Question: Let \( V \) be the vector space over \( \mathbb{C} \) of all complex polynomials of degree at most 10. Let \( D: V \rightarrow V \) be the differentiation operator, so \( D f(x) = f'(x) \). Find all eigenvalues and eigenvectors of the operator \( e^D \) on \( V \).

Answer: 1


Question: Evaluate the double integral \(\iint_D (\sqrt{x} + \sqrt{y})^{100} \, dx \, dy\), where \(D\) is the region bounded by \(\sqrt{x} + \sqrt{y} = 1\), \(x = 0\), and \(y = 0\).

Answer: \dfrac{1}{156}


Question: Given a positive harmonic function $u$ on the unit disc $U$ with $u(0) = 1$, determine the maximum possible value of $u(1/2)$. Provide your answer as a bound or specific value.

Answer: 3


Question: Evaluate the double integral \( \iint_D \left(\sqrt{a^2-x^2-y^2}-\sqrt{x^2+y^2}~\right)dxdy \) where \( D_{xy} \) is the disk \( x^2+y^2\le a^2 \), using a change of variables.

Answer: 0


Question: Given matrices \(A\) and \(B\) of size \(n \times n\) such that \(A^2 = B^2 = 0\) and \(AB = BA\), find the smallest natural number \(x\) such that \((A+B)^x = 0\).

Answer: 3


Question: Determine if the set \(\left\lbrace(x,y,z)\mid x \sin z + y \cos z = e^z \right\rbrace\) can be described as a graph of a function \(f\) in the neighborhood of \((2, 1, 0)\), where \(f(2,1) = 0\).

Answer: Yes


Question: Determine the number of real solutions to the equation \(6x^2 - 77[x] + 147 = 0\), where \([x]\) denotes the greatest integer less than or equal to \(x\).

Answer: 4


Question: Evaluate the integral \( I = \int_0^{1/2} \arcsin(\sqrt{x}) \, dx \) using the substitution \( \sqrt{x} = \sin(u) \).

Answer: \dfrac{1}{4}


Question: Find a natural number \( n \) such that whether or not 11 is a square modulo a prime \( p \) only depends on the congruence class of \( p \) modulo \( n \), apart from finitely many exceptions. Also, determine the congruence classes \( p \equiv a \pmod{n} \) for which \( \left(\frac{11}{p}\right) = 1 \).

Answer: 44


Question: Consider a convergent sequence $(x_1, x_2, \ldots)$ in an arbitrary normed vector space with limit $L$. Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a bijection, and define the sequence $(y_1, y_2, \ldots)$ by $y_i = x_{f(i)}$. Prove or disprove that the sequence $(y_1, y_2, \ldots)$ is convergent with limit $L$.

Answer: L


Question: Compute the residue of the function \( \frac{z^2 + 3z - 1}{z+2} \) at the pole \( z = -2 \).

Answer: -3


Question: Find the coefficient of \(x^{20}\) in the expansion of \((x + x^2 + x^3 + x^4 + x^5)(x^2 + x^3 + x^4 + \cdots)^5\).

Answer: 1876


Question: Determine the values of $x$ for which the series \( \sum_{n=0}^\infty \sin\left(\frac{x}{2^n}\right) \) converges.

Answer: (-\infty, \infty)


Question: Given that \( \lim_{n \to \infty}(a \sqrt[n]{n} + b)^{n/\ln n} = e^{-3} \), find the value of \(4b + 3a\).

Answer: 7


Question: In a party attended by 2015 guests, among any 5 guests, at most 6 handshakes have been exchanged. Determine the maximal possible number of handshakes.

Answer: 1015056


Question: Let $a_1, a_2, \ldots, a_6$ be real numbers such that $a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 2014$. Find the minimum value of \( \sum_{1 \le i < j \le 6} \lfloor a_i + a_j \rfloor \), where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \).

Answer: 10060


Question: Evaluate the limit: \[ \lim_{x \rightarrow 0} \frac{\int_{0}^{2 \sin x} \cos(t^2) \, dt}{2x} \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{a \to 0} \int_0^2 \frac{1}{ax^4 + 2} \, \mathrm{d}x \]

Answer: 1


Question: Find the probability distribution of the sum \( \Phi = \Phi_1 + \Phi_2 \), where \( \Phi_1 \) and \( \Phi_2 \) are independent circular uniform random variables on the interval \([0, 2\pi]\).

Answer: \frac{1}{2\pi}


Question:
Let \( f(x) \) be a 3rd degree polynomial such that the equation \( f(x^2) = 0 \) has exactly 4 distinct real roots. If \( f(k) = 0 \) and \( k \neq \alpha^2, \beta^2 \) where \( \pm \alpha, \pm \beta \) are roots of the equation \( f(x^2) = 0 \), determine which of the following is correct about \( k \):

(a) \( k < 0 \)
(b) \( k > 0 \)
(c) \( k \leq 0 \)
(d) \( k \geq 0 \)

Answer: a


Question:
Find the number of imaginary roots of the equation:
\[ \frac{A_1^2}{x-a_1} + \frac{A_2^2}{x-a_2} + \cdots + \frac{A_n^2}{x-a_n} = k \]
where \(a_1, a_2, \ldots, a_n, A_1, A_2, \ldots, A_n, k\) are all real numbers.

Answer: 0


Question: Let $M$ be a local martingale, and assume that $M$ is continuous. Given that the quadratic variation $\langle M \rangle = 0$, can we conclude that $M = 0$? Justify your answer.

Answer: No


Question: Let $X$ be a dense subset of a set $E$ such that $E \backslash X$ is also dense in $E$. Is $X$ necessarily totally imperfect in $E$? Consider the special case where $E = \mathbb{R}$. Provide a justification for your answer.

Answer: No


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

Answer: -1


Question: Find the limit superior (\(\limsup\)) of the sequence \(x_n = n(\sqrt{n^2+1} - n)\sin\frac{n\pi}{8}\), where \(n \in \mathbb{N}\).

Answer: \dfrac{1}{2}


Question: Given coprime integers $a$, $b$, and $c$ where $c \gg ab$, determine if it is possible for integers $0 < d = e = f < abc$ to exist such that the following congruences hold: $da^2 \equiv a \pmod{abc}$, $eab \equiv ab \pmod{abc}$, and $fb^2 \equiv b \pmod{abc}$. Is $0 < d = e = f < abc$ possible?

Answer: Yes


Question: Find the limit superior of the sequence \( a_j = |\sin j|^{\sin j} \) for all \( j \in \mathbb{N} \). Determine \( \limsup_{j \to \infty} \{ a_j \} \).

Answer: e^{1/e}


Question: Determine whether the following statement is true or false: "If K is a basis for a vector space V, then there exists a subset of K that is a basis for a subspace W of V."

Answer: A


Question: Evaluate the limit: \[ \lim_{t\to 1} \frac{\sqrt[4]{t}-1}{\sqrt[3]{t}-1} \]

Answer: \dfrac{3}{4}


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

Answer: \dfrac{1}{3}


Question: Is the rank of a symmetric or skew-symmetric matrix equal to the number of its non-zero eigenvalues?

Answer: Yes


Question: Find the real number \(a\) such that the equation \(|\log x| = ax\) has exactly 3 distinct solutions.

Answer: \dfrac{1}{e}


Question: Determine whether two matrices $A$ and $B$ with characteristic polynomial $x^5-x^3$ and minimal polynomial $x^4-x^2$ must be similar. Justify your answer.

Answer: Yes


Question: Let $k$ be a separably closed field, such as $k=\mathbb{C}$. Is it true that any finite field extension of $k$ is separably closed?

Answer: Yes


Question: Let $\Omega$ be a bounded domain in $\mathbb{C}^n$ and $f$ be a holomorphic function defined on $\Omega$. Is it possible for the $L^2$-norm of $f$ to be bounded while $f$ itself is unbounded? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the expression \( \frac{(2^n-2)(2^{n-1}-2)\cdots(2^3-2)(2^2-2)}{n!} \) is always an integer for \( n \geq 2 \).

Answer: Yes


Question: Suppose $c_n \geq 0$ for all $n$ and the series $\sum_{n=0}^\infty c_nr^n$ converges for all $r \geq 1$. Is it true that only finitely many $c_n$ are non-zero? Answer with 'True' or 'False'.

Answer: False


Question: Determine the number of ways to arrange 3 red, 4 blue, and 5 green wagons in a line such that no two blue wagons are adjacent. The wagons of the same color are identical.

Answer: 7056


Question: Determine for which values of $n$ the sphere $S^2$ can cover the connected sum of $n$ tori, denoted as $\Sigma_n$. Assume $\Sigma_0$ is the sphere, $\Sigma_1$ is the torus, and $\Sigma_n$ is a connected sum of $n$ tori for $n \in \mathbb{N}$. Provide a justification for your answer.

Answer: 0


Question: Let $O \subseteq \mathbb{R}^n$ be an open set, $K \subseteq \mathbb{R}^{n-1}$ a compact set, and $a \in \mathbb{R}$ such that $\{a\} \times K \subseteq O$. Prove that there exists an $\epsilon > 0$ such that $(a-\epsilon, a+\epsilon) \times K \subseteq O$.

Answer: \epsilon


Question: Evaluate the limit: \( \lim_{n\rightarrow \infty} n(\sqrt[n]{2}-1) \).

Answer: \ln 2


Question: Using Burnside's lemma, determine the number of distinguishable ways to paint the edges of an equilateral triangle if four different colors are available, with the condition that only one color is used per edge and the same color may be used on different edges.

Answer: 20


Question: Determine the correct bounds for the integral \( \int_0^b \mathbb{P}(E_0 > x - s) f_{E_1}(s) \, ds \) to calculate \( \mathbb{P}(E_0 + E_1 > x) \), where \( E_0 \sim \text{Exp}(\lambda) \) and \( E_1 \sim \text{Exp}(1) \) are independent random variables, \( \lambda > 1 \), and \( x > 0 \).

Answer: \infty


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

Answer: f(x) = x


Question: A normal is drawn to the parabola \( y^2 = 4ax \) at the point \( P(at^2, 2at) \). If this normal meets the parabola again at the point \( Q(at_2^2, 2at_2) \), find the value of \( t \) such that the distance \( PQ \) is minimized. Express your answer as \( \pm \sqrt{2} \).

Answer: \pm \sqrt{2}


Question: Find an explicit expression for the function $u_0$ that minimizes the energy functional \[ F(u) = \int^1_0 \frac{1}{2}((u')^2+u^2)\,dx - \int^1_0 fu\,dx \] where $f(x) = -3e^{2x}$, subject to the boundary conditions $u(0) = 1$ and $u(1) = e^2$. Assume $u_0$ is a local minimizer in the class of continuously differentiable functions on $(0,1)$. Use the Euler-Lagrange equation to find $u_0$. \[ (u_0 - f) - \frac{d}{dx} (u_0'') (u_0') = 0 \]

Answer: e^{2x}


Question: Suppose $V$ is a vector space over a finite field $F$ with a quadratic form $N:V \rightarrow F$ and an associated non-degenerate symmetric bilinear inner product $f : V\times V \rightarrow F$. Let $v \in V$ be an arbitrary vector. Is it always possible to find an isotropic vector $u \in V$ such that $N(u) = 0$ and $f(v,u) = 0$?

Answer: No


Question: Find the exact length of the curve defined by \( x = \frac{y^4}{8} + \frac{1}{4y^2} \) for \( 1 \le y \le 2 \). Use the formula for arc length: \( \int \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy \).

Answer: \dfrac{33}{16}


Question: Compute the value of the sum \( S_{2016} = \sum_{k=1}^{2016}\left(\sum_{n=k}^{2016}\frac{1}{n}\right)^2 + \sum_{k=1}^{2016}\frac{1}{k} \).

Answer: 4032


Question: Given the basis \(\{u_1, u_2, u_3\}\) of \(\mathbb{R}^3\), where \(u_1=(1,0,0)\), \(u_2=(1,1,0)\), and \(u_3=(1,1,1)\), and its dual basis \(\{f_1, f_2, f_3\}\) where \(f_1=(1,-1,0)\), \(f_2=(0,1,-1)\), and \(f_3=(0,0,1)\), find the coefficients \((\alpha_1, \alpha_2, \alpha_3)\) such that the linear functional \(f(a,b,c) = a + b + c\) can be expressed as \(f = \alpha_1 f_1 + \alpha_2 f_2 + \alpha_3 f_3\).

Answer: (1, 2, 3)


Question: Let $p = q^2 + q + 1$, where $p$ and $q$ are prime numbers. Determine whether it is true that $p$ divides $(q-1)^2 (q^2+q)$. Provide a justification for your answer.

Answer: No


Question: Determine whether the number of distinct conjugacy classes of the multiplicative group of $3 \times 3$ upper-triangular matrices with all diagonal entries 1, denoted as $U(3,F)$, is a polynomial in $q$ with integer coefficients, where $F$ is a finite field of order $q$. Provide a justification for your answer.

Answer: q^2 + q - 1


Question: Given a convex polygon with 4 or more sides, determine if it is possible to transform this polygon into a triangle with the same area using only a compass and straight edge.

Answer: Yes


Question: Find all continuous functions \( f: (-1, 1) \to \mathbb{R} \) such that \( f(x) = f(x^2) \) for all \( -1 < x < 1 \) and \( f(-0.5) = 2017 \).

Answer: 2017


Question: Determine the number of solutions to the equation \(16 \sin^3x = 14 + (\sin x + 7)^{\frac{1}{3}}\) in the interval \([0, 4\pi]\).

Answer: 2


Question: Find the maximum value of $a$ such that the function $f(w) = w^2 + w$ is univalent in the disk $|w| < a$. Provide your answer as a single number.

Answer: \dfrac{1}{2}


Question: Determine the Big-O complexity of the recurrence relation: \[ T(n) = \sqrt{n} \cdot T(\sqrt{n}) + \sqrt{n} \]

Answer: O(n)


Question:
Let $(X, d)$ be a metric space and $F_\varepsilon, F\colon X\to [-\infty, \infty]$. Suppose $F_\varepsilon$ is an equicoercive sequence of functions on $X$, meaning for all $t\in\mathbb{R}$ there exists a compact set $K_t$ such that $\{x\colon F_\varepsilon(x)\le t\}\subset K_t$. Suppose $F_\varepsilon\xrightarrow{\Gamma} F$, which means for all $u\in X$:

1. For every sequence $\{u_\varepsilon\}$ converging to $u$, it holds that $F(u)\le\displaystyle\liminf_{\varepsilon\to 0}F_\varepsilon(u_\varepsilon)$.
2. There exists a sequence $\{u_\varepsilon\}$ converging to $u$ such that $F(u) = \displaystyle\lim_{\varepsilon\to 0}F_\varepsilon(u_\varepsilon)$.

Is $F$ coercive on $X$?

Answer: Yes


Question:
Consider the initial value problem given by the differential equation \(y' = 2\sqrt{y}\) with the initial condition \(y(0) = a\). Determine the nature of the solution based on the value of \(a\):

1. A unique solution if \(a < 0\)
2. No solution if \(a > 0\)
3. Infinitely many solutions if \(a = 0\)
4. A unique solution if \(a \ge 0\)

Which of the above statements correctly describe the existence and uniqueness of the solution to the initial value problem?

Answer: 3


Question: Given two vectors \( \vec{V}_1 = 2(\sin\alpha + \cos\alpha) \hat{i} + \hat{j} \) and \( \vec{V}_2 = \sin\beta \hat{i} + \cos\beta \hat{j} \), where \( \alpha \) and \( \beta \) satisfy the relation \( 2(\sin\alpha + \cos\alpha)\sin\beta = 3 - \cos\beta \), find the value of \( 3\tan^2\alpha + 4\tan^2\beta \).

Answer: 35


Question:
Determine whether there exists a non-empty set $X$ and a collection ${\cal F} \subseteq {\cal P}(X) \setminus \{\emptyset\}$ of non-empty subsets of $X$ such that:

1. For every $a \in {\cal F}$, $|a| \geq 2$.
2. For any distinct $a, b \in {\cal F}$, $|a \cap b| \leq 1$.
3. There is no function $f: {\cal F} \to X$ such that:
   - $f(a) \in a$ for all $a \in {\cal F}$, and
   - if $a, b \in {\cal F}$ with $a \cap b \neq \emptyset$, then $f(a) \neq f(b)$.

Answer with 'yes' if such a set and collection exist, otherwise answer 'no'.

Answer: yes


Question: Find the arc length of the parametric curve \( \gamma(t) = (-1+2\sin(t), 1 +2\cos(t)) \) for \( t \in [0, 2\pi] \).

Answer: 4\pi


Question: Evaluate the sum: \[ \sum_{n=1}^{1999}\frac{25^{\frac{n}{2000}}}{25^{\frac{n}{2000}}+5} \]

Answer: \dfrac{1999}{2}


Question: Find the arc length of the curve defined by \( x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1 \). Use symmetry with respect to the line \( y = x \) to simplify your calculations.

Answer: 6


Question: Determine the number of solutions to the equation \( \tan x = x - x^3 \) for \( x \) in the interval \((-1, 1)\).

Answer: 1


Question: Evaluate the contour integral \( \int_{\gamma} \frac{1}{z} \, dz \) where \( \gamma \) is the semicircular path from 1 to -1 in the complex plane.

Answer: i\pi


Question: In a right triangle $ABC$ with $\angle A = 90^\circ$, the inradius is $r$. A cevian $AD$ is drawn such that the inradii of triangles $ABD$ and $ADC$ are both $r_1$. Given that $AD = 2$, calculate the value of $\frac{1}{r_1} - \frac{1}{r}$.

Answer: 0.5


Question: Consider the sequence defined by \( a_n = n(\sqrt[n]{ea} - \sqrt[n]{a}) \) for \( n \geq 1 \). Determine \( \lim_{n\to \infty} a_n \).

Answer: 1


Question: Does there exist a finite group $G$ such that its identities are exactly the same as the standard group identities in the signature $\{+,0,-\}$?

Answer: No


Question: Let $f(s,t)$ be a differentiable function of two variables, and define $h(x,y,z) = z \cdot f\left(\frac{x}{z}, \frac{y}{z}\right)$. Simplify the expression $(x,y,z) \cdot \nabla h$.

Answer: h(x,y,z)


Question: Evaluate the limit \( \lim_{n \rightarrow \infty}\int_{0}^{n+1}e^{\frac{x^2}{n+1}}dx \).

Answer: \infty


Question: Let $K$ be the field with exactly $7$ elements. Let $\mathscr{M}$ be the set of all $2 \times 2$ matrices with entries in $K$. How many elements of $\mathscr{M}$ are similar to the matrix \( \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \)?

Answer: 56


Question: Find the maximum surface area of a box with dimensions $x_1, x_2, x_3$ such that the diagonal of the box is 1, i.e., $x_1^2 + x_2^2 + x_3^2 = 1$. The surface area of the box is given by $2(x_1x_2 + x_2x_3 + x_1x_3)$. Assume $x_1, x_2, x_3 > 0$. What is the maximum surface area?

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{1 - \cos{x}\sqrt{\cos{2x}}}{x\sin{x}} \]

Answer: \dfrac{3}{2}


Question: How many different ways can you roll 10 identical dice such that exactly 3 different numbers appear in the results? For example, if 7 dice show the number 2, 2 dice show the number 3, and 1 die shows the number 6, the set of results is \(\{2, 3, 6\}\).

Answer: 720


Question: Find the expectation of \( Y^2 \) where \( Y = \exp(X^2) \) and \( X \sim N(0,1) \).

Answer: \infty


Question: Let \( R = \{a_0 + a_1X + \ldots + a_nX^n \mid a_1, \ldots, a_n \in \mathbb{Q}, a_0 \in \mathbb{Z}, n \in \mathbb{Z}_{\geq 0} \} \) and \( I = \{a_1X + \ldots + a_nX^n \mid a_1, \ldots, a_n \in \mathbb{Q}, n \in \mathbb{Z}^{+} \} \). Prove that \( I \) is not finitely generated as an \( R \)-module.

Answer: I


Question: Evaluate the limit \( \lim_{n \to \infty} n \int_{0}^{100}f(x)g(nx)dx \) given that \( g: \mathbb{R} \to \mathbb{R} \) is a continuous function with \( g(y)=0 \) for all \( y \notin [0,1] \) and \( \int_{0}^{1}g(y)dy=1 \), and \( f: \mathbb{R} \to \mathbb{R} \) is a twice differentiable function. Show that this limit equals \( f(0) \).

Answer: f(0)


Question: Solve the inequality \[ \frac{a^3}{2b^3+ab^2} + \frac{b^3}{2c^3+bc^2} + \frac{c^3}{2d^3+cd^2} + \frac{d^3}{2a^3+da^2} \geq \frac{4}{3} \] for \( a, b, c, d > 0 \).

Answer: \dfrac{4}{3}


Question:
Find all integer values of \(x\) that satisfy the following system of congruences:
\[ 4x \equiv 3 \pmod{21} \]
\[ 3x \equiv 2 \pmod{20} \]
\[ 7x \equiv 3 \pmod{19} \]

Answer: 6474


Question: Determine the number of Sylow bases for the group \( G = \langle a, b \rangle \), where \( a = (1 2 3)(4 5 6 7 8) \) and \( b = (2 3)(5 6 8 7) \). The group \( G \) is soluble and of order 60, with \( \langle a \rangle \) as a normal subgroup. A Hall \( \{2,3\} \) subgroup is \( \langle a^5, b \rangle \), a Hall \( \{2,5\} \) subgroup is \( \langle a^3, b \rangle \), and a Hall \( \{3,5\} \) subgroup is \( \langle a \rangle \). Given that all Sylow bases are conjugate, find the number of Sylow bases for \( G \).

Answer: 15


Question: Asif has cubes of three colors: red, green, and blue. He builds a tower by stacking the cubes, with the restriction that no more than 4 cubes of each color can be used. Asif stops building the tower as soon as it contains 4 cubes of any two colors. How many different towers can Asif build?

Answer: 34650


Question: Use Stoke's Theorem to evaluate the line integral \( \int_{C} \mathbf{F} \cdot \mathrm{d}\mathbf{r} \), where \( \mathbf{F}(x,y,z) = -x^2\mathbf{i} + xz\mathbf{j} + yx\mathbf{k} \) and \( C \) is the boundary of the surface \( S \), which is the graph of \( f(x,y) = 4x - 8y + 30 \) over the rectangle \( R = \{(x,y) \mid -2 < x < 3, 0 < y < 2\} \). The curve \( C \) is oriented counterclockwise when viewed from above.

Answer: 160


Question: Does every invertible matrix $A \in GL_n(\mathbb{C})$ have at least one eigenvector?

Answer: Yes


Question: Let \( f(x) = e^{x+1} - 1 \) and \( g(x) = 100|f(x)| - \sum_{k=1}^n |f(x^k)| \), where \( n \in \mathbb{N} \). Given that \( g(x) \) is differentiable over the entire set of real numbers, find the sum of all possible values of \( n \).

Answer: 39


Question: Given that \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = \frac{1}{2} \), find the maximum value of \( a + b + c + d \) where \( a, b, c, \) and \( d \) are positive integers.

Answer: 1859


Question:
Given a doubly indexed array $a_{mn}$ where $m,n\in\mathbf{N}$, assume that $\lim_{m,n\to\infty}a_{mn} = a$, and for each fixed $m\in\mathbf{N}$, $\lim_{n\to\infty}(a_{mn}) = b_{m}$. Prove that $\lim_{m\to\infty}b_m = a$.

Proof: Let $\epsilon>0$. Since $\lim_{m,n\to\infty}a_{mn} = a$, there exists an $M\in\mathbf{N}$ such that $\forall m,n\ge M\left(|a_{mn}-a|<\frac{\epsilon}{2}\right)$. Now let $m\ge M$. From the hypothesis $\lim_{n\to\infty}a_{mn} = b_m$, there exists some $N\in\mathbf{N}$ such that $|a_{mn}-b_m|<\frac{\epsilon}{2}$ whenever $n\ge N$. Let $L = \max(M,N)$ and $n\ge L$. Then $|b_m-a_{mn}|<\frac{\epsilon}{2}$ and $|a_{mn}-a|<\frac{\epsilon}{2}$. By the triangle inequality, $$|b_m-a| = |(b_m-a_{mn})+(a_{mn}-a)|\leq|b_{m}-a_{mn}|+|a_{mn}-a|<\frac{\epsilon}{2}+\frac{\epsilon}{2} = \epsilon.$$

$\blacksquare$

Note: We define $\lim_{m,n\to \infty}(a_{mn})=a$ if and only if $\forall\epsilon>0\exists M\in\mathbf{N}\forall m,n\ge M(|a_{mn}-a|<\epsilon)$.

Answer: a


Question: Given that \( \int_0^2 f(x) \, dx = f(\alpha) + f(\beta) \) for all polynomials \( f \) of degree at most 3, find the value of \( 3(\alpha - \beta)^2 \).

Answer: 4


Question: Determine whether there exists a continuous onto function \( f: \mathbb{D} \rightarrow [-1,1] \), where \( \mathbb{D} \) is the closed unit disk in \( \mathbb{R}^2 \).

Answer: Yes


Question: Find all integer values of $n$ such that $n^8 + n + 1$ is a prime number.

Answer: 1


Question: Determine the rate at which the professor writes a letter when there are fewer than \( k \) requests in a continuous-time Markov chain model.

Answer: \mu


Question: Determine whether the following statement is true or false: If $f$ is Riemann integrable on $[a,b]$, then the function $F(x)=\int_a^x f(t) \, dt$ is differentiable on $(a,b)$ and $F'(x)=f(x)$ for $x \in (a,b)$. Provide a counterexample if the statement is false.

Answer: False


Question: Let \( y = f(x) \) be a thrice differentiable function defined on \( \mathbb{R} \) such that \( f(x) = 0 \) has at least 5 distinct zeros. Find the minimum number of zeros of the equation \( f(x) + 6f'(x) + 12f''(x) + 8f'''(x) = 0 \).

Answer: 2


Question: Calculate the limit: \[ \lim_{n\to\infty}\left[\frac{ni}{\pi}-\frac{ni}{\pi}e^{i\pi/n}\right]^n \]

Answer: i


Question:
Find the length of the curve defined by the parametric equations:
\[ x(t) = t\sin(2t) \]
\[ y(t) = t\cos(2t) \]
\[ z(t) = \frac{4}{3}t^{1.5} \]
for \( t \) in the interval \([0, 2\pi]\).

Answer: 4\pi^2 + 2\pi


Question: Let $f$ be a continuous concave function on $[0,1]$ with $f(1)=0$ and $f(0)=1$. Determine if there exists a constant $k$ such that a rectangle with sides parallel to the axes can always be drawn in the area bounded by the two axes and the curve $f$, with an area at least $k \cdot \int_0^1 f(x) \, dx$. If so, find the value of $k$.

Answer: \dfrac{1}{2}


Question: An ant starts at one vertex of a regular dodecagon and wants to reach the opposite vertex by moving to adjacent vertices. Let \( p_n \) represent the number of such paths with length \( n \). Compute the sum \( p_1 + p_2 + p_3 + \ldots + p_{12} \).

Answer: 548


Question: Solve for \(x\) in the equation: \[ 2\cos\left(\frac{x^2+x}{6}\right) = 2^x + 2^{-x} \]

Answer: 0


Question: Let $V$ be an open connected subset of $\mathbb{R}^2$. Determine whether it is true that for every compact set $K \subseteq V$, there exist a compact set $A$ and an open connected set $B \subseteq \mathbb{R}^2$ such that $K \subseteq B \subseteq A \subseteq V$. Provide a proof or counterexample.

Answer: True


Question: Calculate the area enclosed between the curves \( r = \frac{1}{\phi} \) and \( r = \frac{1}{\sin\phi} \) for \( \phi \in (0, \frac{\pi}{2}) \) using polar coordinates.

Answer: \dfrac{1}{\pi}


Question: Evaluate the limit: \[ \lim_{(x,y) \to (0,0)} \frac{x^2+y^2}{1-\cos x\cos y} \]

Answer: 2


Question: Is the functor of taking invariants under a pro-$p$-group $P$ of finite-dimensional $\mathbb{Q}_l$-vector spaces (where $l \neq p$) an exact functor? Assume the action of the group on the vector space is continuous.

Answer: Yes


Question: Find the smallest value of \( n \) such that \( P\left(\frac{Y}{n} \geq 1/2\right) \geq 0.95 \), where \( Y \sim B(n, 0.55) \).

Answer: 248


Question: In triangle ABC, angle B measures 120º. Let I be the incenter, O the circumcenter, and E the excenter relative to side BC. Calculate the measure of angle ∡IEO.

Answer: 30^\circ


Question: Determine whether there exists a continuous function \( f: \Bbb R \to \Bbb R \) such that \( f(\Bbb Q) \subseteq \Bbb R \setminus \Bbb Q \) and \( f(\Bbb R \setminus \Bbb Q) \subseteq \Bbb Q \). Answer true or false.

Answer: False


Question: For the sequence \(\{a_n\}\), given that \(\sum^{\infty}_{n=1} (a_n^2 + 4na_n + 4n^2) = 4\), find the value of \(\lim_{n\to\infty}{\frac{a_n}{n}}\).

Answer: -2


Question: What is the maximum number of isolated local maxima of the function \(|p(x)|\), where \(p: \Bbb R^n \to \Bbb R\) is a quadratic polynomial?

Answer: 1


Question: If a complex power series \(\sum_{n = 0}^{\infty} a_n z^n\) converges absolutely for \(|z| \leq 1\), does it necessarily converge absolutely for \(|z| < 1 + \epsilon\), for some \(\epsilon > 0\)? Provide a justification for your answer.

Answer: No


Question:
A continuous-time Markov chain $X_t$ with state space $\{1,2,3\}$ has an infinitesimal generator matrix:
\[ A= \begin{bmatrix}-6 & 2 & 4\\2 &-5 &3 \\2 & 3& -5\end{bmatrix} \]
If the chain starts in state 2, what is the expected total time spent in state 2 before the chain first enters state 3?

Answer: \dfrac{3}{13}


Question: Is every bijective morphism an isomorphism in a small category? Provide a justification for your answer.

Answer: No


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

Answer: a


Question: Consider a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$, where $X$ is an $E$-valued random variable and $Y$ is an $F$-valued random variable for Polish spaces $E$ and $F$. Is it true that if $X$ is $\overline{\sigma(Y)}$-measurable, then $X$ is $\sigma(Y)$-measurable almost surely? Here, $\overline{\sigma(Y)}$ denotes the completed sigma algebra.

Answer: Yes


Question: How many different designs can be created on a 4x4 handkerchief using 4 different colors, assuming each square can be independently colored? Provide the total number of possible designs.

Answer: 4294967296


Question: Find the number of solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 = 26\) where each \(x_k\) satisfies \(0 \leq x_k \leq 10\).

Answer: 8725


Question: Evaluate the limit \( \lim_{(x,y)\to(0,0)}\frac{x^3-xy^3}{x^2+y^2} \) and show that it equals zero.

Answer: 0


Question: Determine the number of field structures that can be defined on the set \(\mathbb{R} \times \mathbb{R}\).

Answer: 1


Question: Let Z be a normal, second countable space and B be an open subset of Z. Determine whether there exists a continuous function \( f: Z \rightarrow \mathbb{R} \) such that \( f(x) > 0 \) for all \( x \in B \) and \( f(x) = 0 \) for all \( x \notin B \).

Answer: Yes


Question: Determine the number of solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 = 18\) where each \(x_i\) is a non-negative integer and \(x_i \neq 3\) and \(x_i \neq 4\) for all \(i\).

Answer: 2355


Question: Calculate the volume of the region defined by the set \( T = \{(x,y,z) \in \mathbb{R}^3 \mid 0 \leq x, x^2 \leq y \leq \sqrt{x}, 0 \leq z \leq xy \} \).

Answer: \dfrac{1}{12}


Question: Let $f \in L^2(0,T;V)$ where $V=H^1_0(\Omega)$ or $V=L^2(\Omega)$. Suppose that $|f| \leq M$ for some constant $M$ almost everywhere. Is it possible to find a sequence of simple functions $f_n(t) = \sum_i^n \chi_{E_i}(t)w_i$ that converges to $f(t)$ such that $w_i \leq M$ for each $i$?

Answer: Yes


Question: Evaluate \( \lim_{a\to 0^+} \int _0 ^{\infty} \dfrac{t^{a-1}}{1+t^4} \, dt \).

Answer: \infty


Question: Given the expansion \((2x^2+3x+4)^{10} = \sum_{i=0}^{20} a_{i}x^{i}\), calculate the value of \(\frac{a_{7}}{a_{13}}\).

Answer: 8


Question: Determine whether polynomial functions with rational coefficients can approximate any continuous function on $[a,b]$ to any degree of accuracy. Specifically, if the Stone-Weierstrass theorem states that for any continuous function $f$ on $[a,b]$, there exists a polynomial function $p$ such that $\|f-p\|<\varepsilon$ for any $\varepsilon>0$, does this statement remain true if "a polynomial function" is replaced with "a polynomial function with rational coefficients"?

Answer: Yes


Question: Let \( A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix} \) and let \( I \) be the \( 2 \times 2 \) identity matrix. Find the value of \( \alpha \) such that \( A + \alpha I \) is positive semi-definite but not positive definite.

Answer: \sqrt{5} - 2


Question:
Determine whether there exists a pair of functions \(f(x)\) and \(g(x)\) such that:
\[ \frac{d}{dx}f(x) = g(x) \]
\[ \frac{d}{dx}g(x) = f(x) \]
and \(f(x) \neq g(x)\).

Answer: Yes


Question:
For a positive integer \( n \geq 2 \), define the function
\[ f(n) = \frac{(\log_3 2)(\log_3 3)(\log_3 4) \cdots (\log_3 n)}{9^n}. \]
Let \( m \) be the minimum value of \( f(n) \), and let \( N \) be the sum of all values of \( n \) for which \( f(n) = m \). Find \( N \).

Answer: 39365


Question: Is it possible to construct a sequence $(\varepsilon_n)_{n\in\mathbb{N}}$ in $(0,1/2)$ such that $\left(\varepsilon_n/\varepsilon_{n+1}\right)_{n\in\mathbb{N}} \notin \ell^\infty$? In other words, does there exist a sequence for which $\sup_{n\in\mathbb{N}} \frac{\varepsilon_n}{\varepsilon_{n+1}} = \infty$?

Answer: Yes


Question: Consider a simple random walk on a four-vertex graph shaped like a square, with vertices labeled A, B, C, and D. The payoff function is defined as follows: \( f(A) = 2, \ f(B) = 4, \ f(C) = 5, \ f(D) = 3 \). There is no cost associated with moving, but there is a discount factor \( a \). Determine the largest possible value of \( a \) such that the optimal stopping strategy is to stop at every vertex, i.e., \( S_2 = \{A, B, C, D\} \).

Answer: \dfrac{4}{7}


Question: Suppose \(\sum_{i = 1} a_i \sum_{i = 1} b_i < \infty\) and \(\sum_{i = 1} a_i = \infty\), where \((a_i)\) and \((b_i)\) are two non-negative sequences. Does it necessarily follow that \(\sum_{i = 1} b_i = 0\)?

Answer: Yes


Question:
In triangle $\triangle ABC$, the base $AB$ is fixed, and the vertex $C$ lies on a fixed circle of radius $r$. Lines are drawn through $A$ and $B$ to intersect $BC$ and $CA$ at $E$ and $F$ respectively, such that $CE:EB=1:2$ and $CF:FA=1:2$. The point of intersection of these lines is $P$, which lies on the median through $C$ for all positions of $C$. Determine the locus of $P$.

(1) Circle of radius $\frac{r}{2}$
(2) Circle of radius $2r$
(3) A parabola of latus rectum $4r$
(4) A rectangular hyperbola

Answer: 1


Question: Find the value of \( p \) such that the function \( f(x,y) = \frac{1}{(x^2+y^3-2)^p} \) is integrable over the region defined by \( 1 \leq x \leq 2 \) and \( 1 \leq y \leq 3 \).

Answer: p < 2


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

Answer: \dfrac{4}{e}


Question: Find the limit \( \lim_{n \to \infty} \frac{a_1 + a_2 + a_3 + \ldots + a_n}{\ln(n)} \) where \( a_{n} = \int_{1}^{e} \ln^{n}(x)dx \).

Answer: e


Question: Let D be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1, splitting D into two halves, D_1 and D_2. Suppose f: D → ℂ is a continuous function that is holomorphic on the interiors of D_1 and D_2. Is f necessarily holomorphic on D?

Answer: Yes


Question: Let $k$ and $n$ be positive integers. Suppose $h: D^k \to S^n$ is an embedding. Is the image $h(D^k)$ closed in $S^n$?

Answer: Yes


Question: Let \(a_1=1\) and \(a_n=n(a_{n-1}+1)\) for \(n=2,3,\ldots\). Define \[P_n=\left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right)\cdots\left(1+\frac{1}{a_n}\right).\] Find \(\lim\limits_{n\to\infty} P_n\).

Answer: e


Question: Let $p > 3$ be a prime number and $q$ be the next prime number after $p$. Prove that $p!$ is divisible by all integers $n$ such that $n < q$.

Answer: p!


Question: Given that \(x^2 + y^2 + xy = 1\) and \(x, y \in \mathbb{R}\), find the minimum value of the expression \(x^3y + xy^3 + 4\).

Answer: 2


Question: Evaluate the surface integral \( \int_F f \, dS \) where \( f(x,y,z) = x^2z \) and \( F \) is the lateral surface of the cylinder defined by \( x^2 + y^2 = 4 \) with \( 0 \leq z \leq 1 \). Use cylindrical coordinates to express and solve the integral.

Answer: 4\pi


Question: Find the value of \( f^{(12)}(0) \) for the function \( f(x) = \log(e^{x^4} - 2x^8) \).

Answer: 958003200


Question: Given the sequences defined by \(x_0 = 2\), \(y_0 = 1\), and the recursive relations \(x_{n+1} = \frac{x_n + y_n}{2}\) and \(y_{n+1} = \frac{2}{x_{n+1}}\) for \(n = 0, 1, 2, \ldots\), prove that both sequences \(x_n\) and \(y_n\) converge to \(\sqrt{2}\).

Answer: \sqrt{2}


Question:
Evaluate the limit of the following Riemann sum as \( n \to \infty \):
\[ \lim_{n\to\infty}\sum_{i=1}^{n} \frac{3}{n}\left[(1+\frac{3i}{n})^3-2(1+\frac{3i}{n})\right] \]

Answer: \dfrac{195}{4}


Question: Evaluate the limit: \[ \lim_{z \to \infty} \frac{z^{2}-(2+3i)z+1}{iz-3} \]

Answer: \infty


Question:
Determine the greatest value of \( b \) such that any real-valued function \( f \) satisfying the following properties must also satisfy \( f(1) < 5 \):

1) \( f \) is infinitely differentiable on the real numbers.
2) \( f(0) = 1, f'(0) = 1, \) and \( f''(0) = 2 \).
3) \( |f'''(x)| < b \) for all \( x \) in \([0,1]\).

Answer: 12


Question: Evaluate the line integral \( \int_R xy^2 \, ds \) where \( R \) is the upper half of the circle defined by \( x^2 + y^2 = 25 \).

Answer: 0


Question: Find the radius of convergence of the Taylor series expansion of the function $e^{-1/x^2}$ centered at $x=4$. Provide your answer as a single number.

Answer: 4


Question: Calculate the size of the conjugacy class of cycle type \([1,2,3]\) in the symmetric group \(S_6\).

Answer: 120


Question: Given the equation \( n = \frac{\sqrt{16m^{2}+x^2}-x}{m} \), where \( x > 0 \), \( m > 0 \), and \( n \) is a positive integer, determine how many values of \( n \) are possible.

Answer: 3


Question: Is the probability distribution of a random variable a measure on the measurable space \(( \mathbb{R}, {\cal R} )\)? Consider a probability space \((\Omega, {\cal B}, P )\) and a random variable \(X: \Omega \rightarrow \mathbb{R}\). Verify if the probability distribution \(Q\) of \(X\) is a measure on \(( \mathbb{R}, {\cal R})\) such that \(Q( A ) = P( \{ w: X(w) \in A \} )\) for \(A \in {\cal R}\).

Answer: Yes


Question: Given that $a+b+c<0$ and the quadratic equation $ax^2+bx+c=0$ has no real roots, determine whether it is necessarily true that $c < 0$. Provide a justification for your answer.

Answer: Yes


Question:
Given the joint density function of random variables $X$ and $Y$:
\[ f(x,y) = \begin{cases} 3(2-x)y, & \text{if } 0<y<1 \text{ and } y<x<2-y \\ 0, & \text{otherwise.} \end{cases} \]
Calculate the probability that $X + Y \leq 1$. Use the bounds $0 < y < 1$ and $y < x < 1-y$ for your integration.

Answer: \dfrac{3}{16}


Question:
Let $G$ be a non-abelian group, and let $a, b \in G$ with $O(a) = 3$ and $O(b) = 4$. Determine the order of the element $ab$. Choose the correct answer from the following options:

(a) 6
(b) 12
(c) of the form $12k$ for some $k \in \mathbb{R}$
(d) need not be finite

Answer: D


Question: Calculate the relative entropy $D(Q\mathrel{\|}P)$ for the probability distributions $P=\{p,0,0,1-p\}$ and $Q=\{p,1-p\} \otimes \{p,1-p\}$. Verify if $D(Q\mathrel{\|}P)$ is infinite for all values of $p \neq \{0,1\}$.

Answer: \infty


Question: Consider the curve \( y = \frac{x^2}{4} \) and the point \( F(0,1) \). Let \( A_1(x_1,y_1), A_2(x_2,y_2), \ldots, A_n(x_n,y_n) \) be \( n \) points on the curve such that \( x_k > 0 \) and \( \angle{OFA_k} = \frac{k\pi}{2n} \) for \( k = 1, 2, 3, \ldots, n \). Find \( \lim_{n\to \infty}\frac{1}{n} \sum_{k=1}^{n} FA_k \).

Answer: \dfrac{4}{\pi}


Question: Evaluate the definite integral: $$\int_0^1\left(\sum_{r=1}^n\frac1{x+r}\right)\left(\prod_{r=1}^n(x+r)\right)dx$$

Answer: n \cdot n!


Question: Find the matrix representation of the operator \(|v_{j}\rangle \langle v_{k}|\) with respect to the orthonormal basis \(|v_{i}\rangle\) for an inner product space \(V\).

Answer: E_{jk}


Question:
Compute the contour integral:
\[ \oint_{|z-1-i| = 5/4} \frac{\log(z)}{(z-1)^2} \, dz. \]

Answer: 2\pi i


Question: Evaluate the definite integral: $$\int_0^{2\pi} \frac{\cos(t) \ |\cos(t)| \ \sqrt{\sin^2(t) + \frac{\cos^2(t)}{4}}}{\sqrt{1+3\sin^2(t)}} \ dt$$

Answer: 0


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

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{1}{x} \log{\sqrt{\frac{1 + x}{1 - x}}} \]

Answer: 1


Question: Find a parametrization of the surface given by the equation \(x^3 + 3xy + z^2 = 2\) with \(z > 0\). Use this parametrization to determine the equation of the tangent plane at the point \((x, y, z) = \left(1, \frac{1}{3}, 0\right)\).

Answer: 4x + 3y = 5


Question: Determine the rank of the matrix $A^n$ if $A$ is a matrix from the zero ring.

Answer: 0


Question: In triangle \( \triangle ABC \) inscribed in a unit circle, the angle bisectors of the internal angles at \( A, B, \) and \( C \) meet the circle at points \( D, E, \) and \( F \) respectively. Find the value of \( \frac{AD \cos\frac{A}{2} + BE \cos\frac{B}{2} + CF \cos\frac{C}{2}}{\sin A + \sin B + \sin C} \).

Answer: 2


Question: Find the maximum distance between any point on the circle defined by \(x^2+(y-4)^2=1\) and any point on the ellipse defined by \(\frac{x^2}{4}+y^2=1\). What is the maximum value of \(|PQ|\)?

Answer: 6


Question: Find the limit: \( \lim_{\alpha\rightarrow0} \left(\alpha\log^2{\alpha}\right) \).

Answer: 0


Question: Determine whether every nonsingular $n \times n$ matrix with real entries can be made singular by changing exactly one entry.

Answer: Yes


Question: Is it true that for any reductive group $G$ over the finite field $\mathbb{F}_q$, there exists a smooth regular embedding $\iota \colon G \rightarrow G'$ such that the center of $G'$ is an induced torus?

Answer: Yes


Question: Evaluate the integral \( \int_0^\pi \exp(-b\cos x)\,dx \).

Answer: \pi I_0(b)


Question: Evaluate the limit: $$\lim_{x\to \frac{\pi}{2}} \tan(x) \log(\sin(x)).$$

Answer: 0


Question: Determine the dimension of the special linear group \(\operatorname{SL}(n,F)\) as an algebraic group, where \(F\) is an algebraically closed field.

Answer: n^2 - 1


Question: Does there exist a regular curve \( \gamma : \mathbb{R} \to \mathbb{R}^{3} \) and a rigid motion \( f \) such that \( f \) is not the identity motion and \( f \circ \gamma \) is a reparametrization of \( \gamma \)?

Answer: Yes


Question: Compute the limit: $$\lim\limits_{n\to\infty}\frac{(2n)!\sqrt{\pi n}}{2^{2n}(n!)^2}.$$

Answer: 1


Question: Given a prime number $p$ and an integer $m \in \mathbb{Z}_p^*$, if $a$ is chosen uniformly at random from $\mathbb{Z}_p^*$, is the product $a \cdot m \bmod p$ uniformly distributed over $\mathbb{Z}_p^*$?

Answer: Yes


Question: Consider the equation \(9x^2e^{-x}=1\). Let \(c\) be the greatest positive root. Find the greatest positive integer \(n\) such that \(n < c\).

Answer: 5


Question: Evaluate the line integral \(\int_C (x+2y)dx + x^2dy\), where \(C\) consists of line segments from \((0,0)\) to \((2,1)\) and from \((2,1)\) to \((3,0)\).

Answer: \dfrac{5}{2}


Question: Fred receives a total of 30 new emails overnight. If the number of legitimate emails, \(X\), follows a Poisson distribution with mean 10, and the number of spam emails, \(Y\), follows a Poisson distribution with mean 40, what is the expected number of legitimate emails given that he has 30 new emails in total?

Answer: 6


Question: Evaluate the limit: \[ \lim_{x\to -\infty} \left(\frac{2x^3+5}{2x^3+3}\right)^{3x^3+5} \]

Answer: e^3


Question: Determine whether there exists a family of open, bounded intervals of rational numbers (intervals of the form \(]x,y[\) where \(x,y \in \mathbf{Q}\)) whose union is the set of all rational numbers, \(\mathbf{Q}\).

Answer: Yes


Question: Express the size $m$ of a plane graph in terms of its order $n$ and the $k$-cycle of its exterior region, given that all interior regions have a triangular boundary and the boundary of the exterior region is a $k$-cycle where $k \geq 3$. Use Euler's formula for planar graphs to derive the expression.

Answer: 3n - k - 3


Question: Find the volume of the simplex defined by the set \( B := \{ (x,y,z) \in \mathbb{R^3} \mid x \geq 0, y \geq 0, z \geq 0, x+y+z \leq 2 \} \).

Answer: \dfrac{4}{3}


Question: Given that the series \( \sum_{n=1}^\infty a_n x^n \) has a radius of convergence \( R = 10 \), determine the interval of convergence for the series \( \sum_{n=1}^\infty \frac{a_n}{a_{n+1}} x^n \).

Answer: (-1, 1)


Question:
On a certain island, the populations of rats and snakes are related by the equation:
$$(R−13)^2+16(S−20)^2=68$$
where \(R\) is the number of hundreds of rats and \(S\) is the number of hundreds of snakes. What is the maximum combined number of snakes and rats that could ever be on this island at the same time? Round your answer to the nearest integer.

Answer: 4150


Question: Calculate the probability of obtaining a sum of at least 100 when rolling 30 dice, each with 10 sides numbered from 1 to 10.

Answer: 0.99998


Question: Given that for all \( x \in \mathbb{R} \), \( f(f(x)) = x^2 - x + 1 \), find the value of \( f(0) \).

Answer: 1


Question: Find the last three digits of \(341^{101} + 309^{201}\).

Answer: 650


Question: Let \( f: \mathbb{C} \rightarrow \mathbb{C} \) be a differentiable function such that \( |f(z)-z|^{4/3} \le 1 + 3|f(z)-z|^{6/5} \). Find \( f'(-i) \).

Answer: 1


Question: Determine the cardinality of the quotient set \( \frac{\mathbb{Z}}{I_{63} \cap I_{84}} \), where for any integer \( n \), \( I_n \) denotes the ideal \( \{m \in \mathbb{Z} \mid \exists r \in \mathbb{N} \text{ such that } m^r \in n \mathbb{Z}\} \).

Answer: 42


Question: Evaluate the integral \( I = \int_0^{2\pi} \log\left|re^{it} - a\right| \, dt \) where \( a \) is a complex number and \( 0 < r < |a| \).

Answer: 2\pi \log |a|


Question: Determine whether it is possible to arrange the numbers from 1 to 81 on a 9x9 board such that the sum of the numbers in each 3x3 sub-square is the same.

Answer: Yes


Question: Determine the Fourier dimension of the interval \([a, b]\) where \(-\infty < a < b < \infty\). Additionally, identify a measure \(\mu\) in \(\mathcal{M}([a, b])\) that achieves this dimension.

Answer: 1


Question: Find the probability that the length of a randomly chosen chord of a circle lies between \(\frac{2}{3}\) and \(\frac{5}{6}\) of its diameter.

Answer: \dfrac{1}{4}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{1+\sqrt{2}+\cdots+\sqrt{n}}{n^{3/2}}. \]

Answer: \dfrac{2}{3}


Question: Compute the derivative of the sign function with respect to $y$ for the expression $\mathrm{sgn}(x-y)$. That is, find \( \frac{\mathrm{d}}{\mathrm{d}y}\mathrm{sgn}(x-y) \).

Answer: -2\delta(x - y)


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0} \frac{\cos x - e^{- \frac{x^2}{2}}}{x^4} \]

Answer: -\dfrac{1}{12}


Question: Determine all values of \( p \in \mathbb{R} \) for which \( |x-y|^p \) defines a metric on \( \mathbb{R} \).

Answer: (0, 1]


Question: Determine the values of \( \alpha \) for which the integral \( \int_E \frac{x^\alpha}{\sqrt{x^4+y^2}} \, dx \, dy \) converges, where \( E \) is the region defined by the area of the circle \( x^2+(y-1)^2=1 \) minus the area of the circle \( x^2+(y-0.5)^2=0.5^2 \).

Answer: \alpha > -1


Question: Find the maximum value of the function \( f(x) = \lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^n \sqrt{k(n-k)} \binom{n}{k} x^k (1-x)^{n-k} \) on the interval \([0, 1]\).

Answer: \dfrac{1}{2}


Question: Calculate the limit: \[ \lim_{x\to0}\,(a^x+b^x-c^x)^{\frac{1}{x}} \] given that \(a > b > c > 0\).

Answer: \dfrac{ab}{c}


Question: Let $A$ be an invertible $n \times n$ matrix and $B$ be an $n \times n$ matrix such that $AB + BA = 0$. Determine the trace of $B$, denoted as $\operatorname{tr}(B)$. Provide your answer as a single number.

Answer: 0


Question: Given that $y_1$ and $y_2$ are linearly independent solutions of the differential equation $ty'' + 2y' + te^ty = 0$ and $W(y_1,y_2)(1) = 2$, find the value of the Wronskian $W(y_1,y_2)(5)$. ($W(y_1,y_2)(t_0)$ denotes the value of the Wronskian at $t = t_0$.)

Answer: \dfrac{2}{25}


Question: A tetrahedron (3-simplex) is given by the vertices \((0,0,0)\), \((1,0,0)\), \((0,2,0)\), and \((0,0,3)\). Using simplex coordinates, evaluate the integral \(\int _T xz+y^2 \, \mathrm{d}V\).

Answer: \dfrac{11}{20}


Question: Calculate the determinant of the matrix \( J = A \left( A^\intercal W A \right)^{-1} A^\intercal W \), where \( A \) is a non-square matrix, and \( W \), \( J \), and \( A^\intercal W A \) are square matrices.

Answer: 0


Question:
Given a joint Gaussian random vector \(X\) with distribution:
\[
X \sim \mathcal{N}\left(\begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 & 1 & 2 \\ 1 & 4 & 4 \\ 2 & 4 & 16 \end{bmatrix}\right)
\]
Calculate the expected value \(E[X_1 \mid X_2 + X_3 = 0]\).

Answer: -\dfrac{3}{14}


Question: Is the one-point compactification $Y = X \cup \{\infty\}$ of a topological space $X$ compact? Justify your answer.

Answer: Yes


Question: Determine whether the following limit exists: \[ \lim_{u\rightarrow 1^{-}} \int_0^u c(u,y) \, dy \] where \( c(x,y) \) is a copula density function and \( X \) and \( Y \) are uniformly distributed random variables. Consider the relationship \( \int_0^u c(u,y) \, dy = P(Y \leq u|X=u) \) and whether \( \lim_{u \rightarrow 1} P(Y \leq u|X=u) = 1 \).

Answer: 1


Question: Two chess players, X and Y, play a series of games with the following rules: The probability of X winning a particular game against Y is \( \frac{1}{3} \), and the probability of Y winning the game is \( \frac{2}{3} \). X wins the series if X wins two consecutive games, while Y wins the series if Y wins four consecutive games. They continue playing until one of them wins the series. What is the probability that Y wins the series?

Answer: \dfrac{64}{129}


Question: Find all real numbers $k$ that minimize the maximum value of $f(x) = |x^2 - x - k|$ over the interval $0 \le x \le 1$.

Answer: -\dfrac{1}{8}


Question: If $B \subset A$ and there exists an injection $f: A \rightarrow B$, do $A$ and $B$ have the same cardinality? Justify your answer.

Answer: Yes


Question: Let \( \tau = \frac{-1+\sqrt{5}}{2} \), and define \( f(x) = \lfloor (x+1)\tau \rfloor \). Let \( s_n = \frac{\tau n (n+1) (n+2)}{6} \) and \( S_n = \sum_{k=0}^{n} (n−2k)f(n−k) = n f(n) + (n-2) f(n-1) + (n-4) f(n-2) + \dots - (n-2) f(1) - n f(0) \). Determine if \( \frac{S_n-s_n}{n \log n} \) is bounded for \( n > 1 \).

Answer: Yes


Question:
Determine the values of \(\alpha\) and \(\beta\) such that the iterative formula \(x_{n+1} = \alpha x_n\left(3-\dfrac{x_n^2}{a}\right) + \beta x_n\left(1+\dfrac{a}{x_n^2}\right)\) has 3rd order convergence to \(\sqrt{a}\). Choose the correct pair from the following options:

1. \(\alpha=\dfrac{3}{8}\) and \(\beta=\dfrac{1}{8}\)
2. \(\alpha=\dfrac{1}{8}\) and \(\beta=\dfrac{3}{8}\)
3. \(\alpha=\dfrac{2}{8}\) and \(\beta=\dfrac{2}{8}\)
4. \(\alpha=\dfrac{1}{4}\) and \(\beta=\dfrac{3}{4}\)

Answer: 2


Question: Determine the number of elements in the quotient ring \( \mathbb{Z}[\sqrt{-19}]/I \), where \( I = (4, 2 + 2\sqrt{-19}) \).

Answer: 8


Question: Find the radius of convergence of the series \( \sum_{n=0}^{\infty}{\ln\left(\cos{\frac{1}{3^n}}\right)x^n} \).

Answer: 9


Question: Determine the number of individual equations needed to solve the geodesic equation in three dimensions on a spherical surface with coordinates $(r, \theta, \phi)$. The geodesic equation in Einstein notation is given by: $$\frac{d^2u^k}{d\lambda ^2}+\Gamma^k_{ij}\frac{du^i}{d\lambda}\frac{du^j}{d\lambda}=0$$ How many equations are there to solve for the geodesic paths on this surface?

Answer: 3


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

Answer: 0


Question: Evaluate the definite integral: \[ \int_{0}^{2}\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}} \, dx \]

Answer: \dfrac{19}{6}


Question: Find the volume of the wedge-shaped solid that lies above the xy-plane, below the plane $z = x$, and within the cylinder $x^2 + y^2 = 4$. Express your answer as a single number.

Answer: \dfrac{16}{3}


Question: Calculate the limit: $$ \lim_{y\to\infty}\int_{0}^{\infty}  \frac{y\cos^2(x/y)}{y+x^4} \, dx $$

Answer: \infty


Question: Determine all values of the real number \( p \) for which the series \( \sum_{k=2}^{\infty} \frac{1}{k (\log{k})^p} \) converges absolutely.

Answer: p > 1


Question: An insect lays eggs such that the number of eggs, denoted by $X$, follows a Poisson distribution with parameter $\lambda$. Each egg hatches independently with probability $p$. Determine the mean number of hatched eggs.

Answer: \lambda p


Question: In triangle ABC, where AB = BC and \( \angle ABC = 90^{\circ} \), let D be the midpoint of AC. Point E is located on the opposite side of AC from B such that \( \angle AEC = 45^{\circ} \). Determine if the ratio \( \frac{EB}{ED} \) is constant, and if so, find its value.

Answer: \sqrt{2}


Question: Let \( \Omega \subseteq \mathbb{R}^n \) be a bounded open set. Suppose that the topological boundary \( \partial \Omega = \bar{\Omega} \setminus \Omega \) has the structure of a topological manifold. Determine the dimension \( m \) of \( \partial \Omega \). Is \( m = n-1 \)?

Answer: n-1


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

Answer: 48


Question:
Given that $f: D \Rightarrow \mathbb{C}$ is analytic in a domain $D$ and $a$ is a singularity of $f$, with the condition that there exist $r > 0$ and positive numbers $M$ and $N$ such that:
\[
N|z-a|^{-\frac{5}{2}} \leq |f(z)| \leq M|z-a|^{-\frac{7}{2}},
\]
for $0 < |z-a| < r$, determine the order of the pole at $a$. The order is known to be higher than 2 and less than or equal to 4. Determine whether the order is 3 or 4.

Answer: 3


Question: Let $U$ be a simply connected bounded open set in $\mathbb{R}^2$ with area $A$. Consider a sequence of smooth simple closed curves $\gamma_n$ within $U$, with perimeters $l_n$ and areas $A_n$. Assume $\gamma_n$ eventually leave every compact subset of $U$, $A_n$ converges to $A$, $l_n$ converges to $l$, and $4\pi A = l^2$. Is $U$ necessarily the interior of a circle?

Answer: Yes


Question:
Consider the sequence \( u_n = \int\limits_{1}^{n} e^{-t^2}\, dt \) for \( n = 1, 2, 3, \ldots \). Determine which of the following statements is TRUE about the sequence \( \{u_n\}_{n=1}^{\infty} \) and the series \( \sum\limits_{n=1}^{\infty} u_n \):

(A) Both the sequence \( \{u_n\}_{n=1}^{\infty} \) and the series \( \sum\limits_{n=1}^{\infty} u_n \) are convergent.

(B) Both the sequence \( \{u_n\}_{n=1}^{\infty} \) and the series \( \sum\limits_{n=1}^{\infty} u_n \) are divergent.

(C) The sequence \( \{u_n\}_{n=1}^{\infty} \) is convergent but the series \( \sum\limits_{n=1}^{\infty} u_n \) is divergent.

(D) \( \lim\limits_{n \to \infty} u_n = \frac{2}{e} \).

Answer: C


Question: Suppose $\Omega$ is an open subset of Euclidean space $\mathbb{R}^{N}$, $r > 0$, and $D = \{ x \in \mathbb{R}^{N} \mid \| x \| \leq r \} \subseteq \Omega$. Is there an $R > 0$ such that $D \subseteq \{ x \in \mathbb{R}^{N} \mid \| x \| < R \} \subseteq \Omega$?

Answer: Yes


Question: Evaluate the integral \( \int \frac{z}{z^2+1} \, dz \) over the contour bounding the rectangle defined by \(-1 < x < 1\) and \(-2 < y < 2\), oriented counterclockwise.

Answer: 2\pi i


Question: Determine whether there exist any odd prime powers that can be represented as \(n^4 + 4^n\). Provide a proof or counterexample to support your answer.

Answer: 5


Question: Evaluate the line integral \(\int_{C} \left(\frac{\sin(3x)}{x^2+1}-6x^2y\right) dx + \left(6xy^2+\arctan\left(\frac{y}{7}\right)\right) dy\) using Green's Theorem, where \(C\) is the circle \(x^2+y^2=8\) oriented counterclockwise.

Answer: 192\pi


Question: Evaluate the limit of the expression \( \frac {-4b^4c^3+4a^2b^2c^3+ab^4c^2\ln\frac{a+c}{a-c}-ab^4c^2\ln\frac{a-c}{a+c}}{2a^6-2a^4c^2} \) as \( c \) approaches \( a \).

Answer: \infty


Question: Evaluate the infinite nested radical: \[ \sqrt{1+\sqrt{5+\sqrt{11+\sqrt{19+\sqrt{29+\ldots}}}}} \]

Answer: 2


Question: Find the remainder when $49!$ is divided by $53$. Use Wilson's theorem to assist in your calculation.

Answer: 9


Question: Simplify the ordinal expression \(((\omega + 1) + (\omega + 1))^\omega\) and express it in Cantor Normal Form.

Answer: \omega^{\omega}


Question: Compute the limit: \( \lim_{x \to \infty} \left( x^2 - x \cot\left(\frac{1}{x}\right) \right) \).

Answer: \dfrac{1}{3}


Question: Let \(x, y, z\) be non-negative real numbers such that \(x^2+y^2+z^2=2(xy+yz+zx)\). Find the maximum value of the expression \[ \frac{x^3+y^3+z^3}{(x+y+z)(x^2+y^2+z^2)}. \]

Answer: \dfrac{11}{18}


Question: Determine whether the sequence of functions \( x_a(t) = e^{t-a} \) for \( t \in [0,1] \) and \( a \geq 0 \) is equicontinuous.

Answer: Yes


Question: Find the volume of the solid bounded by the surface \((x+y+z+5)^2+(x+2y+3z+6)^2+(x+3y+4z+7)^2=9\).

Answer: 36\pi


Question: Evaluate the integral \( \int_0^\infty \frac{\sqrt{x}}{x^2+4} \, dx \).

Answer: \dfrac{\pi}{2}


Question: Construct a subset $N \subseteq \mathbb{R}$ with Lebesgue measure zero such that every point in $\mathbb{R}$ is a condensation point of $N$. Is this possible?

Answer: Yes


Question: If the asymptotic equality \( f(x) \sim g(x) \) holds, does it imply that \( \lim_{x\to\infty} \frac{f(x)}{g(x)} = 1 \)? Answer yes or no.

Answer: Yes


Question: Evaluate the limit \( \lim_{x\to\infty}\frac{\ln(1+e^{ax})}{\ln(1+e^{bx})} \) for the cases where \( a > 0 \) and \( b > 0 \) without using L'Hôpital's rule.

Answer: \dfrac{a}{b}


Question: Calculate the remainder when \(23^{{2020}^{2020}}\) is divided by 37.

Answer: 10


Question: Given a natural number $n$ and an element $k \in \mathbb{Z}_n$, determine the number of solutions in $\mathbb{Z}_n$ to the equation $x^2 + y^2 = k$. For example, find the number of solutions for $k = 1$ modulo 7.

Answer: 8


Question: Compute the limit: \[ \lim_{x\to \infty} x \log\left(\frac{x+c}{x-c}\right) \]

Answer: 2c


Question: Suppose $X_1, X_2, \ldots, X_n$ are independent and identically distributed (i.i.d.) random variables, and for some real coefficients $a_1, a_2, \ldots, a_n$, the linear combination $\sum_{i=1}^n a_i X_i$ follows a standard normal distribution, $\mathcal{N}(0, 1)$. Is it necessary that each $X_i$ has a normal distribution? Provide a justification for your answer.

Answer: Yes


Question:
Evaluate the limit of the sequence defined by the Riemann sum:
\[ a_n = \frac{1}{n^2} \sum _{k=1}^n \left( \sqrt{\left(n+k\right)^2+n+k}\, \right) \]
as \( n \to \infty \).

Answer: \dfrac{3}{2}


Question: Evaluate the limit: $$ \lim_{x\to 3} \frac {xe^x-3e^x-xe^3+3e^3}{\ln^2(x^2+x-11)} $$

Answer: \dfrac{e^3}{49}


Question: Evaluate the limit \( \lim_{k \to \infty} \int_{0}^{\infty} k^{3/2} f(x) e^{-kx} \, dx \), where \( f(x) = \int_{0}^{x} \frac{\sin(t)}{t^{3/2}} \, dt \).

Answer: \sqrt{\pi}


Question: Determine the set of values of \( x \) for which the power series \( \sum_{n=0}^{\infty} \frac{\sqrt{n!}}{n^{1000}}x^n \) converges.

Answer: \{0\}


Question:
Given that $A + B + C + P + Q + R = 0$, find the value of the determinant:
\[
\begin{vmatrix}
\tan (A+P) & \tan(B+P) & \tan(C+P) \\
\tan (A+Q) & \tan (B+Q) & \tan (C+Q) \\
\tan (A+R) & \tan (B+R) & \tan (C+R)
\end{vmatrix}
\] for all values of $A, B, C, P, Q, R$.

Answer: 0


Question: Compute the limit: \[ \lim_{n\to\infty}\sum_{i=1}^{2n^4+n^2}\frac{5n^2+1}{n^4+i} \]

Answer: \infty


Question: Suppose $f(z)$ is holomorphic in a punctured neighborhood of $z_0 \in \mathbb{C}$. If $f'(z)$ has an essential singularity at $z_0$, does $f(z)$ also have an essential singularity at $z_0$?

Answer: Yes


Question: Evaluate the integral \( \int_0^\infty\int_{-y}^y f(x,y)\,dx\,dy \) using the change of variables \( u = x^2 - y^2 \) and \( v = x^2 + y^2 \). Assume \( f(x,y) \) is an odd function of \( y \). Express the integral in terms of \( u \) and \( v \) and find its value.

Answer: 0


Question:
Given a differentiable real-valued function \( g(x) \) satisfying \( g''(x) - 3g'(x) > 3 \) for all \( x \ge 0 \) and \( g'(0) = -1 \), determine whether the function \( g(x) + x \) is an increasing function of \( x \) for \( x > 0 \). Choose the correct option:

(A) Increasing function of \( x \)
(B) Decreasing function of \( x \)
(C) Data insufficient
(D) None of these

Answer: A


Question: Calculate the line integral \( I = \int_{|z|=5} \left( \frac{2}{(z-1)^2} + \frac{3}{(z-1)} + \frac{\sin z}{e^{z}} \right) dz \).

Answer: 6\pi i


Question: Complete the square for the expression \(k^4 + 2k^3 - k^2 - 2k + 1\) and express it in the form of a perfect square.

Answer: (k^2 + k - 1)^2


Question:
Calculate the winding number of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) at the origin using the formula:
\[ \omega(\gamma,0) = \frac{1}{2i\pi}\oint_{\gamma}\frac{1}{z}dz \]
where the ellipse is parametrized by \( \gamma(t) = a\cos(t) + ib\sin(t) \) for \( t \in [0, 2\pi] \).

Answer: 1


Question: Determine the number of real solutions to the equation \((\sin x + \cos x + 2)^4 = 128\sin(2x)\) for \(x\) in the interval \([0, \frac{\pi}{2}].\)

Answer: 0


Question: Consider an algebraic signature $\Omega$. Let $F$ be a finite set of equations in the signature $\Omega$, and let $I$ be an infinite set of equations in the signature $\Omega$ that generates the same equational theory as $F$. Is there necessarily a finite subset $F'$ of $I$ that generates the same equational theory as $F$?

Answer: Yes


Question: Find a function, other than $e^x$ and $\cos x$, whose derivative is a function of itself, similar to how $f'(x) = f(x)$ for $f(x) = e^x$ and $f'(x) = f(x+\pi/2)$ for $f(x) = \cos x$. Provide one example of such a function.

Answer: \sin x


Question: If \( \{W_t\}_{t \geq 0} \) is a standard Brownian motion process, and \( \Delta > 0 \), is the random variable \( W_{n\Delta} - W_{(n-1)\Delta} \) normally distributed with mean zero and variance \( \Delta \) for any \( n \in \mathbb{N} \)?

Answer: Yes


Question:
Let \( f(x) \) be a continuous function on \([0, 1]\) satisfying the following conditions:

1. \( f(1) = 0 \)
2. \( \int_0^1 [f'(x)]^2 \, \mathrm{d}x = 7 \)
3. \( \int_0^{1} x^2f(x) \, \mathrm{d}x = \dfrac{1}{3} \)

Find the value of \( \int_0^{1} f(x) \, \mathrm{d}x \).

Answer: \dfrac{7}{5}


Question: Determine the value of the parameter \( a > 0 \) for which the equation \( a - ax + x^{0.8} - x^{0.2} = 0 \) has multiple real roots, given that \( x > 0 \).

Answer: \dfrac{3}{5}


Question: Let \( F = \mathbb{F}_{p} \) and \( E = \mathbb{F}_{p^{2}} \) be the degree 2 extension over \( F \). Let \( \tau \) be the nontrivial Galois automorphism of \( E \). Consider a degree one representation \( \phi \) of \( E^{*} \). Define \( \phi^{\tau} : E^* \to \mathbb{C}^{*} \) by \( \phi^{\tau}(x) = \phi(\tau(x)) \) for all \( x \in E^{*} \). Determine the number of representations \( \phi \) such that \( \phi = \phi^{\tau} \).

Answer: p - 1


Question: Evaluate the integral \( \int_C \cot^2 z \, dz \) where \( C \) is the positively oriented circle \( |z| = 4 \).

Answer: 0


Question: Determine the limit of the sequence \( \sin^2 (\pi\sqrt{n^2+n}) \) as \( n \to \infty \).

Answer: 1


Question:
Let $A$ be a $5 \times 5$ complex matrix such that $(A^2 - I)^2 = 0$. Assume that $A$ is not a diagonal matrix. Determine which of the following statements is true:

1. $A$ is diagonalizable.
2. $A$ is NOT diagonalizable.
3. No conclusion can be drawn about the diagonalizability of $A$.

Select the correct statement.

Answer: 3


Question: Determine whether the identity function \( f: (\mathbb{R}, d) \to (\mathbb{R}, d_{usual}) \) defined by \( f(x) = x \) is uniformly continuous on a closed and bounded interval, where the metric \( d(x, y) = |x^3 - y^3| \) is used.

Answer: Yes


Question: Find all functions \( f \in L^1(\mathbb{R}) \) such that \( \int_{-\infty}^{\infty} f(t)e^{(-t^2+2xt)}dt = 0 \) almost everywhere.

Answer: 0


Question: Compute the integral of \( g(z) = \frac{1}{z^2+4} \) around the circle \(|z-i|=2\) in the positive sense using Cauchy's integral formula.

Answer: \dfrac{\pi}{2}


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

Answer: 12


Question: Determine the limit of the sequence \( a_n = \frac{3 \cdot 5 \cdot 7 \cdot \ldots \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \ldots \cdot 2n} \) as \( n \to \infty \).

Answer: 0


Question: If $X$ is a random variable that follows a Poisson distribution with parameter $u$, what is the variance of $X^2$?

Answer: 4u^3 + 6u^2 + u


Question: Suppose \(X\) and \(Y\) are jointly normal random variables with means 0, variances 1, and correlation \(\rho\). Find \( \operatorname{Cov}(X^2,Y^2) \).

Answer: 2\rho^2


Question: For what values of $c \in [-1,1]$ does the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) given by \[ f(x) = \begin{cases} \sin \left( \frac{1}{x} \right), & \text{if } x \neq 0; \\ c, & \text{if } x = 0 \end{cases} \] have an antiderivative?

Answer: 0


Question: Calculate the limit: \[ \lim_{n\to\infty}\frac{1}{n}\sum_{i=-\frac{nf}{2}}^{\frac{nf}{2}}e^{-\left(\frac{1}{n^2}+\alpha\right)i^2} \] where \( f \) and \( \alpha \) are positive real-valued constants.

Answer: 0


Question: Using Rouche's theorem, determine the number of roots of the polynomial \( z^9 + z^5 - 8z^3 + 2z + 1 \) that lie in the annular region between the circles \(|z|=1\) and \(|z|=2\).

Answer: 6


Question: Let $A$ be a finitely generated $k$-algebra, where $k$ is an algebraically closed field, and $A$ is of dimension one and integrally closed in its quotient field $K$. Let $R \subseteq K$ be a valuation ring. Is it true that $A \subseteq R$?

Answer: No


Question: Calculate the total work required to pump all the water out of a cylindrical tank with a radius of 2 meters and a length of 10 meters. Assume the tank is filled with water, the acceleration due to gravity is 9.8 m/s², and the density of water is 1000 kg/m³. Express your answer in terms of Joules.

Answer: 784000\pi


Question: Solve the equation for \(x\): \[ 0.2 = 0.819 \exp(-11.5x) + 0.0975 \exp(-70.1x) + 0.0325 \exp(-179x) \] and find the value of \(x\) that is approximately 0.1225.

Answer: 0.1225


Question: Prove that if $a_1, a_2, \ldots, a_n$ are distinct primes with $a_1 = 2$ and $n > 1$, then $a_1 a_2 \cdots a_n + 1$ is of the form $4k + 3$ for some integer $k$.

Answer: 4k + 3


Question: Derive the Laurent series of \( f(z)=\frac{e^z}{(z+1)^2} \) around \(z=-1\). What is the radius of convergence of this series?

Answer: \infty


Question:
Let $T:[0,1]\to [0,1]$ be a piecewise smooth expanding map, meaning $|T'(x)|>1$ for all $x$. Consider a sequence of nested intervals $I_n$ such that $I_{n+1}\subset I_n$ and the length of $I_n$ tends to 0 as $n\to\infty$. Define the survivor set for $I_n$ as:
$$
\mathcal J(I_n)=\{x \in[0,1] : T^k(x)\notin I_n, \forall k\ge0\}.
$$
Is it true that the Hausdorff dimension of $\mathcal J(I_n)$ approaches 1 as $n$ tends to infinity?

Answer: Yes


Question: Evaluate the limit \( \lim_{x\rightarrow \infty}\frac{f^{-1}(1000 x)-f^{-1}(x)}{x^{\frac{1}{3}}} \), where \( f(x) = 27x^3 + (\cos 3 + \sin 3)x \) and \( f^{-1}(x) \) is the inverse of \( f(x) \).

Answer: 3


Question: Evaluate the limit: \[ \lim_{z \to 0} \left(\frac{\sin z}{z}\right)^{\frac{1}{z^2}} \]

Answer: e^{-\frac{1}{6}}


Question: Let \( f(x) \in \mathbb{Q}[x] \) be an irreducible monic polynomial of degree 3 that does not split over \( \mathbb{R} \). Find the degree of the splitting field of \( f(x) \) over \( \mathbb{Q} \).

Answer: 6


Question: If $M$ is a symplectic manifold and $f: W \to M$ is an open immersion, does $f$ pull back a symplectic form on $M$ to a symplectic form on $W$?

Answer: Yes


Question: Evaluate the integral \( \frac{2}{\pi i}\int_C \left(ze^{1/z}+\tan\left(\frac{z}{2}\right)+\frac{1}{(z-1)(z-3)^2}\right)dz \), where \( C \) is the positively oriented circle of radius 2 centered at the origin in the complex plane.

Answer: 3


Question: Consider a closed polytope $P$ in $\mathbb{R}^N$. Is every point of $P$ a linear combination of the vertices of $P$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to\infty} \left( \frac{\sqrt{x^4+x^{3}\tanh(x)+x^2}}{x+1} - x \right) \]

Answer: -\dfrac{1}{2}


Question:
Given the joint probability density function for random variables $X$ and $Y$:
\[ f(x, y)=\begin{cases}
2(x+y)  & \text{if } 0<x<y<1 \\
0 &  \text{otherwise}
\end{cases}. \]
Find the real number $a$ such that the conditional expectation $E(X | Y=aX)=\frac{2}{9}$. \(a\) is a real number.

Answer: 3


Question: A circular metal wire $S$ in 3-space is described by $S=\{x^2+y^2+z^2=4\}$ and has density $\sigma (x,y,z) = \sqrt{x^2+y^2}$ at the point $(x,y,z)$. Find the mass of $S$.

Answer: 8\pi


Question: Evaluate the contour integral \( \oint_{C_r}\frac{\pi^2}{\sin^2\pi w}(w-N)dw \), where \( N \) is a fixed integer and \( C_r \) is a circle centered at \( N \) with radius \( 0 < r < 1 \).

Answer: 2\pi i


Question: Determine the largest base \( a \) such that the exponential function \( y = a^x \) has nonzero fixed points in the real numbers.

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


Question: Determine the character \( \lambda \) that defines the canonical bundle \( \omega_{G/B} \) on the flag variety \( G/B \) for a reductive group \( G \) over a field \( k \) of characteristic zero, with maximal split torus \( T \) and Borel \( B \supset T \). Is it \( -2\rho \) or \( 2\rho \), where \( \rho \) is the half sum of all positive roots with respect to \( B \)?

Answer: -2\rho


Question: Let $A = B - C$, where $B$ and $C$ are $p \times p$ symmetric positive semidefinite matrices, and the rank of $C$ is $r < p$. Determine the maximum number of negative eigenvalues of $A$. Provide your answer as a single integer.

Answer: r


Question: Use the residue theorem to evaluate the integral \( \int_{0}^{2\pi}  \frac {27} {(5+4\sin\theta)^2} d\theta \).

Answer: 10\pi


Question: Find the smallest positive integer k such that the equation \( x^2 + kx = 4y^2 - 4y + 1 \) has integer solutions for \( x \) and \( y \).

Answer: 8


Question: Find the area of the region \( R = \{(x,y) \mid x \leq 0 , e^{x} \leq y \leq e^{\frac{1}{2} x} \} \) using the substitution \( x = \ln(u^2 v) \), \( y = uv \) for \( u,v > 0 \).

Answer: 1


Question:
Solve the Volterra Integral Equation of the First Kind:
\[ \int_{0}^{x} J_{0}(x-t)y(t) \, dt = \sin(x) \]
where \( J_{0}(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(k!)^2} \left(\frac{x}{2}\right)^{2k} \) is the Bessel Function of the First Kind.

Answer: J_0(x)


Question: Suppose $f \in \mathbb{C}\{x_1, \ldots, x_n\}$ is irreducible in the ring of convergent power series over $\mathbb{C}$. Determine if $f$ remains irreducible in the ring of formal power series $\mathbb{C}[[x_1, \ldots, x_n]]$. Provide a justification for your answer.

Answer: Yes


Question: Does there exist a function that has a derivative almost everywhere, but where the derivative function is everywhere discontinuous?

Answer: No


Question: Let $(R,m)$ be a commutative Noetherian local ring with $\operatorname{depth}(R)>0$, and let $M$ be a finitely generated $R$-module with $\operatorname{depth}(M)=0$. Determine if there exists an $R$-regular element $x \in m$ such that $x \notin \bigcup_{p\in \operatorname{Ass}(M)-\{m\}}p$.

Answer: Yes


Question: Let \( A \in M_n \). Can the dimension of the subspace \( L(I, A, A^2, \ldots, A^k, \ldots) \) of \( M_n \) be greater than \( n \)? Justify your answer using the Cayley-Hamilton theorem.

Answer: No


Question:
Consider an abelian variety $A/\mathbb{Q}$ with good reduction at a prime $p$, and let $\mathcal{A}/\mathbb{Z}_{(p)}$ be its integral model at $p$. For any number field $K$ and any prime ideal $\mathfrak{p}$ of $K$ over $p$, the valuation criterion ensures that any $K$-point of $A$ extends uniquely to a $\kappa$-point of the special fiber of $\mathcal{A}$. By fixing a prime ideal of $\overline{\mathbb{Z}}$ over $p$ and taking the injective limit, we obtain a reduction map:
\[ A(\overline{\mathbb{Q}})\rightarrow \mathcal{A}_{\mathbb{F}_p}(\overline{\mathbb{F}_p}). \]
Does this map induce isomorphisms on the $\ell$-primary parts of both groups, given that their $\ell$-primary subgroups are isomorphic to $(\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell})^{\oplus 2g}$?

Answer: Yes


Question: Evaluate the infinite sum \( \sum_{m=1}^{\infty}\frac{(-1)^{m+1}}{2m-1} \) by considering the Fourier series of the function \( f(x) = 1 + x \) on the interval \(-1 < x < 1\). First, find the Fourier series representation of \( f(x) \), and then evaluate it at \( x = \frac{1}{2} \) to determine the value of the sum.

Answer: \dfrac{\pi}{4}


Question: Evaluate the integral \( \int_0^2 \frac{2}{\ln(x)} \, dx \) using the Fundamental Theorem of Calculus, given that \( F'(x) = \frac{1}{\ln(x)} \) and \( F(0) = 1, F(1) = 2, F(2) = 5 \).

Answer: 8


Question: Calculate the closed loop integral \( \int_\gamma Pdx + Qdy \) for the path \( \gamma: (2, 0) \to (-1,1) \to (0, -2) \to (1, 2) \to (-2, -1) \to (2, 0) \) where \( P(x,y)=  \frac{x-y}{x^2+y^2} \) and \( Q(x,y)=  \frac{x+y}{x^2+y^2} \). Use Green's Theorem to determine the value of the integral.

Answer: 2\pi


Question: Let $M$ be a Riemannian manifold such that for all $x \in M$ and large $R > 0$, \( \text{Vol}(B(x, R)) \leq \exp(R) \). Let $p \in M$ and $d$ be the geodesic distance. Suppose that for large enough $r$ and all $x, y \in \partial B_{r}(p)$, \( d(x, y) \leq c \log r \). Is it true that there exist constants $N > 0$ and $C > 0$ such that for all large enough $r$, \( \text{Vol}(B(p, r)) \leq Cr^{N} \)?

Answer: Yes


Question: Is any division algebra over a field extension $E$ of $F$ necessarily of the form $D \otimes E$, where $D$ is a division algebra over $F$? Provide a justification for your answer.

Answer: No


Question: Given three circles $\bigcirc O_1$, $\bigcirc O_2$, $\bigcirc O_3$, and three points $A$, $B$, $C$ on $\bigcirc O_3$, if \[ \frac{\operatorname{power}(A, \bigcirc O_1)}{\operatorname{power}(A, \bigcirc O_2)}= \frac{\operatorname{power}(B, \bigcirc O_1)}{\operatorname{power}(B,\bigcirc O_2)}= \frac{\operatorname{power}(C, \bigcirc O_1)}{\operatorname{power}(C, \bigcirc O_2 )} \] can we conclude that these circles have the same radical axis? Answer yes or no.

Answer: No


Question:
Determine the degree of the minimal polynomial $m_{\alpha}(x)$ for the element $\alpha = \beta^6$ in the Galois field $GF(2^4)$, where $GF(2^4)$ is constructed as $K[x]$ modulo $1 + x^3 + x^4$ and $\beta$ is the class of $x$. The powers of $\beta$ are given as follows:

- $1000$ corresponds to $1$
- $0100$ corresponds to $\beta$
- $0010$ corresponds to $\beta^2$
- $0001$ corresponds to $\beta^3$
- $1001$ corresponds to $\beta^4$
- $1101$ corresponds to $\beta^5$
- $1111$ corresponds to $\beta^6$
- $1110$ corresponds to $\beta^7$
- $0111$ corresponds to $\beta^8$
- $1010$ corresponds to $\beta^9$
- $0101$ corresponds to $\beta^{10}$
- $1011$ corresponds to $\beta^{11}$
- $1100$ corresponds to $\beta^{12}$
- $0110$ corresponds to $\beta^{13}$
- $0011$ corresponds to $\beta^{14}$

What is the degree of the minimal polynomial for $\alpha = \beta^6$?

Answer: 4


Question:
Let $N$ be the number of sequences $a_1, a_2, a_3, \ldots, a_{24}$ that satisfy the following conditions:
1. For all $1 \le i \le 24$, $1 \le a_i \le 24$.
2. For all $i \equiv 0 \pmod{2}$, $a_1 + a_2 + \cdots + a_i \equiv 1 \pmod{2}$.
3. For all $i \equiv 0 \pmod{3}$, $a_1 + a_2 + \cdots + a_i \equiv 2 \pmod{3}$.

If $N = p_1^{a_1} p_2^{a_2}$ where $p_1$ and $p_2$ are prime numbers, find $p_1 + a_1 + p_2 + a_2$. Determine the value of $p_1 + a_1 + p_2 + a_2$.

Answer: 81


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

Answer: 1


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

Answer: \infty


Question: Calculate the limit: \[ \lim_{k\to\infty}\int_{\mathbb{R}^2}\frac{(x^2+y^2)^{k/2}}{1+(x^2+y^2)^{(k+3)/2}}\,dx\,dy \] without using the change of variable theorem.

Answer: 2\pi


Question: Find the limit of the sequence \( a_n = n\left(\sqrt[n]{ea} - \sqrt[n]{a}\right) \) as \( n \to \infty \), where \( a > 0 \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^1 nxe^{-nx} \, dx \]

Answer: 0


Question: Let $A$ be an $n \times n$ matrix. Consider the infinite sum \( B = \sum_{k=1}^\infty \frac{A^k t^k}{k!} \). Does the sum \( B \) always converge for each of the \( n^2 \) entries of the matrix?

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{1}\left(\sqrt[4]{1-x^7}-\sqrt[7]{1-x^4}\right)dx \).

Answer: 0


Question: Suppose there are $n-1$ draws from a uniform distribution $[0, 1]$, followed by your draw from the same distribution. What is the probability that your draw, $x$, is exactly the second highest among all $n$ draws?

Answer: \dfrac{1}{n}


Question: Determine the upper bound of the Ramsey number \( R(3,3,3, \ldots,3) \) where the number 3 is repeated \( k \) times. Prove that \( R(3,3,3, \ldots,3) \leq 3k! \).

Answer: 3k!


Question: How many solutions in positive integers are there to the equation \(x_1 \cdot x_2 \cdot x_3 \cdot x_4 = 2^{20} \cdot 13^{13}\)? Assume that \(x_1, x_2, x_3,\) and \(x_4\) are distinguishable.

Answer: 991760


Question: Evaluate the limit as \( R \to \infty \) of the integral \( \int_C \frac{e^{iz}}{z^2} \, dz \), where \( C \) is a semi-circle of center \( O = (0,0) \) and radius \( R \) in the upper half-plane (\( y \geq 0 \)).

Answer: 0


Question: Given a non-zero sequence \(r_n\) such that \(\lim_{n \to \infty}{r_{2n}} = \frac{1}{2008}\) and \(\lim_{n \to \infty}{r_{2n + 1}} = \frac{1}{2009}\), and a sequence \(a_n\) defined by \(a_1 = x\) and \(a_{n+1} = a_n \cdot r_n\) where \(x \in \mathbb{R}\), find \(\lim_{n \to \infty} a_n\).

Answer: 0


Question: Find the least value of \(a\) for which \(5^{1+x}+5^{1-x}\), \(a/2\), and \(25^x+25^{-x}\) are three consecutive terms of an arithmetic progression.

Answer: 12


Question:
Let $S = \{u_1, u_2, u_3\}$ and $T = \{v_1, v_2, v_3\}$ be two orthonormal bases of the subspace $W$. Define $P = (p_1, p_2, p_3)$ where $p_i = \begin{pmatrix} v_i \cdot u_1 \\ v_i \cdot u_2 \\ v_i \cdot u_3 \\\end{pmatrix}$ for $i = 1, 2, 3$. Determine which of the following statements is true:

1. For any $w \in W$, $P\begin{pmatrix} w \cdot u_1 \\ w \cdot u_2 \\ w \cdot u_3 \\\end{pmatrix} = \begin{pmatrix} w \cdot v_1 \\ w \cdot v_2 \\ w \cdot v_3 \\\end{pmatrix}$.
2. For any $w \in W$, $P\begin{pmatrix} w \cdot v_1 \\ w \cdot v_2 \\ w \cdot v_3 \\\end{pmatrix} = \begin{pmatrix} w \cdot u_1 \\ w \cdot u_2 \\ w \cdot u_3 \\\end{pmatrix}$.
3. Both 1 and 2 are false.

Answer: B


Question: Given an $n \times n$ matrix $A$ such that $(A-I)^2=0$, find the determinant of $2I-A$. Express your answer in terms of $n$ if necessary.

Answer: 1


Question: What is the second smallest non-prime order of a simple group? Consider a simple group $G$ such that $|G|$ is not a prime number. Given that the smallest such order is 60, determine the next smallest order.

Answer: 168


Question: Evaluate the double integral: \[ \int_0^\infty\int_0^\infty xye^{-(x^2+y^2)}\,dx\,dy \]

Answer: \dfrac{1}{4}


Question: Use Green's theorem to calculate the line integral \( \oint\limits_{C2} \frac{-y}{(x^{2}+y^2)^2} \, dx +\frac{x}{(x^{2}+y^2)^2} \, dy \), where \( C1: x^{2}+y^2 = 1 \) and \( C2: r = \frac{2}{\sqrt{2-\cos(\theta)}} \). The region \( R \) is the area between \( C1 \) and \( C2 \), and both curves are oriented positively. Given that \( \oint\limits_{C1} \frac{-y}{(x^{2}+y^2)^2} \, dx +\frac{x}{(x^{2}+y^2)^2} \, dy = 2\pi \) and \( \iint\limits_R \frac{1}{(x^{2}+y^2)^2} \, dx \, dy = \frac{\pi}{2} \), find the value of the line integral over \( C2 \).

Answer: \pi


Question: Calculate the residue at the origin for the function \( \frac{1}{z^2 \sin z} \) within the unit circle.

Answer: \dfrac{1}{6}


Question: Determine the number of values of $a$ for which the quadratic equation \[(a^2-1)x^2-(a^2-3a+2)x+a^2-8a+7=0\] has three distinct roots.

Answer: 0


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

Answer: 1


Question:
Consider a function \( f: \Bbb{C} \setminus \{0\} \to \Bbb{C} \) such that \( f(z) = f(z/|z|) \) and its restriction to the unit circle is continuous. Determine which of the following statements is true:

- \( f \) is continuous but not necessarily analytic.
- \( f \) is analytic but not necessarily constant.
- \( f \) is a constant function.
- \( \lim \limits_{z \to 0} f(z) \) exists.

Answer: A


Question: Prove that for any integer $n$, every odd divisor of $n^2 + 1$ is of the form $4k + 1$.

Answer: 4k + 1


Question: Find the minimum possible value of \(a^{2}+b^{2}\) such that the polynomial \(x^{4}+ax^{3}+bx^{2}+ax+1=0\) has at least one real root.

Answer: \dfrac{4}{5}


Question: Compute the Riemann-Stieltjes integral \( \int_{-1}^{1} (x^2 + e^x) \, d(\text{Sgn}(x)) \).

Answer: 2


Question: Find the maximum value of \( k \) such that the inequality \( \frac{2(a^2+kab+b^2)}{(k+2)(a+b)} \ge \sqrt{ab} \) holds for all positive \( a \) and \( b \). Determine \( k_{\max} \).

Answer: 6


Question: Calculate the following limit: $$\lim_{n\to\infty}\frac{\sqrt[n]{2^n+3^n+\cdots +n^n}}{n}$$

Answer: 1


Question: Determine the smallest degree \( n \) of a Taylor polynomial for \( f(x) = e^x \) centered at \( x_0 = 0 \) such that the polynomial approximates \( f(x) \) within \( 10^{-6} \) on the interval \([0, 0.5]\).

Answer: 7


Question: Evaluate the expression: \[ \sum_{k=0}^{10} {30+k \choose k}{20-k \choose 10} \]

Answer: \dbinom{51}{10}


Question: Let $A$ and $B$ be $n \times n$ complex Hermitian positive semidefinite matrices. If $\mathrm{trace}(AB) = 0$, does it necessarily follow that $AB = 0$? Provide a proof or counterexample.

Answer: AB = 0


Question: Evaluate the residue of the function \( \frac{(z^6-1)^2}{z^5(2z^4 -5z^2 + 2)} \) at \( z = 0 \).

Answer: \dfrac{21}{8}


Question: Find the derivative of the function \( f(x) = x + \frac{1}{x + \frac{1}{x + \frac{1}{x + \ldots}}} \) at \( x = \frac{3}{2} \). What is \( f'\left(\frac{3}{2}\right) \)?

Answer: \dfrac{4}{5}


Question: Find the volume inside the cylinder with radius 1, defined by the equation \(x^2 + y^2 = 1\), that is bounded by the planes \(z = 4 - 2x - y\), \(x = 0\), \(y = 0\), and \(z = 0\).

Answer: \pi - 1


Question: Let $n = 2m$, where $m$ is odd. Determine the number of elements of order 2 in the group $D_n/Z(D_n)$. Provide your answer as a single integer.

Answer: m


Question: Evaluate the integral \( \int_0^3 \frac{x^4}{x^4+(x-3)^4} \,dx \).

Answer: \dfrac{3}{2}


Question: Determine the set of irrational numbers $x$ for which the function $f(x) = (-2)^x$ is a real number.

Answer: \emptyset


Question: Given the polynomial equation \(x^4 + 4x^3 + ax^2 + 4x + 1 = 0\), the set of values of \(a\) for which the equation has real roots is \((-\infty, m] \cup \{n\}\). Find the value of \(\sqrt{n-m}\).

Answer: 4


Question: Evaluate the limit: $$\lim_{x\to0^+}\frac{{\int_0^{x^2}\sin{\sqrt{t}}}~ dt}{x^3}$$

Answer: \dfrac{2}{3}


Question: Let $(u_n)_n$ be a real sequence. Suppose there exists a strictly increasing function $\phi:\mathbb{N}\to\mathbb{N}$ such that for all $k\in\mathbb{N}$, the subsequence $(u_{\phi(n)+k})_n$ converges to the same limit $l$. Does this necessarily imply that the sequence $(u_n)_n$ converges to $l$?

Answer: No


Question: Find the closed-form solution for \( x \) in the equation \( e^x - \frac{x^2}{a} + \frac{x}{b} - 1 = 0 \), where \( a > 0 \) and \( b \geq 1 \).

Answer: 0


Question: Let $A$ be an infinite set of positive integers. For any two distinct integers $a$ and $b$ in $A$, at least one of the numbers $a^b + 2$ or $a^b - 2$ is also in $A$. Must the set $A$ contain a composite number?

Answer: Yes


Question: In a monoidal category, let \( C_1, C_2 \) be monoidal categories with tensor bifunctors \( \otimes_i: C_i \times C_i \to C_i \) and tensor units \( 1_i \). Given a monoidal functor \( F: C_1 \to C_2 \) with an isomorphism \( F(1_1) \simeq 1_2 \) and a natural family \( F(U,V): F(U \otimes_1 V) \to F(U) \otimes_2 F(V) \), and a monoidal equivalence \( G: C_2 \to C_2 \), consider the map \( F(U,V) \otimes_2 \text{id}: F(U \otimes_1 V) \otimes_2 F(W) \to (F(U) \otimes_2 F(V)) \otimes_2 F(W) \). Is the map \( G(F(U,V) \otimes_2 \text{id}) \) equal to \( G(F(U,V)) \otimes_2 \text{id} \)?

Answer: No


Question: Evaluate the integral \( \int_{|C|=2} \frac{dz}{z^2 + 2z + 2} \) using the Cauchy-Goursat theorem, where the contour \( C \) is a circle of radius 2 centered at the origin. Express your answer in terms of \( \pi \).

Answer: 0


Question: Given that $\lim_{n\to \infty} \sqrt[n]{a_n} = e$, determine whether the sequence $(a_n \cdot 3^{-n})$ converges to 0.

Answer: 0


Question: Let $K/\mathbb{Q}$ be a number field, and suppose a prime $p \in \mathbb{Z}$ factors in $\mathcal{O}_K$ as $\prod_{i=1}^r \mathfrak{p}_i^{e_i}$. Given the identity $[K:\mathbb{Q}]=\sum_{i=1}^r e_if_i$, determine if it is true that for each $i$, the local extension degree satisfies $[K_{\mathfrak{p}_i}:\mathbb{Q}_p]=e_if_i$. Justify your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{1}\int_{0}^{1-x_1}...\int_{0}^{1-\sum_{i=1}^{n-1}{x_i}}1 \,dx_n\,dx_{n-1}...\,dx_1 \) over the region defined by \( \sum_{i=1}^{n}{x_i}=1 \). Determine the value of this integral in terms of \( n \).

Answer: \dfrac{1}{n!}


Question: Let \((u_n)\) be a sequence defined by \(u_1 = 3\) and \(u_{n+1} = \frac{1}{5}(u_n^2 + u_n + 4)\) for \(n = 1, 2, \ldots\). Define \(v_n = \sum_{k=1}^n \frac{1}{u_k + 3}\). Determine the value of \(\lim_{n \to \infty} v_n\).

Answer: 1


Question: Find \( \lim_{n \rightarrow \infty} \mathbb{P}(X_1 + \ldots + X_n \leq 0) \), where the independent binary random variables \(X_k\) take values \(\pm 1\) with probabilities \((1 \pm k^{-1/2})/2\) for \(k = 1, 2, \ldots\).

Answer: \Phi(-2)


Question: Calculate the surface integral \( \iint_S E \cdot \vec{n} \, d\sigma \) where \( S \) is the parametric surface given by \( X(s,t) = [st, s^2, t^2]^T \) for \( 0 \leq s \leq t \leq 1 \), and \( E \) is the vector field \( E(x,y,z) = [3yz, zx, 2xy]^T \).

Answer: \dfrac{1}{6}


Question: Determine the necessary condition on \((k, p)\) such that the integral \(\int_{\mathbb{R}^d}\frac{1}{(|x|^{2k}+1)^p}\,dx\) is finite. Is the condition \(2kp > d\) sufficient?

Answer: 2kp > d


Question: What is the probability that 12 indistinguishable calls are distributed randomly over 7 days such that each day receives at least one call?

Answer: \dfrac{11}{442}


Question: Let $g(x)$ be a smooth probability density function with a mean value of $0$ and finite variance, where $g(x) > 0$ for all $x \in \mathbb{R}$. Determine whether the following limit holds: \[ \lim_{x \to \infty} xg(x) = 0. \]

Answer: 0


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

Answer: -\dfrac{e}{2}


Question: Calculate \( \lim_{n\to\infty}\int_0^1\frac{nx}{1+n^2 x^3}\,dx \).

Answer: 0


Question: Find all positive integer pairs \((n, m)\) such that \((125 \times 2^n) - 3^m = 271\).

Answer: (3, 6)


Question: Show that \( \int_0^\infty \frac{e^{-x}-e^{-xt}}{x}dx = \ln(t) \) for \( t > 0 \).

Answer: \ln(t)


Question: Find \( \lim_{\varepsilon\rightarrow 0^+} f_\varepsilon \) in \( D'(\mathbb{R}) \), where \( f_\varepsilon(x)=\frac{\sin{\varepsilon x}}{x} \).

Answer: \pi \delta(x)


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}} \) using Stolz's theorem.

Answer: e


Question: Evaluate the limit: $$\lim_{x \to 0}\frac{\sin(\tan x)-\tan(\sin x)}{x-\sin x}$$

Answer: 0


Question: Determine whether the statement is true or false: Let $V$ be a space of polynomials of degree $\leq 3$. Let $T$ and $D$ be linear operators on $V$ such that $T(f(t))=f(t+1)$ and $D(f(t))=f'(t)$. Then $T$ is a polynomial in $D$. Is this statement true?

Answer: True


Question: Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex on the plane $x+2y+3z=3$. Express the volume in terms of $x$, $y$, and $z$, and determine the maximum volume.

Answer: \dfrac{1}{6}


Question: If a matrix is invertible and diagonalizable, is its inverse also diagonalizable? Justify your answer.

Answer: Yes


Question:
Simplify the logical expression:
\[((A \Rightarrow B) \Rightarrow (B \Rightarrow A)) \Rightarrow (\neg(A \land B) \Leftrightarrow \neg(B \lor A))\]

Answer: A \Rightarrow B


Question: Five chameleons, each of a different color, meet one evening. During the night, each chameleon changes its color to one of the other four colors with equal probability. What is the probability that the next day the five chameleons have all different colors again?

Answer: \dfrac{11}{256}


Question: Let \(K\) be a compact subset of \(\mathbb{R}^n\) and \(F\) be a closed subset of \(\mathbb{R}^n\) such that \(K \cap F = \emptyset\). Prove that there exists a positive number \(a > 0\) such that \(\| y-x\| > a\) for all \(x \in K\) and \(y \in F\).

Answer: a


Question: Given a function $f: \mathbb{R} \to \mathbb{R}^*$ that admits primitives and satisfies the conditions $\cos(f(x)) = 1$ for all $x \in \mathbb{R}$ and $|f(\pi) - \pi| \leq \pi$, determine the value of $f(100)$. \( \text{Provide your answer as a single value.} \)

Answer: 2\pi


Question: Evaluate the determinant of the matrix $aI_n + bJ_n$, where $a, b \in \mathbb{R}$, $I_n$ is the $n \times n$ identity matrix, and $J_n$ is the $n \times n$ matrix with every entry equal to 1.

Answer: a^{n-1}(a + bn)


Question: Determine whether the following assertion from Engelking's book General Topology (1989) p.52 is true: If $x$ is a cluster point of a filter $\mathcal{F}$, then $x$ is a limit of a filter $\mathcal{F'}$ that is finer than $\mathcal{F}$. (Part of Proposition 1.6.8)

Answer: True


Question: Determine all real numbers \(a\) such that the polynomial \(f(x) = x^4 - 2x^3 + x^2 + 4x + a\) has a non-real complex root with a real part equal to 1.

Answer: -6


Question: Calculate the probability \( P(p=0.5|X=7) \) given that \( X \sim \text{Bin}(8, p) \) and \( p \) is either 0.5 or uniformly distributed over [0.5, 1]. Use Bayes' Rule to find the solution.

Answer: \dfrac{36}{287}


Question: Evaluate the limit: \[ \lim_{x\to\infty}\left(1+\frac{\ln x}{f(x)}\right)^{\frac{f(x)}{x}} \] where the function \( f:\mathbb{R}\rightarrow(0,\infty) \) satisfies \( f(x)\cdot \ln f(x)=e^x \) for all \( x \in \mathbb{R} \).

Answer: 1


Question: Evaluate the limit \( \lim_{x \rightarrow \infty} (2-a^{1/x})^{x} \) where \( 0 < a < 1 \).

Answer: \dfrac{1}{a}


Question: Evaluate the limit: $$\lim_{x\rightarrow \infty }\int_{0}^{x}\sin\frac{\pi }{t+x}\, \mathrm{d}t$$

Answer: \pi \ln 2


Question: Let \(a, b, c\) be non-negative real numbers such that \(a^2 + b^2 + c^2 = 1\). Determine the maximum value of the expression \(2\sqrt{6}ab + 8bc\).

Answer: \sqrt{22}


Question: Determine whether the group $A_6$ contains a subgroup isomorphic to $S_4$. Provide a justification for your answer.

Answer: Yes


Question:
Find a function \( f(x) \) such that \( \lim_{x\to \infty} \left(\frac{y}{f(x)}\right) = 1 \), where \( y = \frac{1}{w} \) and \( w \) is a solution to the differential equation:
\[ \frac{d^2w}{dx^2} + 2\frac{dw}{dx} + 5w = -5x^2 - 4x - 2 \]
Given the particular solution:
\[ w = e^{-x}(C \cos 2x + E \sin 2x) - x^2 \]
and the general solution:
\[ w = e^{-x}(C \cos 2x + E \sin 2x) \]

Answer: -\dfrac{1}{x^2}


Question: Find the coefficient of $x^{12}$ in the expansion of the expression: $(x^5+x^6+x^7+	ext{...})^2 \cdot (1+x+x^2+\text{...})^8$.

Answer: 55


Question: Do solutions exist for a 6x6 double Latin square, where each number is a two-digit number with digits ranging from 1 to 6, and each row and column contains each digit exactly once in both the most- and least-significant positions?

Answer: Yes


Question: Given the integer function \( f(z) = u(x,y) + iv(x, y) \) where \( u(x, y) = e^{-x}\sin(y -1) + y \), find the value of \( f'(i) \).

Answer: -2i


Question: Determine whether every open set \( G \subset \mathbb{R}^n \) with finite Lebesgue measure can be approximated by a finite union of non-degenerate compact intervals contained in \( G \). Specifically, for every open set \( G \subset \mathbb{R}^n \) and \( \epsilon > 0 \), does there exist a finite collection of compact intervals \( \{ I_j \mid j=1,2,...,m \} \) in \( \mathbb{R}^n \) such that \( \bigcup_{j=1}^{m} I_j \subset G \) and \( m(G \setminus \bigcup_{j=1}^{m} I_j) < \epsilon \), where \( m \) denotes the Lebesgue measure? Note: A compact interval in \( \mathbb{R}^n \) is of the form \([a_1,b_1]\times[a_2,b_2]\times...\times[a_n, b_n]\), where \([a_i,b_i]\) are real closed intervals for each \( i \).

Answer: Yes


Question: Find all matrices in the group \(\operatorname{GL}_2(\Bbb{F}_5)\) that stabilize the vector \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) under left multiplication. Determine the order of this stabilizer.

Answer: 20


Question: A ball is passed among 10 people seated in a circle, with chairs numbered 0 to 9 in a clockwise direction. Initially, the ball is with the person on chair 0. In the first step, the ball is passed 1 turn (\(1^1\)) to chair 1. In the second step, it is passed 4 turns (\(2^2\)) to chair 5. In the third step, it is passed 27 turns (\(3^3\)) to chair 2, and so on. After 2020 steps, on which chair will the ball be? Provide your answer as the chair number.

Answer: 4


Question:
Evaluate the integral \( \oint_\gamma \frac{dz}{(z-3)(z^5-1)} \), where \( \gamma \) is the circle of radius 2 centered at the origin in the complex plane, oriented in the anti-clockwise direction. Choose the correct answer from the following options:

(a) \( \frac{2\pi i}{3^5-1} \)
(b) \( \frac{2\pi i}{3^4-1} \)
(c) \( -\frac{2\pi i}{3^5-1} \)
(d) 0

Answer: C


Question: Evaluate the line integral \( \int^{(8,3,2)}_{(0,0,0)}(2xy^2-2xz^2)dx+2x^2ydy-2x^2zdz \).

Answer: 320


Question: Evaluate the limit of the infinite product: $$\lim_{n \to \infty} \sqrt[2]{2} \cdot \sqrt[4]{4} \cdot \sqrt[8]{8} \cdot \dots \cdot \sqrt[2^n]{2^n}.$$

Answer: 4


Question: Let \( R = (3+\sqrt{5})^{2n} \). If \( f \) is the fractional part of \( R \), find the value of \( R(1-f) \).

Answer: 16^n


Question:
For a subset $S$ of a topological space $X$, let $\operatorname{cl}(S)$ denote the closure of $S$ in $X$, and let $S' = \{x : x \in \operatorname{cl}(S \setminus \{x\})\}$ denote the derived set of $S$. If $A$ and $B$ are subsets of $X$, which of the following statements are true?

1. $(A \cup B)' = A' \cup B'$
2. $(A \cap B)' = A' \cap B'$
3. If $A'$ is empty, then $A$ is closed in $X$.
4. If $A$ is open in $X$, then $A'$ is not empty.

Choose the correct option:

(A) $(1)$ and $(2)$ only
(B) $(1)$ and $(3)$ only
(C) $(2)$ and $(4)$ only
(D) $(1)$, $(2)$, and $(3)$ only
(E) $(1)$, $(2)$, $(3)$, and $(4)$

Answer: B


Question:
Let \( n \) be a positive integer and \( x_1, \dots, x_n, y_1, \dots, y_n \in [0,1] \). Determine the smallest \( r \) in terms of \( n \) such that the indices \( 1, 2, \dots, n \) can always be divided into two groups \( A \) and \( B \) satisfying:
\[ \left| \sum_{i \in A} x_i - \sum_{i \in B} x_i \right| \le r \quad \text{and} \quad \left| \sum_{i \in A} y_i - \sum_{i \in B} y_i \right| \le r. \]

Answer: 1


Question: Determine whether the limit \( \lim_{k\to\infty}\prod_{m=2^k}^{m=2^{k+1}} {\frac{2m}{2m-1}} \) exists.

Answer: \sqrt{2}


Question: For how many distinct congruence classes \([a]\) modulo 631 are there integer solutions \(x\) to the congruence \(x^2 \equiv a \pmod{631}\)?

Answer: 316


Question: Determine whether there exists a continuous surjective map from \( \mathbb{R}^2 \) to \( S^1 \).

Answer: Yes


Question: Let $(x_k)$ be a sequence of real numbers defined by $x_1 = 1$ and $x_{k+1} = x_k + \sqrt{x_k}$ for $k > 0$. Determine the value of \( \lim_{k \to \infty} \frac{x_k}{k^2} \).

Answer: \dfrac{1}{4}


Question: Suppose \( A_k > 0 \) are positive definite square \( n \times n \) matrices with \( n > 1 \). If \( \sum_{k=1}^\infty A_k \) converges, does \( \sum_{k=1}^\infty \|A_k\| < +\infty \) also converge, where \( \|A\| = \sup_{\|x\| \leq 1}\langle Ax,x\rangle \)? Determine if this statement is true.

Answer: True


Question: Let $f$ be a differentiable function such that $f(f(x)) = x$ for all $x \in [0,1]$. Suppose $f(0) = 1$. Determine the value of \( \int_0^1 (x - f(x))^{2016} \, dx \).

Answer: \dfrac{1}{2017}


Question: Let $f: X \to Y$ be a morphism of varieties over a field $k$, such that $X(\overline{k}) \to Y(\overline{k})$ is bijective. Is $f$ necessarily an affine morphism? Provide a justification for your answer.

Answer: Yes


Question: Is the following a valid definition of the set of rational numbers? $$\mathbb{Q}=\left\{\sum_{n=1}^k f(n)\mid k,n\in\mathbb{N}\land f\text{ is a finite composition of } +, -, \div, \times\right\}$$

Answer: \mathbb{Q}


Question: Find the sum to infinity of the series \( \sum_{k=0}^{\infty}\frac{(k+1)(k+3)(-1)^k}{3^k} \).

Answer: \dfrac{45}{32}


Question: Solve for \(x\) in the equation \(x\ln(1+\frac{b}{x})=b\), where \(b>0\).

Answer: \infty


Question: Does there exist a topological space that is also a linear space such that its vector addition is discontinuous but scalar multiplication is continuous?

Answer: Yes


Question: Does there exist a sequence of non-constant polynomials \(\{p_{n}\}_{n \geq 1}\), each \(p_n\) in \(\mathbb{Z}[X]\), such that \( \lim_{n \to \infty} p_n = c \) pointwise on \([0,1]\), where \(c\) is a constant?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{t \rightarrow 0} \int_{0}^{t} \frac{\sqrt{1+\sin(x^2)}}{\sin t} \, dx \]

Answer: 1


Question: Suppose two non-empty sets \( A \) and \( B \) such that \( A \subset B \) and \( A \neq B \). Is it possible for \( |A| = |B| \) to hold true?

Answer: Yes


Question:
Evaluate the integral \( \frac{\pi^2}{10} \int_{-10}^{10} f(x) \cos{\pi x} \, dx \), where \( f(x) \) is defined on the interval \([-10,10]\) by
\[
f(x)=
\begin{cases}
x-\lfloor{x}\rfloor & \text{if} \ \lfloor{x}\rfloor \text{ is odd} \\
1+\lfloor{x}\rfloor-x & \text{if} \ \lfloor{x}\rfloor \text{ is even}
\end{cases}
\]

Answer: 4


Question: Let $C$ be a smooth projective plane curve, and suppose $f: C' \rightarrow C$ is a double cover of $C$ ramified at $d$ points. If $C'$ is irreducible, the Riemann-Hurwitz formula implies that $d$ must be even. Is it true that if $d$ is odd, then the double cover $C'$ is reducible?

Answer: Yes


Question: Evaluate the double integral \( \iint_D \frac{1}{y} \, dy \, dx \), where \( D \) is the region bounded by \( y = x^2 \), \( y = \frac{1}{2}x^2 \), and \( y = 6x \). Use the change of variables \( x = uv \) and \( y = u^2 \) to convert the integral into a proper form. Calculate the Jacobian and determine the new limits of integration to solve the integral.

Answer: 6


Question: Calculate the probability that the sum of four independent random variables, each uniformly distributed over the interval [0, 1], is at least 3. That is, find \( P(X_1 + X_2 + X_3 + X_4 \geq 3) \) where \( X_1, X_2, X_3, X_4 \sim U(0,1) \).

Answer: \dfrac{1}{24}


Question: Let $f$ be a holomorphic function on the unit disk, continuous on its closure, such that $|f(z)| = 1$ for $|z| = 1$, $f(-1) = 1$, and $f$ has no zeros. Determine the function $f$. \( \text{(Hint: Consider the properties of holomorphic functions and the maximum modulus principle.)} \)

Answer: 1


Question: Find the limit of the sequence \( x_n = \left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2 \) as \( n \to \infty \).

Answer: \dfrac{1}{9}


Question:
Consider two independent and identically distributed random variables $X_1$ and $X_2$ with a normal distribution $N(\theta, 1)$. Additionally, let $V$ be a Bernoulli random variable with $P(V=1) = \frac{1}{4}$, independent of $X_1$ and $X_2$. Define $X_3$ as follows:
- $X_3 = X_1$, if $V=0$
- $X_3 = X_2$, if $V=1$

We are testing the hypothesis $H_0: \theta = 0$ versus $H_1: \theta = 1$. Reject $H_0$ if $\frac{X_1 + X_2 + X_3}{3} > C$. Find the critical value $C$ such that the test size is 0.05.

Answer: 1.226


Question: Compute the integral \( \int_A f \) where \( A = \{(x,y) : 0 \le x + y \le 1, 0 \le x + 3y \le 3\} \subseteq \mathbb{R}^2 \) and \( f(x,y) = \sqrt{x+y} \).

Answer: 1


Question: Determine which is larger: \(i^i\) or \(\pi\).

Answer: \pi


Question: Find the integer part of the maximum value of the expression \(x^4y + x^3y + x^2y + xy + xy^2 + xy^3 + xy^4\) given that \(x + y = 3\). Let \(M\) be this maximum value. Determine \(\lfloor M \rfloor\).

Answer: 36


Question: Is it always possible to color the vertices of any graph such that at least \( \frac{2}{3} \) of the edges touch a blue vertex and at least \( \frac{2}{3} \) of the edges touch a red vertex?

Answer: Yes


Question: Consider the symmetric matrix \(G = \begin{pmatrix} 1 & -3 & 2 \\ -3 & 9 & -6 \\ 2 & -6 & 4 \end{pmatrix}\). Find the pair \((a, b)\) such that \(G\) is similar to the matrix \(H_{(a,b)} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & b & a \end{pmatrix}\).

Answer: (14, 0)


Question: Calculate the probability \( P\left(\bigcup_{i=1}^\infty \{U_i \in [0,x]\}\right) \) where \((U_i)_{i\in\mathbb{N}}\) are independent and identically distributed random variables with \(U_1 \sim U[0,1]\) and \(x \in (0,1]\).

Answer: 1


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

Answer: \alpha > 1


Question: Calculate the value of the infinite series: \[ \sum_{n=0}^{\infty} \frac{(2n+1)(n+1)}{3^n} \]

Answer: \dfrac{27}{4}


Question: Determine the cardinality of the set of all subsets \( A \subset \mathbb{R} \) such that \( \mathbb{R} \setminus A \) is countably infinite.

Answer: \mathfrak{c}