Question: Find the natural number \( m \) such that \( \sum_{n=1}^m |a_n-a_{n+1}| = 2009 \), where the sequence of complex numbers \( \{a_n\} \) is defined by \( a_n = (1 + i) \cdot \left (1+ \frac {i} {\sqrt {2}} \right) \cdot \ldots \cdot \left (1+ \frac {i} {\sqrt {n}} \right) \).

Answer: 2009


Question: Let $G_1$ and $G_2$ be groups with $|G_1| = 2p$ and $|G_2| = 2q$, where $p$ and $q$ are distinct primes. Suppose there is a non-trivial homomorphism $\phi: G_1 \to G_2$. Find the size of the kernel of $\phi$, $|\ker(\phi)|$. \(\text{Proof:}\) We have that $\ker(\phi) \lhd G_1$ (by the first isomorphism theorem). By Lagrange's Theorem, $|\ker(\phi)|$ divides $|G_1| = 2p$. Then $|\ker(\phi)| \in \{2p,p,2,1\}$. Consider the case when $|\ker(\phi)| = 2p$, which implies that $\phi(g_1) = e_{G_2}$ for all $g_1 \in G_1$, contradicting the assumption that $\phi$ is non-trivial. Therefore, $|\ker(\phi)| = p$. \(\therefore |\ker(\phi)| = p.\)

Answer: p


Question:
Given the probability density function of a random variable \(X\) as \(f(x) = ae^{-x^2-bx}\) for \(-\infty < x < \infty\), and the condition that \(E(X) = -\frac{1}{2}\), determine the correct values of \(a\) and \(b\) from the following options:

(A) \(a = \frac{1}{\sqrt{\pi}}e^{-1/4}, b = 1\)
(B) \(a = \frac{1}{\sqrt{\pi}}e^{-1/4}, b = -1\)
(C) \(a = \sqrt{\pi}e^{-1/4}, b = 1\)
(D) \(a = \sqrt{\pi}e^{-1/4}, b = -1\)

Select the correct option.

Answer: A


Question: Let $x$ and $y$ be positive integers satisfying $x^{5x} = y^y$. What is the largest possible value of $x$?

Answer: 256


Question: Calculate \( 3^{{2014}^{2014}} \mod 98 \) using Euler's theorem.

Answer: 25


Question: Calculate the value of the series \(\sum\limits_{n=1}^\infty \frac{1}{n^3}\).

Answer: \zeta(3)


Question: Find the area of the part of the plane $5x+2y+z=10$ that lies inside the cylinder $x^2 + y^2 = 25$. Compute the area of this surface.

Answer: 25\pi\sqrt{30}


Question: Evaluate the limit: \[ \lim_{(x,y)\to(0,0)}\frac{(x^5+y^5)\ln(x^2+y^2)}{(x^2+y^2)^2} \]

Answer: 0


Question: Evaluate the double integral \( \iint_L (xy - x^2) \, dx - (x+y)^2 \, dy \) where \( L \) is the triangle with vertices \( O(0,0) \), \( A(3,1) \), and \( B(2,2) \). Solve this without using Green's Theorem.

Answer: -14


Question: Evaluate the integral \( \int_{\gamma} |z-1| |dz| \), where \( \gamma(t) = e^{it} \) for \( t \in [0, 2\pi] \).

Answer: 8


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

Answer: 10


Question: Determine whether \( \arg{(r\text{e}^{i\theta}+z_{0})} \to \theta \) as \( r \to \infty \), where \( r \in \mathbb{R}^{+} \), \( \theta \in \mathbb{R} \), and \( z_{0} \in \mathbb{C} \).

Answer: \theta


Question: Find the limit: \[ \lim_{(x,y)\rightarrow (0,0)} \frac{\sin(xy)}{\sqrt{x^2+y^2}} \]

Answer: 0


Question: Let \(A\) be a skew-symmetric \(n \times n\) matrix. Determine the least possible value of \(\det(I + \lambda A^2)\) for any real value of \(\lambda\).

Answer: 0


Question: Evaluate the integral \( \int_C \frac{\operatorname{tg}(\frac{z}{2})}{z^2-16}dz \), where \( C \) is the boundary of the square with vertices at 1, -1, \( i \), and \( -i \) in the complex plane. Parametrize the curve \( C \) and use the parametrization to compute the integral.

Answer: 0


Question: Determine if there exists a real number \( b \) such that the sum of its powers over a subset of natural numbers is a whole number. Specifically, is there a \( b \in \mathbb{R} \) such that \( \sum_{i \in I \subset \mathbb{N}} b^i \in \mathbb{W} \)?

Answer: Yes


Question: Determine whether the derivative of the cardioid given by \( r = 1 + \cos(\theta) \) exists at the origin \((0,0)\), and if so, find the slope of the tangent line at that point.

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0} (\cos x)^{\left(\frac{-4}{x^2}\right)} \] without using L'Hospital's rule.

Answer: e^2


Question: Evaluate the integral expression: \[ \int_{\frac{1}{e}}^{\tan x}\frac{t}{1+t^2}dt+\int_{\frac{1}{e}}^{\cot x}\frac{1}{t(1+t^2)}dt \] and find its value.

Answer: 1


Question:
Calculate the out-flux of the vector field \( F(x,y,z)=(-\sin(2x)+ye^{3z},-(y+1)^2,2z(y+\cos(2x)+3) \) from the domain \( D = \{(x, y,z) \in \mathbb{R}^3 : x^2+y^2+z^2 \leq 4, x \leq 0, y \leq 0, z \geq 0\} \). Choose the correct answer from the options below:

(A) \( \frac{16\pi}{3} \)

(B) \( -\frac{16\pi}{3} \)

(C) \( -\frac{\pi}{8} \)

(D) \( \frac{\pi}{8} \)

Answer: A


Question:
Let \( f: [0,4] \to [1,3] \) be a differentiable function such that \( f'(x) \neq 1 \) for all \( x \in [0,4] \). Determine the number of fixed points of \( f \). Choose the correct option:

(A) \( f \) has at most one fixed point.
(B) \( f \) has a unique fixed point.
(C) \( f \) has more than one fixed point.

Answer: B


Question: A plane flying with a constant speed of $19 \,\text{km/min}$ passes over a ground radar station at an altitude of $10 \, \text{km}$ and climbs at an angle of $20^\circ$. At what rate is the distance from the plane to the radar station increasing $2$ minutes later?

Answer: 18.5


Question: Calculate the volume of the solid formed by rotating the region bounded by the curves \(y = x\), \(y = 0\), \(x = 4\), and \(x = 7\) about the line \(x = 1\).

Answer: 153\pi


Question: Let $X$ and $Y$ be independent uniform random variables on the interval $[0,1]$. Find the expectation of $X$ given that $X + Y \geq 1.3$, i.e., compute $E[X \mid X + Y \geq 1.3]$. \( \text{Provide your answer as a simplified expression.} \)

Answer: \dfrac{23}{30}


Question: Given the differential equation \( y'(x, y_0) = y^2(x, y_0) + y(x, y_0) \sin(x) \), where \( y_0 \) is the value of \( y \) at \( x = 0 \), find \( \frac{\partial y}{\partial y_0} \bigg|_{y_0=0} \).

Answer: e^{1 - \cos x}


Question: Compute the limit: \[ \lim_{x \to 0} \left(\frac{x^5 e^{-1/x^2} + x/2 - \sin(x/2)}{x^3}\right) \]

Answer: \dfrac{1}{48}


Question: Given vectors $x$ and $y$ such that $x = f(y)$, and a constant matrix $B$, find the expression for $\frac{\partial (x'Bx)}{\partial y}$ using the chain rule. Assume $\frac{\partial x'Bx}{\partial x} = 2Bx$ and use the matrix $J = \frac{\partial x}{\partial y}$. Is the expression $2(Bx)^T J$ correct?

Answer: 2(Bx)^\top J


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^{\infty} \left(\sqrt{4^n + 3^n} + (-1)^n \sqrt{4^n - 3^n}\right)x^n \).

Answer: \dfrac{1}{2}


Question: Find the minimum size of a subgroup \( H \leq S_{10} \) that contains at least one 9-cycle, one 2-cycle, and acts transitively on the set \( \{1,2,...,10\} \).

Answer: 3628800


Question: Let $k$ be a field, and $A$ a $k$-domain such that the fraction field of $A$ has transcendence degree $n$ over $k$. Is it always true that the Krull dimension of $A$ is less than or equal to $n$?

Answer: Yes


Question: Compute the definite integral \( \int_0^{1/2}\sqrt{1+x^3}dx \) using a power series expansion. Approximate the result to an accuracy of \( 10^{-4} \).

Answer: 0.5077


Question: Determine the number of elements of order 5 in the group $U_{1331}$.

Answer: 4


Question:
Let \( f \) be an analytic function on the unit disk \( D = \{ z \in \mathbb{C} : |z| < 1 \} \) such that \( |f(z)| \le 1 \) for all \( z \in D \). Determine which of the following is not a possible value of \( (e^f)''(0) \):

(a) \( 2 \)
(b) \( 6 \)
(c) \( \frac{7}{9} e^{\frac{1}{9}} \)
(d) \( \sqrt{2} + i\sqrt{2} \)

Answer: B


Question: Evaluate the principal value of the integral \( \int_{-\infty}^\infty \frac{\mathrm{e}^{-2\mathrm{i}x}}{x^2+1} \, \mathrm{d}x \) using the method of residues.

Answer: \pi e^{-2}


Question: Let $(A, \,^\ast,\lVert\cdot\rVert)$ be a Banach $\ast$-algebra with the property that $\lVert a \rVert^2=\rho(a^\ast a)$ holds for all $a\in A$, where $\rho(x)$ denotes the spectral radius of an element $x\in A$. Is $(A, \,^\ast,\lVert\cdot\rVert)$ a $C^\ast$-algebra?

Answer: Yes


Question: Let $f: [0, 1]^2 \to \mathbb{R}$ be a function such that for every $x \in [0, 1]$, the function $y \to f(x, y)$ is Lebesgue measurable on $[0, 1]$, and for every $y \in [0, 1]$, the function $x \to f(x, y)$ is continuous on $[0, 1]$. Is $f$ measurable with respect to the completion of the product $\sigma$-algebra $\mathcal{A} \times \mathcal{A}$ on $[0, 1]^2$, where $\mathcal{A}$ is the Lebesgue $\sigma$-algebra on $[0, 1]$?

Answer: Yes


Question: Simplify the expression \( \frac{1}{a(a - b)(a-c)} + \frac{1}{b(b-c)(b - a)} + \frac{1}{c(c - a)(c-b)} \) using interpolation theorems.

Answer: \dfrac{1}{abc}


Question: Determine the domain of validity for the Laurent series of the function \( f(z) = \frac{e^{-z}}{(z-1)^2} \) around the circle \( |z| = 3 \).

Answer: |z| > 1


Question: Determine the number of conjugacy classes of the group \( G = \langle s,t \mid s^8=t^2=1, st=ts^3 \rangle \).

Answer: 7


Question: Evaluate the expression \( x = \frac{n!\binom{m}{n}}{m^n} \) and determine if it is close to 1 or 0, given that \( m = 52! \) and \( n \approx 10^{14} \).

Answer: 1


Question:
Suppose that $Z_1$ and $Z_2$ are independent random variables with the common density function:
\[
f_Z(z) =
\begin{cases}
e^{-z} & \text{if } z > 0 \\
0 & \text{otherwise}.
\end{cases}
\]
Let $X_1 = \min\{Z_1, Z_2\}$ and $X_2 = \max\{Z_1, Z_2\}$. Compute $\mathrm{E}[X_2 - X_1 \mid X_1 = x_1]$. Provide your answer in terms of $x_1$.

Answer: 1


Question: Simplify the expression \((i \theta)^n + (-i \theta)^n\) and use it to show that the series \(\sum_{n=0}^{\infty} \frac{1}{4} \cdot \frac{2^n ((i \theta)^n + (-i \theta)^n)}{n!} + \frac{1}{2}\) represents the Maclaurin series for \(\cos^2(z)\).

Answer: \cos^2(z)


Question:
Determine whether the pullback functor is exact for locally free sheaves. Let \( f: X \longrightarrow Y \) be a morphism of smooth projective schemes. Consider the exact sequence of locally free sheaves on \( Y \):
\[
0 \longrightarrow \mathcal{F}_{n} \longrightarrow \cdots \longrightarrow \mathcal{F}_{0} \longrightarrow \mathcal{F} \longrightarrow 0
\]
Is the following sequence exact?
\[
0 \longrightarrow f^{*}\mathcal{F}_{n} \longrightarrow \cdots \longrightarrow f^{*}\mathcal{F}_{0} \longrightarrow f^{*}\mathcal{F} \longrightarrow 0
\]

Answer: Yes


Question:
Consider the matrix $A = (a_{ij})$ where $a_{ij} = a_i a_j$ for $1 \leq i, j \leq n$ and $a_1, a_2, \ldots, a_n$ are real numbers. Determine which of the following statements is true:

1. It is possible to choose $a_1, \ldots, a_n$ such that $A$ is non-singular.
2. The matrix $A$ is positive definite if $(a_1, \ldots, a_n)$ is a nonzero vector.
3. The matrix $A$ is positive definite for all $(a_1, \ldots, a_n)$.
4. For all $(a_1, \ldots, a_n)$, zero is an eigenvalue of $A$.

Select the correct statement.

Answer: 4


Question: A fair coin is tossed until heads ($H$) appears, with a maximum of $10^6$ tosses. If $H$ appears on the $n$-th toss, you win $2^{n-1}$ dollars. If all tosses result in tails $(T)$, you lose all your money. The entrance fee for this game is $10^5$ dollars. Calculate the expected value of this game and determine if you should play it.

Answer: 400000


Question: Determine the number of solutions to the congruence equation \(x^2 \equiv x \pmod{m}\) for any positive integer \(m\).

Answer: 2^{\omega(m)}


Question: Consider a vector space \( V \) over \( \mathbb{Z}_2 \) with dimension 4. Determine the number of linearly independent subsets of \( V \) that contain exactly 3 elements.

Answer: 420


Question: In the sequence \( \{a_n\} \), each term is defined as \( a_n = \frac{a_1 + a_2 + \ldots + a_{n-1}}{n-1} \) for \( n \geq 3 \). Given that \( a_1 + a_2 \neq 0 \) and the sum of the first \( N \) terms is \( 12(a_1 + a_2) \), find the value of \( N \).

Answer: 24


Question: Determine the radius of convergence of the power series \( \sum_{n\geq 1} n^n z^{n!} \).

Answer: 1


Question: Determine whether the following statement is true or false: "The number of quadratic nonresidues modulo \(p^2\) is greater than the number of quadratic residues modulo \(p^2\), where \(p\) is a prime." Provide a justification for your answer.

Answer: False


Question: Find all entire functions $f$ such that $|f|$ is harmonic.

Answer: f(z) = c


Question: Find the volume of the region bounded by the equation \((x^{2}+y^{2}+z^{2})^{2}=x\).

Answer: \dfrac{\pi}{3}


Question: Find the center of the group with the presentation \( \langle a, b \mid a^2 = b^3 \rangle \).

Answer: \langle a^2 \rangle


Question: Evaluate the contour integral \( \int |z|^2 \, dz \) over the contour defined by \(|z+3| + |z-3| = 10\).

Answer: 0


Question: Let \( f \) be a real continuous function satisfying \( f(0)=0 \) and for each natural number \( n \), \[ n^2\int_{x}^{x+\frac{1}{n}} f(t)\;\text{d}t=nf(x)+0.5. \] Find the value of \( f(1729) \).

Answer: 1729


Question: Let $V = \mathbb{P^4}$ denote the space of quartic polynomials with the $L^2$ inner product \(\langle p,q \rangle = \int^1_{-1} p(x)q(x)dx.\) Let $W = \mathbb{P^2}$ be the subspace of quadratic polynomials. Find a basis for $W^{\perp}$ and determine its dimension.

Answer: 2


Question: Determine whether the polynomial \( x^2 + 2 \) is irreducible over \( \mathbb{Z}_5 \) and can be used to construct a field of 25 elements.

Answer: Yes


Question: Consider vector spaces $V$ and $W$, where $\beta=\{v_1, \ldots , v_n\}$ is a finite basis for $V$ and $\{w_1, \ldots , w_n\}\subset W$. We know there is a unique linear transformation $T:V\rightarrow W$ such that $T(v_i)=w_i$ for $i=1, 2, \ldots , n$. Is this property valid when $V$ is not finite-dimensional?

Answer: Yes


Question: Determine whether the following statement is true or false: If the neighborhood base at each point of a topological space X is finite, then the topology on X is discrete.

Answer: B


Question: Find the infinitesimal order of \( h(x) = (1+4x)^{1/4} + \ln(1-\sin(x)) - 1 \) as \( x \) approaches 0, and calculate \( \lim_{x\to0}\frac{1-\cos(3x)}{h(x)} \).

Answer: -\dfrac{9}{4}


Question: Consider the field extension \(K(x_1, x_2, \dots, x_n)/K(x_1, x_2, \dots, x_r)\), where \(x_1, x_2, \dots, x_r\) is a transcendental basis of \(K(x_1, x_2, \dots, x_n)\) over \(K\). Is the extension \(K(x_1, x_2, \dots, x_n)/K(x_1, x_2, \dots, x_r)\) a finite extension?

Answer: Yes


Question: Let $M$ and $N$ be von Neumann algebras equipped with faithful normal states. Suppose there is a normal unital injective $*$-homomorphism $\pi \colon M \to N$ that preserves the states and commutes with the modular groups. If $N$ is injective, is $M$ injective?

Answer: Yes


Question:
Given that $X_1, X_2, \ldots, X_{16}$ are observations from a normal distribution $N(\mu, \sigma^2)$, and the statistics are defined as:
\[ \overline{X}=\frac{1}{16}\sum_{i=1}^{16}X_i \]
\[ S^2=\frac{1}{15}\sum_{i=1}^{16}(X_i-\overline{X})^2 \]
Find the conditional probability $P(\overline{X}>\mu|S^2>\sigma^2)$. Express your answer as a decimal.

Answer: 0.5


Question: Let $A \in \mathbb{C}$ be a $2 \times 2$ matrix, and let $f(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$ be any polynomial over $\mathbb{C}$. Which of the following statements is true about $f(A)$?\n\n(a) $f(A)$ can be written as $c_0I + c_1A$, for some $c_0, c_1 \in \mathbb{C}$.\n(b) $f(A)$ is the zero matrix.\n(c) The rank of $f(A)$ is 2.\n(d) $f(A)$ can never be the zero matrix.

Answer: a


Question: Find the maximum value of the expression $6x - 3y - 8z$ subject to the constraint $2x^2 + 3y^2 + 4z^2 = 1$.

Answer: \sqrt{37}


Question: Consider a sequence of functions \( f_k \) in \( L^{2+\frac{1}{k}}(\Omega) \) such that \( \|f_k\|_{L^{2+\frac{1}{k}}(\Omega)} = 1 \) for all \( k \ge 1 \), where \( \Omega \) is a bounded domain in \( \mathbb{R}^n \). Determine whether this sequence can strongly converge to zero in \( L^2(\Omega) \).

Answer: Yes


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $\int_{-\infty}^{\infty} |f(z)| \, dz < \infty$. Determine whether $\lim_{h \to \infty} \int_{|f(z)| > h} f(z) \, dz = 0$. Justify your answer.

Answer: 0


Question: Solve the inequality: \[ \sin^{-1}\left(\sin\left(\frac{2x^2+4}{1+x^2}\right)\right) < \pi - 3 \]

Answer: (-1, 1)


Question: Determine whether the following statement is true: Let $M$ be an $F[x]$-module. Then $M \cong F[x]/(p(x))$ where $(p(x))$ is maximal. Here, $F$ is a field.

Answer: False


Question:
Consider the transformation $T$ given by the equations:
$$x = u + v, \quad y = v - u^2.$$ A triangle $Q$ in the $(u,v)$-plane has vertices at $(0,0)$, $(2,0)$, and $(0,2)$. Let $T(Q) = D$ in the $xy$-plane. Compute the area of $D$ using a double integral over $Q$.

Answer: \dfrac{14}{3}


Question: Evaluate the contour integral \( \int_{\gamma} \frac{1}{z^2-1} \, dz \) where \( \gamma(t) = 2e^{it} \) for \( -\pi \leq t \leq \pi \).

Answer: 0


Question: Determine the number of countable models of ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice).

Answer: 2^{\aleph_0}


Question: Is it possible to devise an algebraic function over a prime field $\operatorname{GF}(p)$, where the prime $p$ has the form $p=kn+1$ for some $n$ that is a power of $2$, that maps a specific number to $1$ and all other numbers to $0$? For instance, can a function $f_{23}(x)$ be constructed such that it evaluates to $1$ for $x = 23$ and to $0$ for all other $x$?

Answer: Yes


Question:
Given vectors \( u, v \in \mathbb{R}^n \) with \( ||u||_2 = \frac{1}{2} \) and \( ||v||_2 = 1 \), and the matrix norm \( ||A||_2 = \rho(A^TA)^{1/2} \) where \( \rho(A) \) is the spectral radius of \( A \), verify the following result:

\[ ||uv^T||_2^2 = \frac{1}{4} \]

Answer: \dfrac{1}{4}


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

Answer: \dfrac{1}{2}


Question:
What is the largest integer $k$ such that \( \frac {1001\times 1002 \times \ldots \times 2008} {11^k} \) is an integer? Choose from the following options:

(A) 100
(B) 101
(C) 102
(D) 103
(E) 105

Answer: B


Question: A function \( f \) is defined by \( f(z) = (4 + i)z^2 + az + b \) for all complex numbers \( z \), where \( a \) and \( b \) are complex numbers. If \( f(1) \) and \( f(i) \) are both real, find the smallest possible value of \( |a| + |b| \).

Answer: \sqrt{2}


Question: In a Poisson process with a rate of \( \lambda = 6 \) customers per hour, the probability of a customer being male is \( \frac{2}{3} \). During the first two business hours, 10 male customers have arrived. Calculate the expected number of female customers that have arrived during this time.

Answer: 4


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^n \left\lfloor \sqrt{\frac{4i}{n}} \right\rfloor \] where \( \left\lfloor x \right\rfloor \) is the greatest integer function.

Answer: \dfrac{3}{4}


Question: Consider the theory $U_k = \text{PA} + \text{Con}(U_k, k)$, where $\text{Con}(T,k)$ encodes the statement that there is no proof of a contradiction in theory $T$ shorter than $k$ bits. Using Gödel's fixed point theorem, each $U_k$ is the theory of PA with an additional axiom claiming that $U_k$ is consistent up to $k$-bit proofs. Assuming ZFC as a meta-theory, are all the $U_k$ consistent?

Answer: Yes


Question: Calculate the probability that for a standard Brownian motion \( W(t) \), the inequality \( W(3) > W(2) > W(1) \) holds.

Answer: \dfrac{1}{4}


Question: Find all positive integer solutions \((y, x)\) to the equation \(y^3 - 3^x = 100\).

Answer: (7, 5)


Question: Evaluate the limit: $$\lim\limits_{n\to\infty}\frac{n^n}{e^nn!}$$

Answer: 0


Question: Find all positive integer solutions \((x, y)\) to the equation \(2x^2 - 1 = y^{15}\).

Answer: (1, 1)


Question:
Determine whether the following inequality is true for events $A_i$ in a probability space:
\[ P\left( \bigcup_{i=1}^n A_i \right) \geq \sum_{i=1}^n P(A_i) - \sum_{1 \leq i < j \leq n} P(A_i \cap A_j) \]

Answer: True


Question: Given the matrix \( A = \begin{bmatrix} 1 & -1 & -5 & 1 & 4\\ -1 & 2 & 8 & -3 & -4\\ 3 & -1 & -9 & 0 & 4 \\ 2 & 2 & 2 & -5 & -10\\ 0 & -3 & -9 & 5 & 13 \end{bmatrix} \), define the subspace \( W_2 \) as \( W_2 = \{ Y \in M_{5 \times 5} \mid YA = 0 \} \). If the rank of matrix \( A \) is 3, determine the dimension of \( W_2 \).

Answer: 10


Question:
Find the smallest integer \( n \) such that:
\[ 2 \cdot \sum_{k = 1}^{n} \frac{\left(\frac{n}{100}\right)^k}{k!} \geq e^{\frac{n}{100}} \]

Answer: 70


Question: Find an upper bound for the sum \( S = \sum\limits_{j=1}^{n+1} \sum\limits_{i=0}^{j-1}{n+1 \choose j}{n \choose i} \).

Answer: 4^n


Question: Determine the cardinality of the set \( A \) of continuous functions \( f: [0,1] \rightarrow \mathbb{R} \) that satisfy the equation \( \int_{0}^{1} f(x)(1-f(x)) \, dx = \frac{1}{4} \).

Answer: 1


Question: Evaluate the limit \( \lim_{x \to 0} \left( x^2 \left( 1 + 2 + \cdots + \left[ \frac{1}{|x|} \right] \right) \right) \), where \([a]\) denotes the greatest integer less than or equal to \(a\).

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int^{1/2}_0 \int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx \).

Answer: -\dfrac{1}{400}


Question: Evaluate the integral \( \int_A y^{-1/2} \sin x \, dm^2 \), where \( A = \{(x,y) \in [0,1]^2 : x+y \not\in \mathbb{Q} \text{ and } xy \not\in \mathbb{Q} \} \) and \( m^2 \) is the Lebesgue measure on \( \mathbb{R}^2 \).

Answer: 2(1 - \cos 1)


Question: Is it true that for any sequence of decimal digits \(x_1x_2 \ldots x_m\), there exists a natural number \(n\) such that this sequence occurs as a substring in the decimal expansion of the fractional part of \(\sqrt{n}\)?

Answer: Yes


Question: For a holomorphic function \(f: \mathbb{C}^n \rightarrow \mathbb{C}\) and a point \(a = (a_1, \dots, a_n) \in \mathbb{C}^n\), consider the function \(f_a: \mathbb{C}^n \rightarrow \mathbb{C}\) defined by \((z_1, \dots, z_n) \mapsto a_1z_1 + \dots + a_nz_n + f\). For a general choice of \(a\), does \(f_a\) have only nondegenerate critical points, where nondegeneracy is defined in terms of the holomorphic Hessian bilinear form?

Answer: Yes


Question: Find the maximum value of the expression \( \frac{1}{2a+b+6} + \frac{1}{2b+c+6} + \frac{1}{2c+a+6} \) given that \( a, b, c > 0 \) and \( abc = 8 \).

Answer: \dfrac{1}{4}


Question: Let $X \subset \mathbb{R}^n$ be a closed convex set, and let $L$ be a straight line such that $X \cap L = \emptyset$. Does there exist a hyperplane containing $L$ that does not intersect $X$?

Answer: Yes


Question: Evaluate the integral \( \int_{\gamma} f(z) \, dz \), where \( f(z) = \pi e^{\pi \bar{z}} \) and \( \gamma \) is the square with vertices \( 0, 1, 1+i, \) and \( i \) in the counterclockwise direction. Determine if the integral is zero.

Answer: 4(e^{\pi} - 1)


Question: Find the number of integer pairs \((x, y)\) such that \(x^2 + y^2 = 5^k\), where \(k\) is a fixed positive integer.

Answer: 4(k + 1)


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{n}\sum\limits_{k=1}^{\lfloor{\frac{n}{2}\rfloor}} \cos\Bigl(\frac{k\pi}{n}\Bigr) \]

Answer: \dfrac{1}{\pi}


Question: Suppose $U = U_1 \times U_2 \times \cdots \times U_n$ is an open set in the topological space $X = X_1 \times X_2 \times \cdots \times X_n$, equipped with the box topology, where $n$ is finite. Are the sets $U_1, U_2, \ldots, U_n$ open in their respective topological spaces $X_1, X_2, \ldots, X_n$?

Answer: Yes


Question: Find the limit: \[ \lim _{n\rightarrow \infty }\frac{(n!)^{2}}{(2n)!} \]

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \to \infty}\frac{n}{\sqrt[n]{n!}}. \]

Answer: e


Question: If \( \beta \) is a root of the equation \( x^3 - x - 1 = 0 \), find the value of \( (3\beta^2 - 4\beta)^{\frac{1}{3}} + (3\beta^2 + 4\beta + 2)^{\frac{1}{3}}. \)

Answer: 2


Question: Solve the equation for x in terms of k: \[ 0 = (x+k)e^{-(x+k)^2}+(x-k)e^{-(x-k)^2} \]

Answer: 0


Question: Evaluate the limit \( \lim_{x \to 0} \sum_{k=1}^{2013} \frac{\{x/\tan x + 2k\}}{2013} \), where \( \{x\} \) denotes the fractional part of \( x \).

Answer: 1


Question: Evaluate the integral \( \int_{C} \log(z) \, dz \) where \( C \) is the positively oriented circular contour with center 2 and radius 1.

Answer: 0


Question: Calculate the volume under the surface defined by the function \( h(x,y) = e^{-x-y} \) over the region where \( 0 \le x < \infty \) and \( 0 \le y < \infty \). Set up the integral correctly to find the volume.

Answer: 1


Question: Evaluate the double integral: $$ \int_0^1 \int_{x^2}^x \frac{e^{\sqrt{y}}}{y-y \sqrt{y}} \, dy\, dx $$

Answer: 2(e - 1)


Question: Calculate the covariance \( \Cov((X^2-1)Y + X^3S, X) \) where \(X\), \(Y\), and \(S\) are independent random variables with distributions \(X \sim U(-1,1)\), \(Y \sim \exp(2)\), and \(S \sim N(4,3^2)\).

Answer: \dfrac{4}{5}


Question: Find the radius of convergence of the power series \(\sum^{\infty}_{n=1}\left(\int^{2n}_{n} \frac{e^t}{t}dt\right)y^n.\)

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


Question: Find the greatest integer \(n\) such that \(2^n\) divides \(63^{2018} - 1\).

Answer: 7


Question: Evaluate the expression \(2\sqrt{2^3\sqrt{2^4\sqrt{2^5\sqrt{2\ldots}}}}\).

Answer: 32


Question: Suppose we have a collection of non-empty bounded open sets, $A_{0}, A_{1}, \dots$ that are subsets of $\mathbb{R}^n$, such that $A_{i} \supseteq A_{i+1}$, and the intersection $A = \bigcap_{i=0}^{\infty} A_{i}$ is a non-empty closed set. If there is a convergent sequence $a_{0}, a_{1}, \dots$ with limit $a$ such that $a_{i} \in A_{i}$, can we conclude that $a \in A$? Assume the standard topology induced by the Euclidean metric.

Answer: Yes


Question: Find the value of \( \lambda \) such that \[ \int^{\infty}_0 \frac{\log(1+x^2)}{1+x^2} \, dx = \lambda \int^1_0 \frac{\log(1+x)}{1+x^2} \, dx. \]

Answer: 8


Question: Compute the derivative of the function \( f(z) = z^{1+2i} \) at \( z = 1 \).

Answer: 1 + 2i


Question: Let $G$ be a finite group and $N \trianglelefteq G$. Assume that $G \lesssim G/G' \times Q_8$, $G/N = Q_8$, and $N \le Z(G)$. Determine whether $G = N \times Q_8$ is true or false.

Answer: True


Question: Calculate the expected number of moves required to get a cat, a fish, a dog, and your lunch across a river, given the following conditions: A troll rolls a die showing one of the four items, and you must move that item across the river. If the cat and fish are left alone on one side, the cat will eat the fish. Similarly, if the dog and lunch are left alone, the dog will eat the lunch. The troll will reroll the die if necessary to prevent any item from being harmed. Determine the expected number of moves needed to get all items safely across the river.

Answer: 18


Question: Given that \(a + b + c = 7\), \(a^2 + b^2 + c^2 = 23\), and \(\frac{1}{a+1} + \frac{1}{b+1} + \frac{1}{c+1} = 31\), find the value of \(a^3 + b^3 + c^3\) where \(a, b, c \in \mathbb{R}\).

Answer: \dfrac{307}{31}


Question: In the vector space \((V, \boxplus, \boxdot)\) where \(V = (8, \infty)\), vector addition is defined by \(u \boxplus v := uv - 8(u+v) + 72\) and scalar multiplication by \(a \boxdot u := (u-8)^a + 8\). Find the additive inverse of 16.

Answer: \dfrac{65}{8}


Question: Evaluate \( \lim_{n \to \infty} \int_{1}^{2}\frac{\sin(nx)}{x}dx \).

Answer: 0


Question: Evaluate the limit: $$\lim\limits_{x \to +\infty} \frac{(x+2)!+4^x}{((2x+1)^2+\ln x)x!}$$

Answer: \dfrac{1}{4}


Question: Let \(X\) and \(Y\) be two random variables with the joint density function \(f(x,y) = x+y\) for \(0 \leq x, y \leq 1\), and zero elsewhere. Calculate the probability \(\mathbb{P}(X < 2Y)\).

Answer: \dfrac{19}{24}


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

Answer: 1


Question: Let $k$ be an infinite field and $K$ be a field extension of $k$ of finite degree. Suppose $f(x_1,x_2,\cdots,x_n)$ is a non-constant polynomial over the field $K$. Is it true that there exist $c_1,\cdots,c_n$ in the smaller field $k$ such that $f(c_1,\cdots,c_n) \neq 0$?

Answer: Yes


Question: Find the residue of the function \( \frac{1}{e^{e^{\frac{1}{z}}}} \).

Answer: -\dfrac{1}{e}


Question: Evaluate the limit: \( \lim_{x \to 0^+} (1+x)^{\ln x} \).

Answer: 1


Question:
Let $f : \Bbb R^2 \longrightarrow \Bbb R$ be defined by $$f(x,y) = x^6 - 2x^2y -x^4y +2y^2.$$ Which one of the following statements is TRUE about the function $f$ at the origin?

$(\text A)$ $f$ has a local maximum at the origin.

$(\text B)$ $f$ has a local minimum at the origin.

$(\text C)$ $f$ has a saddle point at the origin.

$(\text D)$ The origin is not a critical point of $f.$

Answer: C


Question:
Given that \( z = f(x, y) \) where \( x = e^u \cos(v) \) and \( y = e^u \sin(v) \), find the value of \( y\frac{\partial z}{\partial u} + x\frac{\partial z}{\partial v} \). Choose the correct option:

(A) \( e^{2u}\frac{\partial z}{\partial y} \)

(B) \( e^{2u}\frac{\partial z}{\partial x} \)

(C) \( e^{2u}\left(\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y}\right) \)

(D) 0

Answer: A


Question: Find the minimum value of the expression \(8x^3 + 36x + \frac{54}{x} + \frac{27}{x^3}\) for positive real numbers \(x\). Express your answer in simplest radical form.

Answer: 48\sqrt{6}


Question:
Calculate the expected value of the square of the average of a standard one-dimensional Brownian motion $W(t)$ over the interval $[0, T]$:
\[ E\left[\left(\frac{1}{T}\int_0^T W_t\, dt\right)^2\right] \]

Answer: \dfrac{T}{3}


Question: Let $V$ be a vector space of dimension 10, and let $L: V \to V$ be a linear transformation. Determine if it is possible for the dimensions of $L(V)$, $L^2(V)$, and $L^3(V)$ to be 8, 6, and 4, respectively.

Answer: Yes


Question: Determine whether there necessarily exists a unit speed differentiable curve \( \gamma : (-\varepsilon, \varepsilon) \rightarrow \mathbb{R}^2 \) such that \( \gamma'(t) \not \in N_{\gamma(t)} \) for all \( t \in (-\varepsilon,\varepsilon) \), given that for each \( x \in \mathbb{R}^2 \), \( N_x \) is a null set of the unit circle \( S^1 \) and for each \( u \in S^1 \), \( u \not \in N_x \) for almost all \( x \in \mathbb{R}^2 \).

Answer: Yes


Question: Let $(a_n)_{n=0}^\infty$ be an arbitrary sequence in a real Banach space $X$. Determine whether there exists a smooth function $f: \mathbb{R} \rightarrow X$ such that $f^{(n)}(0) = a_n$ for all $n = 0, 1, 2, \ldots$.

Answer: Yes


Question: Consider a grid where you can only move right or down. Starting from the top-left corner, labeled as tile A, how many different paths can you take to reach the bottom-right corner, labeled as tile I, without visiting any tile more than once?

Answer: 6


Question:
Given three variables with specified ranges and step sizes, determine the number of combinations of these variables that sum to 100%.

- Variable 1: Minimum 10%, Maximum 60%, Step size 5%
- Variable 2: Minimum 20%, Maximum 70%, Step size 5%
- Variable 3: Minimum 10%, Maximum 50%, Step size 10%

Calculate the number of possible combinations of these values such that the sum of the three variables equals 100%.

Answer: 41


Question: Given that a complex function $f(z)$ is regular in a neighborhood of $z = 0$ and satisfies the equation \( f(z)e^{f(z)} = z \), write the polynomial expansion of $f(z)$ at $z = 0$ and determine its radius of convergence.

Answer: \dfrac{1}{e}


Question: Let $u:(x_0,\infty)\to\Bbb R$ be a monotonically increasing function that is differentiable everywhere and satisfies $\lim_{x\to\infty}u(x)=l\in\Bbb R$. Does it follow that $\lim_{x\to\infty}u'(x)$ exists? If it exists, what is its value?

Answer: 0


Question: Given three independent random variables $x_1, x_2, x_3$ with the probability density function $f(x_i) = 2x_i$ for $0 < x_i < 1$, find the expected value $E(X)$ for $X = x_2$, where $x_1 < x_2 < x_3$. Express your answer as a fraction.

Answer: \dfrac{24}{35}


Question: Find the smallest real number \( A \) such that for any cubic polynomial \( f(x) \) satisfying \( |f(x)| \leq 1 \) for all \( x \in [0, 1] \), the inequality \( |f'(0)| \leq A \) holds.

Answer: 18


Question: Evaluate the limit: \[ \lim _{x\to \infty }\left(\sum _{n=0}^{499}\:\left(-1\right)^n\frac{x^{1+2n}}{\left(2n+1\right)!}\right) \]

Answer: -\infty


Question: Evaluate the contour integral \( \int_\gamma \frac{1}{1+z^2} \,\mathrm{d}z \) where \( \gamma \) is the circle with radius 1 centered at \( i \).

Answer: \pi


Question: Calculate the line integral of the vector field \([P, Q]^T = [x^2 + y^2, x^2 - y]^T\) over the circle defined by \(x^2 + y^2 = 1\).

Answer: 0


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

Answer: e^{-2}


Question: Given that \(x^{x^{x+1}}=\sqrt{2}\), evaluate \(x^{x^{p}}\) where \(p = 2x^{x+1}+x+1\).

Answer: 2


Question: If $p, q, r,$ and $s$ are in continued proportion, find the minimum value of $\frac{p-s}{q-r}$. Assume $qr > 0$. Provide your answer as a single number.

Answer: 3


Question: Given any real number $x_0$ and a natural number $p$, determine whether there exists a function $f$ in $C_0^p(\mathbb{R})$ such that $f$ has compact support and satisfies $f^{(i)}(x_0)=0$ for all $0\leq i \leq p-2$, and $f^{(p-1)}(x_0)= f^{(p)}(x_0)=1$. Here, $C_0^p(\mathbb{R})$ denotes the set of continuous functions $f$ such that $f, f', f'', ..., f^{(p)}$ are continuous and $f$ vanishes at infinity.

Answer: Yes


Question: In how many ways can you collect a total of $20 from 4 different children and 3 different adults, if each child can contribute up to $6 and each adult can give up to $10, with each individual giving a nonnegative whole number of dollars?

Answer: 112567


Question: Let $X$ and $Y$ be metric spaces such that $X \cong Y$. If $X$ is a complete metric space, does it imply that $Y$ is also complete? Answer yes or no and justify your answer.

Answer: No


Question: Consider an arbitrary non-zero ring $R$ and an $R$-scheme $X$. Is it possible for the ring of global sections $H^0(X, \mathcal{O}_X)$ to be a zero ring?

Answer: Yes


Question: Find $e \in \{0,1,\ldots,22\}$ such that the product $\prod_{i=6}^{18} i$ is congruent to $e$ modulo $23$. Determine the value of $e$. \( \prod_{i=6}^{18} i \equiv e \pmod{23} \)

Answer: 9


Question: Are there infinitely many primes of the form \(2^{2^n}-1\) where \(n\) is a natural number?

Answer: No


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

Answer: 1


Question: The prime numbers $a$, $b$, and $c$ satisfy the equation $a + b^2 = 4c^2$. Determine the sum of all possible values of $a + b + c$. Provide your answer as a single number.

Answer: 31


Question: Given that $x$ and $n$ are both odd positive integers, determine the conditions under which $x^2 \equiv -1 \pmod{2^n}$ holds.

Answer: n = 1


Question: Suppose that \( f\colon\mathbb{R}^2\to\mathbb{R}^2 \) is a continuous map that preserves area in the Euclidean sense. Is \( f \) necessarily an isometry?

Answer: No


Question: If $\phi$ is an endomorphism of an elliptic curve such that $\phi = \hat{\phi}$, does it follow that $\phi = [m]$ for some integer $m$? Here, $[m]$ denotes multiplication by $m$ in the group law, and $\hat{\phi}$ is the dual endomorphism of $\phi$. Answer with 'yes' or 'no' and justify your answer.

Answer: yes


Question: Determine the maximum number of positive roots that the equation \( \sum_{i=1}^N \frac{a_i}{x+b_i} = 0 \) can have, where each \( b_i \) is a positive constant and \( a_i \) can be either positive or negative.

Answer: N-1


Question: Let $\varphi : U \rightarrow X$ be a holomorphic mapping of some open set $U\subseteq\mathbb{C}$ into a complex $n$-dimensional manifold $X$. If this mapping is diffeomorphic onto its image, does it follow that it is also biholomorphic onto its image?

Answer: Yes


Question:
Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix such that $0 \preceq A \preceq I$, where $I$ is the identity matrix. Let $w_k \in \mathbb{R}^n$ be arbitrary vectors with $\|w_k\| \leq 1$ for $k = 0, 1, \ldots$. Find an upper bound for the vector norm:
\[ \left\| \sum_{k=0}^\infty (I-A)^k A w_k \right\|. \]

Answer: 1


Question: Find the maximum possible value of the expression \( 9\sqrt{x}+8\sqrt{y}+5\sqrt{z} \) given that \( x, y, \) and \( z \) are positive real numbers satisfying the constraint \( 9x + 4y + z = 128 \).

Answer: 80


Question: Calculate the fundamental group \( \pi_1(\mathbb{Q}, 0) \).

Answer: 0


Question: If \(\lim_{x \to \infty}\frac{a(2x^3-x^2)+b(x^3+5x^2-1)-c(3x^3+x^2)}{a(5x^4-x)-bx^4+c(4x^4+1)+2x^2+5x}=1\), find the value of \(a+b+c\).

Answer: \dfrac{58}{109}


Question: Given that $\alpha, \beta, \gamma, \delta$ are the roots of the polynomial equation $x^4 - 3x + 1 = 0$, find the value of $\alpha^3 + \beta^3 + \gamma^3 + \delta^3$. Use the relationships between the roots and coefficients of the polynomial to determine the answer.

Answer: 9


Question:
Find the value of the series given by:
\[ S = n - \frac{(n-1)}{1!} + \frac{(n-1)(n-2)^2}{2!} - \frac{(n-1)(n-2)(n-3)^2}{3!} + \cdots \]
up to \( n \) terms, where \( n \in \mathbb{N} \) and \( n \ge 3 \).

Answer: (n-1)(n-2)


Question: Determine whether the local ring $R = k[x]/x^p$, where $k$ is a field of characteristic $p$, is regular.

Answer: No


Question: Find the number of ways in which the number 200 can be expressed as a sum of positive integers, denoted as \( p(200) \).

Answer: 3972999029388


Question: Calculate the value of the definite integral of the function \( f(x, y) \) defined as follows: \[ f(x, y) = \begin{cases} \frac{21}{4}x^2y & \text{for } x^2 \leq y \leq 1, \\ 0 & \text{otherwise.} \end{cases} \] over the region where \( y \leq x \).

Answer: \dfrac{3}{20}


Question: Let $R$ be a discrete valuation ring (DVR) with maximal ideal $xR$, and assume that $R$ is not complete in the $xR$-adic topology. Let $\hat{R}$ be the completion of $R$ in the $xR$-adic topology. Set $K=Q(R)$, the fraction field of $R$, and set $K'=Q(\hat{R})$. Is the degree of the field extension $K'/K$ infinite?

Answer: Yes


Question: Compute the limit: \[ \lim_{x \to a} \frac{x}{x-a} \int_a^x f(t) \, dt \] assuming that $f(x)$ is continuous in the neighborhood of $a$.

Answer: a f(a)


Question:
Compute the following integral for $\sigma > 1$:
\[ \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T}\left|\zeta{(\sigma + it)}\right|^2dt .\]

Answer: \zeta(2\sigma)


Question:
Derive the generating function for the sequence \( a_n = \frac{1}{n} \) starting at \( a_1 \), which is given by:
\[ f(x) = \sum_{n=1}^\infty \frac{x^n}{n} \]
Show that this series simplifies to:
\[ f(x) = -\ln(1-x) \]

Answer: -\ln(1 - x)


Question: Consider the series \( \sum\limits_{ - \infty }^\infty {a_jz^j} \) which is convergent on the annulus \(1<|z|<4\) and vanishes on the annulus \(2<|z|<3\). Is it true that all the coefficients \(a_j\) are zero?

Answer: Yes


Question: Determine the value of $x$ at which the function \( f(x) = \int\limits_{-1}^{x} t(e^t-1)(t-1)(t-2)^3(t-3)^5 \, dt \) has a local maximum.

Answer: 2


Question: Evaluate the expression \( \sum_{k=1}^{3} k f(k) \), where \( f(x) = \lim_{n \to \infty} \sum_{r=1}^{n} \frac{n}{n^2 + x^2 r^2} \).

Answer: \pi


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

Answer: \dfrac{5\pi}{6}


Question: Define the operator \( T: P(\mathbb{R}) \to P(\mathbb{R}) \) by \( T(p) = p' \), where \( p' \) is the derivative of the polynomial \( p \). Find all eigenvalues and eigenvectors of \( T \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{a^n}{(1+a)(1+a^2)\cdots(1+a^n)} \] for \( a > 0 \).

Answer: 0


Question: Calculate the degree of the splitting field of the polynomial \(X^3 + 2\) over \(\mathbb{Q}\).

Answer: 6


Question: Determine whether the following statement is true: For any odd natural number \(x > 2\), there exists a positive natural number \(y\) such that \(x^y = 2^n + 1\) or \(x^y = 2^n - 1\), where \(n\) is a natural number greater than 0.

Answer: False


Question: Is a Kähler manifold necessarily symplectic? Specifically, if $M$ is a $2n$-dimensional Riemannian manifold with holonomy group conjugate into $\text{U}(n)$, does there exist a closed $2$-form $\omega$ on $M$ such that $\omega^n$ is nowhere vanishing?

Answer: Yes


Question: Determine whether it is possible to find $n^2$ full-rank matrices in $\mathbb{F}^{n \times n}$ that are linearly independent when vectorized. If not, find the maximum number of such matrices that can be found.

Answer: n^2


Question: Evaluate the limit: $$\lim_{n\to\infty} n\left[e^{\frac{x}{\sqrt{n}}}-\frac{x}{\sqrt{n}}-1\right]$$

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


Question: Evaluate the integral \( \int_{-4}^{4} \cos{x} \delta(\sin{x}) \,{\rm d}x \).

Answer: -1


Question: Let \( f: \mathbf{R} \to \mathbf{R} \) be an analytic function. Determine whether there exists a harmonic function \( u \) on the circular cylinder \( D \times \mathbf{R} \subset \mathbf{R}^3 \) such that \( u = f \) along the axis \( \{ (0,0) \} \times \mathbf{R} \).

Answer: Yes


Question: Determine the smallest ordinal \(\alpha\) such that \(L \cap P(L_{\alpha})\) is uncountable, given that \(0^\sharp\) exists and assuming ZFC. Note: If \(V = L\), then \(\alpha = \omega\), but if \(0^\sharp\) exists, then \(\alpha > \omega\).

Answer: \omega_1


Question:
Let $R = \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$ and $I = \mathbb{Z} \times \mathbb{Z} \times \{0\}$. Determine which of the following statements is correct:

1. $I$ is a maximal ideal but not a prime ideal of $R$.
2. $I$ is a prime ideal but not a maximal ideal of $R$.
3. $I$ is both a maximal ideal and a prime ideal of $R$.
4. $I$ is neither a maximal ideal nor a prime ideal of $R$.

Select the correct statement.

Answer: B


Question: Evaluate the integral \[ I = \frac{1}{2\pi i} \int_{\vert z \vert = R} (z-3) \sin \left(\frac{1}{z+2}\right) dz \] where \( R \geq 4 \).

Answer: -5


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( \tan\left(\frac{n\pi}{2n+1}\right) \right)^{1/n} \]

Answer: 1


Question: Twelve passengers board a train with six cars, each choosing a car with equal probability. Let $P_1$ be the probability that there is 1 car without passengers, 1 car with one passenger, 2 cars with two passengers each, and the remaining 2 cars with three and four passengers, respectively. Let $P_2$ be the probability that there are exactly 2 passengers in each car. Find the ratio $\frac{P_1}{P_2}$.

Answer: 40


Question: Consider a Brownian motion $B_t$ and real numbers $\mu, \sigma > 0$. Determine whether the following property holds: $$P\left(\liminf\limits_{t\rightarrow \infty}e^{\mu t+\sigma B_t}>1\right)=1.$$

Answer: 1


Question: Let $X_1, X_2, \dots$ be a sequence of i.i.d. random variables with probability density function $f(x) = \frac{8x}{9}$ for $0 < x < 1.5$. Determine the almost sure limit of the product $\prod_{i=1}^n X_i$ as $n$ approaches infinity.

Answer: 0


Question: If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix, and their sum $A+B$ is a normal matrix, does it follow that $A$ and $B$ commute?

Answer: Yes


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

Answer: 3


Question: Find the radius of convergence for the power series \( \sum_{n=0}^\infty \left(\int_0^n \frac{\sin^2t}{\sqrt[3]{t^7+1}} \, dt\right) x^n \).

Answer: 1


Question: If $2i$ is an eigenvalue of a real $2 \times 2$ matrix $A$, find $A^2$. Express your answer in terms of the identity matrix $I$.

Answer: -4I


Question: Calculate the probability \( P(p_1 > p_2 | 2H^1, 1H^2) \) for two independent unfair coins with probabilities of heads \( p_1 \) and \( p_2 \), where \( p_1, p_2 \sim U(0,1) \). Each coin is tossed 3 times, resulting in 2 heads for the first coin and 1 head for the second coin. Use Bayes' theorem to find the solution.

Answer: \dfrac{53}{70}


Question: Let $X$ be a normed vector space, and let $P$ and $Q$ be commutative projections on $X$, meaning $PQ = QP$. Show that the operator $P + Q - PQ$ is a projection that is surjective onto the subspace $\text{im} P + \text{im} Q$. Here, $\text{im} P$ denotes the image of $P$. Prove that $P + Q - PQ$ projects onto $\text{im} P + \text{im} Q$.

Answer: P + Q - PQ


Question: Evaluate the limit: \[ \lim_{n\to+\infty} \frac{1}{n\log(n)}\sum_{k=4}^{n}\frac{2k}{k^2-2k-3} \] Find the value of \( \alpha \) such that \( \lim_{n\to+\infty} \frac{1}{n\log(n)}\sum_{k=4}^{n}\frac{2k}{k^2-2k-3} = \alpha \).

Answer: 0


Question: Consider the function \( f(x) = \frac{1}{\sqrt{x}} \) on the interval \([0,1]\). Without using improper integrals, determine whether there exists a tagged partition of \([0,1]\) and a corresponding Riemann sum that does not converge for this function.

Answer: Yes


Question: Consider the functions \( f(x) = \int_0^x (4t^4 - at^3) \, dt \) and \( g(x) = px^2 + qx + r \), where \( g(0) + 6 = g'(0) - c = g''(0) + 2b = 0 \) and \( a, b, c \) are positive real numbers. If \( y = g(x) \) and \( y = f'(x) \) intersect at four distinct points with abscissae \( x_i = 1, 2, 3, 4 \) such that \( \sum_{i=1}^4 \frac{i}{x_i} = 8 \), determine whether the abscissae of the points of intersection are in Arithmetic Progression, Geometric Progression, or Harmonic Progression.

Answer: Harmonic Progression


Question: Find the value of \(\alpha + \beta\) if the line \(\frac{x}{1} = \frac{y}{2} = \frac{z}{3}\) intersects the line given by the equations \(3\beta^2x + 3(1 - 2\alpha)y + z = 3\) and \(-\frac{1}{2}(6\alpha^2x + 3(1 - 2\beta)y + 2z) = 3\).

Answer: 3


Question: Determine whether the following statement is true or false: A sequence of matrices \( \{A_k=(a_{ij}^{(k)})\} \) in \( M(n, \mathbb{R}) \) converges to \( A=(a_{ij}) \in M(n, \mathbb{R}) \) with respect to the norm \( ||A||=\sqrt{\sum_{i=1}^n \sum_{j=1}^n a_{ij}^2}=\text{Trace}(AA^t) \) if and only if each matrix entry \( a_{ij}^{(k)} \) converges to \( a_{ij} \) as \( k \to \infty \) for all \( i,j=1,\ldots,n \).

Answer: True


Question:
Determine whether the following limit is true for all integrable functions \( f(x) \) on \([0,1]\) and for all constants \( c \) with \( 0 \leq c < 1 \):
\[
\lim_{h\to0^+}\int_0^c\bigl(f(x+h)-f(x)\bigr)\,dx=0
\]

Answer: 0


Question: Calculate the sum \( \sum_{k=1}^{2018} \frac{3^{2k/2019}}{3^{2k/2019} + 3} \).

Answer: 1009


Question:
Find the absolute value of the difference between the two possible values of \( A \) in the solution of the matrix equation:
\[
\left[\begin{matrix}2A+1&-5\\-4&A\end{matrix}\right]^{-1}\cdot \left[\begin{matrix}A-5&B\\2A-2&C\end{matrix}\right]= \left[\begin{matrix}14&D\\E&F\end{matrix}\right]
\]
where \( A, B, C, D, E, F \in \mathbb{R} \). Choose the correct answer from the options below:

- \( \dfrac{8}{3} \)
- \( \dfrac{11}{3} \)
- \( \dfrac{1}{3} \)
- \( \dfrac{19}{3} \)

Answer: D


Question: In \( \triangle ABC \), \( \angle C = 120^\circ \) and \( AH \) and \( BD \) are altitudes. Given that the area of \( \triangle HCD \) is \( \frac{15\sqrt{3}}{4} \), find the area of \( \triangle ABC \).

Answer: 15\sqrt{3}


Question: Let $T_1, T_2 : \mathbb{R}^5 \to \mathbb{R}^3$ be linear transformations such that $\text{rank}(T_1) = 3$ and $\text{nullity}(T_2) = 3$. Let $T_3 : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that $T_3 \circ T_1 = T_2$. Find the rank of $T_3$. \(\text{Note: Use the rank-nullity theorem and properties of linear transformations.}\)

Answer: 2


Question: Compute the surface integral \( \int_M \omega \) where \( \omega = x^4 \, dy \wedge dz + y^4 \, dz \wedge dx + z^4 \, dx \wedge dy \) and \( M \) is the sphere defined by \( x^2 + y^2 + z^2 = R^2 \).

Answer: 0


Question: Determine the number of binary relations \( R \subseteq \mathbb{N} \times \mathbb{N} \).

Answer: 2^{\aleph_0}


Question:
Calculate the magnitude of the gradient \( \left\lVert \nabla \phi(1,2) \right\rVert \) for the function \( \phi(x,z) = f(x, g(x,z)) \), given the following information:

- \( f(1,3) = 2 \)
- \( f_x(1,3) = 4 \)
- \( f_y(1,3) = 1 \)
- \( g(1,2) = 3 \)
- \( g_x(1,2) = 2 \)
- \( g_z(1,2) = -2 \)

Choose the correct answer from the options below:

a. \( \sqrt{34} \)
b. \( \sqrt{36} \)
c. \( \sqrt{38} \)
d. \( \sqrt{40} \)
e. \( \sqrt{42} \)

Answer: d


Question: Let $x_1, x_2, \ldots, x_6$ be numbers such that for each $i = 1$ to $6$, the equation \( \log_a(\log_b(\log_c x_i)) = 0 \) holds, where \(a, b,\) and \(c\) are every possible different arrangement of \(2, 4,\) and \(8\). The product \(x_1 x_2 x_3 x_4 x_5 x_6\) can be expressed in the form \(2^N\). Find the value of \(N\).

Answer: 50


Question: Find the function $f(x)$ that satisfies the equation $f(xy) = e^{xy-x-y}[e^y f(x) + e^x f(y)]$ and the condition $f'(1) = e$. Here, $f'$ denotes the derivative of the function $f(x)$. Determine $f(x)$.

Answer: e^x \ln x


Question: Can the set \( E_c(T) = \{ x \in X : \nu(T, x) \geq c \} \) have isolated points, where \( T \) is a positive current of bidegree 1, \( c \) is a positive real number, \( X \) is a complex variety, and \( \nu(T, x) \) is the Lelong number of \( T \) at \( x \)?

Answer: Yes


Question: Find all integer values of \( n \) such that \( \frac{n^3-1}{5} \) is a prime number.

Answer: 6


Question: Determine the values of $p > 0$ for which the sequence of functions $f_n(x) = \frac{nx}{1+n^2x^p}$ on $[0,1]$ converges uniformly to its pointwise limit $f(x) = 0$. Consider the expression \( \sup_{x \in [0,1]} \left| \frac{nx}{1+n^2x^p} \right| \) to analyze the uniform convergence.

Answer: (0, 2)


Question: Find all pairs of prime numbers \(p\) and \(q\) such that \(1+\frac{p^q-q^p}{p+q}\) is a prime number.

Answer: (2, 5)


Question: Find the value of $b > 1$ such that the equation $b^x = x$ has exactly one root $x$. Provide your answer as a single number.

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


Question:
Find the smallest positive integer \( n \) such that:
\[ 1,000,000^{{1,000,000}^{1,000,000}} < n^{n^{n^n}} \]

Answer: 8


Question: In a scalene triangle with sides measuring 13, 14, and 15 units, two external angle bisectors are drawn from different vertices. The third vertex is then connected perpendicularly to these bisectors. Calculate the length of the segment joining the feet of these perpendiculars.

Answer: 21


Question: Assume that $M \subset \mathbb{R}^n$, $n \ge 3$, is the boundary of an open bounded set $D$ containing $0$, which is starlike with respect to 0. This means that each ray $[0,x]$ from $x \in M$ to $0$ intersects $M$ only once. Is $M$ smooth almost everywhere?

Answer: No


Question: Given that $(I - A)^2 + A = 0$ and $A$ is an invertible square matrix, find a formula for $A^{-1}$ in terms of $A$.

Answer: I - A


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

Answer: 1


Question: Minimize the function $6x + 4y + 8z$ subject to the constraint $xyz = 1000$. Use the method of Lagrange multipliers to find the solution.

Answer: 120\sqrt[3]{3}


Question: Evaluate the double integral \( \int_0^1 \int_0^x \left( 1 - \frac{\sqrt{3}}{2 + \cos(2\pi t)}\right) dt \, dx \).

Answer: 0


Question: Let $G$ be a finite simple undirected graph. Suppose there exist subgraphs $G_1, G_2, \dots, G_n$ of $G$, such that $G_i \cong K_5$ or $K_{3,3}$, $E(G_i) \cap E(G_j) = \emptyset$ and $|V(G_i) \cap V(G_j)| \leq 3$, for $i \neq j$. Is it true that the genus of $G$ is greater than or equal to $n$?

Answer: Yes


Question: Let $G$ be a group and let $H$ and $N$ be subgroups of $G$. Suppose that $[G:H] \leq |N|$. Does this always imply that $[G:N] \leq |H|$? Determine whether this statement is true in general, not just in the finite case.

Answer: True


Question: Let \(x, y, z\) be real numbers such that \(x + y + z = 1\) and \(x^2 + y^2 + z^2 = 1\). Find the minimum value of \(x^3 + y^3 + z^3\).

Answer: \dfrac{5}{9}


Question: Determine the number of special elements in the set $X = \{a + b\sqrt{-5} : a, b \in \mathbb{Z}\}$. An element $x \in X$ is considered special if there exists a $y \in X$ such that $xy = 1$. Find the total number of such special elements in $X$. \(\text{Note: } \sqrt{-5} = i\sqrt{5}.\)

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \to \frac{\pi}{2}} \left( \tan \left( \frac{\pi}{4} \sin x \right) \right)^{1/ ( \tan (\pi \sin x))}. \]

Answer: \sqrt{e}


Question: Evaluate the limit of the integral: \[ \lim_{n\to\infty}\int_{0}^{n} \frac{1}{1+x^{2n}}\,\mathrm{d}x \]

Answer: 1


Question: Prove that if $f: S^1 \to S^1$ is the map $z \mapsto z^n$, then the induced map $f_*: H_1(S^1) \to H_1(S^1)$ is given by multiplication by $n$.

Answer: n


Question: Determine the number of roots of the equation $f(f(x))=0$, where $f(x)=x^3-3x+1$. Provide your answer as a single integer.

Answer: 7


Question: Let $Y$ be the eigenspace corresponding to an eigenvalue $\lambda$ of an operator $T$. If $T_1$ is the restriction of $T$ to $Y$, find the spectrum $\sigma(T_1)$. Prove that $\sigma(T_1) = \{\lambda\}$.

Answer: \{\lambda\}


Question: Find the length of the curve defined by the parametric equation \( \gamma(t) = (\cos^3 t, \sin^3 t) \) for \( t \in [0, 2\pi] \).

Answer: 6


Question: Let \( f: [0,1] \to \mathbb{R} \) be a continuous function such that \( \int_a^b f(x) \, dx = 0 \) for all \( 0 \le a \le b \le 1 \). Prove that \( f(x) = 0 \) for all \( x \in [0,1] \).

Answer: 0


Question:
A line is given by the parametric equations \((-2+2t, -1-t, 3t)\). Perpendiculars are drawn from points on this line to the plane \(x+y+z=3\). Determine the line on which the foot of the perpendicular lies. Choose from the following options:

(A) \(\frac{x}{5} = \frac{y - 1}{8} = \frac{z - 2}{-13}\)

(B) \(\frac{x}{2} = \frac{y - 1}{3} = \frac{z - 2}{-5}\)

(C) \(\frac{x}{4} = \frac{y - 1}{3} = \frac{z - 2}{-7}\)

(D) \(\frac{x}{2} = \frac{y - 1}{-7} = \frac{z - 2}{5}\)

Answer: D


Question: Find a real number $a$ such that $0 < a < 1$ and there is no real number $b \neq a$ for which $a^a = b^b$. Assume $a \neq b$. What is the value of $a$?

Answer: \dfrac{1}{e}


Question: Find the number of integral values of \( k \) such that the inequality \( x^2 + 2y^2 - 2xy - 2 \ge k(x + 2y) \) holds for all real numbers \( x \) and \( y \).

Answer: 0


Question: Prove that the polynomial \(x^{2k} + px + q\), where \(k\) is a natural number, has at most two real roots.

Answer: 2


Question: Consider a matrix $A$ with $n=2m$ and $a_{ii}=a_{n+1-i,n+1-i} = a_{i,n+1-i}=a_{n+1-i,i} = i$ for $i=1,\dots, m$. Find the 2-norm $||A||_2$ of the matrix $A$.

Answer: 2m


Question: Calculate the mass of the region $R$ in the first quadrant bounded by the hyperbolas $x^2-y^2=2$, $x^2-y^2=4$, $xy=2$, and $xy=5$. The density $\\delta(x,y)$ is proportional to the square of the distance to the origin, with a proportionality constant of 5. Use the transformation $u=x^2-y^2$ and $v=xy$ to evaluate the integral for the mass $m=\iint_R \delta(x,y)\,dA$. Determine the Jacobian of the transformation and express the integral in terms of $u$ and $v$.

Answer: 15


Question: Two people start at the same time and move with the same pace on a 5 x 5 square grid, one starting from the bottom left corner and the other from the top right corner. Calculate the probability that they meet on the grid. Express your answer as a fraction in the form \( \frac{a}{b} \).

Answer: \dfrac{1}{70}


Question: Find the maximum value of the expression \(a^3 + b^3 + c^3 + 4abc\) subject to the constraints \(0 \leq a, b, c \leq \frac{3}{2}\) and \(a + b + c = 3\).

Answer: \dfrac{243}{32}


Question:
Maximize the integer value of \( c \) subject to the following constraints:
\[ a + b + c < 0 \]
\[ b^2 - 4ac < 0 \]
where \( a, b \in \mathbb{R} \) and \( c \in \mathbb{Z} \).

Answer: -1


Question: Let \( f \) be integrable on \([0,1]\) and \( \lim_{x\to 0} f(x)=L \). Compute \( \lim_{x\to 0^+} x\int_x^1 \frac{f(t)}{t^2}\, dt \).

Answer: L


Question: Solve the equation \( \sqrt[3]{x^2-1} + x = \sqrt{x^3-2} \). Verify that \( x = 3 \) is a solution.

Answer: 3


Question: Calculate the percentage of prime numbers among all numbers with 100 decimal digits.

Answer: 0.43\%


Question: Calculate the limit: \[ \lim_{n \to +\infty}\frac{1}{n}\sum_{k=1}^{n}\sqrt{\frac{k}{n}\cdot\frac{n-k}{n}} \]

Answer: \dfrac{\pi}{8}


Question: From a point, perpendicular tangents are drawn to the ellipse \(x^2 + 2y^2 = 2\). The chord of contact touches a circle that is concentric with the ellipse. Find the ratio of the minimum and maximum areas of the circle.

Answer: \dfrac{1}{4}


Question: Calculate the covariance $Cov(X, Y)$ for the random vector $(X, Y)$ uniformly distributed over the unit ball in $\mathbb{R}^2$. Provide your answer as a single numerical value.

Answer: 0


Question: Find the remainder when dividing \(13^{3530}\) by 12348.

Answer: 169


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

Answer: \mathbb{Z}(p^\infty)


Question: Find the smallest integer value of \( n \) such that \((1+\sqrt{2})^n\) is within \(10^{-9}\) of a whole number.

Answer: 24


Question: Consider a function $y(x)$ that satisfies the differential equation $y(x) = Ay''(x)$, where $A$ is a positive constant. Without any information about the first derivative, determine if it is possible for $y(x)$ to have both positive and negative values.

Answer: Yes


Question: What is the maximum possible order of an element in the symmetric group \(S_{13}\)?

Answer: 60


Question: Consider a geometric progression with a non-zero natural number as the common ratio. The logarithm of the nth term, with the base equal to the common ratio, is 6. The logarithm of the product of the first n terms, with the same base, is 20. The product of the first and the nth term of the sequence is 243. Determine the sum of the first n terms of this sequence.

Answer: \dfrac{3280}{3}


Question: Evaluate the limit: \[ \lim_{x\rightarrow\infty}(\sqrt{x^2+2x+4}-x)^x. \]

Answer: e^{\frac{3}{2}}


Question:
Let \(f: \mathbb{R} \to \mathbb{R}\) be a continuous function. Which one of the following sets cannot be the image of \((0,1]\) under \(f\)?

- \(\{0\}\)
- \((0,1)\)
- \([0,1)\)
- \([0,1]\)

Answer: B


Question: Evaluate the limit \( \lim_{x\to0}\frac{e^x-1-x}{x^2} \) without using l'Hopital's rule or any series expansion.

Answer: \dfrac{1}{2}


Question: Calculate the sum: $$\sum_{k=0}^n \binom{n}{k}(-1)^{n-k}(x-2(k+1))^n.$$ Verify if the result is equal to $$(-2)^{n}n!.$$

Answer: (-2)^n n!


Question: Four music books, five education books, and two medicine books need to be arranged on a shelf. If only the music books must be kept together and all 11 books are different, in how many ways can this be done?

Answer: 967680


Question:
Let $X$ and $Y$ be two continuous random variables with the joint density function given by
\[ f(x,y) = \begin{cases} \frac{1}{2}, & 0 \leq x \leq y \leq 2 \\ 0, & \text{else} \end{cases} \]
Calculate the expectation $E[XY]$. Provide your answer as a single number.

Answer: 1


Question: Determine over which field \( K \) the greatest common divisor of the polynomial \( p(x) = x^6 - 1 \) and its derivative \( p'(x) \) is not equal to 1. The fields to consider are: \( K = \mathbb{Q} \), \( K = \mathbb{Z}_3 \), \( K = \mathbb{Z}_5 \), and \( K = \mathbb{Z}_7 \).

Answer: \mathbb{Z}_3


Question: Find the splitting field $E$ of the polynomial $x^6 - 4$ over $\mathbb{Q}$ and determine the dimension of $E$ over $\mathbb{Q}$. Provide your answer as a single number representing the dimension.

Answer: 6


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

Answer: \dfrac{1}{6}


Question: Find the derivative of \(f(x) = x^2 \cos \frac{1}{x}\) at \(x = 0\) using the definition of a derivative.

Answer: 0


Question: Determine the radius of convergence of the series \( \sum\limits_{n=1}^{\infty} d_n x^n \), where \( d_n \) denotes the number of divisors of \( n^{50} \). Choose from the following options: 1, 0, 50, \( \frac{1}{50} \).

Answer: 1


Question:
Let $U=\left(0,\frac{1}{2}\right)\times \left(0,\frac{1}{2}\right)$ and $V=\left(-\frac{1}{2},0\right)\times \left(-\frac{1}{2},0\right)$. Let $D$ be the open unit disk centered at the origin of $\mathbb{R}^2$. Suppose $f$ is a real-valued continuous function on $D$ such that $f(U)=0$. Determine which of the following statements is possible:

(A) $f(v)=0$ for every $v\in V$.
(B) $f(v)\neq 0$ for every $v\in V$.
(C) $f(v)=0$ for some $v\in V$.
(D) $f$ can assume every real value in $V$.

Choose the correct option(s).

Answer: ABC


Question: Calculate the limit using definite integrals: \[ \lim_{n\to\infty}2n\sum_{k=1}^n\frac1{(n+2k)^2} \]

Answer: \dfrac{2}{3}


Question: Let $p, q, r$ be non-vanishing non-constant entire holomorphic functions such that $p + q + r = 0$. Determine whether there exists an entire function $h$ such that $p, q, r$ are constant multiples of $h$.

Answer: Yes


Question: Consider the independent random variables $X_1, X_2, \ldots, X_{19}$, each with the probability mass function $p(X_i = -1) = p(X_i = 1) = p$ and $p(X_i = 0) = 1 - 2p$. We use the estimator $T = \frac{1}{38} \sum_{i=1}^{19} X_i^2$ for $p$. Compute the Mean Squared Error (MSE) of this estimator when the true value of $p$ is 0.1. Express your answer as a fraction.

Answer: \dfrac{1}{475}


Question: Is there a metric on a finite-dimensional non-trivial vector space $V$ over $\mathbb{R}$ that makes $V$ compact?

Answer: No


Question: Solve the ordinary differential equation \( x(y^2 - 6) - (x^2 - 8)yy' = 0 \) with the initial condition \( y(\sqrt{8}) = -\sqrt{6} \). Find the function \( y(x) \) that satisfies this equation.

Answer: -\sqrt{6}


Question: Compute the limit: \[ \lim\limits_{x\to \infty} \left(|x|^\beta - |x-c|^\beta\right), \quad c \in \mathbb{R}, \, \beta \in (0,1). \]

Answer: 0


Question: Find a positive integer $n$ such that $a^{17n} - a^2$ is divisible by $311$ for any integer $a$. Provide your answer as a single integer.

Answer: 146


Question: In how many ways can 80 gold bars be distributed among 3 peasants and 3 kings such that each king receives at least 10 bars and each peasant receives at most 10 bars?

Answer: 906411


Question: Evaluate the line integral \( \int_{C} P\,dx + Q\,dy \) where \( P(x,y) = \frac{y}{x^2+y^2} \) and \( Q(x,y) = \frac{-x}{x^2+y^2} \), and \( C \) is a piecewise smooth Jordan curve in the region \( S = \{(x,y) \mid x^2 + y^2 > 0\} \) that encloses the origin \((0,0)\).

Answer: -2\pi


Question: Consider a continuous function \( f(v_1, \dots, v_n) \) of \( n \) arguments, which is defined as an average if it satisfies the following conditions: it lies within the closed interval \([\min(v_{1}, \dots ,v_{n}); \max(v_{1}, \dots ,v_{n})]\), is symmetric for all permutations of its arguments, and is homogeneous of degree 1. Can such an average exist that does not tend to infinity when one of the arguments tends to infinity while the rest are fixed nonnegative reals, and also does not tend to zero when one of the arguments tends to zero while the rest are fixed positive reals? Assume \( n \geq 2 \).

Answer: Yes


Question: Evaluate the integral \( I_{n} = \int_{-1}^{1} \arccos{\left(\sum_{k=1}^{n}(-1)^{k-1}x^{2k-1}\right)} \, dx \).

Answer: \pi


Question:
Given a random sample $Y_1, Y_2, \ldots, Y_{10}$ from a normal distribution $Y \sim N(\mu, \sigma^2)$, consider the statistic:
\[ U^2 = \frac{1}{9} \sum_{i=1}^{10} (Y_i - \bar{Y})^2 \]
where $U^2$ is the empirical variance. Determine the distribution of the statistic \( \frac{9U^2}{\sigma^2} \).

Answer: \chi^2(9)


Question: Let $X_1, X_2, \ldots$ be a sequence of infinite independent and identically distributed random variables, where $X_i \sim U[0,1]$. Define $D_n = \sqrt{X_1^2 + X_2^2 + \ldots + X_n^2}$. Determine the value of the following cumulative distribution function at the point $\frac{1}{\sqrt{3}}$: $$\lim_{n\to \infty} F_{\frac{D_n}{\sqrt{n}}} \left(\frac{1}{\sqrt{3}}\right).$$

Answer: \dfrac{1}{2}


Question: Determine the number of non-isomorphic groups of order 30.

Answer: 4


Question: Determine whether the following statement is true or false: If $X \subset \mathbb{R}$ and the set of limit points $X'$ is finite or countable, then $X$ is finite or countable.

Answer: A


Question: Determine whether there exists a non-trivial linear functional on \((C[0,1], ||\cdot||_2)\) that does not arise from the inner product on \(L^2[0,1]\). Specifically, does there exist an \(F \in X^*\), where \(X = C[0,1]\) with the \(L^2\) norm, such that there is no \(g \in X\) for which \(F(f) = \langle f,g \rangle\) holds for all \(f \in X\)?

Answer: Yes


Question: Find the limit of the infinite product \( \prod_{n=2}^{\infty} \left(1+\frac{1}{n^2}+\frac{1}{n^4}+\frac{1}{n^6}+\ldots\right) \). Simplify the expression and determine the limit of the product.

Answer: 2


Question: If \( u = \displaystyle\int_{0}^{\pi/4}\left(\dfrac{\cos x}{\sin x +\cos x}\right)^2\,dx \) and \( v = \displaystyle\int_{0}^{\pi/4}\left(\dfrac{\sin x +\cos x}{\cos x}\right)^2\,dx \), find \( \dfrac{v}{u} \).

Answer: 4


Question: Determine the number of distinct terms in the expansion of \( \left(x + \frac{1}{x} + x^2 + \frac{1}{x^2} \right)^{20} \).

Answer: 81


Question: A two-digit number in base 11 is one-third of the number formed by reversing its digits when considered in base 19. How many such numbers are possible?

Answer: 5


Question: Given that \(x^3[f(x+1)-f(x-1)]=1\), determine \(\lim_{x\rightarrow \infty}(f(x)-f(x-1))\) explicitly.

Answer: 0


Question: Evaluate the limit \( \lim_{|z| \to 1 } |f(z)| \) for the function \( f(z) = \sum_{k=1}^\infty z^{k!} \).

Answer: \infty


Question: Determine the order of the differential equation for the family of parabolas with a fixed directrix.

Answer: 2


Question: Assign weights to the edges of a complete graph on $n$ vertices such that the sum of the weights is 1 and the weight of the minimum spanning tree is maximized. What is the weight of the minimum spanning tree after this assignment?

Answer: \dfrac{2}{n}


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

Answer: \dfrac{1}{2}


Question: Solve for $x$ if \( \frac{1}{2x} + \frac{1}{2}\left(\frac{1}{2x} + \frac{1}{2}\left(\frac{1}{2x} + \cdots\right)\right) = y \).

Answer: \dfrac{1}{y}


Question: Given a nonempty open set $A$ in $\mathbb{R}^2$ and $n$ rational points $a_1, a_2, \ldots, a_n$, determine whether there exists a rational point $a$ in $A$ such that for all $i \in \{1, 2, \ldots, n\}$, the distance $|a - a_i|$ is an irrational number.

Answer: Yes


Question: Evaluate the integral \( \int_{|z|=3} \frac{\mathrm{e}^{1/(1-z)}}{z} \,\mathrm{d}z \) using the Residue Theorem.

Answer: 2\pi i


Question: Given $n$ groups of people $P_1, \dots, P_n$, each containing $k$ persons, and no person is in more than one group, determine the minimal number $z$ of parties needed to find out if each pair of persons from different groups likes each other. At each party, invite one person from each group. What is $z$ in terms of $k$ and $n$?

Answer: k^2


Question: Find a function \( g(x) \) such that the vector field \( g(x)\textbf{F}(x,y) \) is conservative, where \( \textbf{F}(x,y) = (x^2 + y^2 + x)\textbf{i} + xy\textbf{j} \).

Answer: x


Question: Given a random variable $X$ with probability density function $f(x) = 0.5 e^{-|x-2|}$, calculate the conditional variance $\text{Var}(X \mid X > 8)$. Assume $E(X) = 2$ and $E(X^2) = 6$. Provide your answer in terms of a single numerical value.

Answer: 1


Question: Given seven distinct letters A, B, C, D, E, F, and G, determine the number of ways to form seven sets of three letters each such that each letter appears with every other letter exactly once. What is the total number of such combinations?

Answer: 30


Question: Let \( B \) be a nilpotent \( n \times n \) matrix with complex entries. Define \( A = B - I \). Find the determinant of \( A \).

Answer: (-1)^n


Question: Evaluate the limit without using L'Hôpital's rule: \[ \lim_{x\rightarrow \infty}\left(\frac{2x-5}{2x-2}\right)^{4x^{2}} \]

Answer: 0


Question: Compute the integral \( \int_\gamma |z|^2 \, \mathrm{d}z \) where \( \gamma \) is the contour starting at 0, going vertically to \( i \), then going horizontally to \( i + 1 \), then going vertically to 1, and finally going horizontally to 0.

Answer: 1 - i


Question:
Consider $\mathbb{R}^2$ with the radially open topology, where a subset $U \subset \mathbb{R}^2$ is radially open if for every $x \in U$ and every $v \in \mathbb{R}^2$, there exists $\epsilon > 0$ such that $x + sv \in U$ for every $s \in (-\epsilon, \epsilon)$. Let $X$ be $\mathbb{R}^2$ with this topology, and let $Y$ be $\mathbb{R}^2$ with the standard Euclidean topology. Determine which of the following statements about $X$ are true:

I. $X$ is Hausdorff.
II. $X$ is second countable.
III. The identity map $Y \to X$ is continuous.

Choose the correct option:
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) II and III only.

Answer: A


Question: Determine whether the sequence \( cn\left[\frac{1}{\ln(n)} - \frac{1}{\ln(cn)}\right] \) converges as \( n \to \infty \), where \( c > 1 \) is an integer constant.

Answer: \infty


Question: A rectangular prism of length $L$ slides on a flat surface with an initial speed $v_0$ and encounters a rough section where the coefficient of kinetic friction is $\mu$. Given $L = 0.49\ \mathrm{m}$, $v_0 = 1.96\ \mathrm{m/s}$, $\mu = 0.20$, and $g = 10\ \mathrm{m/s^2}$, determine the time it takes for the object to stop after it first touches the rough section.

Answer: 0.98


Question: There are \( n \) planets, and Luke starts his journey on planet \( A \). Each planet, except for planets \( B \) and \( C \), has two unidirectional roads leading to other planets. From at most one of these roads, Luke can find a sequence of roads that brings him back to the planet he just left. Luke's journey ends when he reaches planet \( B \) or \( C \). When leaving a planet, he chooses one of the two roads with equal probability. If the probability that he arrives at planet \( B \) is \( \frac{1}{2016} \), what is the minimum value of \( n \)?

Answer: 2017


Question: Let $C$ be a curve of degree $n$ in the projective plane. Determine an upper bound for the number of points at infinity on the curve $C$. Provide a justification for your answer.

Answer: n


Question: Given the differential equation \( y' = y(1-y)e^{y} \) with the initial condition \( y(2016) = 2 \), determine \( \lim_{t \to \infty} y(t) \).

Answer: 1


Question: Determine the order of the poles of the function \( \frac{z}{\sin \pi z^2} \) at the points \( z = 0 \), \( z = \pm \sqrt{n} \), and \( z = \pm i\sqrt{n} \), where \( n \in \mathbb{Z}^+ \).

Answer: 1


Question: Evaluate the limit: $$ \lim_{x \to 0} \frac {e^x+e^{-x}-2}{\cos(4x)-\cos(x)} $$ without using L'Hopital's rule or infinite series.

Answer: -\dfrac{2}{15}


Question: Evaluate the integral \( \int_{C}\frac{z^2}{z^2+9}dz \) where \( C \) is the circle \( |z|=4 \), using Cauchy's Integral Formula.

Answer: 0


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

Answer: 0


Question: A 10-sided die with numbers 1 to 10 is rolled $n$ times. Determine the smallest value of $n$ such that the probability of rolling each of the numbers 1 to 5 at least once is 50%.

Answer: 21


Question: Determine the multiplicity of the zero of the function \( f(z) = (z^2+1)^3(1+e^{\pi z})^2 \) at the point \( z = i \).

Answer: 5


Question: Is it true that if a densely defined operator \( T: H \to \mathbb{C}^m \) on a Hilbert space \( H \) is closable, then \( T \) is bounded?

Answer: Yes


Question:
Evaluate the surface integral of the vector field \( v(x,y,z) = \begin{pmatrix} -x^3-2y \\ 3y^5z^6 \\ 3y^6z^5-z^4 \end{pmatrix} \) over the hemisphere defined by \( z \geq 0 \) and \( x^2 + y^2 + z^2 = 4 \). The integral is given by:
\[ \int_{\partial F} v \cdot \, d\mathbf{r} \]
where \( \partial F \) is the boundary of the hemisphere. Use appropriate methods to evaluate this integral.

Answer: 8\pi


Question: Is it true that every non-empty open subset of a perfect Polish space is uncountable? Provide a justification for your answer.

Answer: Yes


Question: Prove that for all natural numbers \( n \), the following limit holds: \[ \lim_{x\rightarrow\infty} \frac{x^n}{\exp(x^2)} = 0. \]

Answer: 0


Question:
Given the joint probability density function of random variables $X$ and $Y$:
\[
f_{X,Y}(x,y) = \begin{cases}
\frac{1}{y}, & 0 < x < y, \ 0< y< 1 \\
0, & \text{otherwise}
\end{cases}
\]
find the variance of $X$. Use the law of total variance: \(V(X)=E(V(X|Y))+V(E(X|Y))\).

Answer: \dfrac{7}{144}


Question: Find all the singular points and corresponding residues of the function \( \sin \frac{1}{z} \).

Answer: 1


Question:
Consider a function defined as \( g(w^T f(Wx)) = y \), where \( w \) and \( W \) are matrices with real entries, and \( f \) and \( g \) are functions. Determine whether this function is a linear method. Choose the correct option:

1. True
2. False
3. Depends on \( f \)
4. Depends on \( g \)
5. Depends on \( f \) and \( g \)

Answer: 5


Question:
Suppose \(X\) is a discrete random variable with the following probability distribution function (p.d.f):
\[
\begin{array}{c|cccc}
X & 0 & 1 & 2 & 3 \\
\hline
p(x) & \frac{2\theta}{2} & \frac{\theta}{3} & \frac{2(1-\theta)}{3} & \frac{1-\theta}{3}
\end{array}
\]
where \(0 \leq \theta \leq 1\). Given the sample \((3,0,2,1,3,2,1,0,2,1)\), find the maximum likelihood estimate (MLE) of \(\theta\).

Answer: \dfrac{1}{2}


Question: For a fixed \(n \in \mathbb{N}\), consider a convex polyhedron \(C\) in \(\mathbb{R}^n\), defined as the convex hull of finitely many points with a nonempty interior. Define a function \(F\) from the set of convex polyhedra to \(\mathbb{R}^{n+2}\) by \(F(C) = (a^C_{-1}, \ldots, a^C_n)\), where \(a^C_j\) is the number of \(j\)-faces of \(C\) for \(j = -1, \ldots, n\). Let \(W\) be the affine subspace of \(\mathbb{R}^{n+2}\) generated by \(\operatorname{im} F\). Given that \(a^C_{-1} = 1\), \(a^C_n = 1\), and Euler's formula \(\sum_{j=-1}^n (-1)^j a^C_j = 0\), is it always true that \(\operatorname{dim} W = n-1\) for any \(n\)?

Answer: Yes


Question: Let \( F \) be a field and consider the column vector \( \vec{x} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \end{pmatrix} \in F^m \). Define the set \( W = \{ A \in M_{n \times m}(F) : A\vec{x} = \vec{0} \} \). Determine if \( W \) is a linear subspace of \( M_{n \times m}(F) \). If it is, find the dimension of \( W \).

Answer: n(m - 1)


Question: Find the number of solutions to the congruence equation \( y^2 \equiv 3x^2 - x - 9 \pmod{109} \).

Answer: 217


Question: Find the shortest distance from the point \((0, b)\) to the parabola defined by the equation \(x^2 - 16y = 0\) using the Lagrange multiplier method. Consider the case when \(b < 8\).

Answer: |b|


Question: Find the sum of the series: \[ \sum_{n=1}^{\infty} (-1)^{n+1} \frac{2n+1}{n(n+1)} \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0}{\left(\frac{a^x + b^x + c^x}{3}\right)^{1/x}} \]

Answer: \sqrt[3]{abc}


Question: Let $I$ be an ultraweakly closed ideal in a von Neumann algebra $M$. Is there another ideal $J \subset M$ such that $M = I \oplus J$? Justify your answer.

Answer: Yes


Question: Solve the differential equation \((1 + x)y' = py\) with the initial condition \(y(0) = 1\), where \(p\) is an arbitrary constant. Verify if the solution is \(y = (1 + x)^p\).

Answer: (1 + x)^p


Question: Evaluate the integral \( \int _0^{2\pi}\sin^2 \left(\frac{-\pi}{6}+3e^{it} \right)dt \).

Answer: \dfrac{\pi}{2}


Question: Determine whether the following statement is true or false: If there exists a constant \( K > 0 \) such that for all \( x, y \in \mathbb{R} \), \( |f(x) - f(y)| \leq K|y-x|^2 \), then for all \( x \in \mathbb{R} \), \( f'(x) = 0 \).

Answer: True


Question: Evaluate the line integral \( \int_{0,0}^{1,\frac{\pi}{2}} e^x \sin(y)\,dx + e^x\cos(y)\,dy \).

Answer: e


Question: Consider a 100-dimensional hypersphere with a radius of 4000 miles. What fraction of its volume lies within 40 miles of its surface?

Answer: 0.634


Question: Evaluate the contour integral \( \int_{C} 1 \, dz \) where \( C \) is a straight line contour from a fixed point \( z_1 \) to another fixed point \( z_2 \) in the complex plane. Express your answer in terms of \( z_1 \) and \( z_2 \).

Answer: z_2 - z_1


Question: Calculate the limit \( \lim_{n \to \infty} \frac{1 + 2\sqrt{2} + \cdots + n\sqrt{n}}{n^2\sqrt{n}} \) using Stolz's theorem.

Answer: \dfrac{2}{5}


Question: Evaluate the expression \( \frac{u^{2n+1}\ln(u)}{2n+1} \bigg|^{u=1}_{u=0} \) at \( u=0 \).

Answer: 0


Question: Find the area of the cone defined by the equation \(z=\sqrt{2x^2+2y^2}\) that is inscribed in the sphere \(x^2+y^2+z^2=12^2\).

Answer: 48\sqrt{3}\pi


Question: Let $K$ be an infinite field, and $P \in K[x_1, \dots, x_n]$ be a non-zero polynomial. Is there an $\bar{a} \in K^n$ such that $P(\bar{a}) \neq 0$?

Answer: Yes


Question: Let $A$ be an $m \times n$ matrix of real numbers with $m \leq n$. If Rank$(A) = m$, does there always exist a generalized inverse $A^+$ of size $n \times m$ such that $A^+A = (I_m \ \mathbf{0})$?

Answer: No


Question: Evaluate the integral \( \int^{\pi/2}_0 \left( \int^{\pi/2}_y \frac{\sin x}{x} \, dx \right) \, dy. \)

Answer: 1


Question: Determine the number of positive integer values of \( m \) such that \( m^4 - 4m^3 + 22m^2 - 36m + 18 \) is a perfect square.

Answer: 2


Question: Given a random sample of size \( n = 72 \) from a distribution with probability density function \( f(x) = \begin{cases} 1/x^2,& 1 < x < \infty \\ 0,&\text{otherwise} \end{cases} \), compute approximately the probability that more than 50 of the observations are less than 3.

Answer: 0.266


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}P\left(\sum_{i=1}^n X_i \leq \frac{n}{2}+ n^{3/4} \right) \] where \(X_1, X_2, \ldots, X_n\) are i.i.d. random variables with a uniform distribution on \([0, 1]\).

Answer: 1


Question: There are 100 towns labeled from 1 to 100. The cost to build a bridge between town "i" and town "j" is \( \max\{|i - j|, 4\} \). However, if a bridge is built between town "i" and town "2i", the cost is 2. What is the minimum cost to connect all the towns in a single connected graph?

Answer: 296


Question: Let $(R, \frak m)$ be a Cohen-Macaulay local ring. If there exists a maximal Cohen-Macaulay $R$-module $M$ with finite injective dimension, does $R$ necessarily have a canonical module?

Answer: Yes


Question: Let \(A \in \mathbb{C}^{4 \times 4}\) be a matrix satisfying the following conditions: 1) \(\text{trace}(A) = 0\), 2) \(|A-iI| = 0\), 3) \(\text{rank}(A-I) = 3\), 4) \(Av = -v\) for some non-zero vector \(v\). Calculate \(A^{80}.\)

Answer: I


Question:
Given the system of equations:
\[\begin{cases}
x = u + \ln(v) \\
y = v - \ln(u) \\
z = 2u + v
\end{cases}\]
Find the partial derivative of \(z\) with respect to \(x\) at \(u = 1\) and \(v = 1\).

Answer: \dfrac{3}{2}


Question:
Let $a$ and $b$ be positive real numbers such that $2(a^2 + b^2) - (a + b) = 2ab$. Find the minimum value of the expression:
\[ \frac{a^3 + 2020}{b} + \frac{b^3 + 2020}{a} \]

Answer: 4042


Question: For what values of \( p > 0 \) is the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) defined by \( f(x) = |x|^p \) if \( x \in \mathbb{Q} \) and \( f(x) = 0 \) if \( x \in \mathbb{R} \setminus \mathbb{Q} \) differentiable?

Answer: p > 1


Question: Given the recursive relation \( b_{n+1} = b_{n} + \frac{1}{b_{n}} \) with the initial condition \( b_{1} = 1 \), find the value of \( \lfloor b_{100} \rfloor \).

Answer: 14


Question: Let \( f: D_{18} \to GL(2, \mathbb{R}) \) be a homomorphism where \( |r| = 18 \) and \( f(r) = \begin{pmatrix} 1 & 1 \\ -1 & 0 \end{pmatrix} \). Determine the order of the kernel of \( f \), denoted as \( |\ker(f)| \).

Answer: 3


Question: Find the smallest natural number \(x\) such that \(x \equiv 2 \pmod{6y+1}\), \(x \equiv 3 \pmod{12y+1}\), and \(x \equiv 3 \pmod{18y+1}\) for some natural number \(y\).

Answer: 744


Question: Let $(a(m,n))_{m,n \in \mathbb{N}}$ be a double sequence of positive numbers. Suppose that $\lim\limits_{m\to \infty}\lim\limits_{n \to \infty}a(m,n)=L$. Determine whether there always exist increasing functions $f_1:\mathbb{N} \to \mathbb{N}$ and $f_2:\mathbb{N} \to \mathbb{N}$ such that $\lim\limits_{n \to \infty}a\left(f_1(n),f_2(n)\right)=L$. Provide a proof or counterexample.

Answer: Yes


Question: Let $D$ be an integral domain and $P$ be an associated prime of $D$. Determine whether $P[[X]]$ is also an associated prime of $D[[X]]$. Provide a justification for your answer.

Answer: Yes


Question: In the ordered set \( S = \{1, 5, 6, 25, 26, 30, \dots\} \), each member is generated by adding any number of different powers of 5 (such as 1, 5, 25, 125, etc.). Determine the position \( n \) such that 15756 is the \( n \)th member of this set.

Answer: 75


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{\arctan(1+x^2) - \arcsin\left(\frac{\cos x}{\sqrt{2}}\right)}{x^2} \] without using L'Hopital's rule.

Answer: 1


Question: Determine if there exists a function $f$ such that for all $a, b, c, d$, the following equations hold: \[ f(a)f(b)-f(c)f(d) = (f(a)-f(c))(f(b)-f(d)) \] and \[ f(a)f(b)+f(c)f(d) = (f(a)+f(c))(f(b)+f(d)). \]

Answer: f(x) = 0


Question: Let \(a\) and \(b\) be elements of a group such that \(|a| = 2\), \(b \neq e\), and \(aba = b^2\). Determine the order of \(b\), denoted as \(|b|\).

Answer: 3


Question: Evaluate the line integral \( \int_{C} x^2 \, dl \), where \( C \) is the curve of intersection between the sphere \( x^2 + y^2 + z^2 = 1 \) and the plane \( x + y + z = 0 \).

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


Question: Consider a compact subset $K$ of a Banach space $E$. It is known that for every $\epsilon > 0$, there exists a finite-dimensional subspace $V \subset E$ such that $d(x,V) < \epsilon$ for all $x \in K$. Is the converse true? Specifically, if $K$ is a closed, bounded set in a Banach space $E$ such that for every $\epsilon > 0$, there exists a finite-dimensional subspace $V \subset E$ with $d(x,V) < \epsilon$ for all $x \in K$, is $K$ necessarily compact?

Answer: Yes


Question: Player 1 and Player 2 are playing a game with the function \( f = -20p \cdot q + 9p + 9q \). Player 1 chooses \( p \) and Player 2 chooses \( q \), where both \( p \) and \( q \) are in the interval \([0, 1]\]. Player 1 wants to maximize \( f \) while Player 2 wants to minimize it. Player 1 goes first. What is the optimal value of \( p \) that Player 1 should choose, knowing that Player 2 will choose \( q \) in response to Player 1's choice of \( p \)?

Answer: \dfrac{9}{20}


Question: Consider the function \( f(x,y) = \frac{1}{2}\log(x^2+y^2) \) defined on \( \mathbb{R}^2 \setminus \{(0,0)\} \). Determine if the limit of the normal derivative of \( f \) with respect to the unit circle \( \mathbb{S} = \{(x,y): x^2+y^2=1\} \) as \((x,y)\) approaches a point on \( \mathbb{S} \) is equal to 1.

Answer: 1


Question:
Given the transition matrix \( P = \begin{pmatrix} 0 & 0.5 & 0.5 & 0 & 0 \\
 0 & 0 & 0 & 0.2 & 0.8 \\
 0 & 0 & 0 & 0.4 & 0.6 \\
 1 & 0 & 0 & 0 & 0 \\
 0.5 & 0 & 0 & 0 & 0.5 \end{pmatrix} \), compute the probability that a Markov chain will enter state 5 before it enters state 3, starting from state 1.

Answer: \dfrac{4}{9}


Question: Let $p$ be a prime and $M$ be an $n \times m$ matrix with integer entries such that for any column vector $\vec{v} \neq \vec{0}$ with entries 0 or 1, $M\vec{v} \not\equiv \vec{0} \text{ (mod }p\text{)}$. Determine if there exists a row vector $\vec{x}$ with integer entries such that no entry of $\vec{x}M$ is $0 \text{ (mod }p\text{)}$. Provide a justification for your answer.

Answer: Yes


Question: Use the Fourier sine series expansion to show that the following formula holds for the range \([0, \pi]\): \[ \sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{2k-1} = \frac{\pi}{4} \]

Answer: \frac{\pi}{4}


Question: Find the distribution of the statistic \( T = \frac{S_{n-1}^2/\sigma_{1}^2}{S_{m-1}^2/\sigma_{2}^2} \), where \( X \sim N(\mu_1, \sigma_1) \) and \( Y \sim N(\mu_2, \sigma_2) \). Given two simple random samples of sizes \( n \) and \( m \) from \( X \) and \( Y \) respectively, determine the distribution of \( T \).

Answer: F(n-1, m-1)


Question: A fair six-sided die is rolled repeatedly until a 6 appears. The sum of all numbers rolled, including the 6, is denoted as $S$. Calculate the standard deviation of $S$. Express your answer as a decimal rounded to two decimal places.

Answer: 16.73


Question: Let $f:\mathbb{R} \to \mathbb{R}$ be a function with a continuous derivative such that $f(\sqrt{2}) = 2$ and \[ f(x) = \lim_{t \to 0} \frac{1}{2t} \int_{x-t}^{x+t} s f'(s) \, ds \] for all $x \in \mathbb{R}$. Find $f(3)$. \( \text{(Use the Fundamental Theorem of Calculus to solve this problem.)} \)

Answer: 3\sqrt{2}


Question: Find the maximum value of \(-x-y-z\) for \((x,y,z)\) in the set \(S = \{(x,y,z) : (x-2)^2 + (y-2)^2 + (z-2)^2 \leq 1\}\).

Answer: \sqrt{3} - 6


Question: Is it consistent within ZF set theory for there to exist a set $X$ with more than one element such that every injective map $f: X \to X$ has a fixed point?

Answer: Yes


Question: Given that \( \frac{2}{x} = 2 - x \), find the value of \( \left[x^9 - (x^4 + x^2 + 1)(x^6 + x^3 + 1)\right]^3 \) without using complex numbers.

Answer: 1


Question: Evaluate the limit: \[ \lim_{n\to\infty}n^2\left(\int_{-1/n}^0u(x-s)\,ds -\int_0^{1/n}u(x-s)\,ds\right) \] where \( u(x) \) is an infinitely differentiable function on \( \mathbb{R} \).

Answer: u'(x)


Question: Is it possible to find a norm on \(\mathbb{R}^2\) such that the linear isometry group consists only of \(\pm \operatorname{id}\)? Provide a justification for your answer.

Answer: Yes


Question: Let \( \{X(t); t \ge 0\} \) be a Poisson process with rate \( \lambda = 2 \). Determine \( E[X(1)X(2)] \).

Answer: 10


Question: Determine the number of non-isomorphic well-orderings of the set of natural numbers \( \mathbb{N} \).

Answer: \aleph_1


Question: Let \( f \in \mathbb{Z}[x] \) be an irreducible polynomial. Suppose \( f(x) \) has two roots in \( \mathbb{C} \) with a product of 1. Is it true that the degree of \( f \) is even?

Answer: Yes


Question:
Determine the nature of the set \( A = \left\{ \sum_{i=1}^{\infty} \frac{a_i}{5^i} : a_i \in \{0,1,2,3,4\} \right\} \subset \mathbb{R} \). Choose the correct option:

A. Finite
B. Countably infinite
C. Uncountable but does not contain an open interval
D. Contains an open interval

Answer: D


Question: Determine the values of \(a \in \mathbb{R}\) for which the series \(\sum_{n=0}^{\infty} \frac{1}{u_{n}^{a}}\) converges, given that \(u_0\) is an arbitrary positive real number and the sequence \(\{u_n\}_{n \ge 0}\) is defined by \(u_{n+1} = u_n e^{-u_n}\) for \(n \ge 0\).

Answer: a < -1


Question: Determine the exact topological complexity (TC) for one or two robots moving on a number 8, using Farber's theorem. The configuration space for two robots moving on the number 8 is a bouquet of seven circles (seven petals).

Answer: 3


Question: Consider a natural number \( m = \overline{a_k a_{k-1} \ldots a_1 a_0} \), where each \( a_i \) is a digit from 0 to 9 and \( a_k \neq 0 \). Define the function \( f(n) \) as the number obtained by reversing the digits of \( n \). Is it true that if \( f(m \times m) = f(m) \times f(m) \), then all digits \( a_i \) of \( m \) are in the set \{0, 1, 2, 3\}? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to 1^+}\frac{x^{x^x}-x^x}{x-1}$$

Answer: 0


Question: Is it true that any basis for the topology on \(\mathbb{R}\) generated by the collection \(\{(a,b] : a,b \in \mathbb{R}\}\) must be uncountable?

Answer: Yes


Question: Consider the differential equation \( \frac{d^2}{dx^2}w + \{u(x) + k^2\}w = 0 \) with the asymptotic condition \( \lim_{x\rightarrow \infty}w(x)e^{ikx}=1 \). Given that \( u \in L^1_1(\mathbb{R}) = \{f: \int_\mathbb{R}(1+|x|)|f|dx<\infty\} \) and \( k > 0 \), determine if the solution \( w \) is unique.

Answer: Yes


Question: Is the set $C_f = \{(x,y) \in \mathbb{R}^2 \mid x > 0, y > 0, x^2 + y^2 = 1, (x,y) = f(a,b) \text{ for some } (a,b) \in P\}$ dense in the segment of the unit circle $C = \{(x,y) \in \mathbb{R}^2 \mid x > 0, y > 0, x^2 + y^2 = 1\}$?

Answer: Yes


Question: Let $X=(X_1,X_2,X_3)$ be a continuous random vector with the joint probability density function given by \( f(x_1,x_2,x_3)= 12x_2 \) for \( 0<x_3<x_2<x_1<1 \) and \( 0 \) elsewhere. Calculate the probability of the event \( B = \{ x_3 \leq \frac{1}{3} \} \).

Answer: \dfrac{49}{81}


Question: If $X$ is a compact $R_1$ space, does this imply that $X$ is regular? Provide a justification for your answer.

Answer: Yes


Question: Suppose a probability density function (pdf) $f$ is differentiable almost everywhere, continuous, and has a bounded derivative. Is the pdf $f$ itself bounded? Provide a justification for your answer.

Answer: Yes


Question: Let \( \psi \) be a non-negative continuous function on \([0,\infty)\) that is differentiable on \((0, \infty)\), with \( \psi(0)=0 \) and \( \psi^{'}(x)\leq \psi(x) \) for all \( x\in (0, \infty) \). Prove that \( \psi \equiv 0 \).

Answer: 0


Question: Is there a simple $C^*$ algebra $A$, not isomorphic to $M_2(\mathbb{C})$, such that for every commutator element $x = ab - ba$, $x^2$ is a scalar element?

Answer: \mathbb{C}


Question: Determine the values of $x$ for which the series \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-2)^n}{n2^n} \) is convergent.

Answer: (0, 4]


Question: Determine whether the covariance of the random variables X and Y exists for the joint density function \( f_{(x,y)} (x,y) = \frac{1}{4}(y-x)e^{-y} \) defined for \(-y < x < y\) and \(y > 0\).

Answer: -1


Question: Determine the asymptotic behavior of the inverse function $f^{-1}(y)$ for the function $f(x) = \frac{x}{1+\log_2(x)}$ as $y$ becomes large. Specifically, is $f^{-1}(y)$ in $O(y \log y)$ or $O(y \log^k y)$ for some integer $k$?

Answer: O(y \log y)


Question: Compute the limit: \[ \lim_{n \to \infty} n \int_{0}^{1} (\cos x - \sin x)^n \, dx \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{\ln(1+x^2)-x^2+\frac{x^4}{2}}{\sin x^6} \]

Answer: \dfrac{1}{3}


Question: Calculate the limit \( \lim\limits_{x\to2} (x-2)\tan\left(\frac{\pi}{x}\right) \).

Answer: \dfrac{4}{\pi}


Question: In triangle ABC, segments BD, DF, and FE divide the triangle into four smaller triangles of equal area. Points E and D lie on sides CA and AB, respectively, and point F lies on side BC. Given that BF = 2DE, determine the ratio of AC to BC.

Answer: 2


Question: Evaluate the integral \( \int_{0}^{2\pi}\frac{x\sin^{2n}x}{\sin^{2n}x+\cos^{2n}x}dx \) for \( n > 0 \).

Answer: \pi^2


Question: Find the largest integer \( m \) such that \( 2^m \) divides \( 3^{2n+2} - 8n - 9 \) for natural numbers \( n \).

Answer: 6


Question: Determine the conditions under which the identity \(A^2 - B^2 = (A+B)(A-B)\) holds true for matrices \(A\) and \(B\).

Answer: AB = BA


Question: Given the equation \( {(g^{XB} \bmod P)}^{RA} \bmod P = 12 \), where \( XB = 4 \), \( g = 9 \), and \( P = 23 \), find the value of \( RA \).

Answer: 6


Question: What is the probability that a dart, thrown randomly at a unit square, will land on the diagonal of the square?

Answer: 0


Question: Determine the number of labeled trees on 7 vertices, where the vertex set is \( V = \{1, 2, 3, 4, 5, 6, 7\} \), and the vertices \( 1, 2, 3, \) and \( 4 \) are exactly the leaves of the tree.

Answer: 150


Question: Determine the number of positive integer solutions \((a, b)\) to the equation \(a^{2014} + 2015\cdot b! = 2014^{2015}\).

Answer: 0


Question: Is a symmetric real matrix similar to a diagonal matrix through an orthogonal matrix?

Answer: Yes


Question: Given a separating graph $G=(V,E)$, does there exist a subset $E_1 \subseteq E$ such that $(V,E_1)$ is minimally separating?

Answer: Yes


Question:
Determine the domain of convergence for the series:
\[ \sum_{n=1}^{\infty} \frac{e^{in}}{(z+1)^n} + \sum_{n=0}^{\infty} \frac{(z+1)^n}{e^{\frac{1}{2}+in}} \quad (z \in \mathbb{C}) \]

Answer: \emptyset


Question: Determine the limit points of the set \( S = \left\{ \left(m+\dfrac{1}{4^{|p|}},n+\dfrac{1}{4^{|q|}}\right) : m,n,p,q \in \mathbb{Z} \right\} \).

Answer: \mathbb{Z} \times \mathbb{Z}


Question: Determine the minimum possible non-integral value of \( a \) such that the graphs of the functions \( f(x) = ax^2 + 2bx + b \) and \( g(x) = 5x^2 - 3bx - a \) intersect at some point for all real values of \( b \).

Answer: \dfrac{5}{26}


Question: If $2^\mu$ and $2^\lambda$ are isomorphic as Boolean algebras, prove that $\mu = \lambda$. A Boolean algebra morphism is a function that preserves suprema and infima.

Answer: \mu = \lambda


Question: Determine if there exists a real number $b$ and natural numbers $n$ and $k$ such that the sum \( \sum_{i = n}^{n+k} b^i \) is a rational number.

Answer: Yes


Question: Let $f(x)$ be a quadratic polynomial with a positive leading coefficient. Define $g(x) = 1 - \frac{2}{x-5}$ with the domain $x < 5$. For any real number $t < 3$, let $h(t)$ be the minimum of $f(g(x))$ for $t \leq x \leq t+2$. It is given that $h(t) = f(g(t+2))$ when $t < 1$, $h(t) = 6$ for $1 \leq t < 3$, and $h(-1) = 7$. Determine the value of $f(5)$. Assume that continuity cannot be used in the solution.

Answer: 42


Question: Evaluate the integral: \[ \int_{0}^{2\pi} e^{R \cos t} \cos(R \sin t + 3t) \, dt \]

Answer: 0


Question: Let \( A \in M_{n}(R) \) and \( f(x) \) be the characteristic polynomial of \( A \). Is it true that \( f'(x) = \sum_{i=1}^{n}\sum_{j=1}^{n}\det(xI-A(i\mid j)) \), where \( A(i\mid j) \) is a submatrix of \( A \) obtained by canceling the \( i \)-th row and \( j \)-th column?

Answer: No


Question: A tank in the shape of a frustum of a cone is 2 feet high, with a base radius of 3 feet and a top radius of 5 feet. The tank is filled with water weighing 62.4 pounds per cubic foot. Calculate the work required to pump all the water to a height of 2 feet above the top of the frustum.

Answer: 5782.4\pi


Question: Suppose \(1 = x_1 < x_2 < x_3 < \cdots\) is an increasing sequence such that \(x_{nm} = x_n x_m\) for all \(n, m\). Determine whether there must exist a constant \(c > 0\) such that \(x_n = n^c\) for all \(n\).

Answer: Yes


Question: Determine the best constant \( c \) such that \( n^{\pi(2n) - \pi(n)} \le c^n \) holds for all positive integers \( n \).

Answer: e


Question:
Consider the nonlinear oscillator described by the differential equation \( \frac{d^2 u}{dt^2} + u = |u|^{p-1}u \) for \( p > 1 \), where the solution \( u(t) \) blows up at time \( T > 0 \), i.e., \( \lim_{t \nearrow T} u(t) = +\infty \). The energy associated with this system is given by \( E = \frac{1}{2}\left(\frac{du}{dt}\right)^2 + \frac{u^2}{2} - \frac{|u|^{p+1}}{p+1} \), which is constant over time.

Determine whether the following limit holds true:
\[ \lim_{t \nearrow T} \frac{u(t)}{C(T-t)^{-\alpha}} = 1, \]
where \( C = \left(2\frac{p+1}{(p-1)^2}\right)^{\frac{1}{p-1}} \) and \( \alpha = \frac{2}{p-1} \).

Answer: 1


Question: Determine whether the following statement is true or false: For a multivariable polynomial $f(x_1,\ldots,x_n) \in \Bbb C[x_1,\ldots,x_n]$, if $f(a_1,\ldots,a_n) = 0$ for some $(a_1,\ldots,a_n) \in \Bbb C^n$, does it follow that $f(x_1,\ldots,x_n) = (x_1-a_1)\cdots(x_n-a_n)g(x_1,\ldots,x_n)$ for some polynomial $g \in \Bbb C[x_1,\ldots,x_n]$?

Answer: B


Question: Does the punctured non-orientable surface of genus $g$ deformation retract to a wedge of $g$ circles? Consider $N_g$, the non-orientable surface of genus $g$, obtained by gluing pairs of adjacent sides in a $2g$-gon. Let $X$ be the space $N_g$ without a point. Determine if $X$ deformation retracts to a wedge of $g$ circles.

Answer: Yes


Question: Let $G$ be a finite group, and let $P$ be a Sylow $p$-subgroup of $G$ such that $P = HK$, where $H$ and $K$ are subgroups of $P$. Suppose $Q$ is a Sylow $q$-subgroup of $G$ and that $H^a Q = Q H^a$ and $K^b Q = Q K^b$ for some $a, b \in G$. Determine if there exists an element $t \in G$ such that $P Q^t = Q^t P$. Provide a justification for your answer.

Answer: Yes


Question: Let \(|a|<1\) and let \((x_k)_{k\ge 1}\) be a sequence that converges to zero. Define a sequence \((y_k)_{k\ge 0}\) by the recurrence relation \(y_k = x_k + ay_{k-1}\). Determine whether \(y_k \to 0\).

Answer: 0


Question:
Consider the equation \( \sin\theta_1 z^3 + \sin\theta_2 z^2 + \sin\theta_3 z + \sin\theta_4 = 3 \), where \( \theta_1, \theta_2, \theta_3, \theta_4 \) are real numbers. Determine which inequality any root \( z \) of this equation satisfies if \( |z| < 1 \):

A) \( |z| < \frac{2}{3} \)

B) \( |z| > \frac{2}{3} \)

C) \( |z| < \frac{1}{2} \)

D) None of these

Answer: B


Question: Evaluate the limit: \[ \lim_{x \to 1} \left(\frac{f(x)}{f(1)}\right)^{1/\log(x)} \] given that \( f(x) \) is differentiable at \( x=1 \) and \( f(1) > 0 \).

Answer: e^{\frac{f'(1)}{f(1)}}


Question: Evaluate the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n} \, d\alpha \) using either Stokes' Theorem or the Divergence Theorem, where \( S \) is the truncated cone defined by \( y=2\sqrt{x^2+z^2} \) with \( 2 \le y \le 4 \), \( \mathbf{n} \) is the outward-pointing normal, and \( \mathbf{F}(x,y,z)=(x,-2y,z) \). Consider whether the end caps of the cone should be included in the evaluation.

Answer: 28\pi


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{x^x}{(2x)!} = 0 \]

Answer: 0


Question: Consider a directional vector \(\boldsymbol{u} \in \mathbb{R}^d\) with \(||\boldsymbol{u}||_2 = 1\). Let \(Q_{\boldsymbol{u}} \in \mathbb{R}^{d \times d}\) be an orthogonal matrix such that \(Q_{\boldsymbol{u}} \boldsymbol{u} = \boldsymbol{e}\), where \(\boldsymbol{e} = \frac{1}{\sqrt{d}} (1, \ldots, 1)\). Define the set \(A = \{ \boldsymbol{x} \in \mathbb{R}^d: Q_{\boldsymbol{u}} \boldsymbol{x} \geq 0 \}\), which is obtained by rotating the non-negative orthant of \(\mathbb{R}^d\) in the direction of \(\boldsymbol{u}\). If \(Q'_{\boldsymbol{u}}\) is another orthogonal matrix with \(Q'_{\boldsymbol{u}} \boldsymbol{u} = \boldsymbol{e}\), does it follow that \(\{ \boldsymbol{x} \in \mathbb{R}^d: Q'_{\boldsymbol{u}} \boldsymbol{x} \geq 0 \} = A\)?

Answer: No


Question: In a Hilbert space $H$, let $T$ be a positive operator on $H$ with $\|T\|_\infty \le 1$. Does the sequence $T^p$ converge to $T$ in the strong operator topology as $p$ decreases to 1?

Answer: Yes


Question: Suppose \(f(z)\) is an analytic function that maps the unit disc into itself and has zeros at \(\frac{i}{2}\) and \(-\frac{i}{2}\). Determine the largest possible value of \(|f(1/2)|\) and provide an example of a function \(f\) that attains this upper bound.

Answer: \dfrac{8}{17}


Question: In an equilateral triangle \( ABC \), point \( P \) is located inside the triangle such that \( AP^2 = BP^2 + CP^2 \). Determine the measure of \( \angle BPC \).

Answer: 150^\circ