R}$, the limit \( \lim_{\mathbb{Q} \ni h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} \) exists. Is the function $f$ differentiable at $x_0$?

Answer: Yes


Question: Does the condition \(E[e^{it(aX + bY)}]=E[e^{itaX}]E[e^{itbY}]\) for every \(a,b\in\mathbb{R}\) imply that the random variables \(X\) and \(Y\) are independent? Assume \(X\) and \(Y\) are two random variables such that for every \(\alpha, \beta \in \mathbb{R}\), \[E[e^{it(\alpha X + \beta Y)}]=E[e^{it\alpha X}]E[e^{it\beta Y}]\] for all \(t\in\mathbb{R}\). Determine if \(X\) and \(Y\) are independent.

Answer: Yes


Question: Given that H is the orthocenter of an acute-angled triangle ABC, and the circumcircle of triangle ABC is given by the equation \(x^2 + y^2 = 16\), find the circumdiameter of triangle HBC.

Answer: 8


Question: For a binomial distribution with a mean of 4 and a variance of 3, calculate the third central moment.

Answer: \dfrac{3}{2}


Question:
Given two sets $X$ and $Y$ and a function $f: X \to Y$, consider the following definitions: for $C \subseteq X$, $f(C) = \{f(x) : x \in C\}$, and for $D \subseteq Y$, $f^{-1}(D) = \{x : f(x) \in D\}$. Determine which of the following statements is true:

(A) $f(f^{-1}(B)) = B$

(B) $f^{-1}(f(A)) = A$

(C) $f(f^{-1}(B)) = B$ only if $B \subseteq f(X)$

(D) $f^{-1}(f(A)) = A$ only if $f(X) = Y$

Answer: C


Question: Compute the integral \( I = \int_0^\infty \frac{\sin^4 x}{x^3} \, dx \) using the Laplace transform method.

Answer: \ln 2


Question: Evaluate the integral \( \int_{0}^{\infty} (5x)^4 e^{-2x} \, dx \) using the gamma function.

Answer: \dfrac{1875}{4}


Question: Consider a continuous-time simple random walk $X_t$ on $\mathbb{Z}$ starting at $0$. Determine whether the limit \( \lim_{t \to \infty}\mathbb{P}(X_t>k) \) exists, and if it does, calculate its value.

Answer: \dfrac{1}{2}


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

Answer: 0


Question: Determine the number of elements in the quotient ring \( \mathbb{Z}[\sqrt{2}]/(1+3\sqrt{2}) \).

Answer: 17


Question: Let $A=(a_{ij})$ and $B=(b_{ij})$ be two $n \times n$ real symmetric matrices such that $a_{ij} \leq b_{ij} + \alpha$ for all $i, j$, where $\alpha > 0$. Can we conclude that $A \leq B + \alpha \mathbf{1}$, where $\mathbf{1}$ is the matrix with all elements equal to 1?

Answer: No


Question: Let \( f(x) = (\tan x)^{\sin 2x} \) for \( x \in (0, \frac{\pi}{2}) \). Let \( i \) be the argument where the function takes its infimum and \( s \) be the argument where it takes its supremum. Find \( i + s \).

Answer: \dfrac{\pi}{2}


Question: A particle of mass 2 kg is subjected to a two-dimensional conservative force given by \( F_x = -2x + 2y \) and \( F_y = 2x - y^2 \), where \( x \) and \( y \) are in meters and \( F \) is in Newtons. If the particle has a kinetic energy of \( \frac{8}{3} \) Joules at the point \((2,3)\), find the speed of the particle when it reaches the point \((1,2)\).

Answer: 2


Question: Consider the matrix \(D=\begin{pmatrix}1&0\\0&e^{i\theta}\end{pmatrix}\). For the commonly used norms \(\|\cdot\|\) on \(\mathbb{C}^2\), or for \(\theta=0\), the associated subordinate norm is 1. Is it always true that the subordinate norm is 1, or can it be strictly greater than 1?

Answer: 1


Question: Evaluate the double integral \( \iint_D \sqrt{x^2+y^2}\,dx\,dy \) where \( D=\{(x,y)\in\mathbb{R^2},\ x^2+y^2-4x<0\} \) using polar coordinates. Determine the appropriate limits for \( \rho \) and \( \theta \) and compute the integral.

Answer: \dfrac{256}{9}


Question: What is the best known bound on the index of an abelian subgroup in a discrete subgroup of the Euclidean isometry group \( \isom(\mathbb{R}^n) \)?

Answer: 2^n n!


Question: Find the distance d(A, B) between the set A, defined as the rectangular hyperbola xy = 1, and the set B, defined as the union of the coordinate axes xy = 0. The distance d(A, B) is given by the infimum of the distances d(a, b) for all a in A and b in B.

Answer: 0


Question: Compute the determinant of the \( n \times n \) matrix with entries \( a_{ij} = \begin{cases} 2 & \text{if } i=j \\ -1 & \text{if } i=j \pm 1 \\ 0 & \text{otherwise} \end{cases} \) using the matrix tree theorem.

Answer: n+1


Question: Determine whether $M_{2}(K)$, the ring of all $2 \times 2$ matrices over a field $K$, is a polynomial identity ring (PI ring).

Answer: Yes


Question: Let \( \{x_n\} \) be a non-decreasing sequence of integers such that \( \lim_{n\to\infty} \left( \frac{\text{number of elements in } \{x_i\}_{i\leq n} \text{ that are } \leq n }{n} \right) = \frac{1}{2} \). Determine if \( \lim_{n\to\infty} \left( \frac{\sum_{i=1}^n x_i}{n^2} \right) = 1 \).

Answer: 1


Question: Calculate the sum \( \sum_{k=1}^{2002} r_k \), where \( r_k \) is the remainder when \( 2^k \) is divided by 2003.

Answer: 2005003


Question:
Determine the asymptotic bound in terms of $n$ for the summation:
\[ \sum_{i=0}^{n}{\frac{P(n, i)}{n^i}} = \sum_{i=0}^{n}{\frac{n!}{(n-i)! \cdot n^i}}. \] Is it $O(n)$, $O(\log n)$, or another bound?

Answer: O(\sqrt{n})


Question: Let \( \Omega \subset \mathbb{R}^n \) be a bounded subset, and let \( f \in C(\bar{\Omega}) \) with \( f|_{\partial \Omega} = 0 \). Can we find a sequence \( \{f_n\} \subset C_c^\infty(\Omega) \) such that \( \|f_n - f\|_{C(\bar{\Omega})} \to 0 \)?

Answer: Yes


Question: Evaluate the line integral \( \int (x-y) \, dx + xy \, dy \) over the circle of radius 2 centered at the origin.

Answer: 4\pi


Question: Determine the number of solutions to the equation $f(z) = z$ within the open unit disc $|z| < 1$, given that $f(z)$ is analytic and satisfies $|f(z)| < 1$ for $|z| \leq 1$. Use Rouche's Theorem to justify your answer.

Answer: 1


Question: Let $X$ and $Y$ be Banach spaces, and let $T: X \to Y$ be a surjective continuous linear transformation. Is it true that for every convergent sequence $\{y_n\}$ in $Y$, converging to $y \in Y$, there exists a sequence $\{x_n\}$ in $X$, converging to $x \in X$, such that $T(x_n) = y_n$ for all $n \in \mathbb{N}$ and $T(x) = y$?

Answer: Yes


Question: Suppose $Q$ is a rectangle in $\mathbb{R}^n$ and $\{Q_1,Q_2,Q_3,\ldots\}$ is a countable collection of boxes such that \( Q \subset \bigcup_{i=1}^{\infty} Q_i \). Does it follow that \( v(Q) \leq \sum_{i=1}^{\infty} v(Q_i) \), where \(v\) is the Euclidean volume on $\mathbb{R}^{n}$? Determine if this statement is true or false.

Answer: True


Question: Determine whether there exist polynomials with rational coefficients \( P(x) \) and \( Q(x) \) such that \( P(x)x^3 + P(x)x^2 + Q(x)x + 2Q(x) = 1 \).

Answer: Yes


Question: Find all polynomials $P(x, y)$ with real coefficients such that $P(x, y) = P(x+y, x-y)$ for all real numbers $x$ and $y$. Provide the general form of such polynomials.

Answer: P(x, y) = c


Question: Suppose $D$ is a tournament with the vertex set $V = \{1, 2, 3, 4, 5\}$. Let $\rho$ be the relation on $V$ where $u\rho v$ means "either $u=v$ or the directed edge $uv$ is an edge of $D$". If $\rho$ is transitive, determine whether $D$ has exactly one Hamiltonian path. Answer with true or false.

Answer: True


Question: Let $f(x) > 0$ for all real positive $x$. If $\lim_{x \rightarrow \infty} \frac{f(x)}{x} = c > 0$, does this imply that $f(x) \gg x$? In other words, is there a constant $c'$ such that $f(x) \geq c'x$ for all real positive $x$?

Answer: No


Question: Find the equation of the tangent line to the curve \( x^4y^2 - 2\frac{y}{x} = \ln(y-1) \) at the point \( P = (-1, 0) \).

Answer: y = 0


Question: Does there exist a non-unital $C^*$ algebra $A$ such that the center of the quotient $A/I$, denoted as $\mathcal{Z}(A/I)$, is non-zero?

Answer: Yes


Question: Suppose that for every $\alpha \in \mathbb{R}$, there is a subset $A_\alpha \subseteq \mathbb{R}$ with $|A_\alpha| = \mathfrak{c}$. Does there always exist a set $D \subseteq \mathbb{R}$ such that $D \cap A_\alpha \neq \varnothing$ and $A_\alpha \nsubseteq D$ for each $\alpha \in \mathbb{R}$? Equivalently, do there always exist two choice functions $f, g: \mathbb{R} \to \mathbb{R}$ such that $f(\alpha), g(\alpha) \in A_\alpha$ for all $\alpha \in \mathbb{R}$, while $\{f(\alpha)\}_{\alpha \in \mathbb{R}}$ and $\{g(\alpha)\}_{\alpha \in \mathbb{R}}$ are disjoint?

Answer: Yes


Question: Consider the bicomplex $(A_1^\bullet, \partial_1)$ and $(A_2^\bullet, \partial_2)$ of $\mathbb{C}$-vector spaces, each equipped with an action of a discrete group $G$ that preserves the grading and commutes with the differentials. Assume $A_1$ is a free $\mathbb{C}[G]$-module. Define the product complex $(A^\bullet, \partial)$ by $A^k = \bigoplus_{p+q=k} A_1^p \otimes_\mathbb{C} A_2^q$ and $\partial = \partial_1 + (-1)^q \partial_2$. Let $F^p = \bigoplus_{p'\geqslant p} A_1^{p'} \otimes_\mathbb{C} A_2^q$ be the classical filtration. Consider the spectral sequence $(E_r)$ associated with $((A^\bullet)^G, \partial, F^\bullet)$. Does $(E_r)$ converge at $r=2$?

Answer: Yes


Question: Determine the number of invertible elements in the quotient ring \( \mathbb{Z}[i]/(220+55i)\mathbb{Z}[i] \).

Answer: 30720


Question: Consider a finitely generated nilpotent group $G$. Is it true that all members of the lower central series $\gamma_i G = [\gamma_{i-1} G, G]$, with $\gamma_0 = G$, are finitely generated as well?

Answer: Yes


Question: A uniform rod of length \( l \) has an initial temperature distribution given by \( u(x, 0) = \sin\left(\frac{\pi x}{l}\right) \) for \( 0 \leq x \leq l \). The temperature \( u(x, t) \) satisfies the diffusion equation \( u_t = ku_{xx} \), with insulated ends \( u_x(0, t) = u_x(l, t) = 0 \). Find the general solution \( u(x, t) \) using separation of variables. What happens to the solution as \( t \to \infty \)?

Answer: \dfrac{2}{\pi}


Question:
Compute the total heat in a fully insulated bar described by the one-dimensional heat conduction equation:
\[ \frac{∂^2u(x,t)}{∂x^2}=D\frac{∂u(x,t)}{∂t} , \quad D>0, \quad 0 < x < L \]
with insulated boundary conditions \( \frac{∂u}{∂x}=0 \) at \( x=0 \) and \( x=L \), and initial temperature distribution \( u(x,0)=f(x)=-x(x-L) \).

Calculate the total heat in the bar for all time \( t \) using the integral:
\[ \int_{0}^{L} u(x,t) \, dx, \quad t \ge 0 \]

Answer: \dfrac{L^3}{6}


Question: Calculate the surface area of the region of intersection of the two cylinders given by the equations \(y^2 + z^2 = 1\) and \(x^2 + y^2 = 1\).

Answer: 16


Question: Consider two randomly oriented and independent vectors in 3D space. The angle between these two vectors is \( \theta \). What is the expected value of \( \cos^2\theta \)?

Answer: \dfrac{1}{3}


Question: Calculate the limit: $$\lim_{x \to 0^+}\frac{\tan x \cdot \sqrt {\tan x}-\sin x \cdot \sqrt{\sin x}}{x^3 \sqrt{x}}$$

Answer: \dfrac{3}{4}


Question: Verify that \( \sqrt{12\sqrt[3]{2} - 15} + \sqrt{12\sqrt[3]{4} - 12} = 3 \).

Answer: 3


Question: Compute the limit: \[ \lim_{n\to\infty} \left(\frac{3n^2-n+1}{2n^2+n+1}\right)^{\large \frac{n^2}{1-n}} \]

Answer: 0


Question: Find the value of the infinite series \( \sum\limits_{n=1}^{\infty} \frac{(7n+32) \cdot 3^n}{n(n+2) \cdot 4^n}. \)

Answer: \dfrac{33}{2}


Question: Determine the support of the Dirichlet function, defined as follows: \[ f(x) = \begin{cases} 0, & \text{if } x \text{ is rational} \\ 1, & \text{if } x \text{ is irrational} \end{cases} \]

Answer: \mathbb{R}


Question:
Find the minimum value of \(a\) such that there exists a differentiable function \(f: \mathbb{R} \to \mathbb{R}\) satisfying the equation:
\[ e^{f'(x)} = a \frac{|f(x)|}{|1+f(x)^2|} \] for every \(x \in \mathbb{R}\).

Answer: 2


Question: Evaluate the integral \( \int_{-1}^{3}\int_{0}^{\sqrt{\pi }}\int_{0}^{\sqrt{\pi-y^2}} \sin(x^2 + y^2 ) \,dxdydz \).

Answer: 2\pi


Question: Does every compact operator in $\mathcal{B}(l_2)$ have a reducing subspace?

Answer: No


Question: Evaluate the surface integral \( \iint_S z^2 \, dS \), where \( S \) is the part of the outer surface of the cylinder \( x^2 + y^2 = 4 \) between the planes \( z = 0 \) and \( z = 3 \).

Answer: 36\pi


Question: Are finite-dimensional normed vector subspaces closed with respect to any norm?

Answer: Yes


Question: Evaluate the integral \( \int_0^\pi \cos(x) \cos(2x) \cos(3x) \cos(4x)\, dx \).

Answer: \dfrac{\pi}{8}


Question: Let $G$ be a group with identity $e$ and distinct nonidentity elements $a$ and $b$ such that $b^3 = e$ and $(ba)^2 = e$. What is the order of the element $aba$ in the group $G$?

Answer: 3


Question: Find the smallest integer $n$ such that the symmetric group $S_n$ contains an element of order 30.

Answer: 10


Question: Evaluate the line integral \( \oint_\mathcal{C} \textbf{F}\:d\textbf{r} \) for the curve \( \mathcal{C} \) given by the parametric equations \( \textbf{r}(t) = (1 + 2\cos(t))\:\hat{i} + (3\sin(t) - 2)\: \hat{j} \), where \( t \in [0, 2\pi] \). The vector field is \( \textbf{F}(x,y) = y^2\:\hat{i} + x\: \hat{j} \).

Answer: 30\pi


Question: Given that \(3^a = 5^b = m\) and \(\frac{1}{a} + \frac{1}{b} = 2\), find the value of \(m\).

Answer: \sqrt{15}


Question: Let \( f \in C^1([0, \infty)) \) be a function such that \( \int_0^\infty t|f'(t)|^2\,dt < \infty \) and \( \lim_{T \to \infty} T^{-1} \int_0^T f(t)\,dt = L \). Determine whether \( f(t) \to L \) as \( t \to \infty \).

Answer: L


Question: Find all natural numbers \( n > 3 \) such that the product of all prime numbers less than \( n \) is equal to \( 2n + 16 \).

Answer: 7


Question: Let $S$ be a subset of $\mathbb{N}$ such that $2^k \in S$ for all $k \in \mathbb{N}$, and if $k \in S$ then $k-1 \in S$ for all $k \ge 2$. Prove by induction that $S = \mathbb{N}$.

Answer: S = \mathbb{N}


Question: You sample a number from the Uniform(0,1) distribution. Your friend repeatedly samples from the same distribution until she obtains a number higher than yours. What is the expected number of samples she will make?

Answer: \infty


Question: Compute the surface integral \( \iint_S z^2 \, dS \) over the octant of a sphere with radius 1, centered at the origin, where \( S: x^2 + y^2 + z^2 = 1 \). Use polar coordinates to evaluate the integral and find the value of the surface integral. The expected result is \( \frac{\pi}{6} \).

Answer: \dfrac{\pi}{6}


Question: Determine the number of 3x3 orthogonal matrices where all elements are integers.

Answer: 48


Question: Determine the number of distinct similarity classes of real matrices of size $9 \times 9$ such that $A^2 = I$. Provide your answer as a single integer.

Answer: 10


Question:
Given the equations:
\[ 11\sin x + 7\sin y = 3(\sin z + 3\sin w) \]
\[ 11\cos x + 7\cos y = 3(\cos z + 3\cos w) \]
find the value of \( \frac{\cos(x-z)}{\cos(w-y)} \).

Answer: \dfrac{21}{11}


Question: Given a list of integers of size \( n \), determine the minimum number of moves required to convert it into a permutation of the numbers from 1 to \( n \). In one move, you may increase or decrease any element of the list by one. For example, if \( n = 3 \) and the list is \((-1, -1, 2)\), the minimum number of moves required is 6 to convert it to the permutation \((1, 3, 2)\).

Answer: 6


Question: Let $R = \mathbb{Z}[\sqrt{-5}]$ and $I = (2, 1+\sqrt{-5})$. Prove that $I \not\cong R$ as $R$-modules.

Answer: I \not\cong R


Question: Calculate the limit: \[ \lim_{n\to \infty} \left[(\ln n)^n - n^{\ln n}\right]. \]

Answer: \infty


Question: Determine the least ordinal \( \beta \) such that it is provable in \( \mathsf{ZFC} \) that \( 2^{\aleph_0} \leq \aleph_{\aleph_\beta} \).

Answer: 1


Question: Evaluate the triple integral \( \int \int \int_{V} z \sin(x^2 + y^2) \, dx \, dy \, dz \), where \( V \) is the region bounded by the surfaces \( z = \cos(x^2 + y^2) \) and \( z = 0 \) with \( z \geq 0 \). Determine the appropriate limits of integration for \( V \).

Answer: \dfrac{\pi}{6}


Question: Suppose \( f \) is a continuous function on \([0,1]\) such that \( \int_0^1 f(x)x^k \, dx = 0 \) for any nonnegative integer \( k \). Prove that \( f \equiv 0 \) on \([0,1]\).

Answer: 0


Question: Does there exist a function on the interval \([0,1]\) that has unbounded variation and is everywhere differentiable?

Answer: Yes


Question: Evaluate the triple integral \( \iiint_{E} (10xz+3) \, dV \), where \( E \) is the portion of the unit ball above the \( xy \)-plane, i.e., \( E = \{(x,y,z) \mid x^2+y^2+z^2\leq 1, \; \; z\geq 0\} \).

Answer: 2\pi


Question:
Let \( \mu \) be a continuous measure on \([0,1]\). Denote by \( \hat\mu(n) = \int_0^1 e^{2\pi inx} d\mu(x) \) the Fourier transform of \( \mu \), and let \( \lfloor x \rfloor \) denote the floor of \( x \in \mathbb{R} \). Determine whether the following statement is true:

\[ \lim_{N \to \infty} \sup_{M \in \mathbb{N}} \frac{1}{N} \sum_{n=M}^{M+N} \left| \hat\mu\left(\left\lfloor n^{3/2} \right\rfloor\right) \right| = 0. \]

Answer: 0


Question: Find the coefficient of \(x^{47}\) in the expansion of the product \((x^{10}+x^{11}+\cdots+x^{25})(x+x^{2}+\cdots+x^{15})(x^{20}+\cdots+x^{45})\).

Answer: 149


Question: Determine the set of values for \( \alpha \) for which the improper integral \( \int_1^{+\infty} \frac{x^{\alpha} \cdot \sin x}{1+x^3} \, dx \) converges.

Answer: (-\infty, 3)


Question: Evaluate the integral \( \int_0^a \cos x \, \mathrm{d}x \) using Riemann sums.

Answer: \sin a


Question: Twelve positive real numbers \(s_1 \leq s_2 \leq \ldots \leq s_{11} \leq s_{12}\) have the property that no three of them can be the side lengths of an acute triangle. Compute the minimum value of \(\frac{s_{12}}{s_1}\).

Answer: 12


Question: Let $(X,d_1)$ and $(Y,d_2)$ be metric spaces with $X \cap Y = \emptyset$. Is there a metric on $X \cup Y$ that induces $d_1$ on $X$ and $d_2$ on $Y$?

Answer: Yes


Question: Let $p$ be an odd prime number and $r$ be a natural number greater than 1. Calculate the remainder when $2^{p^r-p^{r-1}+1}$ is divided by $p^r$. Provide your answer as the remainder.

Answer: 2


Question: Consider the ordinary differential equation \( \dot{x}(t) = v - AA^Tx(t) \), where \( A \) is an \( m \times n \) matrix and \( v \) is an \( m \times 1 \) vector. Determine whether the condition that \( AA^T \) is non-singular guarantees that the equation has a unique globally asymptotically stable equilibrium.

Answer: Yes


Question: Evaluate the integral \( \frac{1}{2\pi i}\int _\gamma \frac{1}{z}dz \) where \( \gamma \) is a closed contour with a winding number of 2 around the origin. What is the value of the integral?

Answer: 2


Question: Evaluate the integral \( \int_{0}^{1}\int_{0}^{1-x}\int_{0}^{2-x} xyz\, dz\, dy\, dx \).

Answer: \dfrac{13}{240}


Question: Calculate the flux through a circle of radius 1 centered at the origin (0,0) in a counterclockwise direction.

Answer: 2\pi


Question: Given that \( \lim_{x \to -1} \left(\frac{f(x)+2}{x+1} - \frac{x}{x^2-1}\right)=4, \) find \( \lim_{x \to -1} f(x) \) using limit laws.

Answer: -\dfrac{3}{2}


Question: Evaluate the series: \[ \sum_{n=2}^\infty \frac{\prod_{k=1}^{n-1} (2k-1) }{2^n n!} = \frac{1}{8} + \frac{1}{16} + \frac{5}{128} + \frac{7}{256} + \ldots \]

Answer: \dfrac{1}{2}


Question: Let $M$ be a Riemannian manifold and $\Omega \subset M$ be a connected open set. For $u \in W^{1, 2}(\Omega)$, is it true that if $\nabla u = 0$ almost everywhere, then $u$ is constant almost everywhere?

Answer: Yes


Question:
Evaluate the integral \( I = \int_1^2 \int_1^2 \int_1^2 \int_1^2 \frac{x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4 \). Choose the correct answer from the following options:

\(\boxed{1} \)

\(\boxed{\frac{1}{2}} \)

\(\boxed{\frac{1}{3}} \)

\(\boxed{\frac{1}{4}} \)

Answer: \frac{1}{2}


Question: Evaluate the integral \( \int_{C(0,1)} z^3\exp\left(\frac{1}{z^2}\right)dz \), where \( C(0,1) \) is the contour of the unit circle centered at the origin.

Answer: \pi i


Question: Evaluate the integral \( \int_{\alpha} \frac{1}{z} \, dz \) where \( \alpha \) is the closed path along the square with vertices at \( 1, i, -1, -i \).

Answer: 2\pi i


Question:
An analytic function \( f(z) \) satisfies \( \Re\{f'(z)\} = 2y \) and \( f(1+i) = 2 \). Determine the imaginary part of \( f(z) \). Choose the correct answer from the following options:

- \(-2xy\)
- \(x^2-y^2\)
- \(2xy\)
- \(y^2-x^2\)

Answer: D


Question: Consider the vector spaces in \(\mathbb{R}^2\) with the basis \(B=\{ (2,1), (5,3)\}\) and \(C=\{ (1,1), (2,3)\}\). Let the linear transformation \(f: \mathbb{R}^2 \to \mathbb{R}^2\) be defined by \(f(x,y)=(x+y,-2x+y)\). The matrix representation of \(f\) with respect to the bases \(B\) and \(C\) is \(Cf_B=\begin{pmatrix} \alpha_{11} & \alpha_{12}\\ \alpha_{21} & \alpha_{22} \end{pmatrix}\). Compute the value of \(\alpha_{12} + 2\alpha_{22}\).

Answer: 8


Question: A function $f$ satisfies the equation \( f(x) = x f(x^2 - 3) - x \) for all \( x \). Determine the value of \( f(2) \).

Answer: -\dfrac{4}{3}


Question: Assume that $\lim_{(nt) |z|\to 1}|f(z)|(1-|z|)^p=0$, where $f$ is analytic in the unit disk and $p>0$, and $(nt)|z|\to 1$ nontangentially. Does this imply that $\lim_{|z|\to 1}|f(z)|(1-|z|)^p=0$ uniformly?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{n}{\ln \ln n}\cdot \left(\sqrt[n]{1+\frac{1}{2}+\cdots+\frac{1}{n}}-1\right) \] without using Taylor series.

Answer: 1


Question: Let $G$ be a graph. For every $X \subseteq E(G)$, denote $G[X] = (V(G), X)$. Let $I, J \subseteq E(G)$. Let $\mathcal{P}$ be a partition of $V(G)$ such that each part of $\mathcal{P}$ corresponds to the vertices that are connected in $G[J]$. Suppose $G[I \cup J]$ is a forest and $G[I]$ is also a forest. If $G'[I]$ is obtained from $G[I]$ by identifying vertices of $G[I]$ inside the same part of $\mathcal{P}$ (i.e., replacing each part of $\mathcal{P}$ by one vertex and maintaining the original incidence), is $G'[I]$ still a forest?

Answer: Yes


Question: Evaluate the surface integral \( \oint_S (x \hat{i} + y \hat{j} + z^2 \hat{k}) \cdot \hat{n} \, dS \) where \( S \) is the surface bounded by \( x^2 + y^2 = z^2 \) and the plane \( z = 1 \).

Answer: \dfrac{7\pi}{6}


Question: Determine whether the inequality \( \log_{4}{5} + \log_{5}{6} + \log_{6}{7} + \log_{7}{8} \ge 4.4 \) holds true.

Answer: True


Question: Find a closed form for the greatest common divisor (GCD) of the polynomials $x^3+1$ and $3x^2 + 3x + 1$.

Answer: 1


Question: Calculate the Laplace Transform of the function \(-e^{-at}u(-t)\).

Answer: \dfrac{1}{s + a}


Question: Find the limit of \(\left(\frac{x^{2}+3x}{x^{2}+2x+3}\right)^{2x\cdot\lfloor x \rfloor} \) as \(x\rightarrow\infty\).

Answer: \infty


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}(1+2^n+3^n+...+n^n)^{1/n}\sin(1/n). \]

Answer: 1


Question: Determine the dimension of the vector space formed by the boundaries of subsets of vertices in a connected graph $G = (V, E)$, where the space is a subspace of $\mathbb{F}_2^{|E|}$. The dimension is at most $|V| - 1$. What is the exact dimension of this subspace?

Answer: |V| - 1


Question: Evaluate the limit without using Taylor expansion or L'Hospital's rule: \[ L = \lim_{x \to 0} \frac{e^x - 1 - x - \frac{x^2}{2}}{x^3} \]

Answer: \dfrac{1}{6}


Question: Determine if the function \( f(z) = \frac{z^2-2i}{z^2-2z+2} \) can be defined at \( z_0 = 1 + i \) such that it remains continuous at that point. Specifically, find \( \lim_{z \to 1 + i} f(z) \) if it exists.

Answer: 1 - i


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-1}^{1}g(x)e^{-nx^2}\,dx \] where \( g \) is a continuous bounded function on \([-1,1]\). Show that this limit equals \( g(0) \).

Answer: g(0)


Question: Determine whether any interval of real numbers, such as \([5, 6]\), has the same cardinality as the entire set of real numbers \(\mathbb{R}\). Justify your answer.

Answer: Yes


Question:
Evaluate the following limit, given that \( \rho: \mathbb{R} \to \mathbb{R} \) is a continuous function such that \( \rho(x) \ge 0 \) for all \( x \in \mathbb{R} \), \( \rho(x) = 0 \) if \( |x| \ge 1 \), and \( \int_{-\infty}^{\infty} \rho(t) \, dt = 1 \). Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function.

\[ \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{-\infty}^{\infty} \rho\left(\frac{x}{\epsilon}\right) f(x) \, dx \]

Answer: f(0)


Question: Find all functions \( f(x) \) such that \( f(x) \geq 0 \) for all \( x \) and \( f(x+t) = f(x) + f(t) + 2\sqrt{f(x)f(t)} \) for all real numbers \( x \) and \( t \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{k \rightarrow \infty} \int_{-\infty}^\infty \frac{1}{1 + x^2 + x^4 + \cdots + x^{2k}} \, \text{dx} \]

Answer: \dfrac{4}{3}


Question: Suppose $a_2, a_3, a_4, \cdots, a_7$ are integers such that \[ \frac{5}{7} = \sum_{k=2}^7 \frac{a_k}{k!} \] where $0 \le a_j < j$ for $j=2,3,4,5,6,7$. Find the value of $a_2 + a_3 + \dots + a_7$. \( \text{(Express your answer as a single integer.)} \)

Answer: 9


Question: Let \(\mathbb{F}_9\) be the field with nine elements. Determine the number of triples \((a, b, c)\) of elements in \(\mathbb{F}_9\) such that \(a^2 + b^2 = c^2\).

Answer: 81


Question: In a metric space $(X,d)$, let $U$ be an open subset and $x \in U$. Determine whether the following statement is true: If $d(x, X\setminus U)<\epsilon$, then $d(x, \text{Bd} U)< \epsilon$. Specifically, is it true that $d(x, X\setminus U)=d(x, \text{Bd} U)$?

Answer: False


Question: Let \( f \in \operatorname{Hol}(\mathbb{C} \setminus \{1\}) \) be a holomorphic function such that \( \int_{|z|=4} f(\xi) \, d\xi = 0 \). Prove that for every closed and continuous curve \( \gamma \) in \( \mathbb{C} \setminus \{1\} \), the integral \( \int_\gamma f(z) \, dz = 0 \).

Answer: 0


Question: Determine whether an abelian subgroup of $SO(4)$ can be diagonalized over $\mathbb{C}$. Provide a justification for your answer.

Answer: Yes


Question: Determine all real numbers \( \alpha \) such that for every continuous function \( f: [0,1] \to [0,1] \), there exists a \( c \in [0,1] \) satisfying \( f(c) = \alpha \cdot c \).

Answer: [1, \infty)


Question: Given a measurable function $f$ on a set $E$ of finite measure, does there exist a sequence of simple functions that converge uniformly to $f$? A simple function is defined as a finite linear combination of characteristic functions $\chi_{E_i}$, where $E_i$ is measurable. Answer whether this statement is true or false.

Answer: B


Question: Consider an $m \times n$ matrix $X$ where each entry $x_{ij}$ is an integer chosen uniformly at random from the set $\{1, 2, \ldots, n\}$, independently of other entries. As $n$ approaches infinity, determine the probability that there exists at least one number that appears in every row of $X$. Express your answer as a limiting probability.

Answer: 1


Question:
Let \( f \) be a continuous, strictly decreasing, real-valued function such that \( \int_{0}^{+\infty} f(x) \, dx \) is finite and \( f(0)=1 \). In terms of \( f^{-1} \) (the inverse function of \( f \)), \( \int_{0}^{+x} f(x) \, dx \) is:

(A) less than \( \int_{1}^{+\infty} f^{-1}(y) \, dy \)
(B) greater than \( \int_{0}^{1} f^{-1}(y) \, dy \)
(C) equal to \( \int_{1}^{+\infty} f^{-1}(y) \, dy \)
(D) equal to \( \int_{0}^{1} f^{-1}(y) \, dy \)
(E) equal to \( \int_{0}^{+\infty} f^{-1}(y) \, dy \)

Choose the correct option.

Answer: D


Question: Evaluate the integral \( \iiint_W z\ dxdydz \) where \( W = \{(x,y,z) \in \mathbb{R}^3: x^2+y^2+z^2 \le 4; \frac{1}{\sqrt{3}}x \le y \le \sqrt{3}x\} \). Use spherical coordinates to solve the integral.

Answer: 0


Question: Determine the number of ways to arrange $K$ elements, numbered from $1$ to $K$, in $N$ places such that no two identical numbers are adjacent, except for the number specified as $X$. For example, if $N = 3$, $K = 2$, and $X = 1$, the possible arrangements are $\{1,1,1\}$, $\{1,1,2\}$, $\{1,2,1\}$, $\{2,1,1\}$, and $\{2,1,2\}$. Find a general formula for this arrangement for any given $N$, $K$, and $X$. Provide the number of possible arrangements for $N = 5$, $K = 4$, and $X = 3$.

Answer: 469


Question: Suppose an integral domain has a factorization basis. Is it necessarily a unique factorization domain (UFD)?

Answer: Yes


Question: If a Jordan curve separates the plane into two components, is the bounded component of its complement simply connected?

Answer: Yes


Question: Determine whether the following statement is true or false: If a function $f(x)$ is continuous and differentiable on an interval $[a, b]$, then $f'(x)$ is continuous on $[a, b]$. Provide a justification for your answer.

Answer: B


Question: Evaluate the contour integral of the function \( f(z) = e^{-\frac{1}{(z-1)^2}} \) over the circle of radius 3 centered at \( z=1 \).

Answer: 0


Question: Evaluate the contour integral \( \int_{|z|=1}\frac{e^z-1}{z^2}dz \).

Answer: 2\pi i


Question: Determine the minimal integer \( n \) such that any finite abstract graph \((V,E,\varphi)\) can be realized as a topological graph in \( \mathbb{R}^n \).

Answer: 3


Question: What is the expected number of tosses needed to get 100 consecutive heads with a fair coin?

Answer: 2^{101} - 2


Question: Given the sequence defined by $a_1 = 7$, $a_2 = 7^7$, $a_3 = 7^{7^7}$, and so on, find the remainder when $a_{1000}$ is divided by 1000.

Answer: 343


Question: Let $D \subset \mathbb{C}$ be the open unit disk centered at the origin, and let $I \subset [-\pi,\pi]$ be an open interval. Suppose $f$ is a holomorphic and bounded function on $D$ satisfying $\lim_{r\to1^-}f(re^{i\theta}) = 0$ for almost every $\theta \in I$. Prove that $f = 0$ on $D$.

Answer: 0


Question: Solve the inequality: \((x^2+4x+3)\sqrt{x+2}+(x^2+9x+18)\sqrt{x+7} \geq x^3+10x^2+33x+36\) for \(x\).

Answer: [-2, 2]


Question: Minimize \(xy\) given the constraints \(x^2 + y^2 + z^2 = 7\) and \(xy + xz + yz = 4\). Find the smallest possible value of \(xy\).

Answer: \dfrac{1}{4}


Question: Determine the maximum value of \( \frac{1}{2\pi}\int_{0}^{2\pi}\left|\sum_{n = 1}^{\infty}x_{n}e^{in\theta}\right|\, d\theta \) for \( x \in \ell^{2} \) such that \( \sum_{n = 1}^{\infty}n|x_{n}|^{2} \leq 1 \).

Answer: 1


Question: Does there exist a graded algebra $A = \oplus_{i \in \mathbb{Z}} A_i$ such that $A_i.A_{-i} = A_0$ for all $i \in \mathbb{Z}$, but the algebra is not strongly graded?

Answer: Yes


Question:
Determine the value(s) of \( \varrho \) for which the following equality holds, given that \((X,Y)\) are normally distributed with \(\mathrm{Cov}(X,Y)=\varrho\) and \(\mathrm{Var}(X)=\mathrm{Var}(Y)=1\):
\[ \min (E(X),E(Y)) = E(\min(X,Y)) \]

Answer: 1


Question:
Determine the limit of the following expression as \( n \to \infty \):
\[ \lim_{n \to \infty} \left( p_1 \times p_2 \times p_3 \times \cdots \times p_n \right)^{\frac{1}{p_n}} \]
where \( p_i \) is the \( i \)-th prime number. What does this limit approach?

Answer: e


Question: Let $u$ be a subharmonic, real-valued, and continuous function in the open disk $D \subset \mathbb{C}$, which is non-constant on any open set in $D$. Determine whether the set $\{z \in D : u(z) = 0\}$ has measure zero.

Answer: Yes


Question: Find the product of all possible values of $a$ such that the system of equations \(|2x| - y = 5\) and \(x - |2y + 2| = a\) has an odd number of distinct solutions.

Answer: 32


Question: Determine all functions \( f(x) \) that satisfy the equation \( f(x) = \int_{-1}^{1} f(t) \text{d}t \).

Answer: 0


Question: Given the elliptic curve $y^2 = x^3 + ax + a^2$ defined over $\mathbb{Z}$, find a point of infinite order on $\mathcal{E}(\mathbb{Q})$. Use the Nagell-Lutz theorem to assist in your solution.

Answer: (0, a)


Question: Calculate the residue of the function \( \frac{e^z - 1}{z^2 \sin(z)} \) at \( z_0 = 0 \).

Answer: \dfrac{1}{2}


Question: A character in a game has 195 hit points. You can shoot at them with a gun that has the following probabilities for each shot: a 40% chance to deal 100 damage (critical hit), a 40% chance to deal 50 damage (body hit), and a 20% chance to miss (0 damage). The gun fires 1 bullet per second. Calculate the average time, in seconds, required to eliminate the character using this gun.

Answer: \dfrac{115}{32}


Question: In triangle $ABC$, cevians $AD$, $BE$, and $CF$ are drawn such that $BD:DC = CE:EA = AF:FB = 3:1$. Given that the area of triangle $ABC$ is $100$ square units, find the area of triangle $HIG$, where $H$, $I$, and $G$ are the intersection points of the cevians $AD$, $BE$, and $CF$ respectively.

Answer: \dfrac{400}{13}


Question:
Given the equations of two lines:
\[ n(m-1)x + m(n-1)y = amn \]
and
\[ mx + ny = amn \]
where \( m \) and \( n \) are real constants satisfying \( 0 < m < n < 1 \), determine the condition under which these two lines intersect. Show that this condition implies \( x + y = 0 \).

Answer: x + y = 0


Question: Evaluate the limit \( \lim_{n\to\infty} nt_n \), where \( t_n=\sum_{i=1}^{2n}\frac{(-1)^{i+1}}{2n+i} \). Relate the given limit to suitable Riemann sums for the function \( f(x)=(1+x)^{-2} \).

Answer: \dfrac{1}{8}


Question: Let $(X_n)_n$ be a sequence of random variables in $L^2$. Suppose there exists a constant $M$ such that for all $n$, $$P(X^2_n \ge M) < P(X^2_n \ge E(X^2_n)).$$ Determine whether it follows that $E(X_n^2) \le M$ for all $n$.

Answer: E(X_n^2) \le M


Question: Identify the quotient group $G/H$ where $G = GL_n(\mathbb{R})$ and $H$ is the subgroup of $G$ consisting of all matrices with positive determinant.

Answer: \mathbb{Z}/2\mathbb{Z}


Question: Determine the limit of \( \frac{\hat\sigma_n}{\log(\log(n))} \) as \( n \to \infty \), where \( \hat\sigma_n = \sqrt{\frac{1}{n}\sum_{i=1}^n(X_i - \bar X)^2} \) and \( X_i \) is a sequence of iid random variables with finite variance.

Answer: 0


Question: Evaluate the infinite sum \( \sum_{k=1}^{\infty} \frac{1}{1+a_{k}} \), where the sequence \( \{a_{n}\} \) is defined by \( a_{1}=3 \) and \( a_{n+1}=\frac{1}{2}(a^2_{n}+1) \) for all \( n \ge 1 \).

Answer: \dfrac{1}{2}


Question: Find the limit of the integral \( \int_1^{n+1} \frac{\ln x}{x^n} \, dx \) as \( n \to \infty \).

Answer: 0


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

Answer: 0


Question: Let $(\Omega, \mathcal{A}, P)$ be a probability space, and let $X$ be a random variable such that $P(X > t) = t^{-3}$ for $t > 1$ and $P(X > t) = 1$ for $t \leq 1$. Compute the expectation $\mathbb{E}(X)$. Provide the correct expression for the cumulative distribution function $F_X(t)$ and the integral that evaluates to it.

Answer: \dfrac{3}{2}


Question:
Let \( \{U_n\} \) be a sequence of i.i.d Uniform\((0,1)\) random variables and \( \{a_k\} \) be a sequence of i.i.d random variables such that \( P(a_k = \pm1) = \frac{1}{2} \) and are independent of the \( \{U_n\} \). Define the quantities:
\[ X_n = \sum\limits_{k=1}^nU_ka_k, \quad Y_n = \sum\limits_{k=1}^n(U_k^2 - 1/2)a_k \]
Find the limit:
\[ \lim\limits_{n\rightarrow\infty}P\left(|X_n| > Y_n\sqrt{\frac{20}{7}}\right) \]

Answer: \dfrac{3}{4}


Question: Let \( \Omega \) be a connected and simply-connected open subset of \( \mathbf{C} \) that does not contain the origin. Consider a simple smooth curve \( \gamma :[0,1] \to \Omega \) with \( \zeta := \gamma(0) \). Let \( \phi:[0,1] \to \mathbf{C} \) be a smooth map such that \( e^{\phi(t)} = \gamma(t) \) for all \( t \in [0,1] \). Given that there exists a unique single-valued branch \( f \) of the logarithm defined on \( \Omega \) with \( f(\zeta) = \phi(0) \), does it necessarily follow that \( \phi \) is the restriction of \( f \) to the curve \( \gamma \)?

Answer: Yes


Question: What is the smallest infinite ordinal that is not order isomorphic to a reordering of the natural numbers?

Answer: \omega_1


Question: Find the greatest common divisor of the numbers \( a = 2^{2^{35}} + 1 \) and \( b = 2^{2^{21}} + 1 \).

Answer: 1


Question: Given the differential equation \( \frac{dP}{dt} = cP \ln\left(\frac{K}{P}\right) \), where \( c \) is a parameter and \( P(t) \leq K \) for all \( t \geq 0 \), determine \( \lim\limits_{t \to \infty} P(t) \).

Answer: K


Question: Let $f'(0)=f''(0)=1$, and $f^{(12)}(x)$ is differentiable. Define $g(x) = f(x^{10})$. Find the value of $g^{(11)}(0)$.

Answer: 0


Question: Evaluate the limit \( \lim_{n\rightarrow \infty} \int_0^1 \frac{n^{(3/2)}x}{1+n^2x^2} \,\mathrm{d}x \).

Answer: 0


Question: Evaluate the integral: $$\int_0^{2\pi}\arctan \biggl( \frac{\sin \theta} {\cos \theta +3}\biggr)d\theta$$

Answer: 0


Question: Suppose $K$ is a linear operator in a separable Hilbert space $H$ such that for any Hilbert basis $\{e_i\}$ of $H$, we have $\lim_{i,j \to \infty} (Ke_i,e_j) = 0$. Is it true that $K$ is compact?

Answer: Yes


Question: Determine the weight of the Sorgenfrey line $S$. Recall that the weight of a topological space $X$ is defined as $\text{Weight}(X) = \min\{|\mathcal{B}| : \mathcal{B} \text{ is a base for } X\}$. Provide your answer as a cardinal number.

Answer: \mathfrak{c}


Question: Calculate the radius of convergence of the power series \( \sum_{n>0} \frac{z^{n!}}{n} \).

Answer: 1


Question: A Lie group $G$ acts smoothly, properly, and freely on a smooth manifold $M$. Define $S = \bigcup_{p \in M} T_p(G \cdot p) \subset TM$. Is $S$ a sub vector bundle of the tangent bundle $TM \to M$?

Answer: Yes


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

Answer: 4\pi i


Question: Evaluate the surface integral \( \iint_S (4x \hat{i} - 2y^2 \hat{j} + z^2 \hat{k}) \cdot \hat{n} \, ds \) over the curved surface of the cylinder defined by \( x^2 + y^2 = 4 \) and \( z = 0 \) to \( z = 3 \).

Answer: 48\pi


Question: Determine the dimension of the vector space of functions \( f: \mathbb{R} \to \mathbb{R} \).

Answer: 2^{\mathfrak{c}}


Question: Determine the number of subgraphs of the bipartite graph $K_{7,9}$ that contain exactly 16 nodes and 16 edges.

Answer: \dbinom{63}{16}


Question: Find the limit of the sequence \( B_n = \left( \frac{n^2 -1}{n^2 +2} \right)^{2n^2-3} \) as \( n \to \infty \).

Answer: e^{-6}


Question:
Find the conditions on the function \( f(t) \) such that the limit \( \lim_{t \to \infty} h(t) \) exists and is nonzero, where:
\[
 h(t) = \frac{1}{f(t)} \int_0^t f(s) \, ds
\]
Assume the following conditions hold:
1. \( f(t) \to \infty \) as \( t \to \infty \).
2. \( \frac{f(t)}{f'(t)} \to \infty \) as \( t \to \infty \).
3. \( f(t) \) is continuous and differentiable.

Determine the value of \( \lim_{t \to \infty} h(t) \) under these conditions.

Answer: \infty


Question: Find the largest integer $r$ and the smallest integer $q$ such that $\exp(r) < \int_{1}^{2} \exp(x^3 + x) \, dx < \exp(q)$. Provide your answer as a pair $(r, q)$.

Answer: (7, 8)


Question: Determine the group to which the smallest group containing the functions \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{x-1}{x} \), under composition, is isomorphic. The functions are defined on the set \( A = \mathbb{R} \setminus \{0,1\} \).

Answer: S_3


Question: Find the mean of the probability distribution given by the probability density function (PDF): \[ f(x) = \frac{e^{-x}}{(1+e^{-x})^2} \] using the integral \[ \int_{-\infty}^{\infty} x f(x) \, \mathrm{d}x. \] Determine if the mean exists or if the integral diverges.

Answer: 0


Question: Evaluate the limit using Riemann sums and the Fundamental Theorem of Calculus: \[ \lim_{n \rightarrow \infty}\sum_{k=1}^n \frac{n}{n^2+k^2} \]

Answer: \dfrac{\pi}{4}


Question: Find the point $(r, \theta)$ on the curve defined by $\theta - \ln r = c$, where $-\pi < \theta \leq \pi$ and $c$ is a fixed real constant, that minimizes the function $1/r$. Alternatively, determine the minimum value of $1/|z|$ for $z \in \mathbb{C}$ on the curve $\operatorname{Arg} z - \ln |z| = c$.

Answer: e^{c - \pi}


Question: Evaluate the double integral \( F_{XY} = \int_0^\infty \int_0^\infty xye^{-\frac{x^2 + y^2}{2}} \, dx \, dy \).

Answer: 1


Question: Suppose $f(x)$ is continuous in a neighborhood of $a$, and $f'(a)$ exists. Determine whether the limit \( \lim_{x \rightarrow a} \frac{f(x) - f(2a -x)}{2(x-a)} \) always exists.

Answer: f'(a)


Question: Find the conditional expectation $E[X|X>Y]$ where $X$ and $Y$ are independent standard normal variables.

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


Question: Determine the value(s) of \(\alpha\) in the matrix \(A\) such that the geometric multiplicity of the eigenvalue \(\lambda = 4\) is 2, where \(A\) is a \(4 \times 4\) matrix.

Answer: 0


Question: Let $k$ be a non-algebraically closed field and $I \subset k[x_1, \dots, x_n]$ be a maximal ideal. Is the set $V_{\bar{k}}(I) = \{x \in \bar{k}^n \mid \forall f \in I, f(x) = 0\}$ necessarily finite?

Answer: Yes


Question: Given the equations \( x^2 - y^2 - u^3 + v^2 + 4 = 0 \) and \( 2xy + y^2 - 2u^2 + 3v^4 + 8 = 0 \), find \( \frac{\partial u}{\partial x} \) at \((x,y) = (2,-1)\). Assume that these equations determine functions \( u(x,y) \) and \( v(x,y) \) near the point \((x,y,u,v) = (2, -1, 2, 1)\).

Answer: \dfrac{13}{32}


Question: Let $G$ be the symmetric group $S_5$ of permutations of five symbols. Consider the set $\mathscr{S}$ of subgroups of $G$ that are isomorphic to the non-cyclic group of order $4$. Two subgroups $H$ and $K$ in $\mathscr{S}$ are called equivalent if they are conjugate in $G$. How many equivalence classes are there in $\mathscr{S}$?

Answer: 2


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{2^{\log_3 n}}{3^{\log_2 n}} \]

Answer: 0


Question: Determine the number of integer solutions to the equation \((4a^2 - 4a - 4)^2 + b = 2ab\).

Answer: 6


Question: Given the equations \(x^3 - 3xy^2 = 3\) and \(y^3 - 3x^2y = 3\sqrt{2}\), where \(x\) and \(y\) are real numbers, find the value of \(x^2 + y^2\).

Answer: 3


Question: Determine whether the integral \( \int_0^\infty \frac{\sin ax}{x} \, dx = \frac{\pi}{2} \) holds true when \( a = 0 \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{1}{n}\sqrt[n]{(n+1)(n+2)\ldots (n+n)}. \]

Answer: \dfrac{4}{e}


Question: Let \( A \) and \( B \) be similar matrices. Is it necessarily true that for every pair of invertible \( n \times n \) matrices \( P \) and \( Q \), the matrices \( P^{-1}AP \) and \( Q^{-1}BQ \) are similar?

Answer: Yes


Question: Find the sum of the digits of \( l \), where \( l \) is the number of integers \( n \) such that the expression \( k^2 + \left\lfloor \frac{n}{k^2} \right\rfloor \) has a minimum value of 1991, with \( k \) being any positive integer.

Answer: 7


Question: If $A$ is a matrix with entries in the quaternions and $q$ is a quaternion, determine whether $(qA)^T = q A^T$ or $(qA)^T = \overline{q}A^T$.

Answer: (qA)^T = q A^T


Question: Given that the equation \( dx^3 + 7x^2y + bxy^2 + y^3 = 0 \) represents three straight lines passing through the origin, and two of these lines make complementary angles with the x-axis, find the value of \( |d^2 - bd| \).

Answer: 6


Question: Let $G=(V,E)$ be a simple, undirected graph with $|V|$ and $|E|$ both infinite. Determine whether there exists a subset $B \subseteq E$ such that $|B| = |E|$ and the graph $(V,B)$ is bipartite.

Answer: Yes


Question: Determine the integer value of \( \alpha = \angle BAC \) for which it is possible to find a right-angled triangle ABC with at least two integer-valued sides.

Answer: 45


Question: Determine which is larger: $e^{3\pi}$ or $3^{e\pi}$.

Answer: e^{3\pi}


Question: Determine the minimum value of the parameter $a$ such that the quadratic form \( Q(x) = ax_1^2 + 5x_2^2 + x_3^2 + 7x_1x_2 + 4x_2x_3 \) is positive semi-definite.

Answer: \dfrac{49}{4}


Question: Let \((R, \mathfrak{m})\) be a valuation ring of finite Krull dimension \(d\), where \(\mathfrak{m}\) is a non-principal maximal ideal such that \(\mathfrak{m}^2 = \mathfrak{m}\). Determine if there exists a principal ideal \(J = (r)\) in \(R\) such that the height of \(J\) is \(d\). Specifically, can you find \(r \in R\) such that \(\mathfrak{m}\) is the only prime ideal containing \(r\)?

Answer: Yes


Question:
Let $f$ be a continuous function with bounded variation on $[0,1]$. Determine whether the following equation is true:
\[
\int_{[0,1]}f'(x)dx=f(1)-f(0).
\]

Answer: False


Question: Determine the value of $k$ that maximizes the expression $k \cdot \binom{99}{k}$.

Answer: 50


Question: Let \( f(x) = \sin x \), and denote by \( f^n = f \circ f \circ \ldots \circ f \) the \( n \)-th iteration of the function \( f \). Find the limit (if it exists): \[ \lim_{n \to \infty} n \cdot f^n(n^{-1}) \]

Answer: 1


Question:
Find a continuous, twice differentiable function \( f(x) \) defined on \((\epsilon, \infty)\) for some \( \epsilon > 0 \) that satisfies the following conditions:

1. \( f(x) < x \) for all \( x > \epsilon \).
2. \( f''(x) < 0 \) for all \( x > \epsilon \).
3. \( \lim_{x \to \infty} (x - f(x)) = 0 \).

Provide an example of such a function.

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


Question: Determine the value of the constant \( \alpha \) such that the line integral \( \int_C [x \ln{(x^2 + 1)} + (x^2 + 1)y]dx + \alpha[\frac{x^3}{3} + x + \sin{y}]dy \) is independent of the path \( C \).

Answer: 1


Question: Find the mass of the solid inside the paraboloid \( z = x^2 + y^2 \) and beneath the plane \( z = 1 \), given that the density of the solid is 1.

Answer: \dfrac{\pi}{2}


Question: Find the asymptotic upper bound for the recurrence relation: \( T(n) = (T(n-1))^2 \) with the initial condition \( T(n) = n \) for \( n \leq 2 \).

Answer: O(2^{2^n})


Question: Suppose that $F(z)$ is an analytic function with $f(z)$ as its derivative. Consider the complex line integral \( \int_\gamma f(z)\, dz \), where \( \gamma \) is a closed curve. Does this integral equal zero? Justify your answer.

Answer: 0


Question: Determine whether the functions \( e_n(x) = \frac{1}{\sqrt{2N}}\exp\left( \frac{i \pi n}{N}x \right) \), where \( n \in \mathbb{Z} \), form an orthonormal basis for the space \( L^2(-N,N) \). If they do, provide a proof.

Answer: Yes


Question: Determine whether the following statement is true or false: If $x$ is a self-adjoint compact operator acting in a Hilbert space $H$, and $y$ is a positive compact operator acting in the same Hilbert space $H$, with $-y \leq x \leq y$, then does it necessarily follow that $|x| \leq |y|$? Here, $|x| = \sqrt{x^*x}$. Provide a justification for your answer.

Answer: True


Question: Determine whether the sum of two uniform random variables is uniformly distributed. Specifically, if \(X_1, X_2 \sim U[0,t]\), does it imply that \(2X_1 + X_2 \sim U[0,3t]\)? Provide a justification for your answer.

Answer: No


Question: Given a linear transformation $L: \mathbb{R}^4 \rightarrow \mathbb{R}^4$ with characteristic polynomial $t^4+1$, determine the total number of invariant subspaces of $L$. Consider the factorization of the polynomial and the structure of the module over PIDs to identify all possible invariant subspaces.

Answer: 4


Question: Calculate the expected value of the product of the number of balls in each urn when K balls are uniformly distributed into S urns. For example, if K = 8 and S = 3, one possible distribution is (4, 2, 2), resulting in a product of 16. Determine the expected value for K = 8 and S = 3.

Answer: \dfrac{112}{9}


Question: Given nonnegative real numbers $a, b, c, d, e, f$ such that $a^2 + b^2 + c^2 + d^2 + e^2 + f^2 = 6$ and $ab + cd + ef = 3$, find the maximum value of $a + b + c + d + e + f$. Provide your answer as a single number.

Answer: 6


Question: Are the fibers of a surjective polynomial submersion \(\mathbb{C}^n \to \mathbb{C}\) all homeomorphic?

Answer: Yes


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

Answer: 2


Question: Determine the value of \( m \) such that the polynomial \( p(x) = (x + 1)^m - x^m - 1 \) is divisible by \( q(x) = (x^2 + x + 1)^2 \) without a remainder.

Answer: 7


Question: Find the continuous least squares trigonometric polynomial \( S_{2}(x) \) for \( f(x) = 2x - 1 \) on the interval \([-\pi, \pi]\). Express \( S_{2}(x) \) in the form \( \frac{a_{0}}{2} + a_{1}\cos(x) + a_{2}\cos(2x) + b_{1}\sin(x) \).

Answer: -1 + 4\sin(x)


Question: Let $f:X\rightarrow Y$ be a morphism of schemes over $\mathbb{C}$. Assume that $Y$ and the general fiber $F_y = f^{-1}(y)$ of $f$ are irreducible. Does there exist an irreducible component $X'$ of $X$ such that the restricted morphism $f' := f_{|X'}:X'\rightarrow Y$ satisfies $(f')^{-1}(y) = f^{-1}(y)$ for a general point $y \in Y$?

Answer: Yes


Question: Let \( X_1 \) and \( X_2 \) be independent and identically distributed random variables from the exponential distribution with parameter \( \lambda \). Find the probability density function of the random variable \( Y = \frac{X_1}{X_1 + X_2} \).

Answer: 1


Question: Determine the positive integer $X$ such that the determinant of the matrix \( \begin{vmatrix} b+c & c+a & a+b \\ b_1+c_1 & c_1+a_1 & a_1+b_1 \\ b_2+c_2 & c_2+a_2 & a_2+b_2 \end{vmatrix} \) is equal to $X$ times the determinant of the matrix \( \begin{vmatrix} a & b & c \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} \).

Answer: 2


Question: In a non-commutative ring with unity, if an element $a$ can be expressed as $a = bu$ for some invertible element $u$, must there exist an invertible element $v$ such that $a = vb$? Provide a justification for your answer.

Answer: No


Question: Consider the analogue of the Dirichlet $L$-function for $\mathbb{F}_q[T]$. Let $g = T$ in $\mathbb{F}_q[T]$, and let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a non-trivial homomorphism. Define $$L_c(s, \chi) = \sum_f \chi(f) \cdot c^{\text{deg}(f)} \cdot \#(\mathbb{F}_q[T]/(f))^{-s}$$ where $f$ ranges over all monic polynomials in $\mathbb{F}_q[T]$ that are coprime to $g$, and $\text{deg}(f)$ denotes the degree of $f$. Does $L_c(s, \chi)$ necessarily equal 1?

Answer: 1


Question: Solve the equation \(2^x = -2x + 11\) for \(x\).

Answer: 2.557


Question: Determine whether the function \( f(v) = v^T Q v \) is strongly convex with respect to the norm \( ||u|| = \sqrt{u^T Q u} \), given that \( Q \) is a positive definite matrix.

Answer: Yes


Question: Find the coefficient of \(s^{14}\) in the expansion of \((s+s^2+s^3+s^4+s^5+s^6)^7\).

Answer: 1667


Question: Determine the distribution function of an exponential random variable with mean 2, given that its outcome is larger than 2. Also, calculate the expected value of this conditional distribution.

Answer: 4


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sqrt[n^2]{2^n + 4^{n^2}} \]

Answer: 4


Question: Determine the cardinality of the set $A = \{f: \mathbb{N} \to \{0,1\} : |f^{-1}\{0\}|<\infty\}$. Express your answer in terms of cardinal numbers.

Answer: \aleph_0


Question: Evaluate the limit of the integral \( \int_x^{3x}\frac{1}{\arctan t}dt \) as \( x \rightarrow +\infty \).

Answer: \infty


Question: A spinner is divided into 4 sectors with areas corresponding to probabilities of 10%, 20%, 30%, and 40%. Calculate the expected number of spins required for the spinner to land on each sector at least once.

Answer: \dfrac{445}{36}


Question: Determine the number of solutions to the equation \( \int\limits_x^{x+\frac{1}{2}} \cos \left( \frac{t^2}{3} \right) dt = 0 \) on the interval \([0, 3]\).

Answer: 1


Question: Find all positive integers \(n\) such that \(n\) divides \(a^{25} - a\) for all positive integers \(a\).

Answer: 2730


Question: Find the second last digit of the number \(7^{41^{17}+1}\).

Answer: 4


Question: Evaluate the infinite series \( \sum_{n=1}^{\infty} \frac{2^{f(n)} + 2^{-f(n)}}{2^n} \), where \( f(n) = \left[ \sqrt{n} + \frac{1}{2} \right] \) and \([\cdot]\) denotes the floor function.

Answer: 3


Question: Given the equation \(x^2 = (2^{2^y} + 1)^s + 2^t\) where \(x, y, s, t \in \mathbb{N}\), determine if \(t\) can be odd when \(s\) is odd.

Answer: Yes


Question: Consider the function $k(x)=\frac{x_1}{|x|^3}$ for $x=(x_1,x_2)$ with $|x|>0$. Let $x,y\in\mathbb{R^2}$, $\xi=|x-y|$, and $\tilde{x}=\frac{x+y}{2}$. Determine if the inequality $|k(x-t)-k(y-t)|\leqslant C\frac{|x-y|}{|\tilde{x}-t|^3}$ holds when $|\tilde{x}-t|\geqslant\xi$.

Answer: Yes


Question: Find the spectral radius of the matrix \( A = \begin{pmatrix} 3 & 1 & 1 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{pmatrix} \) without computing its characteristic polynomial. The spectral radius \( \rho(A) \) is defined as \( \max \{ |\lambda| : \lambda \text{ is an eigenvalue of } A \} \).

Answer: 4


Question: Let $E$ be a complex Hilbert space, and let $B \in \mathcal{L}(E)^+$ such that $\langle Bx, x \rangle \geq 0$ for all $x \in E$. If $B$ is invertible, is $B^{1/2}$ also invertible?

Answer: Yes


Question: Determine whether the statement $E \circ E \subseteq E$ holds for every entourage $E$ in every uniform space. Answer with 'True' or 'False'.

Answer: False


Question: A point $P$ inside the tetrahedron $OABC$ is equidistant from its four plane faces. The position vectors of the vertices of the tetrahedron are $O(0,0,0)$, $A(0,0,2)$, $B(0,4,0)$, and $C(6,0,0)$. Find the distance $r$ from point $P$ to each of the faces of the tetrahedron.

Answer: \dfrac{2}{3}


Question: Consider countably many urns \( U_k \), where urn \( U_k \) contains 1 red ball and \( 2^k - 1 \) blue balls. An urn is selected according to the Poisson distribution with parameter 4, i.e., the probability of selecting urn \( U_k \) is \( P_4(\{k\}) \). A ball is then drawn from the selected urn. What is the probability of drawing a red ball?

Answer: e^{-2}


Question:
Let \( \{v_{1},v_{2},v_{3},v_{4}\} \) be a basis of \( \mathbb{R}^{4} \) and \( v = a_{1}v_{1} + a_{2}v_{2} + a_{3}v_{3} + a_{4}v_{4} \) where \( a_{i} \in \mathbb{R}, i=1,2,3,4. \) Determine the condition under which the set \( \{v_{1}-v, v_{2}-v, v_{3}-v, v_{4}-v\} \) is a basis of \( \mathbb{R}^4 \). Choose the correct option:

(a) \( a_{1} = a_{2} = a_{3} = a_{4} \)
(b) \( a_{1}a_{2}a_{3}a_{4} = -1 \)
(c) \( a_{1} + a_{2} + a_{3} + a_{4} \neq 1 \)
(d) \( a_{1} + a_{2} + a_{3} + a_{4} \neq 0 \)

Answer: C


Question: Find the minimum value of the product \((x+y)(y+z)\) given that \(xyz(x+y+z) = 1\) and \(x, y, z > 0\).

Answer: 2


Question: Calculate the flux of the vector field \( \mathbf{F}(x,y,z) = x\,\mathbf{i} + y\,\mathbf{j} + z^4\,\mathbf{k} \) through the surface \( S \) given by the parameterization \( \mathbf{r}(u,v) = 3\,\sin(u)\,\cos(v)\,\mathbf{i} + 3\,\sin(u)\,\sin(v)\,\mathbf{j} + 3\,\cos(u)\,\mathbf{k} \) for \( 0 \le u \le \frac{\pi}{2} \) and \( 0 \le v \le 2\pi \).

Answer: 279\pi


Question: Compute the integral \( I = \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \frac{\pi + 4x^5}{1-\sin(|x|+\frac{\pi}{6})} \, dx \).

Answer: 4\pi


Question: Let $A$ and $B$ be subgroups of a group $G$ such that $|A| = |B| = k$ and $A \cap B = \{id\}$. If $A$ and $B$ commute, what is the order of the subgroup $AB$?

Answer: k^2


Question: Evaluate the surface integral \( \iint_S \vec{F} \cdot \vec{dS} \) where \( \vec{F} = 4x\mathbf{i} - 2y^2\mathbf{j} + z^2\mathbf{k} \) and \( S \) is the surface bounding the region defined by \( x^2 + y^2 = 4 \) with \( z = 0 \) and \( z = 3 \). Do not use the Gauss Divergence Theorem or cylindrical coordinates in your solution.

Answer: 84\pi


Question: Evaluate the limit: $$\lim_{(x,y,z) \to (0,0,0)} \frac{\sin(x^2+y^2+z^2)}{x^2+y^2+z^2+xyz}$$

Answer: 1


Question: Find all integer solutions to the congruence \( f(x) \equiv 0 \pmod{17} \), where \( f(x) = x^4 + 2x^3 + 3x^2 + x + 1 \).

Answer: 8


Question:
Consider the function:
\[ f(N) = 1^1 \cdot 2^2 \cdot 3^3 \cdots (N-1)^{N-1} \cdot N^N. \]
Determine whether the expression \[ \frac{f(N)}{f(r) \cdot f(N-r)} \] is always an integer for all positive integers \(N\) and \(r\).

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to0}(x\tan x)^x$$

Answer: 1


Question:
Determine the number of real solutions \((x, y, z, w)\) to the system of equations:
\[ x + y = zw \]
\[ z + w = xy \]

Answer: \infty


Question: Calculate \( 73^{1567} \mod 990 \).

Answer: 127


Question: Are there any groups with exponential growth that have faithful Schreier graphs exhibiting polynomial growth?

Answer: Yes


Question: Calculate the limit \( \lim_{n \to \infty}{\frac{\sqrt[n]{n!}}{n}} \) using the Stolz-Cesáro theorem.

Answer: \dfrac{1}{e}


Question: Let \( f(x) \) be a fifth-order polynomial with real and distinct roots. Determine the number of roots of the equation \( f''(x)f(x) - (f'(x))^2 = 0 \).

Answer: 0


Question: Given the equations $m^3 = n^4$, $p^5 = q^6$, and $n - p = 61$, find the value of $m - q$.

Answer: 593


Question: Determine the values of \( \alpha \) such that the equation \( n^{\frac{1}{\alpha \log n}} = 1 \) holds as \( n \to \infty \).

Answer: \emptyset


Question: Suppose $S$ is an embedded submanifold of $M$ with $\mathrm{dim}\ S = s < \mathrm{dim}\ M$. Let $U$ be an open subset of $M$, and consider $S' = U \cap S$. If $S' \neq \varnothing$, what is the dimension of $S'$?

Answer: s


Question: Evaluate the integral \( I = \int_C \frac{f(z)}{(z-1)(z-2)} \, dz \), where \( f(z) = \sin\frac{\pi z}{2} + \cos\frac{\pi z}{2} \) and \( C: |z| = 3 \). Choose the correct value of \( I \) from the following options: \( 1. \, 4\pi i, \, 2. \, -4\pi i, \, 3. \, 0, \, 4. \, 2\pi i \).

Answer: 2


Question: Determine the Lebesgue measure of the set \( A \subseteq [0,1] \) consisting of real numbers that do not have the digit zero in their decimal expansion.

Answer: 0


Question: Find the integer value of \( X \) such that \( 22 \mid 11^{2n-1} + X^{3n+1} \).

Answer: 11


Question: Evaluate the integral \( \int_{\partial B(1,2)} \frac{1}{(z-2)^3}dz \), where \( B(1,2) \) is a ball centered at 1 with radius 2 in the complex plane.

Answer: 0


Question: Determine whether the inequality \( \pi(x+200) - \pi(x) \leq 50 \) holds for all natural numbers \( x \), where \( \pi(x) \) is the number of prime numbers less than or equal to \( x \).

Answer: Yes


Question: Evaluate the integral \( \int_{-\pi}^\pi \left( \sum_{n=1}^\infty \frac{\cos(nx)}{2^n} \right)^2 \, dx \).

Answer: \dfrac{\pi}{3}


Question: Evaluate the limit: $$\lim_{x \to \frac{\pi}{3}} \frac{\ln{(2\sqrt{3}\sin{x}-2})}{4\cos^2{x}-1}$$ without using L'Hospital's rule.

Answer: -\dfrac{1}{2}


Question: Let $X$ and $Y$ be jointly distributed continuous random variables, where $Y$ is positive valued and $E[Y^2] = 6$. The conditional distribution of $X$ given $Y = y$ is uniform on the interval $(1-y, 1+y)$. Find $\text{Var}(X)$. \(E[X] = 1\).

Answer: 2


Question:
Evaluate the following limit using the definition of the constant \( e \):
\[ \lim_{n\to\infty} \left(1 - \left(1 - \frac{1}{6^n}\right)^{4 \times 6^{n-1}}\right) \]

Answer: 1 - e^{-2/3}


Question: Evaluate the contour integral \( \int_{|z|=1}\frac{\log z}{z}dz \).

Answer: -2\pi^2


Question: If a sequence \( \{f_n\} \) of nonnegative functions satisfies \( \lim_{n\to\infty}\int_{X}\sqrt{f_n}dm=0 \), does it follow that \( \lim_{n\to\infty}f_n=0 \) almost everywhere on \( X \)?

Answer: No


Question: Find the limit: \(\lim_{|z|\rightarrow \infty}\left(1+\frac{1}{z}\right)^z\).

Answer: e


Question: Find an explicit formula for the Moore-Penrose pseudoinverse of the centering matrix $H_n = I_n - \frac{1}{n}1_n 1_n^{T}$, where $1_n$ is the column vector of all ones.

Answer: H_n


Question: Find the limit of the sequence \(a_n = \int_n^{n+1} f(x) \, dx\), where \(f: [0, +\infty) \to \mathbb{R}\) is a continuous function satisfying \(0 < f(x+1) < f(x)\) for all \(x \geq 0\) and \(\lim_{x \to \infty} f(x) = 0\).

Answer: 0


Question: Given two independent and identically distributed random variables \(X\) and \(Y\) with the probability distribution \(P(n) = 2^{-n}\) for \(n = 1, 2, 3, \ldots\), calculate \(P(X \geq 2Y)\).

Answer: \dfrac{2}{7}


Question: Evaluate the integral \( \int \limits_{-\infty}^{\infty} \frac{\sin^2(x)}{x^2} \operatorname{d}\!x \).

Answer: \pi


Question: Given the equation of the conic $5x^2 + 6xy + 5y^2 = 8$, find the equation of its auxiliary circle.

Answer: x^2 + y^2 = 4


Question: Calculate \( \lim_{n \to \infty} \frac{1^4 + 3^4 + \ldots + (2n-1)^4}{n^5} \).

Answer: \dfrac{16}{5}


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

Answer: \infty


Question: Find the generator for the ideal in \(\mathbb{Z}[i]\) generated by \(3 + 4i\) and \(4 + 7i\).

Answer: 2 + i


Question:
Given the recurrence relation:
\[\begin{align}
T(1) &= 1\\
T(n) &= T(\log n) + \mathcal O(1)
\end{align}\]
Determine the solution to this recurrence relation. Choose the correct option:

(a) \(O(\log n)\)
(b) \(O(\log^* n)\)
(c) \(O(\log^2 n)\)
(d) \(O\left(\frac{n}{\log n}\right)\)

Answer: B


Question: Find the coefficient of \(t^{17}\) in the expansion of the generating function \((t^2 + t^3 + t^4 + t^5)(t^3 + t^4 + t^5 + t^6)(t^4 + t^5 + t^6 + t^7)\). This coefficient represents the number of integer solutions to the equation \(x_1 + x_2 + x_3 = 17\) where \(2 \leq x_1 \leq 5\), \(3 \leq x_2 \leq 6\), and \(4 \leq x_3 \leq 7\).

Answer: 3


Question: Is there an infinite simple, undirected graph $G = (V,E)$ such that there exists an edge $e \in E$ for which $G \cong (V, E \setminus \{e\})$?

Answer: Yes


Question: Find the probability that a random variable \( Z \) is greater than the sum of two other random variables \( X \) and \( Y \), where \( X, Y, Z \sim U(0,1) \).

Answer: \dfrac{1}{6}


Question: Evaluate the sum of the series \( \sum_{k=1}^{+\infty}(-1)^{k-1}\frac{1}{k(4k^2-1)} \).

Answer: 1 - \ln 2


Question:
Consider a matrix \(A\) of size \((n-1) \times n\) with the following form:
\[
\begin{pmatrix}
n-1 & 1 & 0 & \cdots & 0 \\
0 & n-2 & 2 & \cdots & 0 \\
0 & 0 & n-3 & 3 & \cdots \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
\end{pmatrix}
\]
where all entries are non-negative integers. Let \(b\) be an \((n-1) \times 1\) vector where each entry does not exceed \(n^{n^4}\). Determine if there exists a natural number \(n\) such that the equation \(Ax = b\) has a unique solution \(x\) consisting only of non-negative integers.

Answer: Yes


Question:
Consider two real polynomials $f(x)$ and $g(x)$ of degrees $m (\ge 2)$ and $n (\ge 1)$ respectively, satisfying the equation $f(x^2+ 1) = f(x)g(x)$ for every $x \in \mathbb{R}$. Determine which of the following statements is true:

(A) $f$ has exactly one real root $x_0$ such that $f '(x_0) \ne 0$.
(B) $f$ has exactly one real root $x_0$ such that $f '(x_0) = 0$.
(C) $f$ has $m$ distinct real roots.
(D) $f$ has no real root.

Answer: D


Question: Determine the limit of the sequence \( x_n = \frac{\frac{2}{1} + \frac{3^2}{2} + \frac{4^3}{3^2} + \ldots + \frac{(n+1)^n}{n^{n-1}}}{n^2} \).

Answer: \dfrac{e}{2}


Question: Evaluate the integral \( \int_{\gamma}\frac{e^{z^2}-1}{z\sin(z)}dz \) along the circle \( \gamma \) of radius \( r=\frac{1}{4} \) centered at the origin. Determine if the integral equals zero.

Answer: 0


Question:
Consider the set \(S = \mathbb{R} \times \mathbb{C}\). For elements \(a = (x_a, z_a)\) and \(b = (x_b, z_b)\) in \(S\), define the binary operation \(\circ\) as:
\[ a \circ b = (x_a + x_b, x_a + ix_b + z_a + z_b) \]
where \(i\) is the imaginary unit with \(i^2 = -1\). We say \(a \sim b\) if and only if \(a \circ b = b \circ a\). Let \([a]\) denote the equivalence class of \(a\). Determine which of the following is the correct description of \([a]\):

(a) \([a] = \{(x, z_a) \mid \forall x \in \mathbb{R}\}\)
(b) \([a] = \{(x_a, z) \mid \forall z \in \mathbb{C}\}\)
(c) \([a] = \{(x_a + \Re(z_a), z) \mid \forall z \in \mathbb{C}\}\)
(d) \([a] = \{(x_a + \Im(z_a), z) \mid \forall z \in \mathbb{C}\}\)
(e) \([a] = \{(x_a + |z_a|^2, z) \mid \forall z \in \mathbb{C}\}\)

Select the correct option.

Answer: B


Question:
Consider the sequences of functions $f_n:[0,1] \to \mathbb{R}$ and $g_n: \mathbb{R} \to \mathbb{R}$ defined by:

\[ f_n(x) = x^n \]

and

\[ g_n(x) = \begin{cases}
 \cos\left(\frac{\pi}{2}(x-n)\right) & \text{if } x \in [n-1,n+1] \\
0 & \text{otherwise}.
\end{cases} \]

Determine which of the following statements is true:

1) Neither $f_n$ nor $g_n$ is uniformly convergent.
2) $f_n$ is not uniformly convergent but $g_n$ is.
3) $g_n$ is not uniformly convergent but $f_n$ is.
4) Both $f_n$ and $g_n$ are uniformly convergent.

Answer: 1


Question: Identify a polynomial \( f(x) \) such that the quotient ring \( \mathbb{R}[x]/(f(x)) \) is isomorphic to \( \mathbb{C} \).

Answer: x^2 + 1


Question: Calculate \( \lim_{x \rightarrow 0}{\frac{\int_{0}^{x} \frac{\sin(t)}{t}-\tan(x)}{2x(1-\cos(x))}} \) using Maclaurin polynomials.

Answer: -\dfrac{7}{18}


Question: Let $b$ be an odd integer greater than $1$. Determine if it is possible for the multiplicative orders of $2$ modulo two different powers of $b$ to be equal.

Answer: Yes


Question: Determine the value of \( x \) such that the Taylor series expansion \( e^{\left(\frac{d}{dx}\right)} = 1 + \frac{d}{dx} + \frac{1}{2} \left(\frac{d}{dx}\right)^2 + \cdots \) applied to a function \( f(x) \) at \( x = 0 \) results in \( f(x) \).

Answer: 1


Question: Determine the number of 5-element subgroups in the symmetric group $S_7$, the group of permutations on 7 elements.

Answer: 126


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

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


Question: In a chess marathon with 2000 players, each pair of players plays exactly one game against each other. For every pair of players, at least one of them plays no more than 30 games during the marathon. What is the maximum number of games that can be played in this marathon?

Answer: 59100


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int^n_0 t^{x-1}(1-t/n)^n \, dt \]

Answer: \Gamma(x)


Question: Evaluate the line integral \( \int_\gamma F \cdot dr \) using Stokes' Theorem, where \( F = (ye^x, e^x + x^3, z^5) \) and \( \gamma \) is the curve of intersection between the surfaces \( x^2 + y^2 = 1 \) and \( z = 2xy \). The curve \( \gamma \) is oriented such that its orthogonal projection onto the \( xy \)-plane is counterclockwise.

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


Question: Let $X_1, X_2, X_3$ be random variables with a normal distribution, mean vector $\mu = (-1, 1, 0)^T$, and covariance matrix $\Sigma = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 3 & 1 \\ 1 & 1 & 2 \end{pmatrix}$. If $Y = X_1 + 2X_2 - 3X_3$, find the variance of $Y$. \( \text{Express your answer as a single number.} \)

Answer: 13


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

Answer: e


Question: In the space $L^2([0,1])$, if a sequence $(f_n)$ converges weakly to $f$ and additionally satisfies $\|f_n\| \to \|f\|$, does this imply that $(f_n)$ converges strongly to $f$?

Answer: Yes


Question: Calculate the residue of the function \(-\frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s}\) at \(s = 1\).

Answer: x


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{n^{\lg\lg n}}{(3/2)^n} \]

Answer: 0


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

Answer: -\dfrac{1}{3}


Question: Determine the values of \(\alpha\) for which the series \(\sum_{n = 2}^\infty (\sqrt{n + 1} - \sqrt{n})^\alpha \ln \frac{n + 1}{n - 1}\) converges.

Answer: \alpha > 0


Question: Determine the almost sure convergence of the sequence \((Y_n)_{n \ge 1}\), where \(Y_n = \max_{k \le n} X_k\) and \(X_k\) is the result of the \(k\)-th toss of a balanced die.

Answer: 6


Question: Let $n = pq$, where $p$ and $q$ are distinct odd primes, and let $a$ be an integer such that $\gcd(a, n) = 1$. Prove that if the congruence $x^2 \equiv a \pmod{n}$ has any solutions, then it has exactly four solutions.

Answer: 4


Question: Calculate the integral \( \int_{S(a,\frac{|a|}{2})} \frac{1}{|x|} \, dx \), where \( S(x,r) = \{ x \mid x_1^2 + x_2^2 + x_3^2 = r^2 \} \) and \( 0 \neq a \in \mathbb{R}^3 \).

Answer: \pi |a|


Question: Determine whether the expression \( \frac{\pi\displaystyle\prod_{n\geq1}\cos\left(\frac{\pi}{k2^n}\right)}{\sin(\frac{\pi}{k})} \) is always an integer for any integer value of \( k \).

Answer: Yes


Question: Determine whether the infinite power tower \( \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdots}}} = 4 \) is correct. Justify your answer.

Answer: 2


Question:
Do there exist positive definite matrices $A$ and $B$ such that for $0 \prec X \prec Y \prec \lambda X$, where $\lambda = 2^{1-1/p}$ with $p>1$, the following conditions hold?
\[ X = \dfrac{A+B}{2}, \qquad Y = \left(\dfrac{A^p+B^p}{2}\right)^{1/p}. \]

Answer: Yes


Question: Determine the dimension of \(\mathbb{Q}(t)\) as a \(\mathbb{Q}(t^2)\)-vector space.

Answer: 2


Question: Suppose $M$ is a $C^k$-manifold embedded into a Hilbert space $H$, and $g$ is the induced Riemannian metric on $M$. Let $g_H(x,y) = \sqrt{\langle (x-y),(x-y)\rangle_H}$ be the Euclidean distance in $H$. Is it necessarily true that for all $x, y \in M$, $g_H(x,y) \leq g(x,y)$?

Answer: Yes


Question: Determine the values of \( \alpha \) for which the series \( \sum_{n \geqslant 1} n^{\alpha} \cdot x^{n} \cdot (1-x) \) uniformly converges on the interval \([0,1]\).

Answer: \alpha < 0


Question: Let $R$ be a commutative ring with unity, $M$ be an $R$-module, and $I$ be an ideal of $R$ such that $IM$ is not a finitely generated submodule. Let $J \subseteq I$ be a finitely generated ideal such that $JM$ is a finitely generated submodule. Does there exist $a \in I$ such that $JM \subsetneq \langle a,J\rangle M$?

Answer: Yes


Question:
Let $f_1(x), f_2(x), \ldots, f_n(x)$ be polynomials in $\mathbb{R}_{n-2}[X]$, and let $a_1, a_2, \ldots, a_n$ be real numbers. Consider the matrix:
\[ A = \begin{bmatrix} f_1(a_1) & f_1(a_2) & \cdots & f_1(a_n) \\
\vdots & \vdots & & \vdots \\
f_n(a_1) & f_n(a_2) & \cdots & f_n(a_n) \end{bmatrix} \]
What is $\det(A)$?

Answer: 0


Question: Given the function \( F(x) = \int_{\tan x}^{\cot x} \sqrt{1+t^2} \, dt \), find \( F'(\pi/4) \).

Answer: -4\sqrt{2}