Question: Solve the nonlinear partial differential equation using the method of characteristic curves: $$\left\{\begin{matrix} -xu_x+uu_y=y & \\ u(x,2x)=0& \end{matrix}\right.$$

Answer: u(x, y) = y - 2x


Question: Find the probability distribution of the random variable $T = X^2 + Y^2$, where $(X, Y)$ is uniformly distributed over the unit disk.

Answer: 1


Question: Determine whether there exists a metric $d$ such that the metric space $(\mathbb{Q},d)$ is complete.

Answer: Yes


Question: Find all continuous functions \( f: (c,d) \to \mathbb{R} \), where \( c,d \in \mathbb{R} \), that satisfy the equation: \[ f \left( \frac{x+y}{2}\right )=\frac{f(x)+f(y)}{2} \]

Answer: f(x) = ax + b


Question: Find the last two digits of \(77777^{77777}\).

Answer: 97


Question: Given the matrix \( A=\left[\begin{matrix}1&-2&3&-2\\1&1&0&3\\-1&1&1&-1\\0&-3&1&1\end{matrix}\right] \), determine the smallest disc in the complex plane \(\mathbb{C}\) that contains all its eigenvalues. The options are: \( |z-1|\le 7 \), \( |z-1|\le 6 \), and \( |z-1|\le 4 \). The characteristic polynomial of \( A \) is \( x^4-4x^3+21x^2-48x+46 \). Without explicitly computing the eigenvalues, use an alternative method to identify the correct interval.

Answer: |z-1|\le 6


Question: Given that $f(x) \leq x$ for all $x$, $f(0) = 0$, and $f'(x)$ exists for all $x$, determine if it is possible for $f'(x)$ to be greater than 1 for some values of $x$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the double integral \( \iint_\Omega \left(\sqrt{\frac xa}+\sqrt{\frac yb}\right)\,dxdy \) where \(\Omega\) is the region bounded by the curve \(\sqrt{\dfrac xa}+\sqrt{\dfrac yb} = 1\) and the coordinate axes.

Answer: \dfrac{2ab}{15}


Question: If there exists an injection $f: X \to Y$ for some sets $X$ and $Y$, does this imply there exists an injection $g: P(Y) \to P(X)$? Provide a yes or no answer.

Answer: No


Question: In the space \(\mathbb{R}^3\), let \(S\) be the set of all spheres and \(C\) be the set of all circles. Consider the topologies \(\tau_3\) and \(\tau_4\) on \(\mathbb{R}^3\) generated by the subbases \(S\) and \(C\), respectively. Determine the relationship between the topologies \(\tau_3\) and \(\tau_4\).

Answer: \tau_3 = \tau_4


Question: Determine whether there exists a rational number $r$ such that $\ln(r)$ is also a rational number. Provide a proof or justification for your answer.

Answer: 1


Question: Consider a finite group $G$ with $p$ as the smallest prime divisor of its order. If $x \in G$ has order $p$ and for some $h \in G$, it holds that $hxh^{-1} = x^{10}$, does this imply that $p = 3$? Provide a justification for your answer.

Answer: 3


Question: Evaluate the limit: $$\lim_{x \to 0}\left({\frac{\cos(ax)-\cos(bx)}{\sin^2(ax)-\sin^2(bx)}}\right)$$ without using L'Hopital's rule or Taylor expansions.

Answer: -\dfrac{1}{2}


Question: Let \( A \) and \( B \) be two \( d \)-dimensional vectors with all elements initially equal to 1. We randomly select one element from each vector with replacement and change it to 0, repeating this process \( m \) times. If \( 2m \leq d \), what is the expected number of elements in \( A - B \) that are equal to 2?

Answer: d \left(1 - \frac{1}{d}\right)^{2m}


Question: Calculate \( \frac{\partial^2 g}{\partial v \, \partial u}(2,2) \) for \( g(u,v)=vf(v^2, 2uv) \) given that \( f(4,8)=2 \), \( \nabla f(4,8)=(5,1) \), \( \frac{\partial^2f}{\partial x^2}(4,8)= \frac{\partial^2f}{\partial y^2}(4,8)=2 \) and \( \frac{\partial^2f}{\partial x \, \partial y}(4,8)=-2 \).

Answer: 8


Question: Let $X$ be a Banach space that is not reflexive. Determine whether there exists a linear functional $f \in X'$ such that there is no element $x \in X$ satisfying $f(x) = \|f\| \cdot \|x\|$. Provide a justification for your answer.

Answer: Yes


Question: Determine all finite sets \( A \) such that \( A \times \mathcal{P}(A) = \mathcal{P}(A) \times A \), where \( \mathcal{P}(A) \) denotes the power set of \( A \).

Answer: \emptyset


Question: Find the Laurent series expansion of the function \( f(z) = \frac{z}{(z+1)(z+2)} \) around \( z = -2 \). Determine the residue of \( f(z) \) at \( z = -2 \) and state the region of convergence for the series expansion.

Answer: 2


Question: Determine the limit of the sequence $t_1, t_2, t_3, \ldots$ where $t_1 = \ln(n)$, $t_2 = \ln(n - t_1)$, $t_3 = \ln(n - t_2)$, and so on, given that $n$ is a positive integer.

Answer: n - W(e^n)


Question: In a general measure space, can a non-empty measurable set have zero measure?

Answer: Yes


Question: Determine the unit digit of the sum of all divisors of \(20^{90} + 1\) that are of the form \(2^{p} \cdot 5^{q}\), where \(p\) and \(q\) are positive integers.

Answer: 0


Question: Determine the set \(\bigcap_{x \in A} \bigcup_{n \in \Bbb N} \left(\frac{1}{n}, x-\frac{1}{n^2}\right)\) where \(A=\{ x \in \Bbb R \mid x > \pi\}\).

Answer: (0, \pi]


Question: Find the value of \(\alpha \in \mathbb{R}\) such that the function \(f(x) = \begin{cases} x^\alpha \sin\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}\) is twice differentiable at \(x = 0\).

Answer: \alpha > 3


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

Answer: J_0(2\sqrt{x})


Question: Determine whether there exists a permutation matrix $P$ such that for a given $n \times n$ matrix $A$ with nonnegative integer entries, where each row and column sums to $m \in \mathbb{N}$, the matrix $A - P$ consists only of nonnegative entries.

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{x^{2n+1}}{(x^2 + 1)^{n + 2}}\ dx \) using the Ramanujan-MT method.

Answer: 0


Question:
Evaluate the derivative of the following double integral with respect to $x$, given that $f(x,y)$ is a joint density function such that $f(\pm \infty, y) = 0$ for all $y$:
\[ \frac{\partial}{\partial x} \left( \int_{-\infty}^\infty \int_{-\infty}^{g(x)} f(x,y) \, dy \, dx \right) \]

Answer: 0


Question: Given integers $a, b, c,$ and $d$ that satisfy the conditions $ab + cd = 44$ and $ad - bc = 9$, find the minimum possible value of $a^2 + b^2 + c^2 + d^2$. Express your answer as an integer.

Answer: 2018


Question: Find the limit of the sequence \(\left(\frac{N_{2r}}{N_r}\right)_{r\in\mathbb{N}}\), where \(N_r\) represents the number of unit squares with integer coordinates that intersect a circle of radius \(r\) centered at the origin.

Answer: 2


Question: Let $f$ be a continuous function over the interval $[-1,1]$. Prove that \( \lim_{x \to 0}\frac{1}{x}\int_0^x f(t)\,dt = f(0) \).

Answer: f(0)


Question:
Evaluate the expression:
\[ \left(a_{0}-4 a_{2}+16 a_{4}+\cdots-2^{14} a_{14}\right)^{2}+\left(2 a_{1}-8 a_{3}+32 a_{5} \cdots-2^{15} a_{15}\right)^{2} \]
where for a complex number \(z\), we have the binomial expansion \((1+z)^{15}=a_{0}+a_{1} z+\cdots+a_{15} z^{15}\).

Answer: 5^{15}


Question: Find the remainder when the polynomial $f(x)$, which satisfies the conditions $f(x) \equiv 5 \pmod{(x-1)}$, $f(x) \equiv 3 \pmod{(x+1)}$, and $f(x) \equiv 5 \pmod{(x+2)}$, is divided by $x^3 + 2x^2 - x - 2$. The polynomial $x^3 + 2x^2 - x - 2$ can be factored as $(x-1)(x+1)(x+2)$. Determine the remainder.

Answer: x^2 + x + 3


Question: Determine the limit of the series \( y_n = \frac{\lfloor x\rfloor + \lfloor 4^2x\rfloor + \lfloor 7^2x\rfloor + \ldots + \lfloor (3n-2)^2x\rfloor}{n^3} \) as \( n \) approaches infinity, where \( \lfloor \cdot \rfloor \) denotes the floor function.

Answer: 3x


Question: Find the maximum perimeter of a rectangle that can be inscribed in the region enclosed by the $x$-axis, the $y$-axis, and the parabola $y=(x-1)^2$ in the first quadrant.

Answer: 2


Question: Evaluate the triple integral \(\int_{0}^{6} \int_{0}^{12-2y} \int_{0}^{4-\frac{2y}{3}-\frac{x}{3}} x \, dz \, dx \, dy\) and find the correct value. Identify and correct any mistakes in the simplification process to ensure the result is \(144\).

Answer: 144


Question: Given a set $A = \{1, \dots, n\}$ and $n$ linearly independent maps $f_1, \dots, f_n: A \to \mathbb{R}$, determine if it is always possible to find elements $x_1, \dots, x_n \in A$ such that the matrix \( \begin{pmatrix} f_1(x_1) & \cdots & f_1(x_n) \\ \vdots & \ddots & \vdots \\ f_n(x_1) & \cdots & f_n(x_n) \end{pmatrix} \) is invertible.

Answer: Yes


Question: Determine whether the series \( S_{N} = \sum_{n=0}^{N-1} \left( \frac{c_{n}^{N}}{c_{N}^{N}} \right) \) converges as \( N \to \infty \), where \( c_{n}^{N} \) is the coefficient of \( x^{n} \) in the Chebyshev polynomial \( T_{N}(x) = \sum_{n=0}^{N} c_{n}^{N} x^n \) for \( x \in [-1, 1] \).

Answer: -1


Question: The number of emails received at work per hour follows a Poisson distribution with a rate of 4, denoted as $X \sim Po(4)$. Tomorrow, you will work for a uniformly distributed duration of T hours, where T can be 2, 3, 4, or 5 hours, independent of email arrivals. Find the variance of the total number of emails received during this time.

Answer: 34


Question:
Given a matrix $A$ of a strongly k-regular graph $G(srg(n,k,\lambda,\mu);\lambda ,\mu >0;k>3)$, the matrix $A$ can be divided into 4 submatrices based on adjacency of vertex $x \in G$. Assume:

1. $C$ is always the adjacency matrix of a regular graph and bigger than $D$.
2. There exists an algorithm that always orders $D$ (for a vertex $x \in G$) in $O(K)$ time.
3. Each row of $E$ has a different permutation.

For $n$ vertices, there will be a total of $n$ numbers of $C,D$, each taking $O(K)$ time to sort. If each $C$ takes time $O(f(n))$ to sort, then the total complexity will be $O(n \cdot K \cdot f(n))$.

Does there exist a polynomial time algorithm to sort $C$ so that $B = A$? In other words, is there a polynomial $f$?

Answer: Yes


Question: Evaluate the integral: $$\int_{-\pi/2}^{\pi/2} \frac{\sec\theta}{2\tan^2\theta+1}\,\mathrm{d}\theta$$

Answer: \dfrac{\pi}{2}


Question:
Calculate the determinant of the matrix \(A\) of size \(2n\) given by:

\[A = \begin{pmatrix} a & 0 & 0 & \cdots & 0 & 0 & b \\\ 0 & a & 0 & \cdots & 0 & b & 0 \\\ 0 & 0 & a & \cdots & b & 0 & 0 \\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\\ 0 & 0 & b & \cdots & a & 0 & 0 \\\ 0 & b & 0 & \cdots & 0 & a & 0 \\\ b & 0 & 0 & \cdots & 0 & 0 & a \end{pmatrix}\]

using the traditional formula:

\[\det A = \sum_{\sigma \in S_{2n}} \text{sgn}(\sigma) \prod_{i=0}^{2n} A_{i\sigma(i)}\]

where \(\text{sgn}(\sigma)\) is \((-1)^{inv(\sigma)}\) and \(inv(\sigma)\) is the number of inversions in \(\sigma\).

Answer: (a^2 - b^2)^n


Question: Determine whether the following statement is true: "Between two distinct zeroes of a polynomial \( p \), there exists a number \( c \) such that \( p'(c) = 0 \)." If the statement is false, provide a counterexample or an explanation.

Answer: True


Question: Given a finite dimensional vector space and an invertible linear operator $T$ with a minimal polynomial $p_T(x) = x^m + a_{m-1}x^{m-1} + \dots + a_1x + a_0$, determine if the minimal polynomial of the inverse operator $T^{-1}$ is $p_{T^{-1}}(x) = x^m + \frac{1}{a_0}(a_{1}x^{m-1} + \dots + a_{m-1}x + 1)$. Is this statement true?

Answer: True


Question: Let $Y_1 < ... < Y_8$ be the order statistics of $n$ independent observations from a continuous distribution with a 70th percentile of $\pi_{0.7} = 27.3$. Calculate the probability $P(Y_7 < 27.3)$. Provide your answer as a probability value.

Answer: 0.2553


Question: Suppose $X_1, X_2, X_3$ are independently and identically distributed random variables following an exponential distribution with rate $\beta$. Calculate the probability $P(X_1 < X_2 < X_3)$. Express your answer as a single probability value.

Answer: \dfrac{1}{6}


Question: If \(A\) is a non-measurable set in \(\mathbb{R}^n\) (in the sense of Lebesgue), does it necessarily contain a positive measurable subset?

Answer: No


Question: Evaluate the integral of the 1-form \( \alpha = \frac{xdx+ydy}{x^2+y^2} \) along the curve \( c(t) = (t\cos\phi, t\sin\phi) \) for \( 1 \le t \le 2 \), where \( \phi \) is a constant. Verify that the integral evaluates to \( \ln{2} \).

Answer: \ln 2


Question: Evaluate the integral \( \int_{0}^{\infty} x^{9} e^{-x^{2}} \, dx \).

Answer: 12


Question: Calculate the expected value \( \mathbb{E}(X^3) \) for the stick breaking problem.

Answer: \dfrac{1}{4}


Question: Determine whether an antiderivative \( F(x) \) of a function \( f(x) \) necessarily has an antiderivative itself. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{\sin(\sin(\sin(\sin x))) - x}{x^3}$$

Answer: -\dfrac{2}{3}


Question: Find the value of the infinite series: \[ S = \frac{2^2}{3} + \frac{3^2}{3^2} + \frac{4^2}{3^3} + \cdots \] without using calculus.

Answer: \dfrac{7}{2}


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{\sqrt[5]{1+\sin(x)}-1}{\ln(1+\tan(x))}$$ without using L'Hôpital's rule.

Answer: \dfrac{1}{5}


Question: Determine the dimension of the range space of the linear operator \( T \) on the vector space \( C[0, 1] \) of continuous real-valued functions on \([0, 1]\), where \( T \) is defined by \((Tf)(x) = \int_{0}^{1} \sin(x + y)f(y) \, dy\) for \( x \in [0, 1] \).

Answer: 2


Question: Consider two subspaces U and W of a vector space V. If U + W = V and U ∩ W contains more than just the zero vector, can U ⊕ W be a direct sum of V? Answer with 'Yes' or 'No'.

Answer: No


Question: Let $(S, \Sigma, \mu)$ be a measure space and let $f: S \to [0, \infty]$ be a measurable function. If $\mu\{f>n\} \ge \epsilon > 0$ for all $n \in \mathbb{N}$, does it follow that $f = \infty$ almost everywhere? Answer yes or no and justify your answer.

Answer: No


Question: Determine the set of isolated singularities for the function \( f(z)=\frac{1}{(1+z^2)(\cos 2z)} \), and find the radius of convergence of its Taylor series centered at \( 0 \).

Answer: \dfrac{\pi}{4}


Question: Determine whether a non-biconnected outerplanar graph is necessarily a tree. Answer 'yes' or 'no'.

Answer: no


Question: Calculate the value of $c$ in the congruence equation $31^{(41^{59})} \equiv c \pmod{25}$ using Euler's theorem.

Answer: 6


Question: Evaluate the limit: $$\lim\limits_{x\to\infty} \left(\frac{x^2+5x+3}{x^2+x+3}\right)^x$$

Answer: e^4


Question: In an arithmetic progression $a, b, c, d, e$, the sum $a + b + c + d + e$ is a cube of an integer, and the sum $b + c + d$ is a square of an integer. What is the least possible number of digits for the natural number $c$?

Answer: 3


Question: Find the average shortest distance between a random point uniformly distributed inside the circle \((x-a)^2+(y-b)^2=r^2\) and the circle's center.

Answer: \dfrac{2r}{3}


Question: In a queue of $2n$ people at a theater box office, $n$ people have only $20$-zloty banknotes, and the other $n$ people have only $10$-zloty banknotes. Each person buys one ticket costing $10$ zlotys. Initially, the box office has no money. A person with a $20$-zloty banknote must wait for someone with a $10$-zloty banknote to pay first, as the box office cannot provide change. What is the probability that no one will have to wait for change?

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


Question:
Given $Y_1 \sim \Gamma(2,1)$ and $Y_2|Y_1 \sim U(0, Y_1)$, where $y_1 \geq y_2 \geq 0$, find the variance of $Y_2 - Y_1$. Use the following information:

$$f_{Y_1}(y_1) = y_1 \exp(-y_1)$$
$$f_{Y_2|Y_1}(y_2) = \frac{1}{y_1}$$

Answer: 1


Question: Find the radius of convergence for the power series \( \sum_{n>0} \frac{z^{2^n}}{n^2} \).

Answer: 1


Question: Calculate the volume of the solid bounded by the surface \(64x^2 - 4y^2 + 16z^2 = 0\) and the plane \(y = 1\) in three-dimensional space.

Answer: \dfrac{\pi}{24}


Question: Determine whether the limit \( \lim_{y\rightarrow\infty}\left(\zeta\left(\frac{1}{2}+iy\right) + \frac{1}{2}\left(\frac{1}{y/\pi}\right)^{1/2+iy} - \sum_{n=1}^{[y/\pi]}\frac{1}{n^{1/2+iy}}\right) \) converges to 0.

Answer: 0


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

Answer: 0


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{12^n\cdot(n!)^5}{(n^2+n)^{2n}\cdot(n+1)^n}$$

Answer: 0


Question: Determine whether the Hahn-Banach extension of a sublinear functional is always continuous.

Answer: No


Question: In a 6/49 lottery game, where 6 numbers are drawn from 1 to 49 without replacement, calculate the expected sum of the 3 smallest winning numbers.

Answer: \dfrac{300}{7}


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

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


Question: What is the necessary and sufficient condition for the existence of a homomorphism between the finite fields $\mathbb{F}_{p^a}$ and $\mathbb{F}_{p^b}$?

Answer: a \mid b


Question: Evaluate the line integral \( \int_C F \cdot dr \) using Green's Theorem, where \( C \) is the circle with radius 2 oriented clockwise, and \( F = (3y+\frac{e^{80/x^2}}{x^{2017}},-4x-\frac{e^{2017}}{e^{2018}}) \). Determine whether the integral should include the polar coordinate component \( r \) in the area element when converting to polar coordinates.

Answer: 28\pi


Question: Evaluate the limit: $$\lim_{n \to \infty} \sqrt[n^2]{1! + 2! + 3! + \dots + n!}$$

Answer: 1


Question: Compute the sum \( \sum_{k=0}^n k {n \choose k }p^k(1-p)^{n-k} \) for given \( p \in (0,1) \) and \( n \in \mathbb{N} \).

Answer: np


Question:
Determine the values of \(\alpha\) and \(\beta\) such that the iterative sequence \(x_{n+1} = \alpha x_n\left(3-\dfrac{x_n^2}{a}\right) + \beta x_n\left(1+\dfrac{a}{x_n^2}\right)\) converges to \(\sqrt{a}\) with a third-order rate. Choose the correct option:

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

Answer: 2


Question: Determine whether there exists a differentiable function $f(x)$ that satisfies the following conditions: $f(0) = 0$, $f(x) > 0$ for $x > 0$, $f'(0) = 1$, and $f'(q) = \frac{1}{f(q)}$ for some $q > 0$. Provide a justification for your answer.

Answer: Yes


Question: Find the minimum value of the function \(g(x) = \sqrt{x + 16} + \sqrt{25 - x} + \sqrt{x - 9}\) for \(9 \leq x \leq 25\) without using calculus or a calculator.

Answer: 9


Question: Evaluate the limit \( L = \lim_{x \to 0} \left( \frac{\ln((1+x)^{1+x})}{x^2} - \frac{1}{x} \right). \)

Answer: \dfrac{1}{2}


Question: In a tournament with 2n teams, each day features n matches with no team playing more than once per day. Over 2n - 1 days, each team plays every other team exactly once, and there are no ties. Determine if it is possible to select one winning team from each day such that no team is chosen more than once.

Answer: Yes


Question: Compute the limit \( \lim_{n \to \infty}\int_{0}^{\pi} \frac{\sin(x)}{1 + 3\cos^{2}(nx)}\,\mathrm{d}x \).

Answer: 1


Question: Three circles are given by the equations: \( S_1: x^2+y^2+4y-1=0 \), \( S_2: x^2+y^2+6x+y+8=0 \), and \( S_3: x^2+y^2-4x-4y-37=0 \). These circles touch each other at points \( P_1, P_2, \) and \( P_3 \). Let \( C_1, C_2, \) and \( C_3 \) be the centers of these circles. Calculate the ratio \( \frac{\Delta P_1P_2P_3}{\Delta C_1C_2C_3} \).

Answer: \dfrac{2}{5}


Question: Calculate the value of the power series \( f(x)=\sum _{n=1}^{\infty }\frac{\left(-1\right)^{n+1}x^{2n+1}}{2n-1} \) at \( x=\frac{\sqrt{3}}{3} \).

Answer: \dfrac{\pi}{18}


Question: Let $f$ be a continuous, periodic function defined on $[0,2\pi]$, with $f'$ also continuous. The Fourier series representation of $f$ is given by $\frac{a_0}{2} + \sum_{n=1}^\infty(a_n\cos nx + b_n \sin nx)$. Determine if the $L_2$ norm of the difference between $f$ and its mean, $||f-\frac{a_0}{2}||$, is less than or equal to the $L_2$ norm of the derivative, $||f'||$. Is this statement true?

Answer: True


Question: In the Sobolev space $H^1(\mathbb R^n)$, let $g_n \in H^1(\mathbb R^n)$ such that $0 \leq g_n \leq 1$ and $g_n = 1$ on $[n+\frac 14, n+\frac 34]$. If $g_n$ converges weakly to 0 in $H^1(\mathbb R^n)$, does $\int_{\mathbb R^n} g_n \, dx$ converge to 0?

Answer: No


Question: Consider the unit disk $D^n$ and the unit sphere $S^{n-1}$ in $\mathbb{R}^n$. Define an equivalence relation $\sim'$ on $S^{n-1}$ by $(x_1,x_2,\dots,x_n) \sim' (x_2,x_3,\dots,x_n,x_1)$. Determine if the quotient space $D^n/\sim'$ is homeomorphic to any known topological space.

Answer: D^n


Question:
Determine the correct form of the particular solution for the recurrence relation:

\[ a_{n}-7a_{n-1}+10a_{n-2}=2n^{2}+2 \]

with initial conditions:

\[ a_0=0, \quad a_1=1 \]

Choose the correct form of the particular solution:

1. \( a_{p} = (cn^{2}+bn+a) \, 1^{n} \)
2. \( a_{p} = (cn^{2}+bn+a) \, 2^{n} \)

Answer: 1


Question: Let $m, n, p \in \{1,2,\dots\}$, let $q, r \in \{0,1,\dots\}$, let $D$ be a non-empty, open subset of $\mathbb{R}^m$, let $E$ be an open subset of $\mathbb{R}^n$, let $f : D\rightarrow E$ be $q$ times partially differentiable, and let $g : E\rightarrow\mathbb{R}^p$ be $r$ times partially differentiable. Determine the greatest lower bound $s \in \{0,1,\dots\}\cup\{\infty\}$ on the number of times $g\circ f$ is necessarily partially differentiable. Additionally, if $f$'s $q$-degree partial derivatives and $g$'s $r$-degree partial derivatives are all continuous, are $g\circ f$'s $s$-degree partial derivatives continuous (assuming $s<\infty$)?

Answer: \min(q, r)


Question: Determine whether the following statement is true or false: If a function $f$ is continuous on the set of real numbers ($f \in C(\mathbb{R})$) and the limit of $f(x)$ as $x$ approaches infinity is zero, then the limit of its antiderivative $F(x)$ as $x$ approaches infinity is also zero. (Note: $C(\mathbb{R})$ denotes the set of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$. An antiderivative, or primitive function, of $f$ is a function $F$ such that $F'(x) = f(x)$ for all $x$ in the domain.)

Answer: B


Question: Given a binary matrix $A$, determine the maximum possible difference between the rank of $A$ and the rank of NOT$(A)$. Provide your answer as a single integer.

Answer: 1


Question: Given that $f(x - y) = f(x) + f(y) - 2f(xy)$ for all real numbers $x$ and $y$ and $f(1) = 1$, find the value of $f(2023)$. Express your answer as a single number.

Answer: 1


Question: Evaluate the limit: $$\lim_{(x,y)\to (0,0)} \frac{x^2+2x-4y^2+4y}{x+2y}$$. Determine if the limit exists and, if so, find its value.

Answer: 2


Question: Evaluate the expression \( \log_{2}3^4\cdot\log_{3}4^5\cdot\log_{4}5^6\cdot\ldots\cdot\log_{63} {64}^{65} \) and find the value of \( x \) such that it equals \( x! \).

Answer: 65


Question: Calculate the conditional expectation \( E[X_1|X_1 + X_2 + \cdots + X_n = x] \), where \( X_1, X_2, \ldots, X_n \) are independent and identically distributed (i.i.d.) random variables.

Answer: \dfrac{x}{n}


Question: Let \( A \) be an \(8 \times 8\) square matrix with entries \( a_{12} = a_{24} = a_{33} = a_{41} = a_{58} = a_{65} = a_{77} = a_{86} = 1 \) and all other entries \( 0 \). Determine the smallest positive integer \( n \) such that \( A^n = I \).

Answer: 3


Question: Given the ideals $I = (2, x)$ and $J = (3, x)$, determine whether the product $I \cdot J$ is equal to the ideal $(6, x)$ or if $(6, x)$ is contained in $I \cdot J$.

Answer: I \cdot J = (6, x)


Question: Compute the limit of the sequence: \( \sqrt{2^\sqrt{2^\sqrt{2^\sqrt{\ldots}}}} \)

Answer: 2


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

Answer: \dfrac{4}{e^2}


Question: Find the remainder when \(5^{5^{5^{5^{5^{5^5}}}}}\) is divided by 125.

Answer: 0


Question: In a complete normed function space, is it possible for a Cauchy sequence to converge in the norm but not converge pointwise?

Answer: Yes


Question: Determine the values of \(\alpha\) for which the series \(\sum_{n = 1}^{\infty}\left(2^{n^{-\alpha}} - 1\right)\) converges.

Answer: \alpha > 1


Question: Calculate the arc length of the function \( y = \int_1^x \sqrt{t^2-1} \, dt \) for the interval \( 4 \leq x \leq 7 \).

Answer: \dfrac{33}{2}


Question: Evaluate the limit using power series: \[ \lim \limits_{z \to \infty}{z^2 \left( \ln\left(1+\frac{2}{z}\right) - \frac{2}{z} \right)} \]

Answer: -2


Question: Use the Divergence Theorem to evaluate the surface integral \( \iint\limits_S F\cdot dS \) where \( F(x,y,z)=xi+y^2j+(z+y)k \) and \( S \) is the boundary of the region enclosed by the cylinder \( x^2+y^2=4 \) and the planes \( z=x \) and \( z=8 \).

Answer: 64\pi


Question: Evaluate the sum \(\sum_{k=2}^{n}{(k-1)(n-k)!{n \choose k}}\) using properties of binomial coefficients.

Answer: n! - 1


Question: Evaluate the contour integral \( I = \frac{1}{2\pi i} \oint_{\gamma} z^7 \cos\left(\frac{1}{z^2}\right) dz \), where \( \gamma = \{ z \in \mathbb{C} : |z| = 2 \} \).

Answer: \dfrac{1}{24}


Question: Determine the probability that a random polynomial of degree \( n \), with integer coefficients between \(-L\) and \(L\), is irreducible over the rationals and has the symmetric group \( S(n) \) as its Galois group.

Answer: 1


Question: Find the coefficient of \(x^{30}\) in the expansion of \((1+x^5+x^{12})^{12}\) using a combinatorial approach.

Answer: 924


Question: Evaluate the contour integral \( \oint_{C_2(0)} \frac{\sin(z)}{z^2+1} \, dz \) where \( C_2(0) \) represents the open disk of radius 2, centered at the origin, in the complex plane.

Answer: 2\pi i \sinh(1)


Question: An ellipsoid in \(\mathbb{R}^3\) is given by the equation \(2x^2+2y^2+z^2=338\). Determine the radius of a sphere that touches the ellipsoid at points of the form \((x,y,10)\), with the sphere's center located on the \(z\)-axis.

Answer: 12


Question: Evaluate the limit: $$\lim_{x\rightarrow 0}\left(\cos(x)-1\right)^{\tan(x)}$$ using the exponential representation method.

Answer: 1


Question: Given a compact set $A$ with positive measure, determine if the set of accumulation points of $A$, denoted as $A'$, can have zero measure.

Answer: No


Question: Evaluate the limit: $$\lim_{x\to0^{+}}\left(\frac{(e^{\sin x}-e^{\tan x})}{\sin^3 x}\right)$$

Answer: -\dfrac{1}{2}


Question: Determine the values of \(x\) that satisfy the equation: \[ \sum_{n=1}^{\infty} \sum_{k=1}^{\infty} \frac{1}{n} \left(\frac{x}{n}\right)^k = 0 \]

Answer: 0


Question: Does the continuity of a function \(f\) in \(L^p(\mathbb{R})\) for some \(1\leq p < \infty\) guarantee that \(\lim_{|x| \to \infty} f(x)=0\)?

Answer: No


Question: Given the equation $2x = y^{\frac{1}{5}} + y^{\frac{-1}{5}}$, find the value of $k$ such that $(x^2-1)\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + x\frac{\mathrm{d}y}{\mathrm{d}x} = ky$. Determine the value of $k$.

Answer: 25


Question: Determine whether the limit \( \lim_{n \to \infty} \sum_{k=1}^n \ln\left(1-\frac{x^2\sin^2k}{2n}\right) \) exists for \( x \in \mathbb{R} \). If it exists, find its value.

Answer: -\dfrac{x^2}{4}


Question: Evaluate the limit \( \lim_{x\to0}\frac{f(x) - f(-x)}{2x} \) where \( f(x)=\left( 1+x\right)^\frac{1}{x} \).

Answer: -\dfrac{e}{2}


Question: For the relation \(x^3 + 3x^2 + y^3 - y^2 - 15y + 20 = 0\), find the sum of all possible values of \(x\) for a given \(y\) using Vieta's formulas.

Answer: -3


Question: Evaluate the limit: $$\lim_{n\to\infty} \int_0^{\infty} \frac{\ln(n+x)}{n}e^{-x} \cos(x)\, dx.$$

Answer: 0


Question: Evaluate the double integral \(\int_R \exp\left({\frac{x^2}{25}+\frac{y^2}{16}}\right)\,\mathrm dA\) over the region bounded by the ellipses \(\frac{x^2}{25}+\frac{y^2}{16}=1\) and \(\frac{x^2}{25}+\frac{y^2}{16}=4.\)

Answer: 20\pi(e^4 - e)


Question: Given the function $f(x) = \cos{2x}$ defined on the interval $(0, \pi)$, find its $2\pi$-periodic, even extension and compute the cosine Fourier series of this extension.

Answer: \cos(2x)


Question: Let $a = (10^n - 1)^{10^n}$ and $b = (10^n)^{10^n - 1}$. Determine which of these numbers is greater as $n$ approaches infinity.

Answer: a


Question: Evaluate the integral \( I = \int_0 ^{10} [x]^3\{x\}dx \), where \([x]\) denotes the greatest integer less than or equal to \(x\) and \(\{x\}\) denotes the fractional part of \(x\).

Answer: \dfrac{2025}{2}


Question: Evaluate the integral \( \int_0^{\frac{\pi}{2}}\frac{\mathrm{d}x}{1+\tan^{2013}x} \).

Answer: \dfrac{\pi}{4}


Question: Let $f:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ be a function such that for every sequence $(x_n)_{n\geq 1}$ diverging to infinity, we have $\displaystyle\lim_{n\to\infty} f(x_n)=+\infty$. Does it follow that $\displaystyle\lim_{x\to\infty} f(x)=\infty$? Answer yes or no.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{N\rightarrow\infty}\sum_{k=0}^{N-2}N^{3}\left( \frac{k}{N}\right) ^{N\ln N} \]

Answer: \infty


Question: Find the minimum value of the function \( f(x) = xe^{\lfloor x \rfloor} + 7x^2 - 3x \) for \( x \in [-1, \infty) \).

Answer: -\dfrac{1}{7}


Question: Let $f$ be a Lebesgue measurable function from $[0,1]$ to $\mathbb{R}$. Determine whether there exists a measurable set $B$ with $\mu(B) > 0$ and a real number $M > 0$ such that for all $x \in B$, $f(x) < M$. Provide a justification for your answer.

Answer: Yes


Question: Assume we have a problem $R \in \mathit{NP}$ that can be solved with $M(x,y)$ in $O(n^3)$ time, where $y$ is additional information no longer than $3\log n$ bits. For what function $f(n)$ can we say that $R \in \mathit{TIME}(f(n))$?

Answer: n^6


Question: Given two normally distributed random variables $X$ and $Y$ with mean $0$ and variance $1$, and correlation coefficient $\rho$ where $-1 \leq \rho \leq 1$, find the value of $$\mathbb{E}[X+2Y|X-Y=a].$$

Answer: -\dfrac{a}{2}


Question: Given $a > 0$ is a fixed real number, and $a, b, x \in \mathbb{R}^n$, suppose that for all $x > 0$, if $a^T x \leq 1$ holds, then $b^T x \leq 1$ must also hold. Does this imply that $a \geq b$ componentwise? Answer "yes" or "no."

Answer: yes


Question:
Calculate the determinant of the following \(2n \times 2n\) matrix \(A\):
\[
A=\left( \begin{array}{ccccccc}
a & 0 & 0 & \ldots & 0 & 0 & b \\
0 & a & 0 & \ldots & 0 & b & 0 \\
0 & 0 & a & \ldots & b & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
0 & 0 & b & \ldots & a & 0 & 0 \\
0 & b & 0 & \ldots & 0 & a & 0 \\
b & 0 & 0 & \ldots & 0 & 0 & a \\
\end{array} \right)
\]
Use the formula for the determinant:
\[
\det A = \sum_{\sigma \in S_{2n}}{\text{sgn}(\sigma)\prod_{i=1}^{2n}{A_{i\sigma(i)}}}
\]
where \(\text{sgn}(\sigma)\) is \((-1)^{inv(\sigma)}\), \(inv(\sigma)\) is the number of inversions in \(\sigma\), and \(S_{2n}\) is the set of all permutations of \(2n\) elements. Consider that non-zero contributions to the determinant occur only when \(\sigma(i) \in \{i, 2n - i + 1\}\).

Answer: (a^2 - b^2)^n


Question:
Let \(b_n\) be a sequence defined by \(b_{n}=2+\frac{1}{b_{n-1}}\) with \(b_{0}=2\). Given that \(\lim_{n \rightarrow \infty} b_{n}=1+\sqrt{2}\), find the value of:

\[\lim_{n \rightarrow \infty} \left|\frac{b_{n+2}-(1+\sqrt{2})}{b_{n+1}-(1+\sqrt{2})}\right|.\]

Answer: 3 - 2\sqrt{2}


Question: Given two matrices $M_1, M_2 \in \mathbb{R}^{k \times l}$, and two orthogonal matrices $U \in \mathbb{R}^{k \times k}$, $V \in \mathbb{R}^{l \times l}$ such that $M_2 = UM_1V$. If the sum of the elements of $M_1$ and $M_2$ are the same, i.e., \( \sum_{i,j} m^{(1)}_{i,j} = \sum_{i,j} m^{(2)}_{i,j} \), where $m^{(1)}$ are the elements of $M_1$ and $m^{(2)}$ are the elements of $M_2$, can we conclude that $M_1 = M_2$?

Answer: No


Question: Given the piecewise function \(f(x)=\begin {cases} a^2 + e^x & -\infty <x<0 \\ x+2 & 0\le x \le 3 \\ c -\frac{b^2}{x} & 3<x<\infty \end{cases}\), where \(a,b,c\) are positive constants, determine the value of \(\frac{a+b+c}{2}\) if \(f(x)\) is differentiable for all real numbers.

Answer: 6


Question: Let \( N = \min\{n \in \mathbb{N}_0: X_1 + \ldots + X_n > T\} \). Calculate \( E[N] \) for \( T > 1 \).

Answer: T + 1


Question: Determine the number of zeroes (counting multiplicity) of the polynomial \( p(z) = z^4 - z^2 + 2z - 5 \) within the annulus defined by \( 2 < |z| < 3 \). Use Rouché's theorem to justify your answer.

Answer: 0


Question: Evaluate the limit: $$\lim\limits_{x \to 1^+} \frac{x-\sqrt{\arctan(x)-\frac{\pi}{4}}-1}{x-1}$$

Answer: -\infty


Question: Calculate the limit \( \lim_{n\rightarrow\infty}\frac{\int_{0}^{1}f^n(x)\ln(x+2)dx}{\int_{0}^{1}f^n(x)dx} \) where \( f(x)=1-x^2+x^3 \) and \( f^n(x) \) is the function \( f(x) \) raised to the power of \( n \).

Answer: \ln 2


Question:
Calculate the moment of inertia about the $x$-axis for the region bounded by the curves $x = 2y - y^2$ and $x = y^2$, given that the density at a point $(x, y)$ is $
ho(x, y) = y + 1$. Set up the double integral correctly, considering the conversion of $x = 2y - y^2$ to $y$ for the limits of integration.

Answer: \dfrac{1}{6}


Question: Find the greatest constant \( k > 0 \) such that the inequality \( \max\left(\frac{5}{5-3c}, \frac{5b}{5-3d}\right) \geq k \cdot \frac{2+3b}{5-c-2d} \) holds for all \( 0 \leq b \leq 1 \) and \( 0 \leq c \leq d \leq 1 \).

Answer: \dfrac{15}{16}


Question: In a topological space, consider a sequence $(x_n)$. If every subsequence of $(x_n)$ fails to converge, does this imply that the set $\{x_n: n \in \mathbb{N}\}$ is discrete?

Answer: No


Question: Evaluate the integral \( \oint_{\gamma} \frac{1}{z - 2} \, dz \) where \( \gamma \) is the circle centered at the origin with radius 3.

Answer: 2\pi i


Question: Determine the values of \(x \in \mathbb{R}\) for which the series \(\sum_{n=1}^\infty \frac{\log(n)}{n}\{x^n+x^{-n}\}\) converges.

Answer: -1


Question: Evaluate the infinite nested radical: \(\sqrt{1+ 2 \sqrt{1+3 \sqrt{1+\dots\infty}}}\).

Answer: 3


Question: Calculate the Laplacian of the function \( u(x,y,z) = \frac{1}{\sqrt{x^2 + y^2 + z^2}} \).

Answer: 0


Question: Solve the differential equation \( y'(x)=\frac{2x-y(x)}{2y(x)-x} \) with the initial condition \( y(2)=2 \).

Answer: y = x


Question: Find the maximum value of \( \sum_{i=1}^{12} \sin(2x_i) \) given that \( \sum_{i=1}^{12} \sin(x_i) = 0 \) for real numbers \( x_1, x_2, \ldots, x_{12} \).

Answer: 12


Question: Given two strings $x,y \in \{0,1\}^n$, the distance between them is defined as the number of coordinates in which they differ. For example, the strings $00\ldots 00$ and $11\ldots 11$ have a distance of $n$, while $00\ldots 00$ and $00\ldots 01$ have a distance of $1$. Suppose $n \geq 10$. Determine whether there exist three strings $x,y,z \in \{0,1\}^n$ such that the distance between any two of them is strictly greater than $n/2$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to+\infty} \frac{x^4e^{-4x} +2x^2}{x^{-2}\ln(x)-x}$$

Answer: -\infty


Question: Determine whether the following relation holds for any length $k$ in symmetric polynomials: $$\sum _{{i=0}}^{m}(-1)^{i}e_{i}(X_{1},\ldots ,X_{n})h_{{m-i}}(X_{1},\ldots ,X_{k})=0$$

Answer: 0


Question: Determine whether the function \( d(x,y) = \sqrt{|x-y|} \) is a metric on \( \mathbb{R} \).

Answer: Yes


Question: Evaluate the integral \( \int _C e^z \, dz \) for any closed loop \( C \).

Answer: 0


Question: Find the minimum value of the function \(S(x,y)=\max(1-2x+y, \, 1+2x+y, \, x^2-y^2)\) on the domain \(\mathbb{R}\times\mathbb{R}_{\geq 0}\).

Answer: 1


Question: Calculate the difference between \( \lim_{n \rightarrow \infty} \sup (-1)^n\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)} \) and \( \lim_{n \rightarrow \infty} \inf (-1)^n\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)} \).

Answer: \infty


Question: Given that $A$ is a nilpotent matrix with minimal polynomial $\lambda^5$, determine the rank of $A^3$.

Answer: 2


Question: Evaluate the limit: \[ \lim_{n \rightarrow +\infty} \int_{1}^{n} \frac{\cos(nx)(x-1)}{\sqrt{\ln^3(x)(1+x^4)}} \, dx. \]

Answer: 0


Question: Determine whether it is possible to find matrices \(X\), \(Y\), and \(Z\) such that \(X = Z^{-1} Y Z\) but \(X \neq Y\), given that \(Z\) is invertible.

Answer: Yes


Question: Given positive integers $a$ and $b$, find the sum $a + b$ when $a, x, y, z, b$ are in both arithmetic and harmonic progression. It is known that $xyz = 55$ and $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{343}{55}$. The relations for arithmetic progression are $x = \frac{3a + b}{4}$, $y = \frac{2a + 2b}{4}$, $z = \frac{a + 3b}{4}$, and for harmonic progression, $x = \frac{4ab}{a + 3b}$, $y = \frac{4ab}{2a + 2b}$, $z = \frac{4ab}{b + 3a}$. It is derived that $ab = 7$. Calculate $a + b$.

Answer: 8


Question: Calculate the variance of \((X+Y)^2\), given that \(X\) follows a Poisson distribution with parameter 2 and \(Y\) follows a Geometric distribution with parameter \(\frac{1}{3}\), and \(X\) and \(Y\) are independent.

Answer: 1792


Question: Are all sets with measure zero measurable in the context of Lebesgue measure?

Answer: Yes


Question: Let $\{M_{i}(t), t \geq 0\}$, $i=1,2,3$ be independent Poisson processes with rates $\lambda_{i}$, $i=1,2,3$. Define the processes $$N_{1}(t)=M_{1}(t)+M_{2}(t), \quad N_{2}(t)=M_{2}(t)+M_{3}(t).$$ The stochastic process $\{(N_{1}(t), N_{2}(t)),t \geq 0\}$ is called a bivariate Poisson process. Calculate $\text{Cov}(N_{1}(t),N_{2}(t))$. \(\text{Express your answer in terms of } \lambda_1, \lambda_2, \text{ and } \lambda_3.\)

Answer: \lambda_2 t


Question: Determine the number of different square roots of the matrix \( A = \begin{pmatrix} 1 & 4 \\ 0 & 9 \end{pmatrix} \).

Answer: 4


Question: Given an integer lattice $\mathcal{L}(B) = \sum_{i=1}^n x_i B_i$ where $x_i \in \mathbb{Z}$ and $B = [b_1, b_2, ..., b_n]$ is a basis, determine whether a point $t \in \mathbb{R}^n$ that lies within the span of the lattice vectors is inside the fundamental parallelepiped of the lattice. Can this determination be made in polynomial time?

Answer: Yes


Question: Determine the value of \(|a|+|b|\) such that the equation \(|ax+by|+|bx+ay|=|x|+|y|\) is satisfied for all real numbers \(x\) and \(y\).

Answer: 1


Question: Determine whether it is possible to construct a sequence of functions $\psi_n = \psi \circ f_n(r)$, where $\psi \in C^\infty((0,\infty); [0,1])$ is supported away from zero and identically one near $[1, \infty]$, such that $\int^\infty_0 (\partial_r \psi_n)^2 r \, dr \to 0$ as $n \to \infty$. The functions $f_n$ should be chosen so that $\psi_n \in C^\infty((0,\infty); [0,1])$ and are supported away from zero and identically one near $[1, \infty]$. Note that a simple scaling like $\psi_n = \psi(2^n r)$ does not work, as it results in $\int^\infty_0 (\psi'_n)^2 r \, dr = \int^\infty_0 (\psi')^2 r \, dr$. A more clever choice of $f_n$ is required.

Answer: Yes


Question: Determine the dimension of the real vector space \(W\) consisting of all functions \(g: [0, 2] \to \mathbb{R}\) such that the restriction of \(g\) to the interval \([0, 1]\) is a polynomial of degree at most 4, the restriction of \(g\) to the interval \([1, 2]\) is a polynomial of degree at most 2, and \(g(1) = 1\).

Answer: 6


Question: If two rings $R$ and $S$ are isomorphic, and $S$ does not have a unit element, does $R$ also lack a unit element?

Answer: Yes


Question: Let $a, b, c$ be positive real numbers. Find the minimum value of \[ P = \frac{a^2}{(a+b)^2} + \frac{b^2}{(b+c)^2} + \frac{c}{4a}. \]

Answer: \dfrac{3}{4}


Question: Determine whether the Gauss-Bonnet theorem applies to the real projective plane.

Answer: Yes


Question: Determine the radius of convergence of the series \(\sum \frac{(-1)^n z^{2n+1}}{\log n} \).

Answer: 1


Question: Determine the order of the distribution $S$ defined by $S(\psi)=\psi^{(n)}(y_0)$ for $\psi \in C_c^\infty (\mathbb R)$ and $n\in \mathbb N$. Provide your answer as a single integer.

Answer: n


Question:
Compute the following limit using only elementary properties of definite integrals:

$$\lim_{h \to 0} \int_0^h \frac {\sqrt {9 + t^2}}{h}dt $$

Answer: 3


Question: Evaluate the limit: $$\lim\limits_{n \to \infty} \frac{\sqrt{1} + \sqrt{2} + \dots + \sqrt{n}}{n\sqrt{n}}.$$

Answer: \dfrac{2}{3}


Question: Determine the commutator subgroup of the symmetric group $S_5$, given that the commutator subgroup of $S_4$ is $A_4$. Provide your answer as a subgroup of $S_5$.

Answer: A_5


Question: Determine whether there exists a Sylow 2-subgroup of the group $G = SL_2(\mathbb{Z}/8\mathbb{Z})$ that does not contain any elements of order 3.

Answer: Yes


Question: Can every positive definite real symmetric matrix $S$ be expressed as $S = A^T A$ for some matrix $A$? Provide a simple proof if possible.

Answer: Yes


Question: Determine whether the power series \( \sum_{n=0}^\infty \frac{z^n}{(n+1)!} \) converges uniformly on the closed unit disk \( |z| \leq 1 \).

Answer: Yes


Question: Provide a specific example of a group where every proper subgroup is normal, but the group itself is not abelian.

Answer: Q_8


Question: Evaluate the surface integral \( \int_{\tau} z \, d\tau \) over the surface \( S: x^2+y^2+z^2=a^2 \) with \( z \geq 0 \) using Gauss's Divergence Theorem.

Answer: \pi a^3


Question: Let $G$ be an abelian group. If $a$ and $b$ are elements of $G$ with orders $m$ and $n$ respectively, where $m$ and $n$ are coprime, determine the order of the element $a^{-1}b$.

Answer: mn


Question: Determine the smallest finite field in which the polynomial \((x^2+x+1)(x^5+x^4+1)\) is reducible.

Answer: \mathbb{F}_2


Question: Evaluate the limit: $$\lim_{x \to 0}\left(\frac{x}{\tan x}\right)^{\frac{3}{x^2}}$$

Answer: \dfrac{1}{e}


Question: Calculate the quadratic covariance \(<t^{3}B^{1}_{t},t^{2}B^{2}_{t}>_{T}\) for independent Brownian motions \(B^{1}\) and \(B^{2}\).

Answer: 0


Question: In a game with two players, each player chooses a non-negative effort level, $x$ and $y$, respectively. The cost of effort for each player is $x^2$ and $y^2$. The joint reward from their efforts is $xy$, which is split equally between them. Determine the Nash equilibria of this game.

Answer: (0, 0)


Question: In an undirected and weighted graph $G$, if a vertex $v$ is a leaf in all minimum spanning trees (MSTs) of $G$, does $v$ have the same predecessor in all such MSTs? Answer 'yes' or 'no' and justify your answer.

Answer: no


Question:
Determine the radius of convergence for the power series:

$$1+ \frac{a.b}{1.c}z + \frac{a(a+1)b(b+1)}{1.2...c(c+1)}z^2+...$$

Answer: 1


Question: Calculate the surface area of the segment of a sphere enclosed by a cylinder with center at (0, 2) and radius 2, using a double integral.

Answer: 16\pi


Question: Calculate \(555^{555^{555}} \bmod{1000}\).

Answer: 875


Question: Let $X$ and $Y$ be jointly continuous random variables, where $Y$ is positive and has a second moment $E[Y^2]=6$. Given that the conditional distribution of $X$ given $Y=y$ is a uniform distribution $U(1-y,1+y)$, find the variance of $X$, denoted as $Var(X)$. Express your answer as a single numerical value.

Answer: 2


Question: Let $f$ be an entire function with $f(0) = 1$, and for each positive integer $n$, it holds that $f\left(\frac{1}{n}\right) = 1 + \frac{i}{n} - \frac{1}{n^3}$. Determine the value of $f(1 - 2i)$. Consider the relationship between $f(z)$ and $g(z) = f(z) - 1 - iz - z^3$.

Answer: 14 - i


Question: Compute the intersection number of the pullbacks of the line bundles \(\iota^*\mathcal{O}(2,3)\) and \(\iota^*\mathcal{O}(2,3)\) on the smooth subvariety \(X = V_+(f) \subset \mathbb{P}^2 \times \mathbb{P}^1\), where \(f \in H^0(\mathcal{O}(1,2))\).

Answer: 20


Question: Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be an $M$-Lipschitz function with respect to the vector $L^2$ norm, and suppose the Jacobian $J_f(x)$ is symmetric for every $x$. Is it true that $||J_f(x)||_2 \leq M$ for all $x \in \mathbb{R}^n$, where $||\cdot||_2$ denotes the spectral norm?

Answer: Yes


Question: Provide an example of an abelian group $G$ with a subgroup $H$ that is not characteristic. Use only the concepts of group definition, subgroups, Lagrange's Theorem, and homomorphisms.

Answer: \mathbb{Z}_2 \times \mathbb{Z}_2


Question: Calculate the work done by the force \( \vec{F}=y\vec{i}+x\vec{j}+z\vec{k} \) in moving an object from the point \((-1, 2, 5)\) to the point \((1, 0, 1)\) along the curve of intersection of the paraboloid \( z = x^2+y^2 \) and the plane \( x+y=1 \).

Answer: -10


Question: Determine the number of distinct arrangements of the letters in the word "MATHEMATICS" such that no two vowels are adjacent and the letter 'T' appears before the letter 'S'.

Answer: 352800


Question: In a locally Hausdorff topological group $G$, does there necessarily exist a non-trivial compact subgroup?

Answer: No


Question: Evaluate the definite integral: $$\int_0^\infty e^{-ax} x^4 \, \text{dx}$$ where $a > 0$. Express your answer in terms of $a$.

Answer: \dfrac{24}{a^5}


Question: Assuming the Continuum Hypothesis holds, if a set in $\Bbb R^2$ is non-measurable, does it follow that both the set and its complement have cardinality $\frak c$?

Answer: Yes


Question: Use the Lagrange multiplier method to find the extrema of the functional \( S[y_1,y_2]=\int_0^2 (y_1^2+y_2^2)\,dx \) subject to the constraint \( \int_0^2 y_1y_2\,dx=1. \)

Answer: 2


Question: If a power series \(\sum_{n = 0}^{\infty} a_{n} x^{n}\) converges absolutely, meaning \(\sum_{n = 0}^{\infty} |a_{n}|\) converges, does it imply that \(|a_{n}|\) approaches 0 as \(n\) approaches infinity?

Answer: Yes


Question: Determine the number of distinct elements in the set \(\{(a+b\beta)(c+d\beta): a,b,c,d\in\mathbb{F}_3\}\), where \(\mathbb{F}_9=\mathbb{F}_3[\beta]\) and \(\beta^2=-1\).

Answer: 9


Question: Find all values of $n$ such that the sum of the digits of $2^n$ equals the sum of the digits of $5^n$. It is given that $n=3$ is a solution and that $3$ divides $n$ when considering the problem modulo $9$. Provide your answer as a set of integers.

Answer: \{0, 3\}


Question: Evaluate the line integral \( \oint_\gamma f(X) \, dX \) where \( f(x,y,z) = (yz,xz,xy) \) and \( \gamma = \{X(t)| 0 \le t \le 2\pi\} \) with \( X(t) = (\cos(t)\cos(\frac{\pi}{8} + \frac{t(2\pi-t)}{4\pi}),\sin(t)\cos(\frac{\pi}{8} + \frac{t(2\pi-t)}{4\pi}),\sin(\frac{\pi}{8} + \frac{t(2\pi-t)}{4\pi})) \).

Answer: 0


Question: Consider an even polynomial function $f: \mathbb{R} \to \mathbb{R}$. Suppose $x \neq 0$ is a real root of the derivative $f'(x)$, meaning $f'(x) = 0$. Can the root $x$ have a multiplicity greater than 1? Provide a justification for your answer.

Answer: Yes


Question: Evaluate \( \lim_{n \rightarrow \infty} J_n \), where \( J_n = \int_{0}^{1} \frac{g(x)}{1+n^2x^2} \, dx \) and \( g(x) \) is an integrable function on \([0,1]\).

Answer: 0


Question: Evaluate the integral \( \int_{0}^{\pi/2}\ \frac{1}{ 1+(\tan x)^{1/2}}\ dx \).

Answer: \dfrac{\pi}{4}


Question: Let $X$ be a Banach space and $A$ be a subset of $X$. Suppose that for every $\varepsilon > 0$ and every weakly compact set $C \subseteq X$, there exists an element $a \in A$ such that $\|1 - a\| \leq \varepsilon$ for all $x \in C$. Determine whether $1$ lies in the norm closure of $A$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to 0} \left(a^x + b^x - c^x\right)^{\frac{1}{x}}$$ where \(a, b, c\) are positive real numbers.

Answer: \dfrac{ab}{c}


Question: Evaluate the infinite series \( \sum_{n=0}^\infty \frac {\sin ((2n+1)\phi)}{2n+1} \) for \(0<\phi<\pi\).

Answer: \dfrac{\pi}{4}


Question: Determine whether a differentiable function \( f: \mathbb{R}^2 \to \mathbb{R}^2 \) can fail to be holomorphic-analytic.

Answer: Yes


Question: Determine whether the statement $G/H \times H \simeq G$ holds true for a subgroup $H \leq G$ in the context of topological groups and homeomorphisms.

Answer: False


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

Answer: \pi


Question: Find the arc length of the curve defined by the parametric equations \(x(t) = \cos^3(t)\) and \(y(t) = \sin^3(t)\) for \(0 \le t \le 2\pi\).

Answer: 6


Question: Given the equation \(\sum_{i=1}^m A_i^k = 0\) for all \(k \in \mathbb{N}^+\), where \(A_1, A_2, \dots, A_m\) are \(n \times n\) matrices, and knowing that \(\sum_{i=1}^m \text{Tr}(A_i^k) = 0\), determine if this implies that all the eigenvalues of these matrices are zero.

Answer: Yes


Question: Calculate the Lebesgue integral of \( \cos(2x)e^{-x} \) from 0 to infinity.

Answer: \dfrac{1}{5}


Question: Evaluate the limit: \[ \lim_{n\to \infty}\int_{x}^{y} \frac{\cos (nt)}{t} \, dt, \] where \( x > y > 0 \).

Answer: 0


Question: Evaluate the integral \( \int_{\gamma} \frac{dz}{(z-2)^2} \) where \( \gamma \) is a curve homologous to the boundary of \( R_r \) for \( r < 2 \).

Answer: 0


Question: In a measure space, can a sequence of integrable functions converge pointwise to a limit function such that the integral of the limit function is different from the limit of the integrals of the sequence, with both integrals being finite?

Answer: Yes


Question: Calculate the surface integral of the function \( f(x,y,z) = x^2 + y^2 + z^2 \) over the portion of the sphere \( x^2 + y^2 + z^2 = 4 \) that lies in the first octant.

Answer: 8\pi


Question: Determine whether there exists a function $g(x)$ that satisfies the Cauchy functional equation $g(x+y)=g(x)+g(y)$ and is bounded on the interval $[2, 2+\sqrt{2}]$. Provide a justification for your answer.

Answer: Yes


Question: Do all finite groups with the same number of elements of each order have the same exponent? Provide a justification for your answer.

Answer: Yes


Question: Consider the ordinary differential equations $y'(t)=f(t,y(t))$ with initial condition $y(0)=0$ and $y'(t)=-f(t,y(t))$ with initial condition $y(0)=1$. The solutions to these equations are denoted by $y^+$ and $y^-$, respectively. Assume that $f$ is strictly positive and the solution always exists. Let $T^-$ and $T^+$ be the unique positive constants such that $y^-(T^-)=0$ and $y^+(T^+)=1$. Determine whether $T^-=T^+$.

Answer: No


Question: Determine whether the sequence \( Y_n = \left\{\sum_{i=1}^n \frac{1}{(i+1)!}\right\} \) is a Cauchy sequence in \( \mathbb{Q} \) with the Euclidean metric.

Answer: Yes


Question: Determine whether the following statement is true for any homomorphism $f: D_4 \rightarrow C_{24}$: For all $a \in D_4$ with order not dividing 2, we have $a^2 \in \ker f$. Provide a proof or counterexample.

Answer: True


Question: How many strings of length \(2n\) over the alphabet \(\{0,1\}\) contain at most \(n\) ones and have no adjacent ones?

Answer: F_{2n+2}


Question: Find a polynomial \( g(x) \) with integer coefficients such that the number of nonzero coefficients in \( g(x)^2 \) is strictly greater than the number of nonzero coefficients in \( g(x) \).

Answer: 1 + x


Question: Let $x_1, x_2, \ldots, x_n$ be positive real numbers, and consider the power sum $S(k)$ defined as $S(k) = x_1^k + x_2^k + \ldots + x_n^k$. Given $S(3) = 3$ and $S(5) = 5$, determine the best possible lower bound for $S(1)$. \(\text{(Use the Cauchy-Schwarz inequality to justify your answer.)}\)

Answer: \dfrac{9}{5}


Question: Suppose $V$ is a vector space with basis $B$. Let $S$ be a non-zero square matrix. Is $BS$ necessarily a basis for $V? Justify your answer.

Answer: No


Question: Suppose $f_n$ is analytic on $(a, b)$ and $f_n \rightarrow f$ uniformly on $(a, b)$. Is $f$ necessarily analytic on $(a, b)$? Provide a justification for your answer.

Answer: No


Question: Evaluate the integral \( \int_{0}^{\infty} x^{5}e^{-x^2}dx \).

Answer: 1


Question: Does every discrete absolute value on a field imply that the absolute value is also non-archimedean?

Answer: Yes


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

Answer: 2


Question: Does an infinite group necessarily contain an element of infinite order? Answer yes or no.

Answer: No


Question: Find the radius of convergence of the series \( \sum_{k=0}^\infty 3^{k^2}x^k \).

Answer: 0


Question: Evaluate the limit: $$\lim\limits_{x\to 0}\int\limits_0^1 \frac{1}{x}\left(e^{-x^2(1+t)}(1+t)^{-1}-(1+t)^{-1}\right)dt.$$

Answer: 0


Question: Find the residue of the function \( f(z) = \frac{e^{4z} - 1}{\sin^2(z)} \) at \( z = 0 \).

Answer: 4


Question: Determine the minimum number of proper subgroups required to cover a noncyclic group \( G \).

Answer: 3


Question: Determine the limit of the sequence \((a_n)_{n \ge 0}\) defined by the recurrence relation \(a_{n+1}^2 = a_n a_{n-1}\) with initial conditions \(a_0 = 2\) and \(a_1 = 16\), given that all terms of the sequence are positive.

Answer: 8


Question: Evaluate the limit: $$\lim_{x \to +\infty} \left(\frac{\left((x-1)^2-x\ln(x)\right)(x)!}{(x+2)!+7^x}\right)$$

Answer: 1


Question: Evaluate the integral \( \frac{1}{2\pi i}\oint_{|z|=R} \frac{z^2}{z-2} e^{\frac{1}{z}} \, dz \) using the residue theorem, where \( R \) is a sufficiently large number.

Answer: \dfrac{13}{2}


Question: Evaluate the integral \( \int_0^\infty \frac{1}{(1+x^2)(1+x^{2018})}\,dx \).

Answer: \dfrac{\pi}{4}


Question: Determine the values of $p > 0$ for which the function $f(x) = \frac{1}{|x|^p}$ is integrable over the region $\mathbb{R}^n \setminus B_r(0)$, where $n \in \mathbb{N}$ and $B_r(0)$ is the ball of radius $r$ centered at the origin.

Answer: p > n


Question: Evaluate the integral \( \int_\gamma{\frac{\log(z+e)}{z}}dz \), where \( \gamma(t)=e^{2it} \), \( t\in[0,\pi] \), without using Cauchy's integral formula.

Answer: 2\pi i


Question: Let $X$ be a topological space and $Y \subseteq X$. If every subset of $Y$ is closed in $X$, does it follow that $Y$ is a discrete space? Answer yes or no and justify your answer.

Answer: Yes


Question: Find the fundamental group of the space $X = X_1 \cup X_2 \cup X_3$, where $X_1 = \{ (x,y,z) : x^2 +(y-1)^2+z^2=1 \}$, $X_2 = \{ (x,y,z) : x^2 +(y+1)^2+z^2=1 \}$, and $X_3 = \{ (0,y,1) : -1 \leq y \leq 1 \}$.

Answer: 0


Question: Evaluate the limit: $$\lim_{k\to0}{\frac{1}{k}\int_0^k{(1+\sin{2x})^{\frac{1}{x}}dx}}$$

Answer: e^2


Question: Determine whether it is possible to find a logarithm for any invertible matrix over the complex numbers.

Answer: Yes


Question: Evaluate the integral: $$\int_{0}^{\frac{\pi }{4}}(\tan^{9}x+5\tan ^{7}x+5\tan^{5}x+\tan^{3}x)dx$$

Answer: \dfrac{25}{24}


Question: Find the number of integer solutions to the equation \(x_1 + x_2 + \cdots + x_5 = 41\) such that \(0 \le x_i \le 20\) for all \(i\), where \(x_i\) is even when \(i\) is even, and \(x_i\) is odd when \(i\) is odd. Use a generating function to solve this problem.

Answer: 5720


Question: Determine whether every unbounded set contains a sequence that diverges to infinity. Provide a justification for your answer.

Answer: Yes


Question: Let $c$ be an integer, not necessarily positive and not a square. Let $R=\mathbb{Z}[\sqrt{c}]$ denote the set of numbers of the form $$a+b\sqrt{c}, a,b \in \mathbb{Z}.$$ Then $R$ is a subring of $\mathbb{C}$ under the usual addition and multiplication. If $R$ is a UFD (unique factorization domain), does it follow that it is also a PID (principal ideal domain)? Answer yes or no.

Answer: Yes


Question: Determine if the groups $A=\langle x,y;x^3,xyx^{-1}y^{-1}\rangle$ and $B=\langle x,y;x^3,y^3,xyx^{-1}y^{-1}\rangle$ have the same order.

Answer: No


Question: Evaluate the limit \( \lim_{x \to \infty} x \int_{x}^{x+1} \cos(t^3) \, dt \).

Answer: 0


Question: Given a series of \(n\) measurements assumed to follow a uniform distribution on \((0,\theta)\), where \(\theta > 1\), and \(m\) of these measurements are less than 1, derive the maximum likelihood estimate of \(\theta\) using the binomial distribution.

Answer: \dfrac{n}{m}


Question: Determine the total number of extensions of the isomorphic mapping \( \phi: \mathbb{Q}(\sqrt{10}) \to \mathbb{Q}(\sqrt{10}) \), where \( \phi(\sqrt{10}) = -\sqrt{10} \), to an isomorphic mapping of \( E = \mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5}) \) onto a subfield of \( \overline{\mathbb{Q}} \).

Answer: 4


Question: Find the number of critical points of the function \( g(x) = \int_{0}^{x}{\cos (\sin t)dt} \) on the interval \([0, 2\pi]\).

Answer: 0


Question: Let $G$ be a group of order $pqr$, where $p$, $q$, and $r$ are three distinct primes. By Cauchy's theorem, there exist elements $a$, $b$, and $c$ in $G$ whose orders are $p$, $q$, and $r$, respectively. If the subgroup generated by $a$ and $b$ is the whole group $G$, determine if there exists a proper normal subgroup of $G$.

Answer: Yes


Question: Determine the asymptotic behavior of the expression \( \frac{\sum_{p \leq x} \log p}{\pi(x)} \), where \( \pi(x) \) is the number of primes less than or equal to \( x \).

Answer: \log x


Question: Determine a closed formula for the nth value of x such that \( 15^x \equiv x \pmod{1000} \) for \( x \geq 3 \).

Answer: 1000n - 625


Question:
Given a $5\times 5$ matrix $A$ with $\det(A) \neq 0$ and a relationship $4A - 5B = 0$, compute the determinant:
$$\frac{1}{25}\cdot\det(4(A^t)^2B^{-1}A^{-1})$$

Answer: 125


Question: Determine the number of solutions to the expression \(\sqrt[a+bi]{1}\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.

Answer: \infty


Question: Evaluate the limit: $$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$$ where \( p \) is a natural number.

Answer: \dfrac{1}{2}


Question: Given the equation $x^3 + y^3 + (x+y)^3 + 33xy = 2662$, where $x$ and $y$ are real numbers, find the value of $S = x + y$. Express your answer as a single number.

Answer: 11


Question: Determine the values of $t \in \mathbb{R}$ for which the function \( f(x) = \begin{cases} x^t \sin \left(\frac{1}{x^2}\right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \) is differentiable at \( x = 0 \).

Answer: t > 1


Question:
Given a sufficiently large positive integer $n$, consider a set $S=\{M_i\}$ consisting of $n$ matrices with entries in either $\mathbb{C}$ or $\mathbb{Z}/m\mathbb{Z}$. Determine if the following conditions are possible:

1. For all $1 \le i \le n$, $M_i^2 = 0$.
2. The product of any permutation of matrices in $S$ is non-zero: $\prod_{i} M_i \neq 0$.
3. The product of matrices in any subset $t$ of $S$ is independent of the ordering of $t$.

If such a set exists, find the smallest possible dimension for the matrices and provide an explicit construction for the largest possible $n$. Provide your answer in the form of the smallest dimension and a description of the construction.

Answer: 2^n


Question: Determine whether there exists a real sequence \((a_n)_{n\in\mathbb{N}}\subset [0,1)\) such that for each \(n\in\mathbb{N}\), there is a bijection \(f:[n]\to [n]\) satisfying \(a_{f(k)} \in \left[ \frac{k-1}{n}, \frac{k}{n} \right)\) for all \(k\in [n]\).

Answer: Yes


Question: Let $A \in M_n(\mathbb{C})$ be a square matrix of order $n$ with integer elements, such that its characteristic polynomial equals its minimal polynomial. Suppose $B$ is another matrix with integer elements that commutes with $A$. If $B$ can be expressed as $B = c_0 + c_1A + \cdots + c_kA^k$, can we conclude that $c_i \in \mathbb{Z}$ for $i = 1, \ldots, k$?

Answer: Yes


Question: If a sequence of functions \( F_n \) converges uniformly to a function \( F \), and there exists a uniform bound \( M \) for all \( F_n \), does this imply that \( F \) is also bounded? Provide a justification for your answer.

Answer: Yes


Question: Determine whether there exist any other solutions for dividing the first k primes into two groups such that the difference of their products is one, beyond the given examples for k=2, k=3, and k=4.

Answer: 7


Question: Let \( \Omega \subset \mathbb{R}^{n} \) be a bounded and open set. Assume that \( a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} \) is a Caratheodory function, meaning \( a(x,.,.) \) is continuous on \( \mathbb{R} \times \mathbb{R}^{n} \) for almost all \( x \in \Omega \) and \( a(.,s,\xi) \) is measurable in \( \Omega \) for every \( (s,\xi) \in \mathbb{R} \times \mathbb{R}^{n} \). Assume further that \( |a(x,s,\xi)| \leq k(x) + \beta(|s|^{p-1}+|\xi|^{p-1}) \) for almost every \( x \in \Omega \), for every \( (s,\xi) \in \mathbb{R} \times \mathbb{R}^{n} \), and for some \( k \in L^{p'}(\Omega), \beta \geq 0, p > 1 \) where \( p' = \frac{p}{p-1} \). If \( u_{m} \rightarrow u \) in \( L^{\infty}(\Omega) \) and \( u_{m} \rightarrow u \) uniformly, does it follow that \( a(x,u_{m}, \xi) \rightarrow a(x,u,\xi) \) in \( L^{1}(\Omega) \)?

Answer: Yes


Question: Determine whether the integral Mean Value Theorem can be applied to functions with discontinuous derivatives, provided that the integral of the derivative exists.

Answer: Yes


Question: Determine whether the statement "The Taylor series of \(\frac{x}{x+2}\) centered at \(X = 1\) has a radius of convergence \(R = 4\)" is correct. Justify your answer by considering the domain of the function within the radius of convergence.

Answer: 3


Question: Determine the number of 19th power residues modulo 229.

Answer: 12


Question: Evaluate the limit: $$\lim_{x\to\infty}\left(\frac{e^{1/x}-e^{\sin(1/x)}}{x^3}\cdot\frac{1}{(x!)^{1/x}}\right)$$

Answer: 0


Question: Evaluate the limit: $$\lim_{x \to\infty} \frac{x^3 (\ln x)^2}{e^\sqrt{x}}$$ without using L'Hopital's rule.

Answer: 0


Question: Evaluate the integral by reversing the order of integration: \[ \int_{0}^{1}\left(\int_{y=x}^{\sqrt{x}}\frac{\sin y}{y}dy\right)dx \]

Answer: 1 - \sin 1


Question: Evaluate the limit: \( \lim_{x\rightarrow0^+}x^{\tan(x^2)} \)

Answer: 1


Question: What is the maximum value of the Kullback–Leibler divergence \(D_{KL}(P\parallel Q)\) when \(\sum_x P(x)=\sum_x Q(x)=1\)?

Answer: \infty


Question: Find the greatest common divisor (GCD) of $7^{3001} - 1$ and $7^{3000} + 1$. Express your answer as a single integer.

Answer: 2


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

Answer: \dfrac{4}{e}


Question: In a 4-dimensional space divided into 16 equal regions about the origin, determine the maximum number of regions that a plane can intersect.

Answer: 11


Question: Let $X$ and $Y$ be topological spaces. If a function $g: X \to Y$ maps open sets of $X$ to closed sets of $Y$, does that imply that $g$ is continuous? Answer yes or no and justify your answer.

Answer: No


Question: Determine whether the statement is true or false: Homology groups can have torsion, while cohomology groups are always torsion-free.

Answer: B


Question: Evaluate the limit \( \lim_{x\to0}\frac{\ln(\cos(4x))}{\ln(\cos(3x))} \) without using L'Hopital's rule.

Answer: \dfrac{16}{9}


Question: Determine whether the following statement is true or false: The eigenvalues of a self-adjoint compact operator on an infinite-dimensional Hilbert space are always finite in number.

Answer: False


Question: Determine whether the equation $a^2 + b^2 = 2 c^2$ has any integer solutions where $|a| \neq |b|$. Provide a justification for your answer.

Answer: Yes


Question: Suppose $X$ and $Y$ are homeomorphic topological spaces with equivalence relations $\sim_X$ and $\sim_Y$ on $X$ and $Y$, respectively. Let $f: X \rightarrow Y$ be a homeomorphism such that $y_1 \sim_Y y_2$ if and only if $f^{-1}(y_1) \sim_X f^{-1}(y_2)$. Are the quotient spaces $X/\sim_X$ and $Y/\sim_Y$ homeomorphic?

Answer: Yes


Question: Find the remainder when the number \( x \), defined by the equation \( \displaystyle \sum_{r=1}^{13}\frac{1}{r} = \frac{x}{13!} \), is divided by 11.

Answer: 9


Question: Determine whether there is a closed-form expression for the integral: $$\frac{1}{2\pi i}\oint_{x=0} e^{\sqrt{z}(x+\frac{1}{x})} \frac{dx}{x}$$

Answer: I_0(2\sqrt{z})


Question: Determine the number of infinite subsets of the set of natural numbers, \(\mathbb{N}\).

Answer: 2^{\aleph_0}


Question: Evaluate the integral \( \int_{-1}^{2} x g''(x) \, dx \) given that \( g(-1) = 5 \), \( g(2) = -2 \), and \( g'(2) = -4 \).

Answer: -\dfrac{5}{3}


Question: Evaluate the integral: $$ \int_{-5}^{5} \min \{ 2+x,2-x,x^2 \} \,dx $$

Answer: -\dfrac{22}{3}


Question: Evaluate the integral \( \int_{0}^{1} \frac{\ln x}{x^2 + 1} \, dx \).

Answer: -G


Question: Find the Galois group of the polynomial \((x^2-3)(x^3-5)\) over \(\mathbb{Q}\).

Answer: S_3 \times \mathbb{Z}/2\mathbb{Z}


Question: Consider a region $R$ bounded by $y=g(x)>0$, $x=2$, and $x=3$. If the volume of the solid formed by rotating $R$ about the $y$-axis is $2\pi$, and the volume formed by rotating $R$ about the line $x=-2$ is $8\pi$, find the area of $R$.

Answer: \dfrac{3}{2}


Question: Let $R$ be a commutative ring with identity and $N$ be a finitely generated module over $R$. If $R$ is an integral domain, determine whether there exists an element $n \in N$ such that the annihilator of $n$ is the zero ideal. Answer true or false.

Answer: False


Question: Determine whether there exists a binary operation on a countably infinite set such that the number of subgroups closed under this operation is greater than the cardinality of the set.

Answer: Yes


Question: Evaluate the limit of the integral: $$\lim_{n\to\infty}\int_0^{2n\pi}\frac{(n+x)\sin(x/n)}{x(1+x)^2}\,dx$$

Answer: 1


Question: Given a continuous and differentiable function $f(x)$ that satisfies $f(x+y) = f(x)f(y)$ for all real numbers $x$ and $y$, and $f'(0) = 2$, find the value of $f'''(0)$. \( \boxed{} \)

Answer: 8


Question: Evaluate the principal value of the improper integral: $$\int_{-1}^{1} \frac{1}{x^{2}}dx$$.

Answer: \infty


Question: Determine whether the following statement is true or false: An open subspace of a locally compact subspace is locally compact.

Answer: B


Question: Find the number of real solutions to the equation \( g = 0 \), where \( g = \frac{1}{x-f(1)} + \frac{3}{x- f(2) } + \frac{4}{x-f(3)} \) and \( f(x) = x^3 + 6x^2 + 12x + 108 \).

Answer: 2


Question: Determine whether the equation \(n^4 + m^4 = c^3\) has any solutions for positive integers \(n\), \(m\), and \(c\) that cannot be expressed in the form \(n^4 + nm^3 = a\) and \(m^4 + mn^3 = b\) where \(a + b = c^3\).

Answer: Yes


Question: Evaluate the line integral \( \int\limits_C (-y^3dx + x^2dy-z^3dz) \) using Stokes' Theorem, where \( C \) is the intersection of the cylinder \( x^2 + y^2 = 1 \) and the plane \( x + y + z = 1 \). Determine the value of the integral.

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


Question: Let $f$ be a differentiable function of one variable and $u=\frac{x}{y}f(\frac{y}{x})$. If $u$ satisfies the partial differential equation $\frac{x}{y}u_{x} + u_{y}= G$, find the function $G$.

Answer: 0


Question: Determine whether a group with an uncountable number of elements can have a countable number of distinct subgroups. Answer with 'Yes' or 'No'.

Answer: No


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{n}{\ln n}\left(\frac{\sqrt[n]{n!}}{n}-\frac{1}{e}\right).$$

Answer: \dfrac{1}{2e}


Question: Given a sequence of real numbers $(s_n)_{n\geq 0}$ and the corresponding Hankel matrix $H_n=(s_{i+j})_{0\leq i,j\leq n}$, if $\det H_n>0$ for all $n\geq 0$, does it imply that $H_n$ is positive semi-definite as well? Justify your answer.

Answer: Yes


Question: Let $Q(x)$ be a polynomial of degree $12$ such that $Q(x)=0$ for $x=-1,0,1,\ldots,12$. If $(x+1)Q(x)+1$ is divisible by $(x-13)$, find the value of $Q(13)$. \( \boxed{} \)

Answer: -\dfrac{1}{14}


Question: Determine whether every bounded operator on a Hilbert space is weakly compact.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to0}\left(\sin\frac{2}{x}\right)^{x^3}$$

Answer: 1


Question: Find the greatest possible value of the greatest common divisor (gcd) of $4T_n$ and $n-1$, where $T_n$ is the $n$th triangular number defined as $T_n = 1 + 2 + \cdots + n$ for all positive integers $n$. Provide your answer as a single integer.

Answer: 4


Question: Determine the range of values for \( \beta > 0 \) such that the random walk defined by \( S_0 = 0 \) and \( S_n = \varepsilon_1 + \cdots + \varepsilon_n, n \geq 1 \), with i.i.d increments \( \lbrace \varepsilon_n \rbrace \) having the distribution \( P(\varepsilon_1 = -1) = 1 - C_{\beta} \) and \( P (\varepsilon_1 > t) = C_{\beta}e^{-t^{\beta}}, t \geq 0 \), where \( C_{\beta} \in (0,1) \) and \( \mathbb{E}\varepsilon_1 = -1/2 \), is heavy-tailed.

Answer: (0, 1)


Question: Determine the range of values for $k$ such that the equation $|x^2 - 5x + 4| = kx$ has exactly 4 distinct real solutions.

Answer: (0, 1)


Question: Evaluate the double integral: $$ \int_0^2\int_0^{2-x}(x+y)^2 e^{2y\over x+y}dy dx. $$

Answer: 2(e^2 - 1)


Question: Determine the number of elements in the field $F_{2}(a^2+a)$, where $F_{16} = F_{2}(a)$ and the minimal polynomial of $a$ over $F_{2}$ is $m_{a,F_{2}} = a^4+a+1$. \[ \text{How many elements are in } F_{2}(a^2+a)? \]

Answer: 4


Question: Determine the upper bound on the dimension of the vector space $W = \text{span}(I, A, A^1, A^2, \dots)$, where all matrices are $n \times n$.

Answer: n


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

Answer: 0


Question: Evaluate the limit \( \lim_{(x,y) \to (1,2)} \frac{(x-1)(y-2)-\sin(x-1)\sin(y-2)}{(x-1)^2+(y-2)^2} \) using Taylor expansions.

Answer: 0


Question: Let $\{X(t), t \geq 0\}$ be a Brownian motion with drift parameter $\mu = -2$ and variance parameter $\sigma^2 = 16$. If $X(0) = 5$, calculate $P(X(1) < 0)$. \(\text{Express your answer as a probability.}\)

Answer: 0.2266


Question: For a function \( f: \mathbb{R} \rightarrow \mathbb{R} \), is it sufficient for \( \frac{f(x)}{x} \) to be bounded in a neighborhood of \( 0 \) to ensure that \( f \) transforms convergent series into convergent series? Answer "yes" or "no" and justify your answer.

Answer: no


Question: Let $f$ be a bijective function defined on the set of real numbers with the power series representation $f(x) = \sum_{n=1}^{\infty} a_nx^n$, where $a_1 = 2$ and $a_2 = 4$. If $f^{-1}(x) = \sum_{n=1}^{\infty} b_nx^n$, find the value of $b_1$. \( \boxed{} \)

Answer: \dfrac{1}{2}


Question: Determine the supremum of the set \( \mathbb{A} = \{ ab - \left(\frac{a^2}{2} + \frac{b^4}{4}\right) \mid a, b \in \mathbb{R}^{+} \} \).

Answer: \dfrac{1}{4}


Question: For two $2 \times 2$ matrices $A$ and $B$ with $\det(A) = 1$ and $\det(B) = 2$, find the value of the expression $\det(A+\alpha B) - \det(\alpha A + B)$, where $\alpha \in \mathbb{R}$. Choose the correct answer from the following options: (A) $\alpha^2$, (B) $0$, (C) $\alpha^2 - 1$, (D) None of these.

Answer: C


Question: Determine whether an abelian group $G$ with order $|G|=p^k m$ (where $p$ is a prime and $p$ does not divide $m$) has a subgroup of order $p^j$ for every integer $1 \leq j \leq k$. Provide a justification for your answer.

Answer: Yes


Question: Let \(\begin{pmatrix} -4 & -15 \\ 2 & 7 \end{pmatrix}^{100} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\). Calculate the value of \(3a + b + 3c + 4d\).

Answer: 7


Question: Determine the smallest integer $n$ such that a square $ABCD$ can be decomposed into $n$ non-overlapping triangles, each having only acute angles.

Answer: 8


Question: If $X_1, X_2, \dots, X_n$ are $KC$ subspaces of a topological space $X$, is the intersection $X_1 \cap X_2 \cap \dots \cap X_n$ also a $KC$ subspace?

Answer: Yes


Question: Determine the best possible contraction constant \( \lambda \) for the map \( f: X \to X \) defined by \( f(x) = \frac{x}{2} + \frac{1}{x} \) for all \( x \in X \), where \( X = \{ x \in \mathbb{R} : x \ge 1 \} \).

Answer: \dfrac{1}{2}


Question: Let $(X,\mu)$ be a finite measure space. Suppose $\int_X\lvert f_n-f\rvert^p\,d\mu\to0$ for $p\geq1$. For $q\in(0,p]$, does it hold that $$\int_X\lvert\lvert f_n\rvert^q-\lvert f\rvert^q\rvert^{\frac{p}{q}}\,d\mu\to0\ ?$$

Answer: Yes


Question: For all positive $x$, if a function $f(x)$ satisfies $f(x) \geq e^x$, determine whether it must also hold that $f'(x) \geq f(x)$. Justify your answer.

Answer: No


Question: Determine the number of real roots for the polynomial equation: \[bt^{12}-at^{11}+ dt^{10}-ct^{9}+ ft^{8}-et^{7}+ ht^{6}-gt^{5}+ mt^{4}-lt^{3}+ rt^{2}-st+q=0\] where all the coefficients are real and positive.

Answer: 0


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

Answer: e^2 - 1


Question: Given non-degenerate integer matrices $X_1$ and $X_2$ with nonnegative entries, the Perron-Frobenius theorem guarantees a positive real eigenvalue $\lambda_{FP}(A)$, which is greater than or equal to the magnitude of other eigenvalues for any such matrix $A$. If $\lambda_{FP}(X_1) + \lambda_{FP}(X_2)$ is an integer, does it necessarily follow that both $\lambda_{FP}(X_1)$ and $\lambda_{FP}(X_2)$ are integers?

Answer: Yes


Question: Consider two sequences of positive numbers, $(x_n)$ and $(y_n)$. Determine whether the following statement is true: If for every $\epsilon > 0$, the inequality $x_n^{1+\epsilon} > y_n$ holds for all $n$, then it follows that $x_n \geq y_n$ for all $n$. Provide a proof or counterexample.

Answer: True


Question: Determine if there exists a sequence of prime numbers of the form $p_1$, $2p_1+1$, $2(2p_1+1)+1$, and so on, that contains more than four terms.

Answer: Yes


Question: Compute the limit \(\lim_{n\to\infty} \sum_{k=1}^{n} \left(\frac{k}{n^2}\right)^{\frac{k}{n^2} + 1}.\)

Answer: \dfrac{1}{2}


Question: Find the smallest positive integer \( N \) such that the function \( f(x) = \sin \left(\frac{1}{x}\right) \) has exactly 2020 local extrema in the interval \( \left(\frac{1}{N}, 1\right) \).

Answer: 6345


Question: Evaluate the limit: $$\lim\limits_{n\to \infty}\ \left(\frac{n!}{2^n}\right)^{1/n}$$

Answer: \infty


Question: Given the integral equation \( f(x) = \int_{0}^{1} e^{x+t} f(t) \, dt \), find the value of \( f(1) \).

Answer: 0


Question: Given a non-convex domain $\Omega = (-0.5, 0.5)^2 \backslash [0, 0.5]^2$, determine if the vector field $v = \nabla \times (r^{2/3} \cos(\frac{2}{3}\theta - \frac{\pi}{3}))$ is an example of a vector field $v \in H(curl; \Omega) \cap H(div; \Omega)$ that is not in $H^1(\Omega; \mathbb{R}^2)$ and satisfies $\text{div } v = 0$. Here, $H(curl) = \{v \in L^2(\Omega)^2: \nabla \times v \in L^2(\Omega)\}$ with $\nabla \times v = \partial_1v_2 - \partial_2v_1$, and $H(div) = \{v \in L^2(\Omega)^2: \text{div } v \in L^2(\Omega)\}$ with $\text{div } v = \partial_1v_1 + \partial_2v_2$. Provide justification for your answer.

Answer: Yes


Question: Determine whether there exists a strictly decreasing sequence \((x_n)_{n \in \mathbb{N}}\) that approaches zero, such that the ratio \(\frac{x_{n+1}}{x_n}\) converges to 1.

Answer: Yes


Question: Consider a sequence of functions $f_n$ converging to $f$ in measure on a measure space $(X, \mu)$. Is it possible for the inequality $|f(x)| \leq M$ to hold for all $x \in X$ even if $|f_n(x)| \leq M$ does not hold for all $n \in \mathbb{N}$ and $x \in X$? Provide a justification for your answer.

Answer: Yes


Question: Determine a necessary condition on the parameters $a$, $b$, and $c$ such that the equation $a^2 \tan x + b^2 \cot x = c^2$ has at least one solution, where $\tan x = \frac{\sin x}{\cos x}$ and $\cot x = \frac{\cos x}{\sin x}$.

Answer: c^2 \geq 2|ab|


Question: Determine the number of solutions to the equation \(x^3 + 2x^2 + 5x + 2\cos x = 0\) within the interval \([0, 2\pi]\).

Answer: 0


Question: Determine the order of the zero at $z=0$ for the function $f(z)=\frac{d^{50}}{{dz}^{50}}[z^{100}\sin(z^{10})]$. Provide your answer as an integer.

Answer: 60


Question: Let $f: G \to \mathbb{C}$ be a function defined on a domain $G \subset \mathbb{C}$. If $f$ is continuously differentiable on $G$ and complex differentiable on a dense subset $H$ of $G$, determine whether the set of points in $G$ where $f$ is complex differentiable is open in $G$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the surface integral \(\int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S}\), where \(\vec{F}(x,y,z) = y^2\hat{i} + 2xy\hat{j} + xz^2\hat{k}\), and \(S\) is the upward-facing surface of the paraboloid \(z = x^2 + y^2\) in the first quadrant, bounded by the planes \(x=0\), \(y=0\), and \(z=1\). Use Stokes' theorem to find the boundary curve \(C\) and evaluate the integral.

Answer: \dfrac{2}{7}


Question: Evaluate the limit: \( \lim_{x \to 0} \frac{\ln \left(1+\sqrt{x \cdot \sin(x)}\right)}{\sqrt{x}} \) without using L'Hopital's rule.

Answer: 0


Question: Determine whether the expression \(\sum_{k=1}^{r} \begin{bmatrix} A_{1k}B_{k1} & A_{1k}B_{k2}&\dots &A_{1k}B_{kq} \\ A_{2k}B_{k1} &A_{2k}B_{k2} &\dots &A_{2k}B_{kq} \\ \vdots & \vdots &\ddots &\vdots \\ A_{sk}B_{k1} &A_{sk}B_{k2} &\dots &A_{sk}B_{kq} \end{bmatrix}\) is equal to the matrix product \(\mathbf{AB}\), where \(\mathbf{A}\) is an \(s \times r\) matrix and \(\mathbf{B}\) is an \(r \times q\) matrix. Provide a justification for your answer.

Answer: Yes


Question: Let $X$ be a complex matrix and $m$ and $n$ be positive integers such that $m > n$. If $X^m$ is the zero matrix, does it imply that $X^p = 0$ for all $p \geq n$? Provide a justification for your answer.

Answer: No


Question: Let $X$ be a complex manifold of dimension $n$. We know that the $k^{th}$ singular cohomology of $X$ is $0$ for $k > 2n$. The $k^{th}$ sheaf cohomology of $X$ is $0$ for $k > n$. For constant sheaves, the sheaf cohomology agrees with singular cohomology. Does the $k^{th}$ singular cohomology of $X$ vanish for $k > n$?

Answer: No


Question: Determine the number of rational-sided right triangles whose sides are of the form \( \binom{x}{2}, \binom{y}{2}, \binom{z}{2} \).

Answer: 0


Question: Determine the interval of convergence for the power series \( \sum_{n=1}^{\infty}\frac{(-1)^{\lfloor \log_2 n\rfloor}}{n}x^n \), including the behavior at the endpoints.

Answer: [-1, 1)


Question: Determine if the series \( \sum_{n=1}^{\infty}(n+2)x^{n+1} \) is continuous on the interval \([-1/10,1/10]\). If it is, evaluate the integral \( \int_{-1/10}^{1/10}\sum_{n=1}^{\infty}(n+2)x^{n+1}dx \).

Answer: \dfrac{1}{495}


Question: In triangle $ABC$, circle $O$ is inscribed, with points $P$, $Q$, and $R$ being the points of tangency with sides $BC$, $CA$, and $AB$ respectively. Given that $AB = AC = 13$ and $BC = 10$, find the value of $\sin \angle AOR$. \( \boxed{?} \)

Answer: \dfrac{12}{13}


Question: Suppose that X is the sum of m independent Bernoulli random variables that are not necessarily identically distributed, and thus it has a Poisson binomial distribution. Determine whether mX is also a Poisson binomial random variable.

Answer: No


Question: Find the value of \(\lambda\) such that \(\sum_{r=0}^{19} (r+1)^4 \binom{20}{r+1} = \lambda \cdot 2^{16}\).

Answer: 209160


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{\sin^2x - x\tan x}{x^4}$$

Answer: -\dfrac{2}{3}


Question: In the alternating group $A_5$, consider elements $x$ and $y$ where $x$ has order $2$, $y$ has order $3$, and together they generate the entire group $A_5$. Must the order of the product $xy$ be $5$?

Answer: Yes


Question: Calculate the flux of the vector field \( F = 2x^2y \hat{i} + 2yz \hat{j} - z^2 \hat{k} \) across the closed surface of the region \( E \), where \( E \) is the part of the ball \( x^2 + y^2 + z^2 \leq 1 \) with \( z \geq 0 \).

Answer: 0


Question: Given functions \(f_n, g_n \in L^p([0,1])\) for all \(p \in [1,\infty]\), suppose they satisfy the following conditions: \(f_n\) converges weakly to \(f\) in \(L^p\), \(g_n\) converges weakly to \(g\) in \(L^p\), and \(f_n \cdot g_n\) converges weakly to \(h\) in \(L^p\). Is it necessarily true that \(h = fg\)?

Answer: No


Question: Evaluate the expression: $$ \sum_{a=2}^\infty \sum_{b=1}^\infty \int_{0}^\infty \frac{x^{b}}{e^{ax} \ b!} \ dx .$$

Answer: 1


Question: Let $F$ be a number field, and let $\alpha \in \mathcal{O}_F$, the ring of integers of $F$. Is the trace of $\alpha$, denoted $\operatorname{tr}(\alpha)$, an element of $\mathbb{Z}$?

Answer: Yes


Question: Determine the number of solutions to the equation \((x-\lfloor x \rfloor)(x-\{x\})=17\) for \(17 \le x \le 2000\).

Answer: 1982


Question: Suppose the random variable Y has a moment generating function given by \( M(t) = \frac{e^{ct}}{1-dt^2} \) for \(-1 < t < 1\). Given that the mean and variance of Y are 5 and 4, respectively, find the value of \( c + d \).

Answer: 7


Question: Calculate the expected value of $X^2$, denoted as $E[X^2]$, for a random variable $X$ with a probability density function (PDF) given by $f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$, where $x$ is any real number.

Answer: 1


Question: Consider a sequence of continuous functions $f_m: [0,T] \to \mathbb{R}$ that are non-negative and satisfy $\int_0^T f_m(t)dt \to 0$ as $m \to \infty$. Does it follow that $f_m(t)\to 0$ as $m \to \infty$ for all $t \in [0,T]$? Answer yes or no and justify your answer.

Answer: No


Question: Suppose $f: \mathbb{R}_{+} \to \mathbb{R}_{+}$ is a function satisfying the following conditions: \(f(x) > 0\) for all \(x > 0\) and \(\lim_{x \to 0} f(x) = 0\). Is it sufficient to conclude that \(f(x) \leq 2x\) for sufficiently small \(x\)? Justify your answer.

Answer: No


Question:
Solve the matrix equation for matrix $A$:

$$A^T \begin{pmatrix} 10 & 0 & 0 \\ 0 & 20 & 0 \\ 0 & 0 & 25 \\ \end{pmatrix} A = \begin{pmatrix} \frac{35}{2} & \frac{5 \sqrt{3}}{2} & 0 \\ \frac{5 \sqrt{3}}{2} & \frac{25}{2} & 0 \\ 0 & 0 & 25 \\ \end{pmatrix}$$

where $A^T$ is the transpose of $A$. Verify if the following matrix $A$ is a solution:

$$A = \left(\begin{array}{ccc} \frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 1 \end{array}\right)$$

Answer: A


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

Answer: 1


Question: Let \( G \) be a group of order 8 generated by elements \( a \) and \( b \) such that \( a^4 = b^2 = 1 \) and \( ba = a^3 b \). Determine the order of the center of \( G \).

Answer: 2


Question: Determine whether \( \lim_{x \to 0^+} \left(x\lfloor \frac{a}{x} \rfloor\right) = a \).

Answer: a


Question: Find the remainder when \(11^{2402}\) is divided by \(3000\).

Answer: 121


Question: Evaluate the limit \( \lim_{(x, y) \to (0, 0)} (1 + x^2 + y^2)^{\frac{1}{x^2 + y^2 + xy^2}} \) using polar coordinates.

Answer: e


Question: Determine whether the set of $n \times n$ symmetric copositive matrices, defined by $x^T A x \geq 0$ for all $x \in \mathbb{R}^n$ with $x \geq 0$, is convex.

Answer: Yes


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

Answer: 0


Question: Given a differentiable function $f: \mathbb{R} \to \mathbb{R}$ that satisfies the condition $f(x) = f\left(\frac{x}{2}\right) + \frac{x}{2}f'(x)$, find the value of $f''(20) - f''(10) + 4f''(8)$. \( \boxed{?} \)

Answer: 0


Question: Find the fractional part of \( \frac{7^{300}}{10} \).

Answer: \dfrac{1}{10}


Question: Evaluate the double integral \( \iint\limits_{A} y \, dA \) where \( A \) is the region in the \( xy \)-plane defined by \( x^2 \leq y \) and \( x^2 + y^2 \leq 2 \).

Answer: \dfrac{22}{15}


Question: Determine if the sequence of random variables $(Y_n)_{n\geq 1}$, where $E(Y_n) \to 2$ and $Var(Y_n) \to 0$, converges in distribution. If it does, specify the limiting distribution.

Answer: 2


Question: Evaluate the ratio \(\frac{J_{6,3}}{J_{5,2}}\), where \(J_{m, n} = \int_{0}^{\frac{\pi}{2}} \cos^m x \cos nx \; dx\) for natural numbers \(m\) and \(n\). Given that \(J_{m,n}\) can be expressed recursively as \(J_{m,n} = \frac{m}{m+n} J_{m-1,n-1}\), find the value of the ratio.

Answer: \dfrac{2}{3}


Question: Determine the number of ways to color the edges of a regular hexagon using three colors such that each color is used on exactly two edges, and the coloring is invariant under reflection across a diagonal.

Answer: 18


Question: Find all possible values of $m$ such that the equations $\sin^4x+\cos^4x=m$ and $\sin^6x+\cos^6x=m$ have common real solutions, where $m$ is a non-negative real number.

Answer: 1


Question: Find the function $f(x)$ that satisfies $f'(0) = 1$, $-1 < f(x) < 1$, and $f(x+y) = \frac{f(x) + f(y)}{1 + f(x)f(y)}$ for all $x \in \mathbb{R}$. Then, determine the limit $\lim_{x\to\infty} f(x)^x$. Provide your answer as a single value.

Answer: 1


Question: Find the smallest positive integer $x$ such that $149$ divides $x^2 - 69^3$. Provide your answer as an integer.

Answer: 34


Question: Determine whether there exists a non-zero holomorphic function $g(z)$ that satisfies $|g(z)| = |z|^b$ for some real number $b$ in an annulus centered at the origin.

Answer: Yes


Question: Let $R$ be a commutative integral domain with $I$, $J$, and $K$ as three ideals, where $I$ is a non-zero finitely generated ideal. Determine whether the equality $IJ = IK$ implies that $J = K$. Provide a proof or counterexample.

Answer: J = K


Question: Determine the value of \( f(0) \) for the function \( f(x) = \frac{2}{x}(e^{\frac{x}{2}}-1) \) by considering the series representation \( \sum_{k=0}^\infty \frac{x^k}{2^k(k+1)!} \).

Answer: 1


Question: Determine whether a quotient ring \( R/I \) can be isomorphic to the ideal \( I \), where \( I \) is a proper ideal of \( R \).

Answer: Yes


Question: Suppose $X \in SO(3)$ is embedded in $SO(4)$ by placing it in the bottom right of a $4 \times 4$ matrix $Y$, with $0$'s in the first row and column except for the top left entry, which is $1$. Let $x \in S^3$ be a unit quaternion. If each column of $Y$ is right-multiplied by $x$, will the resulting matrix still belong to $SO(4)$?

Answer: Yes


Question: Consider the power series $f(z) = \sum_{n \geq 0} a_nz^n$ with convergence radius $R$. If $f(z)$ cannot be continued continuously to any point on the circle $|z|=R$, does it imply that there are no regular points of $f$ on this circle? Answer yes or no.

Answer: Yes


Question: Find the derivative of the function \( f(x) = \Phi^{-1}(1 - \Phi(x)) \), where \( \Phi \) is the cumulative distribution function of the standard normal distribution.

Answer: -1


Question: Evaluate the integral \( \int_0^\pi \frac{\sin \theta}{1+\cos^2 \theta} d\theta \) using an infinite series.

Answer: \dfrac{\pi}{2}


Question: Determine the number of solutions to the equation $e^z - z = 0$.

Answer: \infty


Question: Calculate \( \int_{[0,1]} g \, dm \), where the function \( g : [0,1] \to \mathbb{R} \) is defined by \( g(x) = 0 \) if \( x \) is rational, and \( g(x) = e^2 \) if \( x \) is irrational, and \( m \) is the Lebesgue measure.

Answer: e^2


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{\ln\left(\frac{n+2}{n+1}\right)}{\ln\left(\frac{n+1}{n}\right)} \) using the Stolz-Cesaro Theorem.

Answer: 1


Question: Evaluate the integral \( \int_{|z|=10} \frac{\sin(z)}{z(z-1)} \, dz \) using the residue theorem, where the contour is a circle of radius 10 centered at the origin.

Answer: 2\pi i \sin(1)


Question: Suppose $D$ is an open subset of $\mathbb{R}^n$ containing the origin $0$, such that every ray emanating from $0$ intersects the boundary of $D$ in at most one point. Is it true that the boundary $\partial D$ has Lebesgue measure 0?

Answer: Yes


Question: Determine whether there exists an infinite group \( G \) and an infinite subgroup \( H \) such that the conjugate of \( H \) by an element of \( G \) is a proper subgroup of \( H \).

Answer: Yes


Question: Rewrite the expression \(X^2 - 2X - 1 + \frac{2}{X} + \frac{1}{X^2}\) in terms of \(Y\), given that \(Y = X - \frac{1}{X}\).

Answer: (Y - 1)^2


Question: Find all positive integers \( n > 1 \) such that \( n^2 \) divides \( 2^n + 1 \).

Answer: 3


Question: Determine whether for every closed subset $A$ of $\mathbb{R}$, there exists a real, continuous, infinitely differentiable function $f$ such that the zero set of $f$, denoted by $f^{-1}(0)$, is exactly $A$. Provide a justification for your answer.

Answer: Yes


Question: Express the area of the portion of the surface \( z = x^2 + y^2 + 2 \) that lies above the circle \( x^2 + y^2 \leq 1 \) as a single integral and evaluate it to four decimal places.

Answer: 5.3304


Question: Determine the number of solutions for the equation \(x^y - y^x = x + y\) where both \(x\) and \(y\) are prime numbers.

Answer: 1


Question: A coin is randomly selected from an infinite bag where each coin has a probability of landing heads that is uniformly distributed between 0 and 1. If the coin lands heads on the first flip, what is the probability that it will also land heads on the second flip?

Answer: \dfrac{2}{3}


Question: Given the premises $\lnot A \lor \lnot B \lor C$, $\lnot A \lor B \lor D$, and $A \lor E$, prove that $C \lor D \lor E$ follows using natural deduction.

Answer: C \lor D \lor E


Question: Evaluate the limit: \[ L = \lim_{x \to 0} \left(1 + \arctan\left(\frac{x}{2}\right)\right)^{\frac{2}{x}} \]

Answer: e


Question: Determine whether the set $S$ of continuous functions $f:[0,1]\to[0,\infty)$ satisfying the equation \[\int_{0}^{1} x^2f(x) \, dx=\frac{1}{2}\int_{0}^{1} xf^2(x) \, dx+\frac{1}{8},\] is a singleton set containing only one function.

Answer: Yes


Question: Solve for $x$ in the equation: $$ x^{(x-1)^2} = 2x + 1 $$ without using guessing methods.

Answer: 1 + \sqrt{2}


Question: If $a$ is an odd perfect number, does there exist an even perfect number $b$ such that the product $ab$ is a perfect number?

Answer: No


Question: Find the area enclosed by the curve $y = f(x)$ from $x = 1$ to $x = 3$, given that the function $f(x)$ satisfies the equation $\int_0^1 (x - f(x))f(x)\,dx = \frac{1}{12}$. Determine the function $f(x)$ and calculate the area.

Answer: 2


Question: Determine whether a continuous function on the real numbers can have a periodic point of period 48 without having a periodic point of period 24.

Answer: Yes


Question: Calculate the surface area of the region defined by the equation $x^2 + y^2 = 9$, bounded by $z = 0$ and $z = y$, and considering only the region where $y \geq 0$. Use the parameterization $r(u, v) = \langle 3\cos v, 3\sin v, u \rangle$ for $0 \leq u \leq 3\sin v$ and $-\frac{\pi}{3} \leq v \leq \pi$. Given that $|r_u \times r_v| = 3$, find the surface area.

Answer: 18


Question:
Determine the value of $d$ for which the generalized Cantor set, constructed by retaining intervals of size $d<1$ at both ends and removing an interval of size $1-2d$ at each stage, has Lebesgue measure zero. Specifically, find the value of $d$ that satisfies the equation:

$$ \sum_{n=1}^{\infty} 2^{n-1} (1-2d^n) = 1. $$

Answer: \dfrac{1}{3}


Question: Find the maximum area of an ellipse that passes through the point \((3,1)\) and is expressed in the form \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\).

Answer: 6\pi


Question: In circle $\Gamma_1$, let $CD$ be a chord and $AB$ be a diameter perpendicular to $CD$ at point $N$. A second circle $\Gamma_2$ is centered at $C$ and intersects $\Gamma_1$ at points $P$ and $Q$. Given that the radii of $\Gamma_1$ and $\Gamma_2$ are $61$ and $60$ respectively, and $AN = 72$, find the length of $CM$, where $M$ is the intersection point of $PQ$ and $CD$. Provide your answer as a single numerical value.

Answer: 30


Question: Let $h$ and $k$ be functions of $u$ and $v$ such that $h(2, 1) = 1$ and $k(2, 1) = 3$. At the point $(2, 1)$, the differentials are given by $dh = 3du - 2dv$ and $dk = -du + 4dv$. The function $w$ is defined implicitly by the equation $w^3h - wk + u + v = 0$, with the condition $w(2, 1) = 2$. Find the value of $\frac{\partial w}{\partial v}$ at the point $(2, 1)$. \(\boxed{\text{Answer}}\)

Answer: \dfrac{23}{9}


Question: Determine if the set \(\mathcal{N}\) of subsets of \(Y\) such that the subset or its complement is countable forms a \(\sigma\)-algebra when it is intersected with the \(\sigma\)-algebra generated by the singletons of \(Y\).

Answer: Yes


Question: Evaluate the limit: $$\lim\limits_{(x,y) \to (0,0)} \frac{\arctan(xy)}{\sqrt{x^2+y^2}}$$

Answer: 0


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

Answer: 0


Question: Find the integer value of \( K \) for the expression \[ K = \left\lfloor\sum_{r=1}^{80}\int_0^1x^{\sqrt r-1}dx\right\rfloor \] after evaluating the integral and simplifying the sum.

Answer: 16


Question: Determine the values of $n$ for which there exists a one-to-one, bounded, and continuous function $f: B(0,1) \subset \mathbb{R}^n \rightarrow \mathbb{R}$ that is not uniformly continuous.

Answer: \emptyset


Question: Given a countably infinite set $H$, find the maximum number of subsets that can be selected from the power set $P(H)$ such that every $n$ subsets have infinitely many elements in common, but every $n+1$ subsets have only finitely many elements in common, where $n$ is a positive integer greater than or equal to 2.

Answer: 2^{\aleph_0}


Question: Consider a triangle $ABC$ in the $X-Y$ plane and a set of $m$ points lying strictly inside the triangle. Determine the number of line segments with endpoints at the vertices of the triangle that intersect all the $m$ points.

Answer: 0


Question: In a directed graph $G=(V,E)$ where each vertex has equal in-degree and out-degree, and self-loops are not allowed, if there are $k$ mutually vertex-disjoint paths from vertex $v_1$ to vertex $v_2$, must there also be $k$ mutually vertex-disjoint paths from $v_2$ to $v_1$? Justify your answer.

Answer: Yes


Question: Calculate the sum: \( \tan(\pi/16) + \tan(5\pi/16) + \tan(9\pi/16) + \tan(13\pi/16) \).

Answer: -4


Question: Let $f:\mathbb{R} \to \mathbb{R}$ be a measurable, integrable function. Suppose that the function \( x \mapsto \int_{-\infty}^x f(t) \, \mathrm{d}t \) is non-decreasing throughout \( \mathbb{R} \). Does it necessarily follow that \( f \) is non-negative almost everywhere with respect to the Lebesgue measure?

Answer: Yes


Question: In hyperbolic geometry, determine the maximum possible sum of the angles of a triangle $\Delta ABC$. Provide your answer in degrees.

Answer: 180


Question:
Determine which of the following formulas correctly represents the orientation quaternion of a joint connecting frame A (upper arm) and frame B (forearm) in the context of unit orientation quaternions:

1. \( q_{joint} = q_A ^* \otimes q_B \)
2. \( q_{joint} = q_A \otimes q_B ^* \)

Answer: 1


Question: Determine the smallest value of the parameter \(\alpha\) for which the equation \(\sin^2 x \cdot \cos 2x + \alpha(\cos^4 x - \sin^4 x) = -10(2\alpha + 1)^2\) has at least one real solution. Simplify the expression to obtain \(2\sin^4 x - \sin^2 x(1-2\alpha) - (\alpha + 40\alpha^2 + 40\alpha + 10) = 0\) and solve the resulting equation.

Answer: -\dfrac{5}{8}


Question: Calculate the limit: $$ \lim_{x\to 0}\frac{2-x}{x^3}e^{(x-1)/x^2} $$

Answer: 0


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

Answer: 2e


Question: Provide an example of a bounded sequence that is not convergent, but where the differences of consecutive terms converge to zero.

Answer: a_n = \sin(\sqrt{n})


Question: Is it possible to find a quartic polynomial with real coefficients and distinct real roots \((r_1, r_2, r_3, r_4)\) such that its first derivative has roots \(r_1, r_2, r_3\) and its second derivative has roots \(r_2, r_3\)? Answer with 'Yes' or 'No'.

Answer: No


Question: Can a polynomial of degree $n$ have more than $n$ roots in an integral domain?

Answer: No


Question: Determine whether the additive group of real numbers \((\mathbb{R}, +)\) has a proper subgroup that is isomorphic to itself.

Answer: Yes


Question: Let \( \{a_n\} \) be a fixed sequence with \( a_n \ge 0 \). Assume that the series \( \sum_{n=1}^{\infty} n^q a_n \) converges for any arbitrarily small \( q \). Given \( p \ge 1 \), determine if there exists a \( q \) such that the series \( \sum_{n=1}^{\infty} n^p \log(n) \sqrt{a_n} \) also converges.

Answer: Yes


Question: Evaluate the integral \( \int_0^{\frac{\pi}{4}} Re^{-R^2 \sin x} \, dx \) as \( R \to \infty \) using a suitable substitution.

Answer: 0


Question: Find the points of continuity for the function \( f(x,y) = \frac{\sin(xy)}{\sqrt{4-x^2-y^2}} \).

Answer: (x, y) \mid x^2 + y^2 < 4


Question: In a collection of 20 sets denoted by \(X_r\) (\(r = 1, 2, \ldots, 20\)), each set contains 5 elements. Similarly, there is another collection of sets denoted by \(Y_r\) (\(r = 1, 2, \ldots, n\)), where each \(Y_r\) contains 2 elements. It is given that \(\bigcup_{r=1}^{20} X_r = S = \bigcup_{r=1}^{n} Y_r\). Every element in the set \(S\) is found in exactly 10 of the \(X_r\) sets and in exactly 4 of the \(Y_r\) sets. Determine the value of \(n\).

Answer: 20


Question: Evaluate the contour integral \( \int_C \frac{z+2}{(z^2-2z+1)^2}dz \) using the residue theorem, where \( C \) is the semi-circle in the right half-plane with radius \( R > 1 \) and its diameter lies on the imaginary axis. The contour is traced counterclockwise.

Answer: 0


Question: Evaluate the integral \( \int_{1}^{5} \delta(x^2-4)x^3 \,dx \) using the properties of the Dirac delta function.

Answer: 2


Question: Find the value of $c$ such that the random variables $3X - 2Y$ and $-X + cY$ are independent, given that $(X,Y)$ is a bivariate normal distribution with means $\mu_x=2$, $\mu_y=-1$, variances $\sigma_x^2=4$, $\sigma_y^2=9$, and correlation $\rho=-0.5$. \( \text{Express your answer as a single value of } c. \)

Answer: -\dfrac{2}{3}


Question: Find the value of the limit \(\lim_{x \to 1}\frac{f(x)}{x-1}\), given that the polynomial function \(f(x)\) satisfies the differential equation \(f(x) + f'(x) + f''(x) = x^5 + 64\).

Answer: -15


Question: Let \((X, d)\) be a metric space and \(Y \subseteq X\). Suppose there exists an \(x \in X\) and \(r > 0\) such that \(Y \subseteq B(x, r)\). Determine the smallest \(r''\) such that \(conv(Y) \subseteq B(x, r'')\), where \(conv(Y)\) denotes the convex hull of \(Y\).

Answer: 2r


Question: Evaluate the integral \( I = \int_{0}^{\pi} \sin(x)\cos(x)\cos(2x)\cos(4x)\dots \cos(2^{n-1}x) \, \mathrm{d}x \).

Answer: 0


Question: Find the minimum value of the expression \(x^3y + xy^3 + 4\) given that \(x\) and \(y\) are real numbers satisfying the equation \(x^2 + y^2 + xy = 1\). Assume \(x = r\sin(\omega)\) and \(y = r\cos(\omega)\).

Answer: 2


Question: Let $X$ and $Y$ be random variables with hazard functions $h_X(x)$ and $h_Y(y)$, respectively, such that $h_X(x) = k h_Y(x)$, where $k > 0$. Determine the points of intersection of the survival functions $S_X(t)$ and $S_Y(t)$. Provide your answer as the value(s) of $t$ where $S_X(t) = S_Y(t)$.

Answer: 0