Question: Evaluate the limit: \[ \lim_{x\to 0^+}(\sin^2(4x))^{\sin^{-1}(2x)}. \]

Answer: 1


Question: In the complex plane, consider all continuous curves with $a$ as the starting point and $b$ as the endpoint. If two curves are homotopic, they are regarded as the same curve. How many distinct such curves are there?

Answer: 1


Question: Let \( f : [0,+\infty) \to [0,+\infty) \) be a continuous function such that \( \lim_{x \to +\infty} f(x)=0 \). Determine whether there exists a convex function \( g : [0,+\infty) \to [0,+\infty) \) such that \( g \geq f \) and \( \lim_{x \to +\infty} g(x)=0 \).

Answer: Yes


Question: Calculate the remainder when \(31^{(41^{59})}\) is divided by 25 using Euler's theorem. Find \(c\) such that \(31^{(41^{59})} \equiv c \pmod{25}\).

Answer: 6


Question: Determine the total number of matrices that are similar to the matrix \( \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \) over the finite field \( \mathbb{Z}_7 \).

Answer: 56


Question: Is there a type of pseudorandom binary sequence that satisfies the "Window property" but is unbalanced, having a specified percentage of ones and zeros?

Answer: Yes


Question: Compute the counterclockwise circulation of the vector field \( \mathbf{F} = (-y - e^y \cos x)\mathbf{i} + (y - e^y \sin x)\mathbf{j} \) around the closed curve \( C \), where \( C \) is the right lobe of the lemniscate given by \( r^2 = \cos 2\theta \), using Green's Theorem.

Answer: \dfrac{1}{2}


Question: A line segment AB of length 1 meter is broken at a random point P, where the length of AP follows the probability density function \( f(x) = 6x(1-x) \) for \( 0 < x < 1 \). A point Q is uniformly selected from AP. Given that the length of AQ is 0.5 meters, find the expected length of AP.

Answer: \dfrac{2}{3}


Question: Let $X_1$ and $X_2$ be jointly distributed with the joint density function $f(x_1, x_2) = 2e^{-x_1 - x_2}$ for $0 < x_1 < x_2 < \\infty$. Compute the joint density of $Y_1 = X_1$ and $Y_2 = X_1 + X_2$ under this transformation.

Answer: 2e^{-y_2}


Question: Let $S$ be a totally ordered set such that any nonempty subset of $S$ bounded above has a greatest element and any nonempty subset of $S$ bounded below has a least element. Is $S$ order isomorphic to a subset of the integers?

Answer: Yes


Question: Suppose $L/F$ is a normal extension and $E/F$ is a subextension, i.e., $F \subset E \subset L$. Determine whether every $\sigma \in \operatorname{Hom}_{F}(E,L)$ can be extended to a $\sigma' \in \operatorname{Hom}_{F}(L,L)$. Provide a justification for your answer.

Answer: Yes


Question: Given triangle ABC with \( \angle CAB = \alpha = \frac{\pi}{4} \) and \( \angle ABC = \beta = \frac{2\pi}{3} \), a point M lies inside the triangle. Perpendiculars are drawn from M to the sides BC, AC, and AB, with lengths 5, 3, and 4, respectively. Find the area of triangle ABC. If the answer is not an integer, round it to the nearest integer.

Answer: 162


Question:
Find the limit of the sequence as \( n \to \infty \):
\[ \lim_{n \to \infty} \dfrac{1}{\sqrt{n}}\left(1+\dfrac{1}{\sqrt{2}}+\cdots+\dfrac{1}{\sqrt{n}}\right) \]

Answer: 2


Question: Determine the number of paths from point $x_0$ to point $v_n$ in a hexagon structure.

Answer: 2^{n-1}


Question:
Consider a 4-dimensional vector space $V$ over a finite field with $p$ elements, with basis \( \{e_1, e_2, e_3, e_4\} \). Determine the number of 2-dimensional subspaces of $V$ that do not contain $e_4$ and are not contained in \( \text{Span}(\{e_1, e_2, e_3\}) \). Choose the correct answer from the following options:

1) \((1+p+p^2)(p^2-1)\)
2) \((1+p+p^2)(p^2+1)\)
3) \((1+p+p^2+p^3)\)
4) \((1+p)^3\)

Answer: 1


Question: Evaluate the series \( \sum\limits_{x=0}^{\infty} \binom{4+x}{x} \frac{1}{2^x} \).

Answer: 32


Question: Determine the number of group homomorphisms from \(\mathbb{Z}_8\) to \(S_3\).

Answer: 4


Question: Compare the numbers $e^\pi$ and $\pi^e$. Given that the function $f(x) = \frac{\ln(x)}{x}$ has a maximum at $(e, \frac{1}{e})$, determine which of the two numbers is larger.

Answer: e^\pi


Question: Evaluate the integral \( \int_{S^2} \sin(\theta) \, d\theta \wedge d\phi \).

Answer: 4\pi


Question: An insurance company pays $10,000 for the first loss, $7,500 for the second loss, and $5,000 for each successive loss. If the number of losses follows a Poisson distribution with a mean of 2.5, calculate the expected payment due to losses.

Answer: 18871.33


Question: Assume the observed data is normally distributed. Is it correct to conclude that the best linear unbiased estimator (BLUE) obtained through least square adjustment is also a maximum likelihood estimator (MLE)?

Answer: Yes


Question: Determine whether the group presented by \( \langle a, b \mid ab = 1 \rangle \) is isomorphic to the free group generated by \( a \).

Answer: Yes


Question: Let $X$ be a non-zero real normed linear space, and let $x, y \in X$. If $B(x, r) \subseteq B(y, s)$, is it true that $r \leq s$? Justify your answer.

Answer: Yes


Question: Given two real-valued sequences \( \{x_n\} \) and \( \{y_n\} \) such that \( \lim_{n\to\infty} (x_{n}^{3}-y_{n}^{3})=0 \), determine whether it is always true that \( \lim_{n\to\infty} (x_n-y_n) =0 \). Justify your answer.

Answer: Yes


Question: Determine the number of units in the ring \( R = \mathbb{Z}_8 \times \mathbb{Z}_9 \times \mathbb{Z}_5 \). Provide a proof for your answer.

Answer: 96


Question: Determine the total number of analytic functions that vanish only on the set \( S = \{0\} \cup \{ \frac{1}{4n+1} : n = 1, 2, 3, 4, \dots \} \).

Answer: 0


Question: If $X$ is a finite set, determine the smallest cardinality of a family of open subsets $\mathcal{U} \subseteq 2^X$ such that $\mathcal{U}$ generates the discrete topology on $X$. In other words, find the smallest $|\mathcal{U}|$ such that if $\mathcal{U} \subseteq \tau \subseteq 2^X$ and $\tau$ is a topology, then $\tau = 2^X$. Provide your answer as a single number.

Answer: |X|


Question: In a Banach space $E$, consider biorthogonal basic sequences $e_{n} \in E$ and $g_{n} \in E^{*}$, where $\langle e_n, g_m \rangle = \delta_{mn}$, and both sequences are weakly null. Can we always find $\alpha > 0$, an infinite-dimensional subspace $H \subset E$, and a weakly compact set $D \subset E^{*}$ such that $\sup_{d \in D} |\langle h, d \rangle| \ge \alpha \|h\|$ for every $h \in H$?

Answer: Yes


Question: Given the joint probability density function (PDF) \( f_{XY}(x,y) = 8xy \) with support \( 0 \leq x \leq y \leq 1 \), let \( U = X \) and \( V = XY \). Find the PDF of \( (U, V) \).

Answer: \frac{8v}{u}


Question: Find the least value of \(b\) such that at least one permutation of the numbers consisting of one \(1\), two \(2\)'s, three \(3\)'s, ..., nine \(9\)'s can be expressed in the form \(a^b\), where \(a\) and \(b\) are natural numbers.

Answer: 1


Question:
Prove that the number of permutations of the set \( \{1, 2, 3, \dots, n\} \) in which none of the patterns 12, 23, 34, \dots, (n-1)n occurs, denoted by \( Q_n \), can be expressed as:

\[ Q_n = (n-1)!\left(n - \frac{n-1}{1!} + \frac{n-2}{2!} - \frac{n-3}{3!} + \cdots + (-1)^{n-1}\frac{1}{(n-1)!}\right) \]

Answer: Q_n = (n-1)!\left(n - \frac{n-1}{1!} + \frac{n-2}{2!} - \frac{n-3}{3!} + \cdots + (-1)^{n-1}\frac{1}{(n-1)!}\right)


Question: Let \( E = \mathbb{R}^n \) and \( f : E \rightarrow E \) be a function such that \( f \in \mathcal{C}^2 \) and for all \( x \in E \), the differential of \( f \) is an isometry. Define \( \varphi(x) = \langle \text{d}f(x)h, \text{d}f(x)k \rangle \). Calculate the differential of \( \varphi \).

Answer: 0


Question: Find the value of \( m \) such that \( \lim_{x \to 0} \frac{(e^x + e^{-x}) \sin(mx)}{e^x - 1} = 4 + m \).

Answer: 4


Question: Determine whether there exist two matrices \(A, B \in \mathbb{Z}^{2 \times 2}\), both with determinant 2, such that \(AB^{-1} \in \mathbb{Z}^{2 \times 2}\) but \(B^{-1}A \not\in \mathbb{Z}^{2 \times 2}\).

Answer: Yes


Question: Find \( \lim_{n\to \infty} a_n \) where \( a_n = \left(1-\dfrac{1}{\sqrt{2}}\right)\left(1-\dfrac{1}{\sqrt{3}}\right)\cdots\left(1-\dfrac{1}{\sqrt{n+1}}\right) \) for \( n \ge 1 \).

Answer: 0


Question: Find the value of the function \( u(x, y) \) at the point \( \left( \frac{1}{4}, \frac{1}{\sqrt{2}} \right) \), where \( u(x, y) \) is the solution to the equation \( u_{xx} + u_{yy} = 64 \) within the unit disc \( \{(x, y): x^2 + y^2 < 1\} \) and \( u \) vanishes on the boundary of the disc.

Answer: -7


Question: Is the product space of any collection of metric spaces always normal?

Answer: No


Question: Determine the dimension of the vector space of alternating n-linear functions on the set of n x n matrices over a field F.

Answer: 1


Question: Solve the trigonometric inequality: \( \sin^{-1} \frac{2}{x} > \cos^{-1} \frac{2}{x} \). Determine the range of values for \( x \).

Answer: [2, 2\sqrt{2})


Question:
Let f be a twice differentiable function on \(\mathbb{R}\) such that \(f''(x) > 0\) for all \(x \in \mathbb{R}\). Determine which one of the following statements is true:

(i) \(f(x) = 0\) has exactly 2 solutions.
(ii) \(f(x) = 0\) has a positive solution if \(f(0) = 0\) and \(f'(0) = 0\).
(iii) \(f(x) = 0\) has no positive solution if \(f(0) = 0\) and \(f'(0) > 0\).
(iv) \(f(x) = 0\) has no positive solution if \(f(0) = 0\) and \(f'(0) < 0\).

Answer: iii


Question: In a figure with 28 blank hexagons, real numbers are written such that the number in each inner hexagon is the arithmetic mean of the numbers in the six adjacent hexagons. Determine the value in the central hexagon.

Answer: 0


Question: How many $2010 \times 2010$ matrices $A$ are there such that the entries of $A$ are either $0$ or $1$, and the number of $1$s in each row and in each column of $A$ is even?

Answer: 2^{2009^2}


Question: Calculate \( \lim_{n \rightarrow \infty} \int_{0}^{\infty} \left(1+\frac{x}{n}\right)^{-n}\sin\left(\frac{x}{n}\right) dx \).

Answer: 0


Question: Calculate the sum \( \sum_{n=1}^{2015} \phi(n) \cdot \left\lfloor \frac{2015}{n} \right\rfloor \), where \( \phi(n) \) is the Euler phi function and \( \lfloor x \rfloor \) is the floor function.

Answer: 2031120


Question: Let \(f : \mathbb{R} \rightarrow \mathbb{R} \) be a polynomial function satisfying the equation \(f(f(x) - 2y) = 2x - 3y + f(f(y) - x)\) for all \(x, y \in \mathbb{R}\). Find the value of \(f(21) - f(14)\).

Answer: 7


Question: In a Poisson process with intensity \( \lambda = 2 \), let \( S_{n} \) denote the time of the \( n^{th} \) event. Given that \( N(2) = 4 \), find \( E(S_{6}|N(2)=4) \).

Answer: 3


Question: Let $X$ and $Y$ be random variables with the joint probability density function given by $p_{X,Y}(x,y) = \frac{5e^{-5x}}{x}$ for $0 < y < x < \infty$, and $p_{X,Y}(x,y) = 0$ otherwise. Find the covariance Cov$(X,Y)$.

Answer: \dfrac{1}{50}


Question: Let $a_{n(n \geq 1)}$ be a sequence of real numbers such that the sequence $1 + \frac{1}{\sqrt 2} + \cdots + \frac{1}{\sqrt n} - a_n \sqrt n$ is bounded. Find $\lim_{n \rightarrow \infty} a_n$. \[ b_n = 1 + \frac{1}{\sqrt 2} + \cdots + \frac{1}{\sqrt n} - a_n \sqrt n \]

Answer: 2


Question:
Let $x, y, z \ge 0$ such that $xy + yz + zx = 1$. Find the minimum value of the expression:
\[ A = \dfrac{1}{x^2 + y^2} + \dfrac{1}{y^2 + z^2} + \dfrac{1}{z^2 + x^2} + \dfrac{5}{2}(x+1)(y+1)(z+1) \]

Answer: \dfrac{25}{2}


Question: Find the maximum value of \( a \) such that \( \mathbb{E}[\exp(Z^a)] < +\infty \), where \( Z \sim \mathcal{N}(0,1) \).

Answer: 2


Question: Find the limit of the sequence given by: \[ \sum_{k=1}^n e^{\frac{k}{n^2}} \times \ln\left(k+\frac{1}{k}\right) \] as \( n \to \infty \).

Answer: \infty


Question: Calculate the sum of the series \( \sum_{n=1}^\infty \frac{(n^2+n)3^{n-1}}{4^n} \). Determine if the series converges or diverges, and if it converges, find its sum.

Answer: 32


Question: Determine whether there exist two infinite sets of non-negative integers, \(A\) and \(B\), such that every non-negative integer can be uniquely expressed as a sum of one element from \(A\) and one element from \(B\).

Answer: Yes


Question: Determine whether the function \( f(x,y) = \begin{cases} xy \sin\left(\frac{1}{x^2+y^2}\right) & (x,y) \neq (0,0) \\ 0 & (x,y)=(0,0) \end{cases} \) is differentiable at the point \((0,0)\).

Answer: Yes


Question: Use Stirling's Formula to approximate the radius of convergence of the series \( \sum_{k = 0}^{\infty} \frac{x^k}{k!} \). Stirling's Formula is given by \( k! \sim \left(\frac{k}{e}\right)^k\sqrt{2\pi k} \).

Answer: \infty


Question: Determine the number of solutions to the equation \( \cos^5x + \cos^5\left( x+\frac{2\pi}{3}\right) + \cos^5\left( x+\frac{4\pi}{3}\right) = 0 \) in the interval \([0, 2\pi]\).

Answer: 6


Question: Determine the splitting field and its degree over \( \mathbb{Q} \) for the polynomial \( x^6 - 4 \).

Answer: 6


Question: Evaluate \( g'(\frac{1}{2}) \) where \( g(a) = \lim_{h \to 0^+} \int_h^{1-h} t^{-a}(1-t)^{a-1} \, dt \).

Answer: 0


Question: Let \( f: \mathbb{R}^n \to \mathbb{R} \) be defined by \( f(x) = |x| \), where \(|x|\) denotes the Euclidean norm of \(x\). Determine the set of points at which \(f\) is differentiable and find the derivative at those points.

Answer: \frac{x}{|x|}


Question: Determine the constant \( c \) such that the function \( f(x, y) = c \frac{2^{x+y}}{x! y!} \) is a valid probability density function for \( x = 0, 1, 2, \ldots \) and \( y = 0, 1, 2, \ldots \).

Answer: e^{-4}


Question: Let $R = k[x_1 , \ldots , x_n]$ be a polynomial ring over a field $k$ and $M$ be a finitely generated $R$-module. Is it true that the projective dimension of $M$, denoted as $\text{pd }M$, satisfies $\text{pd }M \leq n$?

Answer: Yes


Question:
Evaluate the limit:
\[ \lim_{n \to \infty} \frac{\sqrt{a(4^0n)}+\sqrt{a(4^1n)}+\cdots+\sqrt{a(4^{10}n)}}{\sqrt{a(2^0n)}+\sqrt{a(2^1n)}+\cdots+\sqrt{a(2^{10}n)}} \]
where \( a(k) = (2k+3)(k-1) \) for every \( k \in \mathbb{N} \).

Answer: 683


Question: Calculate the limit: \[ \lim _{n\to \infty }\left(\frac{e^n n!}{n^n}\right) \]

Answer: \infty


Question: In how many ways can 12 people be divided into any number of groups such that person A and person B are not in the same group?

Answer: 3535027


Question: Find the value of the limit: \[ \lim_{z\rightarrow \infty}\dfrac{\Gamma(k,iz)}{\Gamma(k)} \] where $z$ is a real number.

Answer: 0


Question: Find the maximum value of the function \( F(x,y,z) = 2x+y-z \) subject to the constraint \( 4x^2 + 2y^2 + z^2 = 40 \).

Answer: 10


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

Answer: \dfrac{\pi}{4}


Question: Find the derivative \( \partial J / \partial U \) for the function \( J = (I-Ur)^T(I-Ur) \), where \( U \) is a matrix and \( r \) is a constant vector.

Answer: -2(I - Ur)r^T


Question: Determine the order of the pole at \( z = \pi \) for the function \( f(z) = \frac{\sin z}{z-\pi} \).

Answer: 0


Question: Calculate the Lebesgue integral \( \int_{\mathbb{R}^n} 1_S \, d\mathbf{x} \) for the set \( S = \{ (x_1, \ldots, x_n) \in \mathbb{R}^n \mid x_i \ge 0, \sum_{i=1}^n x_i \le a \} \), where \( n \ge 2 \) and \( a > 0 \). Use Fubini's Theorem to find the volume of the standard \( n \)-simplex \( S \).

Answer: \dfrac{a^n}{n!}


Question: If a complex number \(z\) satisfies \(|z+3| + |z-3| = 10\), find the value of \(\frac{60|z+3|}{|z+\overline{z}+\frac{50}{3}|}\).

Answer: 18


Question: How many 6-letter words can be formed using the letters a, b, c, and d if each letter must appear at least once, and 'a' must appear exactly once?

Answer: 900


Question: Determine whether the inequality \( h(a_0,\cdots,a_n) \le \max_{0\le i\le n}\log(\max(1,|a_i|)) \) holds for algebraic integers \( a_0,\cdots,a_n \), where \( h(a_0,\cdots,a_n) \) denotes the logarithmic Weil height.

Answer: Yes


Question: Compute the value of the triple integral \( \int\int\int_{E}{|xyz|dxdydz} \) where \( E=\{(x,y,z)|\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\leq1\} \).

Answer: \dfrac{a^2 b^2 c^2}{6}


Question: Determine the number of real solutions to the equation \( x = 99 \sin(\pi x) \).

Answer: 199


Question: Transform the expression \( \sum^{n}_{i=0}{(-1)^{i}\binom{n}{ i}\binom{n +m - i}{j-i}} \) into a form without a summation, using a combinatorial interpretation and the Principle of Inclusion and Exclusion. Here, \( n, m, \) and \( j \) are natural numbers with \( n \leq j \leq n+m \).

Answer: \dbinom{m}{j}


Question: The periodic function \(g\) is defined on \(\mathbb{R}\) by \(g(x) = f(x)\) for \(0 \le x < a\) and \(g(x) = g(x+a)\) for all \(x\), where \(a > 1\). Given that \(g\) is a continuous function, determine the exact value of \(a\).

Answer: 2


Question: Given that \(t^3\) and \(t e^t \sin(t)\) are solutions to an nth-order linear homogeneous differential equation with constant coefficients, determine the minimum possible value of \(n\).

Answer: 8


Question:
Which of the following statements about a specific $X_{t}$ is formally correct?

1. $X_t$ is a $\mathcal{F}_{t}$-measurable random variable on $ \left ( \Omega, \mathcal{F}, \mathbb{P} \right ) $.
2. $X_t$ is a $\mathcal{F}_{t}$-measurable random variable on $ \left ( \Omega, \mathcal{F}_{t}, \mathbb{P} \right ) $.

Select the correct statement.

Answer: 1


Question: Consider a Jordan curve $L$ in the complex plane $\mathbb{C}$, and let $G$ be the exterior of $L$. There exists a conformal mapping $\Phi: G \rightarrow \overline{\mathbb{C}} \setminus \{z: |z| \leq 1\}$ such that $\Phi(\infty) = \infty$ and $\Phi'(\infty) > 0$. Is the $n$-th derivative of the reciprocal of this mapping, $\left(\frac{1}{\Phi}\right)^{(n)}(z)$, non-zero for all $z \in G$?

Answer: Yes


Question: Find the supremum of the expression \( ab - \left(\frac{a^2}{2} + \frac{b^4}{4}\right) \) for all positive real numbers \(a\) and \(b\).

Answer: \dfrac{1}{4}


Question: Let $F$ be a field and $\langle a_1, \ldots, a_n \rangle \subset F$. Given a non-zero polynomial $f \in F[X_1, \ldots, X_n]$, is it true that if $f(a_1, \ldots, a_n) = 0$, then $(X_i - a_i)$ divides $f$ for some $i \leq n$? Answer with "True" or "False".

Answer: False


Question: Determine the values of \( \alpha > 0 \) for which the integral \( \int_{1}^{\infty} \frac{\ln(x)}{(x-1)^{\alpha}} \, \mathrm{d}x \) is convergent.

Answer: (1, 2)


Question: Is it possible for an analytic function $f(x)$, represented by its power series $\sum_{n=0}^\infty a_n x^n$ with radius of convergence $R$, to have a point $p \in (-R, R)$ such that $f(p) \neq \sum_{n=0}^\infty a_n p^n$?

Answer: No


Question: Determine the distribution of the second greatest value among three independent random variables X1, X2, and X3, each uniformly distributed over the interval (0,1).

Answer: 6x(1 - x)


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

Answer: 0


Question: Find the number of divisors of the number \(2^2 \times 3^3 \times 5^5 \times 7^4 \times 11^3\) that are of the form \(6k+1\), where \(k\) is a non-negative integer.

Answer: 60


Question: Evaluate the limit: \[ \lim_{n\to+\infty} \left(\frac{(3^n-1)^2-9^n}{\sqrt{9^n+n^9}}\right) \]

Answer: -2


Question: Compute the volume enclosed by the lemniscate defined by the equation \(x^4 = x^2 - y^2\).

Answer: \dfrac{4}{3}


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

Answer: \dfrac{\pi}{4}


Question: Calculate the area enclosed by the parametric curve given by \( x = a(t - \sin t) \) and \( y = a(1 - \cos t) \) for \( 0 \leq t \leq 2\pi \), and the line \( y = 0 \).

Answer: 3\pi a^2


Question:
Verify if the following integral correctly represents the volume of the region in the first octant under the surfaces $z=4-x^2$ and $y=4-x^2$, where the projection of the region $R$ onto the $xy$-plane is bounded by $0 \leq y \leq 4$ and $0 \leq x \leq \sqrt{4-y}$, and the cylinders intersect along the plane $z=y$. The integral is given by:
$$\iiint_R dV= 2 \int_0^4 \int_0^{\sqrt{4-y}} \int_0^y dzdxdy$$

Answer: \dfrac{256}{15}


Question: Find the maximum value of \( \frac{x^2+y^2+z^2}{xy+xz+yz} \) where \( x, y, z \in [1, 2] \).

Answer: \dfrac{6}{5}


Question: Using the sequential definition of a limit, prove that \( \lim_{x\to 1} \frac{x^2 - 1}{\sqrt{x} - 1} = 4. \)

Answer: 4


Question: Consider a Brownian motion $W_t$ and the Bachelier model $S_t = 1 + \mu t + \sigma W_t$ for the stock price $S_t$. Find the value of an option that pays $1(S_1 > 1)$. Use the risk-neutral valuation argument to determine the option's value.

Answer: \dfrac{1}{2}


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

Answer: 0


Question: If $n$ and $m$ are positive integers such that $n^3 + (n+1)^3 + (n+2)^3 = m^3$, is it true that $4$ divides $n+1$?

Answer: Yes


Question: Evaluate the integral: \[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \left(\cos(x)+\sqrt{1+x^2}\sin^3(x)\cos^3(x)\right) dx \]

Answer: \sqrt{2}


Question: Evaluate the double limit: \[ \lim\limits_{(x, y) \rightarrow (+\infty, +\infty)} \left(1 + \frac{1}{xy}\right)^{x\sin y} \]

Answer: 1


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

Answer: (-1, 1)


Question: Find the smallest integer $k$ such that there exists an integer $n$ with $2^{10}\cdot3^4\cdot5\cdot k = n^6$. Determine the value of $k$. \( \boxed{} \)

Answer: 112500


Question: Given two 3×3 matrices P and Q such that P is not equal to Q, P³ = Q³, and P²Q = Q²P, find the determinant of the matrix P² + Q².

Answer: 0


Question: Given a finite set $S$ of primes, determine if there exists an imaginary quadratic field $K$ such that all primes in $S$ split completely in $K$. Provide a justification for your answer.

Answer: Yes


Question: Let \( y = \frac{x}{1+x} \), where \( x = {{\omega^{2009}}^{2009}}^{\ldots \text{up to } 2009 \text{ times}} \) and \( \omega \) is a complex cube root of unity. Determine the value of \( y \).\n\n(a) \( \omega \); \quad (b) \( -\omega \); \quad (c) \( \omega^2 \); \quad (d) \( -\omega^2 \).

Answer: B


Question: Find the limit of the sequence \( a_n = \frac{7^n + 6^n - n^{100}}{(7.1)^n - 7^n + n^{101}} \) as \( n \to \infty \).

Answer: 0


Question: Compute the limit of the product \( \prod_{k=2}^{n}\left( 2-\sqrt[k]{2}\right) \) as \( n \) approaches infinity.

Answer: 0


Question: Given the differential equation \( \frac{dy}{dx} = \frac{\sin x}{x} \) and the initial condition \( f(1) = 2 \), find the value of \( f(2) \).

Answer: 2.6593


Question: Determine whether there exists a graph $G$ with chromatic number $\chi(G) \geq n$ that does not contain a complete subgraph $K_n$, for $n \geq 3$. Provide a proof or a counterexample.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{1} (\{2x\} - 1)(\{3x\} - 1) \, dx \), where \( \{x\} \) denotes the fractional part of \( x \).

Answer: \dfrac{19}{72}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)}. \]

Answer: 0


Question: Determine the number of roots of the polynomial equation \(1 + 2x + 3x^2 + 4x^3 + \, ... \, + (n+1)x^n = 0\), where \(n\) is an even integer.

Answer: 0


Question: Find the splitting field of the polynomial \(x^6 + x^3 + 1\) over \(\mathbb{Q}\) and determine the degree of this field extension.

Answer: 6


Question: Determine the value of \( c \) such that the function \( f(x,y) = \begin{cases} \frac{xe^{-y^{2}}}{y}, & y \neq 0 \\ c, & y = 0 \end{cases} \) is continuous at \( y = 0 \).

Answer: 0


Question: Given a basis \((X_1,...,X_m)\) for \(\mathbb{R}^m\) and a basis \((Y_1,...,Y_n)\) for \(\mathbb{R}^n\), do the \(m \cdot n\) vectors \(X_iY_j^t\) form a basis for the vector space \(\mathbb{R}^{m \times n}\) of all \(m \times n\) matrices? Justify your answer.

Answer: Yes


Question: Let $L: V \to V$ be a linear transformation where $V$ is a subspace of $\mathbb{R}^n$ for some $n \in \mathbb{N}$. Given that $L^5 = 0$ and $\dim(\ker(L)) = 7$, determine the largest possible value of $\dim(V)$. Use the rank-nullity theorem to justify your answer.

Answer: 35


Question: Determine the minimum value of \( k \) such that \( \lfloor \log_{2}(k) \rfloor > \frac{2n^{2}}{2n^{3}-n^{2}-n} \), where \( n > 100 \) and \( n, k \in \mathbb{N} \).

Answer: 2


Question: Find the greatest common divisor (GCD) of the complex numbers \(5-3i\) and \(7+i\) in the ring of Gaussian integers \(\mathbb{Z}[i]\).

Answer: 1 + i


Question:
Given three positive real numbers $a$, $b$, and $c$ such that $a + b + c = 3$, find the minimum value of the expression:
\[ A = \frac{2-a^3}{a} + \frac{2-b^3}{b} + \frac{2-c^3}{c}. \]

Answer: 3


Question: Given a matrix \( A \) with \( \text{tr}(A) = 10 \), \( \text{tr}(A^2) = 28 \), and \( \text{tr}(A^3) = 46 \), find the determinant of \( A \).

Answer: 42


Question:
Let $\omega$ be a nonreal root of $z^3 = 1$. Let $a_1, a_2, \dots, a_n$ be real numbers such that
\[\frac{1}{a_1 + \omega} + \frac{1}{a_2 + \omega} + \dots + \frac{1}{a_n + \omega} = 2 + 5i.\]
Compute
\[\frac{2a_1 - 1}{a_1^2 - a_1 + 1} + \frac{2a_2 - 1}{a_2^2 - a_2 + 1} + \dots + \frac{2a_n - 1}{a_n^2 - a_n + 1}.\]

Answer: 4


Question: Determine the maximum dimension of the annihilator \( W^\circ \) of the subspace \( W = \{ A^m ; m \geq 0 \} \) within the vector space \( V = M_n(\mathbb{R}) \).

Answer: n^2 - 1


Question: Let $K$ be a number field, and $H_{K}^{i}, i=1,2,\cdots$ be its Hilbert class field tower. Suppose the tower is finite, and let $L=H_{K}^{n}$ be the top of the tower. Must $L$ be Galois over $K$?

Answer: Yes


Question: Verify the Divergence Theorem for the vector field \( F = xi + y^2j + (z+y)k \) over the region bounded by the cylinder \( x^2 + y^2 = 4 \) between the planes \( z = x \) and \( z = 8 \).

Answer: 64\pi


Question: Let $f: X \to Y$ be a morphism of smooth surfaces, and let $C \subset Y$ be a contractible exceptional curve on $Y$. Assume that $f^{-1}(C)$ is a contractible exceptional curve as well. Denote by $X'$ the blow down of $f^{-1}(C)$ and by $Y'$ the blow down of $C$. Determine whether there exists a morphism $g: X' \to Y'$ such that $X$ is the fiber product $Y \times_{Y'} X'$ and the natural morphism from $Y \times_{Y'} X'$ to $Y$ is the same as $f$. Here, the morphism from $Y$ to $Y'$ is the natural blow-up morphism.

Answer: Yes


Question: Suppose $X$ and $Y$ are random variables with standard deviations $S_X = 2$ and $S_Y = 3$, respectively. Let $Z = 3X - 2Y$, and assume that the standard deviation of $Z$ is $S_Z = 6$. Find the covariance $\text{cov}(X, Y)$. \( \text{cov}(X, Y) = \boxed{?} \)

Answer: 3


Question: Determine whether the following statement is true or false: If $f$ is convex on an open interval $(0, 1)$, then $f$ is continuous on $(0, 1)$. Provide a justification for your answer.

Answer: A


Question: Given a linear equation with two variables \(x\) and \(y\) and three integer coefficients \(a\), \(b\), and \(c\), where \(a\) and \(b\) are prime numbers greater than zero, determine the maximum value of \(c\) such that for any \(c\) greater than this value, the equation \(ax + by = c\) always has a positive integer solution for both \(x\) and \(y\).

Answer: ab


Question: Compute the limit: \[ \lim_{x \to 0}\left(\frac{1+x}{1+2x} \right)^{1/x} \] without using de L'Hopital's rule or Taylor series expansions.

Answer: \dfrac{1}{e}


Question: Consider the sequence defined by \(x_1 = 1\) and \(x_n = \sup\{x \in [0, x_{n-1}): \sin(1/x) = 0\}\) for \(n \geq 2\). Determine the value of \(\limsup x_n\).

Answer: 0


Question:
Let \( v_1, \dots, v_k \) be non-zero vectors of zeros and ones in \( V^n \) over \( \mathbb{R} \), such that for each \( v_i \), there exist indices \( 1 \leq j_1 \leq \dots \leq j_s \leq k \) and \( q > 0 \) satisfying \( v_i + v_{j_1} + \dots + v_{j_s} = q\overline{\mathbf{1}} \), where \( \overline{\mathbf{1}} \) is the vector of all ones. Define a vector \( a = (a_1, \dots, a_n) \) as \( \textit{positive} \) if it is non-zero and \( a_i \geq 0 \) for all \( i \). Let \( M \) be the module over \( \mathbb{Z} \) generated by \( v_1, \dots, v_k \) and \( f: M \rightarrow \mathbb{R} \) a function such that:

1. \( f(\sum k_i v_i) = \sum k_i f(v_i) \)
2. \( f(q\overline{\mathbf{1}}) = q \)
3. If \( a \) is positive, then \( f(a) > 0 \).

Is there an \( \epsilon > 0 \) such that \( f(a) > a_i \epsilon \) for every positive vector \( a = (a_1, \dots, a_n) \) in \( M \) and every \( 1 \leq i \leq n \)?

Answer: Yes


Question:
Calculate the limit of the sequence:
\[
\lim_{n \to \infty} \frac{\left(2\,\sqrt[\Large n]{\, n\,}\, - \,\sqrt[\Large n]{\, 2\,}\right)^{n}}{n^2}
\]

Answer: \dfrac{1}{2}


Question: Calculate the volume of the region enclosed by the plane \( z = 3 - 2y \) and the paraboloid \( z = x^2 + y^2 \).

Answer: 8\pi


Question: Find the maximum value of the expression \( \frac{\cos x - \cos y}{x - y} \) for real numbers \( x \) and \( y \).

Answer: 1


Question: Determine the interval of convergence for the series \(\sum_{n=1}^{\infty}\frac{((n+1)x)^n}{n^{n+1}}\). Specifically, evaluate the convergence at the endpoints of the interval.

Answer: [-1, 1)


Question: Prove that \((\log(n))^{\log(\log(n))} = O(n)\).

Answer: (\log(n))^{\log(\log(n))} = O(n)


Question: Let $f:[0,1] \longrightarrow \mathbb{R}$ be a $C^{\infty}$ function. Define $p(f)= \inf \{ x : f(x)>0 \}$ and $n(f)= \inf \{ x : f(x)<0 \}$, with the convention that $\inf \emptyset = 1$. If $n(f)=p(f)$, is $f$ the constant function $0$?

Answer: No


Question: Let $E$ be a real vector space and $r$ be a positive integer. Determine if the following statement is true: \( \dim \text{Hom}(E \times \stackrel{(r)}{\ldots} \times E, E) = r(\dim E)^2 \).

Answer: True


Question: Consider the curve $C$ obtained by intersecting the surfaces defined by \(x^2+y^2 + z^2=3\) and \(x^2-y^2+z^2 =1\). At the point \((1,1,1)\), which of the following is a tangent vector to the curve? \((2, 0, -2)\)

Answer: (2, 0, -2)


Question: Find the operator norm \( \lVert T \rVert \) for the operator \( T: \ell^2 \to \ell^2 \) defined by \( T(x) = \left( \frac{x_i}{i} \right) \) for \( (x_i) \in \ell^2 \).

Answer: 1


Question: Let \( \rho \in \mathbb{R} \setminus \mathbb{Q} \) be an irrational number, and let \( \varepsilon > 0 \) be arbitrarily small. Define the set \( M = \{ m \in \mathbb{N} : \exists k \in \mathbb{N} \text{ such that } |\rho m - k| < \varepsilon \} \). Do there exist natural numbers \( a \) and \( b \) such that both \( a \) and \( ab+1 \) are elements of \( M \)?

Answer: Yes


Question: Determine whether the radius of convergence of the Taylor series of the function \( f(z) = \frac{1}{1+e^z} \) centered at \( z = 0 \) is \( \pi \).

Answer: \pi


Question:
A bus must visit three cities, labeled A, B, and C, exactly four times each. Determine the number of ways the bus can complete its route if it is not allowed to start and end in the same city. Choose the correct answer from the following options:

(A) 1260
(B) 1120
(C) 980
(D) None of the above.

Answer: D


Question: Evaluate the definite integral \( I = \int\limits_{-\infty}^{\infty} x (\arctan x)' \, dx \).

Answer: 0


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

Answer: 0


Question: Determine whether the sequence of random variables \(X_n = X + I\{0 < X < \frac{1}{n^{4/5}}\}\), where \(X\) is uniformly distributed on \((0,1)\), converges almost surely.

Answer: Yes


Question: Compute the value of the following expression involving definite integrals: \[ 5050\left(\frac{\int_0^1 (1-x^{50} )^{100} \, dx}{\int_0^1 (1-x^{50})^{101} \, dx}\right) \]

Answer: 5051


Question: Find the minimum value of \(x+y\) given the inequality \((2x+\sqrt{4x^2+1})(\sqrt{y^2+4}-2) \geq y > 0\).

Answer: 2


Question: Consider a random walk on the set of natural numbers. If the current position is $M$, with probability $\frac{1}{4}$, the position remains at $M$; with probability $\frac{5}{12}$, the position moves to a random number less than or equal to $M/2$; and with probability $\frac{1}{3}$, the position moves to a random number greater than $M$ but less than or equal to $(3M+1)/2$. Determine whether it is true that almost every random walk of this type eventually reaches 0.

Answer: True


Question: Is it always possible to choose a linear map $f : (\mathbb{Z}/2\mathbb{Z})^n \rightarrow \mathbb{Z}/2\mathbb{Z}$ such that $f$ is nonzero on at least half of a given sequence of nonzero vectors $v_1, \ldots, v_m \in (\mathbb{Z}/2\mathbb{Z})^n$? Specifically, can we ensure that \(|\{i \mid 1 \leq i \leq m \text{ and } f(v_i) \neq 0\}| \geq \frac{1}{2}m\)?

Answer: Yes


Question: Evaluate the integral \( \int _{ 0 }^{ \pi/2 }{ \frac { \sin^n(x) }{ \sin^n(x)+ \cos^n(x) }  } dx \) for \( n = 1, 2, \ldots, n \). Use the substitution \( x = (\pi/2) - t \) to simplify the calculation.

Answer: \dfrac{\pi}{4}


Question: If a sequence \( \{f_{n}\}_{n\in I} \) in a Hilbert space \( H \) is not a Bessel sequence, does it imply that for any given \( B > 0 \), there exists a non-zero \( f \in H \) such that \( \sum_{n\in I}|\langle f,f_{n}\rangle|^{2} > B\|f\|^{2} \)?

Answer: Yes


Question: Let $F$ be a field of characteristic $p$, and let $f(x)$ be a non-constant irreducible polynomial over $F$. Let $E$ be a splitting field of $f(x)$. Determine whether $E$ can be expressed as $F(\alpha)$ for some $\alpha \in E$. In other words, does $E$ contain a primitive element?

Answer: Yes


Question: Find the moment generating function (MGF) of the random variable $X$ with the probability density function $f(x) = \frac{1}{2\theta}e^{-|x|/\theta}$, where $\theta > 0$. Then, use the MGF to determine the second moment $E[X^2]$.

Answer: 2\theta^2


Question: Given a convex polygon with $M$ vertices and $N$ points inside the polygon, where no three points are collinear, determine the maximum number of triangles that can be formed using these $M+N$ points as vertices, such that the sides of the triangles intersect only at the vertices.

Answer: M + 2N - 2


Question: Compute the integral \( \int_{|z|=2} z^n(1 - z)^m\ dz \) for negative integers \( n \) and \( m \).

Answer: 0


Question: Evaluate the flux integral \( \int_S F \cdot \hat{n} \, dS \) where \( F = \langle y, x, z \rangle \), \( S \) is the surface \( z = 1 - x^2 - y^2 \) for \( 0 \leq z \leq 1 \), and \( \hat{n} \) is the upward unit normal to the surface.

Answer: \dfrac{\pi}{2}


Question: Given a $3 \times 3$ matrix $A$ with the characteristic polynomial $p(x) = x^3 - 7x^2 + 5x - 9$, express $\text{adj}(A)$ as a linear combination of $A^2$, $A$, and the identity matrix $I$.

Answer: A^2 - 7A + 5I


Question: Is the local fundamental group of an elliptic singularity virtually solvable? An elliptic singularity is defined as a germ of a normal surface \((X,x)\) such that \(X\) is Gorenstein (\(K_X\) is Cartier) and \(R^1\pi_* \mathcal{O}_Y=\mathbb{C}_x\), where \(\pi:Y\to X\) is a resolution of the singularity. Equivalently, \(\pi_*\omega_Y=m_x\omega_X\).

Answer: Yes


Question: Given positive integers $a, b, c$ that are pairwise relatively prime, determine whether there exist positive integers $x, y, z$ such that $a x + b y = cz$ and $ax, by, cz$ are pairwise relatively prime.

Answer: Yes


Question: Suppose that \( f: \mathbb{R}^2 \to \mathbb{R}^2 \) is an injective mapping from the 2-dimensional Euclidean plane into itself, which maps lines into lines and whose range contains three non-collinear points. Is \( f \) necessarily an affine transformation?

Answer: Yes


Question: Given the function \( L = f \cdot y \) where \( f = X \cdot W + b \), with matrix dimensions \( X.shape=(1, m), W.shape=(m,10), b.shape=(1, 10), y.shape=(10, 1) \), find the derivative \( \frac{\partial L}{\partial W} \).

Answer: X^\top y^\top


Question: Given a surface $S_{0,4}$ (a disk with 3 holes) and two closed curves $A$ and $B$ that fill the surface, consider an element $\phi \in \text{MCG}(S_{0,4})$, the mapping class group of the surface. Let $A' = \phi(A)$ and $B' = \phi(B)$. Do the curves $A'$ and $B'$ fill the surface $S_{0,4}$?

Answer: Yes


Question:
Consider three $n \times 1$ vectors, $a, b, c$. Each element of $a$ and $c$ is in $[0,1]$, and the sum of $a$'s elements is one. Fix $a$ and solve the following optimization problem:
\[
P: \hspace{1cm} \max_{b \in \mathbb{R}^n} \min_{c \in [0,1]^{n}}  b^T(a-c)\\
\hspace{1cm}\text{s.t.} \hspace{1cm} b^Tb \leq 1\\
\hspace{1cm}\hspace{1cm} \sum_{k=1}^n c_k=1
\]
where $c_k$ is the $k$-th element of $c$. Determine if the value of $P$ is always positive.

Answer: No


Question:
Consider the series defined by \(x_n = ac + (a + ab)c^2 + \ldots + (a + ab + \ldots + ab^n)c^{n+1}\). Given that \(|c| < 1\), \(b \neq 1\), and \(|bc| < 1\), determine \(\lim \limits_{n \to \infty} x_n\). Choose the correct answer from the following options:

A) \((x_n)\) does not converge
B) \(\lim \limits_{n \to \infty} x_n = 0\)
C) \(\lim \limits_{n \to \infty} x_n = \frac{(a+bc)}{(1-ab)c}\)
D) \(\lim \limits_{n \to \infty} x_n = 1\)
E) \(\lim \limits_{n \to \infty} x_n = \frac{ac}{(1-bc)(1-c)}\)

Answer: E


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

Answer: f(x) = ax + b


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

Answer: e^{1 - \alpha}


Question:
Determine whether the orthogonal group can be described as a manifold that is the solution to a system of polynomial equations. Specifically, consider the system of equations given by:

If $X$ denotes the $n \times n$ real-valued matrix whose $ij$th entry is the variable $x_{ij}$, then the orthogonal group is proposed to be the zero set of the system of polynomial equations:
$$X^T X - I_n = 0$$

Verify if this system of equations correctly describes the orthogonal group as a manifold. Consider the cases for $n=1$ and $n=2$ to check the validity of this description.

Answer: Yes


Question: Consider the polynomial with integer coefficients: \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \). If all the roots of this polynomial are integers and \( a_n \neq 0 \), must it always be true that \( -\frac{a_0}{a_n} \in \mathbb{Z} \)?

Answer: Yes


Question: Evaluate the infinite series: \[ \sum_{n=1}^{\infty}\frac{1}{(2n-1)^2} \] Choose the correct answer from the following options: \(\text{a) } \frac{\pi^2}{2} \quad \text{b) } \frac{\pi^2}{4} \quad \text{c) } \frac{\pi^2}{6} \quad \text{d) } \frac{\pi^2}{8}\)

Answer: D


Question: Evaluate the limit: \[ \lim_{(x, y) \rightarrow (0,0)} \frac{xy(x^2-y^2)}{(x^2+y^2)^{3/2}} = 0. \]

Answer: 0


Question: Given that the function \( f \) satisfies the equation \( f(a + b) = f(a) + f(b) + ab \) for all real numbers \( a \) and \( b \), and that \( \lim_{h \to 0} \frac{f(h)}{h} = 7 \), determine \( f'(x) \).

Answer: x + 7


Question: Find the limit: \[ \lim_{n \to \infty} \frac{\frac{\sin(1)}{1} + \frac{\sin(2)}{2} + \frac{\sin(3)}{3} + \cdots + \frac{\sin(n)}{n}}{n} \]

Answer: 0


Question: You are given a 100-sided die. After rolling it once, you can choose to either receive the dollar amount shown on the die or pay one dollar to roll again. There is no limit on the number of rolls. What is the expected value of this game?

Answer: \dfrac{1223}{14}


Question: Compute the last four digits of \(2013^{2014^{2015}}\).

Answer: 9361


Question: In a right triangle \( \triangle ABC \) with \( \angle A = 90^\circ \), the triangle is inscribed in a circumcircle \( \mathcal{O} \) with radius 3. Another circle \( \mathcal{K} \) is internally tangent to \( \mathcal{O} \) and also tangent to the sides \( AB \) and \( AC \) of the triangle. The radius of \( \mathcal{K} \) is 2. Find the area of \( \triangle ABC \).

Answer: 7


Question: Evaluate the limit: \[ \lim_{n \to \infty}\left(\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots+\frac{1}{1+a_n}\right), \] where \( a_1 = 3 \) and \( a_{n+1} = a_n^2 + a_n \) for \( n = 1, 2, \ldots \).

Answer: \dfrac{1}{3}


Question: Compute the sum: \[ \sum_{n=1}^{2019} (-1)^{\Omega(n)} \left\lfloor \frac{2019}{n} \right\rfloor, \] where \( \Omega(n) \) is the number of prime factors of \( n \) with multiplicity.

Answer: 44


Question: Evaluate the limit: $$\lim_{n \to \infty}\frac{1}{n}\sum_{k=0}^{n}{\frac{1}{1+\frac{k^6}{n^3}}} $$

Answer: 0


Question:
Evaluate the triple integral \( \iiint_V (3x^{2}+2y+z) \, dxdydz \) over the region defined by the inequalities:
\[
\begin{array}{c}
|x-y| \le 1 \\
|y-z| \le 1 \\
|x+z| \le 1
\end{array}
\]

Answer: 3


Question: Let $L \subset \mathbb{C}$ be a number field such that $L / \mathbb{Q}$ is a Galois extension. Determine if it is true that $[L:L\cap \mathbb{R}] \leq 2$. Justify your answer.

Answer: True


Question:
Determine whether the following statement is true:

\[ e^A = \lim_{j \to +\infty} \left(I + \frac{1}{j}A\right)^j \]

Answer: True


Question: Determine the number of zeros of the polynomial equation \( z^6 - 5z^3 + 1 = 0 \) that lie outside the disk \(|z| > 2\). Use Rouche's theorem to justify your answer.

Answer: 0


Question: Let $R$ be a ring (not necessarily with 1) containing a non-zero element $a$ such that $a^4 + a = 2a^3$. Determine whether $R$ must have at least one non-zero idempotent element.

Answer: Yes


Question: Let $f(x) \in F[x]$ be a polynomial over a field $F$. Let $K$ be the splitting field of $f(x)$ over $F$, and let $G(K/F)$ be the Galois group of $K$ over $F$. The roots of $f(x)$ are $A = \{\alpha_1, \alpha_2, \ldots, \alpha_n\}$. Is it true that for all $\sigma \in G(K/F)$, $\sigma$ is a bijective function from $A$ to $A$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{e^{-x} \cos(x)}{\sin{\frac{1}{x}}} \]

Answer: 0


Question:
Find the minimum value of the expression:
\[
\left| \log_{x_1} \left( x_2 - \frac{1}{4} \right) + \log_{x_2} \left( x_3 - \frac{1}{4} \right) + \cdots + \log_{x_{2017}} \left( x_1 - \frac{1}{4} \right) \right|
\]
where \(x_1, x_2, \ldots, x_{2017} \in \left(\frac{1}{4}, 1\right)\).

Answer: 4034


Question: Determine whether the following statement is true or false: For any square matrix $A$, if $A^4 = \text{Id}$, then $A^2 = -\text{Id}$ or $A^2 = \text{Id}$. Provide a justification for your answer.

Answer: False


Question: Prove that the set of all subsequential limits of a sequence in a metric space is closed. Let $X$ be a metric space and $\{x_n\}$ be a sequence in $X$. Let $E$ be the set of all subsequential limits of $\{x_n\}$. Show that $E$ is closed.

Answer: E


Question: Find the limit: \[ \lim_{n\to\infty} \mathbb{P}\left(\frac{1}{n}\sum_{i=1}^n U_i \in \left[-\frac{\varepsilon}{n^{2/3}},\frac{\varepsilon}{n^{2/3}}\right]\right) \] where \((U_i)_{i\ge 0}\) is an i.i.d. sequence of uniformly distributed random variables on \([-1,1]\) and \(\varepsilon > 0\) is arbitrary.

Answer: 0


Question:
Given the system of equations for real numbers $a, b, c, d, e$:
\[ ac = 1, \]
\[ ad + bc = -1, \]
\[ ae + bd = 1, \]
\[ be = -6, \]
find the value of $a + b + c + d + e$. Provide your answer as a single number.

Answer: 4


Question: Find the maximum value of $x_6 - x_1$ subject to the constraints $\sum_{j=1}^{6} x_j^2 \leq 1$ and $\sum_{j=1}^{6} x_j = 0$, where $x_1, x_2, x_3, x_4, x_5, x_6$ are variables.

Answer: \sqrt{2}


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

Answer: 6240


Question: Evaluate the infinite series: \[ S_2 = \sum_{n=1}^\infty \frac{(-1)^{n-1} (2n+1)}{n(n+1)} \]

Answer: 1


Question: Consider a supersingular elliptic curve $E$ over the finite field $F_q$, where $q = p^n$. The endomorphism ring $\operatorname{End}(E)$ is an order in a quaternion algebra, making it a non-commutative ring. Determine whether there exists an endomorphism $\psi$ of $E$ such that $\psi$ does not commute with the Frobenius morphism $\phi_q: x \mapsto x^q$. Provide a justification for your answer.

Answer: Yes


Question: Determine the values of \( c \in \mathbb{R} \) for which the sequence defined by \( a_1 = c \) and \( a_{n+1} = 1 + \frac{a_n^2}{4} \) converges. If the sequence converges, it converges to the limit \( L = 2 \).

Answer: [-2, 2]


Question:
Let $S$ be a unit sphere in a Hilbert Space $H$ with $\dim H \ge 3$. Suppose $A \subset S$ satisfies the following conditions:

- $A$ is connected.
- The affine hull of $A$ is the whole space $H$.
- For every $x, y \in A$, there exists a unitary operator $T: H \to H$ such that $TA = A$ and $Tx = y$.

Is it true that $A = S$? Assume $A$ is locally compact and locally linearly connected with the induced topology if needed.

Answer: A = S


Question: Given 100 i.i.d. Poisson random variables each with an expected value of 1, approximate the probability that their sum is greater than 85. Use the central limit theorem for your approximation.

Answer: 0.9265


Question: Determine whether the recurrence \( T(n,i) = \left(\left\lfloor\frac{n-i}{i}\right\rfloor \cdot i\right) + T\left(i + (n \operatorname{rem} i), (n \operatorname{rem} i)\right) \) is \( O(n) \), given that \( i < n/2 \).

Answer: O(n)


Question: Let $f$ be a real-valued differentiable function such that $|f(x) - f(y)| < (x - y)^2$ for all real numbers $x$ and $y$, and $f(0) = 0$. Determine the value of $f(1)$. \( \boxed{} \)

Answer: 0


Question: Find the limit superior (limsup) of the series \( \sum \limits _{n=1}^\infty \frac{1+\sin{n}}{4} \).

Answer: \infty


Question: Can a finitely generated subgroup of infinite order be constructed within a group of homeomorphisms that fix the boundary pointwise of a closed unit disc?

Answer: Yes


Question: Is the union of finitely generated submodules of a module also a finitely generated submodule? Specifically, let \( A_i \) be a finitely generated submodule of \( M \) for all \( i \in \mathbb{N} \). Is \( \bigcup_{i\in\mathbb{N}}A_i \) a finitely generated submodule of \( M \)?

Answer: No


Question: Given the equation \( {n \choose 0}^2 + {n \choose 1}^2 + {n \choose 2}^2 + \cdots + {n \choose n}^2 = {x \choose y} \), find the values of \( x \) and \( y \) in terms of \( n \). It is known that the solution is \( {2n \choose n} \).

Answer: \dbinom{2n}{n}


Question: Let $G$ be a graph of order $n$ with degree sequence $d_1, d_2, \\ldots, d_n$ arranged in descending order, such that $d_1 \geq d_2 \geq \\ldots \geq d_n$. Suppose that $d_i + i \geq n$ for all $1 \leq i \leq n$. Does it follow that $G$ is a complete graph?

Answer: No


Question:
If two real polynomials $f(x)$ and $g(x)$ of degrees $m \geq 2$ and $n \geq 1$ respectively satisfy \(f(x^2+1)=f(x) g(x)\) for all real numbers $x$, which of the following statements is true about $f(x)$?

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

Answer: D


Question: Evaluate the limit \( \lim\limits_{x\to 0}{\frac{x+\ln (1-x)}{x^2}} \) given that \( \lim\limits_{x\to 1}{\frac{\ln x}{x-1}}=1 \). Find the value of \( l \) where \( \lim\limits_{x\to 0}{\frac{x+\ln (1-x)}{x^2}}=l\in\mathbb{R} \) without using derivatives.

Answer: -\dfrac{1}{2}


Question: Show that the polynomial \(2z^5 - 6z^2 + z + 1\) has exactly three zeroes in the annular region \(\{z : 1 < |z| < 2\}\).

Answer: 3


Question: Find the general term of the recursive relation \(a_{n+1}=\frac{1}{n+1}\sum\limits_{k=0}^n a_k 2^{2n-2k+1}\) with the initial condition \(a_0=1\).

Answer: \dbinom{2n}{n}


Question: Find the value of \(x > 0\) such that the matrix \(\begin{pmatrix} 100\sqrt{2}&x&0\\-x&0&-x\\0&x&100\sqrt{2}\end{pmatrix}\) has two equal eigenvalues.

Answer: 50


Question: Consider $n$ independent and identically distributed (iid) draws from a cumulative distribution function (CDF) $F$. What is the probability that the $k^{\text{th}}$ highest value is smaller than $x$ and the $(k+1)^{\text{th}}$ highest value is larger than $x$?

Answer: 0


Question:
What is the coefficient of \( \frac{1}{z} \) in the expansion of \( \log\left(\frac{z}{z-1}\right) \), valid in \( |z| > 1 \)?

A. \(-1\)
B. \(1\)
C. \(-\frac{1}{2}\)
D. \(\frac{1}{2}\)

Answer: B


Question: Determine the number of real roots of the equation \(x^{18} + e^{-x} + 5x^2 - 2\cos x = 0\).

Answer: 2


Question: Consider the sequence \( \{a_n\}, n\ge1 \) defined as follows: \[ a_n = \begin{cases} 1989^{1989}, & \text{if } n = 1, \\ \text{sum of digits in }a_{n-1}, & \text{if } n > 1. \end{cases} \] Determine the value of \( a_5 \).

Answer: 9


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function such that \( f(2) = 2 \) and \( |f(x) - f(y)| \leq 5|x-y|^{3/2} \) for all \( x, y \in \mathbb{R} \). Define \( g(x) = x^3 f(x) \). Find \( g'(2) \).

Answer: 24


Question: Find the remainder when the infinite power tower \( 7^{7^{7^{\cdots}}} \) is divided by 5.

Answer: 3


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

Answer: 0


Question: Determine whether the group $\mathbb{R}/\mathbb{Z}$ under addition contains elements of order $n$ for every positive integer $n$. Justify your answer.

Answer: Yes


Question: Is there a cardinal \( \kappa \) with uncountable cofinality such that \( \kappa < \kappa^{\aleph_0} \)?

Answer: Yes


Question: Calculate the integral \( \int_{-a}^{a} \left[ \frac{f(x)}{1+e^x} + f'(x)\log(1+e^x) \right]dx \), where \( f(x) \) is a continuously differentiable even function on \([-a, a]\).

Answer: a f(a)


Question:
Find the greatest common divisor (GCD) of the polynomials:

1. \( y^2 - 10y + 24 + 6x - 9x^2 \)
2. \( 2y^4 - 18x^2y^2 - 48xy^2 - 32y^2 \)

Answer: y - 3x - 4


Question: Determine whether the following statement is true or false: Any function in $L^p$ space, where $1 < p < \infty$, can be expressed as a linear combination of simple functions.

Answer: B


Question: Determine whether the limit \( \lim \limits_{n \to\infty}\ \left[n-\frac{n}{e}\left(1+\frac{1}{n}\right)^n\right] \) exists, and find its value if it does.

Answer: \dfrac{1}{2}


Question: Can an almost unital Banach algebra $A$ be equipped with a modified norm $\|\cdot\|'$ such that it becomes a unital Banach algebra?

Answer: No


Question: Evaluate the limit: \[ \lim _{x \to 1}\frac{\ln\left(\frac{2x-1}{x}\right)}{\sqrt[3]{x}-1} \] without using L'Hopital's Rule.

Answer: 3


Question: If $[F(a): F]=n$, is it true that $a$ is algebraic over $F$?

Answer: Yes


Question: Determine the interval of convergence for the power series \( \sum \frac{3k^2(x)^k}{e^k} \).

Answer: (-e, e)


Question: Determine the values of $q \in \mathbb{C}$ for which the series $\sum_{n=0}^{\infty} p(n)q^n$ converges, where $p(n) = a_k n^k + a_{k-1} n^{k-1} + \ldots + a_1 n + a_0$ is a polynomial with complex coefficients $a_j \in \mathbb{C}$ and $a_k \neq 0$. Use the ratio test to analyze the convergence of the series.

Answer: |q| < 1


Question: Calculate the limit: $$\lim_{x\to 0}\left(\frac{3\sin(x)-3x\cos(x)}{x^3}\right)^{1/x}.$$

Answer: 1


Question: Determine the values of $x$ for which the Taylor series for $e^{-1/x^2}$ around $0$ converges to the function $e^{-1/x^2}$.

Answer: 0


Question: Given a round table with $n$ seats, let $s(n)$ be the smallest number of people such that the probability that at least two people sit next to each other is at least 0.5 when $s(n)$ people choose their seats randomly. Determine whether $\lim_{n \to \infty} \frac{s(n)}{n}$ exists, and if so, find its value.

Answer: 0


Question:
Given that $x + y + z = 1$ where $x, y, z$ are positive real numbers, find the minimum value of the expression:
\[ A = xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2 \]

Answer: 0


Question:
Let $p(z)$ be a polynomial of degree 2 satisfying the following conditions:
\[ \frac{1}{2\pi i}\int_{|z|=2}\frac{zp'(z)}{p(z)}dz = 0 \text{ and } \frac{1}{2\pi i}\int_{|z|=2}\frac{z^2p'(z)}{p(z)}dz = -2. \]
If $p(0) = 2017$, find the explicit form of $p(z)$. Use the Argument Principle to solve the problem.

Answer: 2017(z^2 + 1)


Question: Determine the limit of the function \( f(z) = (z-2)\log|z-2| \) as \( z \to 2 \), or explain why the limit does not exist.

Answer: 0


Question: Let $X_1, X_2, X_3, X_4$ be independent and identically distributed random variables following a standard normal distribution $N(0,1)$. Calculate the expectation of the expression \[ \frac{X_1 - X_2 + X_3}{\sqrt{X_1^2 + X_2^2 + X_3^2 + X_4^2}}. \]

Answer: 0


Question: Let $A$ be a $\mathbb{C}$-algebra and $V$ be an irreducible $A$-module with countable dimension. Determine the dimension of $\text{End}(V)$ as an $A$-module.

Answer: 1


Question:
Find the derivative of the function \( f(x) \) at \( x = 1 \), where \( f: (0, +\infty) \rightarrow \mathbb{R} \) is defined as follows:
\[
f(x) = \begin{cases} \frac{\ln x}{1-e^{x-1}} & x \neq 1 \\ -1 & x = 1 \end{cases}
\]

Answer: 1


Question: Determine if there exists a real number \( x \) such that both \( 2^{\frac{x}{3}} \) and \( 3^{\frac{x}{2}} \) are rational numbers.

Answer: Yes


Question: Find the integer value of $x$ such that $11^x \equiv 15 \pmod{23}$.

Answer: 19


Question: Determine whether the following statement is true: If $X$ and $Y$ are identically distributed random variables, then $E(XY) = E(X^2) = E(Y^2)$. Assume $X$ and $Y$ are not necessarily independent.

Answer: False


Question: Determine whether there exists a simple graph \( \Gamma \) with 16 vertices such that its full automorphism group \( G \) contains a semiregular normal subgroup \( H \) isomorphic to the quaternion group \( Q_8 \).

Answer: Yes


Question: Determine the number of ways to populate \( n \) boxes placed in a row with items such that no two consecutive boxes contain items. Assume the total number of items is greater than or equal to zero.

Answer: F_{n+2}


Question: Evaluate the limit: \[ \lim_{n \to +\infty} n \cdot \cos^2\left(\frac{n \pi}{3}\right) \]

Answer: +\infty


Question: Evaluate the contour integral \( \int_{\lvert z\rvert=1} \mathrm{Log}(z)\,dz \).

Answer: -2\pi i


Question: Calculate the total number of simple cycles of odd length in a complete graph with 17 nodes.

Answer: 16982482829816


Question: Find the smallest integer value of \( N \) such that the area under the curve of \( f(x) = x e^{-x^2} \) from \( N \) to \( N+1 \) is at most 0.01.

Answer: 2


Question: Suppose that $f$ is an entire function and that $|f(z) − z + 2z^2| ≤ \arctan(|z|)$ for all $z ∈ \Bbb C$. Compute $f(2)$.

Answer: -6


Question: For any integer \( n \geq 1 \), provide an example of a finite graph \( G = (V, E) \) such that its spatial dimension \( \text{dim}_s(G) = n \).

Answer: K_{n+1}


Question: Evaluate the double integral \( I = \iint_{[0,1]^2} \frac{dx \, dy}{(1+x^2+y^2)^{3/2}}. \)

Answer: \dfrac{\pi}{6}


Question: Does there exist an algorithm that can solve a system of equations where the equations have a degree of at least 2?

Answer: Yes


Question: Consider a compact oriented simply-connected four-manifold $X$ and an $SU(2)$ bundle $P$ over $X$ with a second Chern class equal to $k$. Let $X$ have a top cell $e^4$, and let $g: X \to S^4$ be the collapsing map that collapses all lower cells to a point. If $P$ is represented by a map $h: X \to BSU(2)$, does $h$ factor through $g$, i.e., does there exist a map $\bar{h}: S^4 \to BSU(2)$ such that $h = \bar{h} \circ g$?

Answer: Yes


Question:
Let \( S = \{(a, b, c) : a, b, c \in \mathbb{R} \} \) such that the function \( f(z) = x + ay + i(bx + cy) \) is analytic. Determine the isomorphism of the linear span \( L(S) \). Choose the correct option:

(a) \( \mathbb{C}(\mathbb{R}) \)
(b) \( \mathbb{R}^2(\mathbb{R}) \)
(c) \( \mathbb{R}(\mathbb{R}) \)
(d) \( \mathbb{R}^3(\mathbb{R}) \)

Answer: B


Question: Suppose \( f(x) \) is differentiable at \( x_0 \) and \( f(x_0) = 1 \). Find \( \lim_{t \to 0} f(x_0 + t)^{\frac{1}{t}} \).

Answer: e^{f'(x_0)}


Question: Let \(X = \min(U, V)\) and \(Y = \max(U, V)\) for independent uniform \((0,1)\) variables \(U\) and \(V\). Calculate the covariance \(\text{cov}(X, Y)\).

Answer: \dfrac{1}{36}


Question: Compute the number of surjective functions \( f : [10] \to [5] \) using the inclusion-exclusion principle.

Answer: 5103000


Question: Let \( \{a_n\} \) be a sequence of strictly positive real numbers such that the series \( \sum_{n=1}^{\infty} a_n \) converges. Does it follow that \( \lim_{n \to \infty} a_n \cdot n = 0 \)? Justify your answer.

Answer: No


Question: Determine the additive upper density \( \operatorname{\delta}_{(u, +)}(A_i) \) for the sets \( A_i = \{n \in \mathbb{N} : L(n) \sim_i c_i \sqrt{n} \} \) where \( i \in \{1, 2\} \), \( L(n) = \sum_{i=1}^{n} \lambda(i) \), and \( \lambda(n) = (-1)^{\Omega(n)} \) with \( \Omega(n) \) being the number of prime divisors of \( n \) (counting with multiplicity).

Answer: 0


Question: Let $(X, \mathcal B, \mu)$ be a measure space where $\mu$ is a positive Radon measure on a locally compact topological space $X$ with Borel $\sigma$-algebra $\mathcal B$. Suppose $A \subset X$ is such that every measurable subset of $A$ has zero measure. Is there a zero measure set $B$ such that $A \subset B$? Provide a justification for your answer.

Answer: No


Question: How many different sets of positive integers \((a, b, c, d)\) are there such that \(a < b < c < d\) and \(a + b + c + d = 41\)?

Answer: 321


Question: Find the residue of \( \frac{z^3}{(z-1)(z-2)(z-3)} \) at \( z = \infty \).

Answer: -6


Question: Let $a > b$ be positive integers. Find the minimum value of \( \frac{a!+1}{b!+1} \).

Answer: \dfrac{3}{2}


Question: Determine the number of complex roots of the equation $x^n = a$, where $a \ge 0$ and $n$ is not an integer.

Answer: \infty


Question:
If $a$ and $b$ are positive integers such that
\[|a-1|+|a-2|+|a-3|+\cdots+|a-2015|=b(b+1)\]
find the sum $a+b$. \(\boxed{?}\)

Answer: 2015


Question: Let $W = \{W_t : t \geq 0\}$ be a Brownian motion. Find $\mathbb{E}(W_t(W_{t+1} + W_{t+2}))$.

Answer: 2t


Question:
In the polynomial ring \(\mathbb{R}[x,y,z]\), determine which of the following ideals is prime:

(a) \(\langle x^{2},y+z\rangle\);
(b) \(\langle x^{2}yz,z\rangle\);
(c) \(\langle xy,xz+yz\rangle\).

Answer: b


Question: Given the equation \( x\cos y + y \cos x = \pi \), find the value of \( y''(0) \).

Answer: \pi


Question: Find the value of the series \( \sum_{n=1}^{\infty}\frac{1}{n^3} \).

Answer: \zeta(3)


Question: Evaluate the contour integral \( \oint_C \frac{1}{(2z+1)^2} \, dz \), where \( C \) is the unit circle centered at the origin, using the Cauchy Integral Formula.

Answer: 0


Question: Prove that any group of order 63 must have an element of order 3, using only Lagrange's theorem and not Cauchy's or Sylow's theorems. Assume that the group cannot consist solely of the identity and elements of order 7. Provide a justification for why this is the case.

Answer: 3


Question: Evaluate the conditional expectation \( E[e^X | Y=1] \) given that \( X \) and \( Y \) are jointly normally distributed with means 0, variances 1, and correlation \( \frac{1}{2} \).

Answer: e^{\frac{7}{8}}


Question: Let \((a_1, a_2, \ldots, a_{2014})\) be a permutation of \((1, 2, 3, \ldots, 2014)\). Define \(S_k = a_1 + a_2 + \cdots + a_k\) for \(k = 1, 2, \ldots, 2014\). Find the maximum possible size of the set \(A = \{S_i \mid S_i \equiv 1 \pmod{2}\}\).

Answer: 1511


Question: Determine the number of non-isomorphic semisimple $\mathbb{C}$-algebras of dimension 8.

Answer: 3


Question: Compute \( 2017^{(2020^{2015})} \mod 13 \).

Answer: 3


Question: Determine the maximal dimension of the eigenvalue 1 subspace of a symmetric zero-diagonal matrix \( A \in \mathbb{F}_2^{n \times n} \).

Answer: n - 1


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

Answer: (1, \infty)


Question: Given that $f(x)$ and $xf(x)$ are particular solutions of the differential equation \(y''+R(x)y'+S(x)y=0\), find the solution of the differential equation \(y''+R(x)y'+S(x)y=f(x)\) in terms of $f(x)$. Assume a solution of the form $(ax^2+bx+c)f(x)$ and determine the suitable value of $a$.

Answer: \dfrac{1}{2}


Question: Calculate the derivative of the function \( f(M) = \operatorname{Trace}(M^pX) \) with respect to the matrix \( M \), where \( M \) and \( X \) are symmetric matrices with \( M \succeq 0 \) and \( X = X^T \), and \( p \) is a nonzero real number with \(|p| \le 10\).

Answer: p M^{p-1} X


Question: Determine the radius of convergence for the power series expansion of \( f(x) = \frac{1}{\cos(x)} \).

Answer: \dfrac{\pi}{2}


Question: Let $M$ be a topological space and $f, g : M \rightarrow \mathbb{R}_+$ be two non-negative functions with $f(x_0) = g(x_0) = 0$ for some $x_0 \in M$. Suppose $f(x) \leq g(x)$ for each $x \in M$, and $g$ is continuous at $x_0$. Is $f$ necessarily continuous at $x_0$?

Answer: Yes


Question: Let $A$ and $B$ be $n$-dimensional complex Hermitian matrices. If $e^{-iA} = e^{-iB}$, does this necessarily imply that $A = B$? Justify your answer.

Answer: No


Question: Consider a random variable Y generated as follows: First, select a value of X = x at random (uniformly) over the interval (0,1). Then, select a value of Y = y at random (uniformly) over the interval (0,x). Find the probability density function fY(y).

Answer: -\ln(y)


Question: What is the maximum number of congruent cylinders (with radius $r$ and height $h$) that can be arranged in space so that each cylinder is externally tangent to every other cylinder?

Answer: 4


Question: Find the value of the infinite sum \( \sum_{n=0}^{\infty} \frac{F_n}{3^n} \), where \( F_0 = 0, F_1 = 1, \) and \( F_n = F_{n-1} + F_{n-2} \) for \( n \geq 2 \).

Answer: \dfrac{3}{5}


Question: Determine the values of \(\alpha\) for which the following conditions hold: \(\lim_{x\to \infty}f(x)=1\) and \(\lim_{x\to \infty}f'(x)=\alpha\), where \(f\) is a differentiable function on \(\mathbb{R}\).

Answer: 0


Question: Calls arrive according to a Poisson process with rate \( \lambda = 15 \). Calculate the expected value \( E(N(2,4]N(3,5]) \), where \( N(a,b] \) denotes the number of arrivals in the interval \((a,b]\).

Answer: 915


Question: Determine the number of roots of the polynomial \( z^6 - 5z^4 + 8z - 1 \) that lie within the region \( |z| < 2 \).

Answer: 4


Question: Find a polynomial $q(x)$ of degree at most 2 such that $\phi^4 = q(\phi)$, given that $\phi : V \rightarrow V$ has a minimal polynomial $x^3 + 2x + 1$. What is $q(x)$?

Answer: -2x^2 - x


Question: Given the function \( f(z) = \frac{2}{z^2 - 1} \), find the sum of the coefficients \( a_{-3} \) and \( a_{2} \) in its Laurent series expansion around \( z = 2 \). The official answer is \( \frac{26}{27} \). Verify this result.

Answer: \dfrac{26}{27}


Question:
Determine which of the following conditions ensure that a continuous function \( f: [-\pi, \pi] \rightarrow \mathbb{R} \) is identically zero:

(a) \( \int_{-\pi}^{\pi} x^n f(x) \, dx = 0 \) for all \( n \geq 0 \)

(b) \( \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx = 0 \) for all \( n \geq 0 \)

(c) \( \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx = 0 \) for all \( n \geq 1 \)

Answer: a


Question:
Let $X$ be a set and $f: X \to P(X)$ a function, where $P(X)$ is the power set of $X$. Define the set $Z = \{x \in X : x \not\in f(x)\}$. Determine which of the following statements is correct:

A. $Z$ is empty
B. $Z$ isn't empty
C. $Z = X$
D. $Z$ doesn't belong to $P(X)$
E. $Z$'s complement belongs to $P(X)$

Answer: E


Question: Consider the function \( f(x) = \frac{1}{x^2+1} \). If we integrate this function over infinitely many disjoint intervals \([x_1, x_2], [x_3, x_4], \ldots, [x_{2k+1}, x_{2k+2}], \ldots\), is the sum of these integrals finite?

Answer: Yes


Question: Let X be a normal topological space, A a closed subset of X, and U an open set in X that contains A. Find an open set V in X such that A is contained in V and the closure of V is contained in U.

Answer: V


Question: Let \( A \) be a unital Banach algebra and \( a \in A \) be an idempotent element such that \( a \neq 0 \) and \( a \neq 1 \). Determine the spectrum of \( a \).

Answer: \{0, 1\}


Question: Evaluate the limit: \[ L_1 = \lim_{x \to 0} \frac{\text{erf}(x^2)}{\text{erf}(x)}. \]

Answer: 0


Question: Find the dimension of the quotient ring \( \mathbb{K}[x,y,z]/(xy,xz,yz) \), where \( \mathbb{K} \) is a field.

Answer: 1


Question:
Let $\mathcal{FG}$ be the category of finite groups, and let $S$ be a full subcategory of $\mathcal{FG}$. Assume that $G \in \mathcal{FG}$ and $P \in S$ is a subgroup of $G$. We say that $P$ is $S$-maximal if there is no object $P' \in S$ with $P \subset P' \subset G$. Assume that $S$ satisfies the following conditions:

1. The subcategory $S$ is closed under taking subgroups, extensions, and isomorphisms. That is, if $P \in S$ and $Q \subset P$, then $Q \in S$. Moreover, for every short exact sequence $1 \to P \to Q \to R \to 1$, we have $Q \in S$ if $P, R \in S$. Additionally, every group isomorphic to an object of $S$ lies in $S$.

2. For every $G \in \mathcal{FG}$, every two $S$-maximal subgroups of $G$ are conjugate. Moreover, for every maximal $S$-subgroup $P$ of $G \in \mathcal{FG}$, we have $N(N(P)) = N(P)$ and $|G|/|N(P)|$ is coprime to $|P|$.

Does it follow that $S$ is the category of $p$-groups for some prime number $p$?

Answer: Yes


Question: Find the value of \( a > 0 \) that maximizes the improper integral \( \int_{0}^{\infty} e^{-ax} \cos x \, dx \).

Answer: 1


Question: Evaluate the integral \( \int \frac{|dz|}{|z-2|^2} \) where \(|z| = 1\) is the domain.

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


Question: Compute the value of the infinite series \( \sum_{n=0}^\infty \frac{(n+1)^2}{n!} \).

Answer: 5e


Question: Determine the cardinality of the set $B$, which consists of all infinite binary strings that contain a finite number of '1's.

Answer: \aleph_0


Question:
Find the radius of convergence of the series whose partial sums are given by:
\[ S_n = \sum \frac{(a+nx)^n}{n!} \]

Answer: \dfrac{1}{e}


Question: Solve the ordinary differential equation \( \dot{x}=\sqrt{x^{2}-\frac{2}{3}x^{3}} \) with the initial condition \( x(0)=0 \).

Answer: 0


Question: Find the equation of the curve that passes through the point (0,0) for the differential equation: \[ y'=\frac{\sin x + e^x (\sin x + \cos x) + x\cos x - y(1 + \cos x + e^x)}{e^x + \sin x + x - 2y} \]

Answer: y = \sin x


Question: Determine the number of different quadratic functions of the form \( x \mapsto ax^2 + bx + c \pmod{12} \).

Answer: 864


Question: Find the volume of the region bounded above by the cylinder \( z = x^2 \) and below by the region enclosed by the parabola \( y = 2 - x^2 \) and the line \( y = x \).

Answer: \dfrac{63}{20}


Question: Find the smallest possible value of $m+n$ for positive integers $m$ and $n$ with $m > n$, such that the last three digits of $2012^m$ and $2012^n$ are identical.

Answer: 104


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

Answer: 0


Question: Find the maximum value of the function \( f(x,y,z) = xyz + xy \) subject to the constraints \( x^2 + y^2 + z^2 \leq 1 \) and \( x, y, z \geq 0 \).

Answer: \dfrac{16}{27}


Question: Solve the second-order differential equation \( y'' - 9 y^2 = 0 \) with the initial conditions \( y(0) = 0 \) and \( y'(0) = 0 \).

Answer: 0


Question: Three stations are distributed uniformly over a distance of 10 units. Calculate the probability that the maximum distance between any two stations is at most 1 unit.

Answer: \dfrac{7}{250}


Question: Does every smooth manifold admit a complete distance metric that is compatible with its topology? Provide a justification for your answer.

Answer: Yes


Question: Find the function \( f(x) \) that satisfies the functional equation \( f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)} \) for \( f(y) \neq 0 \) and the condition \( f'(1) = 2 \).

Answer: x^2


Question: Determine the value of the parameter $a$ for which the polynomial $P(x) = x^3 + ax + 2$ has three distinct real roots.

Answer: a < -3


Question:
Given two integers \(a\) and \(b\) such that:

- \(a\) is even,
- \(8a + 15b = 7\),
- \(a \equiv b \pmod{7}\),

find the greatest common divisor (GCD) of \(2a + 4b\) and \(2a + 8b\).

Answer: 28


Question: Evaluate the limit: $$\lim_{n\to\infty}{n\left(\ln(n+2)-\ln n\right)}$$

Answer: 2


Question: Find the degree of the field extension $[\mathbb{Q}(\sqrt[4]{3},\sqrt[5]{3}):\mathbb{Q}]$. Provide your answer as a single integer.

Answer: 20


Question: Determine whether the statement "If $\kappa$ is an infinite cardinal number, then $[0,\kappa]$ is not homeomorphic to $[0,\kappa+\kappa]$" is true or false, and justify your answer.

Answer: True


Question: Is the uniform limit of a sequence of piecewise continuous functions $f_n:[a,b]\to\mathbb{R}$ also piecewise continuous? If $f_n \rightrightarrows f$, does it imply that $f$ is piecewise continuous?

Answer: No


Question: Calculate the flux of the vector field \( \vec{F}(x,y) = (x^2+y^2)^{-1}\begin{bmatrix} x \\ y \end{bmatrix} \) across the circle \( C \) of radius \( a \) centered at the origin, with positive orientation.

Answer: 2\pi


Question: Evaluate the limit of the sequence: $$\lim_{n\rightarrow\infty}{\frac{\prod_{k=1}^n(2k-1)}{(2n)^n}}$$

Answer: 0


Question: Determine whether there can exist three transitive models of ZFC with the same ordinals, $M_0, M_1, N$, such that there are elementary embeddings $j_i : M_i \to N$ for $i<2$, but there is no elementary embedding from $M_0$ to $M_1$ or vice versa.

Answer: Yes


Question: Let $f \in L^2(\mathbb{R}^3)$ be a function with compact support. Define $v: \mathbb{R}^3 \to \mathbb{R}$ by \[ v(x) = \int_{\mathbb{R}^3} \frac{f(y)}{|x-y|} \, dy. \] Is it true that $v \in W^{2,2}(\mathbb{R}^3)$ and \[ \|v\|_{2,2} \leq C\|f\|_2 \] for some positive constant $C$?

Answer: No


Question: Determine the value of the parameter $a$ such that the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \begin{cases} \sin\frac{1}{x} & \text{for } x \in \mathbb{R} \setminus \{0\} \\ a & \text{for } x = 0 \end{cases} \) has an antiderivative on \( \mathbb{R} \).

Answer: 0


Question: Evaluate \( f^{\prime}(0) \) for the function \( f(x) = \begin{cases} g(x) \sin(1/x) & \text{if} \ x \neq 0 \\ 0 & \text{if} \ x = 0 \end{cases} \), given that \( g(0) = g^{\prime}(0) = 0 \).

Answer: 0


Question: Find the sum of the series \( \sum_{k=1}^{\infty} \frac{a_k}{k} \), where \( a_k = \frac{k+1}{2k+4} - \frac{k}{2k+2} \) for \( k \geq 2 \).

Answer: \dfrac{1}{8}


Question: Let $ f : \mathbb{R} \to \mathbb{R} $ be a continuous function such that $ f(x) f(f(x)) = 1 $. Given that $ f(1000) = 999 $, find the value of $ f(500) $. \( \text{(Express your answer as a single number.)} \)

Answer: \dfrac{1}{500}


Question: Consider $n$ real-valued functions $\{f_{i}(t): t \in \mathbb{R}\}$ for $i=1,2,3,...,n$. If any of the functions $f_{i}(t)$ are discontinuous at $t=a$, can all the nontrivial elementary symmetric polynomials formed with $\{f_{i}(t)\}$ be continuous at $t=a$? Provide a definitive answer with reasoning.

Answer: Yes


Question: Let $p \ge 3$ be a prime number. Suppose you are given $N$ lines $l_1, \dotsc, l_N \subset \mathbb{F}_p^2$. You can translate these lines to obtain new lines $l_1', \dotsc, l_N'$ (which are either parallel or identical to the original lines) such that the entire vector space $\mathbb{F}_p^2$ is covered by these new lines: $l_1' \cup \dotsb \cup l_N' = \mathbb{F}_p^2$. Is it always possible to achieve this if $N \ge 2p - 1$? More generally, given $(p-1)n+1$ affine hyperplanes in $\mathbb{F}_p^n$, can you always translate them so that the resulting translates cover the whole space $\mathbb{F}_p^n$?

Answer: Yes


Question: Consider the standard one-dimensional Brownian motion $\{B_t:t\in [0,1]\}$ on the closed unit interval. Fix $\gamma\in (0,1/2)$. It is known that there exists a positive random variable $K\equiv K(\gamma)$ such that for any pair $s,t\in [0,1]$, we have \(|B_t-B_s|\leq K|t-s|^{\gamma}\) almost surely. Determine whether $K$ can be chosen such that $\mathbb{E}[K]<+\infty$. Provide a justification for your answer.

Answer: Yes


Question:
Given the mass of a thin plate is defined by the double integral:
\[ M = \int_0^1 \left[ \int_0^x (x+y^2) \, dy \right] \, dx + \int_1^2 \left[ \int_0^{2-x} (x+y^2) \, dy \right] \, dx \]
Determine the function that describes the surface mass density of the plate.

Answer: x + y^2


Question:
Let $A_1, A_2, \dots, A_{12}$ be 12 equally spaced points on a unit circle. Calculate the sum of the squares of the distances between all pairs of these points:
\[(A_1 A_2)^2 + (A_1 A_3)^2 + \dots + (A_{11} A_{12})^2.\]

Answer: 144


Question: Determine the values of \(\alpha > 0\) for which the improper integral \(\int_{1}^{\infty} \frac{\sin x}{x^{\alpha}}dx\) is convergent.

Answer: \alpha > 0


Question: Let $C$ be the curve defined by the locus of the intersection points of the lines $x = 2 + m$ and $my = 4 - m$. A circle $S$ is given by the equation $(x-2)^2 + (y+1)^2 = 25$. The circle $S$ intersects the curve $C$ at four points $P, Q, R, S$. If $O$ is the center of the curve $C$, find the value of $OP^2 + OQ^2 + OR^2 + OS^2$. Assume the curve $C$ is given by the equation $xy - 2y + x = 6$.

Answer: 100


Question: Evaluate the integral \( \int_0^1 \frac{2x^3+x}{\tan x+x^{\frac{1}{3}}}\, \mathrm{d}x \).

Answer: 1


Question: Find the natural number $k$ such that the equation $x^3 + y^3 + z^3 = kx^2y^2z^2$ has positive integer solutions for $x$, $y$, and $z$.

Answer: 3


Question:
Determine whether the following identity is true for an indeterminate $x$ and a non-negative integer $n$:
\[ x\prod_{k=1}^n(k^2-x^2)=\frac1{4^n}\sum_{m=0}^n\binom{n-x}m\binom{n+x}{n-m}(x+2m-n)^{2n+1}. \]

Answer: True


Question: Let $X$ be an infinite set, and let $E \subseteq \mathcal{P}(X)$ be a collection of subsets of $X$. We say that $E$ is $T_1$ (with respect to $X$) if for all $x \neq y \in X$, there is $e \in E$ with $x \in e$ and $y \notin e$. The set $E := \{X \setminus \{y\}: y \in X\}$ is an example of a $T_1$-set $E$ with $|E| = |X|$. Is it possible for $|E|$ to be less than $|X|$?

Answer: Yes


Question: Given positive integers \(a, b, c\) that are not all equal, and the condition \(\frac{\sqrt{5}a+b}{\sqrt{5}b+c} \in \mathbb{Q}\), find the minimum value of the expression \(\frac{a^2+b^2+c^2}{a+b+c}\).

Answer: 3


Question: Find the maximum possible area bounded by the parabola \( y = x^2 + x + 10 \) and a chord of the parabola of length 1.

Answer: \dfrac{1}{6}


Question: Determine the range of real values for \( n \) such that the integral \( I(n) = \int^{\infty}_{0} \frac{\ln(x)}{x^n(1+x)}\, dx \) converges.

Answer: (0, 1)


Question: Determine the cardinality of the set of subrings of \(\mathbb{Q}\).

Answer: 2^{\aleph_0}


Question: Compute the value of the infinite product: \[ \prod_{n=1}^\infty\left(1+\frac{(-1)^n}{n+1}\right) \]

Answer: \dfrac{1}{2}


Question: Determine the coefficient of the term $x^{60}$ in the power series expansion of $C(x) = \frac{1}{1-x} \cdot \frac{1}{1-x^2} \cdot \frac{1}{1-x^5}$. This coefficient represents the cardinality of the set $A = \{(N, D, Q) \mid 0.05N + 0.1D + 0.25Q = 3 \text{ and } N, D, Q \in \mathbb{N}\}$.

Answer: 205


Question: Does the existence of a gradient for every point \((x,y) \in \mathbb{R}^2\) for a real function \(f\) imply that the function is continuous?

Answer: No


Question: Evaluate the integral \( \int_{e}^{e^4} \sqrt{\ln x} \, dx \) given that \( \int_1^2 e^{x^2} \, dx = a \).

Answer: 2e^4 - e - a


Question: Determine whether a tangent exists at the point \(x=1\) for the function \(y(x) = \ln\left(\frac{1+\sqrt{1-x^2}}{x}\right) - \sqrt{1-x^2}\), which is defined on the interval \((0,1]\).

Answer: Yes


Question: Evaluate the integral: \[ \int_2^4 \frac{\sqrt{\ln(9-x)}}{\sqrt{\ln(9-x)} + \sqrt{\ln(3+x)}} \, \mathrm{d}x \]

Answer: 1


Question: Evaluate the integral \( \oint_C \frac{z^2}{\exp z + 1} \, dz \) using the residue theorem, where \( C \) is the contour bounded by \( |z| = 4 \).

Answer: 4\pi^3 i


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

Answer: 1


Question: Find a closed form for \(a_n = \sum_{k=0}^{n}\binom{n}{k}(n-k)^n(-1)^k\) using generating functions.

Answer: n!


Question: Determine whether the expression \(\sqrt[3]{\sqrt{52} + 5} - \sqrt[3]{\sqrt{52} - 5}\) is irrational, rational, or an integer.

Answer: 1


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(\frac{n-1}{n+1}\right)^n \] and show that it equals \( \frac{1}{e^2} \).

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


Question: Find the greatest integer less than or equal to \( \sum_{n=1}^{9999}\frac{1}{n^{1/4}} \).

Answer: 1332


Question: What is the probability that five randomly chosen points from a unit square will determine an ellipse when used to form a conic section?

Answer: \dfrac{1}{2}


Question:
Consider the transfer function matrix \(G(s)\) of a continuous-time system given by:
\[G(s) = \begin{bmatrix}\frac{1}{s^2+2s}&\frac{s+1}{s} \\ -\frac{1}{s+1} & \frac{1}{s^2+4s+3} \end{bmatrix}\]
Suppose that \((A,B,C,D)\) is a state-space model that forms a minimal realization. What is the dynamic order of this system (i.e., the dimension of the state-space)?

Answer: 4


Question: Given the probability generating function of a discrete random variable $N$ as \( G_N(z) = e^{-3 + 2z + z^2} \), calculate \( \Pr(N \leq 2) \).

Answer: 6e^{-3}


Question: Can the sample path of a Brownian motion be Hölder continuous on the entire interval $[0, \infty)$?

Answer: No


Question: Given the cubic polynomial \(30x^3 - 50x^2 + 22x - 1\) with distinct real roots \(p, q,\) and \(r\) strictly between 0 and 1, find the value of the sum \(\frac{1}{1-p} + \frac{1}{1-q} + \frac{1}{1-r}\).

Answer: 12


Question:
Find the length of the curve parametrized by the equations:
\[ x(\theta) = r(\theta - \sin\theta) \]
\[ y(\theta) = r(1 - \cos\theta) \]
for \(0 \leq \theta \leq 2\pi\).

Answer: 8r


Question: Let $X$ be a topological vector space, $x \in X$, and $M < X$ be a linear subspace. Does the condition that $x \in M + U$ for every open neighborhood $U$ of $0$ imply that $x$ is in the closure of $M$?

Answer: Yes


Question: Let $S_n=\sum_{i=1}^{n}X_i$ be a random walk on $\mathbb{Z}$ with $S_0=0$, $\mathbb{P}(X_i=1)=\frac{2}{3}$, and $\mathbb{P}(X_i=-1)=\frac{1}{3}$. Determine the strictly positive constant $c\neq1$ for which $M_n:= c^{S_n}$ is a martingale.

Answer: \dfrac{1}{2}


Question: In a triangle $ABC$, it is given that $\cos 3A + \cos 3B + \cos 3C = 1$. Determine the measure of any one angle in the triangle.

Answer: 120^\circ


Question:
Let $X_1, X_2, \ldots, X_{10}$ be independent and identically distributed random variables, each following a uniform distribution $U(-5, 5)$. Determine the distribution of the random variable \( Y = -2\sum\limits_{i=1}^{10}\log\left(\frac{|X_i|}{5}\right) \). Choose the correct option:

(A) $\chi_{10}^2$
(B) $10\chi_{2}^2$
(C) $\chi_{20}^2$
(D) $\frac{1}{2}\chi_{20}^2$

Answer: C


Question: Evaluate the alternating sum: \[ \sum_{k=0}^{49}(-1)^k\binom{99}{2k} \]

Answer: -2^{49}


Question:
Find the maximum value of the cross-entropy function for a discrete random variable with probabilities \( p_1, p_2, \ldots, p_n \), where the cross-entropy is given by:
\[ H(x) = -(p_1\log p_1 + p_2\log p_2 + \cdots + p_n\log p_n) \]
subject to the constraint \( \sum_{i=1}^{n} p_i = 1 \).

Answer: \log n


Question: Determine the automorphism group of the group $\Bbb Z_{2} \times \Bbb Z_{2}$.

Answer: S_3


Question: If $G$ is a compact topological group and $K \subset G$ is closed in $G$, is the quotient space $G/K$ necessarily compact?

Answer: Yes


Question: Determine the number of permutations \( \alpha \in S_9 \) such that \( \alpha \omega \alpha^{-1} = \gamma \), where \( \omega = (13624)(587)(9) \) and \( \gamma = (15862)(394)(7) \).

Answer: 15


Question: Find the remainder when \( P = \sum_{n=3}^{42} \binom{\binom{n}{2}}{2} \) is divided by 10.

Answer: 4


Question: Suppose \( \tau \) is an almost Euclidean refining (AER) topology on \( \mathbb{R} \). Is there a nonempty open set \( U \) in the Euclidean topology \( \eta \) such that the subspace topology on \( U \) induced by \( \tau \) refines the subspace topology on \( U \) induced by \( \eta \)?

Answer: Yes


Question: Verify using the \(\delta-\varepsilon\) definition that \(\lim_{x \to 0^-} e^{1/x} = 0\).

Answer: 0


Question:
Find the mistake in the following summation of the series:
\[ S = \sum_{n=1}^{\infty}\frac{n}{(2n-1)^2(2n+1)^2} \]
The given answer is \( \frac{1}{8} \). The steps taken were:
1. \( \frac{n}{(2n-1)^2(2n+1)^2} = \frac{1}{2}\left(\frac{1}{(n - \frac{1}{2})^2}-\frac{1}{(n + \frac{1}{2})^2}\right) \)
2. From this, \( S_N = \frac{1}{2}\left(4-\frac{1}{(N+\frac{1}{2})^2}\right) \) and \( S = \lim_{N\to\infty}S_N = 2 \).
Identify where the mistake was made in this process.

Answer: \frac{1}{8}


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

Answer: 1


Question: Determine the values of the parameter \( \alpha \) for which the integral \( \int_0^\infty \frac{\ln(1+x^2)}{x^\alpha} \, dx \) converges.

Answer: (1, 3)


Question: Find the radius of convergence for the power series \( \sum_{k=1}^{\infty} a_k x^k \), where \( a_k = \sum_{j=1}^{k} \frac{1}{j!} \binom{k-1}{j-1} \).

Answer: 1


Question: There are 10 balls numbered 1 to 10. Two students each have two identical bags. In how many ways can the students distribute the balls into their bags such that no bag is empty?

Answer: 204630


Question:
Given that \( z + \frac{1}{z} = -1 \) where \( z \) is a complex number, find the value of the sum \( \sum_{r=1}^{99} \left( z^r + \frac{1}{z^r} \right)^2 \). Choose the correct answer from the following options:

A) 198
B) 3
C) 99
D) 0

Answer: A


Question: Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \( f(x + y) + f(z) = f(x) + f(y + z) \) for all \( x, y, z \in \mathbb{R} \).

Answer: f(x) = ax + b


Question: Given a triangle ABC with sides described by the equations: \( x + y = 2 \) for AB, \( 2x + 3y = 5 \) for AC, and \( 5x - y = 7 \) for BC, find the equation of the median AD without determining the vertices of the triangle.

Answer: 29x + 35y = 64


Question: Find the volume of the region bounded by the cylinder \( \{(x,y,z):x^2+y^2\leq 1\} \), above the plane \( z=0 \), and below the plane \( x+z=1 \).

Answer: \pi


Question: Let $(X,\mathcal{M},\mu)$ be a measure space and $1\leq p\leq\infty$. Suppose $f,g\in L^p(X,\mathcal{M},\mu)$ with $f(x)g(x)=0$ for almost all $x\in X$ and $\Vert f\Vert_p=\Vert g \Vert_p=1$. Verify by direct calculation that \(\Vert f+g\Vert_p^2+\Vert f-g\Vert_p^2=2(\Vert f\Vert_p^2+\Vert g\Vert_p^2)\) if and only if $p=2$.

Answer: 2


Question: Determine the value of \( t \) for which the series \( \sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 n } } { ( n + 2 ) ^ { 2 n + t } ( n + t ) ^ { n + 2 t } } \) is convergent.

Answer: t > \dfrac{1}{3}


Question:
Suppose $I$ is an interval, and $g, f, h$ are functions defined on $I$, except possibly at $x_0$. For all $\varepsilon > 0$, for all $x \in I$, and for some constant $L$, the following holds:
\[ (L-\varepsilon)g(x) < f(x) \le Lh(x) .\]
Additionally, assume:
\[\lim_{x\to x_0} g(x) = \lim_{x\to x_0} h(x) = 1.\]
Prove that for all $\varepsilon > 0$,
\[L-\varepsilon < \lim_{x\to x_0} f(x) \le L,\]
and conclude that
\[\lim_{x\to x_0} f(x) = L.\]

Answer: L


Question:
For a random variable $X$ with probability density functions $f_0(x)$ and $f_1(x)$ defined as follows:

\[ f_0(x) = \begin{cases} 1, & \text{if } 0 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases} \]

\[ f_1(x) = \begin{cases} \frac{1}{2}, & \text{if } 0 \leq x \leq 2 \\ 0, & \text{otherwise} \end{cases} \]

Test the null hypothesis $H_0: f = f_0$ against the alternative hypothesis $H_1: f = f_1$ using a single observation of $X$. Determine the power of the most powerful test of size $\alpha = 0.05$. \( \text{Use the Neyman-Pearson lemma to find the critical region and calculate the power.} \)

Answer: 0.525


Question: For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\sqrt{p}|<\frac{1}{2}$. For example, $b(6)=2$ and $b(23)=5$. Calculate the sum $S=\sum_{p=1}^{2007} b(p)$. Determine the value of $S$. \( \boxed{} \)

Answer: 59955


Question: Find the order of the element \(\frac{2}{3} + \mathbb{Z}\) in the quotient group \(\mathbb{Q}/\mathbb{Z}\).

Answer: 3


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

Answer: \infty


Question: Find the coefficient of \(x^{26}\) in the expansion of \((x^2+x^3+x^4+x^5+x^6)^8\).

Answer: 13140


Question: You have $20,000 to invest in 5 different stocks, with the condition that you can only invest in multiples of $1,000 and not all the money can be put into one stock. How many different ways can you invest the $20,000?

Answer: 10621


Question: Given a matrix $A=(a_{ij})_{n\times n}$, let $C_{i,j}$ be the cofactor in position $(i,j)$. By the determinant formula, we have \(\det A=\sum_{i=1}^n a_{i,1}C_{i,1}.\) Evaluate the expression \(\sum_{i=1}^n a_{i,1}C_{i,2}.\) Must this expression evaluate to zero?

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \to \infty} \binom{3n}{n}^{1/n} \]

Answer: \dfrac{27}{4}


Question: Compute the flux of the vector field \( \textbf{F} = \langle x\cos^2z, y\sin^2z, \sqrt{x^2+y^2}\:z \rangle \) through the surface of the solid cone above the \(xy\)-plane and inside \(z = 1 - \sqrt{x^2+y^2}\) using the divergence theorem. Express your answer as a single numerical value.

Answer: \dfrac{\pi}{2}


Question: Find the length of the arc between two concentric circles with radii $R$ and $R + \mathrm{d}R$, given that the arc subtends an angle $\theta$.

Answer: \theta \, dR


Question: Given the equation \(140x^{0.75}y^{0.25} = 213,415\), find the rate of change of \(x\) with respect to \(y\) when \(x = 3000\) and \(y = 200\).

Answer: -5


Question:
Find all functions \( f : \mathbb{R}^{+*} \to \mathbb{R}^{+*} \) such that:
\[ f(x+1) = f(x) + 1 \]
and
\[ f(xy)f\left(\frac{x+y}{xy}\right) = f(x+y) \]
for all \( x \) and \( y \) in \( \mathbb{R}^{+*} \).

Answer: f(x) = x


Question: Determine all values of \(x\) for which the series \(\sum_{n=1}^{\infty}\left[\frac{x(x+n)}{n}\right]^n\) is convergent.

Answer: (-1, 1)


Question: Let \( X \) be a random variable such that \( 0 < X < 2 \) almost surely and \( \mathbb{E}[X] = 1 \). Define \( C_{n} := \{X \geq 2-\frac{1}{n} \} \) for all \( n \in \mathbb{N} \). Determine whether \( \lim_{n\to\infty} \mathbb{P}(C_{n}) = 0 \).

Answer: 0


Question: Given the function \(P(z) = \sum_{n \geq 0} p_n z^n\) with \(0 \leq p_n \leq 1\), and a function \(R(z) \in W^+\) such that \(R(z) \neq 0\) for all \(z\) and \(R(1) = m\), if \((1-z)P(z)R(z) = 1\) for \(|z| < 1\), prove that \(p_n \to 1/m\) as \(n \to \infty\).

Answer: \dfrac{1}{m}


Question: Determine the largest integer \( n \) such that \( 2^n \) divides \( 3^{2008} - 1 \).

Answer: 5


Question: Determine the number of solutions to the congruence equation \( x^3 - 2x^2 - x + 2 \equiv 0 \pmod{143} \) without finding the actual solutions.

Answer: 9


Question: Determine the number of elements \( \alpha \in \mathbb{F}_{3^5} \) such that \( \mathbb{F}_{3}(\alpha) = \mathbb{F}_{3^5} \). In other words, find the number of irreducible polynomials of degree 5 over \( \mathbb{F}_{3} \) that divide \( x^{3^5} - x \).

Answer: 48


Question: Consider a quadratic polynomial \( p(x) = x^2 + bx + c \), where \( b \) and \( c \) are randomly chosen from the interval \([0, 1]\). What is the probability that the polynomial \( p(x) \) has two real roots?

Answer: \dfrac{1}{12}


Question: Find the residue of the function \( f(z) = \frac{\cos(z)}{z^6} \) at its singularity.

Answer: 0


Question: Find all positive integer solutions to the equation \( \log_2(x+1) = \lfloor \log_2(x) \rfloor + 1 \).

Answer: 2^k - 1


Question: If $f \in A(D)$, where $D$ is the open unit disk in $\mathbb{C}$ and $A(D)$ is the disk algebra (the algebra of analytic functions in $D$ that are continuous in $\overline{D}$), is $f'$ continuous in $\overline{D}$?

Answer: No


Question: Provide an example of a sequence of functions that is monotone, bounded, but not uniformly convergent.

Answer: f_n(x) = x^n


Question: Let $S$ be the set of all pairs $(a,b)$ of real numbers satisfying the equations $1+a+a^2+a^3 = b^2(1+3a)$ and $1+2a+3a^2 = b^2 - \frac{5}{b}$. Calculate the value of $A+B+C$, where $A = \prod_{(a,b) \in S} a$, $B = \prod_{(a,b) \in S} b$, and $C = \sum_{(a,b) \in S} ab$. Find $A+B+C$. \( \text{Express your answer as a single number.} \)

Answer: -1


Question: Evaluate the limit: $$\lim_{n\to\infty} \frac{n^{\log(n)}}{2^n}$$

Answer: 0


Question: Can a polynomial with rational coefficients have \(-\sqrt{2}\) as its minimum value?

Answer: Yes


Question: Which is larger: $Ackermann(G_{64}, G_{64})$ or $G_{G_{64}}$? Compare the two expressions and determine which one represents a larger number.

Answer: G_{G_{64}}


Question: Solve the equation: \(\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x-5}}}}=5\). Find the value of \(x\).

Answer: 30


Question: Evaluate the line integral \( \int_C F \cdot dr \) for the vector field \( F = \left(\frac{2xy-2xy^2}{\left(1+x^2\right)^2}+\frac{8}{13}\right)i+\left(\frac{2y-1}{1+x^2}+2y\right)j \), where \( C \) is the path from \((2,0)\) to \((5,6)\). The vector field is path-independent.

Answer: 39


Question: Calculate the limit: \[ \lim_{x \rightarrow 0^+} \frac{\arctan (\log (1+\sqrt x)) \sin^3(x^{3/4})}{(e^{\tan(x)}-1)(1-\sin^2(x))} \]

Answer: 0


Question: How many integer solutions are there to the inequality \(x_1 + x_2 + x_3 + x_4 \leq 50\) with the constraints \(x_i \geq 2\) for all \(i = 1, 2, 3, 4\) and \(x_1, x_2 \leq 7\)?

Answer: 26781


Question: Let $G$ be an infinite, at most countable discrete group acting freely on a standard probability measure space $X$ in a measure-preserving way. Given a natural number $N_0$ and a positive real number $\varepsilon_0$, does there exist an $(\varepsilon, N)$-fundamentalish domain with $\varepsilon < \varepsilon_0$ and $N > N_0$?

Answer: Yes


Question:
Approximate the integral \( \int_3^4 \frac{x}{\sqrt{(7x-10-x^2)^3}} \, dx \) and determine which of the following intervals contains its value:

(a) \( \left(\frac{9}{10}, \frac{19}{20}\right) \)
(b) \( \left(\frac{19}{20}, 1\right) \)
(c) \( \left(1, \frac{21}{20}\right) \)
(d) \( \left(\frac{21}{20}, \frac{11}{10}\right) \)
(e) \( \left(\frac{11}{10}, \frac{23}{20}\right) \)
(f) \( \left(\frac{23}{20}, \frac{6}{5}\right) \)

Answer: D


Question: Isaac is planning a nine-day holiday. Each day, he will choose one of three activities: surfing, water skiing, or resting. He never does different water-sports on consecutive days. How many different schedules are possible for his holiday?

Answer: 3363


Question: Given that \( \frac{\sin \alpha+\sin \beta+\sin \gamma}{\sin(\alpha+\beta+\gamma)} = \frac{\cos \alpha+\cos \beta+\cos \gamma}{\cos(\alpha+\beta+\gamma)}=2 \), find the value of \( \cos(\alpha+\beta)+\cos(\beta+\gamma)+\cos(\gamma+\alpha) \).

Answer: 2


Question: Let $M$ be an $n$-dimensional compact connected smooth manifold without boundary whose universal cover is diffeomorphic to $\mathbb{R}^n$. Must the Euler characteristic of $M$ be zero?

Answer: No


Question: If \( \alpha \) is an irrational number such that \( 0 < \alpha < 1 \), is there a finite game with an honest coin such that the probability of one player winning the game is \( \alpha \)? An honest coin is one for which the probability of heads and the probability of tails are both \( \frac{1}{2} \). A game is finite if with probability 1 it must end in a finite number of moves.

Answer: Yes


Question: Calculate the limit \( \lim_{n\rightarrow\infty} \|T^n\| \), where \( T \) is the left-shift operator on \( l^2 \).

Answer: 1


Question: Let \(p(x) = x^5 - 833x^4 + ax^3 + bx^2 + cx + d\) be a polynomial whose roots are in geometric progression. If the sum of the reciprocals of the roots is 17, determine all possible values of \(d\).

Answer: -16807


Question: Compute the integral \( \lim_{\epsilon\rightarrow 0^{+}} \int_{S_{\epsilon}} \frac{x}{(x^{2}+4)(x-1)}dx \) using the residue theorem, where \( S_{\epsilon} := \{x \in \mathbb{R} : |x-1| > \epsilon\} \).

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


Question: Let \( X \) be a locally connected topological space, and let \( A \subset X \) be an arbitrary subset. Suppose \( S \subset A \) is a connected and open subset in the subspace topology on \( A \). Prove that there exists a connected and open subset \( U \subset X \) such that \( S = U \cap A \).

Answer: S = U \cap A


Question: Calculate the limit of the complex function as \( z \to 0 \): \[ \lim_{z\to 0} \frac{\bar{z}^2}{z} \]

Answer: 0


Question:
Consider the set \( M = \{2, 3, 4, \ldots\} \). The Cartesian product \( M \times M \) is partially ordered such that \((a, b) \leq (c, d)\) if and only if \(c\) is divisible by \(a\) and \(b \leq d\). Which of the following statements is true about the minimal and maximal elements of the partially ordered set \((M \times M, \leq)\)?

1) Each pair \((p, m)\) for \(m \in M\) is a minimal element, and there is no maximal element.
2) Each pair \((p, m)\) for \(m \in M\) is a minimal element, and there is a maximal element.
3) Each pair \((p, 2)\) for \(m \in M\) is a minimal element, and there is no maximal element.
4) Each pair \((p, m)\) for \(m \in M\) is a minimal element, and there is a maximal element.

Here, \(p\) is an arbitrary prime number.

Answer: 3


Question: Verify whether the field $K = \mathbb{Z}_2[x]/(x^5 + x^4 + 1)$, where $x^5 + x^4 + 1$ is irreducible, is correctly constructed to have 32 elements. The provided construction lists 22 elements. Is this construction correct?

Answer: 32


Question: Let $f(x)$ and $g(x)$ be polynomials in $\mathbb{Q}[x]$. If $\deg(f) \geq 2$ and $f$ is irreducible, is the composition $f(g(x))$ always reduced, meaning it has no repeated irreducible factors?

Answer: Yes


Question: Consider the ordinary differential equation (ODE): \( \frac{dy}{dt}=f(y,t) \) with the initial condition \( y(0)=a \). Assume that \( f \) is Lipschitz continuous, ensuring a unique solution \( y(t) \) exists on the interval \([0,2]\). Is \( y(1) \) a monotone increasing function of the initial value \( a \)?

Answer: Yes


Question:
Determine the values of $a$ and $b$ given the following limits:

\[ a = \lim_{x\rightarrow0}\left[\lim_{y\rightarrow0}\frac{1-\cos(x)}{\sin(x)+\sin(y)}\right] \]
\[ b = \lim_{y\rightarrow0}\left[\lim_{x\rightarrow0}\frac{1-\cos(x)}{\sin(x)+\sin(y)}\right] \]

Choose the correct option:
(a) No such $a$ and $b$ exist.
(b) $a=0$ but no such $b$ exists.
(c) $b=0$ but no such $a$ exists.
(d) $a=0$ and $b=0$.

Answer: D


Question: Determine whether a spherically symmetric function \( f: \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R} \) can be expressed as \( f(x,y) = \phi(|x|, |y|, \sphericalangle(x,y)) \) for some function \( \phi: [0,+\infty[ \times [0,+\infty[ \times [0,\pi[ \rightarrow \mathbb{R} \), where \( |x| \) and \( |y| \) are the magnitudes of \( x \) and \( y \), and \( \sphericalangle(x,y) \) is the angle between \( x \) and \( y \).

Answer: Yes


Question: Evaluate the complex line integral \( \int_{\gamma} |z|^2(\bar{z} + i) \, dz \), where \( \gamma \) is the circle given by \( |z - i| = 1 \) with counterclockwise orientation.

Answer: 4\pi i


Question: Let \( f: Y \to X \) be a finite étale cover of smooth projective connected varieties, or a finite degree connected topological cover of connected Riemann surfaces. Let \( y \in Y \) and \( x = f(y) \). Let \( Y_x \) be the fibre \( f^{-1}(x) \), which contains \( y \). Determine the length of the local ring \( \mathcal{O}_{Y_x,y} \). Is it 1 or \( \deg f \)?

Answer: 1


Question: In triangle ABC, where AB = 84, BC = 112, and AC = 98, angle B is bisected by line segment BE, with point E on AC. Angles ABE and CBE are further bisected by line segments BD and BF, respectively. Determine the length of segment FC.

Answer: 32


Question: Is it true that every tilting module in the BGG category $\mathcal{O}$ over a semisimple Lie algebra has a simple socle?

Answer: Yes


Question: Find the value of $f^{-1}(2)$ for the function $f(x) = 2 + x^2 + \tan(\frac{\pi x}{2})$. Provide your answer as a single number.

Answer: 0


Question: Evaluate the integral \( \int_D x\sin(y)\ dA \), where \( D \) is the half circle centered at \((0,0)\) with radius 1 above the \(x\)-axis.

Answer: 0


Question: Given \(\lim\limits_{x\to 0} \frac {x(1+a\cos x)-b\sin x}{x^3}=1\), find the value of \(a+b\).

Answer: -4


Question: Suppose we have $n$ independent normal random variables $X_1, X_2, \ldots, X_n$ and another independent normal random variable $Y$. What is the probability that $Y$ is less than all of $X_1, X_2, \ldots, X_n$?

Answer: \dfrac{1}{n+1}


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

Answer: 1


Question: Compute the residue of the function \( g(z) = \frac{e^z - 1}{z^3} \) at \( z = 0 \).

Answer: \dfrac{1}{2}


Question: Compute the limit of the sequence \( 1 + 6^2 + 6^4 + \cdots \) in the field \( \mathbb{Q}_2 \).

Answer: -\dfrac{1}{35}


Question: Find all analytic functions \( f: \mathbb{C} \to \mathbb{C} \) such that \( f'(z) = -2f(z) \) for all \( z \in \mathbb{C} \) and \( f(0) + f'(0) = 1 \).

Answer: -e^{-2z}


Question: Determine whether the product of a sequence of numbers between 0 and 1 can converge to a positive number. Specifically, if $x_n \in (0,1)$ for all $n$, is it possible that $\prod_{n=1}^\infty x_n > 0$?

Answer: Yes


Question: Let $(X, \mathcal{M}, \mu)$ be a measure space with $\mu(X) = 1$, and let $\mu^{*}$ be the outer measure induced by $\mu$. Suppose $E \subset X$ satisfies $\mu^{*}(E) = 1$. If $A, B \in \mathcal{M}$ and $A \cap E = B \cap E$, prove or disprove that $\mu(A) = \mu(B)$.

Answer: \mu(A) = \mu(B)


Question: Evaluate the infinite series: \[ \sum_{k=0}^{\infty} \frac{2^k}{3^{2^k}+1} \]

Answer: \dfrac{1}{2}


Question: For the system with the transfer function \( W(s) = K\frac{(s+5)}{(s-2)(s+1)} \), determine the range of gain \( K \) for which the system is stable, given the Nyquist diagram for \( K=1 \).

Answer: K > 1


Question: Determine whether a group of order 55 can have exactly 20 elements of order 11.

Answer: No


Question: Find the larger of the two positive integers \(n \ge 3\) for which the expression \(\frac{\log 2 \times \log 3 \times \log 4 \times \cdots \times \log(n-1)}{10^n}\) takes its smallest value.

Answer: 22027


Question: Let $(z_n)$ be the sequence given by \(z_n = (-1)^n + \frac{1}{n}\) for \(n \in \{1, 2, 3, \cdots\}\). Let \(A\subset \mathbb{R}\) be a subset with the properties: \(\inf A = -2\) and \(\sup A = 1\). Define the set \(B := \{ a \cdot z_n \mid a \in A, n \in \mathbb{N}_+\}\). Find \(\sup B\).

Answer: 2