Question: In a two-player game played on a $9 \times 9$ board, each square initially has one mark. Players take turns, with the first player picking a mark adjacent to at least one other mark and the second player removing one mark from an adjacent square. Determine the maximum number of marks the first player can ensure the second player removes, regardless of the second player's strategy.

Answer: 40


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

Answer: 2e - 5


Question: Compute the limit: $$ \lim_{n\to\infty}\frac{\dfrac{n}{1}+\dfrac{n-1}{2}+\dots+\dfrac{2}{n-1}+\dfrac{1}{n}}{\ln(n!)} $$

Answer: 1


Question: Find the remainder when \(3^n + n^3\) is divided by 7, given that \(3^n + n^3\) is a prime number and \(n\) is a positive integer greater than or equal to 2.

Answer: 3


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

Answer: 1


Question: Determine whether the expression \(4^{1536} - 9^{4824}\) is divisible by 35.

Answer: Yes


Question: Calculate the sum \( T = \sum_{n=1}^{\infty} \frac{1}{(2n-1)(2n+1)(2n+3)} \).

Answer: \dfrac{1}{12}


Question: Find the singular part of the function \( f(z) = \frac{\pi^2z^2}{\sin^2(\pi z)} \) at \( z = 0 \).

Answer: 0


Question: Find the right-hand limit of the function \( f(x) = (x^2 - 25)^{\frac{1}{\ln(x-5)}} \) as \( x \) approaches 5.

Answer: e


Question: Compute the integral of the square of the weak derivative of the absolute value function on the interval \([-1, 1]\).

Answer: 2


Question: Evaluate the integral \( I = \int_0^{2\pi}\sum_{n=1}^{N}a_n \sin (nt)\cos(n\xi) \sum_{n=1}^{N} a_n\sin(n\xi)\cos(n\tau) \, d\xi \).

Answer: 0


Question: Determine the number of integral solutions for the equation \(x^4 - ax^3 - bx^2 - cx - d = 0\), given that \(a \geq b \geq c \geq d\) and \(a, b, c, d \in \mathbb{N}\).

Answer: 0


Question: Evaluate the limit: $$\lim\limits_{n\to \infty }\left(\sqrt[n]{\left(\frac{(n+1)!+1}{(n+1)!-1}\right)^{\left(n\right)!}}\right)$$

Answer: 1


Question: Evaluate the integral: $$\int_{-\infty}^\infty \frac{\cos(x)}{x+i}\,dx.$$

Answer: -\pi i e^{-1}


Question: Calculate the limit of the sequence \( \frac{1}{\sqrt{n+1}} + \frac{1}{\sqrt{n+2}} + \frac{1}{\sqrt{n+3}} + \cdots + \frac{1}{\sqrt{n+n^2}} \) as \( n \to \infty \).

Answer: \infty


Question: Find the maximum value of the function \(f(t)=16\cos t \cdot \cos 2t \cdot \cos 3t \cdot \cos 6t\) using logarithmic differentiation.

Answer: 16


Question: Consider a random vector $(x_1, \ldots, x_k)$ with a multinomial distribution characterized by parameters $n$ and $(p_1, \ldots, p_k)$, where $p_i > p_j$ for some indices $i$ and $j$. Determine whether the probability of the event $x_i > x_j$ approaches 1 as $n$ approaches infinity, i.e., evaluate $\lim_{n\to\infty}Pr(x_i > x_j) = 1$.

Answer: 1


Question: Given a continuous and differentiable function $f: [-1, 1] \rightarrow \mathbb{R}$ with $|f(t)| \leq 1$ and $|f'(t)| \leq \frac{1}{2}$ for all $t \in [-1, 1]$, consider the set $A = \{t \in [-1, 1]: f(t) = t\}$. Determine the cardinality of the set $A$.

Answer: 1


Question: Consider a continuous-time, homogeneous diffusion process $X$ and two points $x$ and $y$. A "bump" is defined as the time it takes for $X_t$ to go from $y$ to $x$ and then back to $y$ again. Define the stopping times $t_1, t_2, \ldots$ iteratively as follows: $t_1$ is the first time a bump finishes, $t_2$ is the time it takes from the first bump finished till the second bump finishes, and so on. Are the random variables $t_1, t_2, \ldots$ independent and identically distributed?

Answer: Yes


Question: Find the area bounded by the curve given by the parametric equations \( y = a(1 - \cos t) \) and \( x = a(t - \sin t) \) and the x-axis.

Answer: 3\pi a^2


Question: Find a compact metric space $X$ with at least two points such that $X^2$ is homeomorphic to $X^3$. Provide an example of such a space.

Answer: C


Question: Evaluate the triple integral: $$\int_0^1 \int_0^x \int_y^x x e^{-z^2} \ dz\ dy\ dx$$

Answer: \dfrac{1}{4e}


Question: Let $f:E\rightarrow F$ be a function that is twice differentiable at $x\in E$. Prove or disprove that the following limit holds: $$\lim_{h,k\rightarrow 0}\frac{A_x(h,k)-f''(x)(h)(k)}{\|h\|\|k\|}=0$$ where $A_x(h,k):=f(x+h+k)-f(x+h)-f(x+k)+f(x)$.

Answer: 0


Question: Evaluate the integral \(\displaystyle\int\limits_{\gamma }\frac{e^z}{z^2+1}dz\) where \(\gamma\) is the circle centered at the origin with radius 2.

Answer: 2\pi i \sin(1)


Question: Simplify the expression \(E\left(\left\{\sum\limits_{i=1}^n (Y_i-\mu ) \right\}^2\right)\) where \(Y=(y_1, \dots , y_n)^T\) is a random sample from \(N(\mu,\sigma^2)\).

Answer: n\sigma^2


Question: Find the polynomial $f(x) = x^3 + ax^2 + bx + c$ such that the maximum value of $|f(x)|$ on the interval $[-2, 2]$ is minimized.

Answer: x^3 - 3x


Question: In Euclidean space, determine whether there exists an open but not closed subset \( X \) and a bounded but not open subset \( Y \) such that they are homeomorphic.

Answer: Yes


Question: Let $Y$ be a subset of a metric space $X$. Suppose a subset $E$ of $Y$ can be expressed as $E = Y \cap G$ for some open subset $G$ of $X$. Does this imply that every singleton subset of $E$ is open relative to $Y$? Answer yes or no.

Answer: No


Question: Let $S = \{m + \frac{1}{n} \mid m, n \in \mathbb{N}\}$. Determine the derived set (set of cluster points) of $S$.

Answer: \mathbb{N}


Question: Compute the limit of \( 4\sum_{n=1}^k\frac{(-1)^{n+1}}{2n+1} \) as \( k \to \infty \).

Answer: 4 - \pi


Question: Expand the function \( \frac{1}{z-i} \) into a power series around \( z_0=1 \) and determine its radius of convergence.

Answer: \sqrt{2}


Question: Evaluate the integral \( \int_{|z| = 2}\frac{z^3}{(z-1)^2}dz \) using Cauchy's integral formula, where \(|z| = 2\) is oriented counter-clockwise.

Answer: 6\pi i


Question: Evaluate the integral \( \int_{|z|=1} \sqrt{9-z^2} \, dz \) using the local Cauchy theorem.

Answer: 0


Question: Determine the number of nonreal complex roots of the polynomial \(z^5+\frac{7 z^3}{30}+\frac{17 z^2}{100}+\frac{z}{6000}+\frac{239}{37500}\) without using calculus.

Answer: 4


Question: Evaluate the limit: $$\lim_{x \rightarrow 0} \frac{\cos(x \sqrt{2}) - \frac{1}{1+x^2}}{x^4}$$

Answer: -\dfrac{5}{6}


Question: Evaluate the limit of the expression \[ \frac{\sum_{i=1}^{Q} \left ( \frac{k-i+1}{k}\right )^{j} - \sum_{i=1}^{Q} \left ( \frac{k-i}{k}\right )^{j}}{\left [\left ( \frac{k-i+1}{k}\right )^{j} - \left ( \frac{k-i}{k}\right )^{j} \right ]} \] as $k$ approaches infinity, where $j$ and $Q$ are constants.

Answer: Q


Question: Evaluate the double integral \( \iint_{E} xy \, dA \) where the region \(E\) is bounded by the curves \(xy = 3\), \(xy = 1\), \(y = 3x\), and \(y = x\). Use the substitution \(x = \frac{u}{v}\) and \(y = v\) with the Jacobian \(\frac{\partial(x,y)}{\partial(u,v)} = \frac{1}{v}\). Determine the correct bounds and evaluate the integral to find the correct solution of \(2 \ln(3)\).

Answer: 2 \ln(3)


Question: Determine whether the statement is true or false: Suppose $f(x)$ is an irreducible polynomial in $R[x]$ over an integral domain $R$, and $r$ is a root of $f(x)$ in an algebraic closure of the fraction field of $R$. Is $R[x]/(f(x))$ isomorphic to $R[r]$?

Answer: True


Question:
Consider the following ordinary differential equation (ODE) system in \(\mathbb{R}^2\):

$$\dot{x} = f(x)$$

where \(x = (x_1, x_2)\) and \(f(x) = \alpha e_r(x) + \beta e_{\theta}(x)\), with \(\alpha, \beta \in \mathbb{R}\), \(e_r(x)\) is the radial unit vector, and \(e_{\theta}(x)\) is the tangent unit vector at point \(x \in \mathbb{R}^2\). Given that \(f(0) = 0\), determine the necessary condition for the origin to be a stable equilibrium point:

(a) \(\alpha < 0\) and \(\beta \le \alpha\)
(b) \(\beta < 0\) independently from \(\alpha\)
(c) \(\alpha = 0\) and \(\beta < 0\)
(d) \(\alpha \le 0\) independently from \(\beta\)

Answer: d


Question: Find the Cumulative Distribution Function (CDF) of the absolute difference between two independent uniform random variables $X_1$ and $X_2$, both distributed on the interval [0, 1].

Answer: 2z - z^2


Question: Determine the value of $b \in \mathbb{R}$ for which the equation $|z-1|=|z-i|=|z-(a+bi)|$ has no complex solutions, given $a \in \mathbb{R}$.

Answer: 1 - a


Question: Determine the radius of convergence of the Taylor series expansion of \( f(z) = \sin\left(\frac{1}{z+2}\right) \) centered at \( z = 0 \).

Answer: 2


Question: Evaluate the limit \( \lim_{N \rightarrow \infty} \frac{g(N)}{N^3} \), where \( g(N) \) is the number of quadruples \((x, y, z, w)\) of positive integers satisfying \( x + y + z + w = N \) and \( x \le y \le z \le w \).

Answer: \dfrac{1}{144}


Question: Evaluate the limit: $$\lim\limits_{x\to 0} \frac{x\sin x+1-\cos x}{x^2+x\sin x}$$ without using Taylor expansion.

Answer: \dfrac{3}{4}


Question: Evaluate the limit: $$\lim_{x\to\ \infty} {x - \log(e^x + 1)}$$

Answer: 0


Question: Find the number of elements in the quotient ring \( \mathbb{Z}/8\mathbb{Z}[X] / (x^2 + 2x + 2) \).

Answer: 64


Question: Let \( f(x) = \int_2^x \frac{dt}{\sqrt{1+t^4}} \) and \( g \) be the inverse of \( f \). Find \( g'(0) \).

Answer: \sqrt{17}


Question: Let $g:[a,b]\to\mathbb{R}$ be a Riemann-integrable function. If there exists a dense subset $A\subseteq[a,b]$ such that $g(x)=0$ for all $x\in A$, is it always true that $\int_{a}^{b} g = 0$?

Answer: 0


Question: Determine the highest order of elements in the direct product group \( G = C_{5} \times C_{12} \times C_7 \).

Answer: 420


Question: Evaluate the limit: $$\lim_{x\rightarrow 0} \frac{\sqrt{1+ x\sin(x)} - \sqrt{\cos(2x)}}{ \tan^2(x/2)}$$

Answer: 6


Question: Calculate the limit: $$\lim_{x\to0}{\frac{\sqrt{\cosh{(3x^2)}}\cdot e^{4x^3}-1}{x^2\tan(2x)}}$$ without using L'Hopital's rule.

Answer: 2


Question: Evaluate the integral \( \int_{-1}^1 \frac{x}{x^6+\mathrm{i}} \, dx \).

Answer: 0


Question: Determine whether the sum of two prime numbers, $p$ and $q$, can be an odd prime number, given that $q$ divides $p^2+1$ and $p$ divides $q^2-1$. Provide a justification for your answer.

Answer: Yes


Question:
Determine if there is a single matrix operation that can be applied to the block matrix \( M_2 = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \), where \( A, B, C, D \) are square matrices of the same size, to produce the matrix \( M_4 \):

\[
M_4 = \begin{pmatrix} A & 0 & B & 0 \\ 0 & A & 0 & B \\ C & 0 & D & 0 \\ 0 & C & 0 & D \end{pmatrix}
\]

without explicitly manipulating the sub-matrices. If \( A, B, C, D \) were scalars, the operation would be \( M_4 = M_2 \otimes I_2 \), where \( I_2 \) is the \( 2 \times 2 \) identity matrix.

Answer: M_4 = M_2 \otimes I_2


Question: Determine whether it is true or false that in a Dedekind domain, every ideal can be generated by a lattice basis consisting of at most two elements.

Answer: A


Question: Find the universal cover of the space obtained by removing a single point from the real projective plane, $\mathbb{R}P^{2}$. Provide your answer with justification.

Answer: \mathbb{R}^2


Question: If $A$ is a $3 \times 3$ matrix and $\det(A) = 2$, find the value of $\det(\text{adj}(\text{adj}(\text{adj}(A))))$. \( \text{adj}(A) \) denotes the adjugate of matrix \( A \).

Answer: 256


Question: Determine whether the following statement is true or false: The union of a sequence of non-decreasing convex sets is always convex.

Answer: True


Question: Given the equation \( | \xi |^{2} = 1 - 4p^{2}(1-p^{2})s^{4} \) where \( s = \sin\left(\frac{\omega}{2} \right) \), determine the range of \( p \) for which the method is stable, i.e., \( | \xi| \leq 1 \). Show that the scheme is stable for \( -1 \leq p \leq 1 \).

Answer: -1 \leq p \leq 1


Question: Determine whether a random matrix chosen from a bounded region in $\mathbb{C}^{n \times n}$ is always diagonalizable over the complex numbers. Answer with "True" or "False."

Answer: True


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

Answer: e - 2


Question: Determine the number of real solutions to the equation \( \cos(\pi\sqrt{x-4})\cos(\pi\sqrt{x}) = 1 \) for \( x \geq 4 \).

Answer: 1


Question: In a non-equilateral simplex, is it possible for the altitudes from different vertices to have the same length?

Answer: Yes


Question: Find the minimum number of distinct non-trivial solutions for the Diophantine equation \(\frac{1}{a} = \frac{1}{x_1} + \cdots + \frac{1}{x_n}\) for any positive integers \(a\) and \(n\).

Answer: 1


Question: Determine whether the converse of the statement "If a function f is convex (or strictly convex) on [a, c] and on [c, b], then f is convex (or strictly convex) on [a, b]" is true or false.

Answer: True


Question: Determine the value of $x$ such that \( \lim\limits_{n\to\infty} \sqrt{1+\sqrt{x+\sqrt{x^2+\cdots+\sqrt{x^n}}}} = 2 \).

Answer: 4


Question: Evaluate the integral \( \int_\sigma \omega \) directly, where \( \omega = x_3 \, dx_1 \wedge dx_2 \) and \( \sigma(\theta, \phi) = (\sin\phi\cos\theta, \sin\phi\sin\theta, \cos\phi) \) is defined on the domain \([0, 2\pi] \times [-\pi/2, \pi/2]\).

Answer: 0


Question: Determine the number of cyclic subgroups of order $10$ in the group $\mathbb{Z}_{100} \oplus \mathbb{Z}_{25}$, given that there are $24$ elements of order $10$ and each cyclic subgroup of order $10$ contains $4$ such elements.

Answer: 6


Question: Suppose \(n \geq 0\) is an integer, and the equation \(x^3 + \alpha x + 4 - (2 \times 2016^n) = 0\) has integer roots \(u, v, w\). Determine all possible values of \(\alpha\).

Answer: -3


Question: How many distinct analytic functions exist that vanish exclusively on the set \( S = \{0\} \cup \{1\} \cup \left\{\frac{1}{4n+7} : n = 1, 2, \ldots\right\} \)?

Answer: 0


Question: Determine the number of real solutions for the equation \((x^3-2x^2+5)\exp(-x) = 5\).

Answer: 2


Question: Evaluate the integral \( \iint\limits_{x^2+y^2 \geq b^2} \frac{1}{\sqrt{x^2+y^2}} \,\mathrm{d}A \) using polar coordinates.

Answer: \infty


Question: Determine the Galois group of the polynomial $x^{4}-4x^{2}+5$ over the field $\mathbb{Q}$. Provide your answer in terms of a well-known group, if possible.

Answer: D_4


Question: Given a polynomial $f$ of degree $1007$ such that $f(k) = 2^k$ for $0 \le k \le 1007$, determine the value of $f(2015)$. \( \boxed{} \)

Answer: 2^{2014}


Question: Determine whether the following statement is true or false: For the sequence of functions \( f_{n}(x)=\frac{nx}{e^x+n+x} \) defined for all \( x \ge 0 \), and for any \( b > a \ge 0 \), the equality \( \int_{a}^{b}(\lim_{n \to \infty}f_{n}(x))dx= \lim_{n \to \infty}\int_{a}^{b}f_{n}(x)dx \) holds.

Answer: True


Question: Determine the dimension of the Lie algebra $\mathfrak{so}(3)$ over $\mathbb{R}$ and find an orthonormal basis for it using its isomorphism with $\mathfrak{su}(2)$. Provide your answer as a dimension and a set of basis elements.

Answer: 3


Question: Find the integer $n$ such that $\phi(n^2) = \phi(n) + 10$, where $\phi(n)$ is the Euler's totient function.

Answer: 6


Question: Determine the interval centered around $x = 0$ that guarantees the uniqueness of the solution for the initial-value problem: $$(x - 2)y'' + 3y = x$$ with initial conditions $y(0) = 0$ and $y'(0) = 1$. Provide your answer as an interval.

Answer: (-2, 2)


Question: Given an undirected graph G where every node has an even degree, determine if it is always possible to orient the edges such that the indegree of each node is exactly half of its original degree.

Answer: Yes


Question: Determine whether a continuous function \( f: [0,1] \to [0,1] \) can make the set \( S = \{(a,b) \mid f(a) = b\} \) dense in \([0,1] \times [0,1]\).

Answer: No


Question: In a Markov chain modeling a population's genotype distribution, two individuals with genotype Aa reproduce, resulting in various states: (AA, AA), (AA, Aa), (Aa, Aa), (AA, aa), (Aa, aa), and (aa, aa), with the first and last states being absorbing. If the process starts in the state (Aa, Aa), calculate the probability that it will eventually reach the state (AA, AA). Assume each child receives one allele from each parent with a 50% chance of getting either allele.

Answer: \dfrac{1}{2}


Question: Let $f(x)$ be a differentiable function with $f(0)=1$ and $f(1)=2$. For any $a \neq 0$, it holds that $$\frac{1}{2a}\int_{x-a}^{x+a}f(t)dt=f(x).$$ Determine the function $f(x)$. \( \text{Provide your answer in the form of a function.} \)

Answer: x + 1


Question: Find the independent coefficient in the inverse of the polynomial $2x^2+2$ in the ring $\mathbb{Z}_3[x]/(x^3+2x^2+2)$. Provide your answer as an integer in $\mathbb{Z}_3$. \(\boxed{\phantom{0}}\)

Answer: 2


Question: Evaluate the residue of the function \( f(z) = \frac{e^{z^2}}{z^5(1-z^2)} \) at \( z=1 \).

Answer: -\dfrac{e}{2}


Question:
Let $f:(-1,1) \rightarrow \mathbb{R}$ be a function such that its $n$th derivative $f^{(n)}(x)$ exists and satisfies $|f^{(n)}(x)| \leq 1$ for all $n \geq 1$ and for every $x \in (-1,1)$. Determine for which of the following intervals $f$ has a convergent power series expansion:

(a) Every $x \in (-1,1)$
(b) Every $x \in (-\frac{1}{2},0)$ only
(c) No $x \in (-1,1)$
(d) Every $x \in (0,\frac{1}{2})$ only

Select the correct interval.

Answer: a


Question: Evaluate the limit: $$\lim_{x \to \infty} \frac{(x \log x + x)^5(1+2/x)^x}{(x+1/x)^5 (\log x + 1/\log x)^6}$$

Answer: 0


Question:
Calculate the integral \( \int_0^\infty \left(\frac{e^{-2t}\sin(-t)}{-t} + \frac{e^{-3t}\sin(-t)}{-t}\right) dt \) using the Laplace transform. Use the frequency domain integration theorem, which states:

\[\mathcal{L}\left[\frac{f(t)}{t} \right] = \int_s^\infty F(\omega) d\omega = \int_s^\infty \mathcal{L}(f(t)) d\omega\]

where \(\omega\) is a dummy variable, and \(s\) is the Laplace variable.

Given \(f(t) = e^{-\alpha t}\sin(-t)\), the Laplace transform of \(\frac{f(t)}{t}\) is:

\[\mathcal{L}\left[\frac{e^{-\alpha t}\sin(-t)}{t} \right] = -\frac{i}{2} \ln\left(\frac{s-\alpha-i}{s-\alpha+i}\right)\]

Calculate the integral for \(\alpha = 2\) and \(\alpha = 3\).

Answer: \dfrac{\pi}{4}


Question: Find the integer \( r \), with \( 0 \leq r < 624 \), such that \( 2^{82} \equiv r \pmod{625} \).

Answer: 329


Question: For large $n$, approximate the probability that the expression \( Y = \frac{V_1 + V_2 + \cdots + V_n} {\sqrt{V_1^2 + V_2^2 + \cdots + V_n^2}} \) is greater than 0.5, where \( V_i \) are independent and identically distributed random variables with a uniform distribution over \([-2, 2]\).

Answer: 0.3085


Question: Find the maximum value of the function \( f(x,y) = \sin{x} + \sin{y} - \sin{(x+y)} + \sqrt{3}[\cos{x} + \cos{y} + \cos{(x+y)}] \).

Answer: 3\sqrt{3}


Question: Consider an $n \times n$ array of sites, where each site is either open or blocked. Fluid is poured from the top row, and a site becomes "full" if it has fluid. Fluid can only move downwards and stops at blocked sites. The array is said to percolate if fluid reaches the bottom row through any column. If each site is open with probability $p$ and blocked otherwise, determine the probability that the $n \times n$ array percolates.

Answer: 1 - (1 - p^n)^n


Question: Are fields that are finitely generated as $\mathbb{Z}$-algebras necessarily finite? Provide a justification for your answer.

Answer: Yes


Question: Given a function $f: \mathbb{R} \to \mathbb{R}$ defined as $f(x) = 4x$ for rational numbers $x$ and $f(x) = x + 6$ for irrational numbers $x$, let $E = \{x \in \mathbb{R} \mid f \text{ is continuous at } x\}$. Determine all the limit points of the set $E$. Provide your answer as a set of limit points.

Answer: \emptyset


Question: If $A$ is a function field in one variable over $\mathbb{C}$, and $B$ is a finite extension of $A$ with degree $[B : A]$, such that $B$ is isomorphic to $\mathbb{C}(x)$ over $\mathbb{C}$, does it follow that $A$ must also be isomorphic to $\mathbb{C}(x)$ over $\mathbb{C}$? Consider the property that a function field in one variable over an algebraically closed field is of genus zero if and only if it is isomorphic to the field of rational functions with coefficients in that field.

Answer: Yes


Question: Consider Bessel's equation given by \(t^2y'' + ty' + (t^2-n^2)y=0\) for \(t \in (0, \infty)\). Suppose \(y_1\) and \(y_2\) are two solutions with initial conditions \(y_1(1)=1\), \(y_1'(1)=0\), \(y_2(1)=0\), and \(y_2'(1)=1\). Determine the Wronskian \(W[y_1,y_2](t)\). Given that \(W[y_1,y_2](1)=1\), compute its value for arbitrary \(t\).

Answer: \dfrac{1}{t}


Question: Consider a non-archimedean local field $F$ with finite field $\mathbb{F}_q$ of prime characteristic $p$. Let $L$ be the completion of the maximal unramified extension of $F$, and denote by $\mathcal{O}$ the valuation ring of $L$ with $\varpi$ as a uniformizer. Let $G = \mathrm{GL}_n$, and define $I$ as the inverse image of lower triangular matrices in $G(\mathcal{O})$ under the map sending $\varpi$ to $0$. The Iwahori decomposition states that $G(L) = \bigcup_{w \in \tilde{W}} I\tilde{w}I$. Determine if it is true that for $w = w_1w_2$ with $\ell(w_1) + \ell(w_2) = \ell(w)$, the equality $Iw_1Iw_2I = IwI$ holds.

Answer: True


Question: Evaluate the limit: \[ \lim_{t\rightarrow 0^{+}} \sum_{n=1}^{\infty} \frac{\sqrt{t}}{1+tn^2} \]

Answer: \dfrac{\pi}{2}


Question: Determine the minimum value of $y$ for $n \geq 2$, where $y$ is given by the expression: $$y = \frac{1}{2}(e^x - e^{-x}) + \frac{n}{2}(e^x + e^{-x}).$$

Answer: \sqrt{n^2 - 1}


Question: If $M$ is a free $R$-module of finite rank $n$ over a principal ideal domain $R$, do proper submodules of $M$ have strictly less rank than $M$? Answer yes or no and justify your answer.

Answer: No


Question: In the octal number system, given the multiplication \((12345)_8 \times (54321)_8 = (d17743365)_8\), determine the missing digit \(d\) without performing the full multiplication.

Answer: 7


Question:
Evaluate the limit using Taylor series expansions up to the second order around $x = 0$:

$$\lim_{x\rightarrow 0}\frac{x \cos (x) - \sin (x)}{e^{x} - e^{-x} - 2x} $$

Answer: -1


Question: If $f:X \to Y$ is a local homeomorphism between connected, locally path-connected, compactly generated Hausdorff spaces, and $f$ is proper, is $f$ necessarily a covering map? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_C \frac{dz}{z(z-2)} \) where \( C \) is the contour defined by \( z = 3e^{i\theta} \).

Answer: 0


Question: Let $V$ be a $\mathbb{C}$-vector space with two norms $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$, each induced by an inner product, making $V$ a separable Hilbert space. Assume there exists a positive constant $c$ such that for every $\phi \in V$, $c^{-1}\Vert \phi \Vert_1 \leq \Vert \phi \Vert_2 \leq c \Vert \phi \Vert_1$. Determine if there exists a linear operator $S : V \rightarrow V$ that is bounded with respect to either norm and satisfies $\Vert S\phi\Vert_1 = \Vert \phi \Vert_2$ for every $\phi \in V$. Provide a justification for your answer.

Answer: Yes


Question: Given the function \( f(x) = x^3 - 12x + 3 \), determine the number of real solutions to the equation \( f(f(x)) = 0 \).

Answer: 9


Question: Determine whether the function \( f(z) = \operatorname{Re}(\ln z) \) is harmonic in the complex plane excluding the negative real axis.

Answer: Yes


Question: Determine if there exists a constant $K$ such that for any non-negative martingale $(Y_n)_{n \geq 0}$, the inequality \( \mathbb{E}[\max_{k \leq n}Y_k] \geq K\mathbb{E}[Y_n] \) holds.

Answer: 1


Question: Find the equation of the directrix of a parabola with a focus at (-1, -1) and a tangent at the point (7, 13) given by the equation 3x - y - 8 = 0.

Answer: x + 8y + 19 = 0


Question: Find the number of maximal ideals in the quotient ring \( \Bbbk[x]/\langle x^3 \rangle \).

Answer: 1


Question: Let $X$ be a connected and locally simply connected topological space with a universal cover $\widetilde{X}$ that has a finite number of sheets. Suppose $f:X\to X$ is a continuous involution. Does the lift $\widetilde{f}:\widetilde{X}\to\widetilde{X}$ of $f\circ\pi$ have a finite period?

Answer: Yes


Question: Use differentials to approximate the percentage decrease in mass of an object moving at a velocity of 0.85c as its velocity decreases to 0.83c, according to Einstein's Special Theory of Relativity.

Answer: 6.1\%


Question: Determine the number of integers less than or equal to $10^8$ that are neither perfect squares, perfect cubes, nor perfect fifth powers.

Answer: 99989526


Question: Evaluate the integral \( \int_{1}^{\infty}\left\{\frac{1}{x}\left\lfloor\frac{1}{x}\right\rfloor\right\}dx \) using the method of partial sums.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to\infty} \left(\frac{x^2+5x+3}{x^2+x+3}\right)^x$$ Find the final result.

Answer: e^4


Question: Consider the function \( f(x, a) = \frac{\ln(ax)}{x} \). If the parameter \( a \) increases at a rate of 2 cm/min, determine how fast the x-coordinate of the local maximum changes when \( a = e \).

Answer: -\dfrac{2}{e}


Question: Evaluate the limit: $$\lim_{x\to0} \frac{8}{x^8} \left[ 1 - \cos\frac{x^2}{2} - \cos\frac{x^2}{4} + \cos\frac{x^2}{2}\cos\frac{x^2}{4} \right]$$

Answer: \dfrac{1}{32}


Question: Determine if there is a closed form for the integral \( I = \int_{-\infty}^{\infty} e^{-e^x-e^{-x}+x}~ dx \).

Answer: 2 K_1(2)


Question: Determine the image of the Lemniscate defined by the polar equation \(r^2 = \cos(2\theta)\) under the inversion transformation \(f(z) = \frac{1}{z}\).

Answer: x^2 - y^2 = 1


Question: Evaluate the limit: $$\lim\limits_{x \to 0} \frac{x - \sin (x)}{(x \sin (x))^{3/2} }$$ using L'Hopital's Rule and Taylor series.

Answer: \dfrac{1}{6}


Question: Calculate the product of all values of \((1+i\sqrt{3})^{\frac{3}{4}}\).

Answer: 8


Question: Let $f$ and $g$ be continuous functions from a compact space $X$ to a metric space $Y$. Consider the mapping $h: X \to Y \times Y$ defined by $h(x) = (f(x), g(x))$. Is the image of $X$ under $h$ compact?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to\infty}\int_{\left(0,\frac{n}{2}\right)}x^2e^x\left(1-\frac{2x}{n}\right)^nd\lambda(x)$$

Answer: 2


Question: Find the largest integer $n$ such that both $n$ and $n+1$ are Hamming numbers, where a Hamming number is defined as a number whose prime factors are all less than or equal to 5.

Answer: 80


Question: Determine whether the sequence of functions $v_n''(x)$, where $v_n(x)=\frac{x^{2n}}{2n(2n+1)}$, converges to a continuous function in $L^2([0,1])$ as $n \to \infty$. Provide a justification for your answer.

Answer: Yes


Question: Determine the upper bound for the absolute value of the partial sum of Nth roots of unity, given by \( \Bigl \vert \sum_{l=0}^{Z-1} e^{\frac{j2\pi li}{N}} \Bigl\vert \), where \(N\) is a power of 2, \(Z < N\), and \(1 \le i \le N\).

Answer: Z


Question: Given that for a fixed $m$, the inequality $a_{n,m} < b_{n,m}$ holds for $n \geq 1-m$, and that $\limsup_{n\to\infty}\frac{1}{n}\ln b_{n,m} = c$, determine whether $\limsup_{n\to\infty}\frac{1}{n}\ln a_{n,m} < c$ or $\limsup_{n\to\infty}\frac{1}{n}\ln a_{n,m} \leq c$.

Answer: \limsup_{n\to\infty}\frac{1}{n}\ln a_{n,m} \leq c


Question: Determine if there exists a program that can compute an increasing function \( g: \mathbb{N} \to \mathbb{N} \) and a decreasing function \( h: \mathbb{N} \to \mathbb{N} \) such that \( g(x) \leq h(x) \) for all \( x \in \mathbb{N} \).

Answer: Yes


Question: Calculate the integral \( \int_0^1{\frac{\left(2x^3-3x^2\right)f'(x)}{f(x)}}\,dx \) given that the function \( f(x) \) is differentiable on \([0, 1]\), satisfies \( f(1) = 1 \), and the functional equation \( f(x)f(1-x) = e^{x^2 - x} \) holds.

Answer: -\dfrac{1}{10}


Question:
Determine whether the following limit holds for any \(c \in (0,1)\):
\[ \lim_{n \to \infty} \left(n^c(\log n)^c-(n-1)^c(\log(n-1))^c\right) = 0 \]

Answer: 0


Question: Determine whether there exists an orientation of the edges of a graph such that the absolute difference between the in-degree and out-degree of each vertex is at most 1.

Answer: Yes


Question: What is the theoretical maximum average compression rate of a reversible function applied to a random bit sequence, given that the function satisfies f'(f(s)) = s?

Answer: 1


Question: For any integer \( n \geq 2 \), determine if there always exists a prime number \( p \) such that \( \varphi(n) < p \leq n \), where \( \varphi(n) \) is the Euler totient function.

Answer: Yes


Question: Find the point of intersection of the lines defined by the equations \( \vec{r}\times\vec{a}=\vec{b}\times\vec{a} \) and \( \vec{r}\times\vec{b}=\vec{a}\times\vec{b} \), where \( \vec{a}=\langle 2,-1,3 \rangle \) and \( \vec{b}=\langle 1,0,1 \rangle \).

Answer: \langle 3, -1, 4 \rangle


Question:
Given a circle $\Omega$ with radius $r$, let $Z$ be the random variable representing the distance from the circle's center to a randomly chosen point, and $X$ be the random variable for the $x$-coordinate of the point. Determine the covariance $\text{cov}(X,Z)$. The joint probability density function (pdf) $p_{X,Y}(x,y)$ is given by:

$$ p_{X,Y}(x,y) = \begin{cases} \frac{1}{\pi r^2}, & \text{if } x^2 + y^2 \leq r^2 \\ 0, & \text{otherwise} \end{cases} $$

You have found $E[Z]=\frac{2}{3}r$ and $E[X]=0$. Calculate $E[XZ]$ by defining the joint pdf $p_{X,Z}(x,z)$ and determining the appropriate integration limits.

Answer: 0


Question: Provide an example of an ideal $A \subseteq \Bbb{C}[X]$ such that $A \subsetneq I(Z(A))$, where $Z(A)$ is the set of zeroes of all polynomials in $A$ and $I(Z(A))$ is the ideal of polynomials that vanish on $Z(A)$. What is one such ideal $A$?

Answer: (X^2)


Question: Evaluate the improper integral \( \int^{+\infty}_0\frac{e^{-t}} {\sqrt t} \, dt \) and determine whether it converges.

Answer: \sqrt{\pi}


Question: Calculate the turning number of the curve \(\gamma(t) = (\cos(t), \sin(5t))\) using the formula for curvature.

Answer: 5


Question: If $\mathbf{M}$ and $\mathbf{N}$ are positive semi-definite matrices, $\alpha \in (0,1)$, and $\mathbf{L}=\alpha\mathbf{M} + (1-\alpha) \mathbf{N}$ is positive definite, does it imply that both $\mathbf{M}$ and $\mathbf{N}$ are positive definite? Answer with 'yes' or 'no' and justify your answer.

Answer: no


Question: Evaluate the integral: $$\int_0^\infty \frac{1}{x^2 + 2x + 2} \mathrm{d} x.$$

Answer: \dfrac{\pi}{4}


Question: A particle follows the path given by \( r(t) = (2t^{3/2}, 2t+1, \sqrt{5} t) \) starting at \( t=0 \). Determine the time \( t \) at which the particle has traveled a total distance of 14 units along its path.

Answer: 3


Question: Determine the smallest positive integer $n$ for which the series \( \sum_{k=0}^\infty \frac{1}{\binom{n+k}{n}} \) converges.

Answer: 2


Question: Determine the largest number of 3-element subsets that can be formed from the set \( \{1, 2, 3, \ldots, 9\} \) such that every pair of subsets has at least one element in common.

Answer: 28


Question: Determine whether every closed subgroup of a locally compact, compactly generated group \( G \) is also compactly generated. Answer true or false.

Answer: False


Question: Evaluate the limit \(\lim_{x\to0}\frac{1}{x^3}\left(\left(\frac{2+\cos x}{3}\right)^x-1\right)\) without using L'Hopital's rule.

Answer: -\dfrac{1}{6}


Question: Determine whether the equation \(a^2 + b^2 = 5(c^2+d^2)\) has any integer solutions where not all of \(a, b, c,\) and \(d\) are zero.

Answer: Yes


Question: Evaluate the integral: $$\int_0^a \sin \left(\frac{\pi x}{a}\right)\delta\left(x-\frac{a}{2}\right)\sin \left(\frac{\pi x}{a}\right)dx$$

Answer: 1


Question: Find the degree of the field extension \( \mathbb{Q}(\beta) \) over \( \mathbb{Q} \), where \( \beta = 2 + \sqrt[3]{3} + \sqrt[3]{9} \).

Answer: 3


Question:
In Zermelo-Fraenkel (ZF) set theory, the axiom of infinity ensures the existence of a set \(x\) such that:

$$\exists x ((\emptyset \in x )\land \forall y (y \in x \implies y\cup \{y\} \in x)$$

The set of natural numbers \(\mathbb{N}\) is defined as:

$$\mathbb{N}:= \bigcap\{z \in \mathcal{P}(x): \emptyset \in z \land \forall y(y \in z \implies y \cup \{y\} \in z)\}$$

If another set \(x'\) satisfies the same property, will defining \(\mathbb{N}\) with respect to \(x'\) yield the same set? Provide a justification for your answer.

Answer: Yes


Question: Find the expected value of the random variable $X$ defined over the positive integers, with probability mass function $P[X=i] = C/i^2$ for any integer $i>0$, where $C^{-1} = \sum_{i=1}^\infty 1/i^2$. Express your answer in terms of known constants if necessary.

Answer: \infty


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

Answer: f(x) = ax + b


Question: Find the minimum value of the expression \(x_1^2+y_1^2+x_2^2+y_2^2-2x_1x_2-2y_1y_2\), given that the points \((x_1, y_1)\) and \((x_2, y_2)\) lie on the curve \(xy=1\), with \(x_1 > 0\) and \(x_2 < 0\).

Answer: 8


Question:
Evaluate the following integral and verify if the computation is correct:

\[ \int_{\mathbb{R}} f \, d\mu = \lim_{n\to\infty} \int_{[0,n]} f \, d\mu \]

Given that:

\[ \begin{align}
\int_{\mathbb{R}}f\,d\mu &= \lim_{n\to\infty}\int_{[0,n]}\chi_{[0,\infty)}e^{-x}\,d\mu \\
&=\lim_{n\to\infty}\int_0^n e^{-x}\,dx\\
&= \lim_{n\to\infty} (-e^{-x})\bigg\vert_0^n\\
&= \lim_{n\to\infty} (-e^{-n} + 1)\\
&= 1.
\end{align} \]

Is the final result of the integral equal to 1?

Answer: 1


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

Answer: \dfrac{1}{8}


Question: Calculate the sum of the power series \( S = \sum_{n=0}^{\infty}\frac{(2-x)^{n+1}}{n+1} \) when \( x = 1.5 \).

Answer: \ln 2


Question: Determine the characteristic of a field $\mathbb{F}$ if the function $f(a) = a^2$ is a permutation of the field.

Answer: 2


Question: Evaluate the limit: $$\lim_{N\to\infty} \frac{\sum_{n=0}^N \sin^4 (5n)}{N}$$

Answer: \dfrac{3}{8}


Question: Determine whether any function in $L^2(\mathbb{R})$ that is continuous and vanishes at infinity can be approximated using functions of the form $p(x) e^{- \alpha \vert x \vert}$, where $p(x)$ is a polynomial and $\alpha > 0$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the statement is true or false: There is no finite vector space over the reals.

Answer: B


Question: Solve the ordinary differential equation \( y'' + (y')^2 \cdot e^x = 0 \) with the initial conditions \( y(0) = 1 \) and \( y'(0) = 1 \). Find \( y(x) \).

Answer: 2 - e^{-x}


Question: Calculate the limit of the tail sum: \( \lim\limits_{n \to \infty} \sum_{k=n}^{\infty} e^{-k^2} \).

Answer: 0


Question: Calculate the limit: $$ \lim_{n\rightarrow\infty}\frac{2^n + n\sin{n}}{\log_2{n} + e^n} $$

Answer: 0


Question: Calculate the expected number of coin tosses required to obtain the sequence H-T-H-T-T-H with a fair coin.

Answer: 66


Question: Can an entire function of finite order have infinitely many zeros on a sequence that diverges to infinity? Provide a yes or no answer and justify your reasoning.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x \to 0}\left(\frac{x\ln (1+x)-x\ln x + \arctan\frac{1}{2x}}{x^2\arctan x}\right)^{1/\arctan x}$$

Answer: \infty


Question: Find the minimum value of the expression \( P = \frac{32a^3}{(b+3c)^3} + \frac{32b^3}{(a+3c)^3} - \frac{\sqrt{a^2 + b^2}}{c} \), given that \( a, b, c \) are positive real numbers satisfying \( (a+c)(b+c) = 4c^2 \).

Answer: 1 - \sqrt{2}


Question: Evaluate the integral \( \int_0^{2\pi} \log |1-re^{i\theta}| \, d\theta \) where \( r \) is a real number such that \( |r| < 1 \).

Answer: 0


Question:
Let \( f(z)=\sin(z+\mathrm{e}^{3z}) \). Compute the Wirtinger derivative \( \frac{\partial f}{\partial \bar{z}}(z) \) using the Wirtinger derivative operator:
\[
\frac{\partial}{\partial \bar{z}} = \frac{1}{2} \left( \frac{\partial}{\partial x} + \mathrm{i} \frac{\partial}{\partial y} \right),
\]
where \( x = \Re(z) \) and \( y = \Im(z) \).

Answer: 0


Question: Determine whether there always exists a continuous function from the open unit disk to an open and connected subset of the complex plane that covers the entire subset, even if the function is not injective.

Answer: Yes


Question: Determine if the integral \( \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx \) can be evaluated without using complex analysis methods.

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


Question: Given a cubic Bezier curve, determine the parameter 't' corresponding to a specific point B on the curve.

Answer: t


Question: Find all prime numbers $p$ such that $p$ divides $2^p + 1$. Provide your answer as a list of primes.

Answer: 3


Question: Let $P_1, P_2, \ldots, P_n$ be points on a circle of radius $r$, centered at $O$. Define $G$ as the point such that $\overrightarrow{OG}=\frac{\overrightarrow{OP_1}+\overrightarrow{OP_2}+\cdots+\overrightarrow{OP_n}}{n}$, and let $d=OG$. Express $\sum_{1\le i < j\le n}(P_i P_j)^2$ in terms of $n$, $r$, and $d$. \(\boxed{\text{Expression}}\)

Answer: n^2(r^2 - d^2)


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

Answer: \pi


Question: In a finite-dimensional vector space $V$ over the real numbers $\mathbb{R}$, consider two non-zero linear functionals $f$ and $g$. If $f(x) \geq 0$ implies $g(x) \geq 0$ for all $x \in V$, determine whether it is true that $f = \alpha g$ for some positive scalar $\alpha$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{(x,y)\rightarrow (0,0)}(x^2+y^2)^{x^2y^2}$$

Answer: 1


Question: Calculate the volume under the surface defined by the function \( f(x,y) = 3x - 2y \) over the parallelogram with vertices at \((1,0), (0,1), (-1,0), (0,-1)\) using the integral transformation formula.

Answer: 0


Question: Suppose $T_1$ and $T_2$ are two linear transformations on $\mathbb{R}^4$ such that $T_1(x) = T_2(x)$ for all $x \in \mathbb{R}^4$. Determine the minimum number of vectors $n$ in a set $A = \{v_1, v_2, ..., v_n\} \subset \mathbb{R}^4$ such that if $T_1$ and $T_2$ agree on the span of $A$, then $T_1$ and $T_2$ must be the identity transformations.

Answer: 4


Question: Find the minimum value of \( f(x,y) = \frac{x}{y^2} + \frac{y(xy-1)^2}{(x+y)^2} + \frac{x+y}{x^2 (xy-1)} \) subject to the constraint \( x + y + z = xyz \).

Answer: \sqrt{3}


Question: Let $(X,d)$ be a compact metric space and $f: X \rightarrow X$ be a weak contraction, meaning $d(f(x),f(y))<d(x,y)$ for all $x,y \in X$ with $x \neq y$. Determine if $f$ is also a generalized contraction. Specifically, for any $0<\alpha<\beta<\infty$, does there exist a constant $\lambda \in (0,1)$ such that $d(f(x),f(y)) \leq \lambda d(x,y)$ for all $x,y \in X$ with $\alpha \leq d(x,y) \leq \beta$?

Answer: Yes


Question: For an algebraic group $G$ and a number field $F/\mathbb{Q}$, determine if the Weil restriction of $G(\mathbb{A}_{\mathbb{Q}})$ from $\mathbb{Q}$ to $F$ is equal to $G(\mathbb{A}_{F})$. Provide a justification for your answer.

Answer: Yes


Question: Find the number of non-unital ring homomorphisms from the ring \( \mathbb{Z}/20\mathbb{Z} \) to the ring \( \mathbb{Z}/12\mathbb{Z} \).

Answer: 2


Question: Maximize \( f(x) = x_1 + x_2 \) subject to the constraint \( g(x) = x_1^2 + 3x_1x_2 + 3x_2^2 - 3 = 0 \), where \( x \in \mathbb{R}^2 \).

Answer: 2


Question:
Find the expression for $w$ such that the following equation holds for all $x$:
$$\sin(x) + \sin(2^{\frac{n+12}{12}}x) = \sin(x+w) + \sin(2^{\frac{n+12}{12}}(x+w))$$

Answer: 0


Question: Compute the limit \( \lim\limits_{s\rightarrow0^{+}}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{s}} \).

Answer: -\dfrac{1}{2}


Question: Determine if there exists a vector \( \gamma = (\gamma_1, \gamma_2, \gamma_3) \in \mathbb{R}^3 \) such that the set \( \{(1, 0, 1), (2, 1, 0), \gamma\} \) forms a basis for \( \mathbb{R}^3 \) and the vector \( (7, 3, 5) \) has coordinates \( (3, 1, 2) \) relative to this basis.

Answer: (1, 1, 1)


Question: Let $\phi$ be the set of all random variables $X$ such that $P(X \in [0,1]) = 1$, $E[X] = \frac{1}{5}$, and $P(X \leq \frac{3}{10}) = \frac{1}{2}$. Determine the infimum of the variance, $\mathrm{Var}(X)$, for all $X \in \phi$. Provide a hint to start the problem.

Answer: \dfrac{1}{100}


Question: Determine a constant $d > 0$ such that for all $y \geq 2$, the inequality \( \sum_{n>y} \frac{1}{n^{3}} \leq \frac{d}{y^{2}} \) holds.

Answer: \dfrac{1}{2}


Question: Given a Lie group \( G \) and a left-invariant vector field \( X \) on \( G \), does the local flow of \( X \) commute with left translations? Answer "Yes" or "No".

Answer: Yes


Question: Evaluate the integral \( \int_0^1\int_0^1 \frac {x^2-y^2}{(x^2+y^2)^2} \, dx \, dy \) using the reduction formula method.

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


Question: Find the limit of the sequence \( s_n = \int_0^1 \frac{nx^{n-1}}{1+x} \, dx \) as \( n \to \infty \).

Answer: \dfrac{1}{2}


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

Answer: \infty


Question: Simplify the expression using substitution and find the limit: $$\lim_{x\to\infty}\left(\frac{x e^{\frac{1}{x}} - \sqrt{x^2+x+1}}{x e^{\frac{1}{x}} + \sqrt{x^2+x+1}}\right)$$

Answer: 0


Question: Evaluate the line integral \( \int_C (x+y)^2dx + (x^2+y^2)dy \) where \( C \) is the positively oriented triangle with vertices at \((1,1)\), \((3,2)\), and \((2,5)\).

Answer: -\dfrac{56}{3}


Question: If $D$ is a dense subset (not a subspace) of a Hilbert space $H$, is the orthogonal complement of $D$ equal to $\{ 0 \}$?

Answer: \{0\}


Question: Calculate the fundamental group $\pi_1(X)$ of the space $X = \mathbb{R}^3 \setminus A$, where $A$ is a circle, using van Kampen's theorem.

Answer: \mathbb{Z}


Question: In Lang's Algebra, verify if the definition of a left module includes the condition $(ab)x = a(bx)$. Is this condition missing?

Answer: (ab)x = a(bx)


Question: Given that the number of storms in a rainy season follows a Poisson distribution with a parameter \( \lambda \) that is uniformly distributed between 2 and 7, calculate the probability of having at least 5 storms in the season.

Answer: 0.454


Question: Determine whether the dimension of the symplectic group $Sp(2n, F)$ is $2n^2 + n$ when $F$ is the algebraic closure of a finite field.

Answer: 2n^2 + n


Question:
Evaluate the limit of the sequence as \( n \) approaches infinity:
\[ \lim_{n\to\infty}\frac{1+\sqrt[n]{2}+\sqrt[n]{3}+\cdots+\sqrt[n]{n}}{n} \]

Answer: 1


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^{n^2+n} \).

Answer: 1


Question: Determine the minimum order of a homogeneous linear differential equation with constant coefficients that has the function \( y(x) = x^2 e^x \) as a solution.

Answer: 3


Question: In triangle ABC, points E, F, and C are collinear, D, E, and B are collinear, and A, D, and F are collinear. Given that |CF| = |FE|, |DE| = |EB|, and |AD| = |DF|, with |AC| = |CB| = \( \sqrt{14} \) dm and the area of triangle DEF is 0.01 m², calculate the length of |AB| in centimeters.

Answer: 20\sqrt{7}


Question: Evaluate the limit \( \displaystyle \lim_{x \to 0} \frac{ \log ((k+x)^{b} - (k-x)^{b} )}{\log x} \), where \(k \in (0,1)\) and \(b \in (0,1)\) are constants.

Answer: 1


Question: Let $B_{5} = (b^5_{ij})_{n \times n}$ be a matrix where $b^5_{ij} = \sin(i-j)$ and $n \ge 6$. Calculate the determinant of $B_{5}$. \( \det(B_{5}) \)

Answer: 0


Question: Calculate the limit: $$\lim_{n\rightarrow\infty}\sqrt{n}\int_{-\infty}^{+\infty}\frac{\cos(x)}{(1+x^2)^{n}}dx$$

Answer: \sqrt{\pi}


Question: Determine whether the expression \( n^{-2} \times \sum_{m=n/3}^{n-1} n \bmod m \) converges as \( n \to \infty \), and if so, find its limit.

Answer: \dfrac{11}{72}


Question: Given the polynomial \( f(x) = x^4 + ax^3 + 3x^2 + bx + 1 \), where \( a, b \in \mathbb{R} \), if \( f(x) \geq 0 \) holds for all \( x \in \mathbb{R} \), find the maximum value of \( a^2 + b^2 \).

Answer: 40


Question: Consider a free group \( F_n \) of rank \( n \geq 2 \). Is it possible to construct an epimorphism \( \phi: F_n \to F_2 \) such that \( \phi(g) \neq 1 \) for some nontrivial element \( g \in F_n \)?

Answer: Yes


Question:
Given a strictly positive sequence $(a_n)_{n\ge1}$ with the property that $\lim\limits_{n\to\infty}\frac{a_{n+1}}{na_n} = a > 0$, determine the value of the following limit:
$$\lim_{n\to\infty}\left(\frac{(n+1)^2}{\sqrt[n+1]{a_{n+1}}}-\frac{n^2}{\sqrt[n]{a_n}}\right)$$

Answer: \dfrac{e}{a}


Question: Evaluate the integral \( \int_{|z|=3} \frac{e^{\frac{1}{1-z}}}{z} \, dz \) using the residue theorem.

Answer: 2\pi i


Question: Suppose $X$ and $Y$ are dependent random variables, and the conditional expectation of $Y$ given $X$ is $E(Y\mid X) = a + bX$, where $a$ and $b$ are constants. Find the expected value of the product $XY$, i.e., $E(XY)$. \(\boxed{\text{Your answer here}}\)

Answer: aE(X) + bE(X^2)


Question: Given that a random variable $X$ follows a standard normal distribution ($X\sim N(0,1)$) and $Y = X^2$, and knowing that the density function of $Y$ matches that of a Chi-squared distribution with 1 degree of freedom, can you conclude that $Y$ is a Chi-squared distributed random variable with 1 degree of freedom ($Y\sim \chi^2_1$)?

Answer: Y \sim \chi^2_1


Question: Calculate the contour integral of the function \( f(z) = \frac{e^{1/z}}{z^3} \) around the rectangle with vertices at \(1\), \(-1\), \(1+2i\), and \(-1+2i\).

Answer: 0


Question: Determine the radius of convergence for the Taylor series of \( f(x) = \frac{\cos x}{x} \) centered at \( a=0 \).

Answer: 0


Question: Determine whether a metric space with a finite number of elements can have a number of open sets that is not a power of 2. Provide a justification for your answer.

Answer: No


Question: Does every linear transformation from a finite-dimensional complex vector space to itself have an eigenvector? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \( \lim\limits_{x \to 0^+} (e^{x\log x})^{x^2} \).

Answer: 1


Question: Determine the number of terms required in the series expansion of the definite integral \( \int^{1}_{0}e^{-x^2}dx \) to approximate the integral to within \(0.001\).

Answer: 5


Question: In a convex quadrilateral $ABCD$, it is given that $BC = CD$, $AC = AD$, $\angle BCD = 96^\circ$, and $\angle ACD = 69^\circ$. Let $P_0 = A$ and $Q_0 = B$. Define points $P_{n+1}$ as the incenter of triangle $\triangle CDP_n$ and $Q_{n+1}$ as the incenter of triangle $\triangle CDQ_n$, for all $n \geq 0$. If $\angle Q_{2024}Q_{2025}P_{2025} - 90^\circ = \frac{2k-1}{2^{2024}}$, find the value of $k+n$. Determine the value of $k+n$. \(\boxed{}\)

Answer: 2038


Question: Given the polynomial $P(X) = X^4 + X^3 + X^2 + X + 1$, where $X^5 = 1$, find the value of $x_1^{10} + x_2^{20} + x_3^{30} + x_4^{40}$.

Answer: 4


Question: Evaluate the sum \(\sum_{d|n}\tau(d)\mu(d)\) for \(n\) having an even number of distinct prime factors.

Answer: 1


Question: Let \( n \ge 2 \) be an integer. Determine the largest integer \( k \ge 1 \) such that for any \( k \) matrices \( A_1, A_2, \ldots, A_k \in \mathcal{M}_n(\mathbb{C}) \), if \( I_n - A_1A_2\cdots A_k \) is invertible, then \( I_n - A_{\pi(1)}A_{\pi(2)}\cdots A_{\pi(k)} \) is also invertible for every permutation \( \pi \in S_k \).

Answer: 2


Question: Determine the number of rotations in \(\mathbb{R}^3\) that map the cube \([-1,1]^3\) onto itself, excluding the identity rotation.

Answer: 23


Question: Choose a value for $c$ such that $a < c < b$ to minimize the maximum relative error when rounding $x$ to the nearest of $a$, $b$, or $c$, given that $a \leq x \leq b$. What is the optimal value of $c$?

Answer: \sqrt{ab}


Question: Determine the computational complexity of deciding whether $a^b = c$, given that $a, c \leq 2^p$ and $b \leq 2^q$, without explicitly computing $a^b$, when the size of $q$ is not restricted.

Answer: P


Question: Evaluate the limit: $$\lim_{x\to \infty}\left[(x+1)^{1/(x+1)}\right]^{\frac{20(x+1)}{x}}$$

Answer: 1


Question: Determine if there exists a smooth, 1-periodic, positive function \(c:\mathbb{R}\rightarrow (0,\infty)\) such that the given ODE system has solutions that are defined only for a finite interval of time.

Answer: Yes


Question: In a general topological space, can a point be a limit point of a sequence without any subsequence converging to it?

Answer: Yes


Question: Calculate the residue of the function \( f(z) = \frac{1 - e^{2z}}{z^4} \) at the pole \( z = 0 \) of order 3.

Answer: -\dfrac{4}{3}


Question: In how many distinct ways can fifteen balls, consisting of 3 balls each of 5 different colors, be arranged in a triangular formation if rotations are considered identical?

Answer: 56056080


Question: Determine whether the following statement is true or false: For an integer $n \geq 3$, an $n \times n$ matrix $A$ with entries from a field has an eigenvalue if and only if every monic polynomial of degree $n$ is reducible.

Answer: False


Question: Calculate the integral \( \int_{|z|=2} \frac{z^2}{(z-1)^3} \, dz \) where \(|z|=2\) is traversed counterclockwise.

Answer: 2\pi i


Question: Determine whether the function \( f:[0,1]\to\mathbb{R} \) defined by \[ f(x)=\frac{1}{q} \text{ if } x=\frac{p}{q} \text{ for some } p,q\in\mathbb{N} \text{ with } (p,q)=1, \quad f(x)=0 \text{ otherwise} \] is Riemann-integrable over the interval \([0,1]\).

Answer: Yes


Question: Determine the Galois group of an irreducible polynomial of degree 4 with 2 real roots over \( \mathbb{Q} \).

Answer: D_4


Question: Determine whether there exists a continuously differentiable function \( f: (0, \infty) \rightarrow \mathbb{R} \), with bounded support, such that \( f(x) \sim cx^\alpha \) as \( x \rightarrow 0 \), where \( c \neq 0 \) and \( \alpha > 0 \).

Answer: Yes


Question: Determine whether the following statement is true or false: Acyclicity and completeness of a relation imply transitivity.

Answer: True


Question: Given relatively prime polynomials $P, Q \in \mathbb{Q}[X]$, where $Q(0) = 1$, and the rational function $\frac{P}{Q}$ belongs to $\mathbb{Z}[[X]]$, must it be true that $P$ and $Q$ are elements of $\mathbb{Z}[X]$? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: Let $f$ be a Lebesgue-integrable function on the interval $[0,1]$. Suppose that for every $0 \leq a \leq b \leq 1$, the integral $\int_a^b f(x) \, dx = 0$. Prove that $\int_A f(x) \, dx = 0$ for every measurable subset $A$ of $[0,1]$.

Answer: 0


Question: Given continuous functions \(f, g: \mathbb{R}^n \rightarrow \mathbb{R}^n\) with continuous partial derivatives (\(f, g \in C^1(\mathbb{R}^n,\mathbb{R}^n)\)), consider a sequence of measurable subsets \((A_p)_{p \geq 1}\) of \([0,T]\) such that \(A_p \subset [0,T]\) and \(\mu(A_p) \rightarrow 0\) as \(p \rightarrow \infty\). Suppose there exist functions \(x_p, x \in C^1([0,T],\mathbb{R}^n)\) for which \((x_p)_p\) converges uniformly to \(x\). Determine whether it follows that \(\frac{1}{\mu(A_p)}\int_{A_p} ||g(x_p(s))-f(x_p(s))||_{\mathbb{R}^n} \, ds \rightarrow 0\) as \(p \rightarrow \infty\).

Answer: No


Question: Evaluate the limit \( \lim_{x \to 0} \int^{2x}_x \frac{\sin t}{t^3}dt \) without using approximations.

Answer: \infty


Question: Determine the number of connected components in the graph \( H_n \), where \( H_n \) is defined for each positive integer \( n \) as a graph whose vertices are \( n \)-tuples with binary entries (0 or 1), and two vertices are adjacent if they differ in precisely two coordinates. Find the number of connected components for any \( n \geq 1 \).

Answer: 2


Question: Determine the values of $c$ for which the equation $f(x) = c^x f(2x)$ has non-zero differentiable solutions for the function $f(x)$. Provide your answer as a set of values for $c$.

Answer: (0, \infty)


Question: Given a locally compact Hausdorff space $X$ and subsets $K \subset U$ where $K$ is compact and $U$ is open, find a continuous function $f: X \to \mathbb{R}$ with compact support such that $1_K \leq f \leq 1_U$ on $X$. Ensure that the function $f$ is continuous.

Answer: f


Question: Determine the values of $a$ for which the expression \((x,y) = x_1y_1 + x_2y_2 + x_3y_3 + ax_2y_3 + ax_3y_2\) defines an inner product in $\mathbb{R}^3$. Provide your answer as a set of values for $a$.

Answer: (-1, 1)


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\log(1+x^2) \, dx}{1+x^2} \) using complex analysis and contour integration. Modify the contour to account for branch points and determine the correct value of the integral.

Answer: 2\pi \ln 2


Question: Verify Stokes' Theorem for the vector field \( \mathbf{F} = (y^2, z^2, x^2) \) and the surface \( S \) defined by the hemisphere \( x^2 + y^2 + z^2 = 1 \) with \( z \ge 0 \) and outward-pointing unit normal.

Answer: 0


Question: Determine the annihilator of the tensor $e_1\wedge e_2 + e_3\wedge e_4$ in the vector space $V$ with basis $\{e_1, e_2, e_3, e_4\}$.

Answer: \{0\}


Question: Evaluate the limit \( \lim_{x\to0}{\frac{x^2+2\ln(\cos x)}{x^4}} \) without using L'Hopital's rule or Taylor series.

Answer: -\dfrac{1}{6}


Question: Determine whether there exists a partition of the real numbers \( \mathbb{R} \) into uncountably many uncountable sets.

Answer: Yes


Question: Find the unique triplet of positive integers \((a, b, c)\) such that \(a \leq b \leq c\) and \[ \frac{25}{84} = \frac{1}{a} + \frac{1}{ab} + \frac{1}{abc}. \]

Answer: (4, 6, 7)


Question: Find the gradient of the expression \( \| z - Zx \|^2_2 \) with respect to \( x \), where \( z \in \mathbb{R}^n \), \( Z \in \mathbb{R}^{n\times n} \), and \( x = [x_1,x_2,\ldots,x_n]^T \in \mathbb{R}^n \).

Answer: 2Z^\top (Zx - z)


Question:
Given two independent random variables, $X$ and $Y$, where $Y > 0$, and $Z = \frac{X}{Y}$ with $\mathbb{E}[Z]$ existing and satisfying $0 < \mathbb{E}[Z] < 1$, determine which of the following statements must be true:

1. $\mathbb{P}[X \leq Y] = 1$
2. $\mathbb{P}[0 \leq X] = 1$
3. $\mathbb{P}[0 \leq X \leq Y] > 0$
4. $0 \leq \mathbb{E}[X] \leq \mathbb{E}[Y]$

Select the correct statement.

Answer: 4


Question: Evaluate the limit: $$\lim \limits_{z \to 0} z\sin\left(\frac{1}{z^2}\right)$$

Answer: 0


Question: How many distinct configurations are there of a $4 \times 4$ grid, where each cell contains either a $1$ or a $-1$, such that the sum of the numbers in each row and each column equals $0$?

Answer: 90


Question: Let $K$ be a compact subset of $\mathbb{R}^n$ and let $(O_i)_{i=1}^n$ be $n$ open balls that cover $K$. Is it possible to reduce the radii $r_i$ of each $O_i$ slightly, such that the new open balls $(O_i')_{i=1}^n$, which are concentric with the original balls and have radii $r'_i < r_i$, still cover $K$?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\to\infty} \frac{\ln (n!)}{n^2} \) using the Stolz–Cesàro theorem.

Answer: 0


Question: Evaluate the line integral \( \int_C F \cdot dr \) where \( F(x,y,z) = x\,i+xy\,j+x^2yz\,k \) and \( C \) is the elliptical path given by \( x^2+4y^2-8y+3=0 \) in the xy-plane, traversed once counter-clockwise.

Answer: \dfrac{\pi}{2}


Question: Determine whether the following statement is true or false: The image of a disconnected set under a bijective continuous function is always disconnected.

Answer: B


Question: Suppose three functions, $f(x)$, $g(x)$, and $h(x)$, are solutions to the differential equation $y'' + b_1y' + b_2y = G(x)$, where $b_1$ and $b_2$ are constants. Given that $f(x) + g(x) = h(x)$, determine the values of $b_1$, $b_2$, and $G(x)$. Provide your answer in the form of $b_1$, $b_2$, and $G(x)$. \( \boxed{(b_1, b_2, G(x))} \)

Answer: (b_1, b_2, 0)


Question:
Solve the system of equations by eliminating a variable using the hint provided:

$$\large \left\{ \begin{align} \sqrt{y^2 - 8x + 9} - \sqrt[3]{xy - 6x + 12} &= 1\\ \sqrt{2(x - y)^2 + 10x - 6y + 12} - \sqrt{y} &= \sqrt{x + 2}\end{align} \right.$$

Hint: Square the second equation twice and factor to eliminate one variable.

Answer: (2, 4)


Question: Determine the values of \(a, b > 0\) such that the integral \(\int_{1}^{\infty} \frac{1}{x^a(1+x^b)} \,dx\) is convergent.

Answer: a + b > 1


Question: Let $f:(X, d)\to (X, d)$ be a bijection on a metric space $(X, d)$. For $\epsilon>0$, the bijection $f$ is called an $\epsilon$-self isometry if \(\sup_{x,y\in X}|d(x, y)- d(f(x), f(y))|\leq \epsilon.\) Given $\epsilon>0$, is there an $\epsilon$-self isometry that is not an isometry?

Answer: Yes


Question: Consider the atomic formulae \( \forall xPx \) and \( Pa \), and the logical axiom \( \forall xPx \rightarrow Pa \). Define a function \( T \) from the set of sentences to \( \{0,1\} \), where the value depends on whether the sentence is true in a given interpretation. If we let \( T(\forall xPx)=u \) and \( T(Pa) = v \), where \( u \) and \( v \) are Boolean variables taking values 0 or 1, do they always satisfy the equation: \[ T(\forall xPx \rightarrow Pa)=[1-{u\cdot (1-v)}]=1? \]

Answer: Yes


Question: Determine the value of $k$ for which the limit \(\lim_{x\rightarrow0} \frac{(e^{x^3}-x^3-1)(\sin(x)-x)}{x^k}\) has a finite, non-zero solution.

Answer: 9


Question: Chord AB of a circle is extended through B to an exterior point D, and a line is drawn from D tangent to the circle at C. If AB = CD and triangle BCD has an area of 2√5 square centimeters, then the area of triangle ABC can be expressed as a + b√5 square centimeters. Find the ordered pair (a, b).

Answer: (5, 1)


Question: Compare the values of $a = 211!$ and $b = 106^{211}$. Determine which is greater.

Answer: b


Question: Let $V$ be the inner product space of $P_{3}$ with the inner product defined as $\langle p_1(t),p_2(t) \rangle = \int_{-1}^1 p_1(x)p_2(x) \, dx$. Let $W$ be the subspace of $V$ with basis $\{1, t^2\}$. Find a basis for $W^\perp$. \(W^\perp\) is the orthogonal complement of $W$ in $V$. Determine the polynomials in $P_3$ that are orthogonal to every polynomial in $W$.

Answer: \{t, t^3\}


Question: Determine the number of cyclic subgroups of order $p$ in the group ${\bf Z}_{p^n} \oplus {\bf Z}_p$, where $p$ is a prime number and $n \geq 1$. Provide your answer as a single integer.

Answer: p + 1


Question: Evaluate the double integral \( \int \int_{\Omega} \sqrt{x^2+y^2} \, dA \), where the region \( \Omega \) is defined by \( (x-1)^2+y^2 \leq 1 \) and \( 0 \leq y \).

Answer: \dfrac{16}{9}


Question: Determine whether there exists an analytic unitary function \( g: S^{1} \rightarrow M_{n}(\mathbb{C}) \) that approximates a given continuous unitary function \( f: S^{1} \rightarrow M_{n}(\mathbb{C}) \) with respect to the supremum norm.

Answer: Yes


Question: Given the equations $2^x = 3$, $3^y = 4$, $4^z = 5$, $5^w = 6$, ..., $15^u = 16$, find the value of the product $xyzw...u$.

Answer: 4


Question: Find the limiting value of the sequence \( \sum_{k=1}^{n}\frac{1}{\binom{n}{k}} \) as \( n \) approaches infinity.

Answer: 1


Question: Evaluate the limit \( \lim_{x \to \frac{\pi}{2}}\frac{b(1-\sin x) }{(\pi-2x)^2} \) using trigonometric techniques, without applying L'Hôpital's rule.

Answer: \dfrac{b}{8}


Question: Evaluate the integral \( \int_{\frac{1}{4}}^{\frac{3}{4}} f(f(x)) \, dx \), where \( f(x) = x^3 - \frac{3}{2}x^2 + x + \frac{1}{4} \).

Answer: \dfrac{1}{4}


Question: Determine whether the function \( f(x) = x^3 \cos\frac{1}{x} \) for \( x \neq 0 \) and \( f(0) = 0 \) is differentiable at \( x = 0 \).

Answer: 0


Question: If $L: V \to V$ is a linear mapping, and both $\{v_1, \ldots, v_n\}$ and $\{L(v_1), \ldots, L(v_n)\}$ are bases for the vector space $V$, determine if the rank of $L$ is equal to $n$. Justify your answer.

Answer: Yes


Question: Find the infimum of the set \( B = \left\{ \frac{2x^2+5x-2}{x^2+3x-1} \; \middle| \; x \text{ is a real number} \right\} \) using the ε-characterization.

Answer: -\infty


Question: Determine the number of non-constant polynomial factors with leading coefficient 1 of the polynomial \(x^{2015} + 18\).

Answer: 1


Question: Let $f$ be a real-valued continuous function on the interval $[0, 3]$. Suppose that $f(x)$ takes only rational values and $f(1) = 1$. Determine the value of $f(2)$. \( \boxed{1} \)

Answer: 1


Question: Given sets $A$ and $B$ with a bijection $f: A \to B$, determine if there exists a bijection $g: P(A) \to P(B)$ between their power sets. If such a bijection exists, provide an explicit formula for $g$.

Answer: g(S) = \{ f(s) \mid s \in S \}


Question: Determine the number of 8-tuples \((a_1, a_2, \ldots, a_8)\) from the set \([4]^8 = \{1, 2, 3, 4\}^8\) such that each number in \([4]\) appears exactly twice, and for each \(i \in [4]\), the number \(i\) does not appear in the \(i\)-th position.

Answer: 864


Question: Determine all complex polynomials \( p(x) \) such that \( p(0) = 1 \) and \( |p(z)| \leq 1 \) for all \( z \) on the unit circle.

Answer: 1


Question:
Given the differentiable functions $z$ and $w$ of $x$ and $y$ that satisfy the equations:

$$xw^3 + yz^2 + z^3 = -1$$
$$zw^3 - xz^3 + y^2w = 1$$

Find the partial derivative of $z$ with respect to $x$, $\frac{\partial z}{\partial x}$, at the point $(x, y, z, w) = (1, -1, -1, 1)$. \( \boxed{?} \)

Answer: -\dfrac{5}{4}


Question: Evaluate the limit: $$\lim_{n \to \infty} \left( \frac{\ln(n+1)^{n+1}}{\ln n^n} \right)^n$$

Answer: e


Question: Evaluate the limit: $$\lim _{x\to \infty}\left(1+2x\sqrt{x}\right)^{\frac{2}{\ln x}}$$ without using L'Hôpital's rule.

Answer: e^3


Question: Given the function \(g(x) = \frac{(1+x)^p}{(1+x^p)}, \quad x \geq 1, \quad 0 < p < 1,\) determine the interval for which the derivative \(g'(x) > 0\).

Answer: (1, \infty)


Question: Solve the partial differential equation \(x\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}=0\) with the initial condition \(f(1,y) = e^{-2y}\), using the change of variables \(u = xe^y\) and \(v = xe^{-y}\).

Answer: x^2 e^{-2y}


Question: Determine whether the statement "Any closed set has a point on its boundary" is true or false.

Answer: B


Question: Determine the period of the periodic orbits, excluding the origin, for the differential equations in polar coordinates: \[\dfrac{dr}{dt} = r(1 + a~\cos \theta - r^2) \] \[\dfrac{d \theta}{dt} = 1\] where $a$ is a positive number less than $1$. \(\text{Provide your answer as a single number.}\)

Answer: 2\pi


Question: Determine the value of the constant $a$ such that the equation \( \log_2(a) + \log_2(a)^2 + \log_2(a)^3 + \cdots + \log_2(a)^n = 3n(n+1) \) holds for all positive integers $n$. \( \boxed{} \)

Answer: 64


Question: Given real numbers $a_0, a_1, \ldots, a_k$ such that $a_0 + a_1 + \cdots + a_k = 0$, determine the limit as $n$ approaches infinity of the expression: $$a_0 \sqrt{n} + a_1 \sqrt{n + 1} + \cdots + a_k \sqrt{n + k}.$$

Answer: 0


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

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


Question: Determine the group defined by the presentation \( \langle x, y : xy = yx, x^5 = y^3 \rangle \).

Answer: \mathbb{Z}


Question: Evaluate the double integral \(\int\int (x^2-y^2)^3 \, dA\) over the circular region bounded by the circle with center (0, 0) and radius 2 using polar coordinates.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{\sin\left(e^{1-\cos^3x}-e^{1-\cos^4x}\right)}{x\arctan x}$$

Answer: -\dfrac{1}{2}


Question: Evaluate the double integral \( \int_{1}^{2} \int_{\frac{1}{y}}^{y} (x^2+y^2) \, dx \, dy \) using the substitution \( x = \frac{u}{v}, y = uv \).

Answer: \dfrac{27}{8}


Question: Let $W_{1}$ and $W_{2}$ be subspaces of a finite-dimensional vector space $V$ with $\dim(W_{1}) = \dim(W_{2})$. Determine the necessary and sufficient conditions on $W_{1}$ and $W_{2}$ such that $\dim(W_{1}+W_{2}) = \dim(W_{1})$. Express your answer in terms of the relationship between $W_1$ and $W_2$.

Answer: W_1 = W_2


Question: Calculate the surface area of the part of a sphere with radius 2 that is inside a cone with a 60-degree angle from the vertical axis.

Answer: 4\pi


Question: Compute \( \int_{S^1} i^* \omega \) for the 1-form \( \omega \in \Omega^1(\mathbb{R}^2) \) with \( d\omega = dx \wedge dy \), where \( i: S^1 \to \mathbb{R}^2 \) is the inclusion of the unit circle. Use Stokes' theorem to evaluate the integral over the unit disc \( B \), parameterized by \( Y(r, \theta) = (r\cos\theta, r\sin\theta) \) for \( 0 \leq r \leq 1 \) and \( 0 \leq \theta \leq 2\pi \). Verify the orientation consistency and compute the integral.

Answer: \pi


Question: Find the approximate number of pairs of integers $(a, b)$ such that $0 \leq a \leq b < 64$ and $ab \geq 128$.

Answer: 1756


Question: Evaluate the limit using Stirling's formula: \[ \lim_{n\to\infty} \frac{\frac{\left(3n\right)!}{3^{n}n!}}{\left(\frac{n}{e}\right)^{n} 9^{n} \sqrt{3}} \]

Answer: \infty


Question: Evaluate the integral \( \int_0^{\infty} \frac{2 \sin x \cos^2 x}{x e^{x \sqrt{3}}} \, dx \).

Answer: \dfrac{\pi}{4}


Question: Solve for \( x \) in the equation: \[ 4 \cdot 3^x - 9 \cdot 2^x = 5 \cdot 3^{\frac{x}{2}} \cdot 2^{\frac{x}{2}} \]

Answer: 4


Question: Determine whether the sequence \( \{x_n\}_{n=1}^{\infty} \) defined by \( x_1 > 0 \) and \( x_{n+1} = 1 + \frac{a-1}{1+x_n} \) for \( a > 1 \) is convergent or divergent as \( n \to \infty \). If it is convergent, prove that the limit is \( \sqrt{a} \).

Answer: \sqrt{a}


Question: Let $Y$ be a random variable such that $\ln Y$ follows a normal distribution with mean $\ln 6 - \frac{5}{2}$ and variance $5$. Calculate $\mathbb{E}(\max(Y-6,2))$. Provide your answer as a single numerical value.

Answer: 6.18


Question: Find the integer part of the sum \( \sum^{9999}_{k=1}\frac{1}{\sqrt[4]{k}} \).

Answer: 1332


Question: Find the residue of the function \( \frac{z}{\cos z} \) at \( z = \pi/2 \).

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


Question: Find all continuous functions \( g: \mathbb{R} \to \mathbb{R} \) that satisfy the equation \( g(x) = g(C - x^2) \) for all \( x \in \mathbb{R} \), where \( C > 0 \) is a constant.

Answer: g(x) = k


Question: Can a metric be defined on the open interval \((0,1)\) such that the topology generated by this metric is equivalent to the topology induced by the standard Euclidean metric on this interval?

Answer: Yes


Question: Evaluate the limit of the integral as \( n \) approaches infinity: \[ \lim\limits_{n \to \infty} \int_0^n \frac{1}{1 + n^2 \cos^2 x} \, dx \]

Answer: 1


Question: Determine the real number \( a \) such that the integral \( \int_1^{\infty}\left(\frac{1}{x+2} - \frac{ax}{x^2+1}\right) dx \) converges.

Answer: 1


Question: For a binomial distribution with parameters $p=1/2$ and an odd number of trials $n$, determine if there exists a tighter bound for $|F_n(0)-\Phi(0)|$ than the one provided by the Berry-Esseen theorem, such that the bound approaches zero as $n$ increases.

Answer: 0


Question: Evaluate the integral \( \int_0^{2\pi} \cot(e^{it})ie^{it}dt \).

Answer: 2\pi i


Question: Evaluate the integral \(\int_{C}\frac{\sin(z)}{(z-i)^2}dz\) where \(C\) is the circle \(|z-2|=1\).

Answer: 0


Question: Determine whether the sigma algebra generated by the pi-system \( H = \{f^{-1}((-\infty,x]): x \in \mathbb{R}\} \), where \( f:(X,S) \to (\mathbb{R}, \mathcal{B}(\mathbb{R})) \) is a measurable function, is equal to the sigma algebra \( G = \{f^{-1}(A): A \in \mathcal{B}(\mathbb{R})\} \).

Answer: G


Question: Determine the number of 16-digit integers $t$ such that $12t+5$, $15t+8$, $20t+11$, and $30t+17$ are all prime numbers.

Answer: 0


Question: What is the standard notation or terminology for the set \( S = \{ a \in A \mid \sigma(a) = a \} \) if \( A \) is an algebra over a field \( F \) and \( \sigma: A \rightarrow A \) is an involution of the second kind?

Answer: H(A, \sigma)


Question: Determine the type of singularity and calculate the residue of the function \( g(z) = \frac{\tan(z)}{(z-\frac{3\pi}{2})} \) at \( z = \frac{3\pi}{2} \).

Answer: 0


Question: Evaluate the limit \( \lim_{w\rightarrow 0}\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{y-w}{w}\right)^2\right)\frac{y-w}{w^2} \) for \( y \neq 0 \).

Answer: 0


Question: Determine whether the function $f(n) = n^{\log(n)}$ is $O(\log(n)^n)$. Justify your answer by comparing the growth rates of the functions.

Answer: Yes


Question: Determine whether the inequality \(2\int_{0}^{y^{2}}c(t)\,dt<3\frac{1}{y}\int_{0}^{y^{2}}\sqrt{t}c(t)\,dt\) holds for any positive number \(y\), given that the function \(c(t)\) is smooth, strictly increasing, and positive on the domain \([0, \infty)\).

Answer: Yes


Question: Is the normed linear vector space $\mathbb{R}^n$ with the norm $||x|| = \sum_{i=1}^n |x_i|$ complete?

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to\infty} \left(\frac{1}{x}\frac{b^x-1}{b+1}\right)^{\frac{1}{x}}$$ where $b$ is a constant and $0 < b < 1$.

Answer: 1


Question: In the sequence of integers where no number contains the digits '666' consecutively, determine if there is a method to find the nth term without counting each term sequentially. Provide a yes or no answer and, if possible, describe the method.

Answer: Yes


Question: How many ordered triplets of $n$-dimensional vectors $\langle \textbf{u, v, w} \rangle$ are there, with each vector in $\{-1, 1\}^n$, such that $\|\textbf{u + v + w}\|_{\infty} \leq 1$?

Answer: 6^n


Question: Determine the values of $p$ for which the congruence $s^2 \equiv -1 \pmod{p}$ has exactly one solution.

Answer: 2


Question: Determine if there exists an integer \( N \) such that the number of 2-primes less than \( N \) is equal to the number of 3-primes less than \( N \), given that \( N > 30 \) and the number of 2-primes less than \( N \) is greater than 7.

Answer: Yes


Question: Evaluate the limit \(\lim_{n\to \infty} \frac{e^n \cdot n!}{n^n}\). If the limit has a defined value, find it.

Answer: \infty


Question: Given a sequence $(x_n)$, where $\lim_{{n \to \infty}} (x_{2n} + x_{2n+1}) = 315$ and $\lim_{{n \to \infty}} (x_{2n} + x_{2n-1}) = 2016$, find the value of $\lim_{{n \to \infty}} \frac{x_{2n}}{x_{2n+1}}$. \(\boxed{\text{Answer}}\)

Answer: -1


Question: Let $F$ be a field of characteristic $p > 0$ and let $f(t) = t^p - t - a \in F[t]$. Suppose also that $a \neq b^p - b$ for any $b \in F$. Determine the Galois group $Gal(K/F)$, where $K$ is a splitting field of $f$. \( \text{Provide your answer in terms of group structure.} \)

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


Question: Find the maximum value of the function \( f(x,y) = x^2 + 2y^2 \) subject to the constraint \( y - x^2 + 1 = 0 \) using the method of Lagrange multipliers, given that the equation \( 2x = \lambda(-2x) \) implies \( x = 0 \).

Answer: 2


Question: Find the limit \( \lim_{x\to 0^+}\frac{\ln(1-e^{-x})}{x^{-n}} \) as \( n \) approaches infinity.

Answer: 0


Question: Suppose $H$ and $K$ are groups, and there exists an isomorphism $H \cong K \times H$. Does it necessarily follow that $K$ is the trivial group? Answer yes or no.

Answer: No


Question: Given the bivariate normal distribution $\mathbf{X} = (X_1, X_2)' \sim N(\mu, \Lambda)$, where $\mu = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ and $\Lambda = \begin{pmatrix} 3 & 1 \\ 1 & 2 \end{pmatrix}$, calculate the conditional probability $P(X_1 \geq 2 \mid X_2 + 3X_1 = 3)$. Provide your answer as a probability value.

Answer: 0.0003


Question: Consider a sequence $g_n \in L^1(0,1)$ that is uniformly bounded in the $L^2$-norm. Determine whether there exists a subsequence of $g_n$ that converges weakly in $L^1$. Provide a justification for your answer.

Answer: Yes


Question:
Let $u: \mathbb{R} \rightarrow (-\infty, +\infty]$ be a convex function that admits a point of minimum. Define the mollified function:

$$(\varphi_\epsilon * u)(x) = \int_{\mathbb{R}^n} \varphi_\epsilon(y) u(x-y) \, dy,$$

where $\varphi_{\epsilon}$ is the standard mollifier. Introduce the notation:

$$\tilde{u}_i = \varphi_{1/i} * u, \quad \forall i \in \mathbb{N}.$$

Given that $\tilde{u}_i$ is convex and converges pointwise to $u$ in $\mathbb{R}^n$ and uniformly on compact sets of $\mathbb{R}^n$, determine if the sequence $y_i = \min_{\mathbb{R}} \tilde{u}_i$ converges to $y = \min u$. Answer with 'yes' or 'no' and justify your answer.

Answer: yes


Question: In a group of $N$ boys and $N$ girls, each person shakes hands with $d$ random members of the opposite sex. If one person is initially infected with a disease that spreads through handshakes, what is the probability that everyone in the group will eventually become infected, given that $d > 1$?

Answer: 1


Question: Find a generator of the maximal ideal of the local ring of rational functions on the algebraic curve $C: Y^2 = X + X^3$ at the point $P = (0,0)$. This local ring is a discrete valuation ring and thus a principal ideal domain.

Answer: Y


Question: Determine whether the function $y = \ln(xy)$ satisfies the differential equation $$(xy-x)y'' + xy'^2 + yy' - 2y' = 0.$$

Answer: Yes


Question: Determine if the sequence of random variables \(Z_n\), defined by \(Z_n = X\) for odd \(n\) and \(Z_n = 1 - X\) for even \(n\), converges in distribution to \(X_n = X\) for all \(n\), where \(X\) is uniformly distributed on \([0, 1]\).

Answer: Yes


Question: In a metric space $(X, d)$, for subsets $A$ and $B$ of $X$, is it possible for the distance $d(A, B)$ to be zero even if the closures of $A$ and $B$ intersect? Provide an example or a proof.

Answer: Yes


Question: If the Lindelöf degree of $(R)^I$ is a cardinal $\lambda$, can we conclude that the cardinality of $I$ does not exceed $\lambda$? Answer yes or no.

Answer: Yes


Question: Find a change of variable that will transform the quadratic form \(x_1^2-x_3^2-4x_1x_2+4x_2x_3\) into a sum of squares. Express the quadratic form in terms of the new variables.

Answer: y_1^2 - y_2^2


Question: Let $G$ be a finite group generated by two elements $a$ and $b$ with the following properties: the order of $a$ is $2$, the order of $b$ is $2$, and the order of $ab$ is $8$. If $Z(G)$ denotes the center of $G$, determine the group $G/Z(G)$ to which it is isomorphic.

Answer: D_4


Question: Evaluate the limit: $$\lim_{y\to0} \int_{0}^{2} \frac{e^{(x+y)^2}-e^{x^2}}{y} \, dx$$

Answer: e^4 - 1


Question: In an equilateral triangle $\Delta ABC$, a point $P$ is located outside the triangle but inside angle $ABC$. The perpendicular distances from $P$ to the sides $BC, CA, AB$ are $h_1, h_2, h_3$ respectively, and it is given that $h_1 - h_2 + h_3 = 6$. Find the area of $\Delta ABC$. \(\text{Express your answer in terms of the side length of the triangle if necessary.}\)

Answer: 12\sqrt{3}


Question: Find the solutions to the functional differential equation \( f'(x) - f(\alpha x) = 0 \), given that \( f(0) = 0 \), \( f: \mathbb{R}^+ \rightarrow \mathbb{R} \), and \( 0 < \alpha < 1 \).

Answer: 0


Question: Consider an uncountable index set $I$ and let $\mathcal{A}=\text{Fin}(I)$ be the collection of all finite subsets of $I$ ordered by inclusion. Suppose we have a net $(T_\alpha)_{\alpha\in\mathcal{A}}$ consisting of bounded linear operators between a normed space $E$ and a Banach space $F$. Given that, for each $x\in E$, the net $(T_\alpha x)_{\alpha\in\mathcal{A}}$ converges in $F$ to $Tx$, does this imply that $(T_\alpha)_{\alpha\in\mathcal{A}}$ converges to an operator $T$ in the norm topology?

Answer: No


Question: Determine the set of real numbers $x$ for which the power series $x + \sum_{k \geq 2} (-1)^{k-1} \frac{2kx^{2k-1}}{(2k-1)!}$ converges.

Answer: \mathbb{R}


Question: Find the remainder \( r \) when \( 2^{82} \) is divided by 625, where \( 0 \leq r < 625 \).

Answer: 329


Question: Evaluate the limit \(\lim_{x\to\infty}(x!)^{1/x}-\frac{x}{e}\).

Answer: \infty


Question: Find all finite sets $A$ such that the Cartesian product $A \times \mathcal{P}(A)$ is equal to $\mathcal{P}(A) \times A$, where $\mathcal{P}(A)$ denotes the power set of $A$. Provide your answer as a list of sets.

Answer: \emptyset


Question: Derive an expression for the integral of \( \frac{|d s |^2}{|s|^2} \cos^2(\frac{\epsilon}{2}) \) around a circle, given that \(|\Delta s_{12} |^2 = |\vec s_1|^2 + |\vec s_2|^2 - 2 s_1 \cdot s_2\) and assuming a path around a circle of radius \(|s_1| = |s_2|\).

Answer: \pi


Question: Calculate the flux of the vector field \( F = (x^2, y^2, z^2) \) through the surface \( S \) defined by \( x^2 + y^2 + z^2 = 4 \), oriented outward.

Answer: 0


Question: Let $(x_n)_{n \ge 0}$ be a sequence of real numbers with $x_0 > 0$ and $x_{n+1}=x_n+ \frac {1}{\sqrt {x_n}}$. Find the limit \[ L=\lim_{n \rightarrow \infty} \frac {x_n^3} {n^2}. \]

Answer: \dfrac{9}{4}


Question: Evaluate the sum \(\sum_{k=0}^{\infty} \sum_{j=0}^{\infty} \sum_{i=0}^{\infty} \frac{1}{3^i3^j3^k}\) for \(i \neq j \neq k\).

Answer: \dfrac{81}{208}


Question:
Consider two possible events $a_1$ and $a_2$ with probabilities $p(a_1)$ and $p(a_2)$, where $p(a_1) + p(a_2) = 1$. A detector, affected by noise, has a probability $p^d(a_j \to a_i)$ of observing $a_i$ given that the event was $a_j$. The probability of error for this detector is given by:
\[ p_e^d = p(a_1) \cdot p^d(a_1 \to a_2) + p(a_2) \cdot p^d(a_2 \to a_1) \]

Now, consider two detectors, $d_1$ and $d_2$, each affected by independent noise, making simultaneous measurements. Determine the probability of error for the joint observation of the two detectors. Is the joint probability of error $p_e^J$ given by:

1. \[ p_e^J = p_e^{d_1} \cdot p_e^{d_2} \]

or

2. \[ p_e^J = (p(a_1) \cdot p^{d_1}(a_1 \to a_2) \cdot p^{d_2}(a_1 \to a_2) + p(a_2) \cdot p^{d_1}(a_2 \to a_1) \cdot p^{d_2}(a_2 \to a_1)) + \\
+ (p(a_1) \cdot p^{d_1}(a_1 \to a_1) \cdot p^{d_2}(a_1 \to a_2) + p(a_2) \cdot p^{d_1}(a_2 \to a_2) \cdot p^{d_2}(a_2 \to a_1)) + \\
+ (p(a_1) \cdot p^{d_1}(a_1 \to a_2) \cdot p^{d_2}(a_1 \to a_1) + p(a_2) \cdot p^{d_1}(a_2 \to a_1) \cdot p^{d_2}(a_2 \to a_2)) \]

Determine the correct expression for $p_e^J$. Note: $p(a_i \to a_j) + p(a_i \to a_i) = 1$. \[ \]

Answer: 2


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

Answer: 0


Question: Determine the order of contact between the curves \(y=x^3+x^6\) and \(y=x^3+x^5\) at the point \((0,0)\).

Answer: 4


Question: Determine whether the sequence \(a_n = n^2 \prod_{k=1}^{n} \cos_k(x)\) converges, where \(\cos_k(x)\) is defined as in the original question.

Answer: 0


Question: Find the area of the largest triangle that can be formed in the first quadrant by a line tangent to the curve $y = e^{-x}$. Provide your answer as a single number.

Answer: \dfrac{2}{e}


Question: Evaluate the limit \( \lim_{n\rightarrow +\infty}\int_0^a(\cos\sqrt x)^n\,dx \) for any \( a > 0 \).

Answer: 0


Question: Determine whether there exists a 7-vertex planar graph $G$ such that its complement $\overline{G}$ does not contain a 3-vertex clique.

Answer: Yes


Question: Given that $A$ is a dense subset of the open ball $B$ with radius $m$ and center at the origin, where $0 < m < 1$, determine if $A$ necessarily has a limit point within the closed unit disc $\mathbb{D}$ (i.e., the set of all points with $|z| \leq 1$).

Answer: Yes


Question: Suppose $c$ is an isolated point in the domain $D$ of a function $f$. Determine if the function $f$ has the value $f(c)$ in the delta neighborhood of $c$. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty}\left(1-\dfrac{1}{\eta^{x}} \right)^n \) for all real numbers \(x\), given that \(\eta > 0\).

Answer: 0


Question: Determine whether the interval $[0,1]$ can be partitioned into uncountably many sets, each containing a rational number. Answer with 'yes' or 'no'.

Answer: no


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

Answer: \infty


Question: Given that $X$ and $Y$ are independent random variables with moment generating functions $M_x(t) = \frac{(8+e^t)^2}{81}$ and $M_y(t) = \frac{(1+3e^t)^3}{64}$, respectively, find the probability $P(X+Y = 1)$. Provide your answer as a single probability value.

Answer: \dfrac{37}{324}


Question: Given a differentiable function $f:[a,b]\to\mathbb{R}$, such that $\lim\limits_{x\to a}f(x)=\lim\limits_{x\to a}f^\prime(x)=0$ and $f^\prime(x)\ne 0$ in a neighborhood of $a$, determine whether it necessarily holds that $$\lim_{x\to a}\frac{f(x)}{f^\prime(x)}=0.$$

Answer: 0


Question: In a group $G$, given that $a^5 = e$ and $aba^{-1} = b^2$ for some elements $a, b \in G$, determine the order of the element $b$. \(o(b)\)

Answer: 31


Question: Determine whether there exists a matrix $R$ such that for a given symmetric positive definite matrix $P$, the inequality $P < RPR^T$ holds, even if this inequality does not hold for all matrices $R < -I$. Provide a justification for your answer.

Answer: Yes


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

Answer: S_3


Question: Determine the values of $a$ for which the series \( \sum\limits_{n=1}^\infty \frac{\log ( 1 + 1/n)}{n^{a}} \) converges.

Answer: a > 0


Question: Calculate the residue of the function \( \frac{e^z-1}{z^2\sin(z)} \) at the pole \( z_0=0 \) without using the third derivative of the function.

Answer: \dfrac{1}{2}


Question: Let $G$ be a finite group of order 168. If the number of $7$-Sylow subgroups in $G$ is 8, and the size of the conjugacy class for each element of order $7$ is 24, determine the number of elements of order $7$ in $G$.

Answer: 48


Question: Given random variables X, Y, and Z, where (Y+Z) is also a random variable, if X is independent of Y and X is independent of Z, is it guaranteed that X is independent of (Y+Z)? Answer yes or no.

Answer: No


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

Answer: \sqrt{2}\pi


Question: For finite groups $G$, $H$, and $K$, if $H$ and $K$ are normal subgroups of $G$ and $H \cong K$, does it follow that $G/H \cong G/K$?

Answer: No


Question: Determine the values of \(a\) and \(b\) such that \(L_n = a\phi^n + b\widehat{\phi}^n\), where \(L_n\) is the \(n^{th}\) Lucas number.

Answer: 1


Question: Find the largest possible value of $c$ such that $a! + b! = c!$ holds true without contradiction, given that $1 \le a \le b \le c$. What is this value of $c$?

Answer: 2


Question: Given that the pair of random variables $(X,Y)$ is uniformly distributed on a disc of radius $R$ centered at the origin $(0,0)$, find the probability density function $f_Z(t)$ for the random variable $Z = X^2 + Y^2$ for small values of $t$. Provide your answer in terms of $t$ and $R$.

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


Question:
Given non-negative measurable functions $f(x,y)$ and $g(x,y)$ defined on a bounded open set $G \subset \mathbb{R}^{m} \times \mathbb{R}^{n}$, with $m,n \geq 1$, and compact sets $E \subset \mathbb{R}^{m}$ and $F \subset \mathbb{R}^{n}$ such that $E \times F \subset G$, suppose:
\[ 0 \leq \int_{E} \int_{F} f(x,y) \, dy \, dx \leq \int_{E} \int_{F} g(x,y) \, dy \, dx \]
and $g$ is locally integrable. Can Fubini's theorem be applied to $f$ to switch the order of integration?

Answer: Yes


Question:
Given a differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y), f(x)f(y), f(x-y)$ form an arithmetic progression for all $x, y \in \mathbb{R}$ and $f(0) \neq 0$, determine which of the following is true:

A) $f'(0) = -1$
B) $f'(0) = 1$
C) $f'(1) - f'(-1) = 0$
D) $f'(1) + f'(-1) = 0$

Answer: D


Question: Calculate the work done by the vector field \( \underline{G} = \frac{x}{x^2+y^2}\underline{i}+\frac{y}{x^2+y^2}\underline{j} \) along the ellipse \( C \) parameterized by \( \underline{r}(t) = a\cos(t)\underline{i}+b\sin(t)\underline{j} \) for \( 0\leq t\leq 2\pi \), where \( a \) and \( b \) are positive constants.

Answer: 0


Question: Determine the coefficient of $x^n$ in the expansion of the expression $\frac{(1-x)^n}{1+x}$.

Answer: (-2)^n


Question: Let $(s_1, s_2, s_3, \ldots )$ be an equidistributed sequence on $[0,1]$. Determine whether it is true that for each $0 < \delta < \varepsilon < 1$, there exists an $N$ such that \[ \varepsilon - \delta < \frac{ \left\lvert \{ s_1, \ldots, s_n \} \cap [c, c+\varepsilon] \right\rvert }{ n } < \varepsilon + \delta \] for every $c \in [0,1-\varepsilon]$ and for all $n \geq N$. Answer with True or False.

Answer: A


Question: Can an algebraic field extension have a finite degree? Provide a yes or no answer.

Answer: Yes


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

Answer: 0


Question: Consider the ellipse given by the equation \(\frac{x^2}{16}+\frac{y^2}{9}=1\). A line passing through the point \(A(\sqrt{7},0)\) intersects the ellipse at two distinct points, \(N\) and \(M\). Find the minimum possible value of the ratio \(\frac{AM \cdot AN}{NM}\).

Answer: \dfrac{9}{8}


Question: Find the image of the hyperbola $xy = 1$ in the $uv$-plane under the transformations $u = x^{2} - y^{2}$ and $v = 2xy$. Express the result as an equation in terms of $u$ and $v$.

Answer: v = 2


Question: In how many ways can you fill a $6 \times 6$ grid with the numbers 1 and -1 such that the product of the numbers in each row and each column is positive?

Answer: 33554432


Question: Let $f: [1, \infty) \to \mathbb{R}$ be a differentiable function with $f(1) = 1$ and $f'(x) = \frac{1}{1 + f(x)^2}$. If $f$ is strictly increasing and bounded below by 1, determine $\lim_{x \to \infty} f(x)$. \( \text{Express your answer as a number or expression.} \)

Answer: \infty


Question: Determine the cardinality of the set of all real sequences.

Answer: \mathfrak{c}


Question:
Given real numbers $a, b, c$, and real numbers $x, y, z$ that are not all simultaneously zero, satisfy the system of equations:
$$x = cy + bz$$
$$y = cx + az$$
$$z = bx + ay$$
Determine the value of $a^2+b^2+c^2+2abc$. Choose the correct answer from the following options:
(A) 0
(B) $\frac{1}{2}$
(C) $\frac{\sqrt{3}}{3}$
(D) 1
(E) 5

Answer: D


Question: Compute the determinant \(\begin{vmatrix} bc - a^2 & ac - b^2 & ab - c^2 \\ ac - b^2 & ab - c^2 & bc - a^2 \\ ab - c^2 & bc - a^2 & ac - b^2\end{vmatrix}\) and express it in the form \(k(a^3 + b^3 + c^3 - 3abc)^2\). Given that \(\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b\end{vmatrix} = -(a^3 + b^3 + c^3 - 3abc)\), find the value of \(k\).

Answer: 1


Question: Evaluate the limit of the product as \( n \to \infty \): \[ \lim_{n\to \infty} \prod_{k=1}^{n} \left(1 - \frac{1}{2k}\right) \]

Answer: 0


Question: For the nodal cubic defined by \(y^2 = x^2(x+1)\), consider the parametrization given by \(x(t) = t^3 - 3t\) and \(y(t) = t^4 - 4t^2\). Find the values of \(t\) where this parametrization fails to be regular.

Answer: \emptyset


Question: Compute the degree of the splitting field of the polynomial \(X^{4}+X^{3}+X^{2}+X+1\) over the field \(\mathbb{F}_{3}\).

Answer: 4


Question: Find the value(s) of the constant $k$ for which the integral \( \int_{0}^{\infty} \left(\frac{x+2}{2x^2 + k} - \frac{k}{3x+2}\right)dx \) converges.

Answer: \dfrac{3}{2}


Question: Determine the interval of convergence for the infinite series \(\sum\limits_{n=0}^\infty \dfrac{n!x^n}{n^n}\).

Answer: (-e, e)


Question: Consider a non-abelian group $G$ of order $pq^2$, where $p$ and $q$ are primes. Is it possible for the commutator subgroup $G'$ to have order $q^2$? Provide a justification for your answer.

Answer: Yes


Question: Let $A$ and $B$ be two finite sets with the same number of distinct elements. Define $S_A$ and $S_B$ as the symmetric groups on $A$ and $B$, respectively, with $|S_A| = |A|!$ and $|S_B| = |B|!$. Form the set $X = S_A \times S_B$, where each element is a pair $\langle p_A, p_B \rangle$ with $p_A \in S_A$ and $p_B \in S_B$. Given two permutations $p_A, p_B \in X$, denote the "correspondence bijection" induced by $\langle p_A, p_B \rangle$ as a mapping from $A$ to $B$ where $a_i \mapsto b_j$ if $p_A(a_i) = a_j$ and $p_B(b_j) = b_i$. Find the cardinality of the largest subset $Y \subseteq X$ such that no two pairs in $Y$ induce the same correspondence bijection.

Answer: n!


Question: Given that \( f(x) = 2x^3 + 7x - 1 \) and \( g(x) \) is the inverse of \( f(x) \), find \( g'\left(\frac{191}{4}\right) \) without explicitly solving for \( x \) in the cubic equation.

Answer: \dfrac{2}{89}


Question: Determine the maximal ideal of the ring \( \frac{\mathbb{R}[x]}{(x^2)} \).

Answer: (x)


Question:
Determine whether the following equation holds true as \( x \) approaches infinity, where the angle is measured in degrees:
\[ \pi = \lim_{x \to \infty} (10^{x+2}\sin (1.8\times 10^{-x})) \]

Answer: \pi


Question: Find the smallest power of 2, where the exponent is divisible by 10, that does not start with the digit 1.

Answer: 2^{300}


Question: Determine whether there exists an uncountable set of sets such that each set in the collection has a distinct cardinality.

Answer: Yes


Question: Consider a family of continuous vector-valued stochastic processes, denoted by $x_t^\epsilon$. Suppose that the $L^p$ norm of the supremum over the time interval $[0,T]$ converges to zero as $\epsilon$ approaches zero, i.e., $E \sup_{t \in [0,T]} |x_t^\epsilon|^p \to 0$ for some $p > 0$. Does this imply that $\limsup_{\epsilon \to 0} \sup_{t \in [0,T]} |x_t^\epsilon|^p = 0$?

Answer: No


Question: How many distinct 3-dimensional subspaces are there in \(\mathbb{R}^4\)?

Answer: \infty


Question: Determine the index of the subgroup $H$ in a group, where $H$ is defined as the set of words with exponent sum divisible by $k$.

Answer: k


Question: Determine the values of the limits \(L_{g,h}=\lim_{n \to \infty} \frac {E_g(1,10,n)}{E_h(1,10,n)}\), where \(g,h \in \{1,3,7,9\}\), and \(E_g(1,10,n)\) represents the number of primes in the interval \([1,10^n]\) that end in the digit \(g\).

Answer: 1


Question: Evaluate the contour integral \( \int_{\Gamma} |z|^2 \, dz \), where \( \Gamma \) is the boundary of a square with vertices at \( 0, 1, 1+i, \) and \( i \), traced counterclockwise starting from \( 0 \).

Answer: -1 + i


Question: Evaluate the limit \( \lim\limits_{x\to0}\left(\frac{\sin x}{x}\right)^{\frac{6}{x^{2}}} \) using the Maclaurin series.

Answer: \dfrac{1}{e}


Question: Does there exist a restricted universal enveloping algebra $uL$ for a $p$-Lie algebra $L$ over a field $k$ of characteristic $p > 0$, such that $L$ has no nonzero $p$-algebraic elements and the global dimension of $uL$ is infinite?

Answer: Yes


Question: What is the minimum number of comparisons required to determine if a given number $x$ is present in a matrix $B$, where the matrix satisfies the condition $B_{ij} \le B_{kl}$ for all $i \ge k$ and $j \le l$?

Answer: m + n - 1


Question: Find the minimum value of the expression \(x^{\ln(y)-\ln(z)} + y^{\ln(z)-\ln(x)} + z^{\ln(x)-\ln(y)}\), where \(x, y, z > 0\).

Answer: 3


Question: Evaluate the integral \( I = \int_0^\pi \log(\sin(t)) \, dt \).

Answer: -\pi \ln 2


Question: Given that for any arbitrarily small $\epsilon > 0$, \(\lim_{n\to\infty} P\left(|\frac{S_n}{n} - 1| > \epsilon\right) = 0\), determine whether this implies \(\lim_{n\to\infty} P\left(\frac{S_n}{n} = 1\right) = 1\).

Answer: No


Question: Evaluate the contour integral \( \oint_C \frac{z-3}{e^{2z}+1} \, dz \), where \( C \) is the square contour with vertices at \( 0 \), \( 1 \), \( 1+i \), and \( i \).

Answer: 0


Question: If $g(x)$ is a cubic polynomial with roots $p, q, r$, and its derivative $g'(x)$ divides $g(2x)$, find the value of $C$ such that $Cpq = 16r^2$. Given $g(x) = A(x-p)(x-q)(x-r)$ and $g(2x) = A(2x-p)(2x-q)(2x-r)$, determine the value of $C$.

Answer: 1


Question: Determine the coefficient of \(x^{17}\) in the expansion of \((3x^7 + 2x^5 -1)^{20}\).

Answer: -41040


Question: Determine whether the following claim is true or false: Let $f(x)$ be a $2\pi$-periodic and continuously differentiable function on the real line. Then the Fourier series of $f(x)$ converges to $f(x)$ uniformly on $[-\pi, \pi]$. Provide a justification for your answer.

Answer: True


Question: What is the smallest set of points required to intersect all lines in a projective geometry $PG(d,q)$?

Answer: \frac{q^d - 1}{q - 1}


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

Answer: \dfrac{3}{2}


Question: Is it true that there are only five bounded connected non-selfintersecting polyhedra with identical regular-polygon faces and congruent vertices, namely the Platonic solids?

Answer: Yes


Question: For a 2-parameter Weibull distribution, determine the values of the shape parameter $k$ such that the variance is twice the mean.

Answer: k > 0


Question: Determine the sum of the alternating series \( \sum_{n=1}^{\infty}(-1)^{n+1}\left(\frac{1}{n}\right) = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots \) using Abel's theorem or other suitable methods.

Answer: \ln 2


Question:
Given the joint probability distribution of continuous random variables $X$ and $Y$:

$$f(x,y)=\begin{cases} e^{-(x+y)}, & x>0, y>0 \\ 0, & \text{otherwise} \end{cases}$$

Compute the conditional probability $P(Y > X + 1)$. Assume the region of interest is $D$. Provide your answer as a single value.

Answer: \dfrac{1}{2e}


Question: Evaluate the limit \(\lim_{\epsilon \to 0^+} \left(\operatorname{Ci}(\epsilon) - \ln(\epsilon)\right)\), where \(\operatorname{Ci}(x)=-\int_x^\infty \frac{\cos(t)}{t} \, dt\).

Answer: \gamma


Question: Determine whether a group of order 16 must contain an element of order 4.

Answer: No


Question: Determine the order of the Galois group of the polynomial $f(x) = x^5 - 7$ over the field of rational numbers $\mathbb{Q}$.

Answer: 20


Question: Evaluate the contour integral \(\int_c \frac{dz}{z-3i}\) where \(c\) is the circle with radius \(\pi\) centered at the origin, traced counterclockwise.

Answer: 2\pi i


Question: A function $F$ is defined for all positive integers and satisfies the condition: $F(1) + F(2) + F(3) + \ldots + F(n) = n^2 \cdot F(n)$. Given that $F(1) = 2006$, find the value of $F(2005)$. \( \boxed{} \)

Answer: \dfrac{2}{2005}


Question:
Evaluate the limit:
\[ \lim_{n\to\infty}\frac{\sqrt[3]{n+1}-\sqrt[3]{n+\cos\frac{3}{n}}}{\sqrt[6]{n^2+\sin\frac{2}{n}}-\sqrt[3]{n}} \]

Answer: \dfrac{9}{2}


Question: Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that \( f(xy) = f'(x)f(y) \) for all \( x, y \in \mathbb{R} \).

Answer: 0


Question: Determine the number of elements \( \alpha \in \mathbb{F}_{3^5} \) such that \( \mathbb{F}_{3}(\alpha) = \mathbb{F}_{3^5} \).

Answer: 240


Question:
Solve the system of equations for real numbers $x, y, z$:
$$\begin{cases}
6x^2 - 12x = y^3 \\
6y^2 - 12y = z^3 \\
6z^2 - 12z = x^3
\end{cases}$$
Assume that only basic algebraic operations are allowed.

Answer: (0, 0, 0)


Question:
In the binomial expansion of \((1 + x)^n\), where \(n\) is a positive integer and \(p > 0\) is a real number with \(q = 1 - p\), find the value of the following sum:

\[ \sum\limits_{r=0}^n r^2 \binom nr p^{n-r}q^r \]

Choose the correct expression for the sum from the options below:
(A) \(np^2q^2\)
(B) \(n^2p^2q^2\)
(C) \(npq + n^2p^2\)
(D) \(npq + n^2q^2\)

Answer: D


Question: What is the value of \( \nabla g \) for a metric-preserving connection \( \nabla \) on a smooth manifold \( M \) with a metric \( g \)?

Answer: 0


Question: Determine whether the function \( g(x) = \int_0^1 f(t) \, t^x \, dt \), where \( f : [0,1] \to [0,1] \) is a smooth (class \( C^\infty \)) but not necessarily real-analytic function, is necessarily real-analytic on the interval \((-1, \infty)\).

Answer: Yes


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

Answer: 2 \ln 2


Question: If \(x_1, x_2, \ldots, x_{2019}\) are roots of \(P(x) = x^{2019} + 2019x - 1\), find the value of \(\sum_{i=1}^{2019}\frac{x_i}{x_i - 1}\).

Answer: 2017


Question: Find the limit of the expression \( \lim_{n\to\infty}n\sin(2^n \pi \sqrt{e}\mathrm{n}!) \) as \( n \) approaches infinity, given that \( \lim_{n\to\infty}n\sin(2\pi \mathrm{e}n!)=2\pi \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit of \[ \frac{\binom{2n}{n-2}(2n)^{2n-4}(2n \cdot (2n-1) \cdots (n+1))}{(2n)^{4n}} \] as \( n \to \infty \).

Answer: 0


Question: Determine whether there exists a 4-dimensional anisotropic quadratic form over the 2-adic integers \(\mathbb{Z}_2\), which is diagonal and has coefficients \(a, b, c, d\) such that 2 does not divide \(a, b, c,\) and \(d\).

Answer: Yes


Question: Given the sequence \((a_n)\) defined recursively by \(a_{n+1} = \frac{1}{2}\left(a_n + \frac{3}{a_n}\right)\), which converges to \(\sqrt{3}\), determine the rate of convergence \(\mu\) using the limit formula \[\lim\limits_{n \to \infty} \frac{|a_{n+1}-L|}{|a_{n}-L|} = \mu\] where \(L = \sqrt{3}\). Evaluate this limit without using series expansion when it results in an indeterminate form.

Answer: 0


Question: Calculate the expected value $E(X)$, where $Z$ is a standard normal random variable, and $X = \min(Z+1, 0)$. Provide your answer as a single number.

Answer: -0.0833


Question: Find the area under the curve $x^2 + y^2 = 3\sqrt{2} x - 5\sqrt{2} y + 2xy$ and above the x-axis between $x = 0$ and $x = 3\sqrt{2}$. Provide your answer as a numerical value.

Answer: \dfrac{13}{3}


Question: Find the matrix representation of $\exp(tE_{a,b})$ given that $E_{a,b}^2 = 0$.

Answer: I + tE_{a,b}


Question: Find the residue of the function \( f(z)=\frac{e^z-\sin z-1}{z^5+z^3} \) at \( z=0 \).

Answer: \dfrac{1}{2}


Question: For an uncountable subset $U$ of the real numbers $\mathbb{R}$, determine whether there exists a real number $x$ such that both $U \cap (-\infty, x)$ and $U \cap (x, \infty)$ are uncountable.

Answer: Yes


Question: Let $\langle a_{n}\rangle$ be a sequence such that $\displaystyle\lim_{n\rightarrow \infty}a_{n} = 0$. Find $\displaystyle \lim_{n\rightarrow \infty}\frac{a_{1}+a_{2}+\cdots+a_{n}}{\sqrt{n(n+1)}}$. \(\text{Express your answer as a single value or expression.}\)

Answer: 0


Question: Suppose $(a,b)$ is an open and bounded interval of $\mathbb{R}$. Let $f:(a,b)\to \mathbb{R}$ be a bounded function. If the improper integral of $f$ over $(a,b)$ converges, does there exist a function $\tilde{f}:[a,b]\to\mathbb{R}$ such that $\tilde{f}(x)=f(x)$ for $x \in (a,b)$ and $\tilde{f}$ is Riemann integrable over $[a,b]$?

Answer: Yes


Question: Find the critical points of the function \( f(\psi) = a^T \left( P + \psi^2 Q \right)^{-1} b \), where \(a\) and \(b\) are vectors, \(P\) is an invertible matrix, and \(Q\) is a diagonal matrix.

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\int_{a}^{b} \frac{\sin (nx)}{x} \, dx, \] where \(0 < a < b\).

Answer: 0