Question: Given a Hilbert space $H$ with an orthonormal basis $(e_n)_{n \in \mathbb{N}}$, and a bounded linear operator $T \in \mathbf{B}(H)$ represented by the matrix $A = (\alpha_{nm})$ where $\alpha_{nm} = (Te_n|e_m)$, find the matrix representation of the adjoint operator $T^*$.

Answer: (\overline{\alpha_{mn}})


Question: Determine whether the logarithm of a symmetric, positive definite matrix can always be expressed locally as a power series.

Answer: Yes


Question: Determine whether the order of any quotient ring $\mathbb{Z}[i]/\langle a+ib \rangle$ is $a^2+b^2$. Provide a justification for your answer.

Answer: a^2 + b^2


Question: Calculate the limit \( \lim_{n \to \infty}\frac{\sqrt[n]{(2n)!}}{2n} \) using Stirling's approximation.

Answer: \infty


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

Answer: \sqrt{2}


Question: Is a covering map $q: E \rightarrow X$ with a section necessarily a 1-fold cover?

Answer: No


Question: Find the general formula for the denominator of the series: \(\sum_{n=1}^{\infty}\frac{2^nn!}{(2n)!}\).

Answer: (2n)!


Question: In triangle $\triangle ABC$, angle $A=50^\circ$ and angle $C=65^\circ$. Point $F$ is on $AC$ such that $BF$ is perpendicular to $AC$. Point $D$ is on the line $BF$ (extended) such that $AD=AB$. Point $E$ is on $CD$ such that $AE$ is perpendicular to $CD$. Given that $BC=12$, find the length of $EF$. \( \boxed{?} \)

Answer: 6


Question: Find the 3-digit prime number $p$ such that the decimal expansion of $1/p$ has a period of 5.

Answer: 271


Question: Can a metric function $d_X$ on a set $X$ have exactly three distinct values?

Answer: Yes


Question: Find a set of numbers \( S = \{s_1, s_2, \ldots, s_n\} \) such that the sum of the greatest common divisors of all subsets of \( S \) equals the sum of the elements of \( S \).

Answer: \{1\}


Question: Find the greatest value of the expression \( \dfrac{x^{100}}{1+x+x^2+x^3+\cdots+x^{200}} \) for \( x > 0 \).

Answer: \dfrac{1}{201}


Question: Determine the highest degree of the polynomial for which the quadrature formula \( \int_{-1}^1 |x| f(x) \, dx \approx \frac{1}{2}[f(x_0) + f(x_1)] \) is exact, given that \( x_0 \) and \( x_1 \) are the quadrature points.

Answer: 3


Question: Evaluate the limit \(\lim_{n\to\infty} \prod_{k=2}^n \left(1-\frac1{q_k}\right)\) where \(q_k\) is the \(k\)-th prime number greater than 2.

Answer: 0


Question: In how many ways can the letters of the word "INDIVISIBILITY" be rearranged such that no two 'I's are adjacent?

Answer: 3386880


Question:
Given a sequence $\{a_n\}$ such that $\displaystyle\lim_{n \rightarrow\infty} n^x a_n = a$ for some real number $x$, find the limit:

$$\lim_{n \rightarrow \infty} n^x \left(a_1 a_2 \dots a_n\right)^{\frac{1}{n}}$$

Answer: a e^x


Question: Is it possible to define a distance function on the set of integers such that a nonempty subset is open if and only if its complement is finite?

Answer: No


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\frac{e^{4ix}-1}{x^2}\,\mathrm{d}x \) using the residue theorem.

Answer: -4\pi


Question: Determine if the following statement is true or false: If a sequence of sets $\{A_n\}$ satisfies $A_1 \subset A_2 \subset A_3 \dots$ and a set $B$ is a subset of the union of all $A_n$, then for any infinite subset $E$ of $B$, there exists an $A_i$ such that the intersection of $A_i$ and $E$ is also infinite.

Answer: B


Question: Determine the degree of the field extension \([\mathbb{Q}(i,\sqrt[4]{2}):\mathbb{Q}(\sqrt[4]{2})]\) and justify your answer.

Answer: 2


Question: Does there exist a smooth structure $M'$ on a smooth manifold $M$ such that a given continuous function $f: M \to \mathbb{R}$ becomes smooth when regarded as $f: M' \to \mathbb{R}$?

Answer: No


Question: Find the boundary of the set of irrational numbers, \( \mathbb{R} \setminus \mathbb{Q} \), where \( \mathbb{R} \) is the set of real numbers and \( \mathbb{Q} \) is the set of rational numbers.

Answer: \mathbb{R}


Question: Evaluate the triple integral \( \iiint_V (xy+yz+zx) \, \mathrm{d}x\mathrm{d}y\mathrm{d}z \) over the solid \( V \) defined by \( x \geq 0, y \geq 0, 0 \leq z \leq 1, x^2+y^2 \leq 1 \).

Answer: \dfrac{11}{24}


Question: Solve the equation in the set of real numbers: $$(x+1)\cdot \sqrt{x+2} + (x+6)\cdot \sqrt{x+7}=(x+3)\cdot (x+4).$$

Answer: 2


Question: Evaluate the limit using Taylor series and standard limits: $$\lim_{x\to 0} \frac{(-\frac{x^2_2}{2}+o(x^2_2))(\frac{x^2_3}{2}+o(x^2_3))(x_1+\frac{x^3_1}{3}+o(x^3_1))}{(x^2_1+x^2_2+x^2_3)^2}$$ where $x = (x_1, x_2, x_3)$ is a vector in $\mathbb{R}^3$, and $x_1, x_2, x_3$ approach zero independently.

Answer: 0


Question: Find all positive integers \( n \) such that the sum-of-divisors function \( \sigma(n) \) equals \( n \) itself. Are there any solutions other than \( n = 1 \)?

Answer: 1


Question: Evaluate the limit: $$\lim _{n \rightarrow \infty}\left(n \ln \left(1-\frac{1}{n}\right) \cos \left(\pi \sqrt{4 n^2+10}\right)\right).$$

Answer: -1


Question: Find the shortest distance from the point (1, -1, 1) to the surface defined by the equation \( z = xy \).

Answer: \sqrt{2}


Question: Coffee is poured into a truncated cone-shaped cup at a rate of $2 \text{ cc}/\text{sec}$. The cup has an upper radius of $4 \text{ cm}$, a lower radius of $2 \text{ cm}$, and a height of $6 \text{ cm}$. Determine the rate at which the coffee level is rising when it reaches the halfway point. The volume of the coffee in the cup is given by $V = \frac{h\pi}{3}(R^2 + Rr + r^2)$, where $r < R$ and $r = 2 \text{ cm}$. Note that $R$ varies linearly with $h$ as the coffee level rises.

Answer: \dfrac{2}{9\pi}


Question: Determine the radius of convergence for the series \(\sum_{n=1}^\infty \frac{(2z-3i)^n}{(3n-2i)^2}.\)

Answer: \dfrac{1}{2}


Question:
Given a set of real numbers $x_1, x_2, \dots, x_n$, where $n > 2$, satisfying $x_i = -x_{n-i+1}$ for all $1 \leq i \leq n$, evaluate the sum:
\[ S = \sum_{i=1}^{n}\sum_{j=1, j\neq i}^{n}\sum_{k=1, k\neq i, k\neq j}^{n} x_i x_j x_k \]

Answer: 0


Question: Find a function that has a pole of order 2 at $z=0$ and whose contour integral around the circle $|z|=1$ is zero.

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


Question:
Determine whether the integrals \( I_1 \) and \( I_2 \) are equal, where:
\[ I_1 = \iiiint_{\varphi,\theta,\phi,\beta \in [0, 2\pi]} \iint_{x,y \in (-\infty, \infty)} \exp\left(if_1(x,y,\varphi,\theta,\phi,\beta)\right) \frac{dx\,dy}{(1+x^2+y^2)^2} \,d\varphi\,d\theta\,d\phi\,d\beta \]
with
\[ f_1(x,y,\varphi,\theta,\phi,\beta) = \sin{\varphi} + \frac{1}{\sqrt{1+x^2+y^2}}\left(\sin\theta+x\sin\theta-y\cos\theta\right) + \frac{\sqrt{2}}{\sqrt{1+x^2+y^2}}(\sin\phi-\sin\beta) - \sqrt{2}\frac{1-x^2-y^2}{1+x^2+y^2}\sin(\phi+\beta) \]

and

\[ I_2 = \iiiint_{\varphi,\theta,\phi,\beta \in [0, 2\pi]} \iint_{x,y \in (-\infty, \infty)} \exp\left(if_2(x,y,\varphi,\theta,\phi,\beta)\right) \frac{dx\,dy}{(1+x^2+y^2)^2} \,d\varphi\,d\theta\,d\phi\,d\beta \]
with
\[ f_2(x,y,\varphi,\theta,\phi,\beta) = \sin{\varphi} + \frac{1}{\sqrt{1+x^2+y^2}}\left(\sin\theta+x\sin\theta+y\cos\theta\right) - \frac{2\sqrt{2}}{(1+x^2+y^2)^{3/2}}\left((x^2-y^2)\sin(\theta+\varphi)+2xy\cos(\theta+\varphi)\right) + \frac{\sqrt{2}}{\sqrt{1+x^2+y^2}}(\sin\phi-\sin\beta) - \sqrt{2}\frac{1-x^2-y^2}{1+x^2+y^2}\sin(\phi+\beta) \]

Answer: I_1 \neq I_2


Question: Evaluate the integral \( \int_0^1 \int_0^1 \frac{xy\ln(y)}{xy-1}\ dx \ dy \).

Answer: \zeta(3) - 1


Question: Evaluate the integral \( I = \int_0^1 \frac{dx}{[ax + b(1-x)]^2} \) where \( a \) and \( b \) are real numbers.

Answer: \dfrac{1}{ab}


Question: Find the minimum value of the function \( f(r, s, t) = (r-1)^2 + \left(\frac{s}{r} -1\right)^2 + \left(\frac{t}{s}-1\right)^2 + \left(\frac{4}{t}-1\right)^2 \) subject to the constraints \( 1 \le r \le s \le t \le 4 \), where \( r, s, t \in \mathbb{R} \).

Answer: 12 - 8\sqrt{2}


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty} r(n)x^n \), where \( r(n) \) is the divisor function.

Answer: 1


Question: Is it possible for positive integers to satisfy the equation \(a^n + b^n = c^n\) for any integer \(n > 2\)?

Answer: No


Question:
Determine the maximum possible length of the graph of a function \( f: [0,1] \to [0,1] \) that satisfies the following conditions:

- \( f(0)=0 \)
- \( f(1)=1 \)
- \( f \) is increasing.

Answer: 2


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

Answer: 2


Question: Find the angle between the tangent line to the logarithmic spiral \(\alpha(t) = e^{-t}(\cos(t),\sin(t))\) at the point where \(t = \phi + 2\pi k\) (for some integer \(k\)) and the ray from the origin at angle \(\phi\).

Answer: \dfrac{\pi}{4}


Question: Calculate the value of $i^i$.

Answer: e^{-\pi/2}


Question: Determine the order of the centralizer of a 5-cycle in the symmetric group $S_7$. Provide your answer as a single integer.

Answer: 10


Question: Determine the number of homomorphisms from the group $C_2 \times C_3$ to the symmetric group $S_4$. Provide your answer as a single integer.

Answer: 18


Question: Let $X$ and $Y$ be independent random variables, each uniformly distributed on the interval (0,1). Determine the probability density function (PDF) of the random variable $V = XY$.

Answer: -\ln v


Question: Solve the equation \(3\sqrt{2+x}-6\sqrt{2-x}+4\sqrt{4-x^2}=10-3x\) for real number \(x\).

Answer: \dfrac{6}{5}


Question: Calculate the volume of a right circular cylinder with radius $R$ and height $h$ using cylindrical coordinates and the divergence theorem.

Answer: \pi R^2 h


Question: What is the maximum number of linearly independent elements that can be in a subset of the set of all linear transformations from \(\Bbb R^3\) to \(\Bbb R^3\) that does not span the entire set?

Answer: 8


Question: Determine the number of ring homomorphisms from the ring \( \mathbb{Z}[x,y]/(x^2+y^3-2) \) to \( \mathbb{Z}_5 \).

Answer: 5


Question: Evaluate the limit: $$\lim_{x\to \pi} \frac{\sqrt{1-\tan x}-\sqrt{1+\tan x}}{\sin 2x}$$.

Answer: -\dfrac{1}{2}


Question: Evaluate the integral \( \iint xy\ dx\ dy \) over the region bounded by the circle \(x^2 + y^2 = 25\) and the line \(3x + y = 5\).

Answer: \dfrac{135}{4}


Question: Let \(g\) be a continuously differentiable function on \(\mathbb{R}^3\). Consider the closed ball \(B = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq 1\}\). Since \(g\) is continuous on \(B\), it attains its maximum value at some point \(u \in B\), with \(g(u) < \infty\). If \(u\) lies on the boundary of the ball (\(x^2 + y^2 + z^2 = 1\)), there exists an \(a \in \mathbb{R}\) such that \(\nabla g(u) = au\). Determine the sign of \(a\).

Answer: positive


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

Answer: 0


Question: Is the Riemann integral of the function $F(x)$ over $[0, \infty)$ equal to the Lebesgue integral of $F(x)$ over $(0, \infty)$, given that $F(x)$ is an extension of $f(x) = e^{-x}$ defined on $(0 - \delta, \infty)$ for some $\delta > 0$?

Answer: Yes


Question: Let \( G \) be a connected 5-regular embedded planar graph, where every face has the same degree. Determine the number of faces of \( G \).

Answer: 20


Question: Suppose $f(x,y)$ is a differentiable function such that $f(3,3) = 1$, $f_x(3,3) = -2$, $f_y(3,3) = 11$, $f(2,5) = 1$, $f_x(2,5) = 7$, and $f_y(2,5) = -3$. Let $w$ be a differentiable function of $u$ and $v$ satisfying the equation $f(w,w) = f(uv, u^2 + v^2)$ for all $(u,v)$. Find $\frac{\partial w}{\partial u}$ at $(u,v,w) = (1,2,3)$. \( \boxed{} \)

Answer: \dfrac{8}{9}


Question: Jill and Kelly are doing spring cleaning. Jill takes an exponentially distributed time with a mean of 30 minutes to clean the kitchen, and Kelly takes an exponentially distributed time with a mean of 40 minutes to clean the bathroom. The first one to finish will then rake leaves, which takes an exponentially distributed time with a mean of 60 minutes. If the second person finishes their task, they will join the first person, and the raking will be completed at twice the original rate. What is the expected time until all chores are completed?

Answer: \dfrac{408}{7}


Question: Given a smooth manifold $M$ and a smooth function $f: M \rightarrow \mathbb{R}$, if $p \in M$ is a local extremum of $f$, must $p$ be a critical point of $f$? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the magnitudes of the Fourier transforms of a real function $f(x)$ and its reflection $f(-x)$ are equal.

Answer: Yes


Question: Determine whether the distance function \( d(a,b) = \arccos(a_1b_1 + a_2b_2 + a_3b_3) \) satisfies the triangle inequality for two points \( a = (a_1, a_2, a_3) \) and \( b = (b_1, b_2, b_3) \) on a unit sphere.

Answer: Yes


Question: Let $V$ be a finite-dimensional vector space over $\mathbb{C}$. Let $T:V\to V$ be a linear transformation with the minimal polynomial $x^{13}(x-1)$. What is the minimal polynomial of $T^4$?

Answer: x^4(x - 1)


Question: Evaluate the limit: \[ \lim_{x \rightarrow 0}\left(\frac{1}{\ln(\cos x)}-\frac{2}{\sin ^{2}(x)}\right) \] without using the series expansion method.

Answer: -\infty


Question:
Let $I=[a,b]$ be a closed real interval. Let $f: I \to \mathbb{R}$ be a continuous function. Let $y : I \to I$ be a function such that for all $x \in I$:
\[ \sup_{t \in [a,x]}f(t) = f(y(x)) \]
Is $y$ necessarily a continuous function?

Answer: No


Question: Determine the number of independent 3-cycles in a complete graph $K_n$, where $n \ge 3$. Independent 3-cycles form a minimal set that can generate any cycle within $K_n$. Prove that the number of such 3-cycles is given by $\binom{n-1}{2}$.

Answer: \dbinom{n-1}{2}


Question: Let $(\Omega,\mathcal{F},P)$ be a probability space and $h_n:\Omega\to\mathbb{R}$ $(n\in\mathbb{Z}_+)$ and $h:\Omega\to\mathbb{R}$ be Borel functions. Suppose that for all $\omega_0\in\Omega$, $h_n(\omega_0)$ converges almost surely to $h(\omega_0)$. Does this imply that $h_n$ converges almost surely to $h$? Answer yes or no.

Answer: Yes


Question: In an acute triangle $\triangle ABC$, let $O$ be the circumcenter, $H$ be the orthocenter, and $I$ be the incenter. Given that $IO = IH$ and $\angle BAC = 2 \cdot \angle CAO$, determine the measure of $\angle BHC$. \( \text{(Express your answer in degrees.)} \)

Answer: 120


Question: Given points \(A(0, 1, 0)\) and \(B(0, 0, 2)\), a sphere \(S\) with the equation \((x-2)^2 + y^2 + z^2 = 4\) and a one-parameter family of planes passing through \(A\) and \(B\) with the general equation \(ax + 2y + z = 2\), find the value of \(a\) for which the plane is tangent to sphere \(S\) without using calculus.

Answer: -2


Question: Identify a non-cyclic group such that the intersection of all its non-trivial subgroups is a non-trivial subgroup.

Answer: Q_8


Question: Suppose $(X,d)$ is a metric space. Let $x_1, x_2 \in X$ and $r_1, r_2 \in \mathbb{R}^+$. Consider the open balls $B_1(x_1, r_1)$ and $B_2(x_2, r_2)$. If $B_1(x_1, r_1) \subseteq B_2(x_2, r_2)$, does it necessarily follow that $r_1 \le r_2$? Justify your answer.

Answer: No


Question: Find the orthogonal projection of the vector $(1, 1, 1)$ onto the null space of the matrix $A$.

Answer: (0, 0, 0)


Question: Solve the equation for \(x\): \[5^{x/2} - 2^x = 1\]

Answer: 2


Question: Determine whether the following statement is true for any strong deformation retraction $f : X \times [0, 1] \to X$ from a topological space $X$ to its subspace $X'$: For all $x \in X'$ and $t \in [0, 1]$, the equation $f(f(x, t), 1) = x$ holds.

Answer: True


Question: Determine if the sequence $p_1$, $2p_1+1$, $2(2p_1+1)+1$, ... can have more than 3 terms, where all terms are prime numbers.

Answer: Yes


Question: Does the commutativity of the smooth affine algebraic group $G_{\bar{k}}$ over the algebraic closure $\bar{k}$ of a field $k$ imply that the group $G$ is commutative over $k$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to \infty} \sum_{k=1}^n k \sin \left(\frac{k}{n^3}\right) .$$

Answer: \dfrac{1}{3}


Question: Calculate the quadratic covariance \( \langle t^2B^1_t, t^3B^2_t \rangle_T \) where \( B^1 \) and \( B^2 \) are independent Brownian motions.

Answer: 0


Question: Solve the congruence equation: \(x^{23} \equiv 400 \pmod{401}\).

Answer: 400


Question: In a regular pentagon inscribed in a circle, a pentagram is formed by connecting each vertex to every non-adjacent vertex with a straight line. Inside this pentagram, a smaller regular pentagon is formed. Determine the ratio of the area of the larger pentagon to the area of the smaller pentagon, expressed in terms of the golden ratio.

Answer: \phi^4


Question: Evaluate the limit \( \lim_{n\rightarrow\infty}{\int^{1}_{-1}{e^{-x^n}} \, dx} \) where \( n \) is an odd positive integer.

Answer: 2


Question: Compute the integral \( \int_0^{\infty} \frac{\sin (ax)}{x e^x} \, dx \) without using the power series expansion of \( \sin(x) \).

Answer: \arctan(a)


Question: Determine if the operator \(Ax(t) = \int_{0}^{t^2} x(s)ds\) has any eigenvalues other than 0. If it does, find them.

Answer: 0


Question: Determine the value of $a$ (between 1.3 and 1.5) for which the equation $a^x = \frac{\log(x)}{\log(a)}$ has exactly one solution.

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


Question: Find the coefficient $a_{-2}$ in the Laurent series expansion of the function $$f(z)=\frac{1}{2-e^z}$$ around the singularity $z_0=\log(2)$.

Answer: 0


Question: Evaluate the integral \( \int_{0}^{1} g(x) \, dx \), where \( g(x) = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \\ \left\lfloor \frac{1}{x} \right\rfloor & \text{if } x \notin \mathbb{Q} \end{cases} \).

Answer: \infty


Question: Compute the exterior product \(\phi(X, Y)\) for the 2-form \(\phi = \sum_{j=1}^{m-1} y_j \wedge y_{j+1}\) on \(\mathbb{R}^m\) and the vectors \(X = (2, -1, 3, \dots, 2)\) and \(Y = (0, 2, 0, \dots, 2)\).

Answer: 4


Question: Does the uniform convergence of \(\sum_{n=0}^{\infty} f_n(x)\) on every closed sub-interval of \([0, a)\), along with \(\lim_{x \to a^-} f(x) = L < \infty\), guarantee the uniform convergence of the series on \([0, a)\)? Answer yes or no.

Answer: No


Question: Calculate the residue of \( f(z) = \frac{(z^2 - 1)^4}{z^5} \) at \( z = 0 \).

Answer: 6


Question: Calculate the covariance between $X$ and $X^3$ for a random variable $X$ that follows a normal distribution $N(\mu, \mu^2)$ using the moment generating function.

Answer: 6\mu^4


Question: Calculate the probability of selecting 4 distinct numbers from the set \( \{1, 2, \ldots, n\} \) such that the sum of the smallest two numbers equals the sum of the largest two numbers.

Answer: 0


Question: Evaluate the limit: $$\lim_{x \to \infty} 2\sum _{n=1}^{\infty}\left(\frac{\sqrt{x^2+(2n-1)^2}}{x^2}\right).$$

Answer: \infty


Question: Given the permutation $f = (1, 4, 6)(2, 7, 5, 8, 10)(3, 9)$ in the symmetric group $S_{10}$, find the smallest positive integer $n$ such that the permutation $f^n$ has exactly 7 fixed points.

Answer: 10


Question: Solve the differential equation \( u'(t) + p(t)u(t) = 0 \) with the initial condition \( u(0) = 0 \), where \( p(t) = \begin{cases} 2 & 0 \leq t < 1, \\ 1 & t \geq 1 \end{cases} \). Find the expression for \( u(t) \) using the Laplace transform and its inverse.

Answer: 0


Question: Determine whether the inequality \(a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \leq x^n\) has a solution in the interval \((0, 1)\) for arbitrary positive integers \(n > 2\), given that \(a_i\), \(i=0, \ldots, n-1\) are positive numbers with a sum less than 1.

Answer: Yes


Question: Determine the cardinality of the set \(\{(\aleph_{\omega_1})^\kappa : 1 \leq \kappa < \aleph_{\omega_1}\}\), given that for any \(\alpha, \beta < \omega_1\), there exists a \(\gamma < \omega_1\) such that \(\aleph_{\alpha}^{\aleph_\beta} = \aleph_\gamma\).

Answer: 1


Question: Evaluate the integral \( \int_0^n \frac{f(x)}{f(x) + f(n-x)}dx \), where \( f(x) \) is continuous on \([0,n]\) and \( f(x) + f(n - x) \) does not vanish on \([0,n]\).

Answer: \dfrac{n}{2}


Question:
Let \(x := (x_k)_{k\in\mathbb{N}}\) be a sequence of real numbers in \(l^2(\mathbb{R})\), i.e. \(||x||_2:=\sqrt{\underset{k\in\mathbb{N}}{\sum}x_k^2}<\infty\). For any \(n\in\mathbb{N}\), let \(x^{(n)} := (x_k^{(n)})_{k\in\mathbb{N}}\) be a sequence in \(l^2(\mathbb{R})\) with terms defined as:
\[
x_k^{(n)} := x_k\mathbf{1}_{\{0\leq k \leq n\}}\mathbf{1}_{\{|x_k|\leq n\}}=\begin{cases}
x_k & \text{if } k \leq n \text{ and } |x_k| \leq n \\
0 & \text{otherwise}
\end{cases}
\]
Determine whether \(\lim\limits_{n\rightarrow\infty}||x^{(n)}-x||_2 = 0\).

Answer: 0


Question: Find the value of $k^2 + n^2$ if the line $y = kx + n$ is a common tangent to both the circle $x^2 + y^2 = 4$ and the ellipse $2x^2 + 5y^2 = 10$. Provide your answer as a single number.

Answer: 14


Question: Determine the order of the pole at $z_0=1$ for the function $$f(z) = \frac{\sin^3(z-1)}{\log^4(z)[1-\cos(z-1)]^2}.$$

Answer: 5


Question: Is there a data structure with O(log n) query time that can determine if any points from a set P of 2D points, where no three points are colinear and no two points share the same x or y coordinate, lie within a given half-plane h?

Answer: Yes


Question: Determine the maximum number of unique seating plans possible for 15 people in a row of 15 chairs, where two plans are considered identical if they share the same adjacent quadruples. For instance, if the people are seated left to right as 1, 2, 3, 4, ..., 15, then the quadruples 1234, 2345, ..., 12131415 cannot appear in any other plan. What is this maximum number?

Answer: 2730


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

Answer: e


Question: Determine whether the sequence \( b_n = e^{\frac{1}{n}\ln(7^n+11^n)} \) converges, and if it does, find its limit.

Answer: 11


Question: Solve the equation \(|(g^{-1}(x))^{-1}| = 1 + (g^{-1}(x))^{-1}\), where \(g(x)\) is a function.

Answer: g(-2)


Question: Evaluate the integral \( \int_{-\infty}^{\infty} x^2 e^{-|x|} \, dx \) using integration by parts.

Answer: 4


Question: Does convergence in distribution of two independent sequences \((X_n)\) and \((Y_n)\) to \(X\) and \(Y\) respectively imply that the joint sequence \((X_n, Y_n)\) converges in distribution to \((X, Y)\), assuming \(X\) and \(Y\) are also independent?

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to\pi/4}{10(\tan(x))^{\tan(2x)}}$$

Answer: \dfrac{10}{e}


Question: Determine the number of critical points of the function $f(x) = x^3\ln(x) - x$ without explicitly finding them.

Answer: 1


Question: Find the limit \( \lim_{x\to -\infty} \left(\cfrac{2x^3+5}{2x^3+3}\right)^{3x^3+5} \).

Answer: e^3


Question: Consider a sequence of independent and identically distributed (i.i.d.) Gaussian matrices $A_i \in \mathbb{R}^{p \times n}$ with $p < n$. Each matrix $A_i$ can be decomposed into its reduced singular value decomposition (SVD) as $A_i = V_i \Sigma_i U_i^T$. When we concatenate the column spaces of these matrices by forming $\sum_{i=1}^m U_i U_i^T$, what is the minimum value of $m$ for this sum to be a full-rank matrix, assuming that the matrices $U_i$ follow the Haar measure?

Answer: \left\lceil \dfrac{n}{p} \right\rceil


Question: Find the formula for the number of reflection subgroups of the dihedral group $D_{2n}$, where the group has order $2n$.

Answer: \sigma(n)


Question: Find the number of factors of 3 in the expression \(2^{3^{n+1}}+1\) for any positive integer \(n\).

Answer: n + 2


Question: Determine whether there exists a real and invertible matrix $Y$ of size $n \times n$ for a complex matrix $X$ of size $m \times n$ with $m < n$, such that the product $ZY$ is a real matrix, where $Z$ is the imaginary part of $X$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the contour integral \( \oint_C \frac{e^z-1}{\sin^3(z)}dz \) where \( C \) is the unit circle.

Answer: \pi i


Question: Evaluate the limit \(\lim_{x,y\to 0} \frac{x^2y^2}{x^2+y^2}\left(1-\cos\left(\frac{1}{x}\right)\cos\left(\frac{1}{y}\right)\right)\), if it exists.

Answer: 0


Question: Find a number \( n \) such that \( 100 < n < 2000 \) and \( 2^n + 2 \) is divisible by \( n \).

Answer: 946


Question:
If a meromorphic function $f$ on the extended complex plane $\Bbb C\cup \{\infty\}$ is one-to-one and has a Laurent expansion at the origin given by:

$$f(z) = \frac{b_N}{z^N} + \dots + \frac{b_1}{z} + a_0 + a_1z + \dots,$$

where $b_N \neq 0$, does this imply that $f$ is a Möbius transformation?

Answer: Yes


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

Answer: 1


Question: Determine the values of $n$ for which the limit \( \lim_{(x,y)\to(0,0)}\frac{|x||y|^n}{x^2 + y^2} \) exists.

Answer: n > 1


Question: Determine whether the space of infinite sequences with a norm similar to Euclidean space and the space of square integrable functions are isometrically isomorphic.

Answer: Yes


Question: Determine the dimension of the subspace $S_n$ of $n \times n$ matrices with real entries, where each matrix $A = (a_{ij})$ satisfies $a_{ij} = a_{kl}$ whenever $i + j = k + l$, for $n \geq 2$. Provide your answer as a single integer.

Answer: 2n - 1


Question: Let $f:(a,b) \rightarrow \mathbb{R}$ be a continuous function. If $|f|$ is differentiable on $(a,b)$, is $f$ differentiable on $(a,b)$? Answer yes or no.

Answer: Yes


Question: Determine whether there exists a non-negative function $f \in L^1(\Omega)$, where $\Omega \subset \mathbb{R}^3$ is a bounded domain, such that the Coulomb energy integral \[ E[f] = \iint\limits_{\Omega\times\Omega} \frac{f(x)f(y)dxdy}{|x-y|} \] is infinite, while ensuring that $f \notin L^\infty(\Omega)$. Provide a justification for your answer.

Answer: Yes


Question: Determine the function $y(x)$ that minimizes the functional $$\int_1^2{x^4[y''(x)]^2dx}$$ subject to the boundary conditions $y(1)=1$, $y'(1)=-2$, $y(2)=1/4$, and $y'(2)=-1/4$.

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


Question: Determine whether the limit \( \lim_{N\to\infty} e^{\sqrt{N}} \sum_{k=N}^\infty e^{-\sqrt{k}} (-1)^k = 0 \) holds, indicating that the alternating series \( \sum_{k=0}^\infty e^{-\sqrt{k}} (-1)^k \) converges more rapidly compared to the non-alternating series \( e^{-\sqrt{k}} \).

Answer: 0


Question: For a twice differentiable function \(f(x)\), if \(|f(x)| \leq 3\) for all \(x \in \mathbb{R}\) and there exists a point \(\alpha\) such that \(f(\alpha) + \left(f'(\alpha)\right)^2 = 80\), determine the number of integer values that \(\left(f'(x)\right)^2\) can take within the interval \((0, 77)\).

Answer: 76


Question: Let $A$ be a $3\times3$ matrix with characteristic polynomial $(x-2)^2(x+1)$. If $A$ is diagonalizable and $(A-2I)(A+I)=0$, where $I$ is the identity matrix, find the value of $a$ in the matrix $A= \begin{pmatrix} 2 & 0 & 0\\ a & 2 & 0 \\ b & c & -1\\ \end{pmatrix}$.

Answer: 0


Question: Determine whether the sequence of functions \( g_n(x) = \begin{cases} \left(1+\frac{x^2}{n}\right)^{-n} & 0\leq x\leq \sqrt{n} \\ 0 & \text{otherwise}\end{cases} \) converges uniformly to a limit function on \([0, \infty)\). If so, identify the limit function.

Answer: e^{-x^2}


Question: Consider a $4$ state Markov Chain with the row-stochastic transition matrix $$P = \left(\begin{array}{cccc} 1/4&1/4&1/4&1/4 \\ 1/2&0&0&1/2 \\ 1/3&1/3&1/3&0 \\ 0&0&0&1 \end{array} \right).$$ If the chain starts in state $1$, find the expected length of time intervals until absorption in state $4$.

Answer: 4


Question: Given a random sample $X_1$ and $X_2$ of size $n=2$ from an exponential population, find the efficiency of the estimator $2Y_1$ relative to the estimator $\bar{X}$, where $Y_1$ is the first order statistic and both $2Y_1$ and $\bar{X}$ are unbiased estimators of the parameter $\theta$. Provide your answer as a numerical value.

Answer: \dfrac{1}{2}


Question: Given two distributions $p$ and $q$ over a common finite set $X$ with a total variation distance $TV(p, q) = \epsilon$, and a function $f: X \to \mathbb{R}$ such that $0 \leq \sum_{x \in X} f(x) p(x) \leq 1$ and $0 \leq \sum_{x \in X} f(x) q(x) \leq 1$, determine the tightest upper bound on the expression \( \Big|\sum_x f(x)(p(x)-q(x))\Big| \) in terms of $\epsilon$. Consider how the condition $TV(p, q) = \epsilon$ can be used to refine the bound beyond the trivial upper bound of 2.

Answer: 1


Question: Determine the effect of the map $q: S^n \rightarrow S^n \vee S^n$, obtained by quotienting the equator, on the $n$-th homology group. What is $q_*$ on the $n$-th homology level?

Answer: (1,1)


Question: Consider a state feedback control system described by the equation \(x_{k+1} = Ax_k + Bu_k\) with \(u_k = Kx_k\), where \(A\) and \(B\) are constant matrices, and \(K\) is a designed matrix. The Lyapunov stability criterion for this system is given by \(P - (A + BK)P(A + BK)^\top > 0\), assuming \(P\) is a positive definite matrix. If a bounded noise \(w_k\) is introduced, resulting in the system \(x_{k+1} = Ax_k + Bu_k + w_k\), does the condition \(P - (A + BK)P(A + BK)^\top > 0\) still ensure the stability of the closed-loop system in the presence of noise?

Answer: No


Question: Evaluate the integral \( \int_{-\pi/4}^{\pi/4} (\cos{t} + \sqrt{1 + t^2}\cos^3{(t)}\sin^3{(t)})\;\mathrm{d}t \).

Answer: \sqrt{2}


Question: In a complex vector space, if a normal operator \( N \) can be expressed as \( N = UDU^* \), where \( U \) is a unitary matrix and \( D \) is a diagonal matrix, do the diagonal entries of \( D \) correspond to the eigenvalues of \( N \)?

Answer: Yes


Question: Determine whether every finite group of odd order is isomorphic to a subgroup of the alternating group $A_n$. Provide a justification for your answer.

Answer: Yes


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

Answer: 1


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

Answer: 1


Question: Find the maximum value of the expression \( \left\lvert \frac{1-(-x^2)^{n+1}}{1+x^2} \right\rvert \) for \(-1 \le x \le 1\).

Answer: 1


Question: Calculate the limit \( \lim_{n\to\infty} \frac{\sum_{i=1}^n ia_i}{n^2+n} \), given that \(a_n \to a\). Use the relationship \(\sum_{i=1}^n i = \frac{n(n+1)}{2}\) to simplify the expression and find the limit.

Answer: \dfrac{a}{2}


Question: Evaluate the surface integral \( \int_{S}(z-x^2-y^2)\,dS \) over the surface of the cylinder \( x^2+y^2=4, -2\le z\le2 \) using symmetry and geometry, without parametrizing \( S \).

Answer: -64\pi


Question: Consider a function $g(z)$ that is holomorphic on the complex plane and satisfies $g(x) \in \mathbb{R}$ for all real $x$. Given that $g(1+i)=1-i$, determine the value of $g(1-i)$. \( \boxed{?} \)

Answer: 1+i


Question: Evaluate the integral \(\int_C \sqrt{1+4x^2 z^2} \, ds\), where \(C\) is the curve of intersection between the surfaces \(x^2 + z^2 = 1\) and \(y = x^2\). Provide a complete parametrization of the curve, determine the appropriate sign for \(z\), and establish the limits of integration.

Answer: 3\pi


Question: Determine the winding number of the curve \( \alpha(t) = ((2 \cos t - 1) \cos t, (2 \cos t - 1) \sin t) \) with respect to the point \( p = (1/2, 0) \) using complex analysis.

Answer: 2


Question: Find the gradient of \( \text{trace}(f(BB^\top)) \) with respect to the matrix \( B \), where \( f \) is a differentiable function and \( B \in \mathbb{R}^{p \times q} \).

Answer: 2 f'(BB^\top) B


Question: Evaluate the limit: \( \lim_{n \to \infty} \sin(2\pi(n^2+n^{1/2})^{1/2}) \).

Answer: 0


Question: Find the second solution to the differential equation \(xy'' - y' + 4x^3y = 0\) given that \(y_1(x) = \sin(x^2)\) is a solution and assuming \(x > 0\).

Answer: \cos(x^2)


Question:
Find the smallest possible value of \( p \) such that the following identity holds:

$$(x^3+px^{2}+2x-5)^{19}(x^{2}+qx-41)^{8}(x^4-x^3+x-7)^6=x^{97}+391x^{96}+a_{95}x^{95}+\cdots+a_1x+a_0$$

Answer: 7


Question: Evaluate the Lebesgue integral: $$\int_{\mathbb{R}} 1 \, d\lambda(t).$$

Answer: \infty


Question:
In a 4-state Markov Chain with state space \(\{1,2,3,0\}\), represented by the transition matrix:
\[
\begin{pmatrix}
0.2 & 0.5 & 0.2 & 0.1 \\
0.2 & 0.6 & 0.1 & 0.1 \\
0.2 & 0.3 & 0.3 & 0.2 \\
0 & 0 & 0 & 0.1 \\
\end{pmatrix}
\]
calculate the probability that state 2 is never reached, meaning the process is absorbed into state 0 without transitioning to state 2.

Answer: \dfrac{11}{52}


Question: Evaluate the expression \( \delta_{ij}\delta_{jk}\delta_{ki}\delta_{nm}\delta_{mp}\delta_{np} \) using the Kronecker-delta symbol \( \delta_{rs} \) and the Einstein Summation Convention, given that \( \delta_{ij}\delta_{jk}\delta_{ik}=\delta_{ii}=3 \) and \( \delta_{mn}\delta_{mp}\delta_{np}=\delta_{mm}=3 \) in a 3-dimensional space.

Answer: 9


Question: Given the probability density function (pdf) \( p(x) = xe^{-x} \) for \( x \ge 0 \), find the expectation by first determining the characteristic function and then differentiating it.

Answer: 2


Question: Determine whether the limit of the summation diverges: $$\lim\limits_{n \to +\infty} \sum_{k=1}^n \frac{n\cdot \ln (k)}{n^2+k^2}$$

Answer: \infty


Question: Determine the number of $3 \times 3$ matrices with integer entries such that the product of the matrix and its transpose equals the identity matrix.

Answer: 48


Question: A standard deck of 52 cards is shuffled and dealt to 4 different people. Calculate the probability that at least three people receive exactly 3 aces, 4 kings, and 6 queens.

Answer: 0


Question: Is it possible to divide a convex polygon into two convex polygons of equal area by moving a given straight line parallel to itself on a plane?

Answer: Yes


Question: Determine the values of $x$ for which the series \( \sum _ { n = 1 } ^ \infty \frac { x ^ n } { \sin ( n \theta ) } \) converges when \( \theta = t \pi \), where \( t \) is an irrational number, such as \( t = \sqrt{3} \).

Answer: (-1, 1)


Question: Determine the set of real values of $x$ for which the series \( \sum_{n=1}^{\infty}\frac{n^{n^2}}{(n+x)^{n^2}} \) converges.

Answer: (0, \infty)


Question: Let $f(x)$ be an irreducible quartic polynomial over $\mathbb{Q}$. Let $L$ be the splitting field of $f(x)$. Suppose $\mathbb{Q}(\alpha) \cap \mathbb{Q}(\beta)=\mathbb{Q}$ for any two distinct roots $\alpha$ and $\beta$ of $f(x)$. Calculate the Galois group of $f(x)$ over $\mathbb{Q}$. \( \text{Provide your answer in terms of a known group structure.} \)

Answer: S_4


Question: Calculate the flux of the vector field \( F=(x^3, y^3, z^3) \) through the surface of the sphere defined by \( x^2+y^2+z^2=x \) using the divergence theorem.

Answer: \dfrac{\pi}{5}


Question: Given linear transformations $T_1: \mathbb{R}^n \to \mathbb{R}^m$ and $T_2: \mathbb{R}^m \to \mathbb{R}^n$, where $n \neq m$, and $T_1T_2$ is bijective, determine the ranks of $T_1$ and $T_2$. Note that the rank of a linear transformation is the dimension of its image.

Answer: m


Question: Determine the value of the Legendre symbol \(\left(\frac{2}{n}\right)\) for \(n \equiv 5 \pmod{8}\) using Gauss' Lemma.

Answer: -1


Question: Evaluate the limit \( \lim_{n\to\infty} \int_1^\infty \frac{n\cos x}{1+n^2x^4}\ dx \) using the Dominated Convergence Theorem.

Answer: 0


Question: Given a sigma field $\mathfrak{g}$ and a random variable $X$, consider the conditional expectation $E(X|\mathfrak{g})$, which is $\mathfrak{g}$-measurable. Let $\mathfrak{1}_G$ denote the indicator function for a set $G \in \mathfrak{g}$. If $l \in \mathfrak{g}$ and $l \not\subseteq G$, determine the value of the expression $(E(X|\mathfrak{g})\cdot\mathfrak{1}_G)(l)$, where the multiplication refers to pointwise multiplication.

Answer: 0


Question: In how many distinct ways can 10 children be seated around a circular table if 6 boys must sit together as a block and the 4 girls cannot sit next to each other?

Answer: 0


Question: Determine whether a countably infinite set can be equivalent to one of its proper subsets.

Answer: Yes


Question: Find the second order Taylor approximation for the function \( y(x) \) defined implicitly by the equation \( f(x,y) = e^{x^2+y^2} + 3y - 4x^2 - 1 = 0 \) at the point \((0,0)\). Express \( y(x) \) in the form: \[ y(x) \approx y(0) + y'(0)x + \frac{1}{2!}y''(0)x^2 \]

Answer: x^2


Question: Determine whether the following statement is true or false: If a function $f: \mathbb{R} \to \mathbb{R}$ has the property that for every constant $c \in \mathbb{R}$, the set $\{x \in \mathbb{R} : f(x) = c\}$ is Lebesgue measurable, then $f$ itself is measurable.

Answer: B


Question: Does there exist an infinite torsion group $G$ such that for every finitely generated subgroup $H \le G$, there is some $n \in \mathbb{N}$ with $$(\forall x \in H)(x^{n}=1), ~~(\forall x \in G \setminus H)(x^{n} \ne 1)?$$

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{2\pi} \frac{1}{5-3\sin(\theta)} \, d\theta \) using contour integration.

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit: $$\lim_{n\to\infty}\left(\frac{1}{n}\sum_{i=0}^{n} \sqrt{\frac{i}{n}}\right)$$

Answer: \dfrac{2}{3}


Question:
Evaluate the integral \( \iint\limits_{R} f(x,y) \, dA \) where the function \( f(x,y) \) is defined as:

\[
f(x,y) = \begin{cases} \frac{21}{4}x^2y & \text{for } x^2 \leq y \leq 1, \\ 0 & \text{otherwise.} \end{cases}
\]

and the region \( R \) is defined by \( y \leq x \).

Answer: \dfrac{3}{20}


Question: Given an increasing function \( f: \mathbb{R} \rightarrow \mathbb{R} \) and any real number \( M \), is it always possible to find an \( x \geq M \) such that \( f \) is continuous at \( x \)?

Answer: Yes


Question: For a set of independent and identically distributed continuous random variables $X_1, X_2, \\ldots, X_n$, calculate the probability $P[X_1 > \max(X_2, X_3, \\ldots, X_j)]$, where $j$ is an integer such that $3 \leq j \leq n$. Express your answer in terms of $n$ and $j$.

Answer: \dfrac{1}{j}


Question: Let $f(x)$ be a polynomial such that \( f(x)f\left(\frac{1}{x}\right) + 3f(x) + 3f\left(\frac{1}{x}\right) = 0 \) and \( f(3) = 24 \). Find \( f(2) + f(-2) \).

Answer: 18


Question: Consider the region $\Omega :x^2+y^2<4$ in $\Bbb R^2$ with boundary $\partial \Omega$. Let $u(x,y)$ be the solution of the Dirichlet problem \[\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0, \hspace{1cm} (x,y)\in \Omega\] with the boundary condition $u(x,y)=1+2x^2$ for all $(x,y)\in \partial \Omega$. Find the value of $u(0,1)$. \(\boxed{\text{Answer}}\)

Answer: 4


Question: Evaluate the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) for the vector field \( \mathbf{F} = x\,\mathbf{i} + zy\,\mathbf{j} + zx^2\,\mathbf{k} \), where \( C \) is the circle \( x^2 + y^2 = 16 \) in the plane \( z = 3 \) and is oriented clockwise.

Answer: 0


Question: If the inverse of the function \( y = \sqrt{x + \sqrt{2x - 1}} \) can be expressed as \( y = ax^2 + bx + c \), determine the value of \( a^2 + b^2 + c^2 \). Choose the correct answer: \( A) 3 \) \( B) 4 \) \( C) 5 \) \( D) 6 \).

Answer: B


Question: Determine the number of real values of $x$ that satisfy the equation: \[ \left| 2 - x \right|^{ \log_5(x^2) - 5\log_x (25)} = (2-x)^2 \]

Answer: 3


Question: Given that the probability of \(\gcd(a,b) \in P\) is \(0.2749\ldots\), the probability that \(\gcd(a,b) = 1\) is \(0.6079\ldots\), and the probability that \(\gcd(a,b) \in \{1\} \cup P\) is \(0.8822 \ldots\), find the probability that \(\gcd(a,b) \in 2P\), where \(2P = \{2p : p \in P\}\).

Answer: 0.0687


Question: Determine the infimum of the set \(\left\{\sqrt{x^2+\frac{1}{y^2}}:x,y\in(-\infty,0) \right\}\).

Answer: 0


Question:
Determine the maximum value of \(|f(0.5)|\) for a function \(f(z)\) that is analytic in the unit disk \(D\), continuous in \(\bar{D}\), and satisfies the following conditions:

1. \(\frac{f(z)}{z^2}\) can be extended to be analytic in all of \(D\) (including the origin).
2. \(|f(z)| \leq 3\) in \(D\).

Answer: \dfrac{3}{4}


Question: Given the fourth-order differential equation \( y^{(4)} + a_3 y^{(3)} + a_2 y'' + a_1 y' + a_0 y = 0 \) with a particular solution \( y = 3te^{-t} + e^{-4t}\sin(t) \), determine the constant \( a_0 \). The characteristic polynomial is \( r^4 + a_3 r^3 + a_2 r^2 + a_1 r + a_0 = 0 \). Use the roots of the characteristic polynomial to find \( a_0 \).

Answer: 17


Question: Determine whether the following statement is true or false: For any two vectors $u=(u_1,u_2,\ldots,u_n)$ and $v=(v_1,v_2,\ldots,v_n)$ in $\mathbb{R}^n$ with $|v_i| > |u_i|$ for all $i$, and for any vector norm $\| \cdot\|$ on $\mathbb{R}^n$, it holds that $\|v\| > \|u\|$.

Answer: False


Question: Evaluate the limit \(\lim_{x \to 0}{\frac{1-\sqrt{\cos x}}{1-\cos \sqrt{x}}}.\)

Answer: 0


Question:
Suppose $\{a_n\}$ and $\{b_n\}$ are two real sequences such that $\lim_{n \to \infty} a_n = a$. Determine whether the following statement is true or false:
$$\limsup_{n \to \infty} (a_n + b_n) = a + \limsup_{n \to \infty} b_n$$

Answer: True


Question:
Consider a set $[N] = \{1, \ldots, N\}$ where each item $i$ has a utility $u_i > 0$ and a unit revenue $r_i > 0$, with $r_1 \geq r_2 \geq \ldots \geq r_N$. Under the Multinomial-Logit Model, the probability of purchasing item $i$ is given by:
\[ P(i, u) = \frac{u_i}{\sum_{j=1}^N u_j} \]
The expected revenue is:
\[ R(u) = \sum_{i=1}^N P(i, u) \cdot r_i \]
If utilities are modified to $u'_i = u_i \cdot r_i$, determine if the new expected revenue $R(u')$ is greater than or equal to the original expected revenue $R(u)$. Provide a justification for your answer.

Answer: R(u') \geq R(u)


Question: Calculate the limit \(\lim\limits_{n \to \infty} \left( \frac{\ln(n^n)}{\ln(n+1)^{n+1}} \right)^{1/n}\).

Answer: 1


Question: Determine the number of rational roots of the polynomial \(g(x)=x^{1000}-x^{750}+x^{250}-1.\)

Answer: 2


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^\infty \frac{(2n)!}{(n!)^2}z^n \).

Answer: \dfrac{1}{4}


Question: Evaluate the limit of the function \( f(x,y) = \frac{x^3y^3}{x^2+y^2} \) as \((x,y)\) approaches \((0,0)\).

Answer: 0


Question: Determine whether the following statement is true or false: The free abelian group over the disjoint union of two sets is isomorphic to the direct product of the free abelian groups over each set.

Answer: True


Question:
Determine the number of values of $x$ that satisfy the equation:

$$\alpha_1^3 \left( \frac{\prod_{i=3}^n (x - \alpha_i)}{\prod_{i=3}^n (\alpha_1 - \alpha_i)} \right) + \alpha_2^3 \left( \frac{\prod_{i=1, i \neq 2}^n (x - \alpha_i)}{\prod_{i=1, i \neq 2}^n (\alpha_2 - \alpha_i)} \right) + \sum_{j=3}^{n} \left( \left( \frac{\prod_{i=1}^{j-1}(x-\alpha_i)\prod_{i=j+1}^n(x-\alpha_i)}{\prod_{i=1}^{j-1}(\alpha_j - \alpha_i) \prod_{i=j+1}^n(\alpha_j - \alpha_i) } \right) \alpha_j^3 \right) = x^3$$

where $n \geq 4$. Provide your answer as a single integer.

Answer: n-1


Question: Determine for which complex numbers $z$ with $|z|=1$ the series \( \sum_{n=1}^\infty \frac{z^n}{n} \) converges. It is known that the series diverges for \( z=1 \) and converges for \( z=-1 \). Prove that the series converges for all \( |z|=1 \) except \( z=1 \), without using Dirichlet's test or Abel's test.

Answer: z \neq 1


Question: Given a sequence of random variables $\{X_1, X_2, X_3, \ldots \}$, each with a mean of $1$, and a geometric random variable $N$ with probability mass function $$\mathsf P(N=k) = \frac{1}{2^k}, \quad k \in \{1,2,3,\ldots\}$$ which is independent of all $X_k$, find the expected value of the sum $X_1 + X_2 + X_3 + \cdots + X_N$. \(\boxed{\text{E}(X_1 + X_2 + \cdots + X_N)}\)

Answer: 2


Question: Evaluate the limit: $$\lim_{n \to \infty} \int_0^1 \frac{dx}{x^n + 1}$$

Answer: 1


Question: Does the Lebesgue measurability of a function defined on a closed interval $[a, b]$ with values in the complex numbers imply that the function is integrable?

Answer: No


Question: In a topological space $X$, determine if an open set $A$ and a non-open set $B$ can be homeomorphic.

Answer: Yes


Question: Determine if the integral \( \int_{0}^{1} \frac{1}{x + \sqrt{1-x^2}} \,dx \) can be evaluated without using a trigonometric substitution.

Answer: \dfrac{\pi}{4}


Question: Determine the values of the parameter $p$ for which the improper integral \( \int_{0}^{\pi/2}\tan^{p}(x) \, dx \) is convergent.

Answer: (-1, 1)


Question: Find the real solutions $x \in \mathbb{R}$ to the equation: $$ \sqrt{4^x-6^x+9^x}+\sqrt{9^x-3^x+1}+\sqrt{4^x-2^x+1} = 2^x+3^x+1 $$

Answer: 0


Question: Calculate the expectation $E[B_t \cdot B_s \cdot B_v]$ for a Brownian motion $B_t$ with $0 < t < s < v$.

Answer: 0


Question: Evaluate the limit: \[ \lim _{x\to \infty }\left(\frac{x^2+3x+2}{x^2-x+1}\right)^x \] without using L'Hopital's Rule.

Answer: e^4


Question: Given a conical tent with a fixed volume, determine the ratio of its height to the radius of its base that minimizes the amount of canvas required for its construction.

Answer: \sqrt{2}


Question: Given the equations \(a^3 - 3a^2 + 5a - 17 = 0\) and \(b^3 - 3b^2 + 5b + 11 = 0\), where \(a, b \in \mathbb{R}\), find the value of \(a + b\).

Answer: 2


Question: Determine the density of the set \( S_1 \) in \( \mathbb{N} \), where \( S_1 = \{a \mid \exists m \in \mathbb{N}; \sigma_1(m) = a\} \), and \( \sigma_1(n) \) is the sum of divisors function of \( n \).

Answer: 0


Question: Evaluate the limit \( \lim_{{n\to\infty}} \int_{0}^{1} \frac{n e^{-\sqrt{n x}} \sin(x)}{(1+x^2) \arctan(x)} \, dx \) using the dominated convergence theorem.

Answer: 2


Question: Provide a counterexample to the statement: "If a function has an antiderivative on a punctured disc, then it has an analytic extension to the entire disc."

Answer: \frac{1}{z^2}


Question: Determine whether the following statement is true or false: For a matrix $A$, the singular vectors corresponding to the smallest nonzero singular value with multiplicity $k$ are not necessarily orthogonal.

Answer: B


Question: Find \( \limsup_{n\to\infty}\left(\frac{2\cdot5\cdot8\cdot\cdots\cdot(3n-4)}{3^nn!}\right)^{1/n} \).

Answer: 1


Question: Calculate the flux of the vector field \( G(x,y,z) = (y,z,x) \) through the surface defined by \( x^2+y^2=z^2 \) for \( 0 \leq z \leq 1 \), with the surface oriented by an inward-pointing normal vector field.

Answer: 0


Question: Find the Bayes estimator for the parameter \( \theta \) given the loss distribution \( l(a,\theta)=\frac{(a-\theta)^2}{\theta(1-\theta)} \), a prior distribution \( \pi(\theta) \) that is Uniform from 0 to 1, and a likelihood function \( f(x|\theta) \) for a sequence of \( n \) independent Bernoulli trials, each with a success probability \( \theta \). The total number of ones in the sequence is \( X \sim \text{Bin}(n, \theta) \).

Answer: \dfrac{X}{n}


Question: Determine the set of real numbers $x$ for which the series \( \sum_{n=2}^{\infty} \frac{\sin(nx)}{\log n} \) converges.

Answer: \mathbb{R}


Question: Determine whether every compact submanifold with corners in a Riemannian manifold is locally star-shaped with respect to some point in its boundary.

Answer: Yes


Question: Determine the smallest cardinality of a basis for the topology \( \tau \) on \( \mathbb{R}^2 \), where \( \tau \) consists of the empty set, \( \mathbb{R}^2 \), and all open disks \( \{ (x, y) \in \mathbb{R}^2 \mid x^2 + y^2 < r \} \) for all \( r > 0 \).

Answer: \aleph_0


Question: Find the projection of the function $p_3(x) = x^3$ onto the subspace $M$ generated by the functions $p_0(x) = 1$, $p_1(x) = x$, and $p_2(x) = x^2$ in the space $L^2[-1,1]$ with the usual $L^2$ integral norm. Denote this projection as $P_M(x^3)$. Provide your answer as an expression.

Answer: \dfrac{3}{5}x


Question: Evaluate the limit: $$\lim_{n\to \infty}\frac{\log(p^n+q^n)}{\sqrt{n^2-n+2}}$$ where $0<p<q$.

Answer: \log q


Question: Is it possible to construct a module homomorphism \( \psi: \ker(q) \rightarrow N \) given a module homomorphism \( q: N \rightarrow N \) with \( q^2 = q \)?

Answer: Yes


Question: A cylindrical container has a fixed inner volume \(V\) mm\(^3\), a 2 mm thick solid wall, and an open top. The bottom is a solid circular disc with the same thickness and radius as the outer radius of the container. The volume of the material used to construct the container is minimized when the inner radius of the container is 10 mm. What is the value of \(\frac{V}{250\pi}\)?

Answer: 4


Question:
Given the recursive system of equations:

\[\begin{cases}
a_{n+1} = -2a_n - 4b_n \\
b_{n+1} = 4a_n + 6b_n \\
a_0 = 1, b_0 = 0
\end{cases}\]

find the value of \(\frac{a_{20}}{a_{20}+b_{20}}\).

Answer: -39


Question: Find an expression for the area under the graph of the function \( f(x) = \frac{\log x}{x^2} \) using the definition of a limit of a Riemann sum: \( A = \lim_{x \to\infty} R_n = \lim_{x \to\infty} (\sum_{i=1}^{n} f(x_i)\triangle x) \).

Answer: 1


Question: Let \((a_n)_{n \in \mathbb{N}}\) be a sequence in \(\mathbb{R}\). For every \(r \in \mathbb{R}\), there exists an accumulation point \(b \in \mathbb{R} \cup \{-\infty\}\) of \((a_n)_{n \in \mathbb{N}}\) such that \(b < r\). Prove that \(-\infty\) is an accumulation point of \((a_n)_{n \in \mathbb{N}}\).

Answer: -\infty


Question:
Given the random variable \( \xi \) defined as follows:

\[\xi\overset{\circ}{=}\begin{cases}
\frac{p_{1}\cdot0.3}{p_{1}\cdot0.3+p_{2}\cdot0.7} & \text{with probability } p=p_{1}\cdot0.3+p_{2}\cdot0.7\\
\frac{\left(1-p_{1}\right)\cdot0.3}{\left(1-p_{1}\right)\cdot0.3+\left(1-p_{2}\right)\cdot0.7} & \text{with probability } 1-p=\left(1-p_{1}\right)\cdot0.3+\left(1-p_{2}\right)\cdot0.7
\end{cases}\]

where \( p_{1}, p_{2} \in [0,1] \), find the maximum value of \( \mathbf{P}(\xi > 0.5) \).

Answer: \dfrac{3}{5}


Question: Determine the number of elements in the quotient ring $\Bbb{Z}[i]/\langle3+i\rangle$. Provide your answer as a single integer.

Answer: 10


Question: Find all integer values of \( n \) such that \((1-i)^n\) is a natural number.

Answer: 8k


Question: Find the volume of the $d$-dimensional $\ell_1$-surface $\mathbb{S}_d$ using the parameterization and integral approach.

Answer: \dfrac{2^d \sqrt{d}}{(d - 1)!}


Question: Given two matrices A and B, does the equality \((A^2)(B^2) = (B^2)(A^2)\) always imply that A and B commute, i.e., \(AB = BA\)? Provide a justification for your answer.

Answer: No


Question: How many non-isomorphic Frobenius groups of order 100 exist with a kernel of $C_{25}$?

Answer: 1


Question: Evaluate the limit: $$\lim_{n \to \infty} \frac{1}{n^3} \sum_{i=1}^n \left(e^{\frac{2i}{n}} + \frac{4i^2}{n^2}\right)$$

Answer: 0


Question: Determine the maximum value of the function \( F(y) = \int_{0}^{y} \sqrt{x^4 + (y-y^2)^2} \, dx \) for \( y \) in the interval \([0, 1]\).

Answer: \dfrac{1}{3}


Question:
Find the limit of the following sum as \( n \) approaches infinity:
\[ \lim_{n \to \infty}\sum_{r=1}^{n}\frac{r^2}{n^3 + n^2 + r} \]

Answer: \dfrac{1}{3}


Question: Calculate the limit \(\displaystyle \lim_{(x,y) \to (0,0)} f(x,y)\) for the function \(f(x,y)=\frac{\ln(x+e^y)-x-y}{\sqrt{x^2+y^2}}\).

Answer: 0


Question: Find the points in the complex plane where the function \( g(z) = \frac{\overline{z}^2}{1+z} \) is complex-differentiable.

Answer: 0


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

Answer: \dfrac{\pi}{4}


Question: Determine whether the following statement is true or false: If $X$ is an infinite set and $T$ is the finite-closed topology, then $(X, T)$ is a door space.

Answer: B


Question: Calculate the residue of the function \( f(z) = \frac{1}{z - \sin z} \) at \( z = 0 \).

Answer: \dfrac{3}{10}


Question:
Let \( f:\Bbb R \to \Bbb R \) be a function defined by:
\[
f(x) = \begin{cases} x^2 \left\lfloor \dfrac{1}{x} \right\rfloor & x \neq 0 \\
0 & x = 0 \end{cases}
\]
Find the value of \( \lim_{x \to 0} f'(x) \).

Answer: 2


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), the following equation holds: \[ f(y^2f(x) + x^2 + y) = x(f(y))^2 + f(y) + x^2. \]

Answer: f(x) = x


Question: Evaluate the series \( \sum_{n=1}^\infty\frac{1}{(n-1)!(n+1)} \) and determine whether it converges.

Answer: 1


Question: Consider a topological space $(X, \tau)$ where $D$ is a closed subspace of $X$. Suppose $D$ is equipotent to $\mathbb{R}$ with a discrete induced topology, and $X$ is normal. Determine the number of continuous functions from a countable subset of $D$ to the interval $[0,1]$. Express your answer in terms of cardinality.

Answer: 2^{\aleph_0}


Question: Find the maximum value of the function $f(x,y,z)=xy^2z$ subject to the constraint $x+y+z=4$ in the first octant ($x>0, y>0, z>0$).

Answer: 4


Question: Determine whether the converse of Euler's Totient Theorem is true: If $a^{\varphi(n)} \equiv 1 \pmod{n}$ for positive integers $a$ and $n$, must $a$ and $n$ be coprime?

Answer: Yes


Question: In a finite abelian group \( G \), two subgroups each have an index of 3 in \( G \). Determine the index of their intersection in \( G \).

Answer: 9


Question: Find the limit of the sequence $y_n$ defined recursively by $y_n = 2x_{n+1} + x_n$, where $x_n = \frac{x_{n-1} + x_{n-2}}{2}$ for $n \geq 3$, with initial conditions $x_1 = a$ and $x_2 = b$. Determine the limit as $n$ approaches infinity.

Answer: a + 2b


Question: Let $Y_{1}, Y_{2}, Y_{3}$ be independent random variables, each uniformly distributed over the interval $(0, 2)$. Let $Y_{(1)}, Y_{(2)}, Y_{(3)}$ denote the order statistics of these variables. Compute the probability $P(Y_{(1)} + 2Y_{(3)} \le 3)$. \( \boxed{?} \)

Answer: \dfrac{9}{32}


Question: Evaluate \( \lim_{n\to \infty} \left(\frac{\sin(1)}{1+n}+\frac{\cos(1)}{1+n^2}\right)^n \).

Answer: 0


Question: Let $f: \mathbb R \to \mathbb R$ be a non-constant continuous function such that $(e^x-1)f(2x)= \left(e^{2x}-1\right)f(x)$. If $f'(0) =1 $, find $\lim_{x\to 0}\left(\dfrac {f(x)}{x}\right)^{\frac 1x}$. \( \text{Express your answer as a single number.} \)

Answer: \sqrt{e}


Question: Determine the determinant of a square matrix \( A \) of size \( n \times n \) with the property that if \( a \neq b \), each row is either identical to the first row or the sum of the first row and a scalar multiple of the second row.

Answer: 0


Question: Given the equation \(4x^3 = y^2 + 27\) with \(x\) and \(y\) being integers, determine if it is always true that \(y = 3k\) for some integer \(k\).

Answer: Yes


Question: Given two random variables $x$ and $y$, and a positive function $h(x)$, suppose the inequalities $-h(x) < f(x,y) < h(x)$ hold with probabilities $P[-h(x) < f(x,y)] = \beta$ and $P[f(x,y) < h(x)] = \beta$. Find the probability that both inequalities hold simultaneously.

Answer: 2\beta - 1


Question: Find the range of values of \(n\) for which the equation \(x^n - x - n = 0\) has more than one solution in the interval \((1,2]\).

Answer: \emptyset


Question: Find the smallest value of $U$ such that \( g(X_n) < U < 1 \), where \( g(X_n) = \frac{\mu^2(X_n)}{\mu^2(X_n) + \sigma^2(X_n)} \) and \( \mu, \sigma \) are the mean and standard deviation of a random sample \( X_n \), respectively.

Answer: 1


Question: In a variant of the dominating set selection game, player $P_2$ chooses $n/2$ vertices first instead of player $P_1$. Determine if this change in the order of moves affects the outcome of the game.

Answer: Yes


Question: Find the infimum of the set of distances \(\left\{\int_A |A'(t)|\,dt + \int_B |B'(t)|\,dt:(A,B)\in S\right\}\), where \(S\) is the set of pairs of smooth paths \(A,B: [0,1]\rightarrow\mathbb{R}^2\) satisfying: i) \(A(0)= (0,0)\) and \(B(0)=(1,0)\), and ii) \(B(1) - A(1) = (0,1)\in\mathbb{R}^2\).

Answer: \sqrt{2}


Question: Determine if there exists a constant \(C_h\) such that \(\|\partial_h x\| \leq C_h \|x\|\) for all \(x \in \ell^2(\mathbb{N})\), where \(\partial_h\) is the finite difference operator defined as \((\partial_h(x))_j = (x_{j+1}-x_j)/h\) for \(h>0\). If such a constant exists, find the optimal value of \(C_h\).

Answer: \dfrac{2}{h}


Question:
Calculate the limit of the following expression as \( h \) approaches 0, given that \( f(x) \) is a differentiable function and there exists an \( a \) such that \( f'(a) \neq 0 \):

\[ \lim_{h\rightarrow0}\frac{f(a+3h)-f(a-2h)}{f(a-5h)-f(a-h)} \]

Answer: -\dfrac{5}{4}


Question: Given a joint distribution where $X$ has a standard normal marginal distribution and $Y|X$ follows a uniform distribution $U\left[|X|-\frac{1}{2},|X|+\frac{1}{2}\right]$, find the conditional distribution of $X$ given $Y$ and the conditional expectation $E(X|Y)$. Provide your answers in terms of $Y$ and $X$.

Answer: 0


Question: Using Rouche's theorem, determine the number of zeros of the polynomial \(1+z+\frac{z^2}{2}+ \ldots + \frac{z^{n}}{n!}\) inside the circle \(|z| = R\).

Answer: n


Question: Let $f:X\to X$ be a contraction mapping with constant $q$. If $B(a,r)$ is a ball such that $f(B(a,r)) \subseteq B(a,r)$, determine the maximum possible value of $d(f(a),a)$.

Answer: r(1 - q)


Question: Determine the number of square roots of matrices in a neighborhood of the identity matrix $I$ that are not reflections, for the function $\mathbf{f}: A \mapsto A^2$ where $A \in \mathbb{R}^{n \times n}$. Provide your answer as a single number.

Answer: 1


Question: Determine whether the infinity of the Mandelbrot fractal is countable or uncountable.

Answer: uncountable


Question: Determine whether the following statement is true or false: "If $R$ is a field, then $R[x]$ is an Euclidean Domain (ED)."

Answer: A


Question: Evaluate the line integral \(\int_{C} y \,dx + x^2 \, dy\) where \(C\) is the parabola defined by \(y=4x-x^2\) from \((4,0)\) to \((1,3)\).

Answer: \dfrac{69}{2}


Question: Evaluate the limit: $$\lim_{x\rightarrow 0}\frac{(4^x-1)^3}{\sin\left(\frac{x}{a}\right)\log\left(1+\frac{x^2}{3}\right)}$$ as x approaches 0.

Answer: 3a (\ln 4)^3


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

Answer: 2


Question: In a gambler's ruin Markov chain with equal probability of winning and losing each round, consider a condition where the game stops if the player earns N dollars, making x=N an absorbing state along with x=0. Compute the probability $P_x(N(0)<\infty)$, where $N(0)$ is the number of visits to state 0, starting from state x.

Answer: \dfrac{x}{N}


Question: If a real polynomial $f(x)$ has a root $c$ with multiplicity $r$, and its derivative $f'(x)$ has $c$ as a root with multiplicity $k$, does it follow that $c$ is a root of $f(x)$ with multiplicity $k+1$? Answer with 'yes' or 'no' and justify your answer.

Answer: yes


Question: Calculate \( \mathbb{E}\left[\sinh \left(B_{t}\right)\right] \) where \( B_t \) is a Brownian motion, without using Itô's formula.

Answer: 0


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

Answer: 2\pi \ln \pi


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{(4(n3^n + 3))^n}{(3^{n+1} (n+1)+3)^{n+1}}. \]

Answer: 0


Question: Determine the range of the function \( f(x,y) = \frac{|x|-|y|}{|x-y|} \) for \( x,y \in \Bbb R \) with \( x \neq y \).

Answer: [-1, 1]


Question: Let $a, b, c \in \mathbb{R}$ such that $a + b + c = 3$. Find the minimum value of \( \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}} + \frac{1}{\sqrt{c}}. \)

Answer: 3


Question: Determine whether the following statement is true or false: If an automorphism of a finite group fixes more than half of the group's elements, then the automorphism must be the identity function.

Answer: True


Question: An insurance company observes that imposing a deductible of $d$ on car insurance claims results in a 10% decrease in the expected claim payments. If the amount of a claim follows an exponential distribution, what is the percentage reduction in the variance of the claim payment?

Answer: 1\%


Question: Determine whether the order of 5 in \( \mathbb{Z}_{2^k} \) is \( 2^{k-2} \) for \( k \ge 3 \).

Answer: 2^{k-2}


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

Answer: 1


Question: Determine the number of possible transformed z-axes that can correspond to a given transformed x-axis in a rotation transformation, given the projections of these axes onto their original counterparts.

Answer: 2


Question: Given the function \( g(x) = x + \frac{1}{x} \), if \( \lim \limits_{x \to b}\left(g(x)+\frac{1}{g(x)}\right)=4 \), find \( \lim \limits_{x \to b}g(x) \).

Answer: 2 + \sqrt{3}


Question: Determine the value of $a+b$ such that the function $f(x)=x^2|x-2|+a|x+b|$ is differentiable at every point.

Answer: -6


Question: Evaluate the contour integral \( \int_{\gamma} \frac{e^{iz}}{z^2} \, dz \) where the contour \( \gamma(t) = e^{it} \) for \( 0 \le t \le 2\pi \).

Answer: -2\pi


Question: Evaluate the integral \( \int_{|z|=1} |z-1|^2 \, dz \), where \( z \) is a complex number and the path of integration is the unit circle.

Answer: -2\pi i


Question: Determine whether the product of two positive semidefinite matrices is always a positive definite matrix. Answer with 'yes' or 'no'.

Answer: no


Question: Suppose $X$ and $Y$ are Banach spaces, $U \subset X$ is open, and $f: U \to Y$ is continuously differentiable. If $Df(x)$ is not invertible for some $x \in U$, does it imply that $f$ is not locally invertible at $x$? Provide a justification for your answer.

Answer: No


Question: Evaluate the expression \( f(e^2) - f\left(\frac{1}{e^2}\right) \) for the function \( f(x) = \int_1^x \frac{\arctan(t)}{t} \, \mathrm{dt} \).

Answer: \pi


Question: Let \( f(x) = x^{2n+1} + (2n + 1) x + a \) be a polynomial where \( n \) is a natural number and \( a \) is a real number. Determine the number of real roots of \( f(x) \) in the interval \([0, 2]\) if \( a > (2n + 1)^2 \).

Answer: 0


Question: A parabola has its focus at point $F$ and its vertex at point $V$, with $VF = 10$ units. A chord $AB$ of length $100$ units passes through $F$. Given the parabola equation $y^2 = 4ax$, determine the area of triangle $VAB$.

Answer: 100\sqrt{10}


Question: Let $A \in M_{7}(\mathbb{R})$ be a matrix that satisfies the polynomial $x(x-3)(x+5)$. Given that $\operatorname{tr}(A) = -1$, determine the possible values of $\operatorname{rank}(A)$. \( \boxed{} \)

Answer: 5


Question: Find the intersection points of the function \( f(x) = x^3 + x - 1 \) and its inverse \( f^{-1}(x) \).

Answer: (1, 1)


Question: Evaluate the limit: $$\lim_{n\to \infty} \frac{1^{n}+2^{n}+\ldots+n^{n}}{n^{n+1}}$$

Answer: 0


Question: Let $F \subseteq E$ be a field extension. Let $\alpha \in E$ be a root of an irreducible polynomial $f(x)$ over $F$ with multiplicity 3. If $\beta$ is any other root of $f(x)$ in $E$, what is the multiplicity of $\beta$?

Answer: 3


Question:
Consider a continuously differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ with $f'(0) = 0$. Let $g(x, y) = f(\sqrt{x^2 + y^2})$ for all $x, y \in \mathbb{R}$. Determine which of the following statements is true:
1. $g$ is differentiable on $\mathbb{R}^2$.
2. $g$ is differentiable on $\mathbb{R}^2$ if and only if $f(0) = 0$.
3. $g$ is differentiable on $\mathbb{R}^2 \setminus \{(0, 0)\}$. \(\boxed{\text{Choose the correct statement.}}\)

Answer: 1


Question: Let $A$ be a $3\times 3$ matrix with eigenvalues $-1, 0, 2$. Calculate the determinant of the matrix $A^4 - 2A^3 + A^2 - 3I$. \( \det(A^4 - 2A^3 + A^2 - 3I) \)

Answer: -3


Question: Find the limit of the integral \( \int_0^{+\infty}\frac{\mathrm{d}x}{(1+x^2)^{1/n}} \) as \( n \) approaches infinity.

Answer: \infty


Question: Does every infinite subspace of the interval \([0,1)\) have a limit point within \([0,1)\)?

Answer: No


Question:
Consider the set of all real functions $f$ satisfying the property $f(f(x)) = 1 - x$ for all $x \in \mathbb{R}$. Let $S_f$ be the sum defined as follows:

$$S_f = \sum_{k=-2017}^{2018} f(k)$$

Determine the possible values of $S_f$ for such functions $f$. \( \boxed{} \)

Answer: 2018


Question: Determine whether the series \( \sum\limits_{n=1}^{\infty}\frac{n^{n-1}}{n!e^n} \) converges to 1.

Answer: 1


Question: Calculate the flux of the vector field $\mathbf{G}=(y^2,z^2,x^2)$ through the surface $T: (y-2)^2+(z-1)^2+(x+2)^2=4$ after translating the surface to the origin.

Answer: 0


Question: Compute the infinite summation: $$\sum_{n=1}^\infty \frac{2n-1}{2\cdot4\cdot6\cdots(2n)}.$$

Answer: 1


Question: Evaluate the integral \( \int_{|z|=1} ze^{-z} \, dz \) over the unit circle in the complex plane.

Answer: 0


Question: Given the expression \( k = |a + bw + cw^2| \), where \( w \) is a cube root of unity (\( w \neq 1 \)) and \( \{a, b, c\} \) are consecutive integers, find the minimum value of \( k \).

Answer: \sqrt{3}


Question: Consider the sequence \((x_n)_{n\geq 0}\) of positive integers defined recursively as follows: \(x_0 = 0\), \(x_1 = a\), and \(x_n = x_{n-2} + bx_{n-1}\) for \(n \geq 2\), where \(a\) and \(b\) are fixed positive integers. Determine \(\gcd(x_n, x_{n+1})\) for all \(n\).

Answer: a


Question: Evaluate the line integral \( \int\limits_C \left(xy + \ln(x^2 + 1) \right) dx + \left(4x + e^{y^2} + 3\arctan(y)\right) dy \), where \( C \) is the curve enclosing the region \( D \) in \( \mathbb{R}^2 \) defined by the points \((x,y) : x^2 + y^2 \leq 1\) and \(y \geq 0\), oriented counterclockwise.

Answer: 2\pi


Question: Evaluate the contour integral \( \oint_{|z| = 2} \frac{1}{\cos(z) \sin(z)} \, dz \) using the substitutions \( \sin(z) = \frac{e^{iz} - e^{-iz}}{2i} \) and \( \cos(z) = \frac{e^{iz} + e^{-iz}}{2} \).

Answer: -2\pi i


Question: Determine whether the following inequality is true: If $a^2+b^2\leq 1$ and $c^2+d^2<1$, then $a^2+b^2+c^2+d^2\leq 1+a^2c^2+b^2d^2+b^2c^2+a^2d^2$. Provide a proof or counterexample.

Answer: True


Question: Determine the number of zeros of the function \( f(z) = z^3 - 2z + 2 \) within the annulus \( \{ 0.5 < |z| < 1.5 \} \).

Answer: 2


Question: In a triangle, determine the ratio of the distance from a vertex to the orthocenter to the distance from the circumcenter to the side opposite that vertex.

Answer: 2


Question: Determine the asymptotic probability that the random matrix $A(p, n)$ is non-singular as $n$ approaches infinity, given that $p = 1/2$. Provide your answer as a probability expression or value.

Answer: 1


Question: Determine whether there exists a four-dimensional smooth manifold that admits a non-vanishing continuous vector field but is not parallelizable.

Answer: Yes


Question: If $X$ is a collection of subsets of $\Omega$ such that $X$ is contained in the $\sigma$-algebra $\mathcal{F}$ and $\mathcal{F}$ is also contained in $X$, does $X$ equal $\mathcal{F}$?

Answer: Yes


Question: In circle $\Gamma_1$, let $CD$ be a chord and $AB$ be a diameter perpendicular to $CD$ at point $N$, with $AN > NB$. A second circle $\Gamma_2$ is centered at $C$ with radius $CN$ and intersects $\Gamma_1$ at points $P$ and $Q$. The segments $PQ$ and $CD$ intersect at $M$. If the radii of $\Gamma_1$ and $\Gamma_2$ are $61$ and $60$ respectively, find the length of $AM$. \( \boxed{} \)

Answer: 78


Question: Given a set \( S \) containing only prime numbers, where \( p_n \) is the largest prime in \( S \), determine the probability that \( 1 + p_1 \cdots p_n \) is also a prime number.

Answer: 0


Question:
Given a function $f: \mathbb{R} \rightarrow \mathbb{R}^+$ that satisfies the following conditions:

1. $f(x + 2) \cdot f(x + 3) = 16$
2. $f(x) + f(-x) = 8$

Evaluate the integral $\int_{-8}^8 f(x - 2019) \, dx$. \( \boxed{?} \)

Answer: 64


Question:
Determine the number of distinct real roots of the equation:
\[ \sum\limits_{i = 1}^{40}|a_i - x| = \sum\limits_{i = 1}^{40}|b_i - x|. \]
Here, \(a_i\) and \(b_i\) are real numbers.

Answer: 40


Question: Determine whether the equation \( \sum_{i=1}^n i^5 + i^7 = 2\left( \sum_{i=1}^n i\right)^4 \) holds true for all natural numbers \( n \).

Answer: True


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

Answer: \dfrac{\pi}{4}


Question: In the context of the Phragmén-Lindelöf theorem, determine if the bound $M$ can be equal to zero.

Answer: Yes


Question: Determine whether there exist any uncountable subrings of the real numbers $\mathbb{R}$ other than $\mathbb{R}$ itself.

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} I_n^n \), where \( I_n = \int_1^{1+\frac{1}{n}}\left\{[(x+1)\ln x +1] e^{x (e^x \ln x +1)}+n \right\}\, dx \) for \( n=1,2,\ldots \).

Answer: e^e


Question: Evaluate the integral \( \int_\Gamma z^{\frac{1}{2}} \, dz \) for the contour \( \Gamma \) in the right half-plane from \( z = -3i \) to \( z = 3i \), choosing appropriate branches of \( \log(z) \).

Answer: 2i\sqrt{6}


Question: Determine whether the space of real-valued processes $X: \Omega\times [0,T] \rightarrow \mathbb{R}$, with the norm defined by $$\Vert X \Vert = \mathbb{E} [\sup_{0 \leq s \leq T} |X_s| ^2] ^{1/2}< \infty,$$ is a closed complete metric space.

Answer: Yes


Question: Calculate the expected straight line distance through a perfect sphere of radius \( R \) between two points chosen randomly on its surface, given that the first point is located at one of the poles.

Answer: \dfrac{4R}{3}


Question: Given that \(\sqrt{2} + \sqrt{3} + \sqrt{6}\) is a root of the polynomial \(x^4 + ax^3 + bx^2 + cx + d = 0\), where \(a, b, c, d\) are integers, find the value of \(|a + b + c + d|\).

Answer: 93


Question: Can every vector space, regardless of its dimension or the underlying field, be equipped with a norm to become a normed vector space?

Answer: Yes


Question: Suppose \( A \) is a real symmetric \( 805 \times 805 \) matrix with eigenvalues \( 1.00, 1.01, 1.02, \ldots, 8.89, 8.99, 9.00 \) and also \( 10, 12, 16, 36 \). What is the minimum number of conjugate gradient iterations required to reduce the initial error \( ||e_0||_A \) by a factor of \( 10^{-5} \)?

Answer: 22


Question: Suppose $(X,\mathcal{M},\mu)$ is a measure space, and $g$ is a bounded measurable function such that $g \in L^{\infty}(X)$. If for every function $f \in L^{2}(X)$, the product $f\overline{g}$ lies in $L^{1}(X)$, does it necessarily follow that $g \in L^{2}(X)$? Provide a justification for your answer.

Answer: Yes


Question: Determine whether it is possible to construct an injective function from the set of all algebraic numbers to the set of natural numbers \( \mathbb{N} \).

Answer: Yes


Question: Consider a polynomial function of degree $n \geq 2$ with arbitrary real coefficients. If a line intersects this function at $n$ distinct points, is the line necessarily parallel to the $x$-axis? Answer yes or no.

Answer: No


Question: Consider the sequence defined by the recurrence relation $x_{n+1} = 2x_n^2 - 1$ with $x_0 = t$. Determine the number of values of $t$ in the interval $[-1, 1]$ such that $x_{11} = 1$. Provide your answer as an integer.

Answer: 1025


Question: Let $A \in \mathbb{C}^{m \times n}$ be a matrix. The index of $A$, denoted by $k$, is the smallest nonnegative integer such that $\text{rank}(A^{k+1}) = \text{rank}(A^{k})$. Does this imply that the range spaces of $A^{k+1}$ and $A^k$ are equal, i.e., $R(A^{k+1}) = R(A^{k})$? Answer yes or no.

Answer: Yes


Question: Evaluate the limit \(\lim_{x\to\infty} (x+2)e^{-1/(x+1)}-(x+1)\) using Taylor series expansion.

Answer: 0


Question: Given a prime number $p$ and an integer $u$ with $0 < u < p$, determine the probability that there exists an integer $t$ such that $u + tp$ is a perfect square and less than $p^2$. Express your answer as a fraction or a decimal.

Answer: \dfrac{1}{2}


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

Answer: 0


Question: Calculate the residue of the function \( g(z) = \frac{1 - \cos(z)}{z^4} \) at \( z = 0 \).

Answer: 0


Question: Find all integer pairs \((x, y)\) that satisfy the equation \(4y^2 = x^7 + 1\).

Answer: (-1, 0)


Question: Given that \(\omega\) is a complex number satisfying \(\omega^7 = 1\) and \(\omega \neq 1\), let \(\alpha = \omega + \omega^2 + \omega^4\) and \(\beta = \omega^3 + \omega^5 + \omega^6\). Determine the ordered pair \((p,q)\), where \(p\) and \(q\) are integers, such that \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2 + px + q = 0\).

Answer: (1, 2)


Question: Given that the joint density of $(X,Y)$ is $f(x,y)=\frac { 1 } { 2 } y ^ { 2 } e ^ { - x }$ for $0 < y < x < \infty$, find the probability $P(Y<1\mid X=3)$. Provide your answer as a single value.

Answer: \dfrac{1}{27}


Question: Determine the number of zeros of the function \( \phi(z) = \alpha + z - e^z \) in the region where \( \text{Re}(z) \leq 0 \), given that \( 1 < \alpha < \infty \). Use Rouche's Theorem to assist in your analysis.

Answer: 1


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty} \sum_{r=0}^{\infty} \frac{^n\text{C}_{r}}{n^r(r+4)}$$

Answer: 6 - 2e


Question: Determine the ideal $I(S_1)$ of the set $S_1 = \{(x,0) \in \mathbb{R}^2 \mid x \in [0, \infty)\}$ in the polynomial ring $k[x,y]$, where $k$ is a field. The ideal $I(S)$ is defined as the collection of polynomials $f \in k[x,y]$ such that $f(a) = 0$ for all $a \in S$. What is $I(S_1)$?

Answer: (y)


Question: Simplify the expression \(((A \Rightarrow B) \Rightarrow (B \Rightarrow A)) \Rightarrow (\neg (A \land B) \Leftrightarrow \neg (B \lor A))\) using the definition of \(\land\) as \(A \land B = \neg (A \Rightarrow \neg B)\) and \(\lor\) as its complement.

Answer: A \Rightarrow B


Question: Does every matrix in \(\operatorname{SL}(2, \mathbb{R})\) have a neighborhood diffeomorphic to \(\mathbb{R}^3\)?

Answer: Yes


Question: Calculate the limit: \[ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} \]

Answer: 0


Question: Given two random variables $X$ and $Y$, does the existence of a function $f: \mathbb{R} \to \mathbb{R}$ such that $Y = f(X)$ imply the existence of a Borel-measurable function $g: \mathbb{R} \to \mathbb{R}$ such that $Y = g(X)$ almost surely?

Answer: Yes


Question: Find a non-recursive equation for the sequence $a_n = 3a_{n-1} - 3a_{n-2} + a_{n-3}$ for $n \geq 3$, with initial terms $a_0 = 2$, $a_1 = 2$, and $a_2 = 4$, using a generating function.

Answer: n^2 - n + 2


Question: Does a subset $M$ of $\mathbb{R}^n$ with Hausdorff dimension $n$ necessarily contain an open ball? Answer yes or no.

Answer: No


Question: Find the limit of the sequence \( a_n = (n^2 + 1)\sin\left(\frac{1}{n^3+1}\right) + (n^2 + 2)\sin\left(\frac{1}{n^3+2}\right) + \dots + (n^2 + n)\sin\left(\frac{1}{n^3+n}\right) \) as \( n \to \infty \).

Answer: 1


Question: Let $X$ be a collectionwise normal and Hausdorff locally convex topological vector space. For each $i \in \{1, \ldots, n\}$, let $X_i$ be a Hausdorff locally convex topological vector space. Assume for every $i \in \{1, \ldots, n\}$ there is an upper semicontinuous multivalued map $\phi_i: X \to P(X_i)$, with $\phi_i(x)$ being a non-empty, convex, and compact subset of $X_i$ for every $x \in X$. Given an open covering $(V_x)_{x \in X}$ of $X$, can a point finite covering of $X$ that is subordinated to $(V_x)_{x \in X}$ be constructed?

Answer: Yes


Question: How many invertible \(3 \times 3\) matrices are there over the residue field modulo 5 (\(\mathbb{Z}_5\))?

Answer: 1488000


Question: Determine the values of \( \beta \) for which the function \( g(x) = \sum_{n=1}^\infty n^{-\beta} f(nx) \) is Lebesgue-integrable for almost all \( x \in \mathbb{R} \), given that \( f: \mathbb{R} \to [0,\infty] \) is a Lebesgue-integrable function.

Answer: \beta > 0


Question: Evaluate the line integral \( \int_C x \, ds \) where \( C \) is the curve parameterized by \( x=\frac{3}{4}\sin(2t) \), \( y=\cos^3 t \), and \( z=\sin^3 t \).

Answer: 0


Question: Find the measure \( \lambda_3(A) \) of the set \( A = \{ (x,y,z) \in \mathbb{R}^3 \mid z^2 < \frac{x^2}{4} + \frac{y^2}{9} < 2z \} \).

Answer: 8\pi


Question: Find the coefficient of \(x^{30}\) in the expansion of \(\left(\frac{1-x^{11}}{1-x}\right)^3\left(\frac{1-x^{21}}{1-x}\right)\).

Answer: 1111


Question: Given the equation \(|x^2| + |2xy| + |y^2| = |(x - y)^2|\), where \(x, y \in \mathbb{C}\) and \(y \neq 0\), find the imaginary part of \(\frac{x}{y}\). Use the triangle inequality and the property \(|x||y| = |xy|\) to solve the problem.

Answer: 0


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

Answer: \dfrac{16}{9}


Question: Using Rouché's theorem, determine the number of zeros of the polynomial \( f(z) = z^{47} - z^{23} + 2z^{11} - z^5 + 4z^2 + 1 \) within the disk \( D(0, \frac{2}{3}) \).

Answer: 2


Question: Evaluate the integral \( \int\limits_{-\infty}^{\infty} x e^{-|(x-u)|} \, dx \).

Answer: 2u


Question: In a path-connected metric space with more than one point, does every open ball contain a path-connected subset with more than one point?

Answer: Yes


Question:
Determine if it is possible to find the centroid of 10 points using the following operations in less than 39 steps:

1. Circle an existing centroid (1 step)
2. Form the centroid of two disjoint point sets of equal size (2 steps)
3. Form the centroid of two disjoint pairs of point sets that have the same union (2 steps)

Answer: Yes


Question: Evaluate the integral: \[ \int_0^{\frac{\pi}{2}}\frac{\sin(x)\cos(x)}{\sin^4(x)+\cos^4(x)}dx \] using the substitution \( \sin x = t \).

Answer: \dfrac{\pi}{4}


Question:
Determine the values of \( p \in (0,\infty) \) for which the function \( f:\mathbb{R} \rightarrow \mathbb{R} \), defined by
\[ f(t) = \begin{cases}
\tan(t) & \text{if } t \in \left(0, \frac{\pi}{2}\right) \\
0 & \text{otherwise}
\end{cases} \]
belongs to the \( L^p \) space with respect to the Lebesgue measure on \( \mathbb{R} \).

Answer: (0, 1)


Question: Determine the number of integers \(1 \leq x \leq 2500\) such that the expression \(\sqrt[3]{x+(x+10)\sqrt\frac{x-1}{3}}-\sqrt[3]{x-(x+10)\sqrt\frac{x-1}{3}}\) is a rational number.

Answer: 1


Question: Compute \(3^{100} \pmod{9797}\) using Euler's Theorem and the Chinese Remainder Theorem.

Answer: 2021


Question: Simplify the expression $E[E(ZX \mid Y) \mid Z,Y]$. Provide the simplified form.

Answer: E(ZX \mid Y)


Question: Determine the radius of convergence of the series \( \sum \frac{h(n)}{n^k}r^n \), where \( h(n) \) is a polynomial of degree \( k \).

Answer: 1


Question: Consider two vector spaces V and W that have the same dimension over a field F. If the cardinality of F is less than the dimension of these vector spaces, are V and W necessarily isomorphic? Provide a justification for your answer.

Answer: Yes


Question: Calculate the limit: $$\lim_{n\to\infty} \frac{n^{n+\frac{1}{n}}}{(n+\frac{1}{n})^n}$$

Answer: 1


Question: Find the radius of convergence for the power series $$\sum_{n=0}^{\infty}\frac{5^{2n+1}+3^{n+2}+7^n}{(3n)!}y^{3n}.$$

Answer: \infty


Question: Determine the values of the parameter \( a \) such that the matrices \( A = \begin{pmatrix} 1 & 4-a-a^2 \\ 2 & -1 \end{pmatrix} \) and \( B = \begin{pmatrix} -a-1 & 3 \\ 3 & -5 \end{pmatrix} \) represent the same bilinear form in different bases.

Answer: -2


Question: Given a matrix $C$, determine a matrix $D$ such that the column vectors of $D$ form a basis for the kernel of the linear transformation represented by $C$, and the kernel of the linear transformation represented by $D$ is trivial.

Answer: D


Question: Evaluate the sum \( \sum_{k=0}^{\frac{n}{2}} {n\choose2k} \).

Answer: 2^{n-1}


Question: Determine whether the direct sum of two exact sequences forms an exact sequence.

Answer: Yes


Question: Find the value of the convergent series: $$\sum_{n=1}^{\infty}\frac{n^2-n+1}{n!}$$

Answer: 2e - 1


Question: Evaluate the integral \( \int_2^\infty \frac{\ln(t-1)}{(t-1)^5} \, dt \) using substitution and integration by parts.

Answer: \dfrac{1}{16}


Question: Determine whether the limit \( \lim_{n \to \infty} \sum_{k=1}^{\infty} \frac{1}{(2k-1)^{2n}} \) converges to 1.

Answer: 1


Question: Evaluate the sum: $$\sum_{j=0}^\infty \left( \sum_{k=j}^\infty \binom{k}{j} \left(\frac{1}{4}\right)^k \right)$$

Answer: 2


Question: Determine the asymptotic behavior of the sequence defined by the recurrence relation \(b_{n+1}=b_n+\left\lfloor\frac{1}{b_n}\right\rfloor\) with the initial condition \(b_1=1\).

Answer: 2


Question: In triangle $\triangle ABC$, the sides $a$, $b$, and $c$ form a geometric progression with a common ratio of $\frac{2}{3}$. Given that the circumradius of the triangle is $6\sqrt{\frac{7}{209}}$, find the length of the longest side, $a$. \(\boxed{\text{Answer}}\)

Answer: \dfrac{7}{4}


Question: Find the order of 2 modulo a prime divisor \( p \) of the Fermat number \( F_n = 2^{2^n} + 1 \), given that \( 2^{2^{n+1}} \equiv 1 \pmod{p} \).

Answer: 2^{n+1}


Question: Evaluate the line integral \( \oint_C (x^2-y^2)dx+2xydy \) using Green's Theorem, where \( C \) is the boundary of the region enclosed by the parabolas \( y=x^2 \) and \( x=y^2 \).

Answer: \dfrac{3}{5}


Question: Calculate the limit: $$ \lim _ {(x,y)\to (0,0)} \frac{\ln (1+3x^2+9y^2)}{\arctan(x^2+3y^2) } $$

Answer: 3


Question: Evaluate the limit: $$\lim_{x\to0} \left(\frac{(1+2x)^{1/x}}{e^2}\right)^{1/x}$$

Answer: e^{-2}


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables such that $Y_i = -2 \log (|X_i|/5)$ follows a $\chi^2_2$ distribution. Determine the distribution of the random variable $Z = \sum_{i=1}^{n} Y_i$. Provide your answer in terms of a standard distribution.

Answer: \chi^2_{2n}


Question: Let $g$ be a function such that $$ \sqrt {x + \sqrt { x - g(x) } } = g(x) , $$ for $x > 1$. If $g(x)$ has the form $\frac{p+\sqrt{qx+r}}{s},$ where $p,q,r,s$ are integers and $p,s$ are relatively prime, find $p+q+r+s$. Determine the value of $p+q+r+s$. \( \boxed{} \)

Answer: 4


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\frac{1}{(1+x^2)^2}dx \) using residue methods.

Answer: \dfrac{\pi}{2}


Question: Determine whether the function \( f(x) = \|x\|_p \), where \( p \ge 1 \), is a Lipschitz function. If it is, find its Lipschitz constant for a general \( p \).

Answer: 1


Question: Determine the Galois group of the polynomial $x^3 - 5$ over the field of rational numbers $\mathbb{Q}$.

Answer: S_3


Question: Determine for which values of \( \alpha \in \mathbb{R} \) the series \( \sum_{n = 1}^{\infty}\left(\sqrt[n]{3\}\, - 1\right)^{\alpha} \) converges.

Answer: \alpha > 1


Question: Suppose $f : (a,b) \rightarrow \mathbb{R}$ is continuously differentiable. Is it true that the number of zeros of $f$ in $(a,b)$ is at most one more than the number of stationary points of $f$ in $(a,b)$?

Answer: Yes


Question: Calculate the area enclosed by the curve $g(x)$, the lines $x=-3$ and $x=5$, and the $x$-axis, where $g(x)$ is the inverse function of $f(x)=x^3+3x+1$. Use techniques of definite integrals to find the area.

Answer: \dfrac{9}{2}


Question: Solve the equation: \[ \sqrt{3x+1} - \sqrt{6-x} + 3x^2 - 14x - 8 = 0 \]

Answer: 5


Question: Find the norm of the linear functional \( \phi(x_n) = \sum_{n\ge 1} (-2)^{1-n}x_n \), where \( \phi: c_0 \to \mathbb{R} \) and \( c_0 \) is the set of sequences converging to 0.

Answer: 2


Question: Find the limit of the series \( \sum_{n=1}^{\infty} \frac{2^{[\sqrt{n}]}+2^{-[\sqrt{n}]}}{2^n} \), where \([\ \ ]\) denotes the nearest integer function.

Answer: 3


Question: Determine the number of ways to factor the number $2^5 \times 3^7$ into three integers that are pairwise coprime.

Answer: 9


Question: Does there exist an infinite group that has only one subgroup of infinite index?

Answer: \mathbb{Z}


Question: Given two real polynomials $p(x)$ and $q(x)$ of degrees $a$ and $b$ respectively, such that $p(x^3 - 1) = p(x)q(x)$ for every $x \in \mathbb{R}$, determine the number of distinct real roots of $p(x)$. Provide a justification for your answer.

Answer: 1


Question: Evaluate the limit: $$\lim_{h\to 0} \frac {1}{h} \int_2^{2+h} \sin(x^2) \, dx $$

Answer: \sin 4


Question: Evaluate the integral \( \int_\Omega xye^{x^2-y^2}dxdy \) over the domain \( \Omega = \{(x,y)\mid 1\leq x^2-y^2\leq 9, 0\leq x \leq 4, y\geq 0\} \).

Answer: 2e^9 - 4e


Question: Evaluate the integral \( \int_{\gamma} \frac{1}{z} \, dz \), where \( \gamma \) is a circle centered at \( z_0 \neq 0 \) with radius \( r < |z_0| \), and the curve \( \gamma \) does not enclose the origin.

Answer: 0


Question:
Determine the tight bound complexity \(\Theta\) for the following nested loop system:

```
for i = n to 2n do
    for j = 2i to 4i do
        for k = 3j to 6j do
            ...
```

Answer: \Theta(n^3)


Question: Evaluate the limit: $$ L=\lim _{k \rightarrow \infty}\left\{\frac{e^{\frac{1}{k}}+2 e^{\frac{2}{k}}+3 e^{\frac{3}{k}}+\cdots+k e^{\frac{k}{k}}}{k^2}\right\} $$ Find the value of \( L \).

Answer: 1


Question: Let $X$ be a finite set and let $\mathcal{P}(X)$ denote the power set of $X$. Suppose $\Gamma$ is a sub-lattice of $\mathcal{P}(X)$, meaning $\Gamma$ is closed under union and intersection. Consider an increasing function $f: \Gamma \to \mathbb{R}$, where for all $A, B \in \Gamma$, if $A \subseteq B$, then $f(A) \leq f(B)$. Determine whether it is always possible to extend $f$ to an increasing function $\widetilde{f}: \mathcal{P}(X) \to \mathbb{R}$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the real convergent improper integral using residue theory: \[ \int_0^\infty \frac{\sin(\pi x)}{x(1-x^2)} \, \textrm{d}x \]

Answer: \pi


Question: Does the set of endomorphisms of an abelian group always form a commutative ring?

Answer: No


Question: Let $F(x,y,z)=(6x,-2y,5z)$ be a vector field and $S$ be the surface of the sphere centered at $(1,0,1)$ with radius $5$. Calculate the flux of $F$ across the surface $S$ using Gauss's theorem.

Answer: 1500\pi


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

Answer: \dfrac{12}{13}


Question: Determine the minimum value of the quadratic form \( Q(\vec{x}) = -2x^2 + 2\sqrt{33}xy + 6y^2 \), where \( \vec{x} = \begin{bmatrix} x \\ y \end{bmatrix} \), given that \( \left \| \vec{x} \right \| = 5 \).

Answer: -125


Question: Determine whether the following convergence in distribution holds for a sequence of i.i.d. random variables $(X_i)_{i \geq 1}$, each with mean 0 and variance 1: $$\left(\frac{1}{\sqrt{n}}\sum\limits_{i=1}^{n^a t}{X_i}\right)_t \xrightarrow{d} 0$$ for $0<a<\frac{1}{2}$. Justify your answer.

Answer: Yes


Question: Determine whether the derivative $f'(0)$ exists for the function $$f\left(x\right)=\begin{cases} x^{2}\sin\left(5/x\right) & \text{if }x\neq0;\\ 0 & \text{if }x=0. \end{cases}$$ If it exists, find its value.

Answer: 0


Question: Consider a non-compact set $A \subset \mathbb{R}$. Determine whether there exists a continuous function $f: A \to \mathbb{R}$ that is bounded but does not attain its extreme values.

Answer: Yes


Question: Evaluate the surface integral \(\int_S \mathbf{G}\cdot\mathbf{dS}\) for the vector field \(\mathbf{G}=\langle 2x,7y,3z\rangle\), where \(S\) is the closed surface that bounds a region of unit volume.

Answer: 12


Question:
Determine the values of the parameter \(a\) for which the solution set of the following nonlinear system of equations forms a subspace in \(\mathbb{R}^3\):

\[x + (1-a^2)y^3 - 2z = 0\]
\[x + y - (2a^2 + 2a)|z| = a^3 - a\]

Answer: -1


Question: In a directed graph $G = (V, E)$ with $n$ vertices, where every vertex has at least one incoming edge, determine if it is always true that there exists a dominating set $D \subseteq V$ such that $|D| \geq \frac{2n}{3}$. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Find the value of $x$ for which the infinite series $$a \cdot a^{2\log(x)} \cdot a^{4\log^2(x)} \cdot a^{8\log^3(x)} \dotsb = \frac{1}{a^7}$$ holds true for all positive real numbers $a$ other than $1$. Provide your answer in terms of $x$.

Answer: e^{\frac{4}{7}}


Question: Determine the minimum value of the expression \(\frac{p^2+ q^2 + r^2}{s^2}\), where \(p\), \(q\), \(r\), and \(s\) are the lengths of the sides of a quadrilateral.

Answer: \dfrac{1}{3}


Question:
Consider a triangular array of row-wise independent random variables $(X_{n,1}, \ldots, X_{n,n})_{n \geq 1}$ and assume $\mu_n = f_n(X_{n,1}, \ldots, X_{n,n})$, for some measurable map $f_n: \mathbb{R}^n \to \mathbb{R}$. Assume that, almost surely, $\mu_n \to c$, where $c$ is a finite constant. For each integer $n \geq 1$ and $i = 1, \ldots, n$, define $\bar{X}_{n,i} := X_{n,i} - \mu_n$ and $\tilde{X}_{n,i} = X_{n,i} - c$; then let $$ \bar{X}_n := (\bar{X}_{n,1}, \ldots, \bar{X}_{n,n}), \\
\tilde{X}_n := (\tilde{X}_{n,1}, \ldots, \tilde{X}_{n,n}). $$ Determine if the total variation distance $d_{TV}$ between $P(\bar{X}_n \in \cdot)$ and $P(\tilde{X}_n \in \cdot)$ converges to zero.

Answer: No


Question: Consider a closed smooth Riemannian manifold $M$ and another closed Riemannian manifold $N$. Let $0 < \alpha < 1$, $f \in C^{\frac{\alpha}{2},\alpha}([0,T]\times M)$, and $u_0 \in C^{2+\alpha}(M)$. Does the existence and uniqueness theorem hold for a solution $u \in C^{1+\frac{\alpha}{2},2+\alpha}([0,T]\times M)$ to the equation $\partial_t u - \tau(u) = f$ on $[0,T]\times M$ with the initial condition $u(0,\cdot) = u_0$ on $M$?

Answer: Yes


Question: Evaluate the line integral \( \int_C \mathbf{F} \cdot \, \mathrm{d} \mathbf{r} \) where \( \mathbf{F} = (e^x + xy) \mathbf{i} + (e^y - xy^2) \mathbf{j} \) and \( C \) is the circle \( x^2 + y^2 = 4 \) oriented clockwise.

Answer: 4\pi


Question:
Evaluate the Lebesgue integral of the function \( f(x) \) defined as follows:
\[ f(x) = \begin{cases} 1 & \text{if } x \text{ is irrational in } [0,1] \\ 0 & \text{if } x \text{ is rational in } [0,1] \end{cases} \]
Determine the value of \( \int_{0}^{1} f(x) \, d\mu \).

Answer: 1


Question: Solve the equation \(3\sqrt{2+x}-6\sqrt{2-x}+4\sqrt{4-x^2}=10-3x\) for real numbers \(x\).

Answer: \dfrac{6}{5}


Question: Let $A$ be a bounded, positive operator on a Hilbert space $H$. Determine whether the following statement is true: $$\|A\| \leq 1 \implies \|Id - A\| \leq 1.$$

Answer: True


Question: Find the number of incongruent integers of order $2^{10}$ modulo $3 \cdot 2^{12}$.

Answer: 2048


Question: Find the natural numbers \( n \) for which the integral \( \int_{0}^{\infty} \cos x \cdot \cos 2x \cdot \cos 3x \cdots \cos nx \cdot \frac{\sin 2020x}{x} \, \operatorname{d}x = \frac{\pi}{2} \).

Answer: 63


Question: In a right triangle $ABC$ with integer side lengths, the area is twice the perimeter. Find the sum of all possible circumradii for triangle $ABC$. Express your answer as a single number.

Answer: \dfrac{87}{2}


Question: Does the functor $X \times -$ preserve small colimits for all locales $X$ in the category of locales $\mathbf{Loc}$?

Answer: No


Question: Determine the natural density of nonnegative integers whose fourth power ends in 0000.

Answer: \dfrac{1}{10}


Question: Determine whether there exist functions $f(x, t)$, where $t$ is a parameter, such that $f(x, t)$ is absolutely continuous in $x$ for each fixed $t$, but there does not exist an integrable function $B(t)$ satisfying $\sup_{x} |f(x, t)| < B(t)$ almost everywhere. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: For every polynomial \( f(x) \in \mathbb{Z}[x] \) of degree greater than 1, there exists a prime number \( p \) such that \( f(x) \) is reducible in the field \( \mathbb{F}_p \).

Answer: True


Question: Determine whether the condition $a > 0$ is necessary in the lemma: If $a, b, c$ are real numbers such that $a > 0$ and $a\lambda^2 + 2b\lambda + c \geq 0$ for all real numbers $\lambda$, then $b^2 \leq ac$. Provide a justification for your answer.

Answer: No


Question: Calculate the sum of the series: \[ \sum_{k=0}^{\min\{m,n\}} {m\choose k}(-1)^k {m+n-k\choose n-k} \]

Answer: 1


Question: Let $\mu$ be a regular Borel measure on a compact subset $K \subseteq [0, \infty[$. If $\inf\{C > 0\mid \mu\{x\in K: x > C\}=0\}$ is equal to $M$, does this imply that the support of $\mu$ is contained in $[0, M]$? Answer yes or no and justify your answer.

Answer: Yes


Question: Evaluate the limit \[ \lim_{x\to 1}\left(\frac{x^{x^{x{^{.^{.^{.}}}}}}-1}{x-1}\right)^{\displaystyle\frac{1}{\sqrt{x-1}}} \] without using the Lambert W function.

Answer: 1


Question: Find the residue of the function \( f(z) = \frac{1}{(z^2+1)^2} \) at the point \( z = i \).

Answer: -\dfrac{i}{4}


Question: Find the value of \( a \) such that the following equation holds: \[ \int_{0}^{\alpha b}(a^x-1)\,dx = \int_{\alpha b}^{b}(a^x-1)\,dx \] where \( 0<\alpha<1 \) and \( b\geq1 \).

Answer: 1


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2} \frac{\sin^2(x)}{1+b^x} \, dx \).

Answer: \dfrac{\pi}{4}


Question: Determine whether the following statement is mathematically correct: Given that \( \lim_{x\to\infty} \sum_{n=1}^{x} a_{n} = \sum_{n=1}^{\infty} a_{n} \) and \( \lim_{x\to\infty} \sum_{n=1}^{x^2} a_{n} \) is known to be convergent, is it true that \( \lim_{x\to\infty} \sum_{n=1}^{x^2} a_{n} = \sum_{n=1}^{\lim_{x\to\infty}x^2} a_{n} \)?

Answer: Yes


Question: Determine whether the following statement is true or false: If $G$ is a group and $N \lhd G$, and both $G/N$ and $N$ are $p$-groups for some prime $p$, then $G$ must also be a $p$-group.

Answer: A


Question: Evaluate the limit: $$\lim_{x \rightarrow \infty} \frac{\int_0^x\frac{\sin(t)dt}{t}-x}{x^3}$$

Answer: 0


Question: Consider three buses whose arrival times are exponentially distributed with rates \( \lambda_1 = 2 \), \( \lambda_2 = 5 \), and \( \lambda_3 = 10 \). Determine the probability distribution of the minimum waiting time required to board any of the three buses. Calculate the expected waiting time for the minimum waiting time.

Answer: \dfrac{1}{17}


Question: In the polynomial ring $R_n = \mathbb{C}[x_1, \ldots, x_n]$ for $n \geq 2$, consider the intersection $I$ of the ideals $\langle x_1 - c \rangle$ as $c$ ranges over $\mathbb{C}$. Is $I = 0$?

Answer: 0


Question: Determine whether the sequence \( \left( \frac{P(n)}{P(n + 1)} \right) \) is convergent, where \( P(x) = ax^3 + bx^2 + cx + d \) and \( a, b, c, d \in \mathbb{R} \). If the sequence is convergent, calculate its limit.

Answer: 1


Question: Determine the order of the numerical scheme given by \(y_{n+1} = y_n + \frac{h}{2}\left(f(x_n, y_n) + f(x_{n+1}, y_{n+1})\right)\) by inserting the exact solution into the scheme.

Answer: 2


Question:
Consider the $2\pi$-periodic function $f:\Bbb{R}\to\Bbb{R}$ defined as follows:

$$f(x) = \begin{cases}
1 & \text{if } x = 0 \\
1 + \sin\left(\frac{\pi^2}{x}\right) & \text{if } x \in [-\pi,\pi]\setminus\{0\}
\end{cases}$$

Determine whether the Fourier series of $f$ converges at $x=0$. If it converges, find the sum at $x=0$. \( \boxed{} \)

Answer: 1


Question: Determine the condition on the parameter $s \in \mathbb{R}$ such that the series \( \sum_{\xi\in\mathbb{Z}^n}\frac{1}{(1+|\xi|^2)^{s/2}} \) converges, where \(|\xi|\) is the Euclidean norm of \(\xi\).

Answer: s > n


Question: Determine if the operator \(T\colon\ell^2\to\ell^2, (a_n)_{n\in\mathbb{N}}\mapsto\left(\frac{a_n+a_{n+1}}{2}\right)_{n\in\mathbb{N}}\) is an isometry. Also, find the norm of this operator.

Answer: 1


Question: Let $E/F$ be a finite extension. Let $L$ and $K$ be two intermediate fields such that $L \cap K = F$ and $[L : F] [K:F] = [E:F]$. Does it necessarily follow that the compositum $LK$ equals $E$?

Answer: Yes


Question: Let $U$ be an open subset of $\mathbb{R}^n$ that is $C^k$ ($0 \leq k \leq \infty$). A set is $C^k$ if, for each $x \in \partial U$, there exists an open neighborhood $V$ of $x$ and a $C^k$ function $f$ defined on $\mathbb{R}^{n-1}$ such that $V \cap U = V \cap \{(x,y):y>f(x)\}$ and $V \cap \partial U = V \cap \{(x,y):y=f(x)\}$. Consider a $C^k$ diffeomorphism $h:\mathbb{R}^n \rightarrow \mathbb{R}^n$. Is the image $h(U)$ also a $C^k$ open set?

Answer: Yes


Question: If $(I-A)^k=0$ for some positive integer $k$, is $A$ invertible?

Answer: Yes


Question: Find the minimum value of the sum \(\sum_{i<j} x_{i}x_{j}\) for a set of numbers \(A = \{x_1, x_2, \ldots, x_n\}\) such that \(0 \leq x_i \leq 1\) and \(\sum_{i=1}^{n} x_i = 1\).

Answer: 0


Question: Evaluate the limit: $$\lim_{b \to 0}\int _0^{\infty}\frac{\left(x^2-1\right)\cos \left(bx\right)}{\left(x^2+1\right)^2}dx$$ using real analysis methods.

Answer: 0


Question: Calculate the contour integral \( \oint_{\gamma}\frac{1}{z}\,\mathrm{d}z \), where \( \gamma \) is the unit circle parameterized by \( z(t) = \cos(t) + i\sin(t) \) for \( 0 \le t \le 2\pi \).

Answer: 2\pi i


Question: Given any sequence of random variables $Y_n$, determine whether there exist constants $d_n$ tending to zero such that $d_nY_n$ converges to 0 almost surely.

Answer: Yes


Question: Given the function \( x^2 = \int_{\tan(x)}^{y(x)}\frac{1}{\sqrt{2+t^2}}\,\mathrm dt \), and knowing that \( y'(0) = 1 \), find \( y''(0) \).

Answer: 2\sqrt{2}


Question: Calculate the quotient group of the chain complex $C_0$ by the image of $C_1$ for the simplicial complex with vertices \( \{x, y, z\} \) and edges \( \{xy, xz, yz\} \).

Answer: \mathbb{Z}


Question: Evaluate the integral \( \iiint\limits_{D} xy \, \mathrm{d}V \) where \( D \) is the region bounded by the planes \( y-x=0 \), \( y-x=2 \), \( z-y=0 \), \( z-y=1 \), \( z=0 \), and \( z=3 \). Use the change of variables \( u = y-x \), \( v = z-y \), and \( w = z \).

Answer: 5


Question: Let X and Y be two independent random variables, each uniformly distributed on the interval [0,1]. Find the probability density function (pdf) of the random variable Z, where Z = XY.

Answer: -\ln z


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

Answer: e


Question: Provide an example of a ring where an ideal generated by a prime element is not maximal.

Answer: \mathbb{Z}[x]


Question: Find the value of \( \liminf_{n\to\infty} f_n \), where \( f_n = \mathbb{1}_{[n,n+1]} \) is the indicator function that equals 1 on the interval \([n,n+1]\) and 0 elsewhere.

Answer: 0


Question: Evaluate the limit: $$ \lim_{n\to\infty}\frac{\sin(1)+\sin\left(\frac{1}{2}\right)+\dots+\sin\left(\frac{1}{n}\right)}{\ln(n)} $$

Answer: 1


Question: Evaluate the integral: \[ \int_0^1 \frac{dx}{\sqrt{x + \sqrt{x^2 + \sqrt{x^3}}}}. \]

Answer: 2


Question: Evaluate the limit: $$\lim_{{x \to 0}} \frac{1-\cos x(\cos2x)^{\frac{1}{2}}(\cos3x)^{\frac{1}{3}}}{x^2}$$

Answer: 3


Question: Determine whether there exists a ring $R$ containing an invertible element $p$ such that $A_p = B_p$, where $p - 1$ is a regular element.

Answer: Yes