Question: Given a bounded self-adjoint operator $T$ on a Hilbert space $H$, the Jordan decomposition provides a unique pair of bounded positive operators $(T_+, T_-)$ such that $T = T_+ - T_-$ and $T_+T_- = 0$. Is $T_+$ the smallest positive operator greater than or equal to $T$? Specifically, if $S$ is a bounded positive operator on $H$ such that $T \leq S$, does it follow that $T_+ \leq S$?

Answer: Yes


Question:
A fair coin is tossed 100 times. Determine which of the following events has the greatest probability:

A. The number of heads is exactly 50.
B. The number of tails is at least 60.
C. The number of heads is between 51 and 55, inclusive.
D. The number of heads is at least 48 and the number of tails is at least 48.
E. The number of heads is 5 or fewer, or 95 or more.

Answer: D


Question: Suppose $X$ is a compact Hausdorff space and $A$ is a subset of $X$. Given a continuous function $f: \overline{A} \to [0,1]$ such that $f(y) = 0$ for each $y \in \overline{A} \setminus A$, and an open set $U$ containing $A$, can $f$ be extended to a continuous function on $X$ that vanishes outside of $U$?

Answer: Yes


Question: Compute the limit: $$\lim_{n\to \infty} \frac{n^n\left(\frac{3}{\sqrt{e}}\right)^{2n}}{n! 9^n}.$$

Answer: 0


Question: Evaluate the integral \( \int_{0}^{\pi} \sin(A\sin(x))\sin(x) \,dx \), where \( A \) is a constant.

Answer: \pi J_1(A)


Question: Let $I$ be an $n$-rectangle in $\mathbb{R}^n$, and let $\mathcal{D}_I$ be the collection of Jordan measurable sets contained in $I$. Suppose $A_n \in \mathcal{D}_I$ are mutually disjoint sets such that $\bigcup_{n=1}^{\infty} A_n \in \mathcal{D}_I$. Is it true that the Jordan content satisfies $c\left(\bigcup_{n=1}^{\infty} A_n\right) = \sum_{n=1}^{\infty} c(A_n)$?

Answer: Yes


Question:
Determine whether the following two statements are equivalent for a real-valued function $f$ defined on $(0,a)$ for some $a>0$:

1. $\lim_{x \to 0^+} f(x) = l$
2. $\lim_{x \to +\infty} f\left(\frac{1}{x}\right) = l$

Answer: Yes


Question: Determine whether the following proposition is true or false: Given two positive and strictly increasing sequences \(a_n\) and \(b_n\), if \(a_{n+1}-a_{n}=c \in \mathbb{R}\) for every \(n\in \mathbb{N}\) and  \(b_{n+1}-b_{n} \rightarrow \infty\), then \(\frac{a_n}{b_n}\) converges to 0.

Answer: True


Question: Evaluate the infinite series \( \sum_{n=2}^{\infty} \frac{2+4+6+\ldots+2n}{1+a_n} \), where \( a_n = 1!\left(1^2+1+1\right) + 2!\left(2^2+2+1\right) + 3!\left(3^2+3+1\right) + \ldots + n!\left(n^2+n+1\right) \).

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{C} \frac{z\cos(z) \, dz}{(z^2 + 5)(z+2)(z-2)} \), where \( C \) is the circle defined by \( x^2 + y^2 = 3 \).

Answer: 0


Question: Find the last 5 digits of the sum: \(1! + 2 \cdot 2! + 3 \cdot 3! + \cdots + 23 \cdot 23! + 24 \cdot 24!\).

Answer: 99999


Question:
Let \((f_n)\) and \((g_n)\) be sequences with exponential generating functions (e.g.f) \(F(t)\) and \(G(t)\) respectively. Prove that the following statements are equivalent:

1. \(g_n = \sum_{k=0}^{n} \binom{n}{k} f_k\)
2. \(G(t) = F(t) e^t\)

Answer: G(t) = F(t) e^t


Question:
Determine the radius of convergence for the Taylor series given by:
\[
1 + \sum_{n=1}^\infty (-1)^n \cdot \frac{(2n)!}{2^{2n} (n!)^2} \cdot (x-1)^{n}.
\]

Answer: 1


Question: Determine whether the following statement is true: If \( f: [a,b] \subset \mathbb{R} \to \mathbb{R} \) is an integrable function, then the inequality \( \left| \int_{a}^{b} f(x)\; dx \right|^2 \leq \int_{a}^{b} |f(x)|^2\; dx \) holds.

Answer: False


Question: Find all values of $N \geq 9$ for which it is possible to write $N$ distinct real numbers on a blackboard, all nonnegative and less than 1, such that for every 8 distinct numbers, there exists a ninth number distinct from the eight, making the sum of these nine numbers an integer.

Answer: 9


Question:
Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables, each following a Uniform(0,1) distribution. The harmonic mean of these variables is defined as:
\[ H_n = \frac{n}{\sum_{i=1}^n \frac{1}{X_i}}. \]
Determine the almost sure limit of $H_n$ as $n \to \infty$. Provide a justification for your answer.

Answer: 0


Question: Find the smallest real number \( k > 0 \) such that for all \( x_1, x_2, x_3 \in \left[0, \frac{1}{2}\right] \) and \( y_1, y_2, y_3 \in (0, \infty) \) with \( x_1 + x_2 + x_3 = y_1 + y_2 + y_3 = 1 \), the inequality \( y_1y_2y_3 \leq k(x_1y_1 + x_2y_2 + x_3y_3) \) holds.

Answer: \dfrac{1}{8}


Question: Calculate the radius of convergence of the power series \( S(x) = \sum_{n=1}^\infty \left(\frac{n^2+4n+3}{5n(n+2)}\right)^n \cdot x^n \).

Answer: 5


Question:
Let \( \alpha \) be a continuous function on \([0,T] \times [0,1]\) where \( T > 0 \) is a fixed positive number. Consider the subspace of \( H^1(0,1) \):
\[ V(t) = \left\{ v \in L^2(0,1) \mid \partial_x v + \alpha(t,x)v \in L^2(0,1), \ t \in [0,T] \right\} \]
Is \( V(t) \) dense in \( H^1(0,1) \)?

Answer: Yes


Question: Determine whether there exists a linear function \( f: \mathrm{End}_K(V) \otimes \mathrm{End}_K(V) \to \mathrm{End}_K(V) \) such that \( f(A \otimes B) = A \circ B \) for all \( A, B \in \mathrm{End}_K(V) \), where \( K \) is a field and \( V \) is a finite-dimensional \( K \)-vector space.

Answer: Yes


Question: Find the last four digits of \(7^{2020}\) using the Chinese Remainder Theorem (CRT).

Answer: 2001


Question: Solve for $x$ in the equation: $$4^{x-1} + \frac{7^x + 8^x + 9^x}{4} = 2016^{x/4}$$

Answer: 0


Question:
Consider the system of differential equations given by:
\[ x' = ax + ay \]
\[ y' = ax - (a + 2b)y \]
where \(a\) and \(b\) are real numbers. Which of the following conditions ensures that each solution \(\begin{pmatrix}x(t)\\y(t)\end{pmatrix}\) satisfies \(\lim_{t\to \infty}x(t)=\lim_{t\to \infty}y(t)=0\)?

(a) \(a^2 + b^2 > 0\)
(b) \(a, b > 0\)
(c) \(-b < a < 0\)
(d) \(a > 0 > b\)

Answer: C


Question:
Evaluate the integral \( \int_{1/2}^1 f(x) \, dx \) where \( f'(x) = \frac{192x^3}{2+\sin^4{\pi x}} \) and \( f\left(\frac{1}{2}\right) = 0 \). Determine which of the following ranges the integral belongs to:

(a) \( m = 13, M = 24 \)
(b) \( m = \frac{1}{4}, M = \frac{1}{2} \)
(c) \( m = -11, M = 0 \)
(d) \( m = 1, M = 12 \)

Answer: D


Question: Find the minimum value of \( q \) such that \( \frac{p}{q} = 0.123456789\ldots \), where the decimal representation is repeating with the first 9 digits after the decimal point being 123456789.

Answer: 111111111


Question: Determine all possible Jordan forms of an $8 \times 8$ matrix given that the minimal polynomial is $t^2(t-1)^3$.

Answer: 9


Question: Let $A \in M_n$ be a nonsingular matrix. Assume $A$ is a nonnegative matrix (i.e., all $a_{ij} \ge 0$). If the inverse matrix $A^{-1}$ is also nonnegative, does it imply that $A$ has exactly one nonzero entry in each column?

Answer: Yes


Question: Evaluate the integral \( \int_{\theta_0}^{\pi} \sqrt{\frac{1-\cos \theta}{\cos \theta_0 - \cos \theta}} \, d\theta \).

Answer: \pi


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

Answer: 1


Question: Calculate the number of ways to distribute 10 different toys among 10 children such that exactly 2 children receive no toys.

Answer: 1360800000


Question: Let $n$ be a sufficiently large integer and $X\subseteq \mathbf{Z}_n$ a finite set of residues modulo $n$ such that $0\in X$ and $2X \neq X$. Fix $y \in 2X\setminus X$ and define $Y=\{x \in X:y\notin X+x\}$. Determine whether $2Y \neq Y$.

Answer: 2Y \neq Y


Question: Determine the cardinality of the set of countable ordinals. Is it \(\aleph_0\) or \(\aleph_1\)?

Answer: \aleph_1


Question: Determine the number of prime numbers of the form \(3^n + n^3\).

Answer: 1


Question: Compute the limit: \[ \lim_{n\to\infty}\int_0^\infty\frac{n^2[\cos(x/n^2)-1]}{1+x^3}dx. \]

Answer: 0


Question: Calculate the value of the infinite series: \[ \sum_{k=2}^\infty \frac{k^2+3k}{k!} \]

Answer: 5e - 4


Question: Evaluate \( \lim_{x\to \infty} \frac{ \cot^{-1} (\sqrt {x+1} -\sqrt x)}{\sec^{-1} \left(\frac{2x+1}{x-1}\right)^x} \).

Answer: 1


Question: Evaluate the improper integral: \[ \int_0^\pi \frac{\sin x}{\sqrt{|\cos x|}} \, dx \]

Answer: 4


Question: Evaluate the line integral \( \int_{C} (x^{4}+5y^{2})\,dx + (xy-y^{5})\,dy \), where \( C \) is the polygonal path joining the vertices \( (-1, 0), (4, 0), (6, 4), (4, 8), (-1, 8), \) and \( (-1, 0) \), oriented counterclockwise.

Answer: -1728


Question:
Let \(x\) and \(y\) be complex numbers such that \(|x| = |y| = 1\). Find the maximum value of the expression:
\[
|1+x| + |1+xy| + |1+xy^2| + \cdots + |1+xy^{2013}| - 1007|1+y|.
\]

Answer: 2014\sqrt{2}


Question: For any given \( m \in [0,1] \), does there exist a dense subset \( A \subseteq [0,1] \) with Lebesgue measure \( m \)?

Answer: Yes


Question: Evaluate the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) where \( \mathbf{F}(x, y, z) = \langle -y^2, x, z^2 \rangle \) and \( C \) is the curve of intersection of the plane \( y + z = 2 \) and the cylinder \( x^2 + y^2 = 1 \). Orient \( C \) counterclockwise when viewed from above.

Answer: \pi


Question: Calculate the line integral of the vector field \( \vec{F} = (2xy + z^3)\hat{\text{i}} + (x^2)\hat{\text{j}} + (3xz^2)\hat{\text{k}} \) from the point (1,-2,1) to the point (3,1,4). The vector field is conservative with the potential function \( \phi = x^2y + xz^3 \).

Answer: 202


Question: Determine the value of \( k \) such that the parabola \( u(x) = x + kx(2-x) \), passing through the points \((0,0)\) and \((2,2)\), generates a solid of revolution with the least possible volume when revolved about the x-axis between \( x = 0 \) and \( x = 2 \).

Answer: -\dfrac{5}{4}


Question: Determine whether there exists a pair of distinct odd prime numbers \(p\) and \(q\) such that \(p^q + q^p + 1 \equiv p + q + 1 \pmod{pq-1}\).

Answer: Yes


Question: Find the sum of the series \( \binom{n}{0}+\binom{n-1}{1}+\binom{n-2}{2}+ \ldots + \binom{\frac{n+1}{2}}{\frac{n-1}{2}} \) for an odd integer \( n \). Express the result in terms of \( n \) and explore its connection to the Fibonacci sequence.

Answer: F_{n+1}


Question: Given an $n_1 \times n_1$ matrix $E$ and a $p_1 \times n_1$ matrix $C$, it is known that the rank of the matrix \( \begin{bmatrix} I_{n_1} - E \\ C \end{bmatrix} \) is equal to the rank of $C$. Prove that the rank of the matrix \( \begin{bmatrix} E \\ C \end{bmatrix} \) is $n_1$.

Answer: n_1


Question:
Let $n$ be a positive integer and let $P_1, \dots, P_n \in M_n(\mathbb{C})$ be a collection of positive semidefinite $n \times n$ matrices satisfying the following properties:

- $\operatorname{Tr}P_i = 1$ for each $i \in \{1, \dots, n\}$
- $P_iP_j = 0$ whenever $i \neq j$.

Is it necessarily the case that there is an orthonormal basis $u_1, \dots, u_n$ such that $P_i = u_i u_i^*$ for each $i \in \{1, \dots, n\}$?

Answer: Yes


Question: Determine the kernel of the homomorphism \( \phi : \mathbb{R}[X] \to M_{2 \times 2}(\mathbb{R}) \) defined by \( \phi\left(\sum_{i=0}^d r_i X^i\right) = \sum_{i=0}^d r_i \mathbf{A}^i \), where \( \mathbf{A} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \), without using the Cayley-Hamilton Theorem.

Answer: (X^2 + 1)


Question: Determine the number of field homomorphisms from \( \mathbb{Q}(\sqrt[3]{2}) \) to \( \mathbb{K} \), where \( \mathbb{K} \) is the splitting field of \( x^3 - 2 \).

Answer: 3


Question: Given $X_1, X_2, \ldots, X_{100}$ are correlated normal random variables with $E(X_i) = 0$, $Cov(X_i, X_j) = 0.2 \times 2$ for $i \neq j$, and $Var(X_i) = 2$. Find the covariance matrix of the multivariate normal distribution for $X = (X_1, X_2, \ldots, X_{100})'$ and determine the variance of $\sum_{i=1}^{100} X_i$. Assume $\rho = 0.2$ and $\sigma^2 = 2$. What is the variance of the sum?

Answer: 4160


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

Answer: e


Question: Consider a number $n$ randomly selected from the first 100 positive integers. Let $f(n)$ denote the product of all the positive divisors of $n$, and let $\varphi(n)$ denote the sum of all the positive divisors of $n$. What is the probability that $f(n) - \varphi(n)$ is an even number?

Answer: \dfrac{43}{100}


Question: Find the smallest natural number \( n \) such that its decimal representation ends with the digit 6, and if the 6 is moved to the beginning of the number, the resulting number is 4 times the original number \( n \).

Answer: 153846


Question: Calculate the mass of the hemisphere with radius 1, centered along the z-axis such that the south pole is at the origin, using the mass function \( p(x,y,z) = 1-z \).

Answer: \dfrac{\pi}{4}


Question: Consider the function \( f : [-\pi, \pi] \to [-\pi,\pi] \) defined by \( f(x)= \frac{\sin(x)}{2} \). This function is a contraction. Using the fixed point iteration \( x_{k+1}= f(x_{k}) \) starting from \( x_{0}=\frac{\pi}{2} \), determine the minimum number of iterations \( k \) required such that the error satisfies \(|x_{k}-x^{*}| \leq \frac{1}{1024}\), where \( x^{*} \) is the fixed point.

Answer: 10


Question: Let $f(x)$ be a polynomial such that $f(\alpha)^2 + f'(\alpha)^2 = 0$. Evaluate the limit: \[ \lim_{x \to \alpha} \frac{f(x)}{f'(x)} \left\lfloor \frac{f'(x)}{f(x)} \right\rfloor. \]

Answer: 1


Question: Given that $m > n$, determine which is larger: $(n!)^m$ or $(m!)^n$?

Answer: (m!)^n


Question: Let $T$ be an invertible linear operator on a finite-dimensional vector space over a finite field. Is it true that there exists a positive integer $n$ such that $T^n = I$, where $I$ is the identity operator?

Answer: Yes


Question: Find the volume of the region bounded by the plane \( z = 2y \) and the paraboloid \( z = x^2 + y^2 \). The expected volume is \( \frac{\pi}{2} \).

Answer: \dfrac{\pi}{2}


Question: Suppose the number of homomorphisms from a group $G$ to an abelian group $H$ is $n$. Determine the number of homomorphisms from the direct sum $G \oplus G \oplus \cdots \oplus G$ (with $s$ copies of $G$) to $H$.

Answer: n^s


Question: Determine the kernel of the operator $A : L^2(\Omega) \to L^2(\Omega)$ defined by $Af = xf$ for $f \in L^2(\Omega)$, where $\Omega \subset \mathbb{R}$ is a bounded subset. Describe the kernel of $A$ and discuss whether the coefficient $x$ affects the kernel.

Answer: \{0\}


Question:
Find the product abc for positive integers a, b, and c that satisfy the following system of equations:

1. \(-10(c-2ab)+\frac{b-5}{a}=52\)
2. \(a-b+c=53\)
3. \(a(b+7)=54\)

Answer: 2013


Question: Let \( G \) be a closed and connected subgroup of \( \mathrm{U}(n) \). Is \( G \) defined by polynomial equations in the real and imaginary parts of the matrix entries?

Answer: Yes


Question: Calculate the triple integral \( \iiint_{x^2+y^2+z^2\leq z} \sqrt{x^2+y^2+z^2} \, dx \, dy \, dz \) using spherical coordinates. Determine the correct limits for \( \rho \), \( \phi \), and \( \theta \) to evaluate the integral.

Answer: \dfrac{\pi}{10}


Question: Find the value of \(x\) that satisfies the equation \(x = \ln(x+1)\).

Answer: 0


Question: Find a closed formula for the series \( \sum_{n\geq 0}{\frac{n^2+4n+5}{n!}} \).

Answer: 11e


Question: Compute the product of the following integrals: \( \int_0^1 \frac{\mathrm{d}x}{\sqrt{1-x^4}} \) and \( \int_0^1 \frac{x^2\mathrm{d}x}{\sqrt{1-x^4}} \).

Answer: \dfrac{\pi}{4}


Question: Find the sum of the series \( \sum_{n=0}^\infty \frac{3n^2 -1}{(n+1)!} \).

Answer: 2e - 2


Question: Given a coin where the probability of getting a head, $P(\text{Head})=x$, is a uniform random variable over the interval $[0, 1]$. After flipping the coin once and observing a head, determine the expected value of $P(\text{Head})$.

Answer: \dfrac{2}{3}


Question: Calculate the limit: \[ \lim_{x \to \infty} \frac{\Gamma(x+1)}{\Gamma(x+1+1/x^2)} \]

Answer: 1


Question: A tank is full of oil weighing 20 lb/ft³. The tank is an inverted right rectangular pyramid with a base width of 2 feet, a depth of 2 feet, and a height of 5 feet. Calculate the work required to pump the oil to a height of 3 feet above the top of the tank.

Answer: \dfrac{1700}{3}


Question: Let $U_1, U_2, \ldots, U_n$ be a sequence of independent random variables such that for every $i$, $P(U_i=1) = P(U_i=-1) = \frac{1}{2}$. Define $X_n = \sum_{i=1}^{n} U_i$. For $m \geq n$, find $E(U_n \mid X_m)$. Provide your answer in terms of $m$ and $n$.

Answer: \dfrac{X_m}{m}


Question: Let \(X_{(1)}, X_{(2)}, X_{(3)}\) be the order statistics of three independent random variables \(X_1, X_2, X_3\) each uniformly distributed on \((0,1)\). Calculate the probability \(P\left(X_{(1)} + X_{(2)} \leq 1\right)\).

Answer: \dfrac{3}{4}


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{x\sin(\sin x) - \sin^2 x}{x^6} \]

Answer: \dfrac{1}{18}


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{x^2}{x+\sin \left(\frac{1}{x}\right)} \] without using any series or expansion.

Answer: 0


Question: Prove that \( \left(\frac{1}{n!}\right)^{1/n} \to 0 \) as \( n \to \infty \).

Answer: 0


Question: Let \( \alpha \) be the closed curve along the square with vertices at \( 1, i, -1, -i \). Calculate \( \frac{1}{2\pi i}\int_\alpha\frac{dz}{z} \).

Answer: 1


Question: Evaluate the limit: $$ \lim_{n \to \infty } \frac{\ln(1+n)-\ln(n^{2})}{\sin(1/n)} $$

Answer: -\infty


Question: Calculate the volume of the set \( M = \{(x,y,z) \in \mathbb{R}^3 : x^2+y^2-z^2 \leq 1, \ 0 \leq z \leq 3\} \).

Answer: 12\pi


Question: Find the limit in distribution of the sequence of random variables \(Z_n = \frac{1}{n} X + \left(1 - \frac{1}{n}\right) Y\), where \(X\) and \(Y\) are random variables.

Answer: Y


Question: Find the limit: \[ \lim_{n\to \infty} \left(1+\frac{\sqrt[n]{a}-1}{b}\right)^n \] where \( a, b \in \mathbb{R} \).

Answer: a^{1/b}


Question: Find all $C^{1}$ functions $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $f(x + iy) = u(x,y) + iu(x,y)$ is holomorphic. Determine if the only such functions are constant functions.

Answer: u(x, y) = C


Question: Let \( K \) be an \( n \)-dimensional vector subspace of \( \mathbb{R}^N \). Given that \( \mathbb{R}^N \cong (\mathbb{R}^N/K) \oplus_e K \), determine if \( \mathbb{R}^N = H \oplus_i K \) for some \( (N-n) \)-dimensional vector subspace \( H \subseteq \mathbb{R}^N \).

Answer: Yes


Question: Given that $f(z)$ satisfies the condition $\Im (f'(z)) = 6x(2y-1)$, with $f(0) = 3-2i$ and $f(1) = 6-5i$, find the value of $f(1+i)$. Provide your answer in the form $a + bi$.

Answer: 6 + 3i


Question: Let $V$ be a finite-dimensional vector space, and let $A$ and $B$ be two linear transformations on $V$ such that $A^2 = B^2 = 0$ and $AB + BA = I$. Determine the dimension of $V$. \( \text{(Express your answer as an integer.)} \)

Answer: 2


Question: Find the sum of the series \(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{12} + \cdots\), where the terms are the reciprocals of positive integers whose only prime factors are 2 and 3.

Answer: 3


Question: Consider the 3-torus \( T = S^1 \times S^1 \times S^1 \). Let \( \alpha \) be the isotopy class of the loop \( S^1 \times \{pt\} \times \{pt\} \). Define \( \mathcal{F}_\alpha \) as the set of all smooth oriented foliations \( F \) of \( T \) by circles such that every leaf of \( F \) lies in the class \( \alpha \). Is the set \( \mathcal{F}_\alpha \) connected?

Answer: Yes


Question: Let $R$ be a ring and $S \subset R$ be a proper subset such that $0_R, 1_R \in S$. Define $I = \langle R \setminus S \rangle$, the ideal generated by elements not in $S$. Is it possible for some left ideal $Rx$ to be contained in $I$ for some nonzero $x \in S$?

Answer: Yes


Question: If $PQRSPQRS$ is an eight-digit number with 56 divisors, find the number of divisors of $PQRS$. Here, $PQRS$ is a four-digit number where $P, Q, R, S$ are the digits of the number. Determine the number of divisors of $PQRS$ (including 1 and $PQRS$).

Answer: 14


Question:
Given a complete weighted graph G and a real number R, a machine can determine if G has a Hamiltonian tour with weight at most R. Using this machine, how many times must you call it to find a Hamiltonian tour with minimum weight in G, assuming the sum of the edges in G is up to M? Choose the correct answer:

1) We cannot do this, because there is an uncountable state.

2) O(|E|) times

3) O(log M) times

4) Because A is NP-Hard, this cannot be done.

Answer: 3


Question: Three independent and identically distributed random variables $X_1, X_2, X_3$ have a probability density function given by $f(x) = 3x^2$ for $0 < x < 1$. Determine the probability density function of the median of these three variables.

Answer: 18x^5(1 - x^3)


Question: Evaluate the integral: \[ \int_0^\infty x^{-\frac{1}{2}}e^{-\frac{x}{2}}\,dx \]

Answer: \sqrt{2\pi}


Question: Compute the residue at \( z = 0 \) of the function \( \frac{z^{n-1}}{\sin^n z} \) for \( n \in \mathbb{N} \).

Answer: 1


Question: Let \( p\colon E\to B \) be a fibration with simply connected fibers that are homotopy equivalent to a compact CW-complex. Must the map \( p_*\colon H_3(E;\mathbb{Q})\to H_3(B;\mathbb{Q}) \) be surjective?

Answer: No


Question: Find the positive value of \( a \) such that the remainder of \( x^3 - x^2 + ax \) when divided by \( x+a \) is twice the remainder when divided by \( x-2a \).

Answer: \dfrac{2}{17}


Question: Evaluate the Lebesgue integral \( \int_{[0,1]} f(x) \, du \), where \( f(x) = 0 \) on the Cantor set \( C \) and \( f(x) = x \) on \( [0,1] \setminus C \).

Answer: \dfrac{1}{2}


Question:
Consider a Markov chain \( \{X_0, X_1, \ldots \} \) with state space \( S \). For any \( n \geq 1 \) and \( i_0, i_1, \ldots, i_n \in S \), determine which of the following conditional probabilities is equal to \( P(X_0 = i_0|X_1 = i_1, \ldots, X_n = i_n) \):

1. \( P(X_0 = i_0|X_1 = i_1) \)
2. \( P(X_0 = i_0|X_n = i_n) \)

Choose the correct option.

Answer: 1


Question: Let \( f \in L^p(\mathbb{R}) \) be a continuous function for some \( 1 \leq p < \infty \). Does it follow that \( \lim_{|x| \to \infty} f(x) = 0 \)? Justify your answer.

Answer: No


Question: Compute the degree of the field extension \([\mathbb{Q}(\sqrt{3}, \sqrt{3 + \sqrt{3}}) : \mathbb{Q}]\).

Answer: 4


Question: Compute the limit: $$\lim_{x \to 0} \int_{0}^{x} \frac{(x-t)f(t)}{x^2} \,dt$$ where $f(x)$ is a continuous function with domain $\mathbb{R}$. Use the Fundamental Theorem of Calculus if applicable.

Answer: \dfrac{f(0)}{2}


Question: Consider a sequence of open sets $A_n \subseteq [0,1]^k$ in the Euclidean topology of $\mathbb{R}^k$ such that $A_{n+1} \subset A_n$ for all positive integers $n$. If the limiting set $A = \lim_{n \rightarrow \infty} A_n$ has an empty interior, i.e., $A^\circ = \emptyset$, is $A$ necessarily a zero measure set?

Answer: No


Question: Let \(\Gamma\) be the set of functions \(f_n: \mathbb{C} \to \mathbb{C}\) defined by \(f_n(z) = \frac{\pi^2}{24}z^2 - \frac{1}{n^2}z + \frac{589}{20} \cdot \frac{1}{n^3}\) for \(n \in \mathbb{N}\). Let \(\Lambda\) be the set of all roots with positive imaginary part of the elements of \(\Gamma\). Find the sum of all elements of \(\Lambda\), rounding the real and imaginary parts to the nearest integer.

Answer: 2 + 22i


Question: Let $X$ be a connected topological space, and let $\pi : Y \rightarrow X$ be a surjective covering space map. Suppose the group of deck transformations of $\pi$ contains a subgroup $\mathbb{Z}_p$, where $p$ is a prime number, such that $\mathbb{Z}_p$ acts freely and transitively on the fibers of $\pi$. If $Y$ is not connected, is $Y$ necessarily homeomorphic to a disjoint union of $p$ copies of $X$?

Answer: Yes


Question:
Consider the stochastic processes $X$ and $X^{\varepsilon}$, for $\varepsilon > 0$, satisfying the following stochastic differential equations (SDEs) on $[0, T]$:
\[ dX = \mu(t, X_t) \, dt + \sigma(t, X_t) \, dW_t, \]
\[ dX^{\varepsilon} = \mu(t, X^{\varepsilon}_t) \, dt + \sigma_{\varepsilon}(t, X^{\varepsilon}_t) \, dW_t, \]
with initial conditions $X_0 = X^{\varepsilon}_0 = x_0$, where $x_0 \in \mathbb{R}$. The functions $\mu$, $\sigma$, and $\sigma_{\varepsilon}$ are uniformly bounded and Lipschitz continuous with a uniform Lipschitz constant. Assume that for every $\varepsilon > 0$, $\sigma(t, X_t) = \sigma_{\varepsilon}(t, X^{\varepsilon}_t)$ for all $t$ in a set of Lebesgue measure $T - \varepsilon$, which may depend on the random outcome $\omega$.

Determine whether $X^{\varepsilon}$ converges to $X$ uniformly in $t$ in probability, i.e., whether
\[ \lim_{\varepsilon \to 0^+} \mathbb{P}\left(\sup_{t \in [0, T]} |X_t - X^{\varepsilon}_t| > \delta\right) = 0 \]
for all $\delta > 0$. Provide a justification for your answer.

Answer: Yes


Question: Suppose $G$ is an affine algebraic group acting on $\mathbb{C}\mathbb{P}^2$. Is the action of $G$ on $\mathbb{C}\mathbb{P}^2$ equivalent to the action of a closed subgroup of $\mathbb{P}\mathbb{G}\mathbb{L}(3,\mathbb{C})$ on $\mathbb{C}\mathbb{P}^2$?

Answer: Yes


Question: Three-bit signals are transmitted through a noisy channel where each bit is flipped (1 to 0 or 0 to 1) with a probability of \( \frac{1}{3} \), independently of the other bits. What is the expected number of signals that are transmitted without any bit being flipped before the first signal with a flipped bit occurs?

Answer: \dfrac{8}{19}


Question: Compute the Lebesgue integral \( \int_E \sin(x) \, d\lambda \) where \( E = [0, \pi] \cup [2\pi, 3\pi] \cup \mathbb{Q} \subset \mathbb{R} \) and \( \lambda \) is the Lebesgue measure on \( \mathbb{R} \).

Answer: 4


Question:
Find the limit as \( h \) approaches 0 of the expression:
\[ \frac{e^{x(h+2)^2} - e^{4x}}{h} \]

Answer: 4x e^{4x}


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

Answer: 0


Question: Given the complex number \( w \) with modulus \( r \), and the equation \( \frac{500}{w} = 3|w| + 40i \), find the value of \( r \) using a non-calculator method.

Answer: 10


Question: Calculate the radius $r$ of a circle that is halfway between being not bent at all (with radius $r = a$) and fully bent (with radius $r = \infty$). Assume the bending is uniform.

Answer: 2a


Question: Let $f:[0,1]\to\mathbb{R}$ be a continuous function such that the map $[0,1]\to \mathbb{R}^2$, given by $t\mapsto e^{2\pi if(t)}$, is smooth ($C^\infty$). Is $f$ also smooth?

Answer: Yes


Question: Find the surface area of the portion of a sphere with radius $R=4$, centered at the origin, that lies inside the cylinder $x^2 + y^2 = 12$ and above the $xy$-plane.

Answer: 16\pi


Question: Find the inverse $z$-transform of the function \( \frac{z}{z+1} \).

Answer: (-1)^n u[n]


Question: Find the maximum value of the function \( f(x) = \frac{e^{\frac{2x}{x+1}} - 1}{x} \) for \( x \ge 0 \).

Answer: 2


Question: Calculate the expected number of failures before the first success, given that there is exactly one success in 15 Bernoulli trials. Express your answer without the parameter \( p \).

Answer: 7


Question: Find the average value of the function \( f(x,y) = x^2 y \) over the triangular region with vertices \((1,1), (2,0), (0,1)\).

Answer: \dfrac{3}{5}


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

Answer: 1


Question: Two poles, each 50 meters tall, are connected by an 80-meter rope that hangs between them. The rope is 20 meters above the ground at its lowest point. Determine the distance between the two poles.

Answer: \dfrac{70}{3} \ln 7


Question: For a matrix $A \in \mathbb{C}^{n \times n}$, determine the maximum number of linearly independent eigenvectors that can be associated with a single eigenvalue.

Answer: n


Question: Given that \( S = \sum_{n=1}^{\infty} \frac{\sin(n)}{n} \) converges, find the value of \( 2S + 1 \).

Answer: \pi


Question: Determine the maximal interval of existence for the differential equation \( y^{\prime}=-\frac{4}{x^{2}}-\frac{1}{x} y+y^{2} \) with the initial condition \( y(1)=1 \), without solving the equation.

Answer: (0, \infty)


Question: Given a sequence \( \{x_n\} \) of real numbers defined by \( x_1 = \frac{2014}{3} \) and \( x_{n+1} = x_n \left(1 + \frac{1}{x_n - \lfloor x_n \rfloor}\right) \) for \( n \in \mathbb{N} \), find \( \lim_{n \to \infty} \left( \sum_{r=1}^n \frac{1}{x_r} \right) \).

Answer: \dfrac{2}{1007}


Question:
Which of the following properties can hold for a map \( f: \mathbb{Z} \to \mathbb{Q} \)?

1) \( f \) is bijective and increasing.
2) \( f \) is onto and decreasing.
3) \( f \) is bijective and \( f(n) \ge 0 \) if \( n \le 0 \).
4) \( f \) has an uncountable image.

Determine the correct property.

Answer: 3


Question: Calculate the volume of the set \(E=\{(x,y,z) \in \mathbb{R}^3:0<y<\pi,z^2+(x-y^5)^2<\sin y\}\).

Answer: 2\pi


Question: Find the remainder when the number formed by concatenating all integers from 101 to 998 is divided by 990.

Answer: 548


Question: Let $(\Omega,\mathcal{A},P)$ be a probability space, and let $(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables that are not integrable. For a fixed $K>0$, define $C_n:=\{|X_n| \geq nK\}$ for each $n \in \mathbb{N}$. Determine whether $P\left(\limsup_{n \to \infty} C_n \right)=1$.

Answer: 1


Question: Find the maximum value of \( k \) such that the inequality \( y^2 - xy + x + 2y + 5 \geq k(\sqrt{3x} + \sqrt{y}) \) holds for all \( x \) and \( y \) in the range \( 0 \leq x, y \leq 3 \).

Answer: 2


Question: Determine whether there exist functions that are $S$-differentiable but not differentiable in the usual sense.

Answer: Yes


Question: A ball with mass 0.15 kg is thrown upward with an initial velocity of 20 m/s from the roof of a building 30 m high. The force due to air resistance is given by \(|v|/30\), directed opposite to the velocity, where \(v\) is the velocity in m/s. Determine the time at which the ball hits the ground.

Answer: 5.13


Question: Given a pentagon ABCDE with a fixed perimeter \( P \) cm, where \( \triangle ABE \) is an equilateral triangle and \( BCDE \) is a rectangle, find the value of \( \frac{P}{x} \) that maximizes the area of the pentagon if the length of \( AB \) is \( x \) cm.

Answer: 6 - \sqrt{3}


Question: An investor needs to cover a liability of $1000 in two years. The investor can invest in a one-year pure discount bond and a three-year pure discount bond, both with a face value of $100. Assume the yield to maturity is 10%. How many units of the three-year pure discount bond must the investor purchase to ensure the liability is covered, even if the yield to maturity changes in the second year? Provide your answer as a decimal number.

Answer: 5.5


Question: Calculate the flux integral of the vector field across the circle $C$ defined by $x^2 + y^2 = a^2$. Given the vector field $f(x,y) = \ln(x^2 + y^2)$, find the value of the integral \( \int_{C} \text{grad} f \cdot n \, ds \).

Answer: 4\pi


Question: In a group $G$, suppose $a^5 = e$ and $aba^{-1} = b^2$ for some elements $a$ and $b$ in $G$. Determine the order of the element $b$. Here, $e$ denotes the identity element in $G$.

Answer: 31


Question: Find the maximum value of \( \frac{ab+ac+bc}{a^2+b^2+c^2} \) for real numbers \(a, b, c\) such that \(abc \neq 0\).

Answer: 1


Question: If $A=\prod_{n=1}^{\infty}M_{k(n)}(\mathbb{C})$ and $x=(x_1, x_2, \ldots)$ is strictly positive in $A$, does this imply that each $x_n \in M_{k(n)}(\mathbb{C})$ is a positive definite matrix?

Answer: Yes


Question: Let $E$ be a compact subset of $\mathbb{R}^n$. Consider a density function $\phi(x,y)$ that is Lipschitz continuous and satisfies \( \int\limits_E \phi(x,y)dy=1 \) for all $x\in E$. Given a non-increasing sequence of non-empty compact sets $A_n$ such that for all $x\in A_{n+1}$, \( \int\limits_{A_n} \phi(x,y)dy=1 \), and the limit set $A = \bigcap\limits_n A_n$, determine if for all $x\in A$, \( \int\limits_A \phi(x,y)dy = 1 \).

Answer: 1


Question:
Given that \( \alpha, \beta \) are two real roots of the quadratic equation \( ax^2 + bx + c = 0 \) and \( \alpha + \beta, \alpha^2 + \beta^2, \alpha^3 + \beta^3 \) are in geometric progression, determine which of the following is correct about the discriminant \( \Delta = b^2 - 4ac \):

(a) \( \Delta \neq 0 \)
(b) \( b\Delta = 0 \)
(c) \( c\Delta = 0 \)
(d) \( \Delta = 0 \)

Answer: C


Question: Evaluate \( \Delta z \) along the path \( y = x^2 \) from the point \((0,0)\) to \((1,1)\). Given the differential \( dz = xydx + xydy \), find the change in \( z \) as you move from \((0,0)\) to \((1,1)\) along the path \( y = x^2 \).

Answer: \dfrac{13}{20}


Question: Compute the derivative of the function \( f(x) = \left(x^2 - \ln^2(x+1)\right)^{\frac{1}{3}} \) at \( x = 0 \) for \( x > -1 \).

Answer: 1


Question: Evaluate the integral \( I_C = \frac{1}{2\pi}\int_{0}^{2\pi}\left(\frac{d}{d\theta}\phi(\theta)\right) d\theta \), where \( \phi(\theta) = \arctan\left[\frac{3\cos(\theta)}{4(\cos(\theta)+\sin(\theta))}\right] \).

Answer: -1


Question: Evaluate the integral \( \int_{0}^{\pi}\ln (1+\cos x)\, dx \).

Answer: -\pi \ln 2


Question: Evaluate the integral \( \int_{\gamma} \frac{1}{z} \, dz \) where \( \gamma \) is the unit circle in the complex plane.

Answer: 2\pi i


Question: Let \(\sum a_n\) be a convergent series of positive terms. Given that \(\lim \inf (na_n) = 0\), prove that if \(\{a_n\}\) is a decreasing sequence, then \(\lim (na_n) = 0\).

Answer: 0


Question: Let \( f \in L_1([0,1], \lambda) \) be a function such that \( \int_0^1 |f|^2 \, dx \leq M \). Does there exist a function \( g \in L_1([0,1], \lambda) \) such that \( \|f-g\|_{L_1} < \epsilon \) and \( \int_0^1 |g|^2 \, dx > M \)?

Answer: Yes


Question: Evaluate the integral \( \int_\gamma e^{z}dz \) where the path \( \gamma(t) = (t, \sin(t)) \) for \( t \in [0, \pi] \).

Answer: e^{\pi} - 1


Question: Calculate the integral of the 2-form \( w = x \, dx \wedge dy + x \, dy \wedge dz - dx \wedge dz \) over the lower hemisphere (\( z \leq 0 \)) of the ellipsoid \( \frac{x^2}{9} + \frac{y^2}{9} + z^2 = 1 \), with the orientation determined by the normal vectors having a negative third component. Use the parametrization \( x = 3\cos\theta\sin\phi, \; y = 3\sin\theta\sin\phi, \; z = \cos\phi \) where \( (\theta, \phi) \in [0, 2\pi] \times [\frac{\pi}{2}, \frac{3\pi}{2}] \).

Answer: 0


Question:
Given two 3x3 real matrices A and B, where:

1. $A^2 - B^2$ is an invertible matrix,
2. $A^5 = B^5$,
3. $A^3B^2 = A^2B^3$,

find the determinant of the matrix $A^3 + B^3$. \( \det(A^3 + B^3) \).

Answer: 0


Question: Adanna grows apples, oranges, avocados, persimmons, and cherymoia in her backyard. She has 10 persimmons and 1 cherymoia, but an unlimited supply of apples, oranges, and avocados. Adanna wants to give 14 pieces of fruit to her neighbor Farida, ensuring that the number of oranges is even. In how many ways can Adanna distribute the fruit to Farida?

Answer: 660


Question: Given the recursive function \( f(n+1) = (-1)^{n+1}n - 2f(n) \) with the condition \( f(1) = f(1986) \), find the sum \( f(1) + f(2) + f(3) + \ldots + f(1985) \).

Answer: 331


Question:
Determine the number of solutions, counting multiplicity, in the domain \(\{z \in \mathbb{C} : 1 < |z| < 2\}\) for the equation:
\[z^9 + z^5 - 8z^3 + 2z + 1 = 0\]

Answer: 6


Question:
If \( p = (8+3\sqrt{7})^m \) and \( f = p - \lfloor p \rfloor \) where \( \lfloor p \rfloor \) denotes the greatest integer function, find the value of \( p(1-f) \). Choose the correct answer from the following options:

(a) 1
(b) 2
(c) \( 2^n \)
(d) \( 2^{2n} \)

Answer: A


Question:
Consider the group homomorphism \( \phi:M_2(\mathbb{R}) \to \mathbb{R} \) given by \( \phi(A) = \text{trace}(A) \). The kernel of \( \phi \) is isomorphic to which of the following groups?

(a) \( M_2(\mathbb{R})/ \{A \in M_2(\mathbb{R}):\phi(A)=0\} \)
(b) \( \mathbb{R}^2 \)
(c) \( \mathbb{R}^3 \)
(d) \( GL_2(\mathbb{R}) \)

Answer: C


Question: Given the polynomial $f = X^6 - 6 \in \Bbb Q[X]$, determine if the information that a splitting field for $f$ over $\Bbb Q$ is $L = \Bbb Q(\zeta_3, \sqrt[6]{6})$ and that $L$ is of degree 12 over $\Bbb Q$ is sufficient to identify the Galois group of $f$. Justify your answer.

Answer: D_6


Question: Determine whether there is only one function \( f: \mathbb{Z}^+ \to \mathbb{Z}^+ \) that satisfies the equation \( f(n) + 2f(f(n)) = 3n + 5 \) for all positive integers \( n \), or if there are multiple functions that satisfy this relation.

Answer: f(n) = n + 1


Question: Given a random sample $X_1, X_2, \ldots, X_{400}$ from a distribution with median $m$ (i.e., $P(X_i \le m) = 0.5$), calculate the probability $P(X_{220:400} \le m)$, where $X_{220:400}$ denotes the 220th order statistic in the sample of 400 observations.

Answer: 0.0256


Question: Consider a projective variety $X \subseteq \mathbb{P}^n$. The degree of $X$ can be defined in terms of the maximal number of intersections with projectivisations $L = \mathbb{P}(\hat{L})$ of linear varieties $\hat{L} \in \mathrm{Gr}(n+1-\dim X, n+1)$, or in terms of the leading coefficient $(\deg X/(\dim X)!)\lambda^{\dim X}$ of the Hilbert polynomial $p_X(\lambda)$. If $X$ is embedded into $\mathbb{P}^{n+1}$ in a 'trivial' way, does this embedding leave the degree of $X$ unchanged? For instance, consider the embedding of $v_2(\mathbb{P}^1) = \{[\lambda^2:(\lambda\mu)^2:\mu^2] \mid \lambda,\mu \neq 0\} \cup \{[1:0:0],[0:0:1]\} \subset \mathbb{P}^2$ into $\mathbb{P}^3$ as $v_2(\mathbb{P}^1) \cong \{[\lambda^2:(\lambda\mu)^2:\mu^2:\mu^2] \mid \lambda,\mu \neq 0\} \cup \{[1:0:0:0],[0:0:1:1]\} \subset \mathbb{P}^3$. Does this embedding change the degree of the variety?

Answer: 2


Question: Evaluate the complex integral \( \oint_L \frac{\cos^2 z}{z^2} \, dz \), where \( L \) is the closed loop that goes counterclockwise around the square with vertices \(-1\), \(-i\), \(1\), and \(i\).

Answer: 0


Question: Derive a reduction formula for the improper integral \( I_n = \int_{0}^{1} x^3(\ln x)^n \, dx \). Then, use this formula to evaluate \( I_4 \).

Answer: \dfrac{3}{128}


Question: Evaluate the limit: \[ \lim_{x \rightarrow 0^{+}} \frac{1}{9}\left(\frac{1}{\tan^{-1}x} - \frac{1}{x}\right) \]

Answer: 0


Question: Evaluate the integral \( \int_{|z|=2} \frac{dz}{z\sqrt{1-\frac{1}{z^2}}} \) over the circle \(|z|=2\), considering the main branch of the square root.

Answer: 2\pi i


Question: Given $n$ different wine bottles numbered $i=1, 2, \ldots, n$, where the $i$-th bottle is $i$ years old and is still good with a probability of $1/i$, find the expected value of the age of a randomly selected bottle that is still good.

Answer: \dfrac{n}{H_n}


Question: Find a solution $u(x,y)$ of Laplace’s equation on the domain $-\infty < x < \infty$ and $0 < y < \infty$ such that $u(x,0) = x^{1/2}$ for $0 < x < \infty$. Determine $u(x,0)$ for $-\infty < x < 0$. Use conformal mapping techniques to solve the problem.

Answer: 0


Question: Evaluate \( \lim_{n\to\infty}\int_n^{n+1}\cos^2(u^2) \, du \).

Answer: \dfrac{1}{2}


Question: Can the norm of the resolvent function \( R_\lambda = (\lambda - A)^{-1} \) be equal to zero for \( \lambda \in \rho(A) \)?

Answer: No


Question: Find the limit of the sequence defined by the infinite product: $$\lim\limits_{n \to\infty}\prod_{k=1}^n \left(1+\frac{k}{n^2}\right)$$

Answer: \sqrt{e}


Question: Evaluate the integral \( \int_{-\pi/4}^{\pi/4} (\cos t + \sqrt{1+t^2} \sin^3 t \cos^3 t) \, \mathrm{d}t \).

Answer: \sqrt{2}


Question: Calculate the probability that the square of a geometric random variable is greater than or equal to a binomial random variable. Let $X$ be a $\operatorname{Binomial}(4, \frac{1}{3})$ random variable and $Y$ be a $\operatorname{Geometric}(\frac{1}{3})$ random variable. Find $P(Y^2 \geq X)$. Provide your answer as a probability value.

Answer: \dfrac{70}{81}


Question: Find the square of the perpendicular distance from the origin to the line L, which intersects perpendicularly both the lines \( \frac{x+2}{2}=\frac{y+6}{3}=\frac{z-34}{-10} \) and \( \frac{x+6}{4}=\frac{y-7}{-3}=\frac{z-7}{-2} \).

Answer: 5


Question:
A rigid body has one point $O$ fixed and experiences no external torque about $O$. Given that the body has equal principal moments of inertia, determine the behavior of the body. Choose the correct option:

(1) Angular velocity of variable magnitude
(2) Angular velocity with constant magnitude
(3) Constant angular momentum but varying angular velocity

Which option is correct?

Answer: 2


Question: Let $V$ be the vector space of $2 \times 2$ matrices over a field $F$. Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in V$ and let $T : V \rightarrow V$ be the linear transformation defined by $T(X) = AX$. Compute $\det(T)$. Provide your answer in terms of the determinant of $A$.

Answer: (\det A)^2


Question: Evaluate the triple integral \( \iiint_{R} (2x+y) \,dx \,dy \,dz \), where \( R \) is the region bounded by the cylinder \( z = 4 - x^{2} \) and the planes \( x = 0 \), \( y = 0 \), \( y = 2 \), and \( z = 0 \). Determine the limits of integration from these conditions and compute the integral.

Answer: \dfrac{80}{3}


Question: Determine whether there exists a positive constant $C$ such that for all sufficiently large integers $n$, the inequality \[ \prod_{i=1}^n \left(1+\frac{1}{\sqrt{p_i}}\right) > C\sqrt{p_{n+1}} \] holds, where $p_k$ denotes the $k$-th prime number.

Answer: Yes


Question: Find the sum of the series \(\sum_{n=0}^\infty \frac{n^{2}}{2^{n}}.\)

Answer: 6


Question: Let $G$ be a compact connected semisimple Lie group and $\frak g$ its Lie algebra. Consider the complex semisimple Lie algebra ${\frak g}_{\Bbb C}={\frak g}\otimes\Bbb C$. If $X\in{\frak g}_{\Bbb C}$ lies in the Cartan subalgebra ${\frak h}_{\Bbb C}\subseteq{\frak g}_{\Bbb C}$ and the Killing form $B(X,X)=0$, does that imply that $X=0$?

Answer: No


Question: Consider a sequence of real \( n \times n \) matrices \( \{A_k\}_{k=1}^{\infty} \) where each matrix \( A_k \) is invertible. Suppose the limit \( \lim_{k \rightarrow \infty} A_k \) exists and is not invertible. Is it true that \( \lim_{k \rightarrow \infty} A_k = 0 \)? Provide a justification for your answer.

Answer: No


Question: Determine the value of \( y_0 \) such that for the initial value problem \( y'+\frac{2x^2-4xy-y^2}{3x^2}=0, x>0, y(1)=y_0 \), the condition \( \lim_{x\to 0}y'(x) \neq 1 \) is satisfied, while \( \lim_{x\to 0}y'(x) = 1 \) for all other values of \( y_0 \).

Answer: -2


Question: Is the number of distinct irreducible characters of a finite group $G$ finite?

Answer: Yes


Question: Find the limit of the sequence \( x_n = \frac{(a)(a+1)(a+2)\cdots(a+n)}{(b)(b+1)(b+2)\cdots(b+n)} \) as \( n \to \infty \), where \( a \) and \( b \) are positive numbers and \( a < b \).

Answer: 0


Question:
For the function \( f(t) = \frac{\pi - t}{2} \) defined on the interval \([0, 2\pi]\), the complex Fourier coefficients are given by:
\[ c_n = \frac{1}{2\pi}\int_0^{2\pi}\frac{\pi - t}{2}e^{-int}dt \]
Determine the value of the \( n=0 \) term in the Fourier series of \( f(t) \).

Answer: 0


Question:
Determine the values of \(a\) and \(b\) such that the polynomial equation \(x^4 - 4x^3 + ax^2 + bx + 1 = 0\) has four real roots. Choose from the following options:

- \((4, -6)\)
- \((6, -4)\)

Answer: (6, -4)


Question: Find the volume of the solid bounded by the surfaces $z = x^2 + y^2$ and $x + z = 0$. Provide your answer in terms of integrals or a numerical value.

Answer: \dfrac{\pi}{32}


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

Answer: 7872


Question: Calculate the limit in probability: \[ \lim_{n\rightarrow\infty} \frac{n}{X_1^2+X_2^2+\ldots+X_n^2} \] where \(X_1, X_2, \ldots, X_n\) are IID random variables with distribution \(N(-1, 2^2)\). Show your work using the definition of convergence in probability.

Answer: \dfrac{1}{5}


Question: Given that the $p$th, $q$th, and $r$th terms of a geometric progression are $27$, $8$, and $12$ respectively, determine how many roots of the quadratic equation $px^2 + 2qx - 2r = 0$ lie in the interval $(0,1)$. Assume $p$, $q$, and $r$ are positive integers.

Answer: 1


Question: Evaluate the sum \( S_n = \sum_{r=0}^{n-1} \arccos\left(\frac{n^2+r^2+r}{\sqrt{(n^2+r^2+r)^2+n^2}}\right) \).

Answer: \dfrac{\pi}{4}


Question: Is it true that for a graph with $n$ vertices and $m$ edges, if $m > cn$ and $c > 1$, then the girth is $O(\log n)$? Specifically, is there a constant $C$ which depends on $c$ but not on $n$ such that $g(n,m) < Cn$?

Answer: O(\log n)


Question: Determine the Banach-Mazur distance between the Banach spaces \(\ell_\infty = \ell_\infty(\mathbb{N})\) and \(L_\infty = L_\infty([0,1], \lambda)\), where \(\lambda\) is the Lebesgue measure on \([0,1]\). The Banach-Mazur distance is defined as \( d(X,Y) = \inf\{ \|\varphi\|\|\varphi^{-1}\| : \varphi\colon X\to Y \text{ isomorphism} \} \). What is \(d(\ell_\infty,L_\infty)\)?

Answer: 2


Question:
Let $U$ be an orthonormal set in a Hilbert space $H$, and let $x \in H$ be such that $\| x \| = 2$. Consider the set
\[ E = \{ u \in U : |\langle x, u \rangle| \geq \frac{1}{4} \} \]
What is the maximal possible number of elements in $E$?

Answer: 64


Question: Let $H$ be a minor of a graph $G$. Is it always true that the genus of the embedding surface of $H$ is less than or equal to the genus of the embedding surface of $G$?

Answer: Yes


Question: Calculate the sum of the series: \[ \sum_{n \geq 1} \frac{C(n,0) + C(n,1) + C(n,2) + \cdots + C(n,n)}{P(n,n)} \] where \( C(n,k) \) is the binomial coefficient and \( P(n,n) \) is the number of permutations of \( n \) items. Express your answer in terms of known constants or functions.

Answer: e^2 - 1


Question: Let $V$ be an $n$-dimensional Euclidean space with inner product $\langle\cdot,\cdot\rangle$, and basis $e_1,\dots,e_n$. The Gram matrix is $A=(a_{ij})$ with $a_{ij}=\langle e_i,e_j\rangle$. Suppose $\langle e_i,e_j\rangle\leq 0$ for $i\neq j$. Determine whether the following statement is true: if $Ax\geq 0$, then $x\geq 0$, where $x$ and $0$ are column vectors, and a vector $y\geq 0$ means each coordinate is $\geq 0$. Provide a justification for your answer.

Answer: True


Question: Find the maximum value of \(|z^2 - iz + 1|\) for \(|z| = 1\).

Answer: \sqrt{5}


Question: Find a number \(a < 0\) such that \(\int_a^0 \frac{e^{\frac{1}{x}}}{x^2} \, dx = \frac{e^{\frac{1}{a}}}{a^2}\).

Answer: -1


Question: Consider a smooth, geometrically connected, projective curve $X$ of genus $g$ over the finite field $\mathbb{F}_q$. The zeta function of $X$ is given by $P(s)/(1 - s)(1 - qs)$, where $P(s)$ is a polynomial of degree $2g$. According to the Riemann hypothesis for curves, the zeros of $P(s)$ have complex absolute value $1/\sqrt{q}$. Is it possible to find a family of curves $\{X_n\}_{n=1}^{\infty}$ with $g(X_n) \rightarrow \infty$ such that the zeros of $P_{X_n}$ become evenly distributed? Specifically, if $Y_n$ denotes the set of phase angle differences (normalized into $[0,2\pi)$) of consecutive zeros of $P_{X_n}$, does $\max{Y_n}-\min{Y_n} = o(1/g(X_n))$ hold?

Answer: No


Question: Find the least possible value of \( k \) such that \((2y - 2017)^2 = k\) has two distinct positive integer solutions for \( y \), one of which is a factor of 100.

Answer: 3301489


Question: Are the trigonometric functions \( \{e^{2\pi i nx}\}_{n \in \mathbb{Z}} \) dense in \( C^k(S^1) \) with respect to the norm \( |f|^{C^k}=\sum_{i=0}^k \text{max}_{x \in S^1}|f^{(k)}(x)| \)?

Answer: Yes


Question: Evaluate the integral: $$\frac{1}{\sqrt{2\pi s^2}} \int_{-\infty}^{\infty} xe^{-(x-m)^2/(2s^2)} \, dx$$

Answer: m


Question:
A square matrix of order 10 has exactly 4 distinct eigenvalues. What is the degree of its minimal polynomial?

(A) At least 4
(B) At most 4
(C) At least 6
(D) At most 6

Answer: A


Question: In a race with 20 runners, the time it takes each runner to finish is an independent exponential random variable with a mean of 60 minutes. What is the expected time that you leave if you leave after 3 runners finish?

Answer: \dfrac{541}{57}


Question: Evaluate the stochastic integral \( \int_0^t dW_s \).

Answer: W_t


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

Answer: -4


Question: Is simply connectedness a topological property? Specifically, if $f: X \to Y$ is a homeomorphism and $X$ is a simply connected space, is $Y$ also a simply connected space?

Answer: Yes


Question: Suppose $f: [0, \infty) \to \mathbb{R}$ is a continuous function that is eventually almost periodic with periods $1$ and $a$, where $a$ is an irrational number such that $0 < a < 1$. Define the function $F: [0, 1) \to \mathbb{R}$ by \( F(x) = \lim_{n \to \infty} f(x + n) \). Is the function $F$ necessarily constant?

Answer: Yes


Question: Calculate the limit: \( \lim_{x\to0} \sum_{n=1}^\infty \frac{\sin x}{4+n^4x^4} \)

Answer: \dfrac{\pi}{8}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{10n! + 10^n}{n^{10} + n!} \]

Answer: 10


Question: Find the limit of the sequence \( S_n = \frac{1}{n}\left(\frac{1}{2}+\sum_{k=1}^{n}\cos(kx)\right) \) as \( n \to \infty \).

Answer: 0


Question: Let $f$ be a measurable function on $\mathbb{R}$ such that $f$ is not almost everywhere zero. Determine whether there exists a set $S \subset \mathbb{R}$ with positive Lebesgue measure such that $|f(t)| \geq c > 0$ for almost every $t \in S$ and some constant $c > 0$. Provide a justification for your answer.

Answer: Yes


Question: Calculate \(56!! \mod 29\) using Wilson's theorem. Here, \(56!!\) represents the product \(2 \cdot 4 \cdot 6 \cdots 56\).

Answer: 28


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

Answer: 1


Question: Simplify the expression: \[ \prod_{k=1}^{100}\left(1+2\cos\left(\frac{2\pi \cdot 3^k}{3^{100} +1}\right)\right) \]

Answer: 1


Question: Evaluate the integral \( \int_0^{a} f(z) \delta(z) \, dz \) for \( a > 0 \). Does it hold that \( \int_0^{a} f(z) \delta(z) \, dz = f(0) \)?

Answer: f(0)


Question: Consider a connected graph $G = (V, E)$ with $n$ vertices, where all edges are part of $3$-cycles. The graph contains exactly $n-2$ $3$-cycles, and every vertex is part of at least one $3$-cycle. The minimum degree of the graph is $2$. Determine the maximum number of vertices in $G$ that have degree $2$. Provide your answer as a function of $n$.

Answer: n - 2


Question: Evaluate the limit: \[ \lim\limits_{n \to \infty}{\frac{\sqrt[n]{n}-1}{\frac{\ln{n}}{n}}}=1 \] without using derivatives.

Answer: 1


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

Answer: e^{\frac{\pi^2}{12}}


Question: Determine the number of solutions to the equation $1 = x^\pi$.

Answer: 1


Question: Evaluate the limit \( \lim_{n\to \infty}\int_0^1 \frac{2nx^{n-1}}{1+x} \, dx \).

Answer: 1


Question:
Given that \( f(0) = 1 \), \( f(2) = 3 \), and \( f'(2) = 5 \), evaluate the integral:
\[ \int_0^2 x f''(x) \, dx \]

Answer: 8


Question: Find the determinant of the linear transformation \( L: \mathbb{R}^{2 \times 2} \to \mathbb{R}^{2 \times 2} \) defined by \( L(A) = A^T \), where \( A \) is a \( 2 \times 2 \) matrix.

Answer: -1


Question: Given that $X, Y, Z$ are independent random variables uniformly distributed over the interval $(0, 1)$, and that $0 < X < Y < Z < 1$, find the probability that $Z - X < \frac{1}{3}$.

Answer: \dfrac{7}{27}


Question:
Verify whether the following claim is true or false:

Let \(f\) be a continuous and nonincreasing function on \([0,+\infty)\) such that
\[\lim_{t\to+\infty}\int_{0}^{t}(f(s))^{2}ds<+\infty.\]
Then,
\[\lim_{t\to +\infty}f(t)=0.\]

Answer: True


Question: Person A selects a number $X$ randomly from the uniform distribution $[0,1]$. Person B then repeatedly and independently draws numbers $Y_1, Y_2, Y_3, \ldots$ from the uniform distribution on $[0,1]$ until he gets a number larger than $\frac{X}{2}$, then stops. What is the expected number of draws that Person B makes?

Answer: 2 \ln 2


Question: Evaluate the double series: $$ \sum_{i=1}^{\infty} \sum_{j=i-1}^{\infty} \frac{3^j}{j!} $$

Answer: 4e^3


Question: Determine the number of duplication formulas that exist for the Mordell curve family $Y^2-X^3=c$. Given that Bachet and Réalis have provided two such formulas, how many additional duplication formulas are there?

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\rightarrow \infty }\dfrac {2} {x}\sum _{k=1}^{x}\ln \left( \dfrac {x+k} {x}\right) \] where \( k \) are positive integers.

Answer: 4 \ln 2 - 2


Question: Given the functional equation \( f(x) + f(y) = f(x+y) \), with \( f'(0) = 3 \) and \( f(5) = 3 \), find \( f'(5) \).

Answer: 3


Question: Evaluate the volume of the solid that lies under the paraboloid \( z = 4 - x^2 - y^2 \) and above the xy-plane using spherical coordinates.

Answer: 8\pi


Question: Find all pairs of positive integers \((m, n)\) such that \( \frac{m}{n} = n.m \).

Answer: (5, 2)


Question: For a random labelled simple graph with n vertices, determine the smallest number of vertices n such that there are more simple labelled graphs with more edges than vertices than there are simple labelled graphs with more vertices than edges.

Answer: 6


Question: What is the boundary of \(\mathbb{Q}^n\) as a subset of \(\mathbb{R}^n\), endowed with the standard topology?

Answer: \mathbb{R}^n


Question: Calculate the limit: $$ \lim_{n \to \infty}  \frac{1+\frac{1}{2}+\ldots +\frac{1}{n}}{\left(\pi^n+e^n \right)^{\frac{1}{n}} \ln{n}} $$

Answer: \dfrac{1}{\pi}


Question: Is it true that the ultraproduct of non-commutative $L^p$-spaces, \( \prod_{\mathcal U}L^p(M_i,\tau_i) \), can be identified as \( L^p(M,\tau) \) for some von Neumann algebra \( M \) with \( \tau \) being a normal faithful semifinite trace on \( M \)?

Answer: Yes


Question: Consider the differential equation \(\frac{dy}{dx}+10y=f(x), x>0\), where \(f(x)\) is a continuous function such that \(\lim_{x\to \infty}f(x)=1\). Find the value of \(\lim_{x\to \infty}y(x)\).

Answer: \dfrac{1}{10}


Question: Does the group $SO(3)$ have an open nontrivial subgroup? Here, $SO(3)$ is the group of all $3 \times 3$ orthogonal matrices with determinant 1, equipped with the operator norm topology.

Answer: No


Question: Consider a power series $P(z) = 1 + a_1 z^2 + a_2 z^4 + a_3 z^6 + \ldots$. It is given that $P$ converges in the open unit disc centered at the origin. Also, for any $\omega$ such that $|\omega|=1$, we have $\lim_{z \to \omega} |P(z)| = 1$. Can we conclude that $a_i = 0$ for all $i \geq 1$?

Answer: Yes


Question: Let $R$ be a commutative ring and let $\varphi: M \to N$ be an injective linear map of finitely generated free $R$-modules. Are the $k$-th tensor power maps $\varphi^{\otimes k}: M^{\otimes k} \to N^{\otimes k}$ also injective for all $k \geq 0$?

Answer: Yes


Question: Let $A$ be a square matrix of order $n$ with real entries such that $A^3 + I = 0$. Determine the value of $\operatorname{trace}(A)$. \(\text{Note: } I \text{ is the identity matrix of the same order as } A.\)

Answer: 0


Question: Suppose $G$ is a finite group and $t$ is an involution in $G$. Given that $\frac{C_G(t)}{\langle t\rangle}\cong \mathbb{S}_{n-2}$ and $C_G(t)$ contains a subgroup isomorphic to $\mathbb{S}_{n-3}$, determine if $\langle t\rangle$ has a complement in $C_G(t)$. A complement is a subgroup $L$ such that $L\cap \langle t\rangle=1$ and $C_G(t)=L\langle t\rangle$. Is this true?

Answer: True


Question:
Let $F$ be the splitting field of $X^7-2$ over $\mathbb{Q}$, and let $z=e^{\frac{2\pi i}{7}}$ be a primitive seventh root of unity. Determine the values of $[F:\mathbb{Q}(z)] = a$ and $[F:\mathbb{Q}(2^{1/7})] = b$. Choose the correct statement:

- $a = b = 7$.
- $a = b = 6$.
- $a > b$.
- $a < b$.

Calculate $a$ and $b$ to find the correct relationship.

Answer: C


Question: Let $G = P \ltimes Q$ be a finite group where $P$ is a maximal subgroup of $G$, non-normal, cyclic, and satisfies $P = C_G(P) = N_G(P)$. The derived subgroup $G' = Q \cong C_q \times C_q \times C_q$, with $p$ and $q$ being distinct primes. Assume that for any two distinct non-normal non-cyclic subgroups $H$ and $K$ of $G$, they are conjugate in $G$. There exists a homomorphism $\varphi : P \to \text{Aut}(Q) \cong GL(3,q)$, and $\text{Ker}(\varphi) = C_P(Q) = Z(G)$. Prove that $P \cong C_p$, where $p = q^2 + q + 1$ is a prime.

Answer: C_p


Question:
Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function such that \( f(\mathbb{Q}) \subseteq \mathbb{N} \). Choose the correct statement from the following options:

(a) \( f(\mathbb{R}) = \mathbb{N} \)
(b) \( f(\mathbb{R}) \subseteq \mathbb{N} \) but \( f \) need not be constant.
(c) \( f \) is unbounded.
(d) \( f \) is a constant function.

Answer: D


Question: Evaluate the surface integral \( \iint_S \mathbf{A} \cdot d\mathbf{S} \) where \( \mathbf{A} = y\hat{i} + 2x\hat{j} - z\hat{k} \) and \( S \) is the surface of the plane \( 2x + y = 6 \) in the first octant, cut off by the plane \( z = 4 \).

Answer: 108


Question: Calculate the first Chern class of the bundle $N \to E$, where $N = \{ (z_0,z_1,z_2) \in S^5 \mid z_0^3+z_1^3+z_2^3 = 0 \}$ and $E = N/U(1)$ is an elliptic curve in $\mathbb{C}P^2$. The circle $U(1)$ acts on $N$ by $e^{i\theta} \cdot (z_0,z_1,z_2) = (e^{i\theta}z_0,e^{i\theta}z_1,e^{i\theta}z_2)$. The answer is known to be $-3$. Explain why this is the case.

Answer: -3


Question: Find the volume in the first octant under the paraboloid \( z = 1 - \frac{x^2}{4} - \frac{y^2}{81} \). Use the substitution \( x = 2r\cos\theta \) and \( y = 9r\sin\theta \) to set up the integral. Determine the correct bounds for \( r \) and \( \theta \) and evaluate the integral to find the volume.

Answer: \dfrac{9\pi}{4}


Question: Find the value of \( m \) if \( m = p_1^2 p_2^2 \) and \( \varphi(m) = 11424 \), where \( p_1 \) and \( p_2 \) are distinct prime numbers.

Answer: 14161


Question: If $f_n(x)$ converges to $f(x)$ uniformly on a domain $E$ and both $f_n$ and $f$ never vanish on $E$ (i.e., $f_n(x) \neq 0$ and $f(x) \neq 0$ for all $x \in E$ and for all natural numbers $n$), does it follow that the functions $1/f_n$ converge uniformly to $1/f$ on $E$?

Answer: No


Question: Calculate \( \lim\limits_{n \to \infty} f_n(x) \) for the sequence \((f_n)\) defined by \( f_n(x) = \sqrt{\int_0^x f_{n-1}(t) \, \mathrm{d}t} \), where \( f_0(x) \) is integrable on \([0,1]\) and \( f_0(x) > 0 \).

Answer: \dfrac{x}{2}


Question: Determine the fundamental group of the space obtained by taking a closed disk $D$ with boundary $C$, removing the interiors of two disjoint closed disks $D_a$ and $D_b$ inside $D$ (with boundaries $C_a$ and $C_b$, respectively), and identifying points on $C_a$ that are 90 degrees apart, points on $C_b$ that are 120 degrees apart, and points on $C$ that are 180 degrees apart.

Answer: S_4


Question: Evaluate the integral \( \frac{1}{\pi^2}\int_{x=0}^1\int_{y=\sin^{-1} x}^{\dfrac{\pi}{2}}f(x,y)\mathop{dy}\mathop{dx} \) where \( f(x,y)=\begin{cases} \frac{y}{\sin y} & \text{if } y\neq 0 \\ 1 & \text{if } y=0 \end{cases} \). Provide the answer correct to three decimal places.

Answer: 0.125


Question: Find the minimum value of the functional \( F(u) = \int_0^1 (1+x^2)u^2(x)\,dx \) for \( u \in L^2(0,1) \) subject to the constraint \( \int_0^1 u(x)\,dx = 1 \).

Answer: \dfrac{4}{\pi}


Question: Find the function \( f(x) \) that satisfies the functional equation \( f(x+y) = f(x) + f(y) + y\sqrt{f(x)} \) for all real numbers \( x \) and \( y \), given that \( f'(0) = 0 \).

Answer: 0


Question: Compute the limit \( \lim_{n \rightarrow \infty}{ \int_0^{\infty}\left(\frac{nx^{1/n}}{ne^x+\sin(nx)}\right)dx} \) using the relation between Riemann and Lebesgue integrals and Lebesgue’s Dominated Convergence Theorem.

Answer: 1


Question: Determine the cardinality of the set \(E(a)\), where \(E(a) = \{(r_1, r_2, \ldots, r_n, \ldots) \mid r_n \in \mathbb{Q} \cap [0,1], r_n \leq r_{n+1}, \forall n \geq 1, \sum_{n=1}^\infty \frac{r_n}{2^n} = a\}\) for any \(a \in (0,1)\). What is \(|E(a)|\)?

Answer: 2^{\aleph_0}


Question: Calculate the limit: $$\lim_{x \to \infty}\left(\frac{x^2+2x+3}{x^2+x+1} \right)^x$$ without using l'Hôpital's rule.

Answer: e


Question: Compute the Legendre symbol \(\left(\frac{77}{257}\right)\) using Euler's criterion.

Answer: -1


Question: Determine the values of \( p \geq 0 \) for which the integral \( \int_0^\infty x \sin(x^p) \, dx \) converges.

Answer: p > 2


Question: Evaluate the limit: \[ \lim_{x \to 0} (\log(x)\log(1-x)) \]

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \pi \dfrac{\sum_{i=1}^{n} \sin\left(\frac{i\pi}{n}\right)}{n} \]

Answer: 2


Question: Evaluate \( \lim_{x \to 0} x^2\left(1+2+3+\ldots+\left[ \frac{1}{|x|} \right] \right) \).

Answer: \dfrac{1}{2}


Question: Given sequences of random variables $(X_n, n \in \mathbb{N})$ and $(Y_n, n \in \mathbb{N})$, where for all $n \in \mathbb{N}$, $X_n = Y_n$ almost surely, determine if $P(X_n = Y_n \forall n \in \mathbb{N}) = 1$.

Answer: 1


Question: In a geometric progression, the first term is 3, and the sum of the first seven terms is 381. What is the common ratio of this progression?

Answer: 2


Question: If $T \in \mathcal{B}(X,Y)$ is not compact, can the restriction of $T$ to an infinite-dimensional subspace of $X$ be compact?

Answer: Yes


Question: Evaluate the integral \( \int_C \frac{e^z+\sin{z}}{z}dz \) where \( C \) is the circle \( |z|=5 \) traversed once in the counterclockwise direction.

Answer: 2\pi i


Question: Find the limit of the sequence \( a_n = \frac{(n^2+n+1)^{10}-(n+1)^{20}}{(n^2+1)^{10}-(n+1)^{20}} \) as \( n \to \infty \).

Answer: \dfrac{1}{2}


Question: Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(x) = f(x^2 + C) \) for all \( x \in \mathbb{R} \), where \( C > 0 \) is a constant.

Answer: f(x) = k


Question: Two random variables $X$ and $Y$ are i.i.d. normal$(\mu, \sigma^2)$. Given that $P(X > 3) = 0.8413$, find the exact value of $P((X+Y)/2 > 3)$. Assume the normal distribution parameters are unknown.

Answer: 0.9213


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

Answer: \pi \ln 2


Question: Find an initial condition \( y(0) \neq (0,0,0) \) for the system of differential equations given by \( y_1'=12y_3 \), \( y_2'=y_1+13y_3 \), and \( y_3'=y_2 \) such that the solution \( y(x) \) approaches \( (0,0,0) \) as \( x \to \infty \).

Answer: (-4, -3, 1)


Question: Find the minimum possible value of \( K \) such that the equation \( 9x + 13y = K \) has exactly five solutions in positive integers \( x \) and \( y \).

Answer: 490


Question:
Evaluate the inverse Fourier transform of the function \( f(w) = \frac{1}{2\pi}e^{iwx_0} \) to show that it results in the Dirac delta function. Specifically, compute:
\[ \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-iw(x-x_0)}dw \]

Answer: \delta(x - x_0)


Question: Calculate the surface integral of the function \( f(x,y,z) = x^2 + y^2 + z^2 \) over the upper hemisphere of the sphere defined by \( x^2 + y^2 + (z-1)^2 = 1 \).

Answer: 6\pi


Question: Let $K$ be a field, $A$ be a finite-dimensional $K$-algebra, and $M$ be a finitely generated $A$-module. Determine whether $M$ admits a projective resolution by finitely generated projective $A$-modules.

Answer: Yes


Question: n balls numbered 1, 2, ..., n are randomly distributed into n cells also numbered 1, 2, ..., n, with each cell holding exactly one ball. Let X be the number of balls placed in cells with matching numbers. Compute the expected value E(X).

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\ln(1+4x) - 4x}{x^2} \] without using L'Hopital's rule.

Answer: -8


Question: Find an element of order 5 in the field \( \mathbb{Z}_3[x]/\langle x^4 + x + 2 \rangle \).

Answer: 2x^3 + x + 2


Question: Use Kronecker's theorem to construct a field with four elements by adjoining a suitable root of the polynomial \(x^4 - x\) to \(\mathbb{Z}_2\).

Answer: \mathbb{Z}_2[x]/(x^2 + x + 1)


Question:
Find the determinant of the following 5x5 matrix $A$:
\[
A = \begin{pmatrix}
d &0 &0 &0 &0 \\
d &d &0 &x &x \\
d &d &d &0 &0 \\
d &d &d &d &0 \\
d &d &d &d &d \\
\end{pmatrix}
\] where $d$ and $x$ are constants.

Answer: d^5


Question: Calculate the integral \( \int_\gamma \frac{1+\sin(z)}{z} \, dz \) where \( \gamma \) is the circle centered at 0 with radius \( \log(\sqrt{2}) \), oriented counterclockwise.

Answer: 2\pi i


Question: Let $X$ and $Y$ be independent random variables, each uniformly distributed on the interval $(0,1)$. Find the probability density function (PDF) of the random variable $V = XY$.

Answer: -\ln v


Question: Find the mass of the area bounded by the polar curves \( r = \cos{t} \) and \( r = \cos{t} + \sqrt{3} \sin{t} \). Use polar coordinates to set up the integral, with \( \theta \) varying from \( 0 \) to \( \pi \) and \( r \) varying from \( \cos{t} + \sqrt{3} \sin t \) to \( \cos{t} \).

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


Question: Let $X \subset \mathbb{R}$ and let $f,g : X\rightarrow X$ be continuous functions such that $f(X)\cap g(X) = \emptyset$ and $f(X)\cup g(X) = X$. Which one of the following sets cannot be equal to $X$?\nA. $[0, 1]$\nB. $(0, 1)$\nC. $[0, 1)$\nD. $\mathbb{R}$

Answer: A


Question: Determine the limit of the function \( f(x,y) = xye^{-x^2y^2} \) as \( x^2 + y^2 \to \infty \) when restricted to a straight line through the origin. Also, evaluate the limit \( \lim_{x^2+y^2 \to \infty} f(x,y) \).

Answer: 0


Question: Evaluate the contour integral \( \int_{\delta D_1(0)} \frac{e^{z}}{z} \, dz \), where \( \delta D_1(0) \) denotes the boundary of the unit disk centered at the origin in the complex plane.

Answer: 2\pi i


Question: Let $X$ and $Y$ be independent random variables such that $X, Y \sim \text{Ber}(0.5)$, and let $Z = X \oplus Y = X + Y \pmod{2}$. Find the conditional entropy $H(Z|X)$. Here, $H$ denotes the entropy.

Answer: 1


Question: Determine the largest possible value of \(|f''(0)|\) for an analytic function \(f\) in the disk \(D = \{z \in \mathbb{C} : |z| < 2\}\) such that \(\iint_{D} |f(z)|^2 \, dx \, dy \leq 3\pi\).

Answer: \dfrac{3}{4}


Question: Determine whether there exist two distinct number fields $E$ and $F$ such that for every prime $p$, the tensor products of their rings of integers with $\mathbb{F}_p$ are isomorphic, i.e., $\mathcal{O}_E \otimes \mathbb{F}_p \cong \mathcal{O}_F \otimes \mathbb{F}_p$. Provide a justification for your answer.

Answer: Yes


Question: Find the smallest natural number \( N \) such that \( 13^N \equiv 1 \pmod{2013} \).

Answer: 30


Question: Find the smallest positive integer \( n \) such that \( 2012 \mid (9^n - 1) \).

Answer: 251


Question:
Compute the expectation of the stochastic process \( \bar{X}_t \), where \( X_t = (1-t)\int_{0}^{t} \frac{1}{1-s} dW_s \) and \( \bar{X}_t = \begin{cases} X_t &\mbox{if } 0\leq t <1 \\
0 & \mbox{if } t=1 \end{cases} \).

Answer: 0


Question: Calculate the following limit using the Lebesgue dominated convergence theorem: \[ \lim_{n\to\infty} \int_{[0,1]} \left(n\left(\sin\left(\frac xn\right)\right)\right)^n \, \mathsf{d}x \]

Answer: 0


Question:
Which of the following statements are true about the open interval $(0,1)$ and the closed interval $[0,1]$?

I. There is a continuous function from $(0,1)$ onto $[0,1]$.
II. There is a continuous function from $[0,1]$ onto $(0,1)$.
III. There is a continuous one-to-one function from $(0,1)$ onto $[0,1]$.

(A) none
(B) I only
(C) II only
(D) I and III only
(E) I, II, and III

Select the correct option.

Answer: B


Question: Let \( f(x) \) be a differentiable function on the interval \([0,1]\) with \( f(0) = f(1) = 0 \). Determine the value of \( \alpha \) such that there exists a point \( c \) in \( (0,1) \) where \( f'(c) + \alpha f(c) = 0 \).

Answer: \mathbb{R}


Question:
Given the recurrence relation for the coefficients $c_n$ of the power series $y = \sum^\infty_{n=0}c_nx^n$:
\[c_{n+1} = \frac{n+2}{5(n+1)}c_n\]
find the radius of convergence of the power series.

Answer: 5


Question: Prove that the boundary value problem \(\begin{cases} y''+2y'+\lambda y=0 \\ y'(0)=y(1)=0 \end{cases}\) admits eigenvalues \(\lambda > 1\).

Answer: \lambda > 1


Question: Given a martingale $M_t$ with $M_0=0$ and $E[M_t^2]<\infty$, where the quadratic variation of $M_t$ is $[M]_t = \ln(1+t)$, calculate $\limsup_{t\to\infty} \frac{M_t}{\sqrt{t}}$. Provide your answer as a single value.

Answer: 0


Question: Determine whether every nice graph contains a copy of $K_4$. A graph $G = (V, E)$ is defined as nice if for every countable partition $\{V_n: n < \omega\}$ of $V$, there exists an $n < \omega$ such that $(V_n, E)$ contains a copy of $K_3$. Here, $K_n$ is the complete graph on $n$ vertices.

Answer: Yes


Question: Given that \( f(\pi) = \pi \) and \( \int_{0}^{\pi} (f(x) + f''(x)) \sin x\, dx = 7\pi \), find the value of \( f(0) \) assuming \( f(x) \) is continuous on \([0, \pi]\).

Answer: 6\pi


Question: Evaluate the integral \( \int_0^{\infty}\frac{\ln(x+1)}{x\cdot(x+1)^\frac{1}{4}} \, dx \).

Answer: \pi^2 + 8G


Question: Determine the signature of the quadratic form \( f(x, y) = xy \).

Answer: (1, 1)


Question: Consider the set $S$ of all $3 \times 3$ matrices whose entries are chosen from the set $\{1, 3, 5, 7, 11\}$. Calculate the average of the determinants of all matrices in $S$. Note that the total number of such matrices is $5^9$. What is the average determinant?

Answer: 0


Question: Find the curve passing through the points (0,0) and (1,1) that is an extremal for the functional \( J(x,y,y') = \int \left\{ [y'(x)]^2 + 12x y(x) \right\} \, dx \).

Answer: y = x^3


Question: Determine whether every finite non-abelian simple group has at least 4 non-identity conjugacy classes.

Answer: Yes


Question:
Given the joint probability density function of the returns on two investments $X$ and $Y$:
\[ f(x,y) = \begin{cases} \frac{1}{2}, & \text{if } 0 < |x| + |y| < 1, \\ 0, & \text{otherwise.} \end{cases} \]
Calculate the variance of $X$, $\text{Var}(X)$. \(\boxed{\frac{1}{6}}\)

Answer: \frac{1}{6}


Question: Calculate the limit \( I_{\infty}(N) = \lim_{x\to\infty} \int \prod_{k=1}^N \left(k-\frac{k}{x}\right) \, dx \).

Answer: \infty


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

Answer: \infty


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^{\infty} \frac{\ln(n+1)}{n!} (z+2)^n \).

Answer: \infty


Question: Let \( \{X_n\} \) be a sequence of independent Gaussian random variables with mean 0. Determine the probability of the event \( \limsup_{n\to \infty} \{ X_n X_{n+1} > 0 \} \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{ x \to 0^- }\frac{2^{\frac{1}{x}}+2^{\frac{-1}{x}}}{3^{\frac{1}{x}}+3^{\frac{-1}{x}}} \] without using L'Hôpital's rule.

Answer: 0


Question: An ellipse has foci at \((1,-1)\) and \((2,-1)\). The line \(x+y=5\) is tangent to the ellipse at the point \((m,n)\). Determine the value of \( \frac{1}{e^2} \), where \( e \) is the eccentricity of the ellipse.

Answer: 41


Question: Use the Divergence Theorem to compute the net outward flux of the vector field \( F = \langle x^2, y^2, z^2 \rangle \) through the sphere \( S \) defined by \( x^2 + y^2 + z^2 = 25 \).

Answer: 0


Question: Determine the number of homomorphisms from the Klein four-group \( V_4 \) to the symmetric group \( S_4 \).

Answer: 52


Question:
Determine the points at which the limit of the function \( f(x) \) exists, where:

\[
f(x)=\begin{cases}0&\text{if} \;x \in \Bbb R \setminus \Bbb Q \\\\\frac{1}{n}&\text{if}\;x=\frac{m}{n} \in \Bbb Q\;\text{where}\;\gcd(m,n)=1\end{cases}
\]

Answer: \mathbb{R}


Question: Evaluate the limit using Taylor expansion: \[ \lim_{x \rightarrow 0} \frac{\exp(2x)-\ln(1-x)-\sin(x)}{\cos(x)-1} \]

Answer: -\infty


Question: Find the limit of the sequence \( x_n = \frac{n^{\frac{n}{2}}}{n!} \) as \( n \to \infty \).

Answer: 0


Question: A set of \(n\) dice is thrown. All dice that land on six are set aside, and the remaining dice are thrown again. This process is repeated until all dice have landed on six. Let \(N\) denote the number of throws needed. Let \(X_{i}\) denote the number of dice rolled on the \(i\)-th throw. Find \(E[\sum_{i=1}^{N} X_{i}]\).

Answer: 6n


Question: Find the limit \( \lim_{||(x,y)||\to \infty} \frac{x+y}{x^2-xy+y^2} \).

Answer: 0


Question: Evaluate the integral \( \int_{\mathbb{R}} e^{-(x+iy)^2} \, dx \) for a fixed \( y \in \mathbb{R} \).

Answer: \sqrt{\pi}


Question:
Let \( N \) be the number of ordered pairs of nonempty sets \( \mathcal{A} \) and \( \mathcal{B} \) such that:
- \( \mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\} \),
- \( \mathcal{A} \cap \mathcal{B} = \emptyset \),
- The number of elements of \( \mathcal{A} \) is not an element of \( \mathcal{A} \),
- The number of elements of \( \mathcal{B} \) is not an element of \( \mathcal{B} \).

Find \( N \).

Answer: 772


Question: Is it possible to approximate a unit vector in \(\mathbb{C}^d\) with complex algebraic vectors in the unit ball in \(\mathbb{C}^d\)? Provide a proof or counterexample.

Answer: Yes


Question:
Evaluate the following limit, where $M$ and $N$ are fixed integers with $M > N > 0$, and $a_i = e^{-N} \cdot \frac{N^i}{i!}$ for $i \ge 0$:
\[ \lim_{k \rightarrow \infty} \sum \{ a_{i_1} a_{i_2} \ldots a_{i_k} \mid i_1 + i_2 + \cdots + i_k \leq kM, i_2 + \cdots + i_k \leq (k-1)M, \ldots, i_k \leq M \}. \]

Answer: 1


Question:
Determine which of the following statements is true for stochastic processes $X_n$ and $Y_n$:

Statement 1: $X_n$ and $Y_n$ are independent $\Rightarrow$ $E[X_nY_n] = E[X_n]E[Y_n]$

Statement 2: $X_n$ and $Y_n$ are independent $\Leftrightarrow$ $E[X_nY_n] = E[X_n]E[Y_n]$

Answer: A


Question: If $f$ is a holomorphic function on the unit disc $\{z:|z|<1\}$, expressed as $f(z)=\sum_{n=0}^{\infty}a_nz^n$, does this series converge uniformly on compact subsets of the disc?

Answer: Yes


Question: In rectangle $ABCD$, point $M$ is the midpoint of side $BC$, and point $N$ lies on $CD$ such that $DN:NC = 1:4$. Segment $BN$ intersects $AM$ and $AC$ at points $R$ and $S$, respectively. If $NS:SR:RB = x:y:z$, where $x$, $y$, and $z$ are positive integers, find the minimum possible value of $x + y + z$.

Answer: 126


Question:
Determine whether the following statement is true or false:

If \( \mathbb{E}[X] < \infty \) and \( \mathbb{E}[Y] < \infty \), then \( \mathbb{E}[XY] < \infty \).

Answer: False


Question: Suppose \((X,\mathbb{X})\) is a measure space and \(f: X \to [0,\infty]\) is a measurable function. Prove that when \(f\) is considered as a function \(f: X \to \bar{\mathbb{R}} := [-\infty,\infty]\), it remains measurable.

Answer: f


Question: For every non-square odd integer \( n > 1 \), does there exist a prime \( p \) such that \( p + n \) is a perfect square?

Answer: Yes


Question: Find the volume of the solid bounded by the $xy$-plane, the cylinder $x^2+y^2=2x$, and the cone $z=\sqrt{x^2+y^2}$. Use spherical coordinates to set up the triple integral for the volume, and evaluate the integral to find the volume of the solid.

Answer: \dfrac{32}{9}


Question: Verify that the set \(SL_n(F)\), consisting of all \(n \times n\) invertible matrices with entries in a field \(F\) and determinant equal to 1, forms a group under matrix multiplication. Ensure that the group axioms of closure, associativity, identity, and inverses are satisfied.

Answer: SL_n(F)


Question: Given the polynomials $p_1 = 40 + 7x + 51x^3$, $p_2 = 8 + x + 24x^2 + 7x^3$, $p_3 = 8x + 4x^3$, and $p_4 = 24 + 3x + 21x^3$, find the value of the Wronskian of the set \( \{p_1, p_2, p_3, p_4\} \) at \( x = 0 \).

Answer: 829440


Question: Let $f: X \to Y$ be a birational morphism where $X$ and $Y$ are projective varieties, and $X$ is smooth. Consider the exceptional locus $\text{Exc}(f) \subseteq X$ and an irreducible divisor $E \subseteq \text{Exc}(f)$. Prove or disprove: For any curve $C \subseteq E$ that is contracted by $f$, the intersection number $C \cdot E < 0$.

Answer: C \cdot E < 0


Question:
Find all real solutions (x, y) to the following system of equations:
\[ x^2 + 3x + \ln(2x + 1) = y \]
\[ y^2 + 3y + \ln(2y + 1) = x \]

Answer: (0, 0)


Question: Evaluate the limit: $$\lim_{x \rightarrow 0_+} (\ln(\cos x))\ln x$$

Answer: 0


Question: Given the functions \( x = 4e^t \) and \( y = 3t \), and the partial derivatives \( f_x(4, 0) = 1 \) and \( f_y(4, 0) = 1 \), find \( \frac{d}{dt} \) at \( t = 0 \).

Answer: 7


Question: Using Burnside's lemma, determine the number of distinct necklaces that can be formed with 12 equally spaced stones, using 4 red, 5 green, and 3 blue beads. Consider the group of symmetries given by the dihedral group \(D_{24}\), which includes rotations and reflections.

Answer: 1170


Question: Find all continuous functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(x+y) = f(x) + f(y) \) for all real numbers \( x \) and \( y \).

Answer: f(x) = kx


Question: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function such that $f(x) = 0$ for all $x$ in an open interval $V \subset \mathbb{R}$. Does there exist an open set $U \subset \mathbb{C}$ such that $f(z) = 0$ for all $z \in U$? Provide a justification for your answer.

Answer: Yes


Question: Determine the boundary of the topological space $S^1 \times I^2$.

Answer: S^1 \times S^1


Question: How many integers \( n \) are there such that \( n^4 - 3n^2 + 9 \) is a prime number?

Answer: 4


Question: Using Stirling's approximation, find \( \lim_{n\to \infty} \frac{n!}{e^n} \).

Answer: \infty


Question: Let $\varepsilon_n \rightarrow 0$ and $b > 1$. Determine whether $\sum_{n<N} \varepsilon_n b^n = o (\sum_{n<N} b^n)$ as $N \rightarrow +\infty$. Justify your answer.

Answer: Yes


Question: Let $\mathscr{C}$ be the class of compact subsets of the Euclidean plane $\mathbb{R}^2$ with connected complement. If $K \in \mathscr{C}$ and $M \subseteq \mathbb{R}^2$ is homeomorphic to $K$, does it follow that $M$ belongs to $\mathscr{C}$? Provide a justification for your answer.

Answer: Yes


Question:
Determine whether the following limit diverges:
\[ \lim_{n \to +\infty} \frac{n+o(n)}{\sqrt{n+o(n)} - \sqrt{n+o(n)}} \cdot \cos(1) \]

Answer: \infty


Question: Evaluate the limit: \[ \lim_{r\to\infty} \int_0^\infty \frac{rx}{\exp(x) + r^2x} \, dx = 0 \]

Answer: 0


Question: Determine the value of $b-a$ such that the integral \( \int_a^b e^{\cos x}(380-x-x^2) \, dx \) attains its maximum value for some interval \((a, b)\) with \(a \leq b\).

Answer: 39


Question: Calculate the residue of the function \( f(z) = \frac{3\sin 4z}{z^4} \) at \( z = \infty \).

Answer: 32


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

Answer: f(1)


Question: Determine the minimum mean value of a Poisson distribution at which it can be approximated by a normal distribution for statistical purposes.

Answer: 10


Question:
Let $X \subseteq \mathbb{R}$ and let $f, g: X \rightarrow X$ be continuous functions such that $f(X) \cap g(X) = \emptyset$ and $f(X) \cup g(X) = X$. Determine which of the following sets cannot be equal to $X$:
1. $[0,1]$
2. $(0,1)$
3. $[0,1)$
4. $\mathbb{R}$

Answer: A


Question: Calculate the double integral \( I = \iint_A y\cos(x) \, dx \, dy \) over the region \( A \), which is the upper half of the disk centered at the origin with radius \( \pi/2 \), using polar coordinates.

Answer: 2


Question: Consider the set \( S = \{0.2, 0.3, 0.5, 0.7, \ldots\} \), where each element is formed by placing a decimal point at the beginning of each prime number. This set is dense in the interval \((0.1, 1)\). Conjecture: For any subinterval \((a, b)\) of \([0.1, 1)\), there exists an integer \(m\) such that for all integers \(k \ge m\), there exists a number \(t \in (a, b)\) such that \(t \cdot 10^k\) is a prime number. Determine if this conjecture is true.

Answer: True


Question: Find the minimum value of \(|5^{4m+3} - n^2|\) for positive integers \(n\) and \(m\).

Answer: 275


Question: Determine whether the following statement is true or false: If a sequence \( \{x_{n}\} \) has no properly divergent subsequence, then the sequence is bounded. Provide a justification for your answer.

Answer: True


Question: Find the units digit of \((3+\sqrt{7})^{2014} + (3-\sqrt{7})^{2014}\).

Answer: 8


Question: Determine the dimension of the space of all linear maps from the vector space $V_5$ to $V_{11}$, where $V_j$ is the real vector space of all polynomials in two variables of degree strictly less than $j$, such that the kernel of the map contains $V_3$ and the image is contained in $V_7$. Provide your answer as a single integer.

Answer: 252


Question: Let $n$ be a positive integer, and let $a_1, a_2, \ldots, a_n$ be rational numbers. Suppose that $a_1^k + a_2^k + \cdots + a_n^k$ is an integer for all positive integers $k$. Must $a_1, a_2, \ldots, a_n$ be integers? Justify your answer.

Answer: Yes


Question: For which values of \( x \in \mathbb{R} \) is the integral \( \int_0^\infty e^{-t}t^{x-1} \, dt \) convergent?

Answer: (0, \infty)


Question: Determine the Picard group of the projective space over the integers, \( \operatorname{Pic}(\mathbb{P}^n_{\mathbb{Z}}) \).

Answer: \mathbb{Z}


Question: Evaluate the integral \( \iint_C \sin (x^2+y^2)\,\mathrm{d}x\,\mathrm{d}y \), where \( C \) is the first quadrant, i.e., \( \int_0^\infty\int_0^\infty \sin (x^2+y^2)\,\mathrm{d}x\,\mathrm{d}y \). Given that \( \int_0^\infty \sin x^2\, \mathrm{d}x = \int_0^\infty \cos x^2\, \mathrm{d}x = \frac{1}{2}\sqrt{\frac{\pi}{2}} \), find the value of the double integral.

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit: \[ \lim_{(x,y)\to(0,0)} \frac{\sin( |x| + |y|) + |y|(e^x - 1)} {|x| + |y|} \]

Answer: 1


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{dz}{\cosh(z)} \) using contour integration. Consider a rectangular contour with vertices at \(-R\), \(R\), \(R + \pi i\), and \(-R + \pi i\), oriented in the positive sense. Show that the value of the integral is \( \frac{\pi}{2} \).

Answer: \dfrac{\pi}{2}


Question: Given that \(X\sim\text{Bin}\left(5,\frac{1}{2}\right)\) and \(Y\sim \text{Unif}(0,1)\), calculate the ratio \(\frac{P(X+Y\leq2)}{P(X+Y\geq5)}\).

Answer: 6


Question: Let $U \subseteq \mathbb{R}^n$ be an open set and $f \in C_c^2(U)$ be a function such that $f \geq 0$ in $U$. Prove that the following estimate holds: \[ \sup_U \frac{|Df|^2}{f} \leq 2 \sup_U |D^2f|. \]

Answer: \sup_U \frac{|Df|^2}{f} \leq 2 \sup_U |D^2f|


Question: Evaluate the integral \( \int_{0}^{6}\int_{-\sqrt{36 - x^2}}^{\sqrt{36 - x^2}}\int_{-\sqrt{36 - x^2 - z^2}}^{\sqrt{36 - x^2 - z^2}} \dfrac{1}{\sqrt{x^2 + y^2 + z^2}} \ dy \, dz \, dx \) by changing to spherical coordinates.

Answer: 36\pi


Question:
Find the smallest value of the sum a + b, where a and b are positive integers such that the following equation holds:
\[ {1999 \choose 0} - {1999 \choose 2} + {1999 \choose 4} - \dots - {1999 \choose 1998} = a^b \]

Answer: 341


Question: Determine the values of $k$ for which the nonlinear system \( \dot{x} = x(x^2 - 1 - k) \) is stable using a Lyapunov function.

Answer: k > -1


Question: How many people do you need in a room so that you expect there to be a shared birthday on a Monday, assuming there are 52 Mondays in a year and 365 days in a year?

Answer: 73


Question: In \(\mathbb{R}^4\), let \(\underline{a} = (1, 2,-2, 0)\), \(\underline{b} = (2, 1, 0, 4)\), and \(\underline{c} = (5, 7, 3, 2)\). Verify that the orthogonal projection of \(\underline{c}\) onto \(\langle \underline{a}, \underline{b} \rangle^\perp\) is \((2, 4, 5, -2)\).

Answer: (2, 4, 5, -2)


Question: Determine the value of the supremum $A = \sup \limits_{f} \{m^*(f[0, 1])\}$, where $f$ are all functions satisfying the following conditions: $f: [0, 1] \rightarrow [0, 1]$; for all $x$, $f(x) - x \in \mathbb{Q}$; and for all $x, y$, if $x - y \in \mathbb{Q}$, then $f(x) = f(y)$. Find the value of $A$, given that $0 < A \leq 1$. \(m^*\) denotes the outer measure.

Answer: 1


Question: Prove the statement $(F \implies E) \land (F \land E \implies R) \implies (F \implies R)$ using the following logical rules: Modus ponens, Modus tollens, Hypothetical syllogism, Disjunctive syllogism, Constructive Dilemma, And-introduction and elimination, Or-introduction, DeMorgan's laws, Commutivity, Associativity, distribution, double negation, Transposition (contrapositive), Material implication, Exportation, and tautologies.

Answer: (F \implies E) \land (F \land E \implies R) \implies (F \implies R)


Question: Determine the automorphism group of the quaternion group \( Q_8 \).

Answer: S_4


Question: Evaluate the expression: \( ^{2n}C_n - ^{n}C_1 \cdot ^{2n-2}C_n + ^{n}C_2 \cdot ^{2n-4}C_n - \ldots \). What is the value of this expression?

Answer: 2^n


Question: Compute the differential of the i-th coordinate map \(x_i: \mathbb{R}^n \to \mathbb{R}\).

Answer: dx_i


Question: Evaluate the integral \( \int_C (x + 2y) \, dx + (2z + 2x) \, dy + (z + y) \, dz \) using Stokes' theorem, where \( C \) is the intersection of the cylinder \( x^2 + y^2 = 2y \) and the plane \( y = z \). Assume the curve \( C \) is oriented counterclockwise when viewed from above.

Answer: 0


Question: Let $L$ be a Dedekind domain and $I = (a) + (b)$ a non-principal proper ideal in $L$. Consider the equation $xa + yb = 0$. Is it true that there exists a proper ideal $J$ such that any $x, y$ that satisfy the equation belong to $J$? Provide a justification for your answer.

Answer: Yes


Question: Let $K_1, \ldots, K_n \subset \mathbb{R}^n$ be convex bodies such that for each subset $I \subset \{1,\ldots,n\}$, the set \(\left( \bigcap_{i \in I} K_i \right) \setminus \left( \bigcup_{i \not\in I} K_i \right)\) has nonempty interior. Does it follow that the intersection of the boundaries \(\bigcap_{i \in I} \partial K_i\) is nonempty?

Answer: Yes


Question: Evaluate the limit: \( \lim_{x\to 0} (1 - \sin x)^{\cot 2x} \)

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


Question: Evaluate the integral \( \int_{0}^{2\pi}\frac{1}{2-\cos^2(x)} \, dx \) using complex analysis.

Answer: \pi \sqrt{2}


Question:
Given an open set $\Omega \subset \mathbb{R}^N$ and a lower semicontinuous function $g: \Omega \to \mathbb{R}^+$ such that $g \geq 1$, consider a sequence of Lipschitz continuous functions $(g_k)$ satisfying $1 \leq g_k(x) \leq g(x)$ and $g_k \nearrow g$ for every $x \in \Omega$. Let $L: C_c(\Omega) \to \mathbb{R}$ be a continuous linear functional. Define:

$$a = \sup\{ L(f) \mid f \in C_c(\Omega), |f(x)| \leq g(x), \operatorname{spt}(f) \subset U \}$$

for each compact $U \subset \Omega$, and

$$a_k = \sup\{ L(f) \mid f \in C_c(\Omega), |f(x)| \leq g_k(x), \operatorname{spt}(f) \subset U \}.$$

Prove or disprove: $a_k \nearrow a$ as $k \to \infty$. Provide justification for your answer.

Answer: a_k \nearrow a


Question: Evaluate the limit \( \lim_{x \to 0} \frac{f(x) - \sqrt{x^2 + 9}}{x} \) given that the function \( f : \mathbb{R} \to \mathbb{R} \) satisfies \( |f(x) - 3| \leq x^2 \).

Answer: 0


Question: Determine the eigenvalues of the linear operator \( T: R[X]_n \rightarrow R[X]_n \) defined by \( T(f(x)) = f(ax+b) \), where \( R[X]_n \) is the vector space of polynomials of degree \( \leq n \).

Answer: 1, a, a^2, \ldots, a^n


Question: Calculate \( \omega(X,Y) \) for the differential form \( \omega = 2x \, dx \wedge dy + y^2 \, dx \wedge dz \) and the vector fields \( X = x^2y \frac{\partial}{\partial y} + x \frac{\partial}{\partial z} \) and \( Y = x \frac{\partial}{\partial y} \).

Answer: 0


Question:
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the following conditions:

1. \( f(x+y) = f(x) + f(y) \) for all \( x, y \in \mathbb{R} \).
2. \( \lim_{x \to \infty} f(x) \) exists.

Answer: 0


Question: Let $X_1, \ldots, X_{735}$ and $Y_1, \ldots, Y_{880}$ be independent random variables such that $P(X_i=0)=\frac{3}{7}$, $P(X_i=1)=\frac{4}{7}$, and $P(Y_i=0)=P(Y_i=1)=\frac{1}{2}$. Using the Central Limit Theorem, find $P\left(\sum_{i=1}^{735} X_i < \sum_{i=1}^{880} Y_i\right)$. Express your answer as a probability.

Answer: 0.8413


Question: Let $(X_n)_{n \in \mathbb{N}}$ be a collection of topological spaces, and let $X$ be a set with injective maps $X_n \hookrightarrow X$. Consider the final topology on $X$ given by these maps, where $G \subseteq X$ is open if and only if $G \cap X_n$ is open in $X_n$ for all $n$. Let $Y \subseteq X$ be a closed subspace, and define $Y_n := Y \cap X_n$. Is the subspace topology on $Y$ the same as the final topology on $Y$ given by the maps $Y_n \hookrightarrow Y$? In other words, is it true that $G \subseteq Y$ is open if and only if $G \cap Y_n$ is open in $Y_n$ for all $n$?

Answer: Yes


Question: Evaluate the limit using Taylor expansions: \[ \lim_{x\to 0} \frac{5^{(1+\tan^2x)} -5}{1-\cos^2x} \]

Answer: 5 \ln 5


Question: Find the 9th derivative at 0 of the function \( \frac{x^3 e^{2x^2}}{(1-x^2)^2} \).

Answer: 5564160


Question: Find the last eight digits of the binary representation of \(27^{1986}\).

Answer: 11011001


Question: Evaluate the line integral \( \int_C (xy) \, dx + \exp(y^2) \, dy \) using Green's Theorem, where \( C \) is the triangle with vertices \((0,0)\), \((2,0)\), and \((2,1)\). Ensure your answer is positive.

Answer: \dfrac{4}{3}


Question: Evaluate the integral \( \int _{-\infty} ^{+\infty} \exp(-(x+i)^2) \, dx \) using complex integration techniques.

Answer: \sqrt{\pi}


Question: Given that \( f(x) + \ln(x^2 + f(x)^2) = 4 \), find \( f'(e^2) \).

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


Question: Let \( A = \begin{bmatrix}2 & -1\\-1 & 2\end{bmatrix} \). Define the linear transformation \( T \) as \( T(X) = AX - XA \), where \( X \) is a \( 2 \times 2 \) matrix. Find the dimension of the null space of \( T \).

Answer: 2


Question: Find the closed form expression for the sequence \(a_n\) defined by the recursive relation \(a_n = 2a_{n+1} - a_{n+2} + 4\) with initial conditions \(a_1 = 3\) and \(a_2 = 9\).

Answer: 2n^2 + 1


Question: Evaluate the integral \( \int_0^1 \left(\frac{1}{x} -  \biggl\lfloor \frac{1}{x}\biggr\rfloor\right) dx \).

Answer: 1 - \gamma


Question: Calculate \( \mathop{\lim}\limits_{z\to 0} \, \frac{e^z+z\mathrm{Log}z}{1-z^2\mathrm{Arg}z} \).

Answer: 1


Question: Calculate the remainder when \( \prod_{i=1}^{1903} (2^i + 5) \) is divided by 1000.

Answer: 931


Question: Find the positive real number \( \alpha \) such that \( \lim_{n\to \infty} \frac{(n^n)(\alpha^n)\sqrt{2\pi n}}{n!} = 1 \). (Hint: use the ratio test.)

Answer: \dfrac{1}{e}


Question: Given the joint characteristic function of the random variables $(X, Y)$, determine the probability $P(X < Y)$. Provide your answer as a numerical value.

Answer: \dfrac{1}{2}


Question: Find the coordinates of the center of mass of the curve defined by the equations \( x^2 + y^2 = 1 \) and \( x + 2y + 3z = 12 \).

Answer: (0, 0, 4)


Question: Given a random variable $X$ with probability density function $f(x)=\frac{5}{x^6}$ for $x > 1$, find the covariance $\text{Cov}(X, \frac{1}{X})$. Express your answer as a simplified fraction.

Answer: -\dfrac{1}{24}


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

Answer: e^6


Question: Determine the values of \(a \in \mathbb{R}\) for which the function \(f_n(x,y,z) = \frac{\arctan^a (|x|+|y|)e^z}{n^4+4x^2+y^2}\) is integrable over the domain \(D = \{(x,y,z) : 1 < x^2 + y^2 + z < 2\}\) for all \(n \geq 1\).

Answer: a > -2


Question: Let $(X, \mathcal{R})$ be a measurable space, where $X = \bigcup_n A_n$ and each $A_n$ is measurable. For each $A_i$, there is a measurable function $f_i: A_i \to \mathbb{R}$. If $x \in A_i \cap A_j$, then $f_i(x) = f_j(x)$. Define $f: X \to \mathbb{R}$ by $f(x) = f_i(x)$ for $x \in A_i$. Is the function $f$ measurable?

Answer: Yes


Question: Determine whether the statement is true or false: A linear transformation $T: V \rightarrow W$ is surjective if and only if there exists a linear transformation $S: W \rightarrow V$ such that $TS = I_W$. Provide a justification for your answer.

Answer: True


Question: Compute the volume of the region bounded by the surfaces \(z=0\), \(y^2=x^3\), and \(\sqrt{x} + z=2\).

Answer: \dfrac{128}{15}


Question: Determine the second last digit of \(7^{(54^{1000} + 2)}\) in base 9.

Answer: 5


Question: Evaluate the limit \( \lim_{k \to \infty} \int_{0}^1 x^{- \frac{1}{2}}\cos(x^k)e^{-\frac{x^2}{k}} \, dx. \)

Answer: 2


Question: For a random variable X that follows a Uniform distribution on the interval (1, 9.9) and a conditional random variable Y given X = x that follows a Normal distribution with mean 1.4 and variance x^2, calculate the covariance Cov(X, Y).

Answer: 0


Question: Consider the diffusion process defined by \(X_t=\int_0^t a(s,X_s)\,dW_s\) for \(t\ge 0\), where \(W_\cdot\) is a standard Wiener process and \(a\) is a smooth positive function bounded away from zero. Does \(X_1\) have a bounded probability density function?

Answer: Yes


Question: Find \( \delta \) such that \( |f(x) - l| < \varepsilon \) for all \( x \) satisfying \( 0 < |x - a| < \delta \), given \( f(x) = \frac{x}{1 + \sin^2(x)} \), \( a = 0 \), and \( l = 0 \).

Answer: \varepsilon


Question: If $\{A_i\}$ is a sequence in the endomorphism of $\mathbb{R}^n$ such that $e^{A_i}$ converges to $e^A$, does it necessarily follow that $A_i$ converges to $A$? Answer yes or no.

Answer: No


Question:
Find the minimum value of the expression \(x_1 + 2x_2 + 3x_3\) subject to the constraints:
\[ x_1 - 2x_2 + x_3 = 4 \]
\[ -x_1 + 3x_2 = 5 \]
and
\[ x_1 \ge 0, \quad x_2 \ge 0, \quad x_3 \ge 0 \]

Answer: \dfrac{76}{3}


Question: Evaluate the line integral of the vector field \( \vec{F} = (5x+y)\mathbf{i} + (3y-2xz)\mathbf{j} + z\mathbf{k} \) along the line segment connecting the points \( A(1,2,0) \) and \( B(3,-4,1) \).

Answer: \dfrac{101}{2}


Question: Evaluate the limit: \[ \lim_{x \to 0} {\left (\frac{1+x\cdot2^x}{1+x\cdot3^x}\right)}^\frac{1}{x^2} \] without using l'Hôpital's rule.

Answer: \dfrac{2}{3}


Question: Let $R$ be a Noetherian domain and $M$ and $N$ be two faithful $R$-modules. Determine if it is true that $\operatorname{Ann}_R(M\otimes_R N)=0$. Provide a justification for your answer.

Answer: 0


Question: Calculate the expectation \( E(X) \) for the random variable \( X \) with the probability mass function \( f(x) = \frac{1}{(x+1)(x+2)} \) for \( x = 0, 1, 2, 3, \ldots \).

Answer: \infty


Question: Count the number of functions \( f: \{1, 2, \ldots, 12\} \rightarrow \{1, 2, \ldots, 12\} \) such that if \((a, b, c)\) is a multiplication pair, then \((f(a), f(b), f(c))\) is also a multiplication pair.

Answer: 7776


Question:
Consider an unweighted undirected graph $G$ with the property that for some integer $r$, for all nodes $v$, the average degree of nodes within the $r$-ball around $v$ satisfies:
$$\frac{\sum \limits_{u \in B(v, r)} \deg(u)}{|B(v, r)|} \le D$$
where $B(v, r) = \{u \, | \, \mathop{dist}(u, v) \le r\}$. Does this imply that the number of edges $|E(G)| = O(nD)$?

Answer: Yes


Question: Compute the line integral \( \int_{C} F \cdot dr \) for the vector field \( F = \frac{2x}{\sqrt{x^2 + y^2}}\hat{i} + \frac{2y}{\sqrt{x^2 + y^2}}\hat{j} \) over the curve defined by \( x(t) = t^3 + 1, y(t) = (1 - t^2)e^{2t} \) for \( t \in [-1, 1] \). Note that the vector field is not defined at the origin, and the curve passes through this point.

Answer: 4


Question:
Find the expectation of the posterior distribution given by:
\[ p_{\theta|x}(\theta) = \frac{\theta^x(1-\theta)^x\cdot6\theta(1-\theta)}{\int_0^1\theta^x\cdot6\theta(1-\theta)d\theta} \] where \( X \) is a random variable from the Bernoulli distribution with parameter \( \theta \), and the prior density \( p(\theta) = 6\theta(1 - \theta) \) for \( \theta \in (0, 1) \).

Answer: \dfrac{1}{2}


Question: Find the moment generating function (m.g.f.) of a standard normal random variable $Z \sim \text{Norm}(0, 1)$. Express the m.g.f. as a function of $t$.

Answer: e^{t^2/2}


Question: Prove that if \( f:\mathbb{R}\to\mathbb{R} \) is a continuous function, then \[ \lim_{t\to0+}\frac{1}{\pi}\int_{-1}^1\frac{tf(x)}{t^2+(x-a)^2}\,\mathrm{d}x=f(a) \] for every \( a\in[-1,1] \).

Answer: f(a)


Question:
In Rabin's cryptosystem, given the public and private keys with \( p = 127 \), \( q = 131 \), \( N = 16637 \), and \( B = 12345 \), the ciphertext \( c \) is computed as follows for the plaintext \( m = 4410 \):

\[ c = m(m + B) \mod N = 4633 \]

To decrypt, compute:

\[ t = \left(\frac{B^2}{4} + c\right) \mod N \]

Find the value of \( t \).

Answer: 1500


Question:
Calculate the expectation \(\mathbb{E}(X^2Y^2)\) for the bivariate normal distribution with the joint density function:
\[
f_{X,Y}(x,y)=\frac{1}{2\pi\sqrt{1-\sigma^{2}}}\exp\left\{ -\frac{1}{2\left(1-\sigma^{2}\right)}\left(x^{2}+y^{2}-2\sigma xy\right)\right\}.
\]

Answer: 1 + 2\sigma^2


Question: Given the vector field \( \vec{F} = ay\hat{i} + z\hat{j} + x\hat{k} \), and the positively oriented closed curve \( C \) defined by \( x^2 + y^2 = 1, z = 0 \), if \( \int_C \vec{F} \cdot \vec{dr} = \pi \), find the value of \( a \).

Answer: -1


Question: Let \( n = km \) where \( k, m \in \mathbb{N} \) and \( m \) is odd. Prove that \( \left(2^{k}+1\right)\left(2^{km-k}-2^{km-2k}+2^{km-3k}-2^{km-4k}+\cdots+1\right) = \left(2^{km}+1\right) \).

Answer: 2^{km} + 1


Question:
Evaluate whether the following sum equals 1:
\[ \sum_{j=1}^{n} \binom{n}{j} \cdot (-1)^{j+1} \cdot \frac{(n-j)^{n-1}}{n^{n-1}} = 1 \]

Answer: 1


Question: Is the space $C([0, 1], [0, 1])$ of continuous functions from $[0, 1]$ to $[0, 1]$ dense in the space $\Pi_{[0, 1]}[0, 1]$ of all such functions with respect to pointwise convergence?

Answer: Yes


Question: Suppose $G$ is the Galois group of an irreducible degree 5 polynomial $f \in \mathbb{Q}[x]$ with $|G|=10$. Determine whether $f$ must have precisely 1 or 5 real roots.

Answer: 1


Question:
Find the coefficient of $x^7y^5$ in the expansion of the expression:
\[(x+y)^{12} + 7(x^2+y^2)^6 + 2(x^3+y^3)^4 + 2(x^4+y^4)^3 + 2(x^6+y^6)^2 + 6(x+y)^2(x^2+y^2)^5.\]

Answer: 912


Question: Find the number of elements in the conjugacy class of the permutation \( \partial = (1\ 10)(2\ 6\ 3\ 7\ 6\ 8\ 12)(4)(9\ 11) \) in the symmetric group \( S_{12} \).

Answer: 4989600


Question: Let $A$ be a noetherian integral local ring with fraction field $K$. Consider an algebraic field extension $L$ of $K$, and let $B$ be the integral closure of $A$ in $L$. If $A$ is regular, is the ring $B$ also regular?

Answer: No


Question: Determine whether the following statement is true or false: If $f$ is a non-constant entire function that satisfies $|f(z)| \leq 2|ze^z|$, then $f$ has an essential singularity at $\infty$. Justify your answer.

Answer: A