Question: Suppose you have $10,000 and invest in the stock market by buying 500 shares of DGCo at $10 each. You trade every day, buying 10 shares when the price increases by $1 and selling 10 shares when it decreases by $1. The stock price is equally likely to go up or down by $1 at any time. Calculate the expected amount of money you will have after trading for one year.

Answer: 10000


Question: Given a number \(n\) in base \(g\), define the function \(\varphi\) as follows: \(\varphi^{(0)}(n) = n\), \(\varphi^{(1)}(n) = \sum_{i=1}^{p_1}a_i^s\), where \(n = (\overbrace{a_1a_2\cdots a_{p_1}})_g\), and \(\varphi^{(k+1)}(n) = \sum_{i=1}^{p_k}b_i^s\) for \(\varphi^{(k)}(n) = (\overbrace{b_1b_2\cdots b_{p_k}})_g\). Is the sequence \(\{\varphi^{(k)}(n)\}\) eventually periodic for all positive integers \(n\), given positive integers \(g\) and \(s\)?

Answer: Yes


Question: Determine the dimension of the range space of the linear operator \( T: C[0, 1] \to C[0, 1] \) defined by \((Tf)(x) = \int_{0}^{1}\sin(x + y)f(y) \, dy\) for \( x \in [0, 1] \).

Answer: 2


Question:
Given that \( \det(A) = \frac{1}{24} \), solve for the determinant of the matrix expression:
\[
\det\left(\left(\frac{1}{3}A\right)^{-1} - 120 A^*\right)
\] where \( A^* \) is the adjugate of \( A \).

Answer: -192


Question: Let $T: P_2(\mathbb{R}) \to P_2(\mathbb{R})$ be a linear transformation defined by the following extensions: $T(x) = x-2$, $T(x^2-x) = x+2x^2$, and $T(x-2+x^2) = -1-x^2$. Compute $T(2+x^2)$. Provide your answer as a polynomial.

Answer: 5x^2 + 5x - 5


Question: Find the volume of the region that lies above the cone \( z = \sqrt{x^2 + y^2} \) and below the paraboloid \( z = 2 - x^2 - y^2 \).

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


Question: Find \( \lim_{x \to 0^{+}} \frac{\pi^{x\ln x} - 1}{x} \) if it exists.

Answer: -\infty


Question: Determine the function to which the series \( f(x) = \sum_{n=1}^\infty \frac{(-1)^n x^{2n-1}}{2n-1} \) converges.

Answer: -\arctan(x)


Question: Show analytically that the expected value of the random variable \( V \), defined as the least number \( n \) such that the sum of \( n \) independent uniform \([0, 1]\) random variables exceeds 1, is equal to \( e \).

Answer: e


Question: Let $f \colon \mathbb{C} \longrightarrow \mathbb{C}$ be an analytic function such that $f(2z) = 4f(z)$ for all $z \in \mathbb{C}$. Given that $f(0) = 0$ and $f(1) = 1 + 2i$, find $f(\sqrt{2} + i\,\sqrt{2})$. \( \text{Provide your answer in the form } \boxed{\text{answer}}. \)

Answer: -8 + 4i


Question:
Let $A$ be a $10 \times 10$ matrix with complex entries such that all eigenvalues are non-negative real numbers and at least one eigenvalue is positive. Determine which of the following statements is always false:

A. There exists a matrix $B$ such that $AB - BA = B$.
B. There exists a matrix $B$ such that $AB - BA = A$.
C. There exists a matrix $B$ such that $AB + BA = A$.
D. There exists a matrix $B$ such that $AB + BA = B$.

Choose the statement that is always false.

Answer: B


Question: Evaluate the limit: $$\lim_{x \to 0} \left( \frac{\tan (\sin (x))-\sin (\tan (x))}{x^7} \right).$$

Answer: \dfrac{1}{30}


Question: Let $G$ be a group of order 27. Prove that if $G$ is not abelian, then $|Z(G)| = 3$, where $Z(G) = \{z \in G: zg = gz, \forall g \in G\}$. Provide a proof or counterexample to support your conclusion.

Answer: 3


Question: What is the maximum number of triangles that can be formed by joining the vertices of a decagon?

Answer: 120


Question: Given that $f(x) + f'(x) - \frac{1}{x+1}\int_{0}^{x}f(t)dt = 0$ and $f(0) = 0$, find the expression for $f'(x)$. Assume $f \in C^{1}[0,\infty)$. What is $f'(x)$?

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to \infty}\left[(n+1)\int_{0}^{1}x^n\ln(x+1)dx\right] \]

Answer: \ln 2


Question: Determine whether the zeroth cohomology group of the 0-sphere with coefficients in a group $G$, denoted $H^0(S^0;G)$, is isomorphic to $G \oplus G$ or $G$. Justify your answer.

Answer: G \oplus G


Question: Determine whether there exists a 6-regular graph on 42 vertices.

Answer: Yes


Question: Determine the number of subfields of a finite field with $2^{15}$ elements, including the trivial subfield and the field itself.

Answer: 4


Question: Evaluate the integral \( \iint_D \left[3-\frac{1}{2}\left( \frac{x^2}{a^2}+\frac{y^2}{b^2}\right)\right] \, dx \, dy \) where \( D \) is the region defined by \( \frac{x^2}{a^2}+\frac{y^2}{b^2} \le 4 \). Use a change of variables to simplify the integration.

Answer: 8\pi ab


Question:
Calculate the following limit using the Central Limit Theorem:
\[ \lim_{n\rightarrow\infty}\int_{0}^{n}\frac{1}{(n-1)!}x^{n-1}e^{-x}dx. \]

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{n \to \infty} n(\sqrt[n]{a} - 1) \] where \( a > 0 \).

Answer: \ln a


Question: Find the largest value of \(a\) such that the function \(f(x) = \frac{1}{6}x^3 - x^2 + \frac{3}{2}x - 5\) is invertible over the domain \((-\infty, a]\).

Answer: 1


Question: Find a simple formula for \( f(n) \), where \( f_n \) is the number of ways to choose a permutation \( \pi \) of \([n]\) and then a subset of the cycles of \( \pi \). For example, \( f(2) = 6 \). Provide a formula for \( f(n) \) that does not involve any sums.

Answer: (n + 1)!


Question: Evaluate the limit \( \lim_{n\to\infty}\frac{n^k(k+1)!(n-(k+1))!}{n!} \) for any fixed integer \( k \).

Answer: 0


Question: Find the least positive integer \( k \) such that \( \cos^k \left(\frac{\pi}{2k}\right) \geq \frac{99}{100} \).

Answer: 123


Question: In a second-countable topological space $(X, \mathcal{T})$, if $S \subset X$, does every limit point of $S$ correspond to the limit of a sequence in $S$? Answer yes or no and justify your answer.

Answer: Yes


Question: Determine the smallest positive integer \( n \) such that the number of distinct groups of order \( n \) is greater than \( n \) itself.

Answer: 32


Question: Given a polynomial \( P(z) = \sum_{n=0}^d a_n z^n \), and the condition that for all \( n = 0, \dots, d \), \( \oint_{|z|=1} \frac{P(z)}{(2z-1)^{n+1}} dz = 0 \), determine the polynomial \( P(z) \).

Answer: 0


Question:
Let $H$ be a subgroup of $S_8$ such that the order of $H$ is $9$. Determine the nature of $H$ from the following options:

A) cyclic
B) not abelian
C) abelian but not cyclic
D) if $H$ is abelian then it must be cyclic

Choose the correct option.

Answer: C


Question: In a set of 1985 positive integers, each integer can be expressed in the form \( p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_9^{\alpha_9} \), where \( p_1, p_2, \ldots, p_9 \) are distinct prime numbers and \( \alpha_1, \alpha_2, \ldots, \alpha_9 \) are non-negative integers. Determine the minimum number of pairs of integers in this set such that their product is a perfect square.

Answer: 2883


Question: Evaluate the integral: \[ \int_0^{\frac{\pi}{2}} \frac{\sin^{1395}x}{\sin^{1395}x + \cos^{1395}x}\, dx. \]

Answer: \dfrac{\pi}{4}


Question: Is the topology on a compact connected Lie group metrizable? Specifically, if $G$ is a compact connected Lie group, does there exist a metric on $G$ that induces the underlying topology, ensuring $G$ is compact and connected with respect to this metric?

Answer: Yes


Question: Is it true that for some ordinal \(\alpha\), the equation \({\aleph _{\alpha+1}}^{\aleph _{\alpha}}={\aleph _{\alpha+1}}\) holds? Justify your answer.

Answer: Yes


Question: Find the area of trapezium $ABCD$ where $AB$ and $DC$ are parallel, $AB = 4$, $DC = 10$, and the diagonals $AC$ and $BD$ are perpendicular. The non-parallel sides $DA$ and $CB$ meet at an angle of $45^{\circ}$. Calculate the area of the trapezium.

Answer: \dfrac{140}{3}


Question: Does there exist a continuous function on the interval $[a, b]$ that is not identically zero, but for every sub-segment $[\alpha, \beta] \subset [a, b]$, there exists a sub-segment $[\alpha ', \beta '] \subset [\alpha, \beta]$ where the function is zero?

Answer: No


Question: Determine whether the sequence of random variables $\frac{N_n}{n}$, where $N_n$ follows a Poisson distribution with parameter $n$, converges to 1 in $L^1$. Specifically, compute $\lim_{n \to \infty} \mathbb{E}\left|\frac{N_n}{n} - 1\right|$.

Answer: 0


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f\left(x^2+f(y)\right) = y + f(x)^2 \) for all real numbers \( x \) and \( y \).

Answer: f(x) = x


Question: Determine the number of zeros of the polynomial \( z^4 - 5z + 1 \) that lie within the annulus \( \{ z \mid 1 < |z| < 2 \} \).

Answer: 3


Question: Find the number of nonnegative integer solutions to the equation \( \sum_{j=1}^{8}x_j=24 \) such that \( \sum_{j=1}^4x_j > \sum_{j=5}^8x_j \).

Answer: 1211275


Question: Given the joint probability density function \( f_{X,Y}(x,y) = C(x+y^3) \) for \( 0 \leq x \leq 1 \) and \( 0 \leq y \leq 1 \), find the probability \( P(X < 0.5 \mid Y \leq 0.5) \).

Answer: \dfrac{9}{34}


Question: Evaluate the limit: \[ \lim_{a,b\to + \infty} \iint_{[0,a]\times[0,b]} e^{-xy} \sin x \,dx\,dy \]

Answer: \dfrac{\pi}{2}


Question: A family buys two insurance policies from the same company. Losses under these policies are independent and follow a continuous uniform distribution on the interval from 0 to 10. One policy has a deductible of 1, and the other has a deductible of 2. The family experiences exactly one loss under each policy. Calculate the probability that the total benefit paid to the family does not exceed 5.

Answer: \dfrac{59}{200}


Question: Find the supremum of the set \( \{4x-2y : x, y \in \mathbb{R}, x^2+y^2<4\} \).

Answer: 4\sqrt{5}


Question: Find all polynomials \( f \in \mathbb{Z}[x] \) such that \( x^6 + x^3 f''(x) = f(x^3) + x^3 \).

Answer: x^2 + x


Question: Find all real numbers \( x \) such that \( \arcsin(2x^2-1) - 2\arcsin x = -\frac{\pi}{2} \).

Answer: [0, 1]


Question: Consider the product measure \((\mu \times \nu)\) defined on \(\mathbb{R} \times \widetilde{\mathbb{R}}\), where \(\mu\) is the Lebesgue measure on \(\mathbb{R}\) and \(\nu\) is the counting measure on \(\widetilde{\mathbb{R}}\), the real line with the discrete topology. Let \(E = \{(x, x) \mid x \in [0, 1]\}\). Determine whether \((\mu \times \nu)(E) = 0\) when \(\widetilde{\mathbb{R}}\) is endowed with the discrete topology and the associated discrete \(\sigma\)-algebra.

Answer: 0


Question: Calculate the work done by the force \( F(x, y, z) = (x^4y^5, x^3) \) along the curve \( C \) given by the part of the graph of \( y = x^3 \) from \((0, 0)\) to \((-1, -1)\).

Answer: \dfrac{11}{20}


Question:
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = \inf\{|x - me| : m \in \mathbb{Z}\} \). Consider the sequence \( \{f(n)\} \) for integer values of \( n \). Determine which of the following statements is true:

(a) \( \{f(n)\} \) is convergent.
(b) The set of subsequential limits of \( \{f(n)\} \) is \([0, \frac{e}{2}]\).
(c) \( \{f(n)\} \) has no convergent subsequences.
(d) The set of subsequential limits of \( \{f(n)\} \) is at most countable.

Answer: B


Question: Let \( X \) be a Hilbert space with scalar product \((\cdot,\cdot)\). For a given angle \( \varphi' \in [0, \pi) \) and a unit vector \( v' \in X \), is there a unit vector \( w' \in X \) such that \((v', w') = \cos(\varphi')\)?

Answer: Yes


Question: Given matrices \( B, C \in M_n(\mathbb{R}) \) with \( \det B \neq 0 \), \( \det C \neq 0 \), and \( \det(B+C) \neq 0 \), let \( A = B^{-1}C(B+C)^{-1}BC^{-1}(B+C) \). Prove that \( A^2 = I_n \).

Answer: A^2 = I_n


Question: Let $f_i$, $i=1,\dotsc,n$, be mutually orthogonal Abelian projections in a von Neumann algebra, and let $e\leq\sum f_i$. Determine whether there exist mutually orthogonal Abelian projections $e_j$, $j=1,\dotsc,m$, such that $e=\sum e_j$. Provide a justification for your answer.

Answer: Yes


Question:
Let \( A \) be an \( n \times n \) real matrix such that:

1. \( a_{ii} = 0 \) for all \( i \), and \( 1 > a_{ij} \geq 0 \) for all \( i, j \).
2. \( \sum_{j=1}^n a_{ij} \in (0,1] \) for all \( i \).
3. There exists at least one row \( i_0 \) such that \( \sum_{j=1}^n a_{i_0j} < 1 \).

Can we conclude that the maximum eigenvalue of \( A \) is less than 1?

Answer: No


Question: Determine the cardinality of the set \(\{(A,B,C) : A \subset B \subset C \subset \{1,2,\ldots,n\}\}\).

Answer: 4^n


Question: Determine the homology class \([i] \in H_2(E;\mathbb{Z})\) to which the inclusion map \(i: S^2 \to E\) belongs, given that \(S^2 \to E \to B\) is a smooth fiber bundle with \(E\) 1-connected and \(B\) 3-connected. Assume \(i\) is an embedding, and use the isomorphisms \(\pi_2(E) \cong H_2(E;\mathbb{Z})\) and \(\pi_2(S^2) \cong \pi_2(E)\) to find \([i]\).

Answer: 1


Question:
Find the expected value of $X$, denoted as $E(X)$, given the joint probability density function (pdf) of $X$ and $Y$:
\[
f(x,y) = \begin{cases}
\frac{e^{-y}}{y}, & 0 < x < y, \ 0 < y < \infty \\
0, & \text{otherwise}.
\end{cases}
\]

Answer: \dfrac{1}{2}


Question: Determine the values of \(\alpha \in \mathbb{R}\) for which the integral \(\int_0^{+\infty} \frac{\ln |1-x^2|}{x^{\alpha}}dx\) is convergent.

Answer: (1, 3)


Question: Find the residue of the function \( f(z) = \frac{1}{(1+\cos z)^2} \) at the point \( z_0 = \pi \).

Answer: 0


Question: Let $f_n, f : [a, b] \to [0, 1]$, $n \in \mathbb{N}$, be continuous functions on $[a, b]$, such that the value $1$ is attained for $f$ and for all $f_n$. Define $g^{-} : [0, 1] \to [a, b]$ by $g^{-}(\alpha) = \inf \{x \in [a, b] ; g(x) \ge \alpha \}$. Is it true that if $f_n \to f$ uniformly on $[a, b]$, then $f_n^{-}(\alpha) \to f^{-}(\alpha)$ almost everywhere in $[0, 1]$?

Answer: Yes


Question: Given a group $G$ and a Sylow $p$-subgroup $P$ of $G$, determine whether the normalizer $N(P)$ of $P$ is a $p$-group.

Answer: No


Question: Find the 9th derivative of the function represented by the power series \( \sum_{n=2}^{\infty}\frac{(x-2)^{2n+1}}{n^22^n} \) evaluated at \( x = 2 \).

Answer: \dfrac{2835}{2}


Question: Determine whether any real continuous function on a compact interval $[0,T]$ can be uniformly approximated by a sequence of step functions. Provide a justification for your answer.

Answer: Yes


Question: Provide an example of a ring $R$ with at least two elements, such that for all $a \in R^*$, there exists a unique $b \in R^*$ such that $aba = a$. The example should be different from $\mathbb{Z}_2$.

Answer: \mathbb{Z}_3


Question: Let $X$ be a random variable with an absolutely continuous cumulative distribution function $F(x)$. Determine the distribution of the random variable $Y = \log(1 - F(X))$. Provide the probability density function of $Y$.

Answer: e^y


Question: Consider a positive-valued random variable $X$ that follows a positively skewed probability distribution. Is there a one-to-one function $f$ such that $f(X)$ follows a symmetric distribution?

Answer: Yes


Question: Evaluate the definite integral \( \int_{1}^{2} \frac{e^x}{x} \, dx \) to five decimal places using the Taylor series expansion for \( \frac{e^x}{x} \).

Answer: 3.05912


Question: A function $f$ satisfies $f(0) = -1$ and its nth derivative at $x = 0$ is given by \( f^{(n)}(0) = 11\frac{n!}{10^n} \) for \( n \ge 1 \). Given that $f$ is equal to its Maclaurin series on the interval $(-10,10)$, find $f(8)$. \( \boxed{} \)

Answer: 43


Question: Determine whether the limit \( \lim \limits_{(x,y)\to(0,0)} \frac{x^2-6y^2}{|x|+3|y|} \) exists. If it does, find its value.

Answer: 0


Question: Given the joint density function \( f(x,y) = cx(y-1) \) for \( 0 < x < y < 1 \) and \( f(x,y) = 0 \) otherwise, determine the distribution of the random variable \( Y-X \).

Answer: 4(1 - z)^3


Question: Evaluate the limit of the double integral: \[ \lim_{n\rightarrow \infty}\int_{-n}^{n}\int_{-n}^{n}\sin(x^2+y^2)\,dx\,dy \]

Answer: \pi


Question: Determine the number of zeros of the function \( f(z) = z^4 - 3z^2 + 3 \) that are located in the first quadrant of the complex plane.

Answer: 1


Question: In triangle ABC, point M is on segment BC such that \( \frac{CM}{MB} = \frac{3}{2} \). Point N is on segment AB. Segments AM and CN intersect at point O, and \( \frac{AO}{OM} = \frac{5}{1} \). If the area of quadrilateral NBMO is 6, what is the area of triangle ABC?

Answer: 40


Question: Evaluate the limit using Taylor expansion: $$\lim_{x\to 0^{+}} (\ln x)^3\left(\arctan\left(\ln\left(x+x^2\right)\right) + \frac{\pi}{2}\right) + (\ln x)^2$$

Answer: \dfrac{1}{3}


Question: Show that \( \int_{0}^{1}{(1+x^2)^n \, dx} \sim \frac{2^n}{n} \) as \( n \to \infty \). Here, \( a_n \sim b_n \) means that \( \lim_{n \rightarrow \infty} \frac{a_n}{b_n} = 1 \).

Answer: \frac{2^n}{n}


Question: Calculate the line integral \( \int_\ell xyz\,ds \), where \( \ell \) is the circle formed by the intersection of the sphere \( x^2+y^2+z^2=1 \) and the plane \( x+y=0 \).

Answer: 0


Question: Determine the values of $b$ for which the sequence of functions $f_n(x) = b\cos\left(\frac{x}{n}\right)$ converges uniformly on the interval $[0,1]$ in the space $C[0,1]$ equipped with the supremum metric $d(f,g) = \sup_{x\in[0,1]}|f(x)-g(x)|$. Provide a justification for your answer.

Answer: \mathbb{R}


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

Answer: 8


Question: In a classroom, there are 10 girls and 5 boys. Each of the 15 students plays against every other student exactly once, with each game having a single winner. Each participant has a 0.5 probability of winning any game. Let X be the total number of wins by the boys. What is the minimal value of Var(X), considering possible dependencies between the game results?

Answer: 0


Question: Find the sum of the 11th powers of all the roots of the polynomial equation \(x^5 + 5x + 1 = 0\).

Answer: 0


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

Answer: 3


Question: Determine the values of \( r > 0 \) for which the limit \( \lim\limits_{n\rightarrow \infty} \sum\limits_{k=1}^{n^2}\frac{n^{r-1}}{n^r+k^r} \) is finite.

Answer: r > 1


Question: Find the real solutions for \( \lambda \) that satisfy the equation \( 1 = e^{-\lambda \tau} \), given that \( \lambda = a \) is one solution to the characteristic equation \((1 - e^{-\lambda \tau})(a - \lambda) = 0\). Assume \( \tau > 0 \) and \( a \) is a constant.

Answer: 0


Question: Given that $f:[0,\infty) \to \mathbb{R}$ is a bounded, Lebesgue measurable function with $\lim\limits_{x\to\infty} f(x) = r$, prove that $\lim\limits_{k \to \infty} \int_{[0,a]} f(kx) \, dx = ar$.

Answer: ar


Question: Determine the Galois group \( G(\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]/\mathbb{Q}) \).

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


Question: Let $f:\mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $|f(x)| \leq 1$ and $|f''(x)| \leq 1$ for all $x \in \mathbb{R}$. Determine whether it is true that $|f'(x)| \leq 2$ for all $x \in \mathbb{R}$.

Answer: True


Question: Determine the highest degree of the polynomial for which the quadrature formula \( \int _{-1}^1\vert x \vert f(x)dx \approx \frac{1}{2}[f(x_0)+f(x_1)] \) is exact, where \( x_0 \) and \( x_1 \) are quadrature points.

Answer: 3


Question:
Given the marginal density function of $X$:
\[
  f_x(x)= \begin{cases} \frac{x^2}{9}, & 0 < x < 3,\\
  0, & \text{otherwise},\end{cases}
\]
and the conditional density function of $Y$ given $X$:
\[
  f_{y \mid x}(y \mid x) = \begin{cases} \frac{4y^3}{x^4}, & 0 < y < x, \\
  0, & \text{otherwise}. \end{cases}
\]
Find the probability $P(Y < 1)$. Use the formula:
\[
P(Y < 1) = \int_{0}^{3} P(Y < 1 \mid X = x) \cdot f_x(x) \, dx
\]
where
\[
P(Y < 1 \mid X = x) = \int_{0}^{1} f_{y \mid x}(y \mid x) \, dy.
\]

Answer: \dfrac{1}{9}


Question: Calculate the double integral of $z$ over the hemispherical surface $S$ defined by $x^2 + y^2 + z^2 = 1$ with $z \geq 0$. Use polar coordinates for the integration.

Answer: \pi


Question: Determine the asymptotic behavior, in terms of the length L, for the number of topologically different knots that can be formed using a perfectly flexible, non-self-intersecting rope of length L and radius 1, with the ends glued together after knotting.

Answer: e^{\Theta(L)}


Question: Let $(B_t)_{t \ge 0}$ be a standard Brownian motion. Define the stopping time \( T := \inf\{t \ge 0 : B_t = 5t - 2\} \). Calculate \( E[T] \).

Answer: \dfrac{2}{5}


Question: Determine whether the function \( f(x,y) = \frac{|x| y \sin(\frac{\pi x}{2})}{x^2+y^2} \) is differentiable at the point \((0,1)\).

Answer: Yes


Question: Find the limit of the sequence: \( \lim \limits_{n\to \infty}\dfrac{n^3}{{((3n)!)^\frac{1}{n}}} \).

Answer: \dfrac{e^3}{27}


Question: Does there exist a closed geodesic on a closed genus 2 orientable surface with a hyperbolic metric that self-intersects at exactly one point three times?

Answer: No


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{\ln(1+x)-\ln(1-x)}{\arctan(1+x)-\arctan(1-x)} \] without using differentiation.

Answer: 2


Question: Given that \( P(x) \) is a polynomial with integer coefficients and degree \( t-1 \), and satisfies the equation \( xP(x) = x^t + P(x-1) \) for any real \( x \), determine the value of \( t \equiv ? \pmod{3} \).

Answer: 2


Question: Evaluate the following expression: \[ \sum_{m=0}^\infty \lim_{n\rightarrow\infty}\int_{2x^2+y^2<n}\left(1-\frac{2x^2+y^2}{n^2}\right)^{n^2}x^{2m}\,dx\,dy \]

Answer: \pi


Question: Determine whether the group homomorphism \( F: \text{SL}_n(\mathbb{Z}) \to \text{SL}_n(\mathbb{Z}/m\mathbb{Z}) \), induced by the reduction map \( \mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z} \), is surjective for \( n, m > 1 \).

Answer: Yes


Question: Let $A$ be an $n \times n$ matrix with rational elements and $p$ a prime number such that $A^p = I$ with $p < n$. If $\det(A-I) \neq 0$, is it true that $p-1$ divides $n$? Justify your answer.

Answer: Yes


Question: Does the Fourier series of any continuous periodic function with period 1 converge pointwise to the function? Answer "yes" or "no."

Answer: no


Question: Calculate the degree of the field extension \(\mathbb{Q}(\sqrt[3]{2}, \zeta_{9})\) over \(\mathbb{Q}\).

Answer: 18


Question: The polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) satisfies the condition \( P(k) = 11k \) for \( k = 1, 2, 3, 4 \). Determine the value of \( c \).

Answer: -39


Question: Evaluate the limit: \[ \lim_{x\to\infty}\frac{f^{-1}(x)}{\ln(x)} \] where \( f(x)=e^x+x^3-x^2+x \).

Answer: 1


Question:
For what values of the constant \( a \) will the function \( f(x,y)=ax^2-0.5y^2+xy-x-y \) have a relative minimum at the critical point \((0,1)\)? Choose the correct option:

\((A) \ a>0\)
\((B) \ a>0.5 \)
\((C) \ a<-0.5 \)
\((D) \ a<0 \)
\((E) \ (A) \) or \((C) \)
\((F)\) real values of \( a \) not precisely described by any of \((A)\) to \((E)\)
\((G)\) no values of \( a \)

Answer: G


Question: Suppose $\{V_t\}$, for $t > 0$, is an uncountable collection of increasing linear subspaces of $\mathbb{R}^n$ such that $\bigcup_{t>0} V_t = \mathbb{R}^n$. Is it true that there exists some $T > 0$ for which $V_T = \mathbb{R}^n$?

Answer: Yes


Question:
Determine whether there exist integers $a, b, c > 1$ such that the following divisibility conditions are satisfied:

1. $a \mid b^{a-1}$
2. $ab \mid c$
3. $\binom{ab}{a} \mid b^{a-1}c$

Answer: Yes


Question: Is the hyperbolic plane convex, meaning that there is a geodesic line between any two points in the plane?

Answer: Yes


Question:
A set contains \(3n\) members. Let \(P_n\) be the probability that the set is partitioned into 3 disjoint subsets with \(n\) members in each subset such that the three largest numbers in the set are in different subsets. Determine \(\lim_{n\to \infty} P_n\).

\((A)\ \frac{2}{7}\hspace{1cm}(B)\ \frac{1}{7}\hspace{1cm}(C)\ \frac{1}{9}\hspace{1cm}(D)\ \frac{2}{9}\)

Answer: D


Question: Determine the values of \( \alpha \in \mathbb{R} \) for which the limit \[ L = \lim_{n \to \infty} \frac{n(n+1)^\alpha}{\sum_{k=1}^n k^\alpha} \] is finite.

Answer: \mathbb{R}


Question:
Evaluate the integral of the cardinal sine function over the entire real line:
\[ \int_{-\infty}^{\infty} \frac{\sin(a)}{a} \, {\rm d} a \]

Answer: \pi


Question:
Let \(\vec{a}, \vec{b},\) and \(\vec{c}\) be vectors such that \(\vec{a} \times \vec{b} = \vec{c}\), \(\vec{b} \times \vec{c} = \vec{a}\), and \(|\vec{a}| = 2\). Determine which one of the following statements is not true:

A) \(\vec{a} \times ((\vec{b} + \vec{c}) \times (\vec{b} \times \vec{c})) = \vec{0}\)

B) The projection of \(\vec{a}\) on \((\vec{b} \times \vec{c})\) is 2.

C) \([\vec{a} \; \vec{b} \; \vec{c}] + [\vec{c} \; \vec{a} \; \vec{b}] = 8\)

D) \(|3\vec{a} + \vec{b} - 2\vec{c}|^2 = 51\)

Answer: D


Question: Determine the number of conjugacy classes in the Dihedral group \( D_{11} \).

Answer: 7


Question: Consider a compact Hausdorff space $K$ and the C*-algebra $C(K)$ acting on a Hilbert space $H$. Suppose $C(K)$ is closed in the $\\sigma$-strong topology in $\\mathcal{B}(H)$. Must $K$ be extremely disconnected?

Answer: Yes


Question: Calculate \(2^{31} \mod 2925\) using the Chinese Remainder Theorem. First, factor 2925 into its prime components and find \(2^{31}\) modulo each of these prime factors. Then, use the Chinese Remainder Theorem to find the solution modulo 2925.

Answer: 1298


Question: Suppose that \(\{x_n\}_{n=1}^{\infty}\) is a bounded sequence with \(x_n > 0\) for all positive integers \(n\). Find \(\lim_{n \to \infty} \frac{x_n}{x_1 + \cdots + x_n}\).

Answer: 0


Question: Evaluate the limit: $$\lim_{(x, y) \to(0, 0)}\frac{(\sin y)(2^x - 1)}{xy}$$

Answer: \ln 2


Question: Given the polynomial \( f(x) = x^4 + ax^3 + bx^2 + ax + 1 \), where \( a \) and \( b \) are real numbers, and \( f(x) = 0 \) has two distinct negative roots and two equal positive roots, find the least integral value of \( a \).

Answer: 1


Question: Determine the number of integer values of \(x\) such that \(\sqrt{x(x+p)}\) is a positive integer for some prime number \(p > 2\).

Answer: \infty


Question: Let $E = \{e_1, \ldots, e_n\}$ and $F = \{f_1, \ldots, f_n\}$ be two orthonormal bases of a vector space $V$. Is the transfer matrix $[Id]_F^E$, which represents the identity transformation from basis $E$ to basis $F$, an orthogonal matrix? Justify your answer.

Answer: Yes


Question: Suppose $A$ and $B$ are commuting invertible matrices with a common generalized eigenvector $v$ and eigenvalues $a$ and $b$, respectively. That is, there exist positive integers $K$ and $L$ such that $(A-aI)^K v = 0$ and $(B-bI)^L v = 0$. Prove or disprove: there exists a positive integer $M$ such that $(AB-abI)^M v = 0$.

Answer: Yes


Question: Provide an example of a non-commutative ring in which all non-zero elements are invertible.

Answer: \mathbb{H}


Question: Let $f:(0,1)\rightarrow\mathbb{R}$ be a function with a nonconstant derivative $f'$ on $(0,1)$. Is it possible for there to exist a real number $t$ such that $f'(x) = f'(x+t)$ for all $x$ with $x, x+t \in (0,1)$, but $f(x) \neq f(x+t)$ for some $x$ with $x, x+t \in (0,1)$? Provide a justification for your answer.

Answer: Yes


Question: Consider an isosceles triangle with a slope of $k$. This triangle is divided into $n$ equidistant parts, where $n$ is an odd number, creating $n$ polygons. The first polygon, a triangle, has an area of $X$. The middle polygon, which is the $(n+1)/2$-th polygon, has an area of $Y$. Determine the maximum ratio $\frac{Y}{X}$. Is this ratio unbounded as the slope $k$ becomes very large?

Answer: n


Question: Evaluate the integral \( \int_{0}^{1000} \frac{e^{-10x}\sin x}{x} \text{d}x \) to within \( \pm 10^{-5} \).

Answer: 0.09967


Question: Evaluate the definite integral \( \int_{1}^{2009} \prod_{r=1}^{2009} (x-r) \, dx \).

Answer: 0


Question: Calculate the limit of \((1 + x^2 + y^2)^\frac{1}{x^2 + y^2 + xy^2}\) as \((x, y) \rightarrow (0, 0)\).

Answer: e


Question: Determine if the matrix \( \mathcal{A} = \begin{pmatrix} a & c \\ c & b \end{pmatrix} \) can be diagonalized using a rotation matrix \( \mathcal{R} = \begin{pmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{pmatrix} \).

Answer: Yes


Question: Evaluate the expression \(\left(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}\right)\left(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c}\right)\) given that \(a + b + c = 0\) and \(a, b, c \in \mathbb{R}\).

Answer: 9


Question: Let \( A \) and \( B \) be two non-singular matrices such that \( B \neq I \), \( A^6 = I \), and \( AB^2 = BA \). Determine the least positive integer \( k \) such that \( B^k = I \).

Answer: 63


Question: Determine whether the following statement is true or false: If \( \sum_{n=0}^\infty f_n(x) \) is uniformly convergent on \( \mathbb{R} \), then there exists a sequence \( \{M_n\} \) with \( M_n > 0 \) such that for all \( x \in \mathbb{R} \), \( |f_n(x)| < M_n \) and \( \sum_{n=0}^\infty M_n \) converges.

Answer: B


Question: Determine the number of non-planar graphs that can be drawn with 5 vertices such that no vertex is isolated.

Answer: 1


Question: Evaluate the integral over all space: \[ \int \delta \left(r^2 - R^2\right) d \vec r \] and determine the correct result.

Answer: 2\pi R


Question: If a function maps every interval to another interval, must the function be continuous? Provide a justification for your answer.

Answer: No


Question: Is the exponential of a linear operator $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ still a linear operator? Justify your answer.

Answer: Yes


Question: Evaluate \( \lim\limits_{x \to 0}\frac{\int_0^{x^2}f(t){\rm d}t}{x^2\int_0^x f(t){\rm d}t} \) given that \( f(x) \) is continuously differentiable, \( f(0)=0 \), and \( f'(0) \neq 0 \).

Answer: 1


Question: Let $A \subseteq U$ be open sets in $\mathbb{R}^n$. Determine whether the following equality holds: $$ \{ f \in C_0^{\infty}(A) \} = \{ f \in C_0^{\infty}(U) : \text{support of } f \subseteq A \} $$

Answer: Yes


Question: Find the points where the curve $x^y = y^x$ intersects itself.

Answer: (e, e)


Question: Evaluate the limit: $$\lim_{x\to\infty} \frac{\sin^2\left( \sqrt{x+1}-\sqrt{x}\right)}{1-\cos^2\frac{1}{x}}$$

Answer: \infty


Question: Prove that \( \lim_{n \to \infty} (3\sqrt{n})^{\frac{1}{2n}} = 1 \) using the Squeeze Theorem, convergence with \(\varepsilon\)-\(\delta\), sequential convergence, or the notion of a monotone increasing/decreasing function, without using derivatives or limit laws.

Answer: 1


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

Answer: -\dfrac{5}{3}


Question: Let $k$ be a field and $K$ be an algebraic extension of $k$. It is known that if $K$ is separable over $k$, then $K$ is geometrically reduced over $k$. Conversely, if $K$ is geometrically reduced over $k$, is $K$ separable over $k$?

Answer: Yes


Question: Let \( x_n \) be the smallest natural number such that \( 1 + \frac{1}{2} + \cdots + \frac{1}{x_n} \geq n \). Determine \( \lim_{n \to \infty} \frac{x_{n+1}}{x_n} \).

Answer: e


Question: Find the value of $k$ for which the integral \( \int_{0}^{\infty} \frac{x^k}{2+4x+3x^2+5x^3+3x^4+4x^5+2x^6} \, dx \) is minimized.

Answer: 2


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f'(q) = 0 \) for all \( q \in \mathbb{Q} \). Is \( f \) necessarily a constant function? Justify your answer.

Answer: Yes


Question: Determine the number of ordered pairs of integers \((m, n)\) such that \(mn \geq 0\) and \(m^3 + n^3 + 99mn = 33^3\).

Answer: 35


Question: Find the absolute minimum of the function \( f(x_1, x_2, \ldots, x_n) = x_1^2 + x_2^2 + \cdots + x_n^2 \) on the boundary defined by the constraint \( x_1^2 + 2x_2^2 + 3x_3^2 + \cdots + nx_n^2 = 1 \). Use the method of Lagrange multipliers to solve this problem.

Answer: \dfrac{1}{n}


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

Answer: 0


Question: Approximate the probability that within 64 days, a player experiences more than 29 "bad days". A "bad day" is defined as a day where the player wins less than 50% of the 9 games played. The probability of a "bad day" is \( \frac{1}{2} \). Use the Central Limit Theorem to approximate this probability.

Answer: 0.734


Question: If points $A$ and $B$ are such that for any point $P$ on a given circle, the ratio $\frac{PA}{PB} = k$ is constant, determine the value of $k$ that cannot occur.

Answer: 1


Question: Find the volume of the region bounded by the surfaces $z = y^2$, $x = 0$, $y = 0$, and $z = 9 - x$. Provide your answer as a single value.

Answer: \dfrac{324}{5}


Question: Calculate the limit: $$\lim_{n\rightarrow\infty}\left(\frac{n-1}{n+1}\right)^{b_{n}}$$ where $b_{n}=(\sqrt{n+1}-\sqrt{n-1})^{-2}$. Provide your answer as a single value or expression.

Answer: e^{-2}


Question: Find a function $f \in L^p(\mathbb{R})$ such that for a given non-negative function $g \in L^1(\mathbb{R})$, the convolution $f * g = ||g||_1 \cdot f$, where $*$ denotes convolution.

Answer: 0


Question: Find an integer \( a \) such that \( a \equiv 19^{10562} \pmod{40320} \). Use Euler's theorem to simplify the computation.

Answer: 361


Question: Determine the set of possible values for the positive constant \( m \) such that the function \( f(x) = \sqrt{(mx + 7)} - 4 \) and its inverse do not intersect, and neither intersect the line \( y = x \).

Answer: (2, 14)


Question: Given a function $f$ defined on $(a, \infty)$ that is bounded on each finite interval $(a, b)$ for $a > b$, and a non-negative integer $k$, prove that if $\lim_{x \to \infty} \frac{f(x+1) - f(x)}{x^k} = l$, then $\lim_{x \to \infty} \frac{f(x)}{x^{k+1}} = \frac{l}{k+1}$. Address how to handle the term $|\frac{l[x]}{x} - \frac{l}{(k+1)}|$ in the inequality to conclude the desired result by the limit definition.

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


Question: Solve the equation \( \frac{\sqrt{4+x}}{2+\sqrt{4+x}} = \frac{\sqrt{4-x}}{2-\sqrt{4-x}} \) given the domain constraints: \( 4+x \ge 0 \), \( 4-x \ge 0 \), and \( 2-\sqrt{4-x} \ne 0 \).

Answer: 2\sqrt{3}


Question: Given vectors \( u \in \mathbb{R}^{n} \) and \( v \in \mathbb{R}^{m} \) with unit \( L^{2} \) norm, i.e., \( \|u\|_{2} = \|v\|_{2} = 1 \), construct a rank-one matrix \( B \in \mathbb{R}^{m \times n} \) such that \( Bu = v \) and \( \|B\|_{2} = 1 \).

Answer: B = v u^\top


Question: Determine the minimum number of weighings needed on a balancing scale to identify the odd ball out of 18 balls, where 17 balls weigh the same and the odd ball may be either heavier or lighter than the others.

Answer: 4


Question: Find all natural numbers \( n \) such that \( n^5 + 4 \) is a perfect square.

Answer: 2


Question: Evaluate the integral \( \int_{1}^{2}\sqrt[3]{\frac{2-x}{x^7}}\,dx \).

Answer: \dfrac{3}{8}


Question: n passengers board an airplane with exactly n seats. The first passenger has lost their ticket and picks a seat randomly. After that, each subsequent passenger will take their own seat if it is available, or pick a seat randomly if their seat is occupied. What is the probability that the nth passenger will sit in their own seat?

Answer: \dfrac{1}{2}


Question: Determine the radius of convergence for the power series \( \sum_{j=0}^{\infty}\frac{i^{j}+(-i)^{j}-i}{j!}z^{j} \).

Answer: \infty


Question: Given the joint distribution of random variables $X$ and $Y$ as $f(x, y) = \frac{8xy}{3}$, where $0 < x < 1$ and $x < y < 2x$, find the covariance of $X$ and $Y$. Use the bounds $x < y < 2x$ when integrating over $Y$ to find the marginal distributions and expected values.

Answer: \dfrac{28}{675}


Question: Let $U$ be an unbounded open subset of $\mathbb{R}$. Does there exist a number $a$ such that $U$ contains infinitely many integer multiples of $a$?

Answer: Yes


Question: A mother with blood type O and a father with blood type AB have twins, both sons, with blood type B. Given that 32% of all twins have different genders, calculate the probability that the twins are identical.

Answer: \dfrac{9}{13}


Question: Find the minimum value of the expression \( P = \sqrt{a^2 + (1-bc)^2} + \sqrt{b^2 + (1-ca)^2} + \sqrt{c^2 + (1-ab)^2} \) given that \( a, b, c \in \mathbb{R} \) and \( a^2 + b^2 + c^2 = 1 \).

Answer: \sqrt{7}


Question: Does there exist a Banach space that satisfies Grothendieck's theorem (G.T. space) whose dual is not a G.T. space?

Answer: Yes


Question: Let \( A \) and \( B \) be \( 3 \times 3 \) matrices such that \( AB = 0 \) and \( BA \neq 0 \). The solution space for the homogeneous equation \( Ax = 0 \) is spanned by \( \{(1,1,1)\} \). Determine the rank of matrix \( B \).

Answer: 1


Question: Solve the differential equation \( f''(x) = 2xf'(x) + 4f(x) \) given the initial conditions \( f(1) = e \), \( f(0) = 0 \), \( f'(0) = 1 \), and \( f''(0) = 0 \).

Answer: x e^{x^2}


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

Answer: 0


Question:
Given the system matrix \( A = \begin{bmatrix} 0&0&0&1\\0&-1&1&3\\0&1&-1&-1\\0&-1&1&2 \end{bmatrix} \), which has eigenvalues \( \lambda_i=0 \) for \( i = 1,2,3,4 \), determine which of the following matrices is the Jordan form of \( A \):

\( A) \quad J=\begin{bmatrix} 0&1&0&0\\0&0&1&0\\0&0&0&1\\0&0&0&0 \end{bmatrix} \)

\( B) \quad J=\begin{bmatrix} 0&1&0&0\\0&0&1&0\\0&0&0&0\\0&0&0&0 \end{bmatrix} \)

\( C) \quad J=\begin{bmatrix} 0&1&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0 \end{bmatrix} \)

\( D) \quad J=\begin{bmatrix} 0&1&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0 \end{bmatrix} \)

Select the correct option.

Answer: B


Question: Let \( X \sim \operatorname{Exp}(\lambda =1) \) and \( Y\sim U(1,2) \) be independent continuous random variables. Calculate \( E\left(\frac{X}{Y}\right) \).

Answer: \ln 2


Question:
Given numbers from 1 to N, find the minimum possible cost of a permutation. For each number p (1 ≤ p ≤ N) placed at the i-th position (1-based index), the cost is Lp*(i-1) + Rp*(N-i), where L[] and R[] are given arrays of size N. Calculate the minimum total cost of such a permutation.

Example: N = 3, L[] = {1, 2, 6}, R[] = {4, 7, 2}. The minimum cost is 13.

Answer: 13


Question: Determine the number of values of $x$ in the interval $0 < x < 1-b$ for which the equality \[ x^{m/2}[m-x(m+2)]=x(1-x)(2b-1+x)(1-x-b)^{-2} \] holds, given that $m > 4$, $0 < b < 1$, and the function on the right is convex for $0 < x < \frac{m^2-2m}{m^2+4m+4}$ and concave otherwise.

Answer: 1


Question:
Consider an infinite sequence of smooth, compactly supported functions \(\{f_{n}\}_{n \geq 1}\) such that each function \(f_{n}\) is orthogonal to all but a fixed absolute constant \(C\) of the other functions in the sequence. That is, \(\int_{\mathbb{R}} f_{m} f_{n} \, dx = 0\) for all but \(C\) of the other functions. Determine if the following inequality holds for some absolute constant \(C'\):
\[\left\|\sum_{n \geq 1} f_{n}\right\|_{L^{2}(\mathbb{R}^n)} \leq C'\left(\sum_{n \geq 1}\|f_{n}\|_{L^{2}(\mathbb{R}^n)}^{2}\right)^{1/2}.\]

Answer: Yes


Question: Determine the cardinality of the quotient ring \( \mathbb{Z}[i]/(a+bi)\mathbb{Z}[i] \) for any non-zero Gaussian integer \( a+bi \).

Answer: a^2 + b^2


Question: Determine the number of real solutions to the equation \((x-2)+2\log_{2}(2^x+3x)=2^x\).

Answer: 2


Question: Through the vertex O of a parabola \( y^2 = 4ax \), chords OP and OQ are drawn at right angles to one another. If the line segment PQ cuts the axis of the parabola at point R, find the distance OR.

Answer: 4a


Question: Find the set of limit points of \( S = \{ \tan(k) : k = 1, 2, 3, \ldots \} \).

Answer: \mathbb{R}


Question: Suppose that an entire function $f$ has uncountably many zeros. Prove or disprove that $f$ must be identically zero.

Answer: 0


Question: Find the inverse Z-transform of the function \( \frac{z}{(z-2)^2} \).

Answer: n \cdot 2^{n-1} u[n]


Question: Let $R$ be a commutative regular local ring. Determine whether it is true that for every prime ideal $\mathfrak{p}$ in the spectrum of $R$, there exists an $R$-regular sequence $\mathbf{x}$ such that the ideal $(\mathbf{x})$ is $\mathfrak{p}$-primary.

Answer: True


Question: Determine whether the sequence $a_n = n\sin n$ has a convergent subsequence.

Answer: Yes


Question: Calculate the probability \( \mathbb{P}\left(\inf_{t\in[\frac{1}{2},1]}W_t>0\right) \) for a standard Wiener process \((W_t)_{t\in[0,1]}\).

Answer: \dfrac{1}{4}


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

Answer: \pi


Question: Determine whether the following statement is true: If \( A \) is a Noetherian local ring with a unique maximal ideal \( M \), and \( I \) is an ideal in \( A \), with \( x \) being an element such that \( x \) is not in any of the associated primes of \( I \), then \([I:x] = I\).

Answer: True


Question: Determine the number of distinct elements in the quotient ring \(\mathbb{Z}[i]/\langle 3+i \rangle\).

Answer: 10


Question: Prove using the definition of a limit that \( \lim_{z \to 2i} \frac{1}{z-i} = -i \).

Answer: -i


Question: Given that $\mathrm{M}$ is a $3\times 2$ matrix and $\mathrm{N}$ is a $2 \times 3$ matrix such that $\mathrm{MN} = \begin{pmatrix} 8 & 2 & -2 \\ 2 & 5 & 4 \\ -2 & 4 & 5 \end{pmatrix}$, and $\mathrm{NM}$ is invertible, find $\mathrm{det(NM)}$. \(\mathrm{det(MN)} = 0\).

Answer: 81


Question: Let $f$ be an infinitely differentiable function such that \(\lim_{x\to 0}\frac{2f(x)+2x^2-6\cos x}{3x^2}=0.\) Calculate the value of \(f(0) + f'(0) + f''(0).\)

Answer: -2


Question: Let $P$ be an irreducible polynomial with rational coefficients and degree $d$. Suppose $\alpha$ and $\beta$ are two real roots of $P$. Define $a_k = P^{(k)}(\alpha)$ and $b_k = P^{(k)}(\beta)$ for $1 \leq k \leq d-1$. If $a_k b_k > 0$ for all $1 \leq k \leq d-1$, does it necessarily follow that $\alpha = \beta$?

Answer: Yes


Question: Can the harmonic series \( \sum_{k=1}^{\infty} \frac{1}{k} \) be rearranged so that its partial sums are integers for infinitely many terms?

Answer: Yes


Question: Let $f$ be a continuous function such that $f(f(x)) = 1 + x$ for every $x$. Find $f'(1)$.

Answer: 1


Question: Let $M$ be a connected smooth manifold and $N$ a closed embedded submanifold of $M$. If $M$ and $N$ have the same dimension, are $M$ and $N$ equal?

Answer: Yes


Question: Suppose the series \( \sum c_n z^n \) has a radius of convergence \( R \). Determine the radius of convergence of the series \( \sum n^p c_n z^n \).

Answer: R


Question: For what values of $c \in \mathbb{R}$ does the matrix \( A = \begin{bmatrix} 3-c & -c & 1 \\ c-1 & 2+c & -1 \\ c+1 & c & 3 \end{bmatrix} \) have eigenvectors that form a basis for $\mathbb{R}^3$?

Answer: 0


Question: Given that \( f(x+y) = 2f(x) + xf(y) + y\sqrt{f(x)} \) for all \( x, y \in \mathbb{R} \) and \( f'(0) = 0 \), find the value of \( f(6) \).

Answer: 0


Question: Let $X$ be a Banach space and $S_X = \{x \in X : \Vert x \Vert = 1\}$ be the unit sphere in $X$. Suppose $x, y, \frac{x+y}{2} \in S_X$. Is it true that $\lambda x + (1-\lambda)y \in S_X$ for every $\lambda \in [0,1]$? Justify your answer.

Answer: Yes


Question: Let \(f: \mathbb{R} \to \mathbb{C}\) be a function and define \(g(x) = \sup_{|y| > |x|} |f(y)|\) for \(x, y \in \mathbb{R}\). Determine whether the inequality \(|f(x)| \leq |g(x)|\) holds for all \(x \in \mathbb{R}\).

Answer: No


Question: Evaluate the expression \( \left(1+\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}\right)^5+i\left(1+\sin\frac{\pi}{5}-i\cos\frac{\pi}{5}\right)^5 \).

Answer: 0


Question: Prove that the sequence of functions $f_n(x) = e^{-nx}$ converges uniformly to 0 on the interval $[r, \infty)$ for $r > 0$.

Answer: 0


Question: Given that \( \int_0^1 f(x) \, dx = 0 \) and \(-1 \leq f(x) \leq 1\), find the maximum value of \( \int_0^1 [f(x)]^3 \, dx \).

Answer: \dfrac{1}{4}


Question: Evaluate the integral \( \int_{dB(0,2)} \left(\frac{z+1}{z-1}\right)^n dz \) using the Cauchy integral formula.

Answer: 4\pi i n


Question: Determine the radius of convergence of the power series \(\sum_{n=0}^{\infty} a_n z^n\), where \(a_n = \frac{f^{(n)}(0)}{n!}\) and \(f(z) = \frac{\sin z}{(z-1-i)^2}\).

Answer: \sqrt{2}


Question: Find the largest prime number $p$ such that when $2012!$ is written in base $p$, it has at least $p$ trailing zeroes.

Answer: 43


Question: Evaluate the integral \( \int_{0}^{2 \pi} \frac{1}{5-3 \sin \theta} \, d\theta \) using Cauchy's integral theorem.

Answer: \dfrac{\pi}{2}


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

Answer: \ln 2


Question: Determine the values of \( a \in \mathbb{C} \) for which the extension \( \mathbb{C}(X)/ \mathbb{C}(Y) \), where \( Y = X^3 + 3X^2 + aX \), is a Galois extension.

Answer: 3


Question: Evaluate the improper integral \( \int_{-1}^0 \frac{e^{\frac{1}{x}}}{x^3} \, dx \).

Answer: -\dfrac{2}{e}


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

Answer: -\dfrac{e}{2}


Question: Let $M=\mathbb{C}^g/ \Gamma$ be a complex torus and $E$ a holomorphic vector bundle of rank $r$ over $M$, characterized by a factor of automorphy $J: \Gamma \times \mathbb{C}^g \to GL(r, \mathbb{C})$ such that $J(\gamma'\gamma, x) = J(\gamma', \gamma x) J(\gamma, x)$. If $f: M \to M$ is a holomorphic diffeomorphism, is the pull-back bundle $f^*(E)$ characterized by the factor of automorphy $J_f(\gamma, x) = J(\gamma, f(x))$?

Answer: Yes


Question: Suppose that \(X\sim B(4,p)\) and \(Y\sim B(6,p)\) where both \(X\) and \(Y\) are binomial random variables. Find the value of \(p\) that satisfies the equation \(P(2\leq X\leq 4) = P(3\leq Y\leq 6)\).

Answer: \dfrac{3}{5}


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

Answer: e^2


Question: Find the expectation \( E[e^{-\alpha t} S(t)] \) for the asset price given by \( S(t) = s \times \exp{((\alpha-\lambda \sigma)t)} (\sigma + 1)^{N(t)} \), where \( s = S(0) > 0 \), \( \alpha > 0 \), \( \sigma > -1 \), \( \lambda > 0 \) are constants, and \( \{N(t) : t \ge 0\} \) is a Poisson process with intensity \( \lambda \).

Answer: s


Question: You roll a fair 10-sided die repeatedly. What is the expected number of rolls until the lowest common multiple (LCM) of all numbers that have appeared is greater than 2000?

Answer: \dfrac{113}{6}


Question:
Consider the improper integral \( \int_a^b f(x) f'(x) \, dx \). Determine which of the following statements is true:

A.) The integral is necessarily zero.
B.) The integral is possibly zero but not necessarily.
C.) The integral is necessarily nonexistent.
D.) The integral is possibly nonexistent but not necessarily.
E.) None of the above.

Provide a justification for your answer.

Answer: E


Question: Determine whether the conjecture that there exists an \( \epsilon \) such that \( \sum_{\underset{\Large p\; prime}{p=1}}^{\epsilon}\frac{1}{p}=\pi \) is true or false. Provide a justification for your answer.

Answer: False


Question: Given the block matrix \( X = \begin{bmatrix} A & B \\ B^T & C \end{bmatrix} \) where \( A, B, C \) are all matrices of dimension \( n \times n \), find an upper bound \( \phi \) for the operator norm of the sub-block \( B \), i.e., find \( \phi \) such that \( \|B\|_{\text{op}} \le \phi \), given that \( \|X\|_{\text{op}} \le \gamma \).

Answer: \gamma


Question: Can 0 be an accumulation point in the spectrum of a finite-rank operator on an infinite-dimensional Banach space?

Answer: No


Question: Compute the integral \( \int_{\mathbb{R}^n} \phi(x,1)|x|^2 \, dx \), where \( \phi(x,1) = (4\pi)^{-\frac{n}{2}} e^{-|x|^2/4} \).

Answer: 2n


Question: Let $H^s(\mathbb{T})$, where $s \in \mathbb{R}$, be the space of $2\pi$-periodic functions $u(x) = \sum_{k \in \mathbb{Z}} \hat{u}_k \, \mathrm{e}^{ikx}$ such that \( \|u\|_{H^s}^2 = \sum_{k \in \mathbb{Z}} (1+k^2)^{s} \lvert \hat{u}_k \rvert^2 < \infty. \) Assume $s \in \left(\frac{1}{2}, \frac{3}{2}\right)$. Determine if there exists a constant $c = c_s$ such that \( \lvert u(x) - u(y) \rvert \le c \|u\|_{H^s} \lvert x-y \rvert^{s-\frac{1}{2}}. \)

Answer: Yes


Question: Given the functions \( f(x) = 3xe^{-x} \) and \( g(x) = \frac{x}{2} \), let \( v \) be the x-coordinate of their second intersection point (not 0). Find the value of \( u \), where \( 0 < u < v \), such that the area of the triangle with vertices \([0, 0], [u, f(u)], [u, g(u)]\) is maximized.

Answer: 1.049


Question: Determine the ratio of the height to the radius of a cone such that the cone's volume is maximized and its surface area is minimized.

Answer: 2\sqrt{2}


Question: Consider the function \( f_0(x) = |x|^{-\alpha} \) for \(|x| < 1\) and \( f_0(x) = 0 \) otherwise, where \( x \in \mathbb{R}^n \). Show that \( f_0 \in L_p(\mathbb{R}^n) \) with respect to the Lebesgue measure if and only if \( p\alpha < n \), where \( 1 \leq p < \infty \).

Answer: p\alpha < n


Question: Let $G$ be a graph such that all of its edge-deleted subgraphs are isomorphic. Is $G$ necessarily edge-transitive?

Answer: Yes


Question:
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with an infinitely smooth boundary. Define the Sobolev norm on $C^\infty(\bar \Omega)$ as follows:
\[||u||_{W^{1,2}}:=\sqrt{\int_\Omega (|\nabla u|^2+u^2)dx}.\]
Let $W_0^{1,2}$ denote the completion of the space of smooth compactly supported functions in $\Omega$ with respect to this norm. If $u\in W^{1,2}_0\cap C(\bar \Omega)$, does $u$ vanish on $\partial \Omega$? Justify your answer.

Answer: Yes


Question: Find the coefficient of $x^{57}$ in the expansion of the polynomial $(x^2 + x^7 + x^9)^{20}$. Provide your answer as a single integer.

Answer: 3420


Question: Evaluate the limit \( \lim _{x\to e}\left(\frac{1-\log _e\left(x\right)}{x-e}\right) \) without using l'Hôpital's Rule.

Answer: -\dfrac{1}{e}


Question: Calculate the sum of the infinite series: \[ 1 - \frac{2}{1!} + \frac{3}{2!} - \frac{4}{3!} + \frac{5}{4!} - \frac{6}{5!} + \dots \]

Answer: 0


Question: Let \( f(x) = x^2 - 3 \). Determine the number of integer values of \( x \) for which \( f(f(f(x))) \) is divisible by \( x \).

Answer: 8


Question: Let $H$ be a real Hilbert space, and let $e_n \in H$ be an orthonormal sequence of vectors. Let $E$ be the closure of the linear span of the $e_n$, and let $x \in H$ be a vector. Suppose that \( \sum_{n=0}^\infty \langle x,e_n\rangle^2 = \lVert x\rVert^2 \). Does it follow that $x \in E$? Justify your answer.

Answer: Yes


Question: Given the joint probability density function \( f(y_1, y_2) = e^{-y_1} \) for \( 0 \leq y_2 \leq y_1 \leq \infty \) and 0 otherwise, find the probability density function of \( U = y_1 - y_2 \) using the method of distribution functions.

Answer: e^{-u}


Question: Suppose $F$ is a finite field. Is it possible to construct an algebraic extension $F'$ of $F$ such that $|F'| = 	ext{infinite}$ and $F' \subsetneq \overline{F}$, where $F'$ is not algebraically closed?

Answer: Yes


Question: Let $X_1, X_2, \ldots, X_{10}$ be independent and identically distributed random variables, each following a standard normal distribution $\mathcal{N}(0,1)$. Calculate the probability $P(X_1 > X_2 X_3 \cdots X_{10})$. Express your answer as a single probability value.

Answer: \dfrac{1}{2}


Question: Determine whether the sequence \( \{\cos(\pi\sqrt{n^2+n})\}_{n=1}^\infty \) is convergent.

Answer: 0


Question: Find a prime number \( p \) such that there exists a sequence of integers \( S_n \) where \( (S_n)^2 \) converges to 21 under the \( p \)-adic norm.

Answer: 5


Question: Given $a < \frac{\pi}{2}$, find the value of $M$ such that $M < 1$ and $|\cos x - \cos y| \leq M|x-y|$ for every $x, y \in [0,a]$. Compute the value of $M$.

Answer: \sin a


Question: Determine whether a graph with 14 vertices, degree 3, and each face bounded by 6 edges can be the skeleton of a torus with one hole (genus 1 surface).

Answer: Yes


Question: If \( V \) is open (or closed) in the product topology \( X \times X \) induced by a uniformity, is \( V(x) \) open (or closed) in \( X \)?

Answer: Yes


Question: Determine whether any of the relations $f(n) \prec g(n)$, $f(n) \succ g(n)$, or $f(n) \sim g(n)$ exist between the functions $f(n) = \sum_{k=0}^{n}k^{\lfloor \cos (k) \rfloor}$ and $g(n) = n^{\frac{3}{2}}$.

Answer: f(n) \prec g(n)


Question: Given the characteristic polynomial of a matrix \( A \) as \( p(\lambda) = (\lambda+1)(\lambda-2)^2(\lambda+3)^2 \), find \( \det(A^{-1}) \).

Answer: -\dfrac{1}{36}


Question: Verify the Pythagorean Theorem for the vectors $f_1 = (1,0,-1)^T$, $f_2 = (0,1,0)^T$, and $f_3 = (1,0,1)^T$ in $\mathbb{R}^3$ with respect to the inner product $\langle u,v \rangle = u^TAv$, where $A = \begin{bmatrix}2&0&1\\0&2&0\\1&0&2 \end{bmatrix}$. Show that $\|f_1 + f_2 + f_3\|^2 = \|f_1\|^2 + \|f_2\|^2 + \|f_3\|^2$. Here, $\|f\| = \sqrt{\langle f, f \rangle}$.

Answer: 10


Question:
Consider the function \( f(z) = \frac{2z+1}{5z+3} \). Determine which of the following correctly describes the mapping of the upper half-plane \( H^+ = \{z : \text{Im}(z) > 0\} \) and the lower half-plane \( H^- = \{z : \text{Im}(z) < 0\} \) under \( f(z) \):

1. \( H^+ \to H^+ \) and \( H^- \to H^- \)
2. \( H^+ \to H^- \) and \( H^- \to H^+ \)
3. \( H^+ \to L^- \) and \( H^- \to L^+ \)
4. \( H^+ \to L^+ \) and \( H^- \to L^- \)

Select the correct option.

Answer: 1


Question: Evaluate the limit without using L'Hôpital's rule: \[ \lim_{x\to 0} \left(\frac{3x+1}{x}-\frac{1}{\sin x}\right) \]

Answer: 3


Question: Determine the radius of curvature for the plane curve \(x^3 + y^3 = 12xy\) at the point \((0, 0)\).

Answer: 6


Question: Find a complex number \( z \neq 0 \) such that \( \operatorname{Ln}\frac{1}{z} \neq -\operatorname{Ln}z \), where \( \operatorname{Ln}z = \operatorname{Ln}|z| + i\operatorname{Arg}z \) and \( -\pi < \operatorname{Arg}z \le \pi \).

Answer: -1


Question: Determine whether the following conjecture is true: \[ \lim_{N \to +\infty}\frac{1}{N}\sum_{k=1}^{N}\frac{\phi(k)}{k}=\frac{6}{\pi^2} \] where \( \phi(k) \) is the Euler's totient function.

Answer: \dfrac{6}{\pi^2}


Question: Determine the VC-dimension of the class of parity functions defined on the Boolean hypercube \( \{0,1\}^n \). The class is given by \( H_{n-parity} = \{h_I : I \subseteq \{1,2,..., n\}\} \), where for a set \( I \subseteq \{1,2,\dots, n\} \), the parity function \( h_I \) is defined as \( h_I(x)=\left(\sum_{i \in I}x_i\right)\mod 2 \) for a binary vector \( x = (x_1, x_2,\dots, x_n) \in \{0,1\}^n \).

Answer: n


Question: Let S be the union of two surfaces, $S_1$ and $S_2$, where $S_1$ is the set of $(x,y,z)$ with $x^2+y^2 =1$ and $0 \le z \le 1$, and $S_2$ is the set of $(x,y,z)$ with $x^2+y^2+(z-1)^2 =1$ and $z \ge 1$. Given the vector field $F(x,y,z) = (zx+z^2y+x)i + (z^3yx+y)j + (z^4x^2)k$, compute the surface integral $\int\int_S \nabla \times F \cdot dS$. Assume the boundary $\Gamma$ is a unit circle on the $xy$-plane.

Answer: 0


Question: Determine whether a space $(X, \tau)$ with $\tau$ being the finite-closed topology has the Fixed Point Property, meaning that every continuous mapping $f: (X, \tau) \to (X, \tau)$ has a fixed point.

Answer: No


Question: Given that the random variable $X$ follows a beta distribution with parameters $a$ and $b$, find the probability density function (PDF) of the transformed random variable $Y = -\log(X)$. Additionally, calculate the expected value $\text{E}(Y)$. The transformation is given by $f_Y(y) = e^y$, and the PDF of $X$ is $f_X(e^{-y}) = e^{-y} \frac{1}{B(a,1)}(e^{-y})^{a-1} = ae^{-ay}$ for $0 \leq y < \infty$. Verify if the expression for $f_Y(y)$ is correct and find $\text{E}(Y)$. Provide your answer for the expected value in a boxed form.

Answer: \dfrac{1}{a}


Question: Let $\Omega \subset \mathbb{R}$ be an open set, and consider the space $C_c^\infty(\Omega)$, which consists of smooth functions with compact support in $\Omega$. For a function $\varphi \in C_c^\infty(\Omega)$, is the Fourier transform $(\mathcal{F}\varphi)(\xi)$ generally an $L^1$ function?

Answer: Yes


Question: Compute the double integral \( \iint_R xe^y \, dx \, dy \) where \( R \) is the region inside \( x^2+y^2 = 1 \) but outside \( x^2+y^2 = 2y \) with \( x \ge 0 \) and \( y \ge 0 \). Use the change of variables \( u = x^2 + y^2 \) and \( v = x^2 + y^2 - 2y \).

Answer: \sqrt{e} - \dfrac{3}{2}


Question: Find the 2-norm of the eigenvalue of the matrix \( A = I - \frac{2}{u^T u} uu^T \), where \( u \) is a non-zero vector in \( \mathbb{R}^n \) and \( A \) is an \( n \times n \) matrix. The 2-norm of an eigenvalue \( \lambda = a + bi \) is given by \( \|\lambda\|_2 = \sqrt{a^2 + b^2} \).

Answer: 1


Question: Calculate the line integral of the scalar field \( x+y \) over the curve \( L \), where \( L \) is the right loop of the polar equation \( r^2 = 2\cos(2\theta) \).

Answer: 2\sqrt{2}


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left( 12x^2 - 2 - 6x \sqrt{3x^2 - 2} \right) \]

Answer: \infty


Question: Determine the minimum distance between the trajectories given by \( r_1(t) = -4t^2 + 8t - 3 \) and \( r_2(t) = 9t^2 - 48t + 62 \).

Answer: \dfrac{61}{13}


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

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


Question: Find the limit: \[ \lim_{x \to \infty} x(e^{1/x}-1)^x \] without using L'Hopital's rule or Taylor expansion.

Answer: 0


Question: Let $R$ be the ring of formal power series in $n$ indeterminates over $\mathbb{C}$, and let $(I_{k})_{k\in \mathbb{N}}$ be a strictly decreasing chain of unmixed radical ideals, all having the same height $s$. Assume that $\bigcap I_{n} = \mathfrak{p}$ is prime and that $I_{1}$ is prime. Is it true that $ht(\mathfrak{p}) = s$?

Answer: s


Question: Determine whether the limit \( \lim_{x\to-\infty}\sum_{n=1}^\infty\lambda(n)\frac{x^n}{\Gamma(n)} \) is finite, where \( \lambda(n) \) is the Liouville function and \( \Gamma(n) = (n-1)! \).

Answer: 0


Question: Consider two independent random variables, $X \sim \mathrm{Exp}(4)$ and $Y \sim \mathrm{Exp}(12)$. Calculate the expected value of the minimum of $4X$ and $Y$, i.e., find $E[\min(4X, Y)]$. Provide your answer as a single value.

Answer: \dfrac{1}{13}


Question: Find an element $a$ in a group $G$ such that the order of $a$ is 6, denoted as $|a|=6$, and the centralizer of $a$ in $G$, $C(a)$, is not equal to the centralizer of $a^3$, $C(a^3)$. Provide an example of such an element.

Answer: r


Question: Four students are giving presentations in four sections of a course, each with a presentation length that is exponentially distributed with an expected value of 10 minutes. What is the expected total time needed until all four presentations are completed?

Answer: \dfrac{125}{6}


Question: Assume that the cumulative distribution functions of random variables $X$ and $Y$ satisfy $F_X(x) > F_Y(x)$ for all $x$. Can we conclude that $E[X] > E[Y]$?

Answer: No


Question: Evaluate the improper integral: $$\int_0^\infty \frac{\sin^4(x)}{x^2} \, dx$$

Answer: \dfrac{\pi}{4}


Question: Let \( p, q, r \) be prime numbers such that \( p < q < r \) and \( p \times q \times r = 19(p + q + r) \). Find the value of \( r - q - p \).

Answer: 5


Question: Calculate the limit: \[ \lim_{n\rightarrow\infty} \left[n\left(1-\frac1{\sqrt[n]{x}}\right)-n(\sqrt[n]{x}-1)\right] \] where \( n \in \mathbb{N} \) and \( n \geq 2 \), \( x \in \mathbb{R} \) and \( x > 1 \).

Answer: 0


Question: Given \( I = 3\sqrt{2} \int_{0}^{x} \frac{\sqrt{1+\cos t}}{17-8\cos t} \, dt \) and \( 0 < x < \pi \), find \( x \) such that \( \tan I = \frac{2}{\sqrt{3}} \).

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


Question: Determine the number of Sylow-3 subgroups in a noncyclic group of order 21.

Answer: 7


Question: Let $A$ be a finite free $R$-algebra where $R$ is a local ring. Determine whether $A$ always has a basis $a_1, \ldots, a_n$, such that $a_1=1$. Provide a justification for your answer.

Answer: Yes


Question: Consider a random permutation of the numbers $1, 2, \ldots, 31415$. Let $A$ be the number of fixed points (numbers that remain in their original position) and $B$ be the number of non-fixed points. Calculate the variance of $B - A$.

Answer: 4


Question: Find the value of \(x\) such that \(7^{50} \cdot 4^{102} \equiv x \pmod{110}\).

Answer: 104


Question: If the Mertens function follows a normal distribution, does this imply the Riemann Hypothesis? Answer yes or no.

Answer: Yes


Question: Calculate \( f'(\sqrt{\pi}) + g'(\sqrt{\pi}) \) for the functions \( f(x) = \left(\int_{0}^{x} e^{-t^2} \, dt\right)^2 \) and \( g(x) = \int_{0}^{1} \frac{e^{-x^2(1+t^2)}}{1+t^2} \, dt \).

Answer: 0


Question: Evaluate the improper integral \( \int_{-\infty}^{\infty} \frac{dx}{x^{2} + 2x + 5} \) using complex analysis techniques.

Answer: \dfrac{\pi}{2}


Question: Let \( f(x) = \sum_{n=1}^{\infty} a_n x^n \) be a bijective function on the set of real numbers, where \( a_1 = 2 \) and \( a_2 = 4 \). The inverse function \( f^{-1}(x) \) is given by \( f^{-1}(x) = \sum_{n=1}^{\infty} b_n x^n \). Find the value of \( b_1 \).

Answer: \dfrac{1}{2}


Question:
Consider the stochastic process defined by:
\[ \tilde{W}(t) := W(t) + \int_{0}^{t} e^{W(s)}\mathbb{1}_{\{|W(s)\le 1\}}\,ds \quad \forall \, t \in [0,T] \]
where \( W(t) \) is a Wiener process. For the stopping time \( \sigma := \inf\{t \ge 0 : W(t) = 1 - \int_{0}^{t} e^{W(s)}\mathbb{1}_{\{|W(s)\le 1\}}\,ds \} \), compute the expectation:
\[ \mathbb{E}_{\mathcal{Q}}\left[e^{-\sigma/2} \right] \]

Answer: e^{-1}


Question: Given a Poisson process \( \{N(t)\}_{t\geq 0} \) with rate \( \lambda=2 \), calculate the probability that the first event occurs before time \( t=1 \) given that four events occur during the time interval \([0,2]\).

Answer: \dfrac{15}{16}


Question: In triangle $ABC$, point $O$ is an interior point. Perpendiculars $OP$, $OQ$, and $OR$ are drawn from $O$ to sides $AB$, $BC$, and $AC$, respectively. Given that $P$, $Q$, and $R$ are midpoints of the sides and the perimeter of triangle $ABC$ is $8$, find the sum $OP + OQ + OR$. Choose the correct answer from the following options: $2, 3, 4, 5, 6$. \(\text{Note: } OP, OQ, \text{ and } OR \text{ are integers.}\)

Answer: 4


Question: Given the function \(f(x) = (-1)^{[x]}(x+a\cdot[\frac{x}{2}]+b)+3\), where \(a, b \in \mathbb{R}\), determine the set \(A = \{(a,b) \in \mathbb{R}^2 \mid f\) is periodic with period 2 and continuous at \(x=1\)\}. Calculate \(S = \sum_{(a,b)\in A}(a+b)\). Here, \([x]\) denotes the integer part of \(x\).

Answer: -3


Question: Let $A$ be an $n \times n$ Hermitian matrix whose elements are continuous functions on $\mathbb{R}^n$. The eigenvalues of $A$ are real and ordered as $\lambda_{1}(x) \leqslant \ldots \leqslant \lambda_{n}(x)$. Is $\lambda_i(x)$ continuous for any $i \in \{1, \ldots, n\}$?

Answer: Yes


Question: Given the functional equation \( f \left(\frac{x+y}{2} \right) = \frac{f(x)+f(y)}{2} \) for all \( x, y \in \mathbb{R} \), and the conditions \( f'(0) = -1 \) and \( f(0) = 1 \), find \( f'(u) \).

Answer: -1


Question:
For 8-bit symbols, Reed-Solomon codes are of the form \((n,k) = (255-z,255-2t-z)\), where:

- \(n\) is the number of symbols in a codeword.
- \(k\) is the number of data symbols that can be encoded.
- \(t\) is the number of errors that can be corrected.
- \(z\) is the number of implicit zeros that can be used if the code is to be shortened.

Determine the maximum value of \(t\) for these codes.

Answer: 127


Question: Let $f: M \to N$ be a submersion from a Riemannian manifold $(M, g)$, with vertical and horizontal distributions $\mathcal{V}$ and $\mathcal{H}$, respectively. The tangent space at each point $p \in M$ decomposes as $T_p M = \mathcal{V}_p \oplus \mathcal{H}_p$. The restriction $\varphi_p = (df_p)|_{\mathcal{H}_p}: \mathcal{H}_p \to T_{f(p)}N$ is an isomorphism for all $p \in M$. Construct a metric $h$ on $N$ by defining $h_{f(p)}(X, Y) = g_p(\varphi_p^{-1}(X), \varphi_p^{-1}(Y))$. Does this construction make $f$ a Riemannian submersion?

Answer: Yes


Question: Evaluate the triple integral \(\int\int\int \frac{1}{x^2+y^2+z^2} \,dV\) by converting it to spherical coordinates, where the region of integration is bounded by the sphere \(x^2 + y^2 + z^2 = 25\).

Answer: 20\pi


Question:
Find the residue of the sum \(1^{965} + 2^{965} + \ldots + 2018^{965}\) modulo 2021. What is the value of \(x\) such that:
\[
1^{965} + 2^{965} + \ldots + 2018^{965} \equiv x \pmod{2021}?
\]

Answer: 1012


Question: Determine the longest sequence of compositions of the prime counting function \( \pi(x) \) that results in perfect squares, starting with \( \pi(100) \).

Answer: 3


Question: Let \( f:[0,\infty)\to [0,\infty) \) be an increasing function satisfying \( \int_0^\infty f(x)\frac{dx}{1+x^2}=\infty \). Determine whether there exists a continuous increasing function \( F \) on \([0,\infty)\) such that \( \int_0^\infty F(x)\frac{dx}{1+x^2}=\infty \) and \( F(x)\leq f(x) \) for all \( x \geq 0 \).

Answer: Yes


Question:
Given that \( \sum_{r=0}^{300}a_rx^r = (1+x+x^2+x^3)^{100} \), if \( a = \sum^{300}_{r=0}a_r \), find \( \sum_{r=0}^{300}ra_r \). Choose the correct answer from the following options:

a) \( 300a \)
b) \( 100a \)
c) \( 150a \)
d) \( 75a \)

Answer: C


Question: Let \(K\) be a compact subset of \(\mathbb{R}^{n}\) and \(F\) a subset of \(K\) which is relatively closed in \(K\). Is \(F\) a compact subset of \(\mathbb{R}^{n}\)?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to \infty} \sum_{k=0}^n \frac {\binom{n}{k}}{(n^k)(k+3)} \]

Answer: e - 2


Question: Given a power series \( \sum_{n=0}^{\infty} a_n z^n \) with a radius of convergence \( R \), determine the radius of convergence of the series \( \sum_{n=0}^{\infty} a_{kn} z^n \) for a fixed positive integer \( k \).

Answer: R^k


Question: Determine the values of \( \delta \) for which the series \( \sum_{n=1}^\infty e^{-n^{\delta}} \) converges using the comparison test.

Answer: \delta > 0


Question: Calculate the residue of the function \( f(z) = z\exp\left(\frac{1}{1-z}\right) \) at \( z = 1 \).

Answer: -\dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{y \to \infty} \frac{-(iy - 1/2) -|iy-1/2|}{(iy - 1/2) -|iy-1/2|} \]

Answer: i


Question: Compute the total Gaussian curvature of the surface defined by the equation \( x^2 + y^4 + z^6 = 1 \).

Answer: 4\pi


Question: Solve the equation \( \sin x = 1 - x \) for \( x \) without using graphical methods.

Answer: 0.511


Question: In triangle \(\triangle ABC\), points \(D\) and \(E\) lie on sides \(CA\) and \(AB\) such that \(BE = 6\) and \(CD = 10\). Let \(M\) and \(N\) be the midpoints of segments \(BD\) and \(CE\), respectively. If \(MN = 7\), find the measure of \(\angle BAC\).

Answer: 120^\circ


Question: Consider a cube $C$ centered at the origin in $\mathbb{R}^3$. How many invertible linear transformations of $\mathbb{R}^3$ map $C$ onto itself?\n(a) $72$\n(b) $48$\n(c) $24$\n(d) $12$

Answer: B


Question: Determine the number of integer values of \( p \) for which the equation \( x^4 + 4x^3 - 8x^2 + p = 0 \) has all four real roots.

Answer: 4


Question: What is the least upper bound for the Hausdorff dimension of the graph of a real-valued, continuous function on an interval? Is this least upper bound attained by some function?

Answer: 2


Question: Find an integer $x$ such that $x^{294408} \not\equiv 1 \pmod{294409}$. Provide a value of $x$ that satisfies this condition.

Answer: 37


Question: Let $K$ be a number field, and $E$ be an elliptic curve defined over $K$. Prove or disprove: If $\text{End}_K(E) \cong O_K$ and $E$ has complex multiplication over $K$, then $K$ is a principal ideal domain (PID).

Answer: True


Question: Find the maximum value of \(x^3 + y^3 + z^3\) given that \(x, y, z \in [0, 2]\) and \(x + y + z = 3\).

Answer: 9


Question:
Calculate the limit:
\[ \lim_{\varepsilon\rightarrow0}\dfrac{1}{\varepsilon}\int_{D(\varepsilon)}\dfrac{1+\sin x+\sin y}{\sqrt{x^2+y^2}}d(x,y), \]
where \( D(\varepsilon)=\{(x,y)\in\mathbb{R}^2|0<x^2+y^2\leq\varepsilon^2\} \).

Answer: 2\pi


Question: Determine the number of different balanced colorings for a $2019 \times 2019$ grid, where a coloring is balanced if, within every square subgrid of size $k \times k$ for $1 \le k \le 2019$, the number of black cells differs from the number of white cells by at most one.

Answer: 2


Question: Find the value of \( \theta \in (0,1) \) such that the sequence \( M_n = \theta^{S_n} \) is a martingale with respect to the natural filtration of \((X_n)\), where \((X_n)\) are i.i.d. random variables with \(P(X_n=0)=1/2\), \(P(X_n=1)=1/3\), and \(P(X_n=-1)=1/6\), and \(S_n = X_1 + \ldots + X_n\).

Answer: \dfrac{1}{2}


Question: Find the value of \( f'(0) \) for the function \( f(x) = \int_{0}^{x} \cos\left(\frac{1}{t}\right) dt \).

Answer: 0


Question: Find all prime numbers of the form \(n^5 - 1\).

Answer: 31


Question: Evaluate the integral \( \int^{\infty}_0 \frac{x}{x^4+1} \, dx \).

Answer: \dfrac{\pi}{4}


Question: Determine the values of \( p > 0 \) for which the improper integral \( \int_0^\infty \frac{\log(1+x)}{x^p} \, dx \) converges.

Answer: (1, 2)


Question: Given that \( \int_{0}^{\pi/2}\ln(\sin x)\,dx = k \), find the value of \( \int_{0}^{\pi/2}\frac{x^2}{(\sin x)^2}\,dx \) in terms of \( k \).

Answer: -2k


Question: Find the degree of the field extension \( \mathbb{Q}(i, \sqrt[4]{3}, \sqrt[6]{3}) \) over \( \mathbb{Q} \).

Answer: 24


Question: Determine if there exists a real number \( x \) such that either \( f(x) + 2x = 2(1 + x^2) \) or \( 2f(x) + 1 = 2x(1 + x) \), where \( f(x) = (1-x)^2(\sin x)^2 + x^2 \) for all \( x \in \mathbb{R} \).

Answer: Yes


Question: Consider the differentiable functions $L^1(x,\theta^1), L^2(x^2,\theta^2), \ldots, L^l(x^l,\theta^l)$, where each $x_k, \theta^k$ are real vectors for $k=1,\ldots,l$. Define $\theta=(\theta^1,\ldots,\theta^l)$ and the composite function $f(x,\theta)=x^{l+1}$ recursively by $x^k= L^{k-1}(x^{k-1},\theta^{k-1})$, with $x^1=x$. Compute $J_\theta f$, the Jacobian of $f$ with respect to $\theta$. Provide your answer in terms of the recursive rule for $G^k$. \[ G^k=\left ( J_{x^k}L^k\cdot G^{k-1}\middle| J_{\theta^k}L^k\right ), \quad G^1=J_{\theta^1}L^1 \]

Answer: G^l


Question: Determine the value of the triple factorial of -2, denoted as \((-2)!!!\).

Answer: 1


Question: What is the greatest possible value of the greatest common divisor of $4T_n$ and $n-1$, where $T_n$ is the $n$th triangular number defined as $T_n = \frac{n(n+1)}{2}$?

Answer: 4


Question:
Find the values of constants \(a\) and \(b\) that minimize the variance of the random variable \(Y = aX_1 + bX_2\), where \(X_1\) and \(X_2\) are independently distributed random variables with the following distributions:

\[ P(X_1=\Theta+1) = P(X_1=\Theta-1) = \frac{1}{2} \]
\[ P(X_2=\Theta-2) = P(X_2=\Theta+2) = \frac{1}{2} \]

Subject to the condition that \(E[Y] = \Theta\). What is the minimum value of this variance?

Answer: \dfrac{4}{5}


Question: Find the 4th derivative of the function \( f(x) = \frac{1+x+x^2}{1-x+x^2} \) at the point \( x = 0 \) using the Taylor series expansion for \( \frac{1}{1-x} \) centered at 0. Do not calculate the 1st, 2nd, or 3rd derivatives of \( f(x) \).

Answer: -48


Question: Given that X is a random variable with the probability density function \( F_X(x) = \frac{1}{2}\sin(x) \) on the interval \([0, \pi]\), and that the random variable Y is defined as follows: \( Y = 3 \) if \( X \in [0, \frac{\pi}{3}] \), \( Y = 2 \) if \( X \in (\frac{\pi}{3}, \frac{2\pi}{3}] \), and \( Y = 1 \) if \( X \in (\frac{2\pi}{3}, \pi] \). Determine the conditional probability density function of X given \( Y = 2 \). What is \( f_{X|Y}(x|Y=2) \)?

Answer: \sin(x)


Question: Evaluate the limit: $$\lim_{n \to \infty }\sqrt[n]{ \frac{\left | \sin 1 \right |}{1}\cdot\frac{\left | \sin 2 \right |}{2}\cdot\cdot\cdot\frac{\left | \sin n \right |}{n}}$$

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to0}\frac{x^2+2\cos x-2}{x \sin^3x}.$$

Answer: \dfrac{1}{12}


Question: How many distinct necklaces can be formed using 6 identical diamonds and 3 identical pearls?

Answer: 7


Question: Let $f$ be an analytic function on the closed unit disc such that $f(0)=0$ and $f'(0)\neq 0$. Does this imply that $\sup \Re(f) > 0$?

Answer: Yes


Question:
Let $I \subset \mathbb{R}$ be an interval, and $f: I \rightarrow \mathbb{R}$ be a function. Consider the following conditions for a fixed integer $n \geq 2$:

1. $f\left(\frac{x_1+x_2}{2}\right) \leq \frac{f(x_1)+f(x_2)}{2}$ for $x_1, x_2 \in I$.
2. $f\left(\frac{x_1+\ldots+x_n}{n}\right) \leq \frac{f(x_1)+\ldots+f(x_n)}{n}$ for $x_1, \ldots, x_n \in I$.

It is known that condition 1 implies condition 2. Does condition 2 imply condition 1?

Answer: Yes


Question: Determine whether the symmetric group \( \mathfrak{S}_n \) always has exactly \( n! \) subgroups for any positive integer \( n \).

Answer: No


Question: Given an infinite simple graph $G=(V,E)$, determine if there exists a topology $\tau$ on $V$ such that the graph associated with the topological space $(V,\tau)$ is $G$. Provide a yes or no answer.

Answer: No


Question: Suppose $\mu=0$ for a random variable with moment generating function $\phi(t)=\mathbb{E}[e^{tX}]$. Define the rate function $$I(a)=\sup_{t>0} \ (at-\log \phi(t))\text{.}$$ Prove or disprove that $I(0)=0$ and that $I(a)$ is strictly positive for $a>0$ and strictly increasing on $[0,\infty)$. Assume $a>\mu$.

Answer: I(0) = 0


Question: Evaluate the integral \( \int_{-\pi}^\pi \left| \sum_{n=1}^\infty \frac{1}{2^n} e^{inx} \right|^2 \, \mathrm{d}x \).

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


Question: Let $V \subset H$ be Hilbert spaces with inner products $(\cdot,\cdot)_V$ and $(\cdot,\cdot)_H$. Suppose $V$ is dense in $H$ and both spaces are separable. If $(\cdot,\cdot)_{V_2}$ and $(\cdot,\cdot)_{H_2}$ are different inner products with norms equivalent to the original inner products, is $V \subset H$ with these new inner products/norms still dense and separable? Justify your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: For every natural number \(n\), there exists an algebraic curve \(C\) and a point \(p\) on that curve such that the tangent plane at \(p\) has dimension \(n\).

Answer: True


Question: Evaluate the contour integral \( \int _{\left|z + 1\right|=4} \frac{5+3z}{-z^3 + 4z} \, dz \) using the residue theorem.

Answer: 0


Question:
Calculate the probability \( P(X+Y \leq 1) \) given the joint density function:
\[
f(x,y)= \begin{cases}
\frac{4x^3}{y^3} & \text{if $0<x<1, x<y$}\\
0 & \text{otherwise}
\end{cases}
\]

Answer: 6 \ln 2 - 4


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

Answer: \dfrac{1}{6}


Question:
Determine the values of \(a\) for which the series \(\sum_{n=1}^\infty \left(\frac{1}{n} - \sin\frac{1}{n}\right)^{2a}\) converges. Choose the correct option:
(a) \(a > \frac{1}{3}\)
(b) \(a < \frac{1}{3}\)
(c) \(a < \frac{1}{2}\)
(d) \(a > \frac{1}{6}\)

Answer: D


Question:
Suppose the group $(\mathbb{Q},+)$ has a non-trivial homomorphism to a group $G$. Determine which of the following can be $G$:

(a) $(\mathbb{Z},+)$
(b) $(\mathbb{Q}^{\circ},\times)$
(c) $(\mathbb{Z}_{m}, +_{(\bmod)})$
(d) None of these.

Answer: D


Question: Evaluate the surface integral \( \int_S \mathbf{F} \cdot d \mathbf{S} \) where \( \mathbf{F}(x, y, z) = (x, y, z) \) and the surface \( S \) is parameterized by \( g(u,v) = \begin{pmatrix} u-v \\ u + v \\ uv \end{pmatrix} \) for \( 0 \le u \le 1 \) and \( 0 \le v \le 2 \). The surface integral is defined as \( \int \mathbf{F}(g(u,v)) \cdot \left(\frac{\partial g}{\partial u}(u,v) \times \frac{\partial g}{\partial v}(u,v)\right) dudv \), where \( \left(\frac{\partial g}{\partial u}(u,v) \times \frac{\partial g}{\partial v}(u,v)\right) = (u-v, -u-v, 2) \). Compute the integral.

Answer: -2


Question: Is it true that for any two distinct real numbers $a$ and $b$ with $a < b$, there exists an uncountable number of transcendental numbers between $a$ and $b$?

Answer: Yes


Question: Evaluate the integral \( \int_0^{2\pi} \frac{1}{\cos^2(x) + 2\sin^2(x)} \,\mathrm{d}x \) using the antiderivative method. Verify that the result is \( \pi\sqrt{2} \).

Answer: \pi\sqrt{2}


Question: Let $R$ be a ring with $1 \neq 0$ that contains noncentral idempotents. If for every noncentral idempotent $e$ of $R$, the corner ring $eRe$ is a division ring and $eR(1-e)Re \neq 0$, is the ring $R$ semiprime?

Answer: Yes


Question: Determine the number of possible solutions to the equation \(a! = 5b! + 27c!\) where \(a\), \(b\), and \(c\) are positive integers. Choose the correct option: \(\text{A. No Solution}\), \(\text{B. 1 Solution}\), \(\text{C. 2 Solutions}\), \(\text{D. 3 Solutions}\), \(\text{E. Infinite Solutions}\).

Answer: A


Question: Evaluate the integral \( \int_{|z|=1} \frac{1}{z-z_0} \, dz \) where \( |z_0| > 1 \).

Answer: 0


Question: Calculate the integral \( \int_X xye^{x^2+y^2}\,dx\,dy \) over the region \( X=\{(x,y)\in\mathbf{R}^2\mid x^2+y^2\leqslant 1,x,y\geqslant 0 \} \) using polar coordinates.

Answer: \dfrac{1}{4}


Question: Let \(a, b, c, d\) be non-negative real numbers such that \(a^5 + b^5 \le 1\) and \(c^5 + d^5 \le 1\). Find the maximum possible value of \(a^2c^3 + b^2d^3\).

Answer: 1


Question: Given the function \( z = f(x, y) \) and \( g(u, v) = uf(u^2, 2uv) \), calculate \( \frac{\partial^2 g}{\partial u \, \partial v}(1, 1) \). Use the following information: \( f(1, 2) = 4 \), \( \nabla f(1, 2) = (3, -1) \), \( \frac{\partial^2 f}{\partial x^2}(1, 2) = \frac{\partial^2 f}{\partial y^2}(1, 2) = 1 \), and \( \frac{\partial^2 f}{\partial x \, \partial y}(1, 2) = -1 \).

Answer: -4


Question: Given the partial differential equation \( f_x(x,y) + f_y(x,y) = 0 \) for all \((x,y) \in \mathbb{R}^2\) and the initial condition \( f(x,0) = \sin x \), find the value of \( f(0,y) \).

Answer: -\sin y


Question: Calculate the mechanical work done by the force \( \vec{F}=(y^2+z^2)\vec{i}+(x+y)\vec{k} \) on a point moving along the curve \( \gamma \) defined by the parametric equations: \( x(t)=2\cos^2t \), \( y(t)=2\cos t \cdot \sin t \), \( z(t)=2\cos t \) for \( t \in \left[-\frac{\pi}{2},\frac{\pi}{2}\right] \). Evaluate the integral \( \int _{\gamma }\:(y^2+z^2)dx+(x+y)dz \).

Answer: -\dfrac{8}{3}


Question: Compute the limit: \[ \lim_{n\to\infty}\left(\frac{n!}{n^n}\right)^{1/n}. \]

Answer: \dfrac{1}{e}


Question: Determine the minimum order $n$ of a linear differential equation with constant coefficients for which $y(x) = x \sin x$ is a solution.

Answer: 4


Question: Find the unique primitive Pythagorean triple \((a, b, c)\) such that the area of the triangle is equal to twice its perimeter. Here, \(a\) and \(b\) are the legs, and \(c\) is the hypotenuse of the triangle.

Answer: (9, 40, 41)


Question: Find the characteristic polynomial of the matrix \( A = \begin{pmatrix}7&1&2&2\\ 1&4&-1&-1\\ -2&1&5&-1\\ 1&1&2&8 \end{pmatrix} \). One eigenvalue of the matrix is given as \( \lambda = 6 \).

Answer: (\lambda - 6)^4


Question: Evaluate the limit: $$\lim_{n\to\infty}\int_0^1 \frac{nx^{n-1}}{1+x}dx$$ Assume that the interchange of limit and integration is possible.

Answer: \dfrac{1}{2}


Question: Determine whether the highest $p$-th power root of unity that lies in a complete discrete valuation ring $\mathcal{O}$, with algebraically closed residue field $k$ of characteristic $p>0$, satisfies the condition that if $n\in\mathcal{O}$ is a $p^a$th root of unity, then $a\leq e$, where $(p)=(\pi^e)$ and $\pi$ generates the maximal ideal.

Answer: a \leq e


Question: Determine whether the following statement is true or false: An $R$-module $M$ is simple if and only if $M \cong R/\mathfrak{p}$ for some prime ideal $\mathfrak{p}$ of $R$. Provide a justification for your answer.

Answer: False


Question: Use the Orbit-Stabilizer Theorem to determine the number of elements in the rotational symmetry group of a cube. Assume the orbit size of a vertex is 8 and the stabilizer size is 3. Calculate the total number of rotational symmetries.

Answer: 24


Question: Consider the Hilbert space $H = L^2[0,1]$ with the norm defined by $||f||^2 = \int^{1}_{0} f^2(x) \, dx$. Let $f_1(x)=1$, $f_2(x)=x$, and $f_3(x)=x^2$ for $0 \leq x \leq 1$. Determine the distance from $f_1$ to the linear subspace spanned by $f_2$ and $f_3$ in $H$. Provide your answer as a single value.

Answer: \dfrac{1}{3}


Question: Consider the map \( A \) of \( \mathbb{R}^3 \) into itself given by the mapping \( Ar = r - 2(n \cdot r)n \), where \( r = (x,y,z) \) and \( n = \left(\frac{2}{3}, \frac{1}{3}, \frac{2}{3}\right) \). Compute \( A^2 \).

Answer: A^2 = I


Question: Determine whether there exists a real function $f$ that satisfies the following properties: \( f \geq 0 \), \( f \) is concave and increasing on \([1, +\infty)\), \( \lim_{n\to\infty}(f(n) - f(n+1)) = 0 \), and \( \lim_{x\to\infty} f(x) \leq 0 \).

Answer: Yes


Question: Suppose we have a surface $M$ embedded in $\mathbb{R}^3$ with an orthogonal parametrization. Assume that the parameter curves $X(u_0, v)$ and $X(u, v_0)$ are geodesics that are unparametrized. What can be concluded about the Gauss curvature of $M$?

Answer: 0


Question: Let $A$ be an $R$-right module, $N$ be a submodule of an $R$-left module $M$, and $\pi: M \rightarrow M/N$ be the natural epimorphism. Determine $\ker(\pi \otimes 1_A)$. Provide your answer in terms of $M$, $N$, and $A$.

Answer: N \otimes_R A


Question: Consider a first-order language $L$ with a finite set of variables. Let $T$ be a consistent set of formulas in $L$. Does Gödel's Completeness Theorem imply that there exists a model for $T$?

Answer: Yes


Question: Evaluate the contour integral \( \int_{c} f(z) \, dz \) where \( c \) is the square with vertices at \( i, 2-i, 2+i, \) and \( -i \), and \( f(z) = \frac{z^2 - iz}{z+1} \).

Answer: 0


Question: Solve the functional equation: \[ f(x+y) = \max(f(x), y) + \min(x, f(y)) \]

Answer: f(x) = x


Question: Let \( f \) be a map from an open subset \( U \) of a normed vector space \( E \) to another normed vector space \( F \). Suppose that \( f \) is differentiable at a point \( a \) in \( U \). Is \( f \) continuous in a neighborhood of \( a \)? Provide a justification for your answer.

Answer: No


Question: If \(2^7\cos^5x \cdot \sin^3x = a\sin8x - b\sin 6x + c\sin 4x + d\sin 2x\) where \(x\) is real, find the value of \(a^4 + b^4 + c^4 + d^4\).

Answer: 1329


Question: Given the graph $G$ represented by the adjacency matrix $A=\begin{pmatrix} 0 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 1 & 0 \end{pmatrix}$, determine the number of minimum spanning trees of the graph $G$. Provide your answer as an integer.

Answer: 21


Question: Determine the cardinality of the set \( \{f:\mathbb{Q}\rightarrow\mathbb{Q} \mid f \text{ is a monotonic function}\} \).

Answer: 2^{\aleph_0}


Question: Determine the distribution that is the limit of the sequence \( \langle f_n, \varphi \rangle = n(\varphi(\frac{1}{n})-\varphi(\frac{-1}{n})) \) as \( n \to \infty \).

Answer: -2\delta'


Question: Find the closed form of the sequence defined by the recurrence relation \( f(n) = 7f(n-1) - 12f(n-2) \) with initial conditions \( f(1) = 1 \) and \( f(2) = 7 \).

Answer: 4^n - 3^n


Question: Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a differentiable function such that \(\lim_{x \to \infty} f(x) = 1\) and \(\lim_{x \to \infty} f'(x) = a\). Find the value of \(a\).

Answer: 0


Question: Let \( f: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) be a function such that for every \( x, y \in \mathbb{R} \), the functions \( f(x, \cdot): \mathbb{R} \to \mathbb{R} \) and \( f(\cdot, y): \mathbb{R} \to \mathbb{R} \) are continuous. Is it true that \( f \) is continuous on \( \mathbb{R} \times \mathbb{R} \)?

Answer: No


Question: Compute the integral \( \int_0^\infty xe^{-kx} \, dx \) for \( k \geq 1 \) using measure theory.

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


Question: Find the remainder when \( 25^{889} \) is divided by 99.

Answer: 70


Question:
Given that \(X_1\) and \(X_2\) are independent and identically distributed discrete random variables, the moment generating function of \(X_1 + X_2\) is given by:
\[ M(t) = 0.01e^{-2t} + 0.15e^{-t} + 0.5925 + 0.225e^{t} + 0.0225e^{2t} \]
Find the probability \(P[X < 0]\).

Answer: 0.1


Question: Determine whether the limit \( \lim_{n\to\infty} \frac{1-2+3-4+\dots+(2n-1)-2n}{\sqrt{n^2+1}+ \sqrt{4n^2-1}} \) converges. If it converges, find its value.

Answer: -\dfrac{1}{3}


Question: Assume \(\{a_n\}\) and \(\{b_n\}\) are two positive series such that \(\sum_{n}a_n=\sum_n b_n=1\). Assume also for all \(n\), \(\sum_{k\geq n}a_k\leq \sum_{k\geq n}b_k\) and \(\lim_{n\rightarrow +\infty}\frac{\sum_{k\geq n}a_k}{\sum_{k\geq n}b_k}=0\). Determine whether \(\lim_{n\rightarrow+\infty}\frac{a_n}{b_n}=0\).

Answer: 0


Question:
Determine which of the following statements is true for a real 2 × 2 matrix \( M \) such that \( M^2 = \begin{pmatrix}-1 & 0 \\ 0 & -1-\epsilon \end{pmatrix} \):

(A) \( M \) exists for all \( \epsilon > 0 \)
(B) \( M \) does not exist for any \( \epsilon > 0 \)
(C) \( M \) exists for some \( \epsilon > 0 \)
(D) None of the above is true.

Select the correct option.

Answer: B


Question: Evaluate the double integral \( \iint_R \cos\left(\frac{\pi}{2}x^2\right)\,dx\,dy \), where \( R \) is the region enclosed by the line \( y=x \), the vertical line \( x=1 \), and the \( x \)-axis.

Answer: \dfrac{1}{\pi}


Question: Determine which number is greater: $2^\sqrt{2}$ or $e$. Justify your answer.

Answer: e


Question:
Given the system of differential equations: \[ \frac{dx}{dt} + \frac{dy}{dt} + 2y = \sin t \] \[ \frac{dx}{dt} + \frac{dy}{dt} - x - y = 0 \] determine which of the following conditions holds for the general solutions $x(t)$ and $y(t)$:

(a) $x(t) + y(t) = ae^t$
(b) $x(t) + y(t) = a\sin t$
(c) $x(t) - y(t) = ae^{-t}$
(d) $x(t) - y(t) = ae^t + \sin t$

where $a$ is an arbitrary constant.

Answer: a


Question: Determine whether the function \( T: (C[0,1],d_{\text{sup}}) \rightarrow (C[0,1],d_{\text{sup}}) \) defined by \( T(f)(x) = 2 \cdot f(1-x) - 3 \) is continuous, where \( d_{\text{sup}}(f,g) = \sup_{x \in [0,1]} |f(x) - g(x)| \).

Answer: Yes


Question: Evaluate the limit: $$\lim_{j,M \to \infty}\frac{1}{M}\prod_{k=1}^{M}\left[\prod_{n=j}^{2j}\left(1+\frac{1}{kn}\right)\right]^{\frac{1}{\ln 2}}$$

Answer: e^{\gamma}


Question: Consider a martingale $Y_n = \sum_{i=1}^n X_i$, where $X_i$ is a martingale difference sequence with $\mathbb{E}[X_n \mid \mathcal{F}_{n-1}] = 0$ for all $n$. We know that $\mathbb{E}[Y_n^2 - Y_{n-1}^2] = \mathbb{E}[X_n^2]$. Does there exist a constant $C > 0$ such that for $1 < r < 2$, the inequality \( \mathbb{E}[|Y_n|^r - |Y_{n-1}|^r] \le C \mathbb{E}[|X_n|^r] \) holds? Assume the distribution of $X_n$ given $Y_{n-1}$ is symmetric about zero. Provide justification for your answer.

Answer: Yes


Question: Calculate the line integral \( \int_{\gamma} \frac{2xy\,\mathrm{dx}-(x^2+z^2)\,\mathrm{dy}+2yz\,\mathrm{dz}}{(x^2+z^2)^2} \) where the path \( \gamma: [0,1] \to \mathbb{R}^3 \) is defined by \( \gamma(t) = (\cos(2\pi t), \sin(2\pi t), t^2-t) \) for all \( t \in [0,1] \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{r=1}^{n-1}\frac{e^{r/n}}{n} \]

Answer: e - 1


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

Answer: 0


Question: Evaluate the surface integral \( \iint_S (yz \, dx \, dy + xz \, dy \, dz + xy \, dx \, dz) \) where \( S \) is the outer surface of the region bounded by the cylinder \( x^2 + y^2 = 4 \) and the planes \( x = 0 \), \( y = 0 \), \( z = 0 \), and \( z = 2 \). Use the Gauss Divergence Theorem to express the integral in the form \( \iint_S \mathbf{F} \cdot \mathbf{n} \, dS \) and calculate \( \nabla \cdot \mathbf{F} \).

Answer: 2\pi


Question: Evaluate the definite integral \( \int_0^{\pi} \cos(ie^{ix}) \, dx \).

Answer: \pi


Question: Evaluate the limit: $$\lim_{x \rightarrow 1^+} (\ln\ x)^{x-1}$$

Answer: 1


Question: Compute the limit: \[ \lim_{n\to \infty} \frac{\tan(\pi/n)}{n\sin^2(2/n)} \]

Answer: \dfrac{\pi}{4}


Question: Determine the smallest number of edges in a 6-vertex-connected graph with 200 vertices.

Answer: 600


Question: Four runners start at the same position on a circular track that is 1 km long, running in the same direction. Their speeds are \( \sqrt{5} \), \( e \), 3, and \( \pi \) meters per second. Determine the first time, in seconds, when the runner with speed \( e \) is at least 250 meters away from each of the other runners.

Answer: \dfrac{250}{3 - e}


Question: Suppose $G$ is a finite group acting via homeomorphisms on the closed $n$-dimensional ball $B^n$. Does $G$ have a fixed point, i.e., a point $p \in B^n$ such that $g \cdot p = p$ for all $g \in G$?

Answer: Yes


Question: Find the operator norm of the operator \(I - S - S^2\), where \(S\) is the shift operator defined on \(\ell^2(\mathbb{Z})\) by \((Sa)_n = a_{n-1}\) for \(a = (a_n)_{n = 1}^\infty \in \ell^2(\mathbb{Z})\).

Answer: \sqrt{5}


Question: Determine whether the function \( H(x) = \int_0^x \left|\sin\left(\frac{1}{t}\right)\right| \, dt \) is differentiable at \( x = 0 \).

Answer: \frac{2}{\pi}


Question: Find the volume of the region bounded above by the paraboloid \( z = 9 - x^2 - y^2 \) and below by the \( x-y \) plane, lying outside the cylinder \( x^2 + y^2 = 1 \).

Answer: 32\pi


Question: Determine the maximum value of \( q \) such that the hyperbolic paraboloid given by \( z(xy) = \frac{y^2}{4} - \frac{x^2}{a^2} \) with the constraint \( x^2 + \frac{y^2}{2} \leq q^2 \) fits inside a cylindrical box with a radius of 2.

Answer: \sqrt{2}


Question: Evaluate the integral: \[ \int_0^y \frac{1}{\sqrt{x(y-x)}} \, dx \]

Answer: \pi


Question: Find \( F''(\pi) \) if \[ F(x) = \int_x^{\sin x}  \left(\int_0^{\sin t} (1+u^4)^{0.5} \,du\right)\,dt \]

Answer: 2


Question: Consider a linear transformation $T$ on a vector space $V$ of dimension $n$. Suppose it is given that for some vector $\mathbf{v}$, the vectors $\mathbf{v}, T(\mathbf{v}), T^{2}(\mathbf{v}), \ldots, T^{n-1}(\mathbf{v})$ are linearly independent. Is it true that every eigenvalue of $T$ has only one corresponding eigenvector up to a scalar multiplication?

Answer: Yes


Question: Let $J$ be a sequentially weakly lower semicontinuous functional on a bounded, closed, and convex subset $C$ of a Hilbert space $H$. Does the functional $J$ attain its minimum on $C$?

Answer: Yes


Question: Calculate the degree of the field extension \([\mathbb{Z}_3(\alpha) : \mathbb{Z}_3]\), where \(\alpha\) is a root of the polynomial \(x^3 + x + 2 \in \mathbb{Z}_3[x]\).

Answer: 2


Question: Determine the pointwise limit of the sequence of functions \((1+\sin(x/n))^n\) for all \(x \in \mathbb{R}\).

Answer: e^x


Question: Does every finite order self-homeomorphism \(\phi: \mathbf{R}^n \to \mathbf{R}^n\) have a fixed point?

Answer: Yes


Question: In triangle $ABC$, $BC = 4$ and $AB = 2\sqrt{19}$. A circle passes through the midpoints of the sides of $\triangle ABC$, and its center lies on the bisector of $\angle ACB$. Determine the length of $AC$. \(\text{Express your answer in simplest form.}\)

Answer: 10


Question: Determine the smallest positive integer \( n \) such that for all integers \( 1 \le a \le 100 \), the inequality \( \left|\sqrt{a}-x\right| \le \dfrac{1}{10^n} \) implies \( \left|a - x^2\right| \le \dfrac{1}{10^{100}} \) for all real numbers \( x \).

Answer: 102


Question: Find the value of \( y(\frac{1}{2}) \) for the function \( y(x) \) that minimizes the integral \( \int_{0}^{1} e^{-(y'-x)} + (1+y)y' \, dx \), given that \( y(0) = 0 \) and \( y(1) = \frac{1}{2} \).

Answer: \dfrac{1}{8}


Question: Is the tangent bundle of the long line, $TM$, isomorphic to its cotangent bundle, $T^{*}(M)$?

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\rightarrow 0}\frac{\arcsin x - \arctan x}{e^x-\cos x -x^2 -x}$$

Answer: 3


Question: Given the points \(A(0,2)\), \(B\), and \(C\) on the curve \(y^2 = x + 4\), where \(\angle CBA = \frac{\pi}{2}\), find the least positive value of the ordinate of \(C\).

Answer: 4


Question: If $f$ is a continuous function that is also a tempered distribution, is it true that $f$ is necessarily bounded by a polynomial function?

Answer: Yes


Question: Find all solutions for \(x\) in the equation: \[ \left(2+\sqrt{3}\right)^{x/2} + \left(2-\sqrt{3}\right)^{x/2} = 2^x. \]

Answer: 2


Question: Compute the expectation \( \sum_{n=0}^{\infty} nP_{n} \) given the following conditions: \( G(n) = e^{-\lambda} \cdot \frac{\lambda^n}{n!} \) for \( \lambda > 0 \), and \( \sum_{j=0}^{n} G(j)P(n-j) + (n+1)[P(n+1)-P(n)] = 0 \) for all \( n \in \mathbb{N} \), with \( \sum_{n=0}^{\infty} P_{n} = 1 \).

Answer: \lambda


Question: Let \(f:[0,1]\longrightarrow\mathbb{R}^2\) be a continuous function such that \(f(1)-f(0)=(1,0)\). Determine if there exist \(0\leq t_1<t_2\leq1\) such that \(f(t_2)-f(t_1)=(\pm1,0)\) and for any \(t_1\leq t<t'< t_2\) or \(t_1< t<t'\leq t_2\), \(f(t')-f(t)=(a,0)\) for some \(a\in \mathbb{R}\) with \(|a|<1\).

Answer: Yes


Question: Prove that the number of odd Catalan numbers $C_{i+j}$ in the $n \times n$ matrix $M_n = \left(C_{i+j}\right)_{i,j=0}^{n-1}$ is $2n-1$. Here, $C_k = \frac{1}{k+1}\binom{2k}{k}$ denotes the Catalan numbers.

Answer: 2n - 1


Question:
Assume that the set of natural numbers $\mathbb{N} = \{0, 1, 2, \ldots\}$ is partitioned into $k \ge 2$ disjoint sets $J(1), \ldots, J(k)$, each with an asymptotic density $d(J(p)) = \lim_{n \to \infty} \frac{1}{n} \frac{|J(p,n)|}{n} > 0$, where $J(p,n) = J(p) \cap \{0, 1, \ldots, n-1\}$. Let $\alpha > 0$ be an irrational number, and define $e(x) = e^{2\pi i x}$ for $x \in \mathbb{R}$. Determine whether the following limit is true:
\[ \lim_{n \to \infty} \frac{1}{n^2} \sum_{p \neq q} \left( \left( \sum_{j \in J(p,n)} e(j\alpha) \right) \cdot \left( \overline{\sum_{j \in J(q,n)} e(j\alpha)} \right) \right) = 0. \]

Answer: 0


Question: Determine whether the sphere $S^2$ can be embedded in $\mathbb{CP}^1 \times \mathbb{CP}^1$ as a totally real submanifold.

Answer: Yes


Question: Let $M$ and $N$ be $n$-dimensional manifolds. Suppose $D_1, D_2, \dots, D_k$ are $n$-dimensional manifolds embedded in $M$ such that $\bigcup_{i=1}^k D_i = M$ and each $D_i$ is homeomorphic to $N$. Determine the minimum value of $k$. For example, if $M$ is a 2-dimensional torus and each $D_i$ is a 2-dimensional disk, the minimum value of $k$ is 3. What is the minimum value of $k$ if $M = L(5,2)$ and each $D_i$ is a 3-dimensional ball?

Answer: 3


Question: Determine whether the following statement is true or false: For any integer $n \geq 4$, there exists an $n \times n$ matrix $A$ such that $A^4 = 0$ and $\text{rank}(A + I_n) < 4$. Provide a justification for your answer.

Answer: False


Question: Evaluate the integral \( \int_C (z+\bar{z})^3 \, dz \), where \( C \) is the unit circle in the complex plane.

Answer: 6\pi i


Question: Given the joint moment generating function \( M(T_1, T_2) = \frac{1}{2} e^{T_1 + T_2} + \frac{1}{4} e^{2T_1 + T_2} + \frac{1}{12} e^{T_2} + \frac{1}{6} e^{4T_1 + 3T_2} \), find the variance of the random variable \( X \).

Answer: \dfrac{25}{18}


Question: Is the inequality \( \left|\left| \sum_{k=1}^\infty x_k \right|\right|_1 \leq \sum_{k=1}^\infty ||x_k||_1 \) necessarily valid for a convergent series \( \sum_{j=1}^\infty x_j \) in the space \( c_{00} \)?

Answer: Yes


Question: Find a 3-digit natural number in base ten such that its representation in base seven is the reverse of its representation in base nine. Explain why 248 is the only possible solution, considering that a 3-digit number in base ten can become a 4-digit number in base seven or nine.

Answer: 248


Question: Given standard Borel spaces \((X, \mathcal{X})\) and \((A, \mathcal{A})\), an analytic set \(D \subseteq X \times A\), and a lower semianalytic function \(c:D \rightarrow [0, \infty]\), consider a stochastic kernel \(p(\cdot|\cdot)\) such that \((x,a) \mapsto p(B | x, a)\) is lower semianalytic for each \(B \in \mathcal{X}\), and \(B \mapsto p(B | x, a)\) is a probability measure on \((X, \mathcal{X})\) for each \((x, a) \in D\). Define \(\eta_u(x, a) := c(x, a) + \int_X u(y)p(dy | x, a)\) for a lower semianalytic \(u:X \rightarrow [0, \infty]\) and \((x, a) \in D\). Let \(\eta^*_u(x) := \inf_{a \in D_x}\eta_u(x, a)\) for \(x \in \text{proj}_X(D)\). Given \(\epsilon > 0\), does there exist a universally measurable function \(\varphi:\text{proj}_X(D) \rightarrow A\) such that \(\varphi(x) \in D_x\) for all \(x \in X\) and, for all \(x \in \text{proj}_X(D)\), \(\eta_u(x, \varphi(x)) \leq \eta_u^*(x) + \epsilon\)?

Answer: Yes


Question: Let \( p(x) = x^4 + ax^3 + bx^2 + cx + d \) be a polynomial where \( a, b, c, \) and \( d \) are constants. Given that \( p(1) = 10 \), \( p(2) = 20 \), and \( p(3) = 30 \), compute \( \frac{p(12) + p(-8)}{10} \).

Answer: 1984


Question:
Calculate \( \frac{\partial x}{\partial w} \) at a constant \( z \) for the system of equations given by:

1) \( x^{2}+y^{2}+z^{2}=6 \)
2) \( w^{3}+z^{3}=5xy+12 \)

Evaluate this derivative at the point \((x,y,z,w) = (1,-2,1,1)\).

Answer: -\dfrac{2}{5}


Question:
Calculate the isothermal compressibility \( k_T \) at the critical temperature \( T = T_c \) and pressure \( p \) using the Van der Waals equation:

\[ p = \frac{R T}{v-b} - \frac{a}{v^2} \]

Given:
- \( p_c = \frac{a}{27b^2} \)
- \( v_c = 3b \)
- \( T_c = \frac{8a}{27bR} \)

The partial derivative of pressure with respect to volume at constant temperature is:

\[ \left(\frac{\partial p}{\partial v}\right)_T = -\frac{R T}{(v-b)^2} + 2\frac{a}{v^3} \]

At \( v = v_c \):

\[ -\frac{R T}{(v-b)^2} + 2\frac{a}{v^3} = -\frac{R(T-T_c)}{4b^2} \]

Hence, the isothermal compressibility is:

\[ k_T(T, v = v_c) = -\frac{1}{v}\left(\frac{\partial v}{\partial p}\right)_T \approx C(T-T_c)^{-\gamma} \]

Determine \( k_T(T = T_c, p) \).

Answer: \infty


Question: Find the limit \( \lim_{t \to \infty} \mathrm{e}^{At} \) for a given square matrix \( A \).

Answer: 0


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

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


Question:
Given a Markov chain $X$ with state space $E = \{1, 2, 3, 4\}$ and transition matrix
$$P = \begin{bmatrix}
0 & 0 & 1 & 0 \\
0 & 0.4 & 0.6 & 0\\
0.8 & 0 & 0.2 & 0\\
0.2 & 0.3 & 0 & 0.5\\
\end{bmatrix}$$
compute the expected value $E[f(X_5)f(X_6)|X_4 = 4]$, where the function $f$ has values $2, 4, 7,$ and $3$ at states $1, 2, 3,$ and $4$ respectively.

Answer: 14.41


Question: What is the largest possible sum of the angle measures of a triangle in hyperbolic space? Determine the maximum value for the sum of the angles in a hyperbolic triangle.

Answer: 180^\circ


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{1}{\arctan(x)} \int_{0}^{1} \left[ \arctan(t + \sin(x)) - \arctan(t) \right] \, dt \]

Answer: \dfrac{\pi}{4}


Question: Solve the equation: \(4\cdot 25^x - 25\cdot 4^{x+1} = 9\cdot 10^x\).

Answer: 2