Question: Does the definition of asymptotic normality imply that there exists an integer $n_0$ such that for all $n > n_0$, the random variables $X_n$ are normally distributed with parameters $(\mu_n, \sigma_n^2)$?

Answer: No


Question: Given that the cubic equation $x^3 - 3x + 1 = 0$ has three real roots $x_1, x_2, x_3$, find the value of the expression $\{x_1\} + \{x_2\} + \{x_3\}$, where $\{x\}$ denotes the fractional part of $x$. Note that the sum of the roots $x_1 + x_2 + x_3 = 0$.

Answer: 1


Question: Determine the number of $k$-tuples of subsets $(S_1, S_2, \ldots, S_k)$ of a set $[n] = \{1, 2, \ldots, n\}$ such that $S_1 \cap S_2 \cap \cdots \cap S_k = \emptyset$. Provide your answer in terms of $n$ and $k$.

Answer: (2^k - 1)^n


Question: Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with the property that there exists $c \in (0,1)$ such that for all $x,y \in \mathbb{R}^n$, $\left\| f(x) - f(y) \right\| \leq c \left\| x-y \right\|$. From Banach's fixed-point theorem, for any $x^{(0)} = x_0 \in \mathbb{R}^n$, the sequence $x^{(k+1)} := f(x^{(k)})$ converges to the unique fixed-point $\bar{x}$ of $f$. Now consider the sequence $y^{(0)} = x_0$ and $y^{(k)} := f\left( \frac{1}{k} \sum_{i=0}^{k-1} y^{(i)} \right)$. Determine the limit $\lim_{k \rightarrow \infty} y^{(k)}$. Does it converge to the fixed point $\bar{x}$?

Answer: \bar{x}


Question: Evaluate the contour integral \( \oint_{|z|=1} z(z+1) \cosh\left(\frac{1}{z}\right) \, dz \) where the contour is the unit circle \(|z|=1\) oriented counterclockwise.

Answer: \pi i


Question: Let $G$ be a group of infinite order. Does there exist an element $x \in G$ such that $x \neq e$ and the order of $x$ is finite?

Answer: No


Question: Evaluate the integral \( I = \int_\gamma \frac{1}{z^2} \, dz \), where \( \gamma \) is the upper half of the unit circle in the complex plane.

Answer: 2


Question: Find the coefficient of \(x^{24}\) in the expansion of the generating function \(f(x) = (x^3 + x^4 + x^5 + x^6 + x^7 + x^8)^4\).

Answer: 125


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

Answer: e


Question: If $n$ is square-free and $n+1$ divides the sum of divisors of $n$, $\sigma(n)$, is $n$ necessarily a prime number? Determine whether these conditions are sufficient to conclude that $n$ is prime.

Answer: Yes


Question: Find the minimum value of the expression \(\sum_{cyc} \sqrt{\frac{a}{2(b+c)}}\) given that \(a, b, c > 0\).

Answer: \sqrt{2}


Question: Evaluate the double integral: \[ \int _0^{\frac{1}{4}}\int _{\sqrt{x}}^{\frac{1}{2}}\:\frac{\cos\left(\pi y\right)}{y^2}~\mathrm dy~\mathrm dx \]

Answer: \dfrac{1}{\pi}


Question: Suppose two functions $f(x)$ and $g(x)$ are both real analytic on an interval $(a,b)$, where $-\infty<a<b<\infty$. There exists a strictly increasing sequence of real values $\{x_{n}\}$ in $(a,b)$ such that $\lim_{n\rightarrow\infty}x_{n}=c\in (a,b)$ and $f(x_{n})=g(x_{n})$ for all $n$. Does there exist an interval $(a^{*},b^{*})\subseteq (a,b)$ such that $f(x)=g(x)$ for all $x\in (a^*,b^*)$?

Answer: Yes


Question: Consider a circle $\mathbb{T}$ and an irrational rotation $f_{\theta} : \mathbb{T} \to \mathbb{T}$ defined by $x \mapsto x+\theta$. Define $f^0_\theta(x)=x$ and $f^{n+1}_\theta = f^n_\theta \circ f_\theta$. Let $F = \{ f^n_\theta : n \in \mathbb{N} \}$ and $F' = \{ f_\theta \circ x : x \in F \}$. Determine if $F = F'$.

Answer: F \neq F'


Question: Determine whether there exists an order-complete field.

Answer: \mathbb{R}


Question: Find the limit of the series \( \lim _{ n\rightarrow \infty }{ \sum _{ k=1 }^{ n }{ { k }^{ \left( \frac { 1 }{ 2k } \right) } } } \). Does the series converge, and if so, what is its limit?

Answer: \infty


Question: Evaluate the limit \( \lim\limits_{x\to 0} \frac{x^3\sin(x)}{[\ln(1+x)-x]^2} \) without using L'Hopital's Rule or series expansions.

Answer: 4


Question: Determine whether the following statement is true or false: An immersion $f:X \rightarrow Y$ in the context of smooth manifolds is an embedding if and only if for every point $y$ in $Y$, there exist arbitrarily small open neighborhoods of $y$ in $Y$ such that $f^{-1}(U)$ is connected.

Answer: True


Question:
Determine whether the function \( f(x,y) = x^6 - 2x^2y - x^4y + 2y^2 \) has a saddle point at the origin or if the origin is not a critical point of \( f \). Choose the correct statement:

1) \( f \) has a saddle point at the origin.
2) The origin is not a critical point of \( f \).

Answer: 1


Question: Consider the points \(A=(0,1)\), \(B=(0,0)\), and \(C=(1,0)\). A random point \((X,Y)\) is chosen on the union of the line segments \(\overline{AB}\) and \(\overline{BC}\). Find the covariance \(\text{Cov}(X,Y)\) of the coordinates of this random point.

Answer: -\dfrac{1}{16}


Question: Evaluate the integral \( \int_{0}^{1} x^{-1/3}\cos(3\arccos(x))\,dx \).

Answer: -\dfrac{39}{55}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{\ln(2^{1/n}) - \ln(n^2)}{1 + 1/2 + 1/3 + \ldots + 1/n} \] without using L'Hôpital's rule, derivatives, or integrals.

Answer: -2


Question: Determine whether every second countable $T_2$ topological space is a developable space.

Answer: Yes


Question: In a metric space, is it true that for any points $x$ and $y$, the equation $d(x,y) = d(B(x,r),y) + r$ holds, where $B(x,r)$ is a ball centered at $x$ with radius $r$, and $d$ is the distance function?

Answer: No


Question: Let $\mathbb{F}$ be $\mathbb{Z}_7$. Consider the matrix $A=\begin{bmatrix} 2 & 5\\ 5 & 3 \end{bmatrix} \in M_{2\times 2}(\mathbb{F})$. Define the subspace $W=\{B \in M_{2\times 2}(\mathbb{F}) \mid (AB)^t = AB\}$. Find a basis and the dimension of the subspace $W$. \(\text{Note: } M_{2\times 2}(\mathbb{F}) \text{ denotes the space of } 2\times 2 \text{ matrices over } \mathbb{F}.\)

Answer: 3


Question: Find the maximum value \( \alpha \) for which the integral \( \int_{0}^{\frac{1}{\pi}} \frac{1}{x^\alpha} \sin{\frac{1}{x}} \, dx \) converges.

Answer: 2


Question: Can the extension $L$ be considered as a vector space over the local field $K_\frak{p}$?

Answer: No


Question: Calculate the flux integral of the vector field \( F=(\cos xyz, \tan xyz, 1+\arctan xyz) \) across the surface defined by \( x^2+y^2=1 \), \( 0\leq z \leq2 \), with the surface oriented by normal vectors pointing away from the z-axis.

Answer: 0


Question: Let $X(t)$ be a stochastic process defined by $X(t) = V + 2t$ for $t \ge 0$, where $V$ follows a Cauchy distribution with parameters $x_0 = 0$ and $\gamma = 1$. Determine the probability that $X(t) = 0$ for at least one $t \in (0, \frac{1}{2}]$. Provide your answer as a single probability value.

Answer: \dfrac{1}{4}


Question: Evaluate the limit: \[ \lim_{x \to \infty} x^2 \left( a^{1/x} - a^{1/(x+1)} \right) \] for \( a > 0 \).

Answer: \ln a


Question: In triangle ABC, point P lies on side AB such that the ratio AP:PB is 1:3. The median from vertex A intersects line segment CP at point Q and side BC at point D. Determine the ratio AQ:QD.

Answer: 2:3


Question: Find the value of \( \frac{m}{n} \) such that \( T = \sin 5^\circ + \sin 10^\circ + \sin 15^\circ + \cdots + \sin 175^\circ = \tan \frac{m}{n} \), where \( m \) and \( n \) are positive integers.

Answer: \dfrac{175}{2}


Question: Solve the differential equation: \[ xx'' = (x')^2, \] where \( x' \neq 0. \)

Answer: x(t) = C_1 e^{C_2 t}


Question: Determine whether the following statement is true or false: Let $f \in L^1(\mathbb{R})$ and $g \in L^1(\mathbb{R}) \cap L^\infty(\mathbb{R})$. If $f \ast g$ is continuous and $f \ast g \in L^1(\mathbb{R})$, then $\lim_{x \to \infty} f \ast g (x) = 0$. Is this statement true or false?

Answer: True


Question: Find the value of \( \lfloor (\sqrt{3} + 1)^5 \rfloor \).

Answer: 152


Question: Find a compactification $K(\mathbb{N})$ for the space $X = \mathbb{N}$ such that the remainder $E(X) = K(\mathbb{N}) - \mathbb{N}$ consists of exactly three isolated points.

Answer: \mathbb{N} \cup \{a, b, c\}


Question: Evaluate the limit: \[ \lim\limits_{x \to 0} \left(x \cdot \left\lfloor \frac {1}{x} \right\rfloor \right) \] where \( \left\lfloor \frac {1}{x} \right\rfloor \) denotes the integer part of \( \frac {1}{x} \).

Answer: 1


Question: Let \((M,g)\) be a Riemannian manifold with \(\nabla\) as its Levi-Civita connection. If the curvature \(R\) is zero, is \(\nabla\) locally trivial, meaning do the Christoffel symbols equal zero?

Answer: Yes


Question: If a Banach space $X$ is isometric to its first dual $X^*$, does it follow that $X$ must be reflexive? Justify your answer.

Answer: Yes


Question: Find the value of \( a \) such that the limit \( \lim _{ x\to 0 } \frac { a-\sqrt { { a }^{ 2 }-{ x }^{ 2 } }-{ x }^{ 2 }/4 }{ { x }^{ 4 } } \) is finite.

Answer: 2


Question: Determine whether the sequence \( t_n = \frac{1}{n} \left( 1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \right) \) is convergent.

Answer: 0


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

Answer: 4


Question: Determine the domain of the solution to the differential equation \( y' = \frac{y + 2x^3 \sinh x}{x} \) with the initial condition \( y(-1) = 0 \).

Answer: (-\infty, 0)


Question: Compute the line integral \( \int_C \textbf{F} \cdot \textbf{ds} \), where \( \textbf{F}(x,y) = \frac{2}{x^2+y^2}(y\textbf{i}-x\textbf{j}) \) and \( C \) is the curve given by \( 3x^{8}+2y^{6}=1 \), oriented counter-clockwise. Discuss whether Green's theorem can be applied given that \( \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} = \frac{2(x^2-y^2)}{(x^2+y^2)^2} = \frac{2\cos2\theta}{r^2} \) and the potential discontinuity at \((0,0)\).

Answer: -4\pi


Question: Find the residue of \(1^5 + 2^5 + \cdots + 1080^5\) when divided by 14 using congruences.

Answer: 12


Question: Let \( f(x) = 1 + \sum_{k=0}^{100}\frac{(-1)^{k+1}}{(k+1)!}x(x-1)\cdots (x-k) \). If \( S \) is the sum of all real roots of \( f \) and \( T \) is the sum of all real roots of \( f' \), find \( S-T \).

Answer: 51


Question: Consider a regular $9$-gon with symmetry group $D_9$. There are three distinct equilateral triangles that can be formed using the nine vertices. Identify the subgroup of $D_9$ that acts trivially on the set of these three triangles.

Answer: C_3


Question: Find the minimum value of the expression \( P = 4(a^3 + b^3 + c^3) + 15abc \) given that \( a + b + c = 2 \) and \( a, b, c \) are the sides of a triangle.

Answer: 8


Question: Given the polynomial \( P(x) \) with real coefficients satisfying the equation \( P(P(x)-2) = 4P(x) + 3 \), find the value of \( P(-1) \).

Answer: 7


Question:
Determine the number of matrices \( A \) in \( \mathrm{Mat}_{M,N}(\mathbb{Z}/2\mathbb{Z}) \) such that the sum of the entries in each row and each column is zero. Specifically, find the cardinality of the set:
\[
\left\{A\in\mathrm{Mat}_{M,N}(\mathbb{Z}/2\mathbb{Z}): \forall j\in\{1,\ldots,N\}\quad \sum\limits_{k=1}^M A_{kj}=0,\quad \forall i\in\{1,\ldots,M\}\quad \sum\limits_{l=1}^N A_{il}=0  \right\}
\]

Answer: 2^{(M-1)(N-1)}


Question: Calculate the outward flux of the vector field \( \mathbf{v} = \begin{pmatrix} x^2yz \\ xy^2z \\ z \end{pmatrix} \) through the surface \( M \) defined by \( M = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = z, \; 0 \leq z \leq 1 \} \) using Gauss's theorem.

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


Question: Consider a 2-dimensional pseudo-Riemannian manifold $(M,g)$ that is topologically a disc. Determine whether it is possible to construct a global coordinate system $(t,x)$ covering the entire manifold such that the metric is conformally flat, i.e., the line element is given by \( ds^2 = \Omega^2(t,x)(-dt^2 + dx^2) \) for some conformal factor \( \Omega \).

Answer: Yes


Question: Find a non-zero function \( f(x) \) such that \( f(2x) = f'(x) \cdot f''(x) \), where \( f' \) and \( f'' \) are the first and second derivatives of \( f \), respectively.

Answer: e^x


Question: Let $\mathbb{K}$ be a field of characteristic 0. Consider $\mathrm{A}$ and $\mathrm{B}$, which are conilpotent cocommutative coaugmented counital dg-coalgebras over $\mathbb{K}$. Is the tensor product $\mathrm{A} \otimes_{ \mathbb{K} } \mathrm{B}$ also conilpotent as a coaugmented counital dg-coalgebra?

Answer: Yes


Question:
Determine the constant \( B \) in the inverse Fourier transform formula given by:
\[ f(x) = B \int_{-\infty}^{\infty} \hat{f}(k) e^{+i2\pi kx} dk \]
where the Fourier transform is defined as:
\[ \hat{f}(k) = \frac{1}{2} \int_{-\infty}^{\infty} f(x) e^{-i2\pi kx} dx \]
Use the condition that the inverse Fourier transform of the Fourier transform of \( f(x) \) should return \( f(x) \), i.e.,
\[ \mathcal{F}^{-1}\mathcal{F}(f)(t) = f(t) \]
and the property of the Dirac delta function:
\[ \delta(t-x) = \frac{B}{2} \int_{-\infty}^{\infty} e^{i2\pi(t-x)k} dk \]
to find \( B \).

Answer: 2


Question: Consider a function $u$ in $C^{0,1}(\overline{B(0,1)})$ such that $u(0) = 0$ and $[u]_{C^{0,1}(B(0,1))} \leq M$ for some $M > 0$. Define $u^r(x) = r^{-1}u(rx)$ for $0 < r \leq 1$. It is known that $[u^r]_{C^{0,1}(B(0,1))} \leq M$ and $u^r(0) = 0$. Using the Ascoli-Arzelà lemma, a subsequence $\{r_j\}$ can be chosen such that $r_j \to 0$ as $j \to \infty$ and $u^{r_j} \to u^0$ uniformly in $C^0(\overline{B(0,1)})$. Is the limit $u^0$ unique?

Answer: No


Question:
Given the alphabet \(\Sigma = \{a, b, c, d, e\}\) with the encoding scheme \(g(a) = 3, g(b) = 5, g(c) = 7, g(d) = 9, g(e) = 11\), and the i-th prime number \(p_i\) where \(p_1 = 2\), for a non-empty string \(s = a_1 \dots a_n\), define \(f(s) = \prod_{i=1}^{n} p_i^{g(a_i)}\). For a non-empty sequence \(\langle s_1, \dots, s_n \rangle\) of strings from \(\Sigma^+\), define \(h(\langle s_1, \dots, s_n \rangle) = \prod_{i=1}^{n} p_i^{f(s_i)}\). Which of the following numbers is the encoding \(h\) of a non-empty sequence of strings?

- \(2^7 3^7 5^7\)
- \(2^8 3^8 5^8\)
- \(2^9 3^9 5^9\)
- \(2^{10} 3^{10} 5^{10}\)

Answer: B


Question: Find the expression for \( E(Z^2 | Z \leq 0) \) where \( Z \sim N(0,1) \).

Answer: 1


Question: Given the generating function \( P(x) = \sum_{n=0}^{\infty} p_n x^n = 1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + 11x^6 + \cdots \), which represents the partition generating function, and \( Q(x) = \sum_{n=0}^{\infty} q_n x^n \), where \( q_n \) is the number of partitions of \( n \) containing no 1s, find the polynomial \( \frac{Q(x)}{P(x)} \).

Answer: 1 - x


Question: Consider a compact complex variety $X$ and a $0$-dimensional analytic subset $A$ of $X$. Can $A$ have infinitely many points?

Answer: No


Question: Let \(\{ f_n \}_{n\in \mathbb{N}}\) be a sequence of biholomorphisms on a bounded region \(\Omega\). If this sequence converges to \(f\) uniformly on every compact subset of \(\Omega\), does it follow that \(\{ f_n^{-1} \}_{n\in \mathbb{N}}\) converges uniformly to \(f^{-1}\), if defined?

Answer: Yes


Question: Consider the function \( f(x,y) = \begin{cases} xy\frac{x^2-y^2}{x^2+y^2} & \text{for } (x,y) \neq (0,0) \\ 0 & \text{for } (x,y) = (0,0) \end{cases} \). Suppose that the second partial derivatives exist at \((a,b)\) and \(\frac{\partial^2f}{\partial x\,\partial y}(a,b) \neq \frac{\partial^2f}{\partial y\,\partial x}(a,b)\). Can the function \( f \) be twice differentiable at \((a,b)\)?

Answer: No


Question: Determine the number of roots of the equation \(4\cos(e^x) = 2^x + 2^{-x}\).

Answer: 4


Question: Calculate the dimension of the vector space \( \dim_k (m^2/m^3) \) where \( k \) is a field and \( m = \langle (x,y) \rangle \) is the maximal ideal in the polynomial ring \( k[x,y] \).

Answer: 3


Question: Consider the region $T = \{(x, y) \in \mathbb{R}^{2}: x \geq 0, y \geq 0\}$. Let $f$ be a continuous function that vanishes outside the disk of radius $R$, i.e., $f(x, y) = 0$ for $x^2 + y^2 > R$. Suppose $u$ is a bounded solution to the equation $\Delta u = f$ in $T$ with the boundary conditions $u(x, 0) = 0$ and $u_x(0, y) = 0$. Determine whether $u \to 0$ as $x^2 + y^2 \to \infty$. Provide a justification for your answer.

Answer: Yes


Question: Assume that $(a_k)$ is a sequence of real numbers such that \( \lim_{n\to \infty}\sum_{i=0}^n a_i\cos(n\pi b_i) = 0 \) for some sequence $(b_k)$ with $1=b_0>b_1>b_2>\dots>0$. Does this imply that all $a_k$ are equal to 0?

Answer: Yes


Question: Determine the radius of convergence for the series \(\sum_{k=0}^{\infty} \frac{-(1)^k (4^k -3)x^{2k}}{k^4+3}\) using the ratio test.

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \sum_{k=1}^{\infty} \frac{\cos{(k^2x)}}{k^p} \, dx \), where \( p > 1 \).

Answer: 0


Question: Prove using the definition of limit that \(\lim_{(x,y)\to(0,0)}\frac{(1-\cos(xy))\sin y}{x^2+y^2} = 0\).

Answer: 0


Question: Determine the value of \( c \) such that the function \( c(2 - |x| - |y|) \) is a probability distribution function. Use the formula for the volume of a pyramid to assist in your calculation.

Answer: \dfrac{3}{16}


Question: Compute the value of the function defined by the following recursive relations: \( f(x,0)=f(x-1,1) \), \( f(0,y) = (y+1) \mod 5 \), and \( f(x,y) = f(x-1, f(x,y-1)) \). Find \( f(333,3) \).

Answer: 1


Question: Let $A$ be an $n \times n$ matrix. Determine whether there exists a diagonal matrix $D$ such that $A + D$ has $n$ distinct eigenvalues.

Answer: Yes


Question: Find the minimum value of \((5+x)(5+y)\) given that \(x^2 + xy + y^2 = 3\) and \(x, y \in \mathbb{R}\).

Answer: 16


Question: Does there always exist an integer \(a\) such that \(a^{p-1} \equiv 1 \pmod{p}\) and \(p-1 = \text{ord}_{p}(a)\) for a prime \(p\)?

Answer: Yes


Question: Given sets A and B where |A| = 7 and |B| = 10, determine the number of onto functions from set B to set A.

Answer: 29635200


Question: Determine the value of \( n \) for which the integral \( I = \int_{n}^{n^2} \frac{1}{x+\sqrt{x}} \, dx \) attains its minimum value.

Answer: 3 - 2\sqrt{2}


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

Answer: e^4


Question: Evaluate the contour integral \( \oint_C \frac{z}{(z+1)(z+3)} \, dz \), where \( C \) is the rectangle with vertices at \( 2 \pm i \) and \( -2 \pm i \).

Answer: -\pi i


Question: Find the maximum integer value of \(x\) such that \(3x^2 - xy - 2x - 5y + 7 = 0\) for integers \(x\) and \(y\).

Answer: 87


Question: Determine the values of \(a\) for which the integral \(\int _0^{\infty }\:\frac{\sin\left(x\right)}{\left(1+x\right)^a}\,dx\) converges.

Answer: a > 0


Question:
Compute the norm of the linear operator \( T \) on the Hilbert space \( \ell^2(\mathbb{Z}) \), defined by:
\[ T(...,x_{-1},x_0,x_1,...)_n = (...,\frac{x_{-2}+x_{0}}{2},\frac{x_{-1}+x_{1}}{2},\frac{x_{0}+x_{2}}{2},...) \]
where \( x_n \rightarrow \frac{x_{n-1} + x_{n+1}}{2} \).

Answer: 1


Question: In $\triangle{ABC}$, the coordinates of point $A$ are $(1,2)$. The line $y=x$ is the perpendicular bisector of $AB$, and the line $x-2y+1=0$ is the angle bisector of $\angle{BCA}$. The equation of line $BC$ is $ax+by-5=0$. Find the value of $a+b$. \(\text{Express your answer as a single number.}\)

Answer: 2


Question: Consider a symplectic manifold $(\mathcal{M}, \omega)$. Determine whether any open subset of $(\mathcal{M}, \omega)$ is a symplectic submanifold.

Answer: Yes


Question: Determine whether the space of $E$-spectral functions of $L^p(G)$, where $G$ is an infinite compact abelian group, $E \subseteq \Gamma$ with $\Gamma$ as the dual group of $G$, and $1<p<\infty$, is a Banach subspace of $L^p(G)$. The space is defined as $L_E^p(G) = \{f \in L^p(G) : \widehat{f}(\gamma) = 0, \forall \gamma \notin E\}$. Is $L_E^p(G)$ a complete space?

Answer: Yes


Question: Determine whether there exists a $T_2$ (Hausdorff) space $X$ and a group $G$ such that the quotient space $X_G$, formed by the action of $G$ on $X$, is not a $T_2$ space.

Answer: Yes


Question: A car starts at a position \( b \), where \( b \) is normally distributed as \( N(0, 6) \). The car then drives a distance \( d \), which is normally distributed as \( N(0, 5) \). If \( d \) is positive, the car moves up; otherwise, it moves down. Given that \( b \) and \( d \) are statistically independent and the car's final position is \( c = 6 \), find the most likely initial position \( b \) of the car.

Answer: \dfrac{36}{11}


Question:
Given that \( n > 3 \), and \( a_1 = a_2 = \cdots = a_{n-1} = 1 \) and \( a_n = -2 \), find the value of the expression:
\[
f_n = \sum_{i=1}^{n}|a_{i}| - \sum_{1 \le i < j \le n}|a_{i} + a_{j}| + \sum_{1 \le i < j < k \le n}|a_{i} + a_{j} + a_{k}| - \cdots + (-1)^{n-1}|a_{1} + a_{2} + \cdots + a_{n}|
\]

Answer: 6 - 2n


Question: Determine whether there exists an invertible matrix $P$ such that for the quadratic form $Q(v) = v'Av$, where $v = \begin{pmatrix} x & y & z & w \end{pmatrix}$ and $A = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$, the expression $Q(Pv)$ equals $x^2 + y^2 - zw$. Assume all terms are in $\mathbb{R}$. Provide a justification for your answer.

Answer: Yes


Question: There are 2011 positive numbers such that both their sum and the sum of their reciprocals equal 2012. Let \(x\) be one of these numbers. Determine the maximum value of \(x + \frac{1}{x}.\)

Answer: \dfrac{8045}{2012}


Question: Calculate the number of solutions to the equation \( x_1 + x_2 + x_3 + x_4 = 15 \) subject to the constraints \( 0 \le x_i < i + 4 \) for each \( i = 1, 2, 3, 4 \).

Answer: 105


Question: Find the point of intersection of two tangents to the curve \(x^2 + 2y^2 + xy + x + y = 10\) that have slopes \(m_1\) and \(m_2\) and pass through the point \((5, 1)\).

Answer: (5, 1)


Question: Evaluate the infinite series \( \sum \limits_{k=1}^{\infty }\frac{\zeta(2k)-1}{k} \), where \( \zeta(s) \) is the Riemann zeta function.

Answer: \ln 2


Question: In triangle $ABC$, points $D$, $E$, $F$, and $G$ are located such that $D$ is on $AC$, $E$ is on $AB$, $F$ is on $BD$, and $G$ is on $CE$. Given that $\frac{BF}{BD} = x$ and $\frac{CG}{CE} = 1-x$, where $0 < x < 1$, determine the ratio of the area of triangle $AFG$ to the area of quadrilateral $BCDE$. Assume $F \neq G$. Find this ratio.

Answer: x(1 - x)


Question: Given that \( f\left(x-\frac{2}{x}\right) = \sqrt{x-1} \), find the value of \( f'(1) \).

Answer: \dfrac{1}{3}


Question: For any integer $n$, define $k(n) = \frac{n^7}{7} + \frac{n^3}{3} + \frac{11n}{21} + 1$. Let $f(n) = 0$ if $k(n)$ is an integer, and $f(n) = \frac{1}{n^2}$ if $k(n)$ is not an integer. Calculate the sum $\sum_{n = -\infty}^{\infty} f(n)$. Provide your answer as a single number.

Answer: 0


Question: Given that the line \( y = -2x + 1 \) is tangent to the curve \( f(x, y) = 2 \) at the point \( (1, 3) \), and \( f_x(1, 3) = 4 \), find the maximum directional derivative of \( f(x, y) \) in the direction of the vector \( (1, 3) \).

Answer: \sqrt{10}


Question: Estimate the error of approximation for \( \ln(1+x) \) using the Taylor series expansion \( \ln(1+x) \approx x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} \) for \(|x| \leq 0.5\). Use the remainder term of the Taylor series to find the error.

Answer: \dfrac{1}{5}


Question: Determine the smallest dimension of a faithful representation of the group $D_8 \times Q_8$ over a field $F$, where $D_8$ is the dihedral group of order 8, $Q_8$ is the quaternion group, and $F$ is a field of characteristic not equal to 2.

Answer: 4


Question:
Let \( f : \mathbb{R} \longrightarrow \mathbb{R} \) be a differentiable function whose derivative is continuous. Evaluate the limit:
\[
\lim_{n \to \infty} (n+1) \int_{0}^1 x^n f(x) \, dx
\]

Answer: f(1)


Question: Find the radius of convergence of the series \( \sum_{n=0}^{\infty} 2^{n} z^{n!} \).

Answer: 1


Question: Evaluate the integral \( \int_{\gamma_r}\sin\left(\frac{1}{z}\right)dz \), where \( \gamma_r:[0,2\pi]\rightarrow \mathbb{C} \) is given by \( \gamma_r(t)=r(\cos t + i\sin t) \) for some \( r>0 \), using power series expansions.

Answer: 2\pi i


Question: Evaluate the integral \( \frac{1}{2\pi i}\int_{\gamma} \frac{1}{z^2(e^{z}-1)}dz \), where \( \gamma=\{z: z=\frac{1}{2}e^{2\pi i t},0\leq t\leq1 \} \). Note that \( z=0 \) is a pole singularity of order 3.

Answer: \dfrac{1}{12}


Question: Determine the value of \( L \) for which the limit \( \lim_{x \to \infty} \frac{2^{x^2}}{L^{x(x-1)/6}} \) transitions from 0 to infinity.

Answer: 64


Question: Evaluate the expression \(6P_{10} - 15P_8 + 10P_6 + 7\), where \(P_n = \sin^n x + \cos^n x\) for a whole number \(n\) and a real number \(x\).

Answer: 8


Question: Find the limit of the series \( \sum_{k=0}^\infty \frac{h^{2k}}{k! k!} \).

Answer: I_0(2h)


Question:
Let \( \omega \) be a root of the polynomial \( x^{2016} + x^{2015} + \cdots + x + 1 = 0 \). Find the value of the sum:
\[ \sum_{k=0}^{2016} (1 + \omega^k)^{2017} \]

Answer: 4034


Question: Find the function \( F(x) \) that satisfies the functional differential equation: \[ 1 - F(x) = F(F'(x)) \]

Answer: \dfrac{1}{2}


Question: Simplify the expression: \[ \int_{-\infty}^{+\infty} f(r-r')\,\Delta_{r'}\delta(r')\,\mathrm{d^3r'} \] where \( \Delta \) denotes the Laplacian operator.

Answer: \Delta f(r)


Question: Calculate the limit \( \lim_{n \to \infty} -n \mathbb{1}_{Z \geq 1 - \frac{1}{n}} \), where \( Z \) is a random variable on the sample space \( \Omega = (0,1) \).

Answer: 0


Question: Given a number \( N = 25 \) and a number of steps \( K = 2 \), determine how many distinct numbers can be obtained by repeatedly multiplying \( N \) by one of its divisors at each step. For example, when \( K = 1 \), the possible numbers are 25, 125, and 625. When \( K = 2 \), the possible numbers are 25, 125, 625, 3125, 15625, 78125, and 390625. Calculate the total number of unique values obtained after \( K = 2 \) steps.

Answer: 7


Question: Given a sequence of complex numbers \(a_n\) such that \(\sum |a_n| < \infty\) and \(\sum n|a_n| = \infty\), determine the radius of convergence \(R\) of the power series \(\sum a_n z^n\).

Answer: 1


Question:
Given that $f, g: \mathbb{C} \rightarrow \mathbb{C}$ are holomorphic functions, and $A = \{x \in \mathbb{R} : f(x) = g(x)\}$, what is the minimum requirement for $f = g$? Choose the correct option based on the identity theorem for holomorphic functions:

1. $A$ is uncountable.
2. $A$ has positive Lebesgue measure.
3. $A$ contains a nontrivial interval.
4. $A = \mathbb{R}$.

Select the correct option.

Answer: 1


Question: Given the differential equation of the curves \( c(y+c)^2 = x^3 \) where 'c' is an arbitrary constant, the differential equation is \( 12y(y')^2 + ax = bx(y')^3 \). Find the value of \( a + b \).

Answer: 35


Question: Is it possible to have a normal, nonprincipal, non-uniform ultrafilter on some uncountable cardinal \( \kappa \)?

Answer: No


Question: Determine the values of \( k \in \mathbb{R} \) such that the equation \( k \cos(x) - (k+1)\sin(x) = k \) has two solutions that differ by \( \frac{\pi}{2} \).

Answer: -\dfrac{1}{2}


Question: Let $\mathcal{A}$ be a Banach $^*$-algebra and $a = a^* \in \mathcal{A}$. Suppose that the spectrum of $a$, $\sigma_{\mathcal{A}}(a)$, is contained in $\{0\} \cup [C,D]$ with $D > C > 1$. Determine whether the sequence $x_n = a(1-a^n)^{-1}$ converges to zero in $\mathcal{A}$. Note that the sequence of polynomials $p_n(z) = \frac{z}{1-z^n}$ converges uniformly to zero on the spectrum of $a$ but not on any compact subset of an open neighborhood of the spectrum.

Answer: 0


Question: Evaluate the integral \( \frac{1}{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta \) using the residue theorem.

Answer: I_n(1)


Question: Calculate the integral \( \int_{[0,1]} f \, dm \) where the function \( f: [0,1] \to \mathbb{R} \) is defined as follows: \( f(x) = 0 \) on the Cantor set \( \mathcal{C} \), and \( f(x) = k \) for every \( x \) in each interval of length \( \frac{1}{3^k} \) that has been removed from \([0,1]\) during the construction of the Cantor set. Express \( f \) as a simple function and compute the integral.

Answer: 3


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

Answer: \dfrac{1}{e}


Question: Find the limit of the sequence: \[ \lim_{n\to\infty}\prod_{n\ge2}\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^{2} \]

Answer: \dfrac{1}{9}


Question: Let $X$ and $Y$ be non-homeomorphic subsets of $\mathbb{R}$, and let $Z$ be an arbitrary subset of $\mathbb{R}$. Is it possible for the product space $X \times Z$ to be homeomorphic to $Y \times Z$?

Answer: Yes


Question:
For independent and identically distributed random variables $X_1, X_2, \ldots, X_n$ with finite expectation, find the value of:
\[ \mathbb{E}\left[\frac{\sum_{i=1}^k X_i}{\sum_{i=1}^n X_i}\right] \]
where $k \le n$. Assume $\mathbb{E}[X_1]$ is finite.

Answer: \dfrac{k}{n}


Question: Let \( \lambda > 0 \) be a non-integer, and define the function \( f(n) = n^{\lambda} \) for \( n \in \mathcal{N} \), where \( \mathcal{N} = \{1, 2, \ldots \} \). A function \( f : \mathcal{N} \to \mathbb{R} \) is considered "almost polynomial" if there exists some \( k \) such that \( \Delta^{k}(f) \) is a bounded function, where \( \Delta(f)(n) = f(n+1) - f(n) \) and \( \Delta^{k} \) denotes the \( k \)-th iteration of \( \Delta \). Is the function \( f(n) = n^{\lambda} \) almost polynomial?

Answer: Yes


Question:
Given the constraints:
\[ x_1 \leq 4, \quad x_1 + x_2 \leq 13, \quad x_1 + x_2 + x_3 \leq 29, \quad x_1 + x_2 + x_3 + x_4 \leq 54, \quad x_1 + x_2 + x_3 + x_4 + x_5 \leq 90 \]
where \( x_1, x_2, x_3, x_4, x_5 \geq 0 \), find the maximum value of the expression:
\[ \sqrt{x_1} + \sqrt{x_2} + \sqrt{x_3} + \sqrt{x_4} + \sqrt{x_5}. \]

Answer: 20


Question: An archer has a probability of hitting a target of 0.4 with each shot. Determine the minimum number of attempts required for the archer to hit the target at least 80 times with a probability of 0.9.

Answer: 223


Question: Find the minimum value of \[ P = \frac{1}{4(x-y)^2} + \frac{1}{(x+z)^2} + \frac{1}{(y+z)^2} \] given that \( x, y, z \) are real numbers such that \( x > y > 0, z > 0 \) and \( xy + (x+y)z + z^2 = 1 \).

Answer: 3


Question: Evaluate the integral \( \int_{0}^{2\pi} e^{\sin (e^{i \theta} )} \, d\theta \).

Answer: 2\pi


Question: Let \( z_{n}=\left(-\frac{3}{2}+\frac{\sqrt{3}}{2}i\right)^{n} \). Find the least positive integer \( n \) such that \( |z_{n+1}-z_{n}|^{2}>7000 \).

Answer: 7


Question: Show that \( \sqrt{n}[\log(X_{(1)}) - \log(\alpha)] \overset{p}{\to} 0 \) for \( X_1, \dots, X_n \sim \text{Pareto}(\alpha, \beta) \) with \( n > \dfrac{2}{\beta} \) being independent. Use only the definitions of convergence in probability and distribution, the Delta method, the Central Limit Theorem, Slutsky's Theorem, the Weak Law of Large Numbers, and the Continuous Mapping Theorem in your solution.

Answer: 0


Question: Evaluate \( \frac{\partial^2 z}{\partial u^2} + \frac{\partial^2 z}{\partial v^2} \) given that \( \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = 0 \) and \( z = z(x, y) \), where \( x = e^u \cos v \) and \( y = e^u \sin v \).

Answer: 0


Question: Calculate the flux of the vector field \( \mathbf{v} = \mathrm{curl}\, \mathbf{u} \) through the surface \( \Sigma \), where \( \Sigma \) is the portion of the paraboloid \( z = 1 - x^2 - y^2 \) with \( x \geq 0 \) and \( z \geq 0 \). The vector field \( \mathbf{u} \) is given by \( \mathbf{u} = (xy, xz^2, x^2y) \). Use Stokes' theorem to compute the flux.

Answer: \dfrac{2}{3}


Question: Find the limit as \( n \to \infty \) of the expression \( \left(\frac{n^{4}11^{n}+n^{9}9^{n}}{7^{2n}+1}\right) \).

Answer: 0


Question: Consider a scenario where a secretary has $n$ letters and $n$ corresponding envelopes. The secretary randomly places each letter into an envelope, not necessarily the correct one. What is the probability that at least one letter is placed in its corresponding envelope? Additionally, determine the limit of this probability as $n$ approaches infinity, without using derangements or conditional probabilities.

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


Question: Find the sum of all distinct values of \( a \) for which the equation \( 4a[X]^2 = a + 12 \) has solutions in the interval \([1, \infty)\), where \([X]\) denotes the greatest integer function.

Answer: 6


Question: Compute the area of the region between the x-axis and the curve given by \( g(t) = \begin{pmatrix} 1 - \sin(t) \\ 1 - \cos(t) \end{pmatrix} \) for \( 0 \le t \le 2\pi \) using Green's Theorem.

Answer: \pi


Question: Determine the smallest integer $m$ such that the wheel graph $W_4$, consisting of a triangle and a central hub, can be embedded in $\mathbb{R}^m$ with every edge having length 1.

Answer: 3


Question: Determine whether there exists a rational univariate polynomial of degree 3 with three irrational roots. Specifically, does there exist a polynomial \( f \in \mathbb{Q}[x] \) such that \( f(x) = (x - \alpha_1)(x - \alpha_2)(x - \alpha_3) \), where \( \alpha_1, \alpha_2, \alpha_3 \in \mathbb{R} \setminus \mathbb{Q} \)?

Answer: Yes


Question: Let $A$ be a unital $C^*$-algebra, and $I_1$ and $I_2$ be closed (two-sided) $C^*$-ideals of $A$ such that $A=I_1+I_2$. Suppose there are two positive elements $a_1 \in I_1$ and $a_2 \in I_2$ such that $a_1^2 + a_2^2 = 1$. Determine if $a_1$ and $1$ define the same class in the quotient $A/I_2$.

Answer: Yes


Question: Let $R$ be a commutative ring with unity. Suppose $M$ is a finitely generated $R$-module with a submodule $N$ such that $M/N$ is a free module. Is it true that $N$ is finitely generated?

Answer: Yes


Question: Find the maximum value of the expression \( x + y + z \) subject to the constraints \( x + y + z = 1 \) and \( 4x^2 + 9y^2 = 6 \).

Answer: 1


Question: Let $P$ be a $p$-group with $|P|=p^n$. Let $M$ be a maximal subgroup of $P$. Determine if it is true that $|M|=p^{n-1}$.

Answer: True


Question: Is there a totally real number field $F$ with class number 1 such that for every prime number $p$, the ideal $pO_F$ is a prime ideal in the ring of integers $O_F$?

Answer: \mathbb{Q}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(\frac{3n^{2}+4n-5}{3n^{2}-7n+9}\right)^{n} \]

Answer: e^{\frac{11}{3}}


Question: Express the antipodal map as a composition of one, two, or three reflections. Determine the minimum number of reflections needed.

Answer: 3


Question: Calculate the limit: \[ \lim_{x \to \infty}\left((x^3-x^2+\frac{x}{2})e^{\frac{1}{x}}-\sqrt{x^6+1}\right) \]

Answer: \dfrac{1}{6}


Question: Evaluate the limit: $$\lim_{x\to-1}\frac{\sqrt[3]{26-x}-\sqrt{8-x}}{3x^2+4x+1}$$ without using L'Hôpital's rule.

Answer: -\dfrac{7}{108}


Question:
Determine the number of ordered triples \((x, y, z)\) of real numbers that satisfy the following system of equations:
\[ x + y^2 = z^3, \]
\[ x^2 + y^3 = z^4, \]
\[ x^3 + y^4 = z^5. \]

Answer: 7


Question: Calculate or estimate the expectation \( \mathbb{E}\left[\left(\sum_{i=1}^n Y_i\right)^4\right] \), where \( Y_i \) are independent and identically distributed random variables with \( \mathbb{P}(Y_i=1) = \mathbb{P}(Y_i=-1) = \frac{1}{2} \).

Answer: 3n^2 - 2n


Question: How many circles of radius \(\frac{\pi}{4}\) can be drawn within a circle of radius \(\pi\) such that they do not intersect one another?

Answer: 19


Question: Evaluate the integral \( \int_{|z|=10} \frac{dz}{z^5+z+1} \).

Answer: 0


Question: Let $P \subset k[x_1, \ldots, x_n]$ be a prime ideal. Is the variety $V(P)$ irreducible? Provide a justification for your answer.

Answer: Yes


Question: Seven pirates stole some rare gems and decided to divide them later. When two pirates tried to divide the gems, one gem was left over. The same happened when they tried to divide the gems among three, four, five, and six pirates. However, when all seven pirates divided the gems, they were divided equally. How many gems did the pirates steal in total?

Answer: 301


Question: Compute the double integral \( \iint\limits_{G} \sqrt{x^{2} - y^{2}} \, dx\, dy \), where the region \( G \) is defined as \( G=\{(x,y) \in \mathbb{R_{+}^{2}}: 0 \leq x + y \leq 2, 0 \leq x - y \leq 2\} \).

Answer: \dfrac{8}{9}


Question: Find the value of \(x\) that satisfies the equation: \((x^2+100)^2=(x^3-100)^3\).

Answer: 5


Question: Find the function \( f: \mathbb{R} \to \mathbb{R}_0^+ \) that satisfies the equation \( f(x+2y) + f(x) = 2(f(x+y) + f(y)) \) for all real numbers \( x \) and \( y \).

Answer: f(x) = ax^2


Question: What is the expected number of $k$-cycles in a randomly selected permutation of the set $[2k] = \{1, 2, \ldots, 2k\}$?

Answer: \dfrac{1}{k}


Question: Find the closed-form solution for the sequence \(a_n\) defined by the recursive relation \(a_n = a_{n-1} + 2n - 1\) with the initial condition \(a_0 = 0\) for \(n \ge 1\), using generating functions.

Answer: n^2


Question: Consider the double integral \( \int_0^{\infty} \int_0^{\infty} f(xy) g(x,y) \, dx \, dy \). Perform the substitution \( y=\frac{t}{x} \) with \( dy=\frac{dt}{x} \). Determine the correct bounds for the variable \( t \) in the transformed integral.

Answer: [0, \infty)


Question: In a supermarket, waiting times at a cashier counter are exponentially distributed with a parameter of 0.1 minutes. If 100 customers visit the counter in a day, approximate the probability that at least 50 of them will wait for more than 10 minutes. Use the Central Limit Theorem for your approximation and assume no continuity correction is needed.

Answer: 0.0031


Question:
Consider the following commutative diagram of Hilbert spaces with exact sequences:
\[
\begin{CD}
0 @>>> H_1 @>f_1>> H_2 @>f_2>> H_3 @>>> 0\\
@V VV @V T_1 VV  @V T_2 VV @V T_3 VV @V VV \\
0 @>>> K_1 @>>g_1> K_2 @>>g_2> K_3 @>>> 0
\end{CD}
\]
where $H_i, K_i$ for $i=1,2,3$ are Hilbert spaces, and $T_1, T_2, T_3$ are trace class operators with dense range. Assume that $S \colon H_2 \to K_2$ is a compact operator with dense range making the squares commute. Is there a constant $C > 0$ such that $C \mathrm{Tr} |T_2| \geq \mathrm{Tr} |S|$? This question relates to whether nuclear operators are closed under extensions.

Answer: No


Question: Given that \( \int(x+1)\cdot f'(x) = x^3+x^2-x+c \) and \( f(0)=\frac{1}{2} \), find the value of \( f(-1) \).

Answer: 3


Question: Let \( f: [0,1] \to \mathbb{R} \) be a Riemann integrable function. Define a function \( F: [0,1] \to \mathbb{R} \) by \( F(x) = \int_0^x f(t) \, dt \). Suppose that \( F \) is differentiable at \( c \in (0,1) \). Is it necessarily true that \( F'(c) = f(c) \)?

Answer: No


Question:
Find the values of $a \in \mathbb{R}$, where $a \neq -1$, such that the following limit holds:
\[ \lim_{n\to\infty}\frac{1^a+2^a+\cdots +n^a}{(n+1)^{a-1}[(na+1)+(na+2)+(na+3)+\cdots+(na+n)]}=\frac{1}{60} \]

Answer: 7


Question: Let $a, b, c$ be positive integers such that both $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ and $\frac{b}{a} + \frac{c}{b} + \frac{a}{c}$ are integers. Prove that $a = b = c$.

Answer: a = b = c


Question: Given two concentric circles with radii \(r\) and \(R\) (where \(R \geq r\)), forming an annulus, and \(n\) equidistant points \(p_0 = p_n, p_1, p_2, \dots, p_n\) on the outer circle, find \(n\) equidistant points \(q_1, \dots, q_n\) on the inner circle such that the triangles \(\Delta(p_i, p_{i+1}, q_i)\) for \(1 \leq i \leq n\) do not overlap and are adjacent to each other. Determine the angle \(\alpha\) that allows for this configuration.

Answer: \dfrac{\pi}{n}


Question:
Let $B = B(0,1) \subset \mathbb{C}$ be the open unit disc and $B' = B \setminus \{ 0 \} \subset \mathbb{C}$ the punctured unit disc. Suppose $u : B' \rightarrow \mathbb{R}$ is a harmonic function that belongs to the Sobolev space $W^{1,2}(B)$, meaning:
\[ \int_B |u|^2 < \infty \quad \text{and} \quad \int_B |\nabla u|^2 < \infty. \]
Does $u$ extend to a harmonic function on $B$?

Answer: Yes


Question:
Given a function \( \varphi: \mathbb{C}^{2 \times 2} \to \mathbb{C} \) with the following properties:

1. Linearity on the columns:
   \[ \varphi\left(\begin{bmatrix} a_1+a_2 & b \\ c_1+c_2 & d \end{bmatrix}\right) = \varphi\left(\begin{bmatrix} a_1 & b \\ c_1 & d \end{bmatrix}\right) + \varphi\left(\begin{bmatrix} a_2 & b \\ c_2 & d \end{bmatrix}\right) \]
   \[ \varphi\left(\begin{bmatrix} a & b_1+b_2 \\ c & d_1+d_2 \end{bmatrix}\right) = \varphi\left(\begin{bmatrix} a & b_1 \\ c & d_1 \end{bmatrix}\right) + \varphi\left(\begin{bmatrix} a & b_2 \\ c & d_2 \end{bmatrix}\right) \]
   \[ \varphi\left(\begin{bmatrix} ka & b \\ kc & d \end{bmatrix}\right) = \varphi\left(\begin{bmatrix} a & kb \\ c & kd \end{bmatrix}\right) = k\varphi\left(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\right) \]

2. Anti-symmetry on the columns:
   \[ \varphi\left(\begin{bmatrix} b & a \\ d & c \end{bmatrix}\right) = -\varphi\left(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\right) \]

3. Maps the identity matrix to 1:
   \[ \varphi\left(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\right) = 1 \]

Prove that \( \varphi\left(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\right) = ad - bc \), the determinant of the matrix.

Answer: ad - bc


Question:
Find the value of the limit:
\[ \lim_{x\to \pi/2} \frac{\cot(x) - \cos(x)}{(\pi-2x)^3} \]

Answer: \dfrac{1}{16}


Question: Find \( \lim\limits_{n \to +\infty}\frac {n}{(\log(n+1))^n} \).

Answer: 0


Question: Use the Banach contraction theorem to find a solution to the equation \(x + e^x = 0\) correct to three decimal places.

Answer: -0.567


Question:
Let \(a_n\) be a sequence of real numbers such that \(\lim_{n\to +\infty}a_n=A\). Determine whether the following limit is true:
\[\lim_{n\to +\infty}\left(1+\frac{a_n}{n}\right)^n=e^A.\]

Answer: e^A


Question: Solve the equation $\sinh(x) = x$.

Answer: 0


Question: Consider a finitely generated field extension $K/k$ where $k=K^G$ for some set $G$ of automorphisms of $K$. Given the extension of polynomial rings $k[X_1,\ldots,X_n]\subset K[X_1,\ldots,X_n]$ and a $G$-invariant ideal $I \lhd K[X_1,\ldots,X_n]$, is it true that $I$ can be defined by polynomials in $k$?

Answer: Yes


Question: Evaluate the integral \( \int_0^1 \frac{1+x^4}{1+x^6} \, dx \).

Answer: \dfrac{\pi}{3}


Question:
Let $\{v_{1},v_{2},\ldots,v_{16}\}$ be an ordered basis for $V=\mathbb{C}^{16}$. If $T$ is a linear transformation on $V$ defined by $T(v_{i})=v_{i+1}$ for $1\leq i\leq 15$ and $T(v_{16})=-(v_{1}+v_{2}+\ldots+v_{16})$, determine which of the following statements is true:

(a) $T$ is singular with rational eigenvalues,
(b) $T$ is singular but has no rational eigenvalues,
(c) $T$ is regular (invertible) with rational eigenvalues,
(d) $T$ is regular but has no rational eigenvalues.

Answer: D


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{|x|^{1/2} \cos(\pi^{1/x^2})}{2 + (x^2 + 3)^{1/2}} \]

Answer: 0


Question: Evaluate the integral: \[ \int_0^\pi \left(\frac{\sin{2x}\sin{3x}\sin{5x}\sin{30x}}{\sin{x}\sin{6x}\sin{10x}\sin{15x}}\right)^2 \, dx \]

Answer: 7\pi


Question: Let \( f(x) = x^4 + x^3 + x^2 + x + 1 \) be a fourth-degree polynomial. If \( p \) is a root of \( f(x) \), determine which of the following cannot be a root of \( f(x) \): \( p^2, p^3, p^4, p^5 \).

Answer: p^5


Question: Compute the fundamental group \( \pi_1(X) \) of the space \( X \) obtained from the 3-sphere \( S^3 \) by identifying antipodal points on the equator \( S^2 \).

Answer: 0


Question: Compute the limit: \[ \lim_{x\to0} \frac{8}{x^8} \left[ 1 - \cos\frac{x^2}{2} - \cos\frac{x^2}{4} + \cos\frac{x^2}{2}\cos\frac{x^2}{4} \right] \] without using L'Hôpital's rule.

Answer: \dfrac{1}{32}


Question: Solve the equation \( \cos\left(\frac{\pi}{x}\right) = \frac{x - 2}{x + 2} \) for \( x \).

Answer: 2


Question: Let $(V, p_V)$ be an irreducible representation of a finite group $G$ on a vector space $V$. Consider $W = \bigoplus \limits_{i = 1}^{n} V$, where $W$ is equipped with the direct-sum representation $p_W(v_1, \ldots, v_n) = p_V(v_1) + \cdots + p_V(v_n)$ for $(v_1, \ldots, v_n) \in W$. Calculate $\dim(\text{Hom}_G(V, W))$. Prove that $\dim(\text{Hom}_G(V, W)) = n$. \(n\) is a positive integer.

Answer: n


Question: Maya has six identical pots, each to be colored either red, yellow, green, or blue, with at least one pot of each color. How many ways can she arrange the pots in a straight line so that no two pots of the same color are adjacent?

Answer: 600


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

Answer: \dfrac{1}{2}


Question: Let $F$ be the vector field $F=4xz\,i+xz^2\,j+x^7y^3\,k$. Let $S$ be the portion of the paraboloid $z=x^2+y^2$ below $z=9$, along with the disk $x^2+y^2\leq9$ in the plane $z=9$, all with outward orientation. Compute the surface integral $\iint_S F \cdot dS$. \(\text{Use the Divergence Theorem if applicable.}\)

Answer: 972\pi


Question: A car is equipped with one spare tire. The lifetimes of the four tires at the beginning of a long-distance journey are independent identically distributed exponential random variables with a mean of 5000 miles. The spare tire has an exponential lifetime with a mean of 1000 miles. Compute the expected number of miles that can be covered without having to go to a tire shop. Note that when one of the regular tires fails, the driver will replace it with the spare one, and then when one of the remaining three regular tires or the spare tire fails, he will go to the tire shop.

Answer: 1875


Question: Determine the value of the prime $p$ for which the polynomial $x^4 + x + 6$ has a root of multiplicity greater than 1 over a field of characteristic $p$. Consider the primes $p = 2, 3, 5, 7$. Provide your answer as one of these primes.

Answer: 3


Question: Given that $a$ and $b$ are positive integers where $b$ is a multiple of $a$, and \[ \log\left(\left(\frac{b}{a}\right)^{\frac{b}{3}}\right) + \log\left(\left(\sqrt[3]{\frac{a}{b}}\right)^{9a}\right) = 1, \] find the value of $a^2 + b^2$.

Answer: 909


Question:
Determine which of the following is true for every integer of the form \((n^{3}-n)(n^{2}-4)\) for \(n = 3, 4, \ldots, K\):

(A) Divisible by 6 but not always divisible by 12;
(B) Divisible by 12 but not always divisible by 24;
(C) Divisible by 24 but not always divisible by 120;
(D) Divisible by 120 but not always divisible by 720.

Answer: D


Question: Does there exist an increasing sequence of $k$-times continuously differentiable functions $(g_n)_n \subset C^k(\mathbb{R}^d ; [0,\infty))$ that converges pointwise to a lower semicontinuous function $f : \mathbb{R}^d \rightarrow [0,\infty)$, such that $g_n(x) \leq g_{n+1}(x)$ for all $x \in \mathbb{R}^d$ and $n \in \mathbb{N}$, and $\lim_{n \rightarrow \infty} g_n(x) = f(x)$ for all $x \in \mathbb{R}^d$?

Answer: Yes


Question: Let $T$ be a linear operator defined by \( T\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = \begin{pmatrix} c & a \\ d & b \end{pmatrix} \) for every \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathbb{R}^{2 \times 2} \). If $A$ is an eigenvector of $T$ corresponding to the eigenvalue $-1$, compute $\det(A)$. \( \boxed{} \)

Answer: 0


Question: Let $a = (a_1, a_2, a_3)$ be a nonzero vector. Define the linear transformation $S: \mathbb{R}^3 \to \mathbb{R}^3$ by $S(x) = a \times x$ for all vectors $x$. Find the determinant of the transformation $S$. \(\text{Note: } a \times x \text{ denotes the cross product.}\)

Answer: 0


Question:
Let \( a, b, c \) be the sides of a triangle where \( a \neq b \neq c \) and \( \lambda \in \mathbb{R} \). Consider the quadratic equation:
\[ x^2 + 2(a+b+c) \cdot x + 3\lambda \cdot (ab+bc+ca) = 0 \]
Determine the correct range for \( \lambda \) such that the equation has real roots.

Options:
(a) \( \lambda < \frac{4}{3} \)
(b) \( \lambda > \frac{4}{3} \)
(c) \( \lambda \in \left(\frac{1}{3},\frac{5}{3}\right) \)
(d) \( \lambda \in \left(\frac{4}{3},\frac{5}{3}\right) \)

Answer: A


Question: Evaluate \( \lim \limits_{n \to \infty} \frac{1}{n}\sqrt[n]{n^5+(n+1)^5+\ldots+(2n)^5} \).

Answer: 0


Question: Let $R$ be a commutative ring with identity, and let $a, b$ be elements of $R$. If $b + (a)$ is not a zero divisor in $R/(a)$, does it follow that $(a, b) = R$?

Answer: No


Question: Find the proportions of a right circular cylinder of greatest volume that can be inscribed inside a sphere of radius $R$. Express the ratio of the height to the radius of the cylinder in its simplest form.

Answer: \sqrt{2}


Question: Suppose \((X, \mathcal{D})\) is a uniform space where every complete subset is closed. Is \((X, \mathcal{D})\) a Hausdorff space?

Answer: Yes


Question:
Given the linear Diophantine equation \(a = 20k + 91806\), find the specific solution \(a = 96646\) or \(k = 242\) using the following information:

1. The prime factorization of \(91806\) is \(2, 3, 11, 13, 107\).
2. \(\gcd(a, 91806) = 22\).
3. \(a \equiv 91806 \pmod{20}\).

Is it possible to find this specific solution using the Chinese Remainder Theorem or a similar method?

Answer: 96646


Question: Suppose $L$ is an extension field of $K$ and $L$ is algebraically closed. Is it true that for every polynomial $f(x) \in K[x]$, $f(x)$ splits over $L$? Provide a justification for your answer.

Answer: Yes


Question: Determine the orbits of the symmetric group $S_n$, where $n \geq 4$, on the set of pairs of 2-subsets of $\{1,2,\ldots,n\}$.

Answer: 2


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

Answer: e - 1


Question: Given a collection of chemicals $C = \{c_1, c_2, \dots, c_n\}$, each with a specific storage temperature range $[l_i, h_i]$, determine the smallest number of temperature-controlled storage units required to store all chemicals. Model this problem using graph theory.

Answer: k


Question: Let $V$ be a vector field on the sphere $S^2$ such that $\operatorname{div} V = 0$. Determine the minimum number of tangency points between this vector field and the equator of the sphere.

Answer: 2


Question: Find the greatest integer $n$ such that \[ n \leq 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \ldots + \frac{1}{\sqrt{2014}}. \]

Answer: 88


Question: In a plane rectangular coordinate system, consider the points $A(0,\frac43)$, $B(-1,0)$, and $C(1,0)$. The distance from a point $P$ to line $BC$ is the geometric mean of the distances from $P$ to lines $AB$ and $AC$. If line $L$ passes through the incenter $D$ of triangle $ABC$ and intersects the locus of point $P$ at exactly 3 points, determine the number of possible values for the slope of line $L$.

Answer: 7


Question:
Given the rotation matrix \( A = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \) in the \((x, y, z)\) basis and its diagonal form \( D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -\left(\frac{1+\sqrt{3} i}{2}\right) & 0 \\ 0 & 0 & \left(\frac{-1+\sqrt{3} i}{2}\right) \end{bmatrix} \) in the \((x', y', z')\) basis, find the angle of rotation about the axis in the direction \( \hat{n} = \frac{\hat{x}+\hat{y}+\hat{z}}{\sqrt{3}} \). The transformation rule between \((x, y, z)\) and \((x', y', z')\) is given by:
\[
x = \frac{1}{\sqrt{3}}\left(x^{\prime}+\left(\frac{-1+\sqrt{3} i}{2}\right) y^{\prime}+\left(\frac{-1-\sqrt{3} i}{2}\right) z^{\prime}\right) \\
y = \frac{1}{\sqrt{3}}\left(x^{\prime}+\left(\frac{-1-\sqrt{3} i}{2}\right) y^{\prime}+\left(\frac{-1+\sqrt{3} i}{2}\right) z^{\prime}\right) \\
z = \frac{1}{\sqrt{3}}\left(x^{\prime}+y^{\prime}+z^{\prime}\right)
\]
Determine the angle of rotation.

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


Question: Determine whether monoids of endomorphisms of nonisomorphic groups can be isomorphic.

Answer: Yes


Question: In triangle ABC, angle C is a right angle. The bisector of angle B intersects ray AC at point D, and the bisector of the exterior angle at B intersects line AC at point E. Given that BD = 15 and BE = 20, find the perimeter of triangle ABC.

Answer: 96


Question: Find the number of integers in the range of the function \( g(x) = \sin x + \cos x \).

Answer: 3


Question:
Determine the volume of the solid defined by the region:
\[ V = \{(x,y,z) : 0 \le x \le 1, x^2 \le y \le \sqrt{x}, -x^{1/3} - y^{1/3} \leq z \leq x^{1/3} + y^{1/3} \} \]

Answer: \dfrac{54}{55}


Question: Determine whether the fundamental group of a compact finite dimensional Alexandrov space with curvature bounded below is finitely generated. Answer with 'true' or 'false'.

Answer: True


Question: Given the partial differential equation \((z^2-2yz-y^2)p+(xy+zx)q= xy-zx\), if the solution to this PDE represents a sphere, determine the coordinates of the center of the sphere. Assume the solution is of the form \(x^2+y^2+z^2= f(z^2+2yz-y^2)\).

Answer: (0, 0, 0)


Question: Determine all values of the parameter \(a\) for which the inequality \(a \cdot 9^x + 4 \cdot (a-1) \cdot 3^x + (a-1) > 0\) is satisfied for all real values of \(x\).

Answer: [1, \infty)


Question: Show that the center of the orthogonal group $O_n(\mathbb{R})$ is $\{I, -I\}$.

Answer: \{\pm I\}


Question: Determine the values of \(a\) and \(b\) such that the integral \(\int \frac{ax+b}{(x-1)(x+1)^2}dx\) is a rational function.

Answer: b = -a


Question:
Evaluate the limit:
\[ \lim_{x\to 0}\frac{\int_0^{x^2}{\frac{1-\cos{t^2}+at^4}{t}}dt}{(1-\cos{(\frac{x}{3})})^4} \]

Answer: 13122(1 + 2a)


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

Answer: \dfrac{1}{e}


Question: Find all positive integers \( n \) such that \( n \mid a^{25} - a \) for all positive integers \( a \).

Answer: 2730


Question: Determine the dimension of a 5x5 matrix where all diagonal entries are \( \frac{1}{2} \) and all off-diagonal entries satisfy \( A_{ij} = -A_{ji} \).

Answer: 10


Question: Determine whether the following asymptotic relation is true for the von Mangoldt function \( \Lambda \): \[ \sum_{1\leq x\leq m} \sum_{1\leq y\leq n}\Lambda(x)\Lambda(y) \sim m n \] given that \( m \neq n \) and \( x \neq y \).

Answer: True


Question: Determine the values of \( p > 0 \) for which the improper integral \( \int_{0}^{1} |\ln{x}|^{p} \, dx \) converges.

Answer: p > 0


Question: Let \( \{a_n\} \) be a sequence of real numbers defined by the recurrence relation \( a_{n+1} = n^2 - a_n \) for \( n = 1, 2, 3, \ldots \). Given that \( a_{101} = 5075 \), find the value of \( a_1 \).

Answer: 25


Question: Determine the conditions on the parameters \(a\) and \(b\) for which the integral \(\int_0^{\infty} e^{(a+bi)x} \, dx\) converges.

Answer: a < 0


Question: Determine the smallest value of the damping coefficient $b$ such that the solutions of the differential equation \( m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 \) lie along a straight line in the position-velocity plane. Provide algebraic justification for your answer.

Answer: 2\sqrt{mk}


Question: Let $E \subset \mathbb{R}^n$ be a set with positive Lebesgue measure. Determine whether there always exists a point $x \in E$ such that for any $r > 0$, the intersection $B(x,r) \cap E$ has positive Lebesgue measure in $\mathbb{R}^n$. Here, $B(x,r)$ denotes the ball of radius $r$ centered at $x$ in $\mathbb{R}^n$. Provide a justification for your answer.

Answer: Yes


Question: Given a random sample \(X_1, X_2, \ldots, X_n\) from a uniform distribution \(U(0, \theta)\) where \(\theta > 0\), find a constant \(c > 1\) such that \((T, cT)\) is a 100\((1-\alpha)\%\) confidence interval for \(\theta\). Assume that \(T = X_{(n)}\), the maximum of the sample.

Answer: \alpha^{-1/n}


Question: Researchers have found a COVID vaccine that causes harmful effects with probability \( p \), where \( p \) is uniformly distributed in the interval [0, 0.5]. To check the effectiveness of the vaccine, the researchers test the vaccine on 10,000 volunteers and find that no one experiences adverse side effects. What is the smallest real number \( \lambda \) such that the researchers can assert \( p < \lambda \) with probability at least 90%?

Answer: 0.00023


Question: Evaluate the integral \( \int\limits_{0}^{\infty} {\frac{2}{3x^{4/3}} \sqrt{\frac{\lambda}{2\pi}} \exp \left( \frac{-\lambda (x^{2/3}-\mu)^2}{2\mu^2 x^{2/3}}  \right) dx} \) for \(\mu>0, \lambda>0\).

Answer: 1


Question: Evaluate the expression \(2I_n + nI_{n-1}\), where \(I_n = \int_{1}^{e^2} (\ln(x))^n \, d(x^2)\).

Answer: 2^{n+1} e^4


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

Answer: 4


Question: Find the minimum value of \( \int_0^1 f(x)^2 \, dx \) subject to the conditions \( \int_0^1 f(x) \, dx = 0 \) and \( \int_0^1 x f(x) \, dx = 1 \). Assume suitable smoothness conditions for \( f(x) \).

Answer: 12


Question: Calculate the mass of fluid flowing through the hemisphere defined by \( x^2 + y^2 + z^2 = 1, z \geq 0 \) in unit time. The fluid flow has a flux density vector \( F(x,y,z) = \langle x, -2x-y, z \rangle \). Let \( n \) denote the unit normal vector pointing out of the sphere. Find the mass of fluid flowing through the hemisphere in the direction of \( n \).

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


Question: Player 1 samples a number from the Uniform(0,1) distribution. Player 2 then repeatedly samples from the same distribution until he obtains a number higher than Player 1's sample. What is the expected number of samples Player 2 needs to make?

Answer: \infty


Question: Evaluate the limit: \[ \lim_{x \to 0} \cot(2x)^{\frac{5}{\ln(3x)}} \]

Answer: e^{-5}


Question: Find the closed form of the integral \( \int_{0}^{\infty} \frac{x^{a} \ln^{2m-1} x}{1+x^{2(a+1)}} d x \) for any natural number \( m \) and \( a > -1 \).

Answer: 0


Question: Determine the number of solutions to the equation \( \{x\} + \left\{ \frac{1}{x} \right\} = 1 \), where \( \{\cdot\} \) denotes the fractional part of a real number \( x \).

Answer: \infty


Question: Determine whether there exists a nearly slender abelian group that is not slender. Provide a justification for your answer.

Answer: Yes


Question: Suppose that $f$ and $g$ are two non-vanishing holomorphic functions on a domain $D$. Prove that if $\frac{f'}{f} = \frac{g'}{g}$, then $f = cg$ for some constant $c \in \mathbb{C}$.

Answer: f = cg


Question: Calculate the limit: $$\lim_{x \to 0} \frac{\sqrt{1+x \sin(5x)}-\cos(x)}{\sin^2(x)}$$

Answer: 3


Question: Determine for which values of \( n \) the integral \( \int_0^1 \frac{x^n}{\sqrt{1-x^4}} \, dx \) is convergent.

Answer: n > -1


Question: Determine the Galois group of the cubic polynomial \( f(x) = x^3 + 6x^2 + 9x + 3 \) over \( \mathbb{Q} \).

Answer: A_3


Question: Random variable $Z$ has a mean of $15$ and a standard deviation of $2$. What is the minimum percentage of $Z$ values that lie between $8$ and $17$?

Answer: 42.45\%


Question: In \( \mathbb{R}^n \) with the usual topology and the Borel sigma-algebra, let \( A \) be an open set, \( B \) be a closed set, and \( C \) be a compact set. Is the set \((A \cap B) \oplus C\) always Borel measurable?

Answer: Yes


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

Answer: \dfrac{1}{2}


Question: Suppose $(M, \omega)$ is a symplectic manifold and $X$ is a smooth vector field defined on $M$. If the Lie derivative $\mathcal{L}_X \omega = 0$, is the flow $\{g_s\}_{s\in\mathbb{R}}$ symplectic, i.e., does $g_s^*\omega = \omega$ hold for all $s$?

Answer: Yes


Question: Suppose a sequence of random variables $Y_n$ converges in $L^2$ to $Y$, i.e., \( E|Y_n-Y|^2\to0 \) as \( n\to\infty \). Additionally, there exist constants \( c_0 \) and \( c_1 \) such that \( 0 < c_0 \leq Y_n, Y \leq c_1 \). Assume further that \( X_n Y_n \) converges in distribution to \( Z \). Is it true that \( X_n \) converges in distribution to \( Z/Y \)?

Answer: Yes


Question: Let $k$ be a number field, $S$ be a finite set of places of $k$ containing all the infinite places of $k$, and $R_S := \{x \in K : |x|_v \leq 1 \;\forall v \notin S\}$ be the ring of $S$-integers of $k$. Suppose $a_1, \ldots, a_t$ are pairwise distinct elements of $R_S$. For any prime number $p$ and any prime ideal $P$ of $R_S$ lying above $p$, we have that for $i=1,\ldots,t$, $a_i \equiv a_i^\prime$ in $R_S / P$, for some rational integer $a^\prime_i$. Is it true that $a_1^\prime, \ldots, a_t^\prime$ are NOT pairwise distinct only for finitely many prime $p$? Answer YES or NO.

Answer: YES


Question: Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth injective function with $\operatorname{det}[f'(x)] \neq 0$ for all $x \in \mathbb{R}^n$. Assume that $f$ has no fixed points. Can $f$ have periodic points, i.e., a point $x$ such that $f^k(x) = x$ for some integer $k > 1$? Provide a justification for your answer.

Answer: No


Question: Determine the one-point compactification of the space $S^n \times \mathbb{R}$. Provide your answer in terms of a well-known topological space.

Answer: S^{n+1}


Question: Let $(\mathbb{X} \times \mathbb{Y}, \mathscr{X} \otimes \mathscr{Y})$ be a measurable product space obtained from measurable spaces $(\mathbb{X}, \mathscr{X})$ and $(\mathbb{Y}, \mathscr{Y})$. Define $\mathscr{A}$ as the $\sigma$-algebra formed by the cylinders of $\mathscr{X} \otimes \mathscr{Y}$ based on $\mathscr{X}$, i.e., $\mathscr{A} = \{X \times \mathbb{Y} : X \in \mathscr{X}\}$, with the agreement that $\emptyset \times \mathbb{Y} = \emptyset$. If $f: \mathbb{X} \times \mathbb{Y} \rightarrow [0, +\infty]$ is $\mathscr{A}$-measurable, is it true that $f(x, y_1) = f(x, y_2)$ for all $x \in \mathbb{X}$ and for all $y_1, y_2 \in \mathbb{Y}$?

Answer: Yes


Question: Determine the number of integer solutions \((a, b)\) for the equation \( \frac{ab}{a+b} = 3600 \).

Answer: 449


Question: Calculate \(\lim_{n\to\infty} \frac{\sum_{i=1}^n ia_i}{n^2+n}\) given that \(a_n \to a\).

Answer: \dfrac{a}{2}


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

Answer: 1


Question: Determine whether the inequality \( \binom{2 \cdot 2^{1000000}}{2^{1000000}} > 2^{2 \cdot 2^{1000000}} \) is true or false.

Answer: False


Question: Let $f: \mathbb{C}^3 \to \mathbb{C}$ be a morphism of varieties that is a smooth fiber bundle. Determine if the fiber is $\mathbb{C}^2$. Provide a justification for your answer.

Answer: \mathbb{C}^2


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x \) and \( y \), the equation \( f(x^2+y) = xf(x) + f(y) \) holds.

Answer: f(x) = kx


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

Answer: \infty


Question: Find the minimum area of a triangle that circumscribes a circle with radius \( r \).

Answer: 3\sqrt{3}r^2


Question: Evaluate the triple integral \( \iiint_K \sqrt{y^2+z^2} \, dV \), where \( K \subset \mathbb{R}^3 \) is defined by the inequalities \( x \geq 0 \), \( y^2 + z^2 \leq 1 \), and \( x+y \leq 2 \).

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


Question: Compute \( 1808^{80085^{2014}} \mod 17 \) using Euler's theorem.

Answer: 11


Question: Find the complex number $z$ in algebraic form such that $z^3 = (3 + i)^6$. Express your answer as $z = a + bi$, where $a$ and $b$ are real numbers.

Answer: 8 + 6i


Question: Let $f$ be a real-valued continuous function defined on $[0,1]$ such that $f(0) = 1$, $f(1/2) = 2$, and $f(1) = 3$. Show that the limit \( \lim_{n \rightarrow \infty} \int_0^1 f(x^n)\,dx \) exists and compute its value.

Answer: 1


Question: Consider a commodity whose unit price evolves randomly from day to day with a general downward drift but with occasional upward jumps due to unforeseen events. Long-term records suggest that, independently of the past, the daily price increases by $1 with probability 0.45, declines by $2 with probability 0.5, and jumps up by $10 with probability 0.05. Let $C_0$ be the price today and $C_n$ the price $n$ days into the future. Compute $\lim_{n \to \infty} P(C_n > C_0)$. How does this probability behave as $n$ approaches infinity?

Answer: 0


Question: Find the limit: \[ \lim_{n\rightarrow\infty}\frac{1}{n^4}\left(\sum_{k=1}^{n}\ k^2\int_{k}^{k+1}x\ln\big((x-k)(k+1-x)\big)dx\right) \]

Answer: -\dfrac{1}{2}


Question: For a finitely generated module $M$ over a principal ideal domain $R$, is it true that $M$ having free rank 0 is equivalent to $M$ being a torsion module? Justify your answer.

Answer: Yes


Question:
Calculate the limit of the sequence of functions as follows:
\[ \lim_{n\to\infty} \frac{1+xn}{(1+x)^n} \]
where \(x\in (0,1]\).

Answer: 0


Question: Determine if the dimension of the symplectic group Sp(2n, \(\tilde{F}\)) over the algebraic closure of a finite field \(F\) is equal to \(2n^2 + n\).

Answer: 2n^2 + n


Question: Consider the space $\mathcal{D}((0,T);V)$ of infinitely differentiable functions with values in a Hilbert space $V$ and compact support. Are functions of the form $\sum_j \psi_n(t)w_n$, where $\psi_n \in \mathcal{D}(0,T)$ and $w_n \in V$, dense in $\mathcal{D}((0,T);V)$ under the norm $\lVert {u}\rVert_{W} = \sqrt{\int_0^T \lVert {u}\rVert_{V}^2 + \lVert {u'}\rVert_{V}^2}$?

Answer: Yes


Question: Suppose \( \sum a_n \) is a series of complex numbers such that \( \sum a_n \) and every rearrangement of it converge to the same sum. Does \( \sum a_n \) converge absolutely?

Answer: Yes


Question: Determine the behavior of the sequence \( \lim_{n \to \infty}\left(\frac{\ln(p_n!)}{\psi(p_n)}\right)_n \), where \( p_n \) is the \( n \)-th prime, \( x! \) denotes the factorial, and \( \psi(x) \) is the second Chebyshev function. Does the sequence diverge to infinity or converge to a constant?

Answer: \infty


Question: Determine the number of positive integers \( N \) between 3 and 1000 (inclusive) such that for any set of \( N \) real numbers \( \{a_i\}_{i=1}^N \) satisfying \( a_1 + a_2 + \cdots + a_N = 0 \), the inequality \( a_1a_2 + a_2a_3 + \cdots + a_{N-1}a_N + a_Na_1 \leq 0 \) holds.

Answer: 2


Question: Is $Z_q[X]/(\phi(X))$ a field if $\phi$ is an irreducible polynomial and $q$ is a prime number? Justify your answer.

Answer: Yes


Question: Find the degree of the field extension \(\mathbb{Q}(\sqrt{1+ \sqrt{2}})\) over \(\mathbb{Q}\).

Answer: 4


Question: Find the minimum distance between the point \((1,1,1)\) and the surface defined by the equation \(z - xy = 10\).

Answer: 3\sqrt{2}


Question: Determine whether the following conjecture is true for infinite sets: Let $A, B$ be two possibly infinite sets, and let $g:B\to A$. For any two functions $f_1,f_2:A\to B$ with $gf_i=id$, is there a bijection between $N(f_1)$ and $N(f_2)$, where $N(f)$ denotes the set of all $g'$ with $g'f=id$ for a given $f:A\to B$?

Answer: True


Question: Find the best approximation of the polynomial $t^3$ in the subspace $W$ of $P_9([-1,1])$, where $W$ is spanned by the polynomials $\{1, t^2, t^4, t^6, t^8\}$. Use the inner product \( \langle f,g\rangle = \int_{-1}^1 f(t)\overline{g(t)} \, \mathrm{d}t \).

Answer: 0


Question: Evaluate the integral \( \int_R x^4 y^7 \, dx \, dy \) over the region \( R \) in the positive quadrant bounded by the hyperbolas \( xy = 1 \), \( xy = 4 \), \( xy^2 = 1 \), and \( xy^2 = 5 \).

Answer: 310


Question: Does there exist an analytic function \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(1) = 1 \) and \( f(n) = 0 \) for all \( n \in \mathbb{N} \) where \( n \neq 1 \)?

Answer: Yes


Question: Given the joint probability density function \( f(x,y) = \frac{6}{7}\left(x^{2}+\frac{xy}{2}\right) \) for \( 0 < x < 1 \) and \( 0 < y < 2 \), find the conditional probability \( P\left\{Y>\frac{1}{2}|X<\frac{1}{2}\right\} \).

Answer: \dfrac{69}{80}


Question: Find the largest possible value of $b$ such that the polynomial \(x^5 - 20x^4 + bx^3 + cx^2 + dx + e = 0\) has only real roots, where \(b, c, d, e\) are real numbers.

Answer: 160


Question: Evaluate the integral \( \int_0^{2\pi} e^{\cos(\phi)}\cos(\phi - \sin(\phi)) \, d\phi \) using contour integration.

Answer: 2\pi


Question: Determine whether the Veronese map \( v_{n,d}: \mathbb{P}^n \to \mathbb{P}^N \) is a morphism.

Answer: Yes


Question: Let $A$ be a Noetherian ring, and let $\mathfrak{p}, \mathfrak{q} \subset A$ be distinct prime ideals of the same height. Suppose $N$ is an $A_\mathfrak{p}$-module of finite length. Is it true that \( \operatorname{Hom}_A(N,E(A/\mathfrak{q})) = 0 \), where $E(A/\mathfrak{q})$ is the injective hull of $A/\mathfrak{q}$?

Answer: 0


Question: Evaluate the limit \( \lim_{x \to 0} \frac{\log(6 - \frac{5}{\cos x})}{\sin^2 x} \) without using l'Hopital's rule.

Answer: -\dfrac{5}{2}


Question: Let $X = A \cup B$ be a topological space where $A$ and $B$ are disjoint subsets. Consider a continuous function $f: X \rightarrow \{-1, 1\}$. Is $f$ necessarily a constant function?

Answer: No


Question: Find a nontrivial unit polynomial \( p(x) \) in \( \mathbb{Z}_4[x] \) such that \( \deg p(x) > 1 \).

Answer: 1 + 2x^2


Question: Evaluate the expression \( \frac{1}{N} \sum_{k=0}^{N-1}|1+\cos(\frac{2k\pi}{N})|^2 \) for \( N \ge 3, N \in \mathbb{N} \).

Answer: \dfrac{3}{2}


Question: Find all analytic functions whose real part is analytic in the unit disk.

Answer: f(z) = c


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \left( x \int_x^1 \frac{f(t)}{\sin^2(t)} \, dt \right) \] where \( f \) is continuous on \([-1, 1]\) and differentiable on \((-1, 1)\).

Answer: f(0)


Question: Determine the number of distinct prime ideals in the ring \( \mathbb{Q}[x]/\langle x^4-1 \rangle \).

Answer: 3


Question: Determine whether the Frechet derivative of the operator \( A \) exists, where \( A \colon C[a,b] \to C[a,b] \) is defined by \( [A(x)](s) = \int_a^b K(s,t) x(t) \, \mathrm{d}t \) for all \( x \in C[a,b] \) and \( s \in [a,b] \). Here, \( K \colon [a,b] \times [a,b] \to \mathbb{R} \) is a continuous kernel function.

Answer: A


Question: Given that $x_1 + x_2 + x_3 = 0$, $y_1 + y_2 + y_3 = 0$, and $x_1y_1 + x_2y_2 + x_3y_3 = 0$, find the value of the expression \[ \frac{x_1^2}{x_1^2 + x_2^2 + x_3^2} + \frac{y_1^2}{y_1^2 + y_2^2 + y_3^2}. \]

Answer: \dfrac{2}{3}


Question:
Let $f(x)$ be a positive differentiable function on $[0,a]$ such that $f(0)=1$ and $f(a)=3^{\frac{1}{4}}$. If $f'(x) \geq (f(x))^3 +(f(x))^{-1}$, what is the maximum value of $a$? Choose from the following options:

(a) $\frac{\pi}{12}$
(b) $\frac{\pi}{36}$
(c) $\frac{\pi}{24}$
(d) $\frac{\pi}{48}$

Answer: C


Question: Find the minimal polynomial of \( e^{2\pi i/3} + 2 \) over \( \mathbb{Q} \).

Answer: x^2 - 3x + 3


Question: Determine the number of positive integers \( n < 180 \) such that \((n^3 + 3n + 1)^{180} \equiv 1 \pmod{180}\).

Answer: 71


Question: Determine whether the following statement is true or false: For a linear operator $T: V \to W$ between Hilbert spaces $V$ and $W$, the norm of the adjoint operator $T^*$ is always equal to the norm of $T$, i.e., $\| T^* \| = \| T \|$. Justify your answer.

Answer: True


Question: Suppose $E$ is a countable set and its derived set $E’$ is at most countable. Determine if the sequence of nested sets $E',E'', E''', E''''\ldots$ converges, where "converge" means there exists an $E_n$ such that $E_n = E_{n+k}$ for all $k=1,2,3,\ldots$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the line integral \( \int_{C} \frac{x}{x^2+y^2} \, dx + \frac{y}{x^2+y^2} \, dy \) where \( C \) is the straight line segment connecting \((1,1)\) to \((2,2)\).

Answer: \ln 2


Question: A person has seven friends and invites a different subset of three friends to dinner every night for one week (seven days). How many ways can this be done so that all friends are included at least once?

Answer: 31169376000


Question: Evaluate the limit: \[ \lim_{{n}\to{\infty}} \left(\dfrac{2^x+3^x+\cdots+n^x}{n-1}\right)^{1/x} \] for \( n > 1 \).

Answer: \infty


Question: Let $a$, $b$, and $c$ be three distinct real numbers satisfying the equation \( \frac{a}{1+ab} = \frac{b}{1+bc} = \frac{c}{1+ca} \). Find the value of the product $abc$.

Answer: -1


Question:
Find the area of the surface generated by revolving the right-hand loop of the lemniscate \( r^2 = \cos 2\theta \) about the vertical line through the origin (y-axis). Use the formula for the surface area of revolution in polar coordinates:
\[ S = 2\pi \int_\alpha^\beta r \cos\theta\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta \]

Answer: 2\sqrt{2}\pi


Question: What is the automorphism group of the symmetric group of degree three, $Sym(3)$?

Answer: S_3


Question: Determine whether the following statement is true or false: Let $f$ be a continuous, non-decreasing function defined on $[0,1]$, and let $E$ be a set of Lebesgue measure zero. Then, $f(E)$ is a set of Lebesgue measure zero.

Answer: B


Question: Find a set of elements in $K[x,y]$ that generate $K[x,y]$ as a module over the subring $A$ of symmetric polynomials, where $A$ consists of elements $f \in K[x,y]$ such that $f(x,y) = f(y,x)$. Describe the generating set.

Answer: \{1, x\}


Question: Does there exist a Borel measure or any valid measure on an infinite dimensional Banach space such that a bounded open set in this space has a positive measure?

Answer: Yes


Question: Given two random vectors \( X = (X_1, \ldots, X_n) \) and \( Y = (Y_1, \ldots, Y_n) \), with a covariance matrix \( G \) such that the covariance between \( X_i \) and \( Y_j \) is \( G_{ij} \), and two \( n \times n \) matrices \( A \) and \( B \), find the covariance matrix between \( AX \) and \( BY \).

Answer: A G B^\top


Question: Determine whether there exists a continuous function \( f: \mathbb{R}^2 \to \mathbb{R}^2 \) that is not one-to-one but still maps open sets to open sets.

Answer: Yes


Question: Find the least non-negative residue of \(3^{252} \mod 610\).

Answer: 131


Question: Evaluate the limit: \[ \lim _ { x \rightarrow \frac { \pi } { 2 } } \frac { \ln \sin x } { \cos ^ { 2 } x } \] without using L'Hopital's theorem.

Answer: -\dfrac{1}{2}


Question: Is there a set $A \subset \mathbb{R}^n$ such that the $n$-dimensional Hausdorff measure $\mathcal{H}^n(A) = 0$ but the Hausdorff dimension $\dim_H(A) = n$?

Answer: Yes


Question: Let $(R, \mathfrak{m})$ be a local ring of dimension $d > 0$, and let $M$ be a finitely generated Cohen-Macaulay $R$-module. If $P$ is a minimal prime ideal of $M$, is $P$ also a minimal prime of $R$? In other words, if $\dim M_P = 0$, does it follow that $\dim R_P = 0$?

Answer: No


Question:
Evaluate the limit given that $f(x)$ is differentiable and $f'(0)=1$:
\[ \lim_{x\to0}\frac {f(x^2)-f(0)}{\sin^2(x)} \]

Answer: 1


Question: Given a labeled clique graph $G$ with $n$ vertices, determine the number of trees that can be formed as subgraphs with exactly $e$ edges, where $e = n-1$. For example, if $G$ is a 4-clique, find the number of trees with 3 edges that can be formed as subgraphs.

Answer: n^{n-2}


Question: Let \( f(x) = x^4 - 2x^2 - 2 \in \mathbb{Q}[x] \) and \( E \) be its splitting field. Determine the degree \( [E: \mathbb{Q}] \).

Answer: 8


Question: Evaluate the limit \( \lim_{n\to \infty} \frac{(-1)^{\frac{(n-1)(n-2)}{2} }}{n} \).

Answer: 0


Question: Evaluate the infinite series: \[ \sum_{n=1}^{\infty} \frac{n^2}{2^n}. \]

Answer: 6


Question:
Determine the intrinsic dimension of the set \( M = \{ (x,y,z) \in \mathbb{R}^3 : (a,b,c,d) \in \mathbb{R}^4\} \) given the parameterized functions:
\[ x(a,b,c,d) = ac \]
\[ y(a,b,c,d) = bc + ad \]
\[ z(a,b,c,d) = bd \]

Answer: 3


Question: Express the complex number \( \dfrac{(\text{cis}\ 75^\circ-\text{cis}\ 155^\circ)(1-\cos 8^\circ+i \sin 8^\circ)}{2-2\cos 8^\circ} \) in the form \( r\ \text{cis}\ \theta \), where \( 0 \le \theta < 360^\circ \). Find \( \theta \) in degrees.

Answer: 111


Question: Given that \( f(x) = \int_0^1 e^{x+t} f(t) \, dt \), find the value of \( f(1) \).

Answer: 0


Question: Consider a sequence \( \{s_n\} \) of positive real numbers where \( s_n > 0 \) for all \( n \). Define its arithmetic means \( \sigma_n \) by \( \sigma_n = \frac{s_0 + s_1 + \cdots + s_n}{n + 1} \) for \( n = 0, 1, 2, \ldots \). Can it happen that \( \limsup s_n = \infty \) while \( \lim \sigma_n = 0 \)? Provide an example or prove why it is not possible.

Answer: Yes


Question: Prove that \( \lim \limits_{n \to \infty}\frac{n^2+4}{(n-1)(n+2)(n-3)}=0 \) using the formal definition of the limit of a sequence.

Answer: 0


Question: Determine whether the equation $n+1 = x^{\varphi(n)}$ has a solution for every positive integer $n$, where $\varphi(n)$ is Euler's totient function.

Answer: No


Question:
Determine the number of integral values of $x$ that satisfy the inequality:
\[ \frac{(e^x-1)(\sin(x)-2)(x^2-5x+4)}{x^2(-x^2+x-2)(2x+3)} \le 0 \]

Answer: 5


Question: If $H$ is a subnormal Hall subgroup of a finite group $G$, is $H$ necessarily normal in $G$? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the symmetric group $S_4$ can be generated by the permutations $(1,2)$, $(1,3)$, and $(1,4)$. Provide a justification for your answer.

Answer: Yes


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

Answer: -\infty


Question: Find the exact minimum value of the function \(f(x,y,z)=(7(x^2+y^2+z^2)+6(xy+yz+zx))(x^2y^2+y^2z^2+z^2x^2)\) subject to the constraint \(xyz=1\).

Answer: 117


Question: Let \(p\) be a prime number. If the topological spaces \(\mathbb{Q}_p^n\) and \(\mathbb{Q}_p^m\) are homeomorphic, must \(n = m\)?

Answer: Yes


Question: Determine whether the function \( f(x) = x^{\sin(x)} \) is uniformly continuous on the interval \((0,1]\).

Answer: Yes


Question: Find the intersection point of the projective lines given by the homogeneous coordinates: Line $a$ through $[1:0:1]$ and $[1:1:1]$, and Line $b$ through $[1:1:0]$ and $[0:1:0]$. Express your answer in homogeneous coordinates.

Answer: [0:1:0]


Question: Calculate the expected number of turns required to jump from the ground floor (floor 0) to the 5th floor in a building with 5 floors. The probability of jumping from the ground to floor 1 is 1. For each floor i (where i ranges from 1 to 4), the probability of jumping to the next floor (i+1) is \( \frac{1}{3^i} \), the probability of staying on the same floor is \( \frac{3^i-1}{2 \times 3^i} \), and the probability of jumping back to the previous floor (i-1) is \( \frac{3^i-1}{2 \times 3^i} \).

Answer: 14543


Question: Find the value of \( \lambda > 0 \) such that the function \( f(x) = \begin{cases} 0 & x = 0 \\ |x|^{\lambda} \sin\left(\frac{1}{x}\right) & x \neq 0 \end{cases} \) is differentiable at zero and has a continuous derivative at zero.

Answer: 2


Question:
Determine which of the following conditions is necessary and sufficient for the set $F$, defined as above, to be a field:

(a) $R$ is a commutative ring but not an integral domain
(b) $R$ is an integral domain but not a commutative ring
(c) $R$ is a commutative integral domain
(d) None of the above

Answer: C


Question: Let $f: X \to Y$ and $g: Y \to (Z, U'')$ be mappings. Let $U'$ be the uniform structure on $Y$ induced by $g$, and $U$ be the uniform structure on $X$ induced by $f$. Let $U_1$ be the uniform structure on $X$ induced by the composition $g \circ f$. Determine if $U = U_1$. Provide a proof or counterexample.

Answer: U = U_1


Question: Consider a set of 10 labeled vertices \( \{1, 2, \ldots, 10\} \). How many labeled trees can be formed such that exactly two of these vertices have a degree of 5?

Answer: 3150


Question: Compute the double limit \( \lim_{\delta \rightarrow 0, k\rightarrow\infty} (1+\delta)^{ak} \), where \( a \) is a positive constant.

Answer: e^{a}


Question: Determine the exponent of convergence \(a\) such that \(\lim_{n\to\infty} n^a \left(\frac{p_{n+1}}{p_n} - 1\right)\) is finite, where \(p_k\) is the \(k\)-th prime number.

Answer: 1


Question: Find the units digit of the expression \(2019^{2018^{2017^{\cdots^{3^{2^{1}}}}}}\).

Answer: 1


Question: Compute the expected value \( E[1/X] \) for a random variable \( X \) that follows an exponential distribution with parameter \( \lambda = 5 \).

Answer: \infty


Question: Suppose that whether or not it rains today depends on the weather conditions over the previous two days. If it has rained for the past two days, then it will rain today with probability 0.7. If it did not rain for any of the past two days, then it will rain today with probability 0.4. In any other case, the weather today will, with probability 0.5, be the same as the weather yesterday. Describe the weather condition over time with a Markov chain. Suppose that it rained today and yesterday. Calculate the expected number of days until it rains three consecutive days for the first time in the future.

Answer: \dfrac{59}{14}


Question: Evaluate the Lebesgue integral of the function \( f(x) = \begin{cases} 3 & x \in \mathbb{Q} \cap [0,1] \\ 2 & x \in \mathbb{Q}^c \cap [0,1] \end{cases} \) over the interval \([0,1]\).

Answer: 2


Question: Consider the non-homogeneous second order differential equation \( t^2y''-3ty'+4y = t \). Find a solution of the form \( y = t^r \) by direct substitution. How many such solutions exist?

Answer: 1


Question: Given three random variables $X$, $Y$, and $Z$ with correlations $\operatorname{corr}(X,Y) = 0.5$ and $\operatorname{corr}(X,Z) = -0.5$, determine the exact range for $\operatorname{corr}(Y, Z)$. Provide your answer as an interval.

Answer: [-1, 0.5]


Question: Given that \( f(f(x)) = x^2 + 2 \) and \( f \) is an increasing function for positive integers, find the value of \( f(3) \).

Answer: 6


Question: A runner is moving around a circular track with a radius of \( \frac{1}{10} \) miles. The runner's speed in miles per hour is given by the function \( v(\theta) = 5 + 3\cos \theta \), where \( \theta \) is the angle in radians. Calculate the time it takes for the runner to complete one full lap around the track.

Answer: \dfrac{\pi}{20}


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

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k = n^3}^{(n+1)^3}\frac{1}{\sqrt[3]{k^2 +4k}} \]

Answer: 3


Question: Calculate the limit: \[ \lim_{x\rightarrow 0^+} \frac{\sqrt{x^2+x^3} - \sin(x)}{2x^2 - e^{-1/x}} \] without using L'Hôpital's theorem.

Answer: \dfrac{1}{4}


Question: Calculate the value of the infinite product: \[ \sqrt{\frac{1}{2}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}+ \frac{1}{2}\sqrt{\frac{1}{2}}}} \times \ldots \] Express your answer as a single numerical value.

Answer: \dfrac{2}{\pi}


Question: Determine whether a continuously differentiable function in a Banach space $X$ is locally Lipschitz in $X$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the Lebesgue integral of the function \(x^2\) over the interval \((0,1]\).

Answer: \dfrac{1}{3}


Question: Given that \( f(x) \) is continuous on \([0,1]\) and \( \int^{1}_{0} f(x) \, \mathrm{d}x = \sqrt{2} \), compute the value of the double integral \( \int_{0}^{1} \int^{1}_{x} f(x)f(y) \, \mathrm{d}y \, \mathrm{d}x \).

Answer: 1


Question: Given any multiset of positive integers of size \(2n-1\), determine if it is always possible to find a multisubset of size \(n\) such that the sum of the integers in this multisubset is divisible by \(n\).

Answer: Yes


Question:
Evaluate the limit:
\[ \lim_{t\to\infty} \frac{1}{t}\exp{\left(\frac{\sqrt[4]{2} \left(\sqrt{\pi }-1\right) t^B e^{-\frac{t^{2 B}}{2 \sqrt{2}}}}{\pi }-\frac{\left(2 \sqrt{2} \left(\sqrt{\pi }-1\right) t^{2 B}+\sqrt{\pi }+2\right) \operatorname{erfc}\left(\frac{t^B}{2^{3/4}}\right)}{4 \sqrt{\pi }}+\frac{1}{2}\right)}, \]
where \( \operatorname{erfc} \left(z\right) \) is the complementary error function given by
\[ \operatorname{erfc}(z) = \frac{2}{\sqrt{\pi}}\int_z^{\infty} \mathrm{e}^{-t^2} \,dt \]
and \(0 \le B \le 1\).

Answer: 0


Question:
Determine the range of \( \alpha \) for which all the points of local extrema of the function \( f(x) = x^3 - 3\alpha x^2 + 3(\alpha^2 - 1)x + 1 \) lie in the interval \((-2, 4)\). Choose the correct option:

(A) \((-1, 3)\)
(B) \((3, 4)\)
(C) \((-4, -2)\)
(D) \((-2, -1)\)

Answer: A


Question: Determine the irreducible elements of the ring of numbers of the form $2^a b$, where $a$ and $b$ are integers.

Answer: \pm 2^k p


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{1}{\sin^3x}\int_0^x{\sin(t^2) dt} \]

Answer: \dfrac{1}{3}


Question: Compute the limit: \[ \lim_{n \to \infty} n^{\frac{1}{n!}} \]

Answer: 1


Question: Use continued fractions to find the minimal solution to the Pell's equation \(x^2 - 11y^2 = 1\).

Answer: (10, 3)


Question: Determine whether the Lebesgue integral \( \int\limits_\Omega d\mu \) over an unbounded domain \( \Omega \) is equal to the measure \(|\Omega|\). Is this statement true?

Answer: True


Question: Determine the order of the group $GL_3(\mathbb{Z_4})$, which consists of all invertible $3 \times 3$ matrices with entries from $\mathbb{Z}_4$. Provide your answer as a single integer.

Answer: 86016


Question: Evaluate the integral \( \int_0^{2\pi} f(e^{it}) \cos t \, dt \) where \( f(z) \) is an analytic function.

Answer: \pi f'(0)


Question: Evaluate the limit \( \lim_{n\to\infty} n\cdot e^{\mathcal{O}(n\cdot z_n^2)} \), where \((z_n)\) is a sequence such that \(z_n\to 0\) as \(n\to\infty\).

Answer: \infty


Question: A block is released from point O on an incline where the x-axis is along the incline. The coefficient of friction varies as \( \mu = kx \) with \( k = \tan\theta \). Given that the acceleration of the block is \( g\sin\theta(1-x) \), find the maximum distance traveled by the block.

Answer: 2


Question: Find the value of \( a \) if \( a = \sqrt[3]{8 \div \sqrt[3]{8 \div \sqrt[3]{8 \div \sqrt[3]{8 \div \cdots} } } } \).

Answer: 2^{\frac{3}{4}}


Question: Evaluate the limit \( \lim_{x \rightarrow 0}\frac{\tan x - x - \frac{x^3}{3}}{\sin^5x} \) using L'Hopital's Rule.

Answer: \dfrac{2}{15}


Question: Calculate the path integral \( S \) of \( \frac{1}{z^2-1} \) along the curve \( |z|=2 \). Use the partial fraction expansion \( \frac{1}{z^2-1} = -\frac{1}{2(z+1)} + \frac{1}{2(z-1)} \) to evaluate the integral.

Answer: 0


Question: Consider a standard Brownian motion \((W_t)_{t \ge 0}\) and a stopping time \(\tau\) that lies in the interval \([1,2]\). For given positive constants \(x\) and \(y\), determine if the expectation \(\mathbb{E}\left[\mathbf{1}_{\{x+\inf_{0 \le t \le 2} W_t > 0\}} (W_{\tau} - y)^+\right]\) is strictly positive.

Answer: Yes


Question:
Evaluate the limit:
\[ \lim_{x \to + \infty} \frac{\mathrm{d}}{\mathrm{d}x} \int_{2 \sin \frac{1}{x}}^{3 \sqrt{x}} \frac{3 t^4 + 1}{(t-3)(t^2 + 3)} \,\mathrm{d}t \]

Answer: \dfrac{27}{2}


Question: Do separable $C^*$-algebras form a set?

Answer: Yes


Question: If $K:F$ is a finite field extension and $F \subseteq E \subseteq K$, is the extension $K:E$ also finite? Justify your answer.

Answer: Yes


Question: Find the mass of a wire formed by the intersection of the sphere \( x^2 + y^2 + z^2 = 1 \) and the plane \( x + y + z = 0 \), given that the density at \((x, y, z)\) is \(p(x, y, z) = x^2\) per unit length of the wire.

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


Question: Prove that if $H$ is a subgroup of the group of rational numbers $\mathbb{Q}$ under addition with finite index, then $H = \mathbb{Q}$.

Answer: \mathbb{Q}


Question: What is the expected value of the number of children, $2n$, that a couple must have in order to have an equal number of sons and daughters for the first time? Assume that each child is equally likely to be a boy or a girl.

Answer: \infty


Question:
Consider the ninth-degree polynomial \((x-3)^9+(x-3^2)^9+(x-3^3)^9+\ldots+(x-3^9)^9=0\). Determine the nature of its roots. Choose the correct option:

A) All real roots
B) Roots are 3, 3^2, ..., 3^9
C) 1 real and 8 imaginary roots
D) 5 real and 4 imaginary roots

Answer: C


Question: Determine whether the following statement is true or false: If $X(t)$ is an Itô process and $p(t)$ is a non-stochastic function, then the ordinary chain rule applies, i.e., \[d(X(t)p(t)) = dX(t)p(t) + X(t)p'(t)dt.\]

Answer: True


Question: Is it true that every $n$-dimensional real Lie algebra is isomorphic to $(\mathbb{R}^n, [\cdot, \cdot])$ for some Lie bracket $[\cdot, \cdot]$ on $\mathbb{R}^n$?

Answer: Yes


Question: Use Stokes' Theorem to evaluate the line integral \( \int_C F \cdot d\mathbf{r} \) where \( F = -y^3 \, dx + x^3 \, dy - z^3 \, dz \) and \( C \) is the intersection of the cylinder \( x^2 + y^2 = 1 \) and the plane \( x + y + z = 1 \).

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


Question: Evaluate the integral \( \int_{\gamma}\frac{\sin(z)}{z^3}\,dz \) where \( \gamma \) is the unit circle centered at the origin. Use the definition \( \sin(z)=\frac{e^{iz}-e^{-iz}}{2i} \) and apply the residue theorem to determine the value of the integral.

Answer: 0


Question: Suppose you have a random vector $Z=(Z_1,Z_2,\ldots,Z_n)^T$, where $Z$ follows the standard Gaussian distribution on $\Bbb R^n$. Are the components $Z_i$, $i=1,2,\ldots,n$, independent? Answer with true or false.

Answer: True


Question: Calculate the flux of the vector field \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \) over the cylindrical surface \( S \) defined by \( x^2 + y^2 = a^2 \) for \( 0 \leq z \leq 2a \). Verify your result using the divergence theorem.

Answer: 4\pi a^3


Question: Evaluate the contour integral \( \int_C \frac{\cos(z)}{z^2} \, dz \), where \( C \) is any circle enclosing the origin and oriented counter-clockwise. Verify if the integral equals zero using residue theory.

Answer: 0


Question: Evaluate the integral \( \int _C \frac{\exp(z^2)}{z^2\left(z-1-i\right)}dz \), where \( C \) consists of the contour \(|z|=2\) oriented counterclockwise and \(|z|=1\) oriented clockwise.

Answer: \pi e^{2i}


Question:
Given the equations:
\[ ax + by = 7 \]
\[ ax^2 + by^2 = 49 \]
\[ ax^3 + by^3 = 133 \]
\[ ax^4 + by^4 = 406 \]
find the value of the expression:
\[ 2014(x+y-xy) - 100(a+b) \]

Answer: 5956


Question: Let \( \{Y_n\}_{n=1}^{\infty} \) be a sequence of random variables, and define \( F_n = \sigma(Y_1,Y_2,\dots,Y_n) \) for each \( n \). Let \( \{X_n\}_{n=1}^{\infty} \) be a sequence of random variables adapted with respect to \( \{F_n\} \), meaning \( X_n \) is \( F_n \)-measurable. Is it true that for each \( n \), there exists a function \( h_n:\mathbb{R}^{n} \to \mathbb{R} \) such that \( X_n = h_n(Y_1,Y_2,\dots,Y_n) \)?

Answer: Yes


Question: Let \( f(x) = x^4 - 6x^2 + 5 \). If \( P(x_0, y_0) \) is a point such that \( y_0 > f(x_0) \) and there are exactly two distinct tangents through \( P \) drawn to the curve \( y = f(x) \), find the maximum possible value of \( y_0 \).

Answer: 8


Question: Determine the positive integers \(x\) for which there exists a positive integer \(X\) such that none of the integers \(1+X, 2+X, \ldots, x+X\) is a power of a prime number.

Answer: \mathbb{N}


Question: Let $f: \mathbb{R}^n \to \mathbb{R}^p$ and $g: \mathbb{R}^n \to \mathbb{R}^q$ be two non-constant functions. If $f(x) = f(y)$ implies $g(x) = g(y)$ for all $x, y \in \mathbb{R}^n$, is $g$ a function of $f$? In other words, does there exist a function $h$ such that $g(x) = h(f(x))$ for all $x \in \mathbb{R}^n$?

Answer: Yes


Question:
Verify the identity for any $c \in C$ in a coalgebra with comultiplication $\Delta: C \rightarrow C \otimes C$ and counit $\varepsilon: C \rightarrow \mathbb{k}$:
\[ \sum_{(c)} \varepsilon(c_{(2)}) \otimes \Delta(c_{(1)}) = \Delta(c) \]
where $\Delta(c) = \sum_{(c)} c_{(1)} \otimes c_{(2)}$ using the sigma notation.

Answer: \Delta(c)


Question: Determine whether there exist distinct natural numbers $a$ and $b$ such that for all non-negative integers $x$, $a + x$ is prime if and only if $b + x$ is prime. Provide a justification for your answer.

Answer: No


Question: Let $a_k \geq 0$ and $b_k \geq 0$ be real sequences such that $\limsup\limits_{k \rightarrow \infty} \ \sqrt[k]{a_k} \leq 1$ and $\limsup\limits_{k \rightarrow \infty} \ \sqrt[k]{b_k} \leq 1$. Determine whether it is true that $\limsup\limits_{k \rightarrow \infty} \sqrt[k]{a_k+b_k} \leq 1$.

Answer: True


Question: Consider a codimension one foliation \(\mathcal{F}\) in a closed 3-manifold \(M\). Determine whether there exists an upper bound for the genus of the compact orientable leaves. Specifically, is there a constant \(G > 0\) such that the genus of every compact orientable leaf is at most \(G\)?

Answer: Yes


Question: Let $G$ be a finite group with a normal Sylow $p$-subgroup $P$ such that $|P|=p^2$, where $p \neq 2$. Assume $P$ is not cyclic, meaning it does not contain an element of order $p^2$. If $g \in G$ has order $k$ with $\gcd(k,p)=1$, is there an element in $G$ of order $kp$? Provide a justification for your answer.

Answer: No


Question: Find the residue of the function \( z^2 \sin\left(\frac{1}{z}\right) \) at \( z = 0 \).

Answer: -\dfrac{1}{6}


Question: Given a Poisson process \( N = \{N(t) \mid t \geq 0\} \) with parameter \( \lambda > 0 \), calculate the conditional probability \( P(N(3) = 2 \mid N(1) = 0, N(5) = 4) \).

Answer: \dfrac{3}{8}


Question: Does there exist a function whose local or absolute minimum is greater than 1 such that its integral from 0 to infinity converges?

Answer: Yes


Question:
In a continuous-time Markov chain, Sleeping Beauty (SB) can be in one of three states while she sleeps: State 0 (flat on her back), State 1 (foetal position), and State 2 (foetal position and sucking her thumb). The transition rate matrix is given by:

\[ G = \begin{bmatrix} -\frac{1}{2} & \frac{3}{8} & \frac{1}{8} \\
\frac{1}{12} & -\frac{1}{3} & \frac{1}{4} \\
\frac{1}{4} & \frac{3}{4} & -1 \end{bmatrix} \]

What is the probability that Prince Charming finds SB in State 0 (looking truly radiant) after a long time? Assume SB has been sleeping for a very long time.

Answer: \dfrac{7}{36}


Question: Evaluate the line integral \( \int_C (x^2-2xy)dx + (y^2-2xy)dy \) where \( C \) is the parabola \( y=x^2 \) for \( -1 \leq x \leq 1 \).

Answer: -\dfrac{14}{15}


Question: A horizontal cylindrical tank has a diameter of 3 feet. The tank is filled with gas, and a dipstick is used to measure the gas level. Determine the height on the dipstick that corresponds to the tank being 3/8 full.

Answer: 1.20


Question: Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be a simple function, defined as $f(x) = \sum_{i = 1}^{n} a_i \mathbb{1}_{A_i}(x)$ for all $x \in \mathbb{R}$, where each $a_i$ is a constant, each $A_i$ is a measurable set, and $\mathbb{1}_{A_i}$ denotes the indicator function of the set $A_i$. If each $A_i$ is an interval, then $f$ is called a step function. Consider a non-decreasing simple function $f$, meaning $x \leq y$ implies $f(x) \leq f(y)$. Is $f$ necessarily a step function in this case?

Answer: Yes


Question: Determine whether the following statement is true or false: Given three non-abelian groups of order 8, two of them must be isomorphic.

Answer: A


Question: Let $P_n$ be the space of polynomials over $\mathbb{R}$ of degree at most $n$. Consider the map $T:P_n\to P_{n+1}$ defined by $T(p(x))=p'(x)-\int_{0}^{x}p(t)dt$. Determine whether $\ker T=\{0\}$.

Answer: \ker T = \{0\}


Question: Determine the number of non-isomorphic tournaments (directed cliques) with 5 vertices.

Answer: 12


Question: Let $R$ be a commutative ring with unity and $I$ be an ideal of $R$. Determine whether $\text{Hom}_R(R/I, R/I) \cong R/I$ as $R$-modules.

Answer: R/I


Question: Suppose that $s = a + bi$ is a complex root of the function $f(s) = 6^s + 2^s - 3^s$. Is the complex conjugate $\overline{s} = a - bi$ also a root of the function?

Answer: Yes


Question: Determine the minimum possible length of a string that contains all two-digit numbers from 11 to 99 as substrings, using only the digits \( \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \). Construct such a string of minimum length.

Answer: 82


Question: Is it true that two birational projective nonsingular curves have the same genus?

Answer: Yes


Question: Let $R$ be a ring of characteristic $0$, and let $G$ be a finite group with $H$ as a subgroup of $G$. If $M$ is a projective $R[G]$-module, is $M$ also projective as an $R[H]$-module?

Answer: Yes


Question: Find all pairs of positive integers \((x, y)\) such that \(x^3 - 5x + 10 = 2^y\).

Answer: (2, 3)


Question: Given that $P(x) - P'(x) = x^n$, where $n$ is a positive integer, find the value of $P(0)$. Assume $P(x)$ is a polynomial.

Answer: n!


Question:
Using the transformation \( u = \frac{w}{y} \) in the partial differential equation:
\[ x\frac{du}{dx} = u + y\frac{du}{dy} \]
find the form of the solution for the transformed equation, expressed as \( w = ? \).

Answer: w = F(xy)


Question: Let $A$ be a bounded sublattice of the bounded lattice $(X, \le)$ with $\max A = \max X$ and $\min A = \min X$. If $a, b \in X$ are complements and $a \in A$, is $b \in A$?

Answer: No


Question: Determine the number of solutions in natural numbers to the equation \( \frac{1}{x} + \frac{1}{y} = \frac{1}{1995} \).

Answer: 81


Question:
Given the joint probability density function of two real-valued random variables $X$ and $Y$:
\[
f_{X,Y}(x,y) = \begin{cases} \frac{1}{4}(x+y), & 0 \leq x \leq y \leq 2 \\ 0, & \text{otherwise} \end{cases}
\]
calculate the probability $\mathbb{P}\{Y < 2X\}$.

Answer: \dfrac{7}{12}


Question: Let \( p \) be a prime such that \( p \equiv 1 \pmod{4} \). Prove or disprove that there always exists a prime \( q \equiv 3 \pmod{4} \) that divides \( p + a^2 \) for some integer \( a \).

Answer: True


Question: Determine the cardinality of the set \( M = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid 2x + y \in \mathbb{N} \text{ and } x - 2y \in \mathbb{N} \} \).

Answer: \aleph_0


Question: Determine whether the following statement is true or false: For a non-negative Riemann integrable function $f$ on $[a,b]$, is it true that $$ \sup_P \sum_{j=1}^{n} |f(c_j)(x_j-x_{j-1})| = \int_{a}^{b} |f(x)|dx $$ where $c_j \in [x_{j-1}, x_j]$?

Answer: B


Question: Find the limit of the sequence as \( L \to \infty \): \[ (1 - \exp(-cL))^{\delta L} \] where \( c \) and \( \delta \) are positive constants.

Answer: 1


Question:
Given a labeled graph with the following adjacency matrix:

\[ A = \begin{pmatrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{pmatrix} \]

and the degree matrix:

\[ D = \begin{pmatrix} 2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 2 \end{pmatrix} \]

Calculate the number of spanning trees in the graph using the matrix tree theorem.

Answer: 8


Question: Determine whether there exist two vectors $x, y$ in a Hilbert space $H$ such that \((x,y) \geqslant \frac{k}{\|x-y\|^{2}}\), where $(x,y)$ denotes the inner product, $\|x\|$ denotes the norm, and $k > 0$.

Answer: Yes


Question: If a function \( f: \mathbb{R} \to \mathbb{R} \) is not monotonic, must there exist three points \( x < y < z \) such that either \( f(y) < f(x), f(z) \) or \( f(y) > f(x), f(z) \)?

Answer: Yes


Question: Compute the area under the curve of the function \( f \) defined on \([0, 1]\), where \( f(0) = 0 \) and \( f(x) = 2^n \) for \( x \in (2^{-2n}, 2^{-2n+2}] \), \( n = 1,2,\ldots \). Determine if the Riemann integral of \( f \) exists formally.

Answer: 3