Question: Determine whether the subgroup \( H = \langle y^2, yx \rangle \) of the free group \( \mathbb{F}_2 = \langle x, y \rangle \) is free. If it is free, find a set of generators for \( H \).

Answer: \{y^2, yx\}


Question: Determine whether the sequence of functions \( g_n(x) = \frac{x^n}{n} \) is Cauchy in the space \( (C[0, 1], ||\cdot||_{\infty}) \).

Answer: Yes


Question: Is the existence of the empty set derivable from the axioms of ZF set theory?

Answer: Yes


Question: If an $n \times n$ matrix $B$ satisfies $B^2 = 0$, are all the diagonal entries of $B$ necessarily zero? Answer yes or no and justify your answer.

Answer: No


Question: Find the residue of the function \( f(z) = (ze^{2-z} + (2-z)e^z)e^{(z-1)^2 + (z-1)^{-2}} \) at \( z = 1 \).

Answer: 0


Question: Determine the number of integers \( x \) such that \( 1 \le x \le 5^k \) and \( 5^k \) divides \( x^4 - 1 \).

Answer: 4


Question: Calculate the limit: \[ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}e^{-\left(\frac{1}{n^2}+\beta\right)i^2} \] where \( \beta \) is a positive real-valued constant.

Answer: 0


Question: Find the value of $g(3)$ for a polynomial function $g(x)$ that satisfies the condition $g(x)g(y) = g(x) + g(y) + g(xy) - 2$ for all real values of $x$ and $y$, and $g(2) = 5$.

Answer: 10


Question: Calculate the Riemann integral of the characteristic function of the Cantor set over the interval \([0,1]\), where the middle third of each interval is removed.

Answer: 0


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

Answer: \dfrac{1}{2}


Question: Determine the result of the tensor product \( \mathbb{Z}_p \otimes \mathbb{Q} \) and explain its relation to the field of p-adic numbers \( \mathbb{Q}_p \).

Answer: \mathbb{Q}_p


Question: Consider the extension \(\mathbb{Q}(\alpha)\) over \(\mathbb{Q}\), where \(\alpha = e^{\frac{2\pi i}{3}}\). Determine if this extension is a normal extension, given that its degree is 2.

Answer: Yes


Question: Determine if the complex function \(g(z)= \sum_{n\geq1}\frac{(z-i)^n}{n}\) is differentiable in the open disk centered at \(i\) with radius \(1\).

Answer: Yes


Question: Evaluate the contour integral \( \frac{1}{2\pi i}\oint_{|z|=1}\frac{e^{\frac{c(z-z^{-1})}{2}}}{z^{n+1}}\,dz \) using the unit circle \( z=e^{i\theta} \), and simplify the expression by exploiting the \( 2\pi \)-periodicity and even-odd symmetries of the integrand.

Answer: J_n(c)


Question: Determine the condition on the parameter \(a\) such that the solution \(u(x, t)\) of the partial differential equation \(\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - au\), with boundary conditions \(u(0, t) = u(\pi, t) = 0\) for \(t \geq 0\) and initial condition \(u(x, 0) = (\sin x)^3\), converges to zero uniformly as \(t \to \infty\).

Answer: a > -1


Question: Determine whether every uncountable subset of the real numbers has a super limit point. Provide a justification for your answer.

Answer: Yes


Question: Let $G$ be a divisible group and $H$ be a subgroup of $G$. If $H$ has finite index in $G$, does it follow that $H=G$? Provide a proof or counterexample.

Answer: H = G


Question: Evaluate the limit: $$\lim_{x\to0}\frac{\log_{\sin^2x}\cos x}{\log_{\sin^2(x/2)}\cos(x/2)}$$ Choose the correct answer from the options: A) 1 B) 2 C) 4 D) none of these.

Answer: C


Question: Let $f(X)$ be a monic and irreducible polynomial of degree $n$ in $\mathbb{Z}[X]$. Determine if the quotient $\mathbb{Z}[X]/ \langle f(X) \rangle$ is a commutative ring with unity, and if so, find its characteristic.

Answer: 0


Question: In an idempotent-complete $\mathsf{Ab}$-enriched category, does every idempotent give rise to a biproduct decomposition? Answer yes or no.

Answer: Yes


Question: Derive the Taylor expansion of \( f(x) = \arctan{\frac{1-x}{1+x}} \) and determine its radius of convergence.

Answer: 1


Question: Find the function $g(z)$ that is analytic on $|z-2|<2$, given that $g'(z)=\frac{1}{z-1}$ and $g(2)=1$. Determine $g(z)$ in this region.

Answer: \log(z - 1) + 1


Question: Calculate the scalar surface integral of the function \( f(x, y, z) = yz \) over the half cylinder \( S \), defined by the equation \( x^2 + z^2 = a^2 \) with \( z \leq 0 \) and \( 0 \leq y \leq h \). Verify if the result is \( -a^2h^2 \).

Answer: -a^2 h^2


Question: Calculate the mass of the solid region bounded by the cone \(z=\sqrt{x^2+y^2}\) and the plane \(z=4\) using cylindrical coordinates, given that the density is \(\rho(x,y,z)=z\).

Answer: 64\pi


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

Answer: \sqrt{22}


Question: Let $X$ be a standard normal random variable and $a > 0$ be a constant. Determine \( \lim_{x \to \infty} \mathbb{P}\left(X > x + \frac{a}{x} \mid X > x\right) \).

Answer: e^{-a}


Question: Find the arc length of the curve defined by the equation \(x^{2/3} + y^{2/3} = 4\).

Answer: 48


Question: Evaluate the integral: \[ \int_{0}^{\infty}\dfrac{1-e^{-2t}}{t}\sin{2t}\operatorname d\!t \]

Answer: \dfrac{\pi}{4}


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\frac{x^4}{x^6+64}dx \) using the residue theorem.

Answer: \dfrac{\pi}{3}


Question: For the differential equation \( y''+P(x)y'+Q(x)y=0 \) on the interval \((0,1)\), given that \( y(x)=x^3 \) is a solution, is it possible for \( P(x) \) to be bounded while \( Q(x) \) is unbounded on \((0,1)\)? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: If a power series \( f(z) = \sum a_nz^n \) has radius of convergence \( R > 0 \) and \( f(\frac{1}{n}) \) is real for all \( n > \frac{1}{R} \), then the coefficients \( a_n \) must be real.

Answer: True


Question: Determine the number of nonequivalent Boolean functions that can be formed using the input variables $x_1, \ldots , x_n$ and the constants and connectives from the set $\{0, 1, \lor, \neg\}$.

Answer: 2^{2^n}


Question: Express the polynomial with roots at the vertices of a regular octagon on the complex plane, with one root at (1, 0), in the form \( p(z) = (z - (a + bi))^8 + q \).

Answer: z^8 - 1


Question: Suppose you have a basket containing a single coin marked with the number 1. At each time interval of noon minus \( \frac{1}{k} \) minutes, where \( k \) is a positive integer, the coin numbered \( k \) is removed from the basket and replaced with two new coins numbered \( 2k \) and \( 2k+1 \). How many coins are in the basket at exactly noon?

Answer: 0


Question: Evaluate \( \lim_{x \to 0^+} \frac{x-\sin x}{(x \sin x)^{3/2}} \).

Answer: \dfrac{1}{6}


Question: Identify the Banach space that is isometrically isomorphic to the quotient space $c/Y$, where $c$ is the space of convergent sequences and $Y$ is the subspace of sequences converging to zero.

Answer: \mathbb{R}


Question: Verify whether the statement $e^{At}e^{Bt}=e^{Bt}e^{At}$ holds if and only if $AB=BA$ for matrices $A$ and $B$.

Answer: AB = BA


Question: Evaluate the limit using a Riemann Sum: $$\lim_{n\to\infty} \sum_{k=1}^{n} \sin\left(\frac{(2k-1)a}{n^2}\right), \ a \in \mathbb{R}$$

Answer: a


Question: Does there exist a semigroup $S$ such that for every element $y \in S$, there exists an element $x \in S$ with $y = xy$, but $S$ does not have a left identity element?

Answer: Yes


Question: Are random variables with identical moment generating functions necessarily identical almost everywhere? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \oint_{\gamma} \frac{z}{\sin z} \, dz \), where \( \gamma \) is the circle \( |z|= 2 \) oriented in the clockwise direction.

Answer: 0


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

Answer: \dfrac{16}{9}


Question: Find all functions $f(t)$ such that for all $x$, the equation \[ \int_0^x f(t) \, dt + \int_{x}^1 f(t) \, dt = \int_x^1 f(t) \, dt + \int_x^1 f(t) \, dt \] holds true.

Answer: 0


Question: Let $f:[0,\infty) \rightarrow[0,\infty]$ be a continuous function on $[0,\infty]$ and differentiable on $(0,\infty)$. Given that $f(x)=\int_{0}^{x}\sqrt{f(t)}dt$, determine the range of possible values of $f(6)$. \( \text{Provide your answer as an interval or set of values.} \)

Answer: [0, 9]


Question: Given two linear projection operators $P_1$ and $P_2$ on a vector space $V$ with the same range $R = P_1(V) = P_2(V)$, determine if any convex combination of $P_1$ and $P_2$ is also a projection operator with $P_3(V) = R$. Is this statement true or false?

Answer: True


Question: Determine whether it is possible for two different nonempty connected open sets, $R_1$ and $R_2$, to have the same closure. Provide a justification for your answer.

Answer: Yes


Question: Find a simple expression or a good upper bound for the expected maximum amount of money found, \(E[\max(x_1, \ldots, x_n)]\), when \(n\) individuals find street money. Each person has a probability of \(\dfrac{1}{2^x}\) to find \(x\) dollars (\(1 \leq x \leq n\)), with probabilities being independent. It is known that the expected value of money found by each person is less than 2 dollars. Initial observations suggest it might be around \(\log(n)\).

Answer: O(\log n)


Question: Find the lowest positive solution of the equation \( x = 254 \times 353^{(\phi(400)-1)} \mod 400 \), where \( \phi \) is the Euler's totient function.

Answer: 318


Question:
Calculate the determinant of the following tridiagonal matrix with size \( n \times n \):
\[
\begin{bmatrix}
2 & -1 & \cdots & 0 & 0 \\
-1 & 2 & \cdots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & 2 & -1 \\
0 & 0 & \cdots & -1 & 2
\end{bmatrix}
\]

Answer: n+1


Question: Find the extreme values of the function \( g(u,v) = \frac{u^2 + v^2}{2} \) subject to the constraint \( 4 \leq u^2 - v^2 \leq 9 \).

Answer: 2


Question: Let $g(x) = (\mathbb{P}(Z>f(x)))^2$, where $Z$ follows a standard normal distribution. Determine if $g(x)$ is maximized at $x_0$, where $x_0 = \operatorname{argmin} f(x)$. Is this statement true?

Answer: True


Question: Suppose $A$ and $B$ are two elements of $M_n(\mathbb{R})$ such that their characteristic polynomials are equal. If $A=C^2$ for some $C\in M_n(\mathbb{R})$, does it follow that $B=D^2$ for some $D\in M_n(\mathbb{R})$? Answer yes or no.

Answer: No


Question: Given a square matrix \( M \) with real entries such that \( \operatorname{tr}(M^{2m})=0 \) for all positive integers \( m \), find an expression for the sum \[ T = \sum_{k=1}^\infty \ln \left| e^{M^{2k+1}} \right|. \]

Answer: 0


Question: Is the product topology on $C \times D$ the same as the topology that $C \times D$ inherits as a subspace of $X \times Y$, given that $C$ is a subspace of $X$ and $D$ is a subspace of $Y$?

Answer: Yes


Question: Determine the locus of the point of intersection of tangents drawn to the circle \(x^2 + y^2 = 1\) at the points where it intersects the family of circles given by the equation \(x^2 + y^2 - (k + 6)x + (8 - 2k)y - 3 = 0\), where \(k\) is a parameter.

Answer: 2x - y + 10 = 0


Question: Determine the values of \(\alpha\) for which the set \(T_{\alpha} = \{(x,y,z) \in \mathbb{R}^3 \mid x^2+z^2=y^2, y-z+\alpha=0\}\) is a subspace of \(\mathbb{R}^3\).

Answer: 0


Question: Using generating functions, find the number of distinct ways to roll five dice, each with three sides labeled $1$ and three sides labeled $2$, such that the sum of the numbers rolled is $9$. The generating function for a single die is $3x+3x^2$. Determine the coefficient of $x^9$ in the expansion of $(3x+3x^2)^5$.

Answer: 1215


Question: Determine whether the following statement is true or false: A stochastic process that lacks independent increments also lacks stationary increments.

Answer: False


Question: Find the general form of all $4 \times 4$ matrices that commute with every other $4 \times 4$ matrix.

Answer: \lambda I_4


Question: Evaluate the limit: $$\lim_{x \to \pi^{-}} \ln(\sin x)$$ without directly substituting values close to $\pi$. Provide a detailed explanation of your approach.

Answer: -\infty


Question: Determine if a group with elements of orders 9 and 11 can be generated by a single element of the form \( f = g^x h^y \), where \( g \) has order 9, \( h \) has order 11, and \( x, y \in \mathbb{Z} \) satisfy \( 1 = 9x + 11y \).

Answer: Yes


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

Answer: -\dfrac{3}{16}


Question: Determine the value of \( k > 0 \) for which the equation \( \ln x - kx^2 = 0.5 \) has exactly one solution.

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


Question: Evaluate the limit: \[ \lim \limits_{n \to \infty}\left( 1 + \sqrt{2} + \sqrt[3]{3} + \dots + \sqrt[n]{n} \right) \ln\left(\frac{n}{2n+1}\right) \]

Answer: -\infty


Question: Determine the cardinality of the set of all well-formed formulas (WFFs) in propositional logic.

Answer: \aleph_0


Question: Find the limit of the sequence \( b_n = \sqrt[n]{7^{3n} - 2^{n+4} - \cos(\sqrt{n})} \).

Answer: 343


Question: Determine if there exist complex number solutions for the equation \( \sum_{i=1}^n z_i^2 = 0 \) when \( n \geq 3 \), where each \( z_i = x_i + iy_i \).

Answer: Yes


Question: Determine whether the following statement is true: For any two equivalent norms \(\|\cdot\|_a\) and \(\|\cdot\|_b\), does the inequality \(\|u\|_a < \|v\|_a\) always imply \(\|u\|_b < \|v\|_b\)?

Answer: False


Question:
Calculate the limit using the definition of the derivative:
\[\lim_{h\rightarrow0}\frac{f(2+5h)-f(2-3h)}{f(2-7h)-f(2-h)}\]
where \(f(x)\) is a differentiable function and \(f'(2) \ne 0\).

Answer: -\dfrac{4}{3}


Question: Find the sum of $a$ and $b$ if $a$ and $b$ are positive integers such that \( |a-1|+|a-2|+|a-3|+\cdots+|a-2015|=b(b+1) \).

Answer: 2015


Question: For a doubly stochastic matrix $A \in M(n,n)$, if $A^k$ is a matrix with all entries equal to $1/n$ for some $k > 1$, does it imply that $A^2$ must also be a matrix with all entries equal to $1/n$? Answer yes or no.

Answer: Yes


Question: Find the remainder when \(37!\) is divided by \(39\) using Wilson's Theorem.

Answer: 0


Question: Calculate the expected value of $X_t$ if $X_t = \sin(B_t)$, where $B_t$ is a Brownian motion and $t \geqslant 0$. Provide your answer as a function of $t$.

Answer: 0


Question: Find the bounded solutions to the differential equation: $$\alpha'''(x) =\alpha'(x) x^2 + \alpha(x) x.$$

Answer: 0


Question: Consider the Banach spaces $l^1(N)=\{\xi:N\rightarrow \mathbb{C}:||\xi||_1=\sum_{n=1}^\infty |\xi(n)|<\infty \}$ and $l^2(N)=\{\xi:N\rightarrow \mathbb{C}:||\xi||_2=(\sum_{n=1}^\infty |\xi(n)|^2)^{1/2}<\infty \}$. Determine if the linear map $J:l^1(N)\rightarrow l^2(N)$, defined by $J(\xi)=\xi$, is continuous. If it is, find its norm.

Answer: 1


Question: Is every GCD-Sequence also a Divisor Product Sequence? Provide a justification for your answer.

Answer: No


Question: Find the coefficient of $x^{11}$ in the power series expansion of $\frac{1}{1-x-x^4}$.

Answer: 19


Question: Find the multiplicative inverse of the polynomial $2x+1$ in the quotient ring $F/(m(x))$, where $F=\mathbb{Z}/3\mathbb{Z}$ and $m(x)=x^2+x+2$ is irreducible over $F$. Provide your answer as a polynomial in $F/(m(x))$. \( \boxed{\text{Your answer here}} \)

Answer: x + 2


Question: Determine the number of different homomorphisms \( \varphi : \mathbb{Z}_{18} \to \mathbb{Z}_{24} \) such that the order of the image of \( \varphi \), denoted as \(|\text{Im}\varphi|\), is 6.

Answer: 2


Question: Given an n-dimensional orthonormal basis in Euclidean space with positive orientation, determine if it is possible to transform it into the standard basis using a sequence of rotations.

Answer: Yes


Question: Determine the order of the zero at $z = 0$ for the function $f(z) = e^{\sin(z)} - e^{\tan(z)}$. Provide your answer as an integer.

Answer: 3


Question: Given a continuous function $f: (0, \infty) \to \mathbb{R}$, for $x, y > 0$ and any $x' > 0$, the property \( \int_{x}^{xy} f(u) \, du = \int_{x'}^{x'y} f(u) \, du \) holds. If $f(17) = 17$, determine $A(x) = \int_{1}^{x} f(t) \, dt$ for $x > 0$. \( \boxed{A(x)} \)

Answer: 289 \ln x


Question: What is the minimum length a sequence must have to guarantee a monotonic subsequence of length $n$, where $n$ is a positive integer?

Answer: (n-1)^2 + 1


Question: Calculate the value of the integral \( \int_{X\times Y}\chi_\Delta d(\lambda\times \nu) \), where \( (X,\mathcal{M},\lambda) \) is a measure space with Lebesgue measure \( \lambda \), \( (Y,\mathcal{N},\nu) \) is a measure space with an arbitrary measure \( \nu \), and \( \Delta=\{(x,y)|x=y,x\in X,y\in Y\} \).

Answer: 0


Question: Evaluate the limit \(\lim_{n \to \infty}\left( \frac{1^a+2^a+3^a+\cdots+n^a}{n^a}-\frac{n}{a+1} \right)\) for \(a \in \mathbb{R}\) with \(a > 1\).

Answer: \dfrac{1}{2}


Question: Let \( f \colon \mathbb{R}^2 \to \mathbb{R} \) be a continuous function such that \( \lim_{|x| \to \infty} f(x) = 0 \). Prove or disprove that \( f \in C_0 \), meaning there exists a compact set \( K_{\epsilon} \) for every \( \epsilon > 0 \) such that \( |f(x)| < \epsilon \) for all \( x \in \mathbb{R}^2 \setminus K_{\epsilon} \).

Answer: f \in C_0


Question: Frank alternately flips two coins. The first coin has a probability of \( \frac{2}{3} \) for heads and \( \frac{1}{3} \) for tails. The second coin has a probability of \( \frac{1}{4} \) for heads and \( \frac{3}{4} \) for tails. Starting with the first coin, what is the probability that Frank sees two consecutive heads before two consecutive tails?

Answer: \dfrac{13}{33}


Question: Find the limit superior of the expression \( \frac{\int_0^x t\ln(1+t) \, dt}{x^2} \) as \( x \to 0 \).

Answer: 0


Question: Given a homomorphism $f:A \to B$, where $A$ and $B$ are algebras in any variety, determine if it is true that if $f(S)$ is isomorphic to $S$ for all subalgebras $S$ of $A$, then $f$ is necessarily injective.

Answer: Yes


Question: Let $a, b, c$ be positive real numbers satisfying the equation \((1+a+b+c)\left(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=16\). Find the value of $a+b+c$. \(\boxed{?}\)

Answer: 3


Question: For any finite group \( G \), determine if there always exist subgroups \( A, B, \) and \( C \) such that \( A \) is a proper subgroup of both \( B \) and \( C \), and \(|A \cap B| \neq |A \cap C|\).

Answer: No


Question: Let $(X, Y)$ be a random pair with the density function given by \[ f(x,y) = \begin{cases} c(y-x)^2, & 0<x<y<1 \\ 0, & \text{elsewhere} \end{cases} \] for some constant $c > 0$. Determine the value of $c$.

Answer: 12


Question: Determine whether a function being of bounded variation on an interval implies that the function is bounded.

Answer: Yes


Question: Compute the limit: $$ \lim_{x\rightarrow 0} \frac{\left( \cosh x \right) ^{\sin x}-1}{\sinh x(\cos \left( \sin \left( x \right) \right)-1)} $$ without using the third derivative.

Answer: -1


Question: Determine the splitting field of the polynomial \(x^4 + x + 1\) over the finite field \(\mathbb{F}_2\).

Answer: \mathbb{F}_{2^4}


Question: Suppose \(a\) and \(b\) are positive real numbers such that \(a\sqrt{a} + b\sqrt{b} = 183\) and \(a\sqrt{b} + b\sqrt{a} = 182\). Find the value of \(\frac{9}{5}(a+b)\).

Answer: 73


Question: Find the minimum value of the function \(f(x) = \frac{1}{288}+\frac{2(\sin{x}+\cos{x})+576}{(\sin{x}+\cos{x})^2-1}\) in the interval \(\left(0,\frac{\pi}{2}\right)\) and round it to the nearest integer.

Answer: 579


Question: Consider an $n \times n$ matrix with all elements in the ring $\mathbb{Z}_n$. Let $p(x)$ be its minimal polynomial with coefficients $a_0, a_1, \ldots, a_L$, where $L$ is the degree of the minimal polynomial. Determine if each coefficient $a_i$ is less than or equal to $n$. Provide a justification for your answer.

Answer: Yes


Question: Suppose $f$ and $g$ are continuous, integrable, increasing, nonnegative-real-valued functions, each defined on some interval of nonnegative real numbers with left endpoint $0$. Given that $\int f = \int g$ and $\int_0^t f \leq \int_0^t g$ for all $t$ for which both functions are defined, does it follow that $\int_0^t g^{-1} \leq \int_0^t f^{-1}$ for all $t$ for which both $f^{-1}$ and $g^{-1}$ are defined? Answer yes or no.

Answer: Yes


Question: Let $z_1$ and $z_2$ be two nth roots of unity that form a right angle at the origin. If the roots are represented as $z_k = \cos\left(\frac{2k\pi}{n}\right) + i\sin\left(\frac{2k\pi}{n}\right)$ for $k = 1, 2$, determine the condition on $n$ such that the angle between $z_1$ and $z_2$ is $\frac{\pi}{2}$ radians.

Answer: 4


Question:
Given the system of equations below, where $q, r, s,$ and $t$ are distinct constants, find the sum $q + r + s + t$.

\[ \frac{1}{qa+1} + \frac{1}{qb+1} + \frac{1}{qc+1} + \frac{1}{qd+1} = 1 \]
\[ \frac{1}{ra+1} + \frac{1}{rb+1} + \frac{1}{rc+1} + \frac{1}{rd+1} = 1 \]
\[ \frac{1}{sa+1} + \frac{1}{sb+1} + \frac{1}{sc+1} + \frac{1}{sd+1} = 1 \]
\[ \frac{1}{ta+1} + \frac{1}{tb+1} + \frac{1}{tc+1} + \frac{1}{td+1} = 1 \]

Answer: 0


Question: Solve the absolute value ordinary differential equation: \( \left|y'(x)\right| + \left|y(x)\right| = 0 \). Determine if \( y(x) = 0 \) is the only solution.

Answer: y(x) = 0


Question: Derive a general formula for the summation: \( \sum_{k=1}^{n} \frac{1}{n+k} \).

Answer: H_{2n} - H_n


Question: Evaluate the integral: \[ \int_{1/4}^{1/2} \frac{dx}{\sqrt{x-x^2}} \]

Answer: \dfrac{\pi}{6}


Question: Determine the values of $r$ and $k$ for which the polynomial $x^{p^k}-1$ divides $x^{p^r}-1$ in the ring of polynomials $\\mathbb{F}_p[x]$, where $p$ is a prime number.

Answer: k \leq r


Question: Find all real values of $b$ such that the polynomial \( r(x) = (x-3)^2(x+5)^2 + b \) has exactly two double roots.

Answer: 0


Question: Evaluate the double integral \(\iint (x^2 \tan(x) + y^3 + 4) \, dx \, dy\) over the domain \(D = \{(x, y): x^2 + y^2 \leq 2\}\), which is a circle of radius \(\sqrt{2}\). Use either the given rectangular domain or convert to polar coordinates to solve the integral.

Answer: 8\pi


Question: Find the last three digits of \(7^{7^{7^7}}\).

Answer: 343


Question: Evaluate the sum \( \sum_{k=1}^n k\cdot(-3)^k{n \choose k} \) for \( n > 0 \) and express it as a function of \( n \).

Answer: -3n(-2)^{n-1}


Question: Evaluate the limit \( \lim_{x\to\infty} \left(1+\frac{1}{x}\right)^{-x} \) without using the formula \( \lim_{n\to\infty}\left(1+\frac{x_n}{a_n}\right)^{a_n} = e^x \).

Answer: \dfrac{1}{e}


Question: Find the Lebesgue measure of the Borel set \( B = \bigcup_{n=2}^{\infty} \left[ n - \frac{1}{n}, n \right] \).

Answer: \infty


Question: Determine whether the function \( f(x,y)=\frac{x^4+2y^4}{x^2+y^2} \), with \( f(0,0)=0 \), is continuously differentiable at the point \((0,0)\).

Answer: Yes


Question: Find the minimum value of the expression \(\frac{(1+x)(x+y)(y+z)(z+16)}{xyz}\) for positive real numbers \(x\), \(y\), and \(z\).

Answer: 81


Question: Let $(X,d)$ be a metric space with the topology induced by the metric $d$. Consider a sequence $\{x_n\}_{n\in \mathbb{N}}$ of points in $X$. Determine whether it is true that if there exists $x\in X$ such that $d(x,x_n)\rightarrow \infty$, then $\{x_n\}_{n\in \mathbb{N}}$ cannot be contained in any compact subset of $X$.

Answer: True


Question: Given a function \( g \in L^1([a,b]) \) with \( g \ge 0 \), determine if there exists a sequence of step-functions that converge monotonically to \( g \) in \( L^1([a,b]) \).

Answer: Yes


Question: Let $F$ be a field such that for every non-zero element $x \in F$, there exists a positive integer $k$ such that $x^k = 1$. Does this imply that $F$ is a finite field? Answer yes or no.

Answer: No


Question: If $A^{-1} = R^{-1}Q^T$ for an invertible matrix $A$ in its QR decomposition, does it follow that $Q^TAR = I$, where $I$ is the identity matrix?

Answer: No


Question: Use Stokes' theorem to evaluate \( \iint_S \operatorname{curl} F \cdot \hat{n}\, dS \) where \( F =\langle xyz, x, e^{xy} \cos(z)\rangle \) and \( S \) is the hemisphere \( x^2+y^2+z^2=25 \) for \( z \geq 0 \) oriented upward.

Answer: 25\pi


Question: Calculate the volume of the solid enclosed by the surfaces given by the equation \(64x^2 - 4y^2 + 16z^2 = 0\) and the plane \(y = 1\).

Answer: \dfrac{\pi}{24}


Question: Find the almost sure limit of the sequence \( Z_n = \frac{1}{n^2} (X_1 + \cdots + X_n) \), where \((X_i)_{i=1}^{\infty}\) are i.i.d. uniform \([0, 1]\) random variables.

Answer: 0


Question: Evaluate the integral \( \int_{0}^{\infty} e^{-x}x^n \, dx \) for non-negative integer values of \( n \).

Answer: n!


Question: Let $\mathcal{G}$ be a countable set of functions from the positive integers to the positive integers. Determine whether there exists a function $h:\mathbb{Z}^+\to\mathbb{Z}^+$ such that $g\in O(h)$ for every $g\in\mathcal{G}$.

Answer: Yes


Question: Let $V$ be the vector space of all real convergent sequences. Consider the transformation $T : V \rightarrow V$ defined as $T(x) = \{y_n\}$, where $y_n = b - x_n$ for $n \geq 1$ and $b$ is a fixed real number. Determine the range of the transformation $T$. \( \text{Express your answer as a set.} \)

Answer: V


Question: Evaluate the limit \(\lim_{n\to \infty} \{(3+2\sqrt 2)^n\}\), where \(\{x\}\) denotes the fractional part of \(x\).

Answer: 1


Question: Find the point on the plane containing point R and spanned by vectors RS and RQ that is closest to point Q, where P(2, -1, 3), Q(-1, 3, 0), R(0, -2, 5), and S(-3, 1, 2) are given points in a three-dimensional coordinate system.

Answer: (-1, 3, 0)


Question: Determine the signature of the quadratic form \( f(x, y, z) = xy + yz + xz \).

Answer: (1, 2)


Question: Calculate the limit of the expression \( x_n = \frac{\ln(1+\sqrt{n}+\sqrt[3]{n})}{\ln(1 + \sqrt[3]{n} + \sqrt[4]{n})} \) as \( n \to \infty \).

Answer: \dfrac{3}{2}


Question: Find the smallest odd number \( n \) such that \( 2^n - 1 \) is divisible by a twin prime pair \( (p, q) \), where \( 3 < p < q \), \( p \equiv -1 \pmod{24} \), and \( q \equiv 1 \pmod{24} \).

Answer: 315


Question: Suppose a function $f(x)$ is defined on the domain $\{x_1, x_2, x_3\}$, such that its graph consists of three points forming a triangle with an area of $32$. If the graph of $y = 2f(2x)$ also comprises three points, what is the area of the triangle formed by these points?

Answer: 32


Question: Given that $a$ and $b$ are the roots of the equation $x^2 - 7x - 1 = 0$, find the value of \( \frac{a^{23} + b^{23} + a^{19} + b^{19}}{a^{21} + b^{21}} \).

Answer: 51


Question: Evaluate the limit: $$ \lim_{x\to+\infty}\frac{(x^{3/2}-2x+1)^{2/3}-x}{\sqrt{x+1}} $$ without using L'Hôpital's Rule.

Answer: -\dfrac{4}{3}


Question: In a right triangle $\triangle ABC$ with $BC = AC = 1$, let $P$ be any point on side $AB$. Perpendiculars $PQ$ and $PR$ are drawn to sides $AC$ and $BC$, respectively. Denote the maximum of the areas of $\triangle BPR$, $\triangle APQ$, and rectangle $PQCR$ as $M$. Find the minimum value of $M$. \( \text{Express your answer as a single number.} \)

Answer: \dfrac{2}{9}


Question: Determine if the 2-adic expansion of \( \frac{2}{9} \) is equal to zero in the 2-adic number system.

Answer: No


Question: Determine the values of $p \in [1, +\infty[$ for which the function $f(x) = |x|$ belongs to the Sobolev space $W^{1,p}(]-1, 1[)$. Recall that $W^{m,p}(I)$ consists of functions $u$ such that $u$ and all its derivatives up to order $m$ are in $L^p(I)$, where $D^{\alpha} u$ denotes the derivative in the distribution sense.

Answer: [1, +\infty)


Question: Determine whether the maximum modulus principle holds for antiholomorphic functions.

Answer: Yes


Question: For $\alpha > 0$, consider a sequence of independent random variables $\left \{ {X_{n}}^{(\alpha )} \right \}_{n\geq 1}$, where $P({X_{n}}^{(\alpha )} = 1) = \frac{1}{n^{2\alpha}}$ and $P({X_{n}}^{(\alpha )} = 0) = 1 - \frac{1}{n^{2\alpha}}$. Define $S = \left \{ \alpha >0 :{X_{n}}^{(\alpha )} \xrightarrow{a.s.} 0, \ n\to \infty \right \}$. Determine the infimum of $S$ using the Borel-Cantelli lemma.

Answer: \dfrac{1}{2}


Question: Determine whether the function \( f(x) = \sum_{n=1}^\infty \frac{\sin(nx^2)}{1+n^4} \) has a continuous second derivative on \( \mathbb{R} \).

Answer: Yes


Question: Determine if the function \( f(x) = \frac{1}{(k-1)!}\int_0^1 e^{xu} (1-u)^{k-1} \, du \) is increasing for all real values of \( x \) and positive integers \( k \).

Answer: Yes


Question: Determine if the recurrence relation \(\sum\limits_{k=0}^{n}\left[{2k\choose k}{2\left(n-k\right)\choose n-k}\right]^3={n^2\choose 2n}\) can be solved using Zeilberger's algorithm.

Answer: Yes


Question: Calculate the line integral of the conservative vector field \( F_1(x,y) = (x^2+y, y^2+x) \) along the path from \((0, 0)\) to \((1, 0)\) and then to \((1, 1)\).

Answer: \dfrac{5}{3}


Question: Evaluate the limit: $$\lim_{n\to \infty}\left(\frac{\ n!}{(mn)^n}\right)^{\frac{1}{n}}$$ where $m$ is a real number.

Answer: \dfrac{1}{e m}


Question: Let $L/K$ be a finite extension of finite fields. Is it true that the image of the norm map $N_{L/K}(L)$ is equal to $K$?

Answer: Yes


Question: Find the natural number \( n \) such that the field \( \mathbf{Q}(z_{5},z_{7}) \) is equal to \( \mathbf{Q}(z_{n}) \), where \( z_{k} = e^{2\pi i / k} \) is a primitive \( k \)th root of unity.

Answer: 35


Question: In a Noetherian ring $R$, if a prime ideal $p$ contains only zero divisors, does it necessarily correspond to an associated prime? In other words, is there an element $x \in R$ such that $p = \text{Ann}(x)$?

Answer: Yes


Question: Determine whether the sequence \( x_n = \frac{1}{(2n+1)^2} + \frac{1}{(2n+2)^2} + \ldots + \frac{1}{(2n+2n)^2} \) converges to zero.

Answer: 0


Question: Find the value of the eighth derivative of \( \arctan\left(\frac{x^3}{7}\right) \) at \( x=0 \) using the Taylor series expansion of \( \arctan(x) \).

Answer: 0


Question: Calculate the surface integral of the vector field \( F=(y(x^2+y^2),-x(x^2+y^2)+\sin(x^2+y^2),e^{(x^2+y^2)}) \) over the portion of the paraboloid \( z=x^2+y^2 \) that lies below the plane \( z=1 \), oriented upward.

Answer: \pi(e - 1)


Question: Let \( \vec{r}=x\hat{i}+y\hat{j}+z\hat{k} \) and let \( r = \|\vec{r}\| \). Let \( \vec{F} = r^p\vec{r} \). Determine all values of \( p \) for which \( \text{div} \vec{F} = 0 \).

Answer: -3


Question: Find the closed-form expression for the sum \( \sum_{k=0}^{n} \binom{n}{k} n^k \).

Answer: (n + 1)^n


Question:
Evaluate the limit of the following integral as \( T \to \infty \), given that \( \omega_r \) is a constant:
\[ \lim_{ T \rightarrow \infty }\frac{1}{T}\int_{-T/2}^{-\tau} \cos (\omega_r(t+\tau)) \, dt \]

Answer: 0


Question: Determine the minimum number of terms required in the Taylor series \( \sum_{n=0}^{\infty}\frac{(-1)^{n} x^{2n + 1}}{n!} \) to approximate \( f(1) = 1e^{-1} \) to within an accuracy of \( 10^{-6} \). Specifically, find the smallest integer \( k \) such that the partial sum \( s_{k, x =1} = \sum_{n = 0}^{k} \frac{(-1)^{n} }{n!} \) satisfies \( |s_{k} - e^{-1}| < 10^{-6} \).

Answer: 9


Question: A 19-digit binary string contains all possible 4-digit binary strings exactly once. If this string starts with 1111, what are the last four digits?

Answer: 0111


Question: Given a field $k$ and a finite-dimensional $k$-algebra $A$ with Jacobson radical $J(A)$, if $A/J(A) \cong k^n$ for some $n \geq 1$, does $A$ possess an ideal with $k$-dimension $1$? Provide a proof or counterexample.

Answer: Yes


Question: Determine whether the equation \( b^{a \sum_x p(x) \log_b p(x)} = \prod_x p(x)^{a p(x)} \) holds true for any base \( b \) and exponent \( a \).

Answer: True


Question: Evaluate the limit as $n$ approaches infinity of the expression: $$\frac{\ln(1+\sqrt{n}+\sqrt[4]{n})}{\ln(1 + \sqrt[4]{n} + \sqrt[5]{n})}$$

Answer: 2


Question: Evaluate the integral \( \int_0^{2\pi} \frac{1}{\sin^4x + \cos^4 x} \,dx \) and verify if the result is \( 2\pi\sqrt{2} \).

Answer: 2\pi\sqrt{2}


Question: Does there exist a non-abelian finite group in which more than 50% of the elements have order 2?

Answer: Yes


Question: Is the subset of \(\mathbb{R}^4\) defined by the equations \(x^2 + y^2 + z^2 + t^2 = 4\) and \(x - y + z - t = 2\) compact?

Answer: Yes


Question: Determine whether it is possible to inscribe a rectangle in an ellipse such that the sides of the rectangle are not parallel to the major or minor axes of the ellipse. Answer with 'Yes' or 'No'.

Answer: No


Question: Given that \(a^2 + b^2 + c^2 = 1\) and \(a^3 + b^3 + c^3 = 1\) for real numbers \(a, b, c\), determine whether it is necessary that two of the numbers \(a, b, c\) must be zero.

Answer: Yes


Question: Consider the graph $G$ obtained by removing a single edge from the complete bipartite graph $K_{7,8}$. Determine the number of spanning trees in the complement graph $\overline{G}$. Provide your answer as a single integer.

Answer: 4405854208


Question: Given a circle and an ellipse, what is the probability that a point sampled uniformly from within the region enclosed by the two shapes will be closer to the circle than to the ellipse?

Answer: \dfrac{1}{2}


Question:
Find the value of \( p \) that satisfies the equation:
\[ \frac{d}{dp}\left[2\int_0^1(p-y)^2{y^{x-1}(1-y)^{-x}}dy\right]=0 \]

Answer: x


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{\cos(x^3)-\cos^3(x)}{x^3\ln(\sin x)} \] without using L'Hopital's rule.

Answer: -\infty


Question: Find the maximum value of $x+y+z$ for real numbers $x$, $y$, and $z$ that satisfy $x^2+y^2+z^2=2$ and $xyz \ge -2$. \( \boxed{} \)

Answer: \sqrt{6}


Question: Evaluate the contour integral \(I=\oint_{\gamma}\frac{dz}{(z^2+4)^2}\) where \(\gamma\) is the circle \(|z-i|=2\). Use the residue theorem to calculate the integral.

Answer: \dfrac{\pi}{16}


Question: Consider the sequence defined by \(a_n = \frac{F_n + F_{n-1}}{F_n - F_{n-1}}\), where \(F_n\) is the \(n\)th term of the Fibonacci sequence. Determine whether this sequence is bounded and, if so, find an upper bound.

Answer: 5


Question: Determine the number of nonnegative integer solutions to the equation \(x_1 + x_2 + x_3 + \dots + x_n = r\) such that \(x_1 \leq x_2 \leq x_3 \leq \dots \leq x_n\).

Answer: p_{\leq n}(r)


Question: In the Game of Life, determine if there exists a structure that remains stable unless a glider is introduced, at which point it launches a spaceship.

Answer: Yes


Question: Determine whether there exist functions $g_k$ such that $D_k = g_k(R_1, \dots, R_k)$ for $1 \leq k \leq n$ under the same setting as the original question.

Answer: Yes


Question: Let $X$ be a continuous random variable with a uniform distribution on the interval $[-1, 1]$. Define $Z = \log(|X|)$. Determine the probability density function of the random variable $Z$. \( \)

Answer: e^z


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{x}{b^2\sqrt{x^2-b^2}} \, dx \), where \( b > 0 \).

Answer: 0


Question: Find the coefficient of \(x^4\) in the expansion of \(\frac{1}{e^{2x} \cdot (1-x)}\).

Answer: \dfrac{1}{3}


Question: Calculate the radius of the symmetric random walk on \(\mathbb{Z}\), given by \(\limsup_k (p^{(k)}(0,0))^\frac{1}{k}\), where \(p^{(k)}(0,0)\) denotes the probability of returning to 0 in \(k\) steps, and the transition probabilities are \(p(n,n+1)=p(n,n-1)=\frac{1}{2}\).

Answer: 1


Question: Given a faithful action of a group $G$ on a set with $n$ elements, determine whether the order of $G$ is less than or equal to $n!$. Justify your answer.

Answer: Yes


Question: Given that $\alpha, \beta, \gamma, \delta$ are the roots of the polynomial equation $x^4 + 4x^3 - 6x^2 + 7x - 9 = 0$, calculate the value of the product $\prod(1 + \alpha^2)$. Express your answer in simplest form.

Answer: 13


Question: Determine the number of $3 \times 3$ matrices $\rm A$ with integer entries such that $\rm AA^{\top} = I_3$, where $I_3$ is the $3 \times 3$ identity matrix.

Answer: 48


Question: Consider an even cycle graph $C_{2n}$. Can contracting an edge in this graph increase its chromatic number by more than one?

Answer: No


Question: Given the parametric equations \(x(\theta)=5\cos \theta\), \(y(\theta)=5\sin \theta\), and \(z(\theta)=\theta\), let \(L(\theta)\) be the arc length from the origin to the point \(P(x(\theta), y(\theta), z(\theta))\), and \(D(\theta)\) be the distance from the origin to \(P\). If \(L(\theta_0)=\sqrt{26}\), find \(D(\theta_0)\).

Answer: \sqrt{26}


Question: Let $G$ be a subset of $\mathbb{R^n}$ and $T$ be an $n \times n$ regular matrix. Define $TG=\{ Tx \mid x \in G \}$. Determine whether the following proposition is true: If $G$ is an open set, then $TG$ is an open set.

Answer: True


Question: How many positive integer solutions less than 71 are there for the equation $x^7 \equiv a \pmod{71}$, given that $a$ is an integer not divisible by 71?

Answer: 7


Question:
Evaluate the integral by changing the order of integration:
\[ \int_{0}^{1} \int_{1-\sqrt{1-y^2}}^{1+\sqrt{1-y^2}} \frac{(x^2+y^2)x}{\sqrt{4x^2-(x^2+y^2)^2}} \, dx \, dy \]

Answer: \pi


Question: Are any two norms on a finite-dimensional vector space equivalent?

Answer: Yes


Question: Determine the maximum number of intersection points possible when three distinct ellipses and four distinct hyperbolas are drawn on a plane, given that each pair of ellipses can intersect in at most 2 points, and each pair of an ellipse and a hyperbola or two hyperbolas can intersect in at most 4 points.

Answer: 78


Question: Find a sequence $b_n$ such that $a_n = \frac{(n!)^n}{n^{n^2}} < b_n$ for all $n$, and $\lim_{n \to \infty} b_n = 0$.

Answer: 2 e^{-n^2/2}


Question: Suppose that $n$ is a given even number, and let $a_1, a_2, \ldots, a_n$ be nonnegative real numbers such that \(a_1 + a_2 + \cdots + a_n = 1.\) Determine the maximum possible value of the expression: \[ \sum_{1 \leq i < j \leq n} \min\left\{ (i-j)^2, (n+i-j)^2 \right\} a_i a_j. \]

Answer: \dfrac{n^2}{16}


Question: Evaluate the integral \( \int_{-\infty}^{\infty} x^3 e^{-x^2/2} \, dx \).

Answer: 0


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

Answer: 1


Question: Calculate the line integral of the function \( f(x, y) = x^2 - y^2 \) along the boundary of a parallelogram with vertices \( A(1, 1) \), \( B(2, 1) \), \( C(3, 3) \), and \( D(2, 3) \).

Answer: -\dfrac{4}{3}


Question: Find the limit of the function \( f(x, y) = |x|^{(1/|y|)} \) as \((x, y)\) approaches the origin \((0, 0)\).

Answer: 0


Question: Using generating functions, find the number of ways to distribute 5 distinct balls among 4 people such that none of them receives exactly 3 balls.

Answer: 664


Question: Determine the orthogonal complement of the subspace \(W = \{f:[0,1] \rightarrow \mathbb{R} ; f \text{ is continuous}, f(1)=0\}\) in the vector space \(V = \{f:[0,1] \rightarrow \mathbb{R} ; f \text{ is continuous} \}\) with the inner product \( \langle f, g\rangle = \int_0^1 f(x) g(x) \, dx \).

Answer: \{0\}


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

Answer: -1


Question: Evaluate the integral: $$\int_{0}^{\infty }\!{\frac {\cos \left( 2x \right)-1 }{{x}^{2}}} \,{\rm d}x$$

Answer: -\pi


Question: Evaluate the limit: $$\lim_{n\to\infty} \frac{\left(\sum_{r=1}^n\frac{1}{\sqrt{r}}\right)\left(\sum_{r=1}^n\sqrt{r}\right)}{\sum_{r=1}^n r}$$ using the concepts of generalized harmonic numbers and Abel's summation.

Answer: \dfrac{8}{3}


Question: Is it possible for an unbounded function $g(x)$ to satisfy the equation $$\int_{-\infty}^{\infty} e^{-x^2} g(x) \, dx = 0?$$

Answer: Yes


Question: Find the smallest rational number $r$ such that the sum of the reciprocals of the first $n$ triangular numbers, $S_n$, is less than $r$ for all positive integers $n$. What is this rational number?

Answer: 2


Question: Compute $H^2(D)$ using the Mayer-Vietoris long exact sequence for cohomology, where $D$ is the union of a 2-dimensional projective space $\mathbb{R}P^2$ and a 2-dimensional manifold $M$ with $\partial M = S^1$. Given that $H^1(M\cap \mathbb{R}P^2) = H^1(S^1) = \mathbb{Z}$, $H^2(\mathbb{R}P^2) = \mathbb{Z}_2$, and $H^2(S^1) = 0$, find $H^2(D)$. \(\boxed{?}\)

Answer: \mathbb{Z}_2


Question: Evaluate the limit: $$\lim_{x\to 0_+}\left(\frac{1}{1+\sqrt{x}}\right)^{\frac{1}{\sqrt{x}}}$$

Answer: \dfrac{1}{e}


Question: Determine whether \( \lim_{n \to \infty}{\frac{p_n}{p_{n+1}}} < 1 \), where \( p_n \) denotes the nth prime number.

Answer: No


Question: Given the function \( h(x) = \int_{0}^{\sin 2x} \sin^{-1}(t) \,dt - \int_{0}^{\sin x} \sin^{-1}(t) \,dt \), find the value of \( h'\left( \frac{\pi}{2}\right) \).

Answer: 0


Question: Given a function \( f \) such that \( \lim_{x\to0^+} f(x)=\infty \), determine if there exists a function \( \varphi \) such that \( \varphi (x) \to 0 \) as \( x \to 0^+ \) and \( \varphi (x) f(x) \to \infty \).

Answer: Yes


Question: Let \( \|x\|= \max|x(t)|+\max|x'(t)| \) be a norm on \( C'[a,b] \). Let \( f(x)=x'(\frac{a+b}{2}) \) be a bounded linear functional on \( C'[a,b] \). Find \( \|f\| \).

Answer: 1


Question: Determine the maximal number of elements in a sigma-algebra generated by a collection of N subsets of a set X.

Answer: 2^{2^N}


Question: Solve the equation \[ \frac{3x^2 + 4x + 5}{\sqrt{5x^2 + 4x + 3}} + \frac{8x^2 + 9x + 10}{\sqrt{10x^2 + 9x + 8}} = 5 \] for \(x \in \mathbb{R}\), given that \(x = -1\) is a solution.

Answer: -1


Question: Determine whether there exists a function $f(x)$, which is both $L^1$ and Riemann-integrable, such that the limit \( \lim_{\omega \to \infty} \int_{-\infty}^{+\infty} f(x) e^{-i x \omega} \, dx \) exists and is non-zero.

Answer: No


Question:
Evaluate the limit of the sequence \( \{y_n\} \) defined recursively by:

\[ y_n = y_{n-1} \cdot \left(3-\frac{\cos^{-1}(n) \cdot n^2 - 2}{4 + n^2}\right) \]

with the initial condition \( y_0 = 1 \).

Answer: \infty


Question: Let \(a, b \geq 2\) be integers and \(N\) the image of the homomorphism \(\mathbb{Z}_a \rightarrow \mathbb{Z}_{ab}\) given by \([k]_a \mapsto [kb]_{ab}\). Identify the quotient group \(\mathbb{Z}_{ab}/N\) with a familiar group.

Answer: \mathbb{Z}_b


Question: For a conditionally convergent series \(\sum_{n=1}^\infty a_n\), determine if there exists an integer \(k \geq 2\) such that the subseries \(\sum_{n=1}^\infty a_{kn}\) converges.

Answer: Yes


Question: Determine whether the following limit approaches infinity or zero: \[ \lim_{n \to \infty}\frac{n!}{n^\sqrt n}. \]

Answer: \infty


Question: Is it possible to define a function $g(x,y)$ in the region $0 < x < L$, $0 < y < L$ with periodic boundary conditions $g(0,y) = g(L,y)$ and $g(x,0) = g(x,L)$ such that the integral \( \int_0^L \int_0^L \partial_x g(x,y) \, dx \, dy \) is non-zero?

Answer: No


Question: Evaluate the integral \( \int_{-1}^{1} \frac{1}{x} \, \mathrm{d}x \) by considering the limit as \( \epsilon \) approaches \( 0^{+} \), where the integral is split into three parts: \( \int_{-1}^{-\epsilon} \frac{1}{x} \, \mathrm{d}x \), \( \int_{-\epsilon}^{\epsilon} \frac{1}{x} \, \mathrm{d}x \), and \( \int_{\epsilon}^{1} \frac{1}{x} \, \mathrm{d}x \).

Answer: 0


Question: Find an element $x_\theta \in \mathfrak{sl}_{n+1}$ such that $[h, x_\theta] = (\alpha_1 + \cdots + \alpha_n)(h) x_\theta$ for all $h \in \mathfrak{h}$, where $\mathfrak{h}$ is the Cartan subalgebra of $\mathfrak{sl}_{n+1}$ and $\alpha_i$ are the simple positive roots.

Answer: E_{1,n+1}


Question: What conditions must the elements of a $3 \times 3$ matrix $\mathbf{R}$ satisfy for the matrix to be orthogonal, thereby preserving the length of any vector $\mathbf{A}$?

Answer: \mathbf{R}^\top \mathbf{R} = \mathbf{I}


Question: Given a knot diagram $K$ in $\mathbb{R}^3$, if $K$ admits only trivial colourings with any quandle, can it be concluded that $K$ is the unknot? Answer with 'Yes' or 'No' and justify your answer.

Answer: Yes


Question: In a Poisson process with intensity \( \lambda > 0 \), the arrival times are \( W_1, W_2, \ldots \). Given that there are exactly 7 arrivals in the time interval \((0, T]\), find the expected value of the second arrival time \( W_2 \).

Answer: \dfrac{T}{4}


Question: Find the vector $x \in \mathbb{R}^n$ that minimizes the function $J(x) = x^T A x - 2 x^T b$, where $A \in \mathbb{R}^{n \times n}$ is a symmetric matrix and $b \in \mathbb{R}^n$. Assume $A$ is positive definite.

Answer: A^{-1}b


Question: Determine the number of distinct elements in the quotient group \(\mathbb{R}/\mathbb{Q}\), where \(\mathbb{R}\) is the set of real numbers and \(\mathbb{Q}\) is the set of rational numbers.

Answer: 2^{\aleph_0}


Question: Evaluate the contour integral: $$ \int _0^{\infty} \frac{\ln x}{x^2+1} \, dx $$

Answer: 0


Question: Evaluate the residue at $z = 0$ of the function $(1 - e^{-z})^{-n}$ for $n \in \mathbb{Z}^+$, where $\mathbb{Z}^+$ represents the set of positive integers. Given that for $n = 1$, the residue is $1$, find the residue for $n > 1$.

Answer: 1


Question: Let $f_n$ be Riemann integrable on $[a,b]$ for all $n$, and let $f, g \in C[a, b]$ be continuous functions. Suppose $f_n \rightarrow f$ in $L^2$ and $f_n \rightarrow g$ pointwise. Is it necessarily true that $f = g$?

Answer: f = g


Question: Evaluate the integral \( \int_{\gamma}\frac{dz}{z} \) where \( \gamma \) is a path from \(-i\) to \(i\) with its image contained in \( \mathbb{C} \setminus \mathbb{R}^- \). Determine if the integral converges.

Answer: i\pi


Question: Calculate the sum \( \sum_{A,B\in\mathcal{P}(\Omega)}\operatorname{card}(A\cap B) \) where \( \Omega \) is a set with \( n > 0 \) elements, and \( \mathcal{P}(\Omega) \) denotes the power set of \( \Omega \).

Answer: n \cdot 4^{n-1}


Question: Evaluate the limit: \[ \lim_{n \to \infty}\left(\frac{12^n-6^n}{3^n-6^n} +\frac{9^n-3^n+\cos(3^n)}{(-1)^n+9^n}\right) \]

Answer: -\infty


Question: Let $X$ be a metric or topological space and $Y \subseteq X$ be a subspace. Let $V \subseteq Y$ be open with respect to the subspace topology on $Y$. Does there always exist an open set $U$ in $X$ such that $U \cap Y = V$? Answer yes or no.

Answer: Yes


Question: Evaluate the integral \( \int_0^\infty \frac{t^2 - \sin^2 t}{t^4} \, dt \).

Answer: \dfrac{\pi}{3}


Question: Determine whether every radical ideal in a Noetherian ring can be expressed as an intersection of prime ideals. Provide a proof or counterexample.

Answer: Yes


Question: Determine the limit of the difference between consecutive positive solutions of the equation \(x = \tan(x)\) as \(n\) approaches infinity.

Answer: \pi


Question: Is the set \( T = \{a^4 + b^4: a,b \in \Bbb N\} \) closed under multiplication?

Answer: No


Question: Determine the minimum number of spheres required to completely shield a point source of light from all directions.

Answer: 4


Question: Does a Lie group $G$ act on its co-adjoint orbit $\Omega$, which is a symplectic manifold, by symplectomorphisms?

Answer: Yes


Question: Determine whether the expression \( \frac{f(x_{0} + h ) - 2 f(x_{0} + h/2 ) + f(x_{0} )}{h^2 / 4} - f^{(2)}(x_{0}) \) is \( O(h^2) \).

Answer: O(h)


Question: Compute the limit \( \lim_{x \to 0} \frac{x^2e^{x^2} + \frac{1}{2}\log(1+2x^2) - 2x^2}{1 + \arctan(x^6) - \cos(x^3)} \) using Taylor series expansions around the origin.

Answer: \dfrac{11}{9}


Question: Compute the limit: $$\lim_{n\to \infty} \frac{\ln^2n}{n}\sum_{k=2}^{n-2}\frac{1}{\ln k\ln(n-k)}$$

Answer: 1


Question: Determine the number of real solutions to the equation \[ \left(|x^2-5x+4|^{\log{x^2}}\right)^{-1+\log x}=|x^2-5x+4|^{1+\log{\frac{1}{x}}} \]

Answer: 6


Question: Evaluate the limit: $$\lim_{n \to \infty}\frac{2^{-n^2}}{\sum_{k=n+1}^\infty 2^{-k^2}}$$

Answer: \infty


Question: Evaluate the integral \( \frac{2}{\pi}\int_{-\pi}^{\pi}{\cos\left(4x\right) \over \sin\left(x/2\right)}\,\mathrm{d}x \) using complex analysis.

Answer: 0


Question: Consider a system of linear equations with rational coefficients. If the system has a solution over an extension field of the rationals, does it necessarily have a solution over the rationals? Answer yes or no.

Answer: Yes


Question: Find the value of \( x \) such that \( 341^{101} + 309^{201} \equiv x \pmod{1000} \).

Answer: 650


Question: Find the limit of the sequence as \( n \to \infty \): \[ \lim_{n \to \infty} \frac{\sum_{k=1}^n k^{1/3}}{n^{4/3}}. \]

Answer: \dfrac{3}{4}


Question: Calculate the limit: \( \lim_{x \to +\infty} \left( x \sin(x) + \cos(x) - x^2 \right) \)

Answer: -\infty


Question: Calculate the volume enclosed by the paraboloid \(x^2 + y^2 = 4z\) and the plane \(z + x = 5\).

Answer: 72\pi


Question:
Evaluate the limit:

$$\lim_{n\to\infty} \frac{a_{n}}{\left(\frac{n}{e}\right)^{n} 4^{n} \sqrt{2}}$$

where $a_{n} = 1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1)$ and can be expressed as $\prod^{n}_{k=1} (2k-1)$. Use Stirling's formula, $n! \approx \left(\frac{n}{e}\right)^{n}\sqrt{2\pi n}$, to find the value of the limit.

Answer: 0


Question: Is there an algorithm that allows any element \( \alpha \in L(x,y) \) to be expressed as \( \sum_{\alpha} x_{\alpha} h_{\alpha} \), where \( h_{\alpha} \) is the Hall basis of the free Lie algebra \( L(x,y) \)?

Answer: Yes


Question: Determine the fixed point of the operator \( T(f) = \int_0^x f(y) \left(\frac{1}{2} + \frac{y}{2}\right) \, dy \) using the Banach fixed point theorem.

Answer: 0


Question: Calculate the conditional probability \( P (Y<X \, | \, X>\frac{a+b}{2}) \) given two independent random variables \(X\) and \(Y\) that are uniformly distributed on the interval \([a, b]\).

Answer: \dfrac{3}{4}


Question: Determine the number of functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the equation \( f(x+y)+f(xy)=f(x)f(y)+1 \) for all real numbers \( x \) and \( y \).

Answer: 2


Question: Determine whether the Gram-Schmidt process can be applied to orthogonalize a linearly independent sequence in an infinite-dimensional Hilbert space.

Answer: Yes


Question: Evaluate the limit: $$\lim \limits_{z \to i}\frac{\log(z)}{z-i}$$ using elementary complex analysis techniques.

Answer: \infty


Question: Determine the conditions that a function $g(x)$ must satisfy for there to exist a real-valued function $f(x)$ such that the functional equation \[ f(x+y) = f(x) + f(y) + g(xy) \] holds for all real numbers $x$ and $y$. Provide the necessary and sufficient conditions on $g(x)$.

Answer: g(x) = ax + b


Question: Consider the bounded solution $u(x,t)$ of the heat equation \(\frac{\partial u}{\partial t} -\frac{\partial^2 u}{\partial x^2} = 0\), where \(u(x, 0) = \frac{e^{2x} - 1}{e^{2x} + 1}\). Determine \(\lim_{t \to \infty} u(1, t)\).

Answer: 0


Question: What is the smallest number of subsets, $n$, into which the set of all integers can be partitioned such that none of the subsets contain an infinite arithmetic progression?

Answer: 2


Question: Determine a rough Big O bound for the summation: \( \sum_{x=2}^{\lg\lg N} 2^{2^x - 1} \).

Answer: O(N)


Question: Let $R$ be an integral domain, and $p \in R$ be a prime element. Determine whether $p$ is also a prime element in the polynomial ring $R[x]$. Provide a justification for your answer.

Answer: Yes


Question: If $F(x)$ is an indefinite integral of $f(x)$ and is differentiable at a point $c$ within the interval $I=[a,b]$, is it always true that $f$ is continuous at $c$? Provide a justification for your answer.

Answer: No


Question: Determine the nature and number of roots for the equation \[x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}\] after making the substitution \(t = \log_{2}{x}\).

Answer: 3


Question: Determine whether there exists a set $E$ in $\mathbb{R}^{m}$ for $m\geq2$ with respect to the Euclidean topology such that: (1) The interior of $E$ is empty, (2) The Lebesgue measure of $E$ is positive, and (3) $E$ is connected.

Answer: Yes


Question: Calculate the length of the curve formed by the intersection of the unit sphere \(x^2+y^2+z^2=1\) and the plane \(x+y=1\).

Answer: \pi \sqrt{2}


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

Answer: \infty


Question:
Given the system of equations:
\[\begin{align*}
x + y &= z^2 + 1 \\
y + z &= x^2 + 1 \\
x + z &= y^2 + 1
\end{align*}\]
find the value of \(2x + 3y + 4z\).

Answer: 9


Question: Using Rouche's theorem, determine the number of zeros of the polynomial \( g(z) = z^6 + 4z^3 + 5 \) inside the circle \( |z| = 2 \).

Answer: 6


Question: Evaluate the sum \(\sum_{k=0}^n \binom{n}{k} \frac{k}{n} x^k (1-x)^{n-k}\) using the binomial theorem. Simplify the expression and explain how the term \(\frac{k}{n}\) affects the evaluation process.

Answer: x


Question: Find the extreme values of the function \(f(x,y,z)=x^2+y^2+z^2\) subject to the constraint \(z=xy+2\).

Answer: 3


Question: Find the probability density function (PDF) of the random variable $X_2 = \max(U_1, U_2, U_3)$, where $U_1, U_2, U_3$ are independent and uniformly distributed on the interval $[0,1]$. Express your answer in terms of $x$ for $0 \leq x \leq 1$. \( \boxed{\text{PDF expression}} \)

Answer: 3x^2


Question: Can the Gram-Schmidt process be used to construct an orthonormal basis from a linearly independent subset of an inner product space?

Answer: Yes


Question: Calculate the integral \( \int_0^{2\pi} \sin^2x\ \cos^4x\ dx \) by hand without using the substitution \( t=\tan(\frac{x}{2}) \).

Answer: \dfrac{\pi}{8}


Question: Let $G$ be a noncyclic group of order $4p$, where $p$ is an odd prime with $p \geq 3$. Prove that $G$ contains an element of order $2$. \(\text{True or False?}\)

Answer: A


Question: Find the minimal polynomial of \( \alpha = 2^{\frac{1}{4}} \) over \( \mathbb{Q}(\beta) \), where \( \beta = e^{\frac{2\pi i}{5}} \), and factor it into irreducibles in \( \mathbb{Q}(\beta)[X] \).

Answer: X^4 - 2


Question: Simplify the expression: $$\sum_{k=1}^{2004}\cos^{2}\left(\frac{k\pi}{2\cdot2005}\right)$$

Answer: 1002


Question: Determine whether there exists an unbounded above function \( f: \mathbb{R} \to \mathbb{R} \) such that there is some \( M > 0 \) for which \( f(x) < \log(x) \) for all \( x > M \).

Answer: Yes


Question: Let $X_1, X_2, \ldots$ be a series of independent random variables that converge weakly to a distribution $\mathcal D$ with finite mean and variance. Suppose that $Y_1, Y_2, \cdots \sim_\text{i.i.d.} \mathcal D$, and the normalized partial sums $(Y_1+\cdots + Y_n)/\sqrt n$ converge weakly to a Gaussian distribution $\mathcal G$. Does it hold that $(X_1+\cdots + X_n)/\sqrt n$ converge weakly to $\mathcal G$? Provide a justification for your answer.

Answer: No


Question: Determine whether there exists a function from \( \mathbb{R}^2 \) to \( \mathbb{R} \) that is not continuous at the origin, but whose restriction to every polynomial curve through the origin is continuous.

Answer: Yes


Question: Calculate the third moment about the mean, \(E(X^3)\), for a Poisson distribution with parameter \(\lambda\).

Answer: \lambda


Question: Find the Laurent series expansion of the function \( f(z) = \frac{1}{1-\cos z} \) around \( z_0=0 \). Determine the residue of \( f(z) \) at \( z=0 \).

Answer: 0


Question: Evaluate the double integral \(\int_0^{1/2}\int_0^{\sqrt{1-x^2}}xy\sqrt{x^2+y^2}\,dy\,dx\) using polar coordinates, given that \(x^2+y^2=r^2\).

Answer: \dfrac{19}{480}


Question: Determine whether the cardinality of $\mathbb{R}^\infty$, the infinite-dimensional Euclidean space, is equal to $\beth_1$. Note: $\beth_1$ represents $2^{\aleph_0}$, where $\aleph_0$ is the cardinality of the set of natural numbers.

Answer: Yes


Question: Suppose the series \( \sum_{n=1}^{\infty} \frac{a_n}{n} \) converges to a finite number \( a \), where \( a_n \geq 0 \) for all positive integers \( n \). Determine if the limit \( \lim_{n \to \infty} \sum_{i=1}^{n} \frac{a_i}{i+n} \) converges to 0.

Answer: 0


Question: Calculate the contour integral \( \int_C \frac{z-3}{(z+1)(z-2)} \, dz \) along the circle \( C \) of radius 5 centered at the origin, oriented counterclockwise, using a shortcut method.

Answer: 2\pi i


Question: Suppose that $a$ has order $h$ modulo $p$ and $a\overline{a} \equiv 1 \pmod{p}$. Prove that $\overline{a}$ also has order $h$ modulo $p$.

Answer: h


Question: Determine the value of $x$ such that $2014^{2014} \equiv x \pmod{1000}$ using Euler's theorem and the Chinese remainder theorem.

Answer: 16


Question:
Let $a,b,c$ be distinct positive integers such that the matrix $A=\text{diag}(a, b, c)$ satisfies $\det A = 10$. Define the sets:

$$D:=\{(x_1,x_2,x_3)\in \mathbb{Z}^3 : x_1^2+x_2^2+x_3^2\leq 100\},$$
$$E:=\{(x_1,x_2,x_3)\in \mathbb{Z}^3 : \frac{x_1^2}{a^2}+ \frac{x_2^2}{b^2}+\frac{x_3^2}{c^2}\leq 1\}.$$

Which of the following statements is true about the number of elements in the sets $D$ and $E$?
1. $|D|=|E|$
2. $|D|<|E|$
3. $|D|>|E|$
4. The relationship between $|D|$ and $|E|$ cannot be determined.

Answer: C


Question:
Given the formulas for the surface area \( S_n(1) \) and volume \( V_n(1) \) of an \( n \)-sphere with radius 1:
\[ S_n(1) = \frac{n \pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}+1\right)} \quad \text{and} \quad V_n(1) = \frac{\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}+1\right)} \]
where \( \Gamma(z) \) is the gamma function defined as:
\[ \Gamma(z) = \int_0^\infty e^{-x} x^{z-1}\, dx \]
Find the integers \( v \) and \( s \) such that \( S_s(1) \) and \( V_v(1) \) are maximum. Determine the value of \( v-s \).

Answer: -2


Question: Calculate the limit \[ l = \lim_{x\to 0}\frac{e^{\sin x}-e^{\tan x}}{e^{\sin 2x}-e^{\tan 2x}} \]

Answer: \dfrac{1}{8}


Question: If sequences $(u_n)$ and $(v_n)$ converge to $l$ and $m$ respectively, with $m < l$, is it true that there exists an $N \in \mathbb{N}$ such that for all $n \ge N$, $u_n > v_n$?

Answer: Yes


Question: Find the last two digits of \(9^{9^9}\).

Answer: 89


Question: Compare the asymptotic growth rates of the functions \(f(n) = e^{\ln^2(n)}\) and \(g(n) = e^{(\ln( \ln(n))-\ln(\ln(2 )))n}\). Determine which function grows faster as \(n\) approaches infinity.

Answer: g(n)


Question: Given a metric space $X$, if for every continuous function $f: X \to \mathbb{R}$, the image $f(X)$ is an interval, is $X$ necessarily connected? Answer yes or no.

Answer: Yes


Question: Find the coefficient $a_4$ in the power series expansion of $\dfrac{1}{1+\ln(1-x)}$ around $x=0$: $$\dfrac{1}{1+\ln(1-x)}=\sum\limits_{n=0}^{+\infty}a_n x^n$$

Answer: \dfrac{11}{3}


Question: Let $Y$ be a non-negative random variable, not necessarily integrable. Show that \[ \lim_{n\rightarrow \infty }\frac{1}{n}\int_{Y<n}Y \, dP=0. \]

Answer: 0


Question: If two graphs, G1 and G2, are isomorphic and share the same adjacency matrix, does a symmetry present in G2 imply the existence of an equivalent symmetry in G1?

Answer: Yes


Question: Determine the number of group homomorphisms from the symmetric group \( S_3 \) to the quotient group \( \mathbb{Z}/6\mathbb{Z} \).

Answer: 2


Question: Given integers \(a, b\) and odd prime numbers \(p, q\), find the total number of distinct integer solutions \((a, b)\) to the equation \(a^2 + 2b^2 = p^{11}q^{13}\).

Answer: 336


Question: Let $(G, \cdot)$ be a centerless group with $n$ elements, where $n$ is an integer not divisible by $4$. Determine whether there exist distinct elements $a, b \in G$ such that their centralizers are equal, i.e., $C(a) = C(b)$. Provide a justification for your answer.

Answer: Yes


Question: Given two bounded sequences $a_n$ and $b_n$, where $a_n$ converges to zero and $b_n$ diverges, determine whether the product $a_nb_n$ converges.

Answer: 0


Question: Determine whether the following statement is true or false: If $x, y, z$ are positive integers such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$, then $hxyz$ is a perfect cube, where $h = (x, y, z)$. Provide a justification for your answer.

Answer: False


Question: Determine the value of $b$ such that the limit \( \lim_{n\to\infty}\left((n^4+bn^3)^\frac{1}{4}-(n^2-bn)^\frac{1}{2}\right) = 2. \)

Answer: \dfrac{8}{3}


Question: Let $A$ be a Banach algebra with an identity element 1, and let $B$ be a closed subalgebra of $A$ that also contains 1. If an element $a \in B$ has an inverse in $A$, must $a$ also have an inverse in $B$? Provide a justification for your answer.

Answer: No


Question: Calculate the conditional expectation \( E(f(x,y)|G) \), where \( f(x,y) = x^2 - y^2 \) and \( G = \sigma(X+Y) \), given the set \( \Omega = [0,1]^2 \) with the Lebesgue measure \( P = dxdy \), and random variables \( X(x,y) = x \) and \( Y(x,y) = y \).

Answer: 0


Question: Determine the number of real solutions for the equation \((y^2-3y+3)^{(y+2)(y-6)}=1\).

Answer: 4


Question: Calculate the limit: \[ \lim_{x \to 0^+} (\sin x)^x \] without using L'Hôpital's rule. Use the fact that \( a^b = e^{b \ln a} \) and the approximation \( \sin(x) \sim x \) as \( x \to 0 \) to find the limit.

Answer: 1


Question: Determine the radius of convergence of the series representation of \( \csc z \) about \( z = \frac{\pi}{2} \).

Answer: \dfrac{\pi}{2}


Question: Is the set of real numbers uncountable?

Answer: Yes


Question:
Given a real number $a$ such that $f(a)=5$ and $f'(a)=2$, calculate the limit:
$$\lim_{n \to \infty}\left(\frac{f\left(a+\frac{1}{n}\right)}{f(a)}\right)^n$$

Answer: e^{\frac{2}{5}}


Question: Let $U$ be a random variable uniformly distributed on $[0,1]$, and let $X_1$ and $X_2$ be identically distributed non-negative random variables. Assume $U$, $X_1$, and $X_2$ are independent. Find the probability $P(U < X_1 \mid X_1 + X_2 = 1)$. Provide your answer as a single probability value.

Answer: \dfrac{1}{2}


Question: Given that \( g(f(x)) = x \) and \( g'(x) = \frac{1}{1 + x^2} \), find \( f'(x) \).

Answer: \sec^2 x


Question: Find the 9th derivative of the function \( f(x) = \frac{\cos(5x^4) - 1}{x^7} \) at \( x = 0 \) using the Maclaurin series.

Answer: 9450000


Question: Let $E$ be a compact subset of an open subset $U$ in the Euclidean plane. Determine whether there exists a positive number $r$ such that for every point $z \in E$, the closed disc $\bar{D}(z,r)$ is contained within $U$. Provide a justification for your answer.

Answer: Yes


Question: Let $G$ be a compact, locally compact abelian (LCA) group, and let $F \subseteq G$ be an open subset with the property that $F + F = G$. Is it true that for every $g \in G$, the intersection $(g + F) \cap F$ is nonempty, which is equivalent to saying that $F - F = G$?

Answer: Yes


Question: Determine whether, for any nonnegative number $h$, there exists a measure-preserving transformation with entropy equal to $h$. Provide a justification for your answer.

Answer: Yes


Question: Suppose the elements of a matrix $A$ are given by $a_{ij} = \cos(i + j)$. Determine the rank of the matrix $A$. \(\operatorname{rank}(A)\)

Answer: 2


Question: Let $f: \mathbb{N} \to \mathbb{N}$ be a strictly increasing function, where $\mathbb{N}$ is the set of all positive integers. Given that $f(f(n)) = 3n$ for all positive integers $n$, determine a possible value of $f(1)$. \( \boxed{} \)

Answer: 2


Question: Determine the minimum number of congruent rectangles needed to dissect a 7x7 square into rectangles using line segments perpendicular to its sides, ensuring all rectangles have integer side lengths.

Answer: 7


Question: Find the smallest value of the expression \( S = \sqrt[5]{\frac{abc}{b+c}} + \sqrt[5]{\frac{b}{c(1+ab)}} + \sqrt[5]{\frac{c}{b(1+ac)}} \) given that \( a \ge 0 \) and \( b, c > 0 \).

Answer: 2


Question: Calculate the expected number of letters needed to be output from a random sequence of the letters \( \{A, B, C, D\} \) until the sequence "ABCD" appears consecutively. Assume each letter is equally likely to appear at each position in the sequence.

Answer: 256


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

Answer: 1


Question: Evaluate the limit: $$\lim_{x\to\infty}\left(\frac{\sinh x}{x}\right)^{\dfrac{1}{x^2}}$$

Answer: 1


Question:
Let $x, y, z > 0$ and $xyz = 1$. Determine the maximum value of the expression:
\[
\frac{2y+3}{(y+3)^2}+\frac{2z+3}{(z+3)^2}+\frac{2x+3}{(x+3)^2}.
\]

Answer: \dfrac{15}{16}


Question: Let $T: \mathbb{R}^4 \to \mathbb{R}^4$ be a linear map with null space $\{(x,y,z,w) \in \mathbb{R}^4 : x+y+z+w=0\}$. The rank of $(T+2I_4)$ is 3. If the characteristic polynomial of $T$ is $x(x+2)^b$, find the value of $b$. \( \boxed{} \)

Answer: 3


Question: Find the limit of the integral \( \lim_{n\rightarrow \infty} \int_0^\infty n f(x) e^{-n|x|} \, dx \), where \( f: [0,\infty) \rightarrow \mathbb{R} \) is Lebesgue integrable and \( \lim_{x\rightarrow 0} f(x) = 2016 \), using the Dominated Convergence Theorem.

Answer: 2016


Question: Let $T$ be a totally ordered family of countable sets under set inclusion. If the cofinality of $T$ is uncountable, determine the cardinality of $\bigcup_{E \in T} E$. \(\text{Provide your answer as a cardinality.}\)

Answer: \aleph_1


Question: Calculate the total number of unique color patterns possible on a stickered Rubik's Cube, taking into account permutations from twisting corners, flipping edges, and reapplying stickers in different color schemes, with colors allowed to be adjacent or opposite.

Answer: 1


Question: Evaluate the integral \( \int _0 ^{2\pi} y e^{-a\cos\left( y\right)}\, \mathrm{d}y \) where \( a \) is a constant.

Answer: 2\pi^2 I_0(a)


Question: Determine the positive integer $n$ such that the sum \( \sum_{k=1}^n \lfloor \log_2 k \rfloor = 1994 \), where \( \lfloor x \rfloor \) denotes the floor function.

Answer: 312


Question: Is it always possible to find a non-irreducible polynomial in $K[x]$ that vanishes at two distinct algebraic elements in an algebraic field extension $E/K$, given a field isomorphism between their subfields that fixes $K$?

Answer: Yes


Question: Calculate the limit: $$\lim_{(x,y) \rightarrow (0,0)}\frac{x^2-y^2}{\sqrt{x^2+y^2}}$$

Answer: 0


Question: Determine the number of positive integer solutions to the equation \(|3^x - 2^y| = 5\).

Answer: 3


Question: Determine the radius of convergence for the power series \( \sum_{n\ge0}\frac{z^{2^n}}{5z^{2^n}+1} \).

Answer: 1


Question:
Evaluate the derivative of the function \( f(x) \) at \( x=0 \), where \( f(x) \) is defined as follows:
\[ f(x) = \begin{cases} x^2 & \text{if } x \in \mathbb{Q}, \\
 x^3 & \text{if } x \notin \mathbb{Q}. \end{cases} \]

Answer: 0


Question: Given a matrix \(A\), determine the value of \(\theta\) such that the sum of the real parts of the eigenvalues of \(A\) equals 1.

Answer: \dfrac{1}{2}


Question: Let $X$ be a Gaussian random variable with mean $\mu$ and variance $\sigma^2$. Define $Y = \max(X, c)$, where $c$ is a constant. Determine the median of $Y$ in terms of $\mu$, $\sigma$, and $c$. Express your answer clearly.

Answer: \max(c, \mu)


Question: Is it possible to find a sequence of functions with uniformly bounded Lipschitz norms that converges to a function with an unbounded Lipschitz norm?

Answer: No


Question: Determine the rational values of \( r > 0 \) for which the derivative \( f'(0) \) exists, where the function \( f: \mathbb{R} \to \mathbb{R} \) is defined by \( f(x) = x^r \cdot \sin\left(\frac{1}{x}\right) \) for all \( x \neq 0 \) and \( f(0) = 0 \).

Answer: r > 1


Question: Given a monic cubic polynomial \( P(x) \) with real roots, where \( P(0) = 0 \) and \( P(1) = 1 \), let \( S \) denote the sum of its roots. How many integer values cannot be taken by \( S \)? The polynomial can be expressed as \( P(x) = x(x - r)(x - s) \), and applying \( P(1) = 1 \) leads to \( rs = r + s \). Determine the number of integer values that \( S \) cannot achieve, considering that there are only two pairs of integer solutions for \( r \) and \( s \), one of which involves \( s = 2 \).

Answer: 3


Question: Calculate the order of the stabilizer of the element \((1 \ 2 \ 3)(4 \ 5 \ 6)\) under the group action of \(S_7\) on itself by conjugation, defined as \(x \cdot y = xyx^{-1}\).

Answer: 18


Question: Evaluate the integral \( \int_0^1 \frac{(\log (1-x))^4}{x}dx \) using Polylogarithm Functions.

Answer: 24\zeta(5)


Question: Suppose $f$ is an increasing function from $\mathbb{N}$ to $\mathbb{N}$, meaning if $x \ge y$, then $f(x) \ge f(y)$. Is there necessarily a program that computes $f$?

Answer: No


Question: Determine whether every linear transformation from a finite-dimensional vector space \( V \) to its dual space \( V^* \) can be represented as an inner product with a vector in \( V \). Answer true or false.

Answer: False


Question: Let $X_i \sim U[0,1]$ be independent random variables. Consider the sequence $Y_n = \max\{X_1, X_2, ..., X_n\}$ and $Z_n = \frac{Y_n}{1+nY_n}$. Does $Z_n$ converge in probability to $0$?

Answer: Yes


Question: Let $A$ be a Hermitian matrix of size $n$ that satisfies the equation $A^5 + A^3 + A = 3I_n$. Prove or disprove that $A = I_n$.

Answer: A = I_n


Question: Calculate the double integral of the function \( f(x, y) = x^3 + y^3 \) over the region \( \mathcal{D} \) defined by the inequalities \( 1 < xy < 2 \) and \( x^2 < y < 2x^2 \). Determine the correct choice among the following options: \((i) \ \ \frac{3}{4},\) \((ii) \ \ 1,\) \((iii) \ \ \frac{36}{37},\) \((iv) \ \ \frac{37}{36}.\) What are the appropriate bounds for the integrals?

Answer: iv


Question: Calculate the radius of convergence of the power series \(\sum_{n=1}^\infty \cos^n(n^2+\ln(n)) x^n\) using Hadamard's formula.

Answer: 1


Question:
Consider the recurrence relation defined by:
\[ \begin{align} a_0 &= 1 \\
 c a_n &= a_{n-1} + 2 a_{n-2} + 3 a_{n-3} + \ldots + n a_0 \end{align} \]
Find the constant \( c < 0 \) that is farthest from zero such that:
\[ \begin{align} a_n &> 0, \quad \text{if } n \text{ is even} \\
 a_n &< 0, \quad \text{if } n \text{ is odd} \end{align} \]

Answer: -\dfrac{1}{4}


Question:
Given the determinants of matrices $P$ and $U$ as $det(P) = 10$ and $det(U) = -3$, determine the value of the determinant of matrix $V$, denoted as $det(V)$. The matrices are defined as follows:

$$P = \begin{pmatrix} a & 2d & 1 \\ b & 2e & -2 \\ c & 2f & -1 \end{pmatrix}, \quad U = \begin{pmatrix} a & b & c \\ 2 & 3 & 2 \\ d & e & f \end{pmatrix}, \quad V = \begin{pmatrix} a & b & c \\ d & e & f \\ 1 & 5 & 3 \end{pmatrix}$$

How can the determinants of $P$, $U$, and $V$ be related to find $det(V)$?

Answer: -2


Question: Is it valid to differentiate both sides of the equation \( \left[z^2-zh(t)\right]\left[\frac{\partial^2 p(r,t)}{\partial r^2}+\frac{\partial p(r,t)}{\partial r} \frac{1}{r}\right] + \frac{dh(t)}{dt} \frac{1}{h(t)} = 0 \) twice with respect to \( z \)?

Answer: Yes


Question: Let \( g \) be a polynomial such that \( g(1)<0 \) and \( g(g(x))=9x-2 \) for all \( x \in \mathbb{R} \). Find \( g(1) \).

Answer: -2


Question: Given that \( \int_0^{\pi}(f(x) + f''(x)) \sin x \, dx = 2 \) and \( f(\pi) = 1 \), find \( f(0) \).

Answer: 1


Question: Given an array of digits $[1,1,2,2,3,3,\ldots,9,9]$ (with no zeros), determine the number of ways to split this array into two non-empty subarrays such that the absolute difference between their sums is a multiple of $11$. Provide your answer as a single integer.

Answer: 3


Question: Evaluate the integral: \[ I = \int_0^{\pi/2} \frac{1}{1+\tan^{n}x} \, dx \]

Answer: \dfrac{\pi}{4}


Question: Determine whether the following statement is true or false: In a normed space $(X, \| \cdot \|)$, if $\|x_n\| \rightarrow \|x\|$, then $x_n \rightarrow x$. Consider the continuity of the norm in the converse direction.

Answer: False


Question: Determine the kernel of the homomorphism from the polynomial ring $\mathbb{F}_{p^n}[X]$ to the function space $\mathbb{F}_{p^n}^{\mathbb{F}_{p^n}}$, where $\mathbb{F}_{p^n}$ is a finite field with $p^n$ elements, that maps polynomials to functions.

Answer: (X^{p^n} - X)


Question: Is the boundary of a set always a closed set if the interior of the set is empty?

Answer: Yes


Question: Determine the average length of a cycle in a permutation of the set \( \{1, 2, 3, \dots, n\} \).

Answer: \dfrac{n}{H_n}


Question: Determine whether the image of the function $f(z) = 2z^{14}\cos^2z$ is $\mathbb{C}$ or $\mathbb{C} \setminus \{a\}$ for some $a \in \mathbb{C}$.

Answer: \mathbb{C}


Question: Calculate $2023^{4046} \mod 1547$ using the Chinese Remainder Theorem.

Answer: 714


Question:
Evaluate the limit:

$$\lim_{n \to \infty} \int _{n}^{2n}\frac{dx}{(f^{-1}(x))^3+f^{-1}(x)}$$

where $f(x)=x^3-x^2+x$.

Answer: \ln 2


Question: Evaluate the integral \( \int_{-\infty}^{\infty} x e^{-\frac{a}{2}x^2} \, dx \) using the substitution \( u = -\frac{a}{2}x^2 \).

Answer: 0


Question: Evaluate the definite integral: $$\int_0^{3\pi} \frac{1}{\sin^4 x + \cos^4 x} \, dx$$

Answer: 3\pi\sqrt{2}


Question: Determine the number of ways to distribute 12 different balls into 4 identical boxes.

Answer: 700075


Question: Determine whether there exists a submodule $I$ of a module $M$ such that $I$ contains no irreducible submodules of $M$, yet $I$ is not the zero module. Consider the case where $M$ is not semisimple.

Answer: Yes


Question: In the context of the geodesic equation using the Einstein summation convention and a symmetric connection, determine the number of times the Christoffel symbols are summed.

Answer: 2


Question: Find the function $f(x)$ that satisfies the functional equation $x^2f(x) + f(1-x) = 2x - x^4$ for all real numbers $x$.

Answer: 1 - x^2


Question: In a game, two players independently choose numbers $x$ and $y$ within the interval $[0,1]$. Player 1 aims to maximize the value of $z = 8xy - 3x - 3y + 1$, while Player 2 tries to minimize it. Both players are unaware of the other's choice. Determine the optimal choice of $x$ for Player 1, given that $z(x, \frac{3}{8}) = -\frac{1}{8}$ for all $x$.

Answer: \dfrac{3}{8}


Question: Find the condition on the parameter \(p\) such that the inequality \(-9 < \frac{3x^2+px-6}{x^2-x+1} < 6\) holds for all real values of \(x\).

Answer: (-3, 6)


Question: In the inner product space \( \mathcal{P}[0,1] \) of all real polynomials on \([0,1]\) with the inner product \( \langle f,g\rangle =\int_{0}^{1}f(t)g(t)dt \), consider the subspace \( V=\operatorname{span}\{t^2\} \). Determine the polynomial \( h(t) \in V \) such that \( \|2t-1-h(t)\| \le \|2t-1-x(t)\| \) for all \( x(t) \in V \). Find \( h(t) \).

Answer: \dfrac{5}{6}t^2


Question: Determine the number of field extensions between the fields $\mathbb{Q}(\zeta^3)$ and $\mathbb{Q}(\zeta)$, where $\zeta$ is a primitive 12th-root of unity.

Answer: 2


Question: In a second-price auction with common values, determine the expected payment of the winner when there are two players with identical valuations. Assume the players' values, represented by random variables $x_1$ and $x_2$, are uniformly distributed on $[0,1]$ and independent. The valuations are given by $v_1 = v_2 = x_1 + x_2$. Given that the bidding strategy at the Bayesian Nash Equilibrium (BNE) is $b_i(x_i) = 2x_i$, calculate the expected payment of the winner.

Answer: \dfrac{2}{3}


Question: Evaluate the limit \( \displaystyle \lim_{n \to \infty} \frac{\tan\left(\frac{\pi}{n}\right)}{n\sin^2\left(\frac{2}{n}\right)} \) using the squeeze theorem.

Answer: \dfrac{\pi}{4}


Question: Let $R$ be an $\mathbb{R}$-algebra. If $A = R \otimes_{\mathbb{R}} \mathbb{C}$ is a finitely generated $\mathbb{C}$-algebra, is $R$ necessarily a finitely generated $\mathbb{R}$-algebra? Provide a justification for your answer.

Answer: Yes


Question: Let $(X, \tau)$ be a topological space with a sub-basis $\beta$. Determine whether the statement $g(\beta)=\tau$ if and only if $\gamma = \beta \cup \{X\}$ forms a sub-basis for $\tau$ is true or false.

Answer: A


Question: Evaluate the limit: \[ \lim_{x\to \infty} \sqrt{x} \left(\sqrt{16x^3+4x-1} - \sqrt{16x^3+3}\right). \]

Answer: \dfrac{1}{2}


Question: Determine the number of $3 \times 3$ skew-symmetric matrices where all non-diagonal elements are distinct and belong to the set $\{-9, -8, -7, \ldots, 7, 8, 9\}$. A skew-symmetric matrix $A$ satisfies $A^T = -A$, meaning the diagonal elements are zero and $a_{ij} = -a_{ji}$ for $i \neq j$. What is the total number of such matrices?

Answer: 4032


Question: Suppose $X, Y, Z$ are independent and identically distributed (iid) random variables. It is given that $\frac{X + Y + Z}{\sqrt{3}} \overset{d}{=} X$ and $E(X^2) = 1$. Determine the distribution of $X$. \( \text{Express your answer in terms of a standard distribution, e.g., } N(\mu, \sigma^2). \)

Answer: N(0, 1)


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

Answer: \dfrac{4}{3}


Question: Evaluate the limit: $$\lim_{x\to0}{\frac{\sqrt[3]{\cos{4x}}-\sqrt[3]{\cos{5x}}}{1-\cos{3x}}}$$ without using L'Hopital's rule.

Answer: \dfrac{1}{3}


Question: Let $n$ be an odd integer. If \( \sin n\theta=\sum_{r=0}^{n} b_r \sin^r\theta \) for every value of \(\theta\), find the value of \(b_1\).

Answer: n


Question: Consider a sequence of nested open intervals $((c_n, d_n))^{\\infty}_{n = 1}$. If $c_n$ approaches infinity, does it follow that $\cap^{\\infty}_{n=1} (c_n, d_n) = \emptyset$? Provide a proof or counterexample.

Answer: \emptyset


Question: Let $f$ be a continuous function from $[0, 1]$ to $[-1, 1]$ such that $|f(x)| \leq x$ for all $x \in [0, 1]$. Determine the minimal value of the expression \( \left| \int_{0}^{1} (f(x))^2 + f(x) \, dx \right| \).

Answer: 0


Question: Evaluate the limit: \[ L = \lim_{x \to 0} \frac{\sin(\ln(x+1)) - \ln(\sin(x)+1)}{\sin^4\left(\frac{x}{2}\right)} \]

Answer: \dfrac{4}{3}


Question: Calculate the expected number of throws required for a ball to land in each of four distinct buckets at least once, given that each bucket has a probability of \( \frac{1}{25} \) of catching the ball. Use the integral \( E[X] = \int^{+\infty}_0 \left(1-\prod_{i=1}^4(1-e^{-0.04x})\right)dx \) to find \( E[X] \), which evaluates to 52.08.

Answer: 52.08


Question: Given the joint density function \( f(y_1,y_2) = \frac{1}{8}y_1 e^{-(y_1 + y_2)/2} \) for \( y_1 > 0 \) and \( y_2 > 0 \), and \( f(y_1,y_2) = 0 \) elsewhere, find the expected value of \( D = 50 + 2Y_1 + 4Y_2 \).

Answer: 66


Question: Find the limit: $$\lim_{x \to \infty}\left(\tan{\frac{\pi x}{2x+1}}\right)^\frac{1}{x}$$.

Answer: 1


Question: Given that $y_1 = x$ is a solution to the differential equation $$(2x^2 + 1)y'' - 4xy' + 4y = 0,$$ find the second linearly independent solution $y_2$.

Answer: 2x^2 - 1


Question: Find the limit of the expression as \( n \) approaches infinity: $$\lim_{n\to \infty}\frac{1}{2n}\log\left(\binom{2n}{n}\right)$$

Answer: \log 2


Question: Evaluate the limit \( \lim_{\epsilon\rightarrow0}\frac{1}{\epsilon}\int_{-\infty}^{\infty}\rho\left(\frac{x}{\epsilon}\right)f(x)dx \) for any continuous function \( f:\mathbb{R}\rightarrow\mathbb{R} \), given that \( \rho:\mathbb{R}\rightarrow\mathbb{R} \) is a continuous function such that \( \rho(x)\geq0 \) for all \( x\in\mathbb{R} \), \( \rho(x)=0 \) for \( |x|\geq1 \), and \( \int_{-\infty}^{\infty}\rho(t)dt=1 \).

Answer: f(0)


Question: Determine whether the following statement is true or false: If the relation $R$ on a set $A$ is transitive, then the relation $S$ on the power set of $A$ defined by $S = \{(X,Y) \in P(A)\times P(A) \mid (\exists x\in X)(\exists y\in Y)((x,y)\in R)\}$ is also transitive.

Answer: False


Question: Evaluate the limit without using L'Hôpital's rule: $$\lim_{x\to 1}\frac{\Gamma(1+x)\Gamma(2+x)\Gamma(3+x)-12}{x-1}$$

Answer: 52 - 36\gamma


Question: Evaluate the contour integral \( \int_{C} z \, dz \) where \( C \) is the path described by \( z = e^{it} \) for \( 0 \le t \le \pi \), a half-circle in the upper half-plane starting from \( z = 1 \).

Answer: 0


Question: Evaluate the integral $$\int\limits_0^1 \big[\chi_{\{u(x)<t\}}(y)\chi_{\{u(y)\geq t\}}(x)+\chi_{\{u(x)\geq t\}}(y)\chi_{\{u(y)<t\}}(x)\big]\mathrm{d}t$$ and determine if the result depends on the values of $x$ and $y$.

Answer: |u(x) - u(y)|


Question: If the concatenation of two numbers X and Y, denoted as XY, is divisible by 23, and the concatenation of (X-a) and (Y-b), denoted as (X-a)(Y-b), is also divisible by 23, must the concatenation of a and b, denoted as ab, also be divisible by 23?

Answer: Yes


Question: Find the inverse Laplace transform of \( \frac{2ks}{(s^2 + k^2)^2} \).

Answer: t \sin(kt)


Question:
Let $x, y, z$ be real numbers such that:

$$x^2+ xy+ yz+ zx = 30$$
$$y^2+ xy+ yz+ zx = 15$$
$$z^2+ xy+ yz+ zx = 18$$

Find the value of $x^2+ y^2+ z^2$.

Express your answer as a single number.

Answer: 21


Question: Evaluate the expression \( \left( \frac{x}{y} \right)^{2015} + \left( \frac{y}{x} \right)^{2015} \) given that \( x \) and \( y \) are non-zero numbers satisfying \( x^{2} + xy + y^{2} = 0 \).

Answer: -1


Question: Determine whether every integer greater than or equal to 12 can be expressed as a sum of distinct primes using a bounded number of summands.

Answer: Yes


Question: Determine whether the following conjecture is true: For any positive numbers $x$ and $M$ with $x > M$, if we define $g(x) = \sqrt{1 + \sqrt{x} \sqrt{1 + \sqrt{x^2} \sqrt{1 + \sqrt{x^3} \sqrt{...}}}}$, then $\lim_{x \to \infty} (g(x+1) - g(x)) = 1$. Provide a proof or counterexample.

Answer: 1


Question: Consider non-negative real numbers \(a, b,\) and \(c\) such that \(a \leq b + c\). For which positive real values of \(n\) does the inequality \(a^n \leq b^n + c^n\) hold?

Answer: (0, 1]


Question: There are 25 individuals numbered $1, 2, \ldots, 25$. Each individual randomly and independently chooses 12 others from this group to call. Let $X$ be the number of calls received by Individual 1, and $Y$ be the number of calls between Individual 1 and Individual 3. Calculate $E(X|Y=1)$.

Answer: 12


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{x^6(1-\cos(x^6))}{(x-\sin(x))^6}.$$

Answer: 23328


Question:
Solve the initial value problem given by the system of equations:
\[\begin{cases}y'=\frac{v(x)-1}{v(x)+2}\\ v(0)=2\\ v(x)=x-y(x)\end{cases}\]
Find the value of \(y(6)\).

Answer: 8 - 2\sqrt{13}


Question: Given a field $F$, determine if it is possible to construct an extension field $K$ by adjoining an element $\alpha$ such that $\alpha^2 + 1 = 0$ and $F \subseteq K$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether there exists a 4-chromatic graph $G$ and a 4-coloring $c$ of $G$ such that every vertex's closed neighborhood $N[v] = \{v\} \cup \{u\ |\ (v,u) \in G\}$ uses at most three distinct colors.

Answer: Yes


Question: Determine the kernel of the ring homomorphism \( f: k[x,y] \to k[t] \) defined by \( f(x) = t \) and \( f(y) = q(t) \), where \( q(t) \in k[t] \).

Answer: (y - q(x))


Question: Determine the smallest integer n such that the inequality \( \prod_{k=1}^n \frac{\arctan\left( \frac{3k+1}{\sqrt{8k^3+8k^2+2k}}\right)}{\arccos \left( \frac{1}{\sqrt{2k+1}} \right)}>1000 \) holds.

Answer: 8


Question: Find the units digit of the sum \(2^{2^2} + 4^{4^4} + 6^{6^6} + \cdots + 2016^{2016^{2016}} + 2018^{2018^{2018}}\).

Answer: 8


Question: Given the equations \( \frac{1}{a} + \frac{1}{b} + \frac{1}{a+x} = 0 \), \( \frac{1}{a} + \frac{1}{c} + \frac{1}{a+y} = 0 \), and \( \frac{1}{a} + \frac{1}{x} + \frac{1}{y} = 0 \), where \(a \neq 0\), \(b \neq 0\), and \(c \neq 0\), find the value of \(a+b+c\).

Answer: 0


Question: Find the limit of the function \( y=\left(x+e^{\frac{x}{3}}\right)^{\frac{3}{x}} \) as \( x \) approaches 0.

Answer: e^4


Question: Evaluate the limit: $$\lim_{x \to a} \left(\frac{\sin(x)}{\sin(a)}\right)^{\frac{1}{x-a}}$$ where $a$ is any real number.

Answer: e^{\cot(a)}


Question: Given that $X$ is a symmetric positive definite matrix, determine whether there exists a symmetric positive definite matrix $Y$ such that $Y^2 - XY - I = 0$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \( \lim _{x\to \infty}\frac{1}{x^2}\cdot\int_0^x \frac{1}{3+\cos t} \, dt \).

Answer: 0


Question: Evaluate the integral \( \int_1^2 \frac{\sqrt{x-1}}{(x+2)\sqrt{x+2}} \, dx \).

Answer: \ln 3 - 1


Question: Evaluate the line integral \( \int_{(1,1,1)}^{(2,2,2)} (y^2z^3\,dx + 2xyz^3\,dy + 3xy^2z^2\,dz) \) using the parametrization \( x = t+1 \), \( y = t+1 \), \( z = t+1 \) for \( t \in [0, 1] \).

Answer: 63


Question: Determine whether the sequence \(x_n = \frac{1}{n}\) is a Cauchy sequence in the metric space \((\mathbb{R}, d)\) where the metric is defined by \(d(x, y) = |\arctan x - \arctan y|\).

Answer: Yes


Question: Evaluate the integral expression: $$\int^1_0 \frac{1}{(1+x^{2015})(1+x^2)}\,dx + \int^1_0 \frac{y^{2015}}{(1+y^{2015})(1+y^2)}\,dy$$

Answer: \dfrac{\pi}{4}


Question: Find the remainder when \(3^{3^{32}}\) is divided by 200.

Answer: 3


Question: Determine whether a nonconstant harmonic function on an open connected set can be constant on a nonempty open subset of its domain. Provide a yes or no answer.

Answer: No


Question: Evaluate the limit: $$\lim\limits_{x\to 0^{+}}\lim\limits_{n\to \infty}\frac{d^n}{dx^n}\left(\frac{e^{\sqrt{x}}+e^{-\sqrt{x}}}{x^n}\right)$$

Answer: 0


Question: Determine whether a measurable function $F:[0,1]\rightarrow \mathbb{R}$ that is bounded almost everywhere necessarily has bounded variation on $[0,1]$. Provide a justification for your answer.

Answer: No


Question: Evaluate the limit: $$\lim_{x\to \infty}\dfrac{e^x\log(1+\frac{1}{x})+\frac{x}{\sqrt{1-2/x}}}{\sin{\frac{1}{x^2}}}$$

Answer: \infty


Question: Determine the degree of the polynomial $Q(x)$ such that $Q(1) + Q(2) + \cdots + Q(n) = n^4 + 2n^2$ for all positive integers $n$.

Answer: 3


Question: Determine whether a group can have two distinct subgroups of the same order whose intersection contains more elements than just the identity. Provide a justification for your answer.

Answer: Yes


Question: Find the limit: $$\lim_{x\to+\infty}(x-1)e^{\pi/2+\tan^{-1} x}-xe^{\pi}.$$

Answer: -2e^{\pi}


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

Answer: Yes


Question:
Calculate the limit as \(x\) approaches infinity for the expression:
\[ \lim_{x\to\infty}\frac{\ln{2^{\frac{1}{x}}}-\ln{x^2}}{1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{x}} \]

Answer: -2


Question: Evaluate the integral \(\int_{|z|=3}\frac{1}{z^2-1}dz\) using the Cauchy integral formula.

Answer: 0


Question: Find the reflection of the point \((2, 1, -3, 4)\) across the hyperplane given by the equation \(x + 2y - z + 5 = 0\).

Answer: (-2, -7, 1, 4)


Question: Determine whether a pointwise limit of continuous functions can be discontinuous on a non-dense subset of the domain. Provide a justification for your answer.

Answer: Yes


Question: Identify the error in the calculation of the limit \( \lim_{x\to0}\frac{\sin\left(x+\frac{x^3}{6}\right)-x}{x^5} \) that results in \(+\infty\), given that the correct limit is \(-\frac{3}{40}\).

Answer: -\dfrac{3}{40}


Question:
Let $A$ be an invertible $n \times n$ matrix with no eigenvalue of magnitude 1, i.e., $|\lambda_i| \neq 1$ for all eigenvalues $\lambda_i$ of $A$. Prove or disprove that the rank of $A - A^{-1}$ is $n$. In other words, is it true that:
\[ \operatorname{rank} \left( A - A^{-1} \right) = n \]?

Answer: n


Question: Consider the norm $\|f\|^2=\int_0^1 f(y)^2 dy$, and let $X$ be a random variable taking values in $L^2[0,1]$. Is bounding $P(X(y)\leq c) \geq 1-\delta$ for every $y\in[0,1]$ sufficient to establish the inequality $P(\|X\|\leq c) \geq 1-\delta$? Answer yes or no.

Answer: No


Question: Evaluate the integral \( \int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx \), where \( n \) is a natural number. Provide your answer in terms of \( n \).

Answer: \dfrac{2\pi}{n!}


Question: Determine whether there exists an invariant non-Lebesgue probability measure for the doubling map \( T:[0,1)\rightarrow [0,1] \) defined by \( T(x)=2x \mod(1) \). Specifically, is there a probability measure different from the Lebesgue measure \( \lambda([a,b))=b-a \) that remains invariant under \( T \)?

Answer: Yes


Question: Find the maximum value of the function \( f(x, y, z) = x + 2y + 3z \) using the method of Lagrange multipliers, subject to the constraints \( x - y + z = 1 \) and \( x^2 + y^2 = 1 \).

Answer: 3 + \sqrt{29}


Question: A bag initially contains 3 white balls and 5 black balls. After each ball is picked, it is returned to the bag along with an additional 4 balls of the same color. What is the probability that the 100th ball picked is black?

Answer: \dfrac{5}{8}


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

Answer: 0


Question: In an abelian category with enough injectives, satisfying (AB3) and (AB4*), and having a generator, does the first derived functor $\varprojlim^1$ vanish on Mittag-Leffler towers?

Answer: Yes


Question: Find all positive integer pairs (x, y) such that \( 7^x - 3^y = 4 \).

Answer: (1, 1)


Question: Determine all values of $a$ at which the function \( h(x) \) is differentiable, where \( h(x) = x^{3}-x^{2} \) if \( x \in \mathbb{Q} \) and \( h(x) = 2x-2x^{2} \) otherwise.

Answer: \emptyset


Question: Evaluate the improper integral \( \int _{ -\infty }^{ +\infty }{ { e }^{ { (ix+b) }^{ 2 } } } \, dx \), where \( x \) and \( b \) are real numbers and \( b \) is a constant.

Answer: \sqrt{\pi}


Question: Let \( R \) be a ring with minimal prime ideals \( p_1, \ldots, p_n \) and define \( D = R/p_1 \times \cdots \times R/p_n \). Find an element \( x \in R \) such that \( \text{ann}_D(x + p_1, \ldots, x + p_n) = \text{ann}_D(1 + p_1, p_2, \ldots, p_n) \).

Answer: x


Question: Determine the Galois group of the polynomial \(x^6 + 2x^3 - 8\) over the field of rational numbers \(\mathbb{Q}\).

Answer: S_3


Question: If the normals at four points \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\), and \((x_4,y_4)\) on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) are concurrent, find the value of \((x_1+x_2+x_3+x_4)\left(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\frac{1}{x_4}\right)\).

Answer: 4


Question: Determine the kernel of the matrix \( A = \begin{pmatrix} x_1^0 & x_1^1 & \ldots & x_1^n \\ x_2^0 & x_2^1 & \ldots & x_2^n \\ \vdots & \vdots & \ldots & \vdots \\ x_m^0 & x_m^1 & \ldots & x_m^n \end{pmatrix} \), where \( n < m-1 \) and the \( x_i \) are pairwise distinct.

Answer: \{\mathbf{0}\}


Question: Find the minimum distance between the curves \( f(x) = e^x \) and \( g(x) = \ln x \).

Answer: \sqrt{2}


Question: Identify a group of order 60 that contains elements of orders 1, 2, 3, 4, and 5. Provide the name or structure of such a group.

Answer: C_{60}


Question: Evaluate the limit: $$\lim_{x\to0} \frac{\ln\left(x+\sqrt{1+x^2}\right)-x}{\tan^3(x)}.$$

Answer: -\dfrac{1}{6}


Question: The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of $d$, the insurance company reduced the expected claim payment by 10%. Calculate the percentage reduction in the variance of the claim payment.

Answer: 1\%


Question: Evaluate the limit: \[ \lim_{x \to \frac{\pi}{2}}{\left[(1+\cot x)^{\tan x}\right]^3} \]

Answer: e^3


Question: Determine the degree of the splitting field of the polynomial \( g(x) = x^4 + 2 \) over the finite field \( \mathbb{F}_3 \).

Answer: 2


Question: Find the smallest positive integer \( n \) such that the quadratic form \( 10x^2 - 12xy + 5y^2 = n \) has a non-negative integer solution \((x, y)\).

Answer: 3


Question: Evaluate the integral \( I = \int_0^\infty \frac{1}{x^2 + b^2} \, dx \) using complex analysis methods, where \( b > 0 \).

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


Question: Find the sum of all natural numbers \( n \) such that \((n^2+n+1)^2\) divides the expression \(n^{2195} + n^{2194} + \ldots + n^2 + n + 1\).

Answer: 14


Question: In a category where all diagrams exist, determine whether the fibred product $A \times_B (B \times_C D)$ is isomorphic to $A \times_C D$ or to $A \times_C (A \times_B D)$. Provide the correct isomorphism.

Answer: A \times_C D


Question: Determine the values of \( \alpha \) for which the matrix \( A = \begin{pmatrix} 1 & 0 & 0 \\ \alpha-3 & 6-2\alpha & 6-3\alpha \\ 2-\alpha & 2\alpha -4 & 3\alpha-4 \\ \end{pmatrix} \) is diagonalizable.

Answer: \alpha \neq 1


Question: Determine whether there exists a bijection from \( \mathbb{N}^2 \) to \( \mathbb{N} \) that is different from the standard pairing function.

Answer: Yes


Question: Given two structures $(P, \cdot)$ and $(Q, \ast)$, and an epimorphism $f: P \to Q$, if the cancellation law holds in $P$, i.e., for all $a, b, c \in P$, $ab = ac$ implies $b = c$, does the cancellation law also hold in $Q$?

Answer: No


Question: Evaluate the limit: $$ \lim_{x \to \infty} x(e^{1/x}-1)^x $$ without using L'Hopital's rule or Taylor series.

Answer: 0


Question: Compute the definite integral \( \int_0^{\frac{\pi}{2}} \sin(x)\, dx \) using the Riemann sum approach and the formula \( \sum_{i=1}^n \sin(i\alpha) = \frac{(\sin(\frac{(n+1)\alpha}{2})\sin(\frac{n\alpha}{2})}{\sin(\frac{\alpha}{2})} \). Given that the length of each subinterval is \( \Delta x = \frac{\pi}{2n} \), the Riemann sum is represented as \( S_n = \left(\frac{(\sin(\frac{(n+1)\pi}{4n})\sin(\frac{\pi}{4})}{\sin(\frac{\pi}{4n})}\right)(\frac{\pi}{2n}) \). Evaluate the limit of \( S_n \) as \( n \to \infty \).

Answer: 1


Question: Determine the precise cardinality of an $\alpha$-generated ($\alpha>\omega$) infinite $\sigma$-algebra $\mathcal{A}$, where $\alpha$-generated means the $\sigma$-algebra is generated by a set of generators of size $\alpha$. Provide your answer in terms of $\alpha$.

Answer: \alpha^{\aleph_0}


Question: Find a Möbius transformation that fixes the origin and maps the interior of the circle $|z - 1| = 1$ to the exterior of the circle $|z + 1| = 1$. This transformation should also map the intersection points of the lines $\operatorname{Im}(z) = \pm 1$ with the circle $|z - 1| = 1$ to their corresponding intersections with the circle $|z + 1| = 1$. Determine the image of the line $\operatorname{Re}(z) = \frac{1}{2}$ under this transformation.

Answer: |z| = 1


Question: Derive a reduction formula for \( I_{n} = \int_{a}^{b} (x-a)^2(\ln (x-a))^n \, dx \) and use it to evaluate \( I_3 \) for \( a=1 \) and \( b=2 \).

Answer: -\dfrac{2}{27}


Question: Let $X=\{(x,y):x^2+y^2<5\}$ and $K=\{(x,y):1\leq x^2+y^2\leq 2 \text{ or } 3\leq x^2+y^2\leq 4\}$. Determine the number of relatively compact connected components of $X\setminus K$ within $X$. \(\text{Provide your answer as a single integer.}\)

Answer: 2


Question: Find the value of the limit \( L = \lim\limits_{n\to\infty} \dfrac{2^{n^{1.1}}}{n!} \).

Answer: \infty


Question: Determine whether the limit \( \lim_{n \to \infty} f^n(x) \) converges to 1 for Lebesgue almost every \( x \in \mathbb{R} \), where \( f(x) = \frac{1 + \cos(2\pi x)}{2} \).

Answer: 1


Question: Determine whether the pointwise limit function \( g(x) = \lim_{n \to \infty} g_n(x) \) of the sequence \( g_n: [0,1] \to \mathbb{R} \) defined by \[ g_n(x) = \begin{cases} 1 & \text{for } x = \frac{2}{i} \text{ with } i \in \{1, \ldots, n\}, \\ 0 & \text{otherwise.} \end{cases} \] is Riemann integrable. If it is, calculate its integral.

Answer: 0


Question: Find the limit of the function \( \frac{\ln(\frac{1+x^2}{1-x^2}) - \arctan(2x^3) + 2x\sin^2(x)}{x\exp(-\frac{1}{x}) - 2x^2 + \ln(\frac{1+x^2}{1-x^2})} \) as \( x \) approaches 0 from the positive side.

Answer: \infty


Question: Evaluate the limit \(\lim_{x\to\infty}\sum_{n=1}^x \frac{x}{n^2(xn+1)^2}\) using the harmonic number formula.

Answer: 0


Question:
Find the value of the following sum:
\[
\left(1+ \frac{1}{2}+\frac{1}{3}+\cdots +\frac 1{2015}+\frac{1}{2016}\right)^2 \\
+\left(\frac{1}{2}+\frac{1}{3}+\cdots +\frac 1{2015}+\frac{1}{2016}\right)^2 \\
+ \left(\frac{1}{3}+\cdots +\frac 1{2015}+\frac{1}{2016}\right)^2 \\
+\cdots \\
+ \left(\frac{1}{2015}+\frac{1}{2016}\right)^2 \\
+ \left(\frac{1}{2016}\right)^2\\
+ \left(1+ \frac{1}{2}+\frac{1}{3}+\cdots +\frac 1{2015}+\frac{1}{2016}\right)\;
\]

Answer: 4032


Question: Compute the expected value of the minimum of the random variables $X$ and $Y$, where $X$ is a discrete random variable with $\mathbb{P}(X=2)=\mathbb{P}(X=4)=\mathbb{P}(X=6)=1/3$, and $Y$ is a continuous random variable uniformly distributed over the interval $[3,5]$. Find $\mathbb{E}(\min(X,Y))$. \( \boxed{} \)

Answer: \dfrac{13}{4}


Question: Calculate the limit of the sequence \(a_n = \frac{\displaystyle \frac{2}{1}+\frac{3^2}{2}+\frac{4^3}{3^2}+\ldots+\frac{(n+1)^n}{n^{n-1}}}{n^2}\) as \(n\) approaches infinity.

Answer: \dfrac{e}{2}


Question: In how many ways can 20 people, labeled A to T, be divided into 4 identical non-empty groups such that A and B are in the same group, and C and D are not in the same group?

Answer: 8460859965


Question: Let $A$ be an $m \times n$ matrix of rank $r$ and $B$ a $p \times q$ matrix of rank $s$. Determine the dimension of the vector space of $n \times p$ matrices $X$ such that $AXB = 0$. \( \text{Express your answer in terms of } m, n, p, q, r, \text{ and } s. \)

Answer: np - rs


Question: Solve the logarithmic equation involving absolute values: \[ \log _{x^{2}}\left | 5x+2 \right | - \frac{1}{2} = \log _{x^{4}}9 \] Consider the cases for \( x \geq -\frac{2}{5} \) and \( x < -\frac{2}{5} \).

Answer: -\dfrac{1}{4}


Question: Given the equation \((\tan\alpha)^2 (\tan\beta)^2 + (\tan\beta)^2 (\tan\gamma)^2 +(\tan\alpha)^2 (\tan\gamma)^2 + 2(\tan\alpha)^2(\tan\beta)^2 (\tan\gamma)^2 = 1\) where \(\alpha, \beta, \gamma\) are within the domain of \(\tan x\), find the value of \(\cos2\alpha +\cos2\beta+\cos2\gamma\).

Answer: 1


Question: Determine the values of $p$ for which the series $\sum_{n=2}^\infty \frac{1}{n(\ln n)^p}$ is convergent.

Answer: p > 1


Question: For the sequence space \((l^2,\|\cdot\|_2)\) and the standard basis vectors \(e_n=(0,0,...,1,0,...)\), let \(\Phi\) be a bounded linear functional. Determine a value of \(p\geq 1\) for which the series \(\sum_{n=1}^\infty |b_n|^p\) converges, where \(b_n=\Phi(e_n)\).

Answer: 2


Question: Calculate the residue of the function \( f(z) = \frac{\sin z}{z^3} \) at \( z = 0 \) using the formula for residues of poles of higher order.

Answer: 0


Question: Given a set \( S \) and a subset \( A \) of \( S \times S \), determine the smallest cardinality of a transitive relation \( B \) on \( S \times S \) that contains all pairs in \( A \) and satisfies the transitive property.

Answer: |A^+|


Question: Find a closed form expression for the average value of the function \( \sin\{t - \sin t\} \) over the entire real line.

Answer: 0


Question: Determine whether every polynomial with integer coefficients that is prime in \( \mathbb{Z}[x] \) is also prime in \( \mathbb{Q}[x] \). Provide a justification for your answer.

Answer: No