Question:
Determine which of the following statements is true about second-order linear differential equations:

1. A second-order homogeneous linear differential equation with continuous coefficients in an open interval can have $x^2$ and $x^3$ as solutions.
2. A second-order non-homogeneous linear differential equation with constant coefficients can have $\sin(x)$ and $\cos(x)$ as solutions.
3. A second-order homogeneous linear differential equation with continuous coefficients in an open interval can have $x$ and $x^2$ as solutions if and only if the interval does not contain $x=0$.

Select the correct statement.

Answer: 3


Question: Calculate the limit: $$\lim\limits_{n\to +\infty}\frac{\left(1+\frac{1}{n^3}\right)^{n^4}}{\left(1+\frac{1}{(n+1)^3}\right)^{(n+1)^4}}$$

Answer: \dfrac{1}{e}


Question: Let $X$ be a compact metric space, and let $Y$ and $Z$ be subspaces of $X$ such that $Y$ is open and $Z$ is closed. Suppose $C$ is a connected subset of $Y \cap Z$ and is both a component of $Y$ and a component of $Z$. Determine whether $C$ is a component of $Y \cup Z$.

Answer: Yes


Question: Evaluate the limit: $$ \lim _{x \rightarrow 0^{+}} \frac{\tan 3 x \sqrt{1-\cos 4 x}}{x^{2}+x^{3}} $$ without using L'Hôpital's rule multiple times.

Answer: 6\sqrt{2}


Question: Determine whether the limit of a continuous positive function on $[1, \infty)$ or $[0, \infty)$ necessarily approaches zero if the integral of the function over the interval converges.

Answer: No


Question: Is it possible to construct a smooth path that passes through any given four points in \(\mathbb{R}^n\)?

Answer: Yes


Question: Let $Y$ be an exponentially distributed random variable with rate parameter 3, i.e., $Y \sim \text{Exp}(3)$. Given $Y = y$, the random variable $X$ is uniformly distributed over the interval $[y, 3y]$. Compute the variance of $X$.

Answer: \dfrac{14}{27}


Question: Find the remainder when $325^{242}$ is divided by 327 without using Euler's function or Fermat's theorem.

Answer: 289


Question: Evaluate the infinite series: $$\sum_{n=1}^{\infty}\frac{g(n)}{n(n+1)}$$ where the function $g(n)$ is defined as follows: $$ g(n) = \begin{cases}0&n=1\\g(z)& n = 2z\\g(z) - 1&n = 2z + 1\end{cases} $$ Find the value of this summation.

Answer: 1 - 2\ln 2


Question: Solve for $x$ in the equation: \[ x\lfloor x + 2 \rfloor +\lfloor 2x - 2 \rfloor +3x =12 \] given that $0 \leq \{x\} < 1$, where $\{x\}$ represents the fractional part of $x$. The equation has been transformed into \[ x^2 +7x-x\{x+2\}-\{2x-2\}=14. \] Find the solution for $x$. \( \lfloor \cdot \rfloor \) denotes the floor function.

Answer: \dfrac{11}{6}


Question: Consider the curve \( C \) given by the equation \( \sqrt{x} + \sqrt{y} = \sqrt{a} \), where \( a > 0 \) is a constant. Let \((x_0, y_0)\) be a point on \( C \) such that \( x_0 > 0 \) and \( y_0 > 0 \). The tangent line to \( C \) at \((x_0, y_0)\) intersects the x-axis at \((x_1, 0)\) and the y-axis at \((0, y_1)\). Prove that \( x_1 + y_1 = a \).

Answer: a


Question: Compute \( \lim_{x\to 0^{+}}x\int_x^1 \frac{g'(t)}{t}dt \) given that \( g \) is differentiable on \([0,1]\) and \( \lim_{x\to 0}g(x) \) exists.

Answer: 0


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

Answer: 0


Question: Evaluate the limit: $$\lim_{\delta\to 0}\int_{\frac{\delta}{1-\delta}}^{\delta}\frac{e^{-x^2}}{x^2}dx$$

Answer: -1


Question: Determine whether the following statement is true or false: For any quandle $Q$, if $y \ast k = y \ast w$ for elements $y, k, w \in Q$, then $k = w$. Provide a justification for your answer.

Answer: False


Question: Evaluate the integral \( \int_{\pi/4}^{3\pi/4}\frac{1}{1+\sin x}\operatorname{d}x \).

Answer: 2\sqrt{2} - 2


Question: In triangle ABC, angle A measures 60 degrees. The angle bisectors of angles B and C, denoted as BE and CF respectively, intersect at point P. Given that BP = 5 and CP = 7, determine the length of side BC.

Answer: \sqrt{109}


Question: Determine the radius of convergence for the complex series \( \sum z^{n!} \), where \( z \) is a complex number and \( n! \) denotes the factorial of \( n \).

Answer: 1


Question: Consider two metric spaces $X$ and $Y$. If a function $g:X\to Y$ is a bijection but does not preserve distance, can the image of a dense subset of $X$ under $g$ be dense in $Y$?

Answer: Yes


Question: Let $f:A\times B\rightarrow C$ be a continuous map from the product of topological spaces $A$ and $B$ to a metric space $C$, with $B$ compact. For any $a\in A$, define $f_a := f(a,\cdot):B\to C$. If a sequence $a_n \to a$ in $A$, does $f_{a_n}$ converge uniformly to $f_a$ on $B$? Provide a justification for your answer.

Answer: Yes


Question: In a class of $n$ students, where $n$ is even, is it possible for the teacher to pair them up each day so that no student is left unmatched and no two students are paired together more than once until all possible pairings have been made?

Answer: Yes


Question: Find the maximum value of the expression \( (\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{c}) \cdot (\vec{c} \times \vec{a}) + (\vec{c} \times \vec{a}) \cdot (\vec{a} \times \vec{b}) \), given that \( \vec{a}, \vec{b}, \vec{c} \) are unit vectors satisfying \( |\vec{a} + \vec{b} + \vec{c}| = \sqrt{3} \).

Answer: 0


Question: In a unital non-commutative ring, given elements $a, b, c$ where $c$ is an inverse of $1-ab$ such that $c(1-ab) = 1$, find an inverse for the element $1-ba$.

Answer: 1 + bca


Question: Let $L/K$ be a finite abelian extension of local fields. If the norm group of the unramified extension of $K$ of degree $n$ is contained in $N(L^*)$, does it imply that $L/K$ is unramified? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_0^\pi \frac{\sin^2 x}{1+\cos x} \, dx \) using the fact that \( \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6} \).

Answer: \pi


Question: Determine the boundary of the set \(\mathbb{Q}^n\) when considered as a subset of \(\mathbb{R}^n\) with the standard topology. Is it true that \(\partial \mathbb{Q}^n = \emptyset\)?

Answer: \partial \mathbb{Q}^n = \mathbb{R}^n


Question: Evaluate the integral \( \int_C \frac{\sin(z)}{z^2} \, dz \), where \( C \) is the unit circle in \( \mathbb{C} \).

Answer: 2\pi i


Question: Find the value of $r$ such that the set $A = \{0, 1, 4, 16, r\}$ is a $\lambda$-difference set in $\mathbb{Z}_{21}$ for some $\lambda$. What is the value of $r$?

Answer: 14


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

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


Question: Evaluate the limit: $$\lim_{x\to0^+} \frac{\cos(x)\ln(1/x)}{\ln(x)}$$

Answer: -1


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(\sum_{k=1}^{n}\frac{I_{k+1}}{k}\right) \] where \( I_n=\int^1_0x^ne^x\,dx \). Is it necessary to compute the values of \( I_n \) first?

Answer: e - 1


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

Answer: 0


Question: Determine whether the following statement is true or false: If a function $f(x)$ is continuous on the interval $[a, \infty)$ and the limit $\lim_{{x\to\infty}} \int_{x}^{2x} f(t)dt$ equals 0, then the improper integral $\int_{a}^{\infty} f(t)dt$ converges. Use Cauchy's test for integral convergence to justify your answer.

Answer: False


Question: Determine whether there exists a Möbius transformation that maps the unit disk $D_1(0)$ to its complement. If such a transformation exists, find its explicit form.

Answer: f(z) = \dfrac{1}{z}


Question: Determine whether a commutative ring can have a polynomial with uncountably many roots, given that the polynomial has a nonzero constant term. Provide a justification for your answer.

Answer: Yes


Question: Given $X \sim N(0,1)$ and $Y \sim N(X, 1)$, where $Y-X$ is independent of $X$, calculate the probability $P(Y<0 \mid X>0)$. This probability is relevant in the context of pricing a Slalom option in a Normal model without drift.

Answer: \dfrac{1}{4}


Question: Consider three mutually interpenetrating solids: two right circular pyramids and one right circular prism. The base of each pyramid and both bases of the prism are sealed by flat circular surfaces. Determine the maximum number of bounded volumes that can be formed, considering only the surfaces of the solids as boundaries and counting only volumes that are not further subdivided.

Answer: 7


Question:
Given two independent random variables, X and Y, with probability density functions:

For X: \( f_X(x) = 4x^3 \) for \( 0 < x < 1 \), and 0 otherwise.

For Y: \( f_Y(y) = \frac{y}{2} \) for \( 0 < y < 2 \), and 0 otherwise.

Calculate the probability \( P(Y > 4X) \).

Answer: \dfrac{1}{48}


Question: Evaluate the double integral \( \int_0^{\frac{\pi}{2}}\int_{x}^{\frac{\pi}{2}} \frac{\cos{y}}{y} \, dy \, dx \).

Answer: 1


Question: Given two matrices A and B, where S denotes the column space of A and T denotes the column space of B, find the matrix M whose column space is the sum of S and T. Assume both A and B have columns in \( \mathbb{R}^m \).

Answer: [A \quad B]


Question: Determine whether the set \( B=\bigcup_{n\in\mathbb{N}} \left[ 2n - \frac{1}{4n}, 2n+\frac{1}{4n}\right] \) is open or closed in \( \mathbb{R} \) with the usual topology. If it is not closed, find its closure.

Answer: B


Question: Evaluate the surface integral \(\iint_S \mathbf{F} \cdot d\mathbf{S}\) using the Divergence Theorem, where the vector field \(\mathbf{F}=(xy^2, yz^2, x^2z)\) and \(S\) is the sphere of radius \(1\) centered at the origin. Compute the divergence \(\text{div}\mathbf{F}\), express the surface integral as a triple integral over the ball \(B\) with the same radius, and convert the triple integral to spherical coordinates to find its value.

Answer: \dfrac{4\pi}{5}


Question: Given equivalence relations $R$ on a set $X$ and $S$ on the quotient set $X/R$, construct an equivalence relation $T$ on $X$ such that the equivalence classes of the quotient set $(X/R)/S$ are in bijection with the equivalence classes of $X/T$. Describe the construction of $T$.

Answer: T = \{(x, y) \in X \times X \mid ([x]_R, [y]_R) \in S\}


Question: Does a map $T:V\to W$ between vector spaces $V$ and $W$ that preserves addition, i.e., $T(v_1 + v_2)=T(v_1)+T(v_2)$ for all $v_1,v_2\in V$, necessarily make $T$ a linear map? Answer yes or no and justify your answer.

Answer: No


Question: Let $A$ be an $n \times n$ complex matrix. Find the eigenvalues of the linear operator $T$ on $n \times n$ matrices defined by $T(B) = AB - BA$, in terms of the eigenvalues of $A$.

Answer: \lambda_i - \lambda_j


Question: Determine the number of roots of the polynomial $z^p + z - 1$, where $p$ is a prime number, that lie in the region $|z| < 1$. If there are any, state how many.

Answer: 1


Question:
Given an even number \( n \) and non-negative real numbers \( a_1, a_2, \ldots, a_n \) such that \( a_1 + a_2 + \cdots + a_n = 1 \), find the maximum possible value of the sum:
\[ \sum_{1 \leq i < j \leq n} \min\left\{\left(i - j\right)^2, \left(n + i - j\right)^2\right\}a_{i}a_{j} \]

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


Question:
In a $4$-state Markov Chain, the transition matrix $P$ is given by:

$$P = \left(\begin{array}{cccc} \frac{1}{4}&\frac{1}{4}&\frac{1}{4}&\frac{1}{4} \\ \frac{1}{2}&0&0&\frac{1}{2} \\ \frac{1}{3}&\frac{1}{3}&\frac{1}{3}&0 \\ 0&0&0&1 \end{array} \right)$$

Starting from state $1$, calculate the expected time until absorption into state $4$. Provide your answer as a single number.

Answer: 4


Question:
Calculate the sum of the power series:

$$S = -\sum_{n=1}^{\infty} \frac{(1-x)^n}{n}$$

where $x = 1.74$. Provide your answer to two decimal places.

Answer: 0.55


Question: Suppose $X$ is a metric space such that the image of every uniformly continuous function $f: X \to \mathbb{R}$ is closed. If $X$ is also complete, is $X$ necessarily compact? Answer yes or no and justify your answer.

Answer: Yes


Question: Let \( \{X_n\} \) be a sequence of i.i.d. random variables with mean \( \mu = 1 \) and variance \( \sigma^2 = 3 \). Determine whether the sequence \( \frac{1}{n}\sum_{i=1}^n X_i \) converges almost surely to 1.

Answer: Yes


Question: Is the set of endpoints in the iterative construction of the Cantor set dense in the Cantor set? Provide a yes or no answer.

Answer: Yes


Question: Determine the number of irreducible representations of the symmetric group $S_n$ over a field of characteristic $0$ that have dimension greater than $1$.

Answer: p(n) - 2


Question: Determine the radius of convergence for the series: $$\sum_{n=1}^\infty \sqrt{2\pi n}\cdot \left(\frac{x}{e}\right)^n.$$

Answer: e


Question: Determine the radius of convergence for the series \( \sum_{n=0}^{\infty} \frac{(\arctan n)^n}{(3 \cdot 2\pi)^n} x^n \).

Answer: 12


Question: Evaluate the infinite sum \( \sum_{n=1}^{\infty} \frac{\phi(n)}{3^n -1} \), where \( \phi(n) \) is the Euler's totient function.

Answer: \dfrac{3}{4}


Question: Determine whether there exists a function \( f: C \to \mathbb{R} \) such that \( f \) is one-to-one and the graph of \( f \) is dense in \( C \times \mathbb{R} \).

Answer: Yes


Question: Evaluate the integral \( \int_{C}(z^2+\bar{z}z+1)dz \), where \( C \) is the semicircular arc of the circle \( |z|=2 \) from \( z=2 \) to \( z=-2 \) in the upper half-plane.

Answer: -\dfrac{76}{3}


Question: Evaluate the integral \( \int_{0}^{\infty} t^3 e^{-3t} \, dt \) using the Laplace transform.

Answer: \dfrac{2}{27}


Question: In an isosceles trapezoid ABCD, where AD = BC = CD = 12 units, a circle with a radius of 8 units is circumscribed around the trapezoid. If AC is the diagonal and AB < DC, find the length of AB.

Answer: 9


Question: Let $X$ be a metric space and $f:X \to X$ be a bijective map. If $f$ carries every connected set of $X$ to a connected set of $X$, does it follow that $f^{-1}$ also carries every connected set to a connected set? Provide a proof or a counterexample.

Answer: Yes


Question: Evaluate the limit \(\displaystyle\lim_{x\to a}\frac{a^{a^{x}}-{a^{x^{a}}}}{a^x-x^a}\) using Lagrange's mean value theorem.

Answer: a^{a^a} \ln a


Question: Evaluate the expected value \( \mathbf{E}[\cos(B(t))] \) where \( B(t) \) is a standard Brownian motion.

Answer: e^{-t/2}


Question: Find the limit of the sequence as \( n \to \infty \): $$\left(\frac{\ln 2}{2}+\frac{\ln 3}{3}+\cdots + \frac{\ln n}{n} \right)^{\frac{1}{n}}$$

Answer: 1


Question: Given the second-order linear homogeneous differential equation \(2x^2y''-(x+x^2)y'+(x^2-2)y=0\) with two linearly independent solutions \(\phi_1(x)\) and \(\phi_2(x)\), where \(\phi_1(0)=0\), find the smallest positive integer \(n\) such that \(\displaystyle \lim_{x \to 0}x^n\frac{\phi_2(x)}{\phi_1(x)}=0\).

Answer: 3


Question: Evaluate \( \lim_{n \rightarrow \infty} \frac{1}{n}\sum_{k=1}^n \sqrt{\frac{n-k+1}{n}} \).

Answer: \dfrac{2}{3}


Question: Find the minimum value of $n + z^3$ for complex numbers $z$ such that both $z$ and $z+1$ are $n^{\text{th}}$ roots of unity, given that $n$ is a multiple of 5 and $z^n = (z+1)^n = 1$. What is this minimum value?

Answer: 31


Question: Suppose $T$ is a self-adjoint operator on a separable Hilbert space $H$ with only point spectrum. Is it true that one can find a basis of $H$ consisting of eigenvectors of $T$?

Answer: Yes


Question: Evaluate the definite integral \( \int_{-\infty}^{\infty} \frac{\sin^2(\pi Rx)}{R(\pi x)^2}dx \) for \( R > 0 \).

Answer: 1


Question: Find the limit of the expression \(\left(\int_{-x}^{x} \cos(t^2)\, dt\right)^{1/x}\) as \(x\to 0\) using the Sandwich theorem.

Answer: 0


Question: Evaluate the limit: $$ \lim_{n \to \infty} \sqrt[n]{\left(\frac{2n+4}{2n-1}\right)^{\frac{n^3+1}{2n}}} $$

Answer: e^{\frac{5}{4}}


Question: Evaluate the integral \( \int_0^b \left(y-\sqrt{b^2-y^2}\right)^4\,dy \) using a trigonometric substitution.

Answer: \dfrac{b^5}{5}


Question: Determine whether the following statement is true or false: Let $T$ be a semigroup and let $\varphi : S \to T$ be a surjective semigroup morphism. Then for every group $H \subseteq S$, there exists a group $G \subseteq T$ such that $\varphi(H) = G$. Provide a justification for your answer.

Answer: True


Question: Evaluate the summation \( \sum^{2020}_{r=0} (-1)^r (r + 1) \binom{2020}{r} \).

Answer: 0


Question: Determine whether there exists a sequence of functions $ f_i: [0,1]^2 \rightarrow \mathbb{R} $ such that $ \int_{[0,1]^2} | f_i | \, {\rm d}x \leq 1$ and $ \int_{[0,1]^2} | f_i - f_j | \, {\rm d}x = c > 0 $ for all $ i \neq j $, where $c$ is a constant.

Answer: Yes


Question: If $A \cap B$ has a one-to-one correspondence with $A$, and $A \cup B \neq A$, does it necessarily follow that $B$ is an infinite set? Answer yes or no and justify your answer.

Answer: No


Question: Evaluate the complex integral \(\int_{C} \frac{|z| e^z}{z^2} \, dz,\) where \(C\) is a circle with radius 2. Use the properties of contour integrals to find the result, which is \(4\pi i\).

Answer: 4\pi i


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

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


Question: In a field extension $F$ of $\mathbb{Z}_5$, suppose $x \in F - \mathbb{Z}_5$. If $x^{7}=3$, determine the degree of the field extension $\deg(x/\mathbb{Z}_5)$. \(\boxed{\text{Your answer here}}\)

Answer: 6


Question: Find the supremum of the total variation of a continuously differentiable function on the interval $[0,1]$.

Answer: \infty


Question: Evaluate the double integral \( \iint \frac{x}{y} \, dA \) over the region where \( 1 < x < 3 \) and \( x < y < 2x \). Determine whether the integral is divergent within its domain or if there is an issue with the integration bounds.

Answer: 4 \ln 2


Question:
Consider the infinite sequence of functions $f_n$, where $f_n(x)=ng_{[0,1/n]}(x)$ for all real numbers $x$ and for each $n = 1,2,3, \dots$. Determine which of the following statements is true:

(A) For every $x$, the sequence of numbers $(f_n(x))_{n=1,2,\ldots\infty}$ has a limit in the space of real numbers.
(B) $\lim_{n\to\infty}f_n(x)$ does not exist, for any x.
(C) when $\lim_{n\to\infty}f_n(x)$ exists, the actual limit depends on the $x$ in question.
(D) $\lim_{n\to\infty}f_n(x)$ exists for all but a finite set of real numbers $x$.

Choose the correct statement.

Answer: D


Question: Determine the order of the group $\mathrm{SL}_2(\mathbb{F}_p)$, which consists of all $2 \times 2$ invertible matrices with determinant $1$ over the finite field $\mathbb{F}_p$. Given that the group $\mathrm{GL}_2(\mathbb{F}_p)$ has $(p^2 - 1)(p^2 - p)$ elements, calculate the order of $\mathrm{SL}_2(\mathbb{F}_p)$. Express your answer in terms of $p$. \( \boxed{} \)

Answer: p(p^2 - 1)


Question: Determine whether the function $g : \ell_2 \to \mathbb{R}$ defined by \[ g(x) = \sum_{n=1}^{\infty} \frac{x_n}{n} \] is continuous.

Answer: Yes


Question: Consider a matrix B with dimensions \((m-1) \times (m-1)\) constructed from a matrix A by removing the last row. If the \(x_i\) values are pairwise distinct, determine the rank of matrix B.

Answer: m-1


Question: Calculate the norm of the linear functional $P$ on the space $C[-1,1]$, defined by $P(f) = f(1) + f(-1) - 2f(0)$. What is the norm of $P$?

Answer: 4


Question: Find the value of $x$ such that $\gcd(a + x, 3a + x - 1) = 2x + 1$, where $a$ is a positive integer.

Answer: 0


Question: Let $A(z_1)$, $B(z_2)$, and $C(z_3)$ be the vertices of a triangle $ABC$ such that $|z_1|=|z_2|=|z_3|=1$. Given that $z_1+z_2\cos\alpha+z_3\sin\alpha=0$ for some $\alpha\in(0,\frac\pi2)$, and $z_2\overline{z_3}=\pm i$, find the length of the longest side of triangle $ABC$. Express your answer in simplest form.

Answer: 2


Question: Consider two monic polynomials, $f$ and $g$, with integer coefficients from $\mathbb{Z}$. If $f$ and $g$ are coprime as polynomials over the rational numbers $\mathbb{Q}$, does there exist a Bezout relation of the form $f(x)u(x) + g(x)v(x) = 1$, where $u$ and $v$ are also polynomials with integer coefficients? Answer "yes" or "no".

Answer: no


Question:
Find the Fourier series representation of the function \(W(t)\) given by:
\[
W(t) = \begin{cases} 2, & \text{for } -2 \leq t < 4 \\ 0, & \text{otherwise} \end{cases}
\]
Use the formulas for \(a_n\) and \(b_n\) extended for an interval of length \(2l\).

Answer: 2


Question: Let $C$ be a smooth projective plane curve, and let $P$ be a point on $C$. The tangent line to $C$ at $P$ is denoted by $\ell$. The dual curve to $C$ is $C^*$, in the dual plane $(\mathbb{P}^2)^*$. The line $P^*$ in $(\mathbb{P}^2)^*$ parameterizes the lines passing through the point $P$. Given that $\ell$ corresponds to a point on $C^*$, determine if $P^*$ is the tangent line to $C^*$ at the point $\ell$.

Answer: Yes


Question:
Which of the following statements is true regarding continuous maps from $\mathbb{R}^2$ to $\mathbb{R}$?

A) There are at most countably many continuous maps.
B) There are at most finitely many continuous surjective maps.
C) There are infinitely many continuous injective maps.
D) There are no continuous bijective maps.

Answer: D


Question: Given three unit vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\), where any two vectors make an angle of \(\arccos\left(\frac{3}{5}\right)\) with each other, and \(\mathbf{a} \times \mathbf{b} = p\mathbf{a} + q\mathbf{b} + r\mathbf{c}\). Find the value of \(55q^2\).

Answer: 9


Question: Let $T_1 : \mathbb{R}^n \to \mathbb{R}^m$ and $T_2: \mathbb{R}^m \to \mathbb{R}^n$ be linear transformations such that $T_1T_2$ is bijective. Determine the ranks of $T_1$ and $T_2$.

Answer: m


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

Answer: \ln 3


Question: Calculate the gradient with respect to the matrix $X$ of the function: $$ \log \mathrm{det}\, X^{-1} $$ where $X$ is a positive definite matrix.

Answer: -X^{-1}


Question: Determine whether the Galois group of an irreducible quintic polynomial with one real root and two complex conjugate pairs of roots always contains a double transposition.

Answer: Yes


Question: If a tree has a unique central vertex, does it necessarily have an even diameter? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \(\int_{-2}^0 (x+1)\,f\left((x+1)^2 \right)\,dx\) given that \(\int_0^2 (x-1)\,f\left((x-1)^2 \right)\,dx=0.\)

Answer: 0


Question: What is the minimum number of queries needed to determine a $k$-dimensional vector $V$ with integer entries, given that you can query the dot product of $V$ with any integer vector of your choice?

Answer: k


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

Answer: \sqrt{e}


Question: In an equilateral triangle $\Delta ABC$, point D is located inside the triangle. Given that $\angle BAD = 12^\circ$ and $\angle DBA = 6^\circ$, determine the measure of $\angle ACD$. Provide your answer in degrees.

Answer: 18


Question: Determine the convergence of the function \( \frac{\sin x}{x(\sin x)^2 +1} \) as \( x \to \infty \) by analyzing its crossing points with the functions \( \pm \frac{1}{x} \).

Answer: 0


Question: Determine the upper bound for numbers such that the sum of the fifth powers of their digits equals the number itself.

Answer: 354294


Question: Given a group \( G \) and two distinct normal subgroups \( H \) and \( K \) such that the quotient groups \( G/H \) and \( G/K \) are simple, determine if it is possible for \( H \) and \( K \) to have different orders, i.e., \( o(H) \neq o(K) \).

Answer: Yes


Question: Let $A$ and $B$ be two $n \times n$ real matrices with strictly positive real eigenvalues. Determine if the matrix $A + B$ can have a strictly negative eigenvalue.

Answer: Yes


Question: Find all positive integers \( n \) such that \( 10 \mid n! - 1 \).

Answer: 1


Question: Evaluate the limit using the Dominated Convergence Theorem: \[ \lim_{n\to\infty}\int_{(\pi,\infty)}\frac{n\sin(x/n)}{x^2} \, dx. \]

Answer: \infty


Question: Evaluate the series: \[ \sum_{k=1}^{\infty} (-1)^{k+1}\frac{\pi^{6k}}{(6k)!} \]

Answer: \dfrac{4}{3}


Question: Evaluate the limit \(\lim_{x\to 0}(\cos x)^{\frac{1}{\ln(\sin^2(x))}}\) without using L'Hopital's rule.

Answer: 1


Question: Calculate \( \lim_{n \to \infty} \int_0^1 nx^n (f(x)-f(0)) \, dx \), where \( f(x) \) is continuous on \([0,1]\).

Answer: f(1) - f(0)


Question: Determine if the partial derivatives of the function \( g(x,y) = \frac{x^2y^2}{(x^2+y^2)^2} \) exist at the point \((0,0)\) given that \( g(0,0) = 0 \).

Answer: 0


Question: Evaluate the infinite series: $$\sum_{n=1}^{\infty}\left(\log\left(\frac{1}{\sqrt[6]{n}}\right) - \log\left(\sin\left(\frac{1}{\sqrt[6]{n}}\right)\right)\right).$$

Answer: \infty


Question: Determine whether the order of a Sylow $p$-subgroup of a group $G$ with $p^nq^m$ elements, where $p < q$ are primes and the order of $[p] \in \mathbb{Z}_q^{\times}$ is larger than $n$, guarantees the existence of a normal Sylow $p$-subgroup.

Answer: No


Question:
Given that \( x, y, \) and \( z \) are distinct positive real numbers, determine which of the following is greater:

Quantity A: \[ \frac{x^2(y+z) + y^2(x+z) + z^2(x+y)}{xyz} \]

Quantity B: \( 6 \)

Choose one of the following options:
- Quantity A is greater
- Quantity B is greater
- Both quantities are equal

Answer: A


Question: Find the smallest positive integer $k$ such that $2^3$ divides $k$, $3^3$ divides $(k+1)$, and $7^3$ divides $(k+2)$. What is the value of $k$?

Answer: 40472


Question: Evaluate the integral \( \lim_{n\to \infty} \int_0^{\infty} \frac{\cos(nx)}{1+x^2} \, dx \) without using complex analysis, specifically the residue theorem.

Answer: 0


Question: Let $F$ be a subfield of the complex numbers, and consider $n \geq 2$ linear functionals on $F^n$ defined by $f_k(x_1, \dots, x_n) = \sum_{j=1}^n (k-j) x_j$ for $1 \leq k \leq n$. Determine the dimension of the subspace of $F^n$ that is annihilated by all $f_1, \dots, f_n$.

Answer: n - 2


Question:
Given the equations:
\[ a^2 x^3 + b^2 y^3 + c^2 z^3 = p^5 \]
\[ ax^2 = by^2 = cz^2 \]
\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{p} \]
find the value of \( \sqrt{a} + \sqrt{b} + \sqrt{c} \) in terms of \( p \).

Answer: \sqrt{p}


Question: Determine if the linear map $L: \mathbb{C}^3 \rightarrow \mathbb{C}^3$ defined by $Lu = (vv^*)u$, where $v$ is a non-zero vector in $\mathbb{C}^3$ with $||v|| = 1$, has a diagonalizable matrix representation.

Answer: Yes


Question: Determine whether the following statement is true or false: If a function $f$ is integrable on $[a,b]$ and its integral $F(x)=\int_a^x f(t) dt$ is continuously differentiable on $[a,b]$, then $f$ must be continuous.

Answer: B


Question: Find the number of distinct ways to color the vertices of a 12-vertex cycle graph using 8 different colors, where two colorings are considered identical if one can be transformed into the other by a rotation or reflection of the cycle.

Answer: 2863912668


Question: Suppose there are 4 bank tellers, and all service times are independent and identically distributed exponential random variables with parameter $\lambda$. If you are waiting in line and no other customers will come in after you, what is the probability that you are the last to leave?

Answer: \dfrac{1}{4}


Question: Calculate the limit: \[ \lim_{n \rightarrow \infty} 2^n(1-\exp(-a/2^n)) \] for some constant \( a > 0 \).

Answer: a


Question: Let $A,B \in M_2(\mathbb{C})$ be matrices such that $AB - BA = B^2$. Determine whether it is true that $AB = BA$. Provide a justification for your answer.

Answer: AB = BA


Question: Determine the complex number $c$ around which the function \( f(z) = \frac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2} \) represents a rotation.

Answer: \sqrt{3} - 5i


Question: Calculate the volume of the wedge-shaped solid that is located above the xy-plane, below the plane defined by $z = x$, and inside the cylinder given by $x^2 + y^2 = 4$. Provide your answer as a single numerical value.

Answer: \dfrac{16}{3}


Question: Is it necessary for the function $V(x)$ to be an even function if both $g(x)$ and $g(-x)$ are solutions to the differential equation $kf_{xx} + V(x)f(x) = cf(x)$, where $k$ is a negative constant and $c$ is a constant?

Answer: Yes


Question: Evaluate the integral \( \oint_{|z| = 3} \frac{\mathrm{d}z}{\sqrt{z^2 - 1}} \) using a branch of the function that is analytic outside the interval \([-1, 1]\) and positive on \((1, \infty)\).

Answer: 2\pi i


Question: Find all continuous functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(f(f(x))) = x \) for all \( x \in \mathbb{R} \).

Answer: f(x) = x


Question: Simplify the expression \(2\pi i\left(\frac{1}{e^{5\pi i/6}} + \frac{1}{e^{3\pi i/6}} + \frac{1}{e^{\pi i/6}}\right)\) to show that it equals \(4\pi\).

Answer: 4\pi


Question: Given a set of unique numbers \(\{1, 2, 3, \ldots, n\}\), determine the number of binary search trees (BSTs) that can be formed with a height not exceeding a given height \(h\). For instance, if \(n = 4\) and \(h = 2\), calculate the number of BSTs with height at most 2.

Answer: 6


Question: Let $p > 2$ be a prime number. Determine the order of the group $G$ of invertible $2 \times 2$ matrices over $\mathbb{Z}_p$ whose determinant is $-1$. It is known that $G \leq GL_2(\mathbb{Z}_p)$ and $|GL_2(\mathbb{Z}_p)| = p(p^2 - 1)(p - 1)$. Find the order of $G$. \( \boxed{\text{Order of } G} \)

Answer: p(p^2 - 1)


Question: Determine the number of positive integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 = x_1 \cdot x_2 \cdot x_3 \cdot x_4 \cdot x_5\).

Answer: 40


Question: Determine the number of bases required to test in order to conclusively determine if any integer less than \(2^{63} - 1\) is prime using the Miller-Rabin test.

Answer: 12


Question: Find the value of $P(1)$ for the polynomial $P(x)$ that satisfies the equation $P(P(x)+1) = 9P(x) + 8$. Provide your answer as a single number.

Answer: 8


Question: Determine if there exists a constant \( C > 0 \) such that for all points \( z = (x, \sqrt{1 - \frac{x^2}{4}}) \) with \( x > 0 \), the inequality \( \delta(z, z_0) \leq C |z - z_0| \) holds, where \( \delta(z, z_0) = \left | \int_x^{x_0} \sqrt{1 - \frac{t^2}{4}} \, dt \right | \) and \( z_0 = (x_0, y_0) \) is a fixed point on the ellipse. If such a constant exists, find its value.

Answer: 1


Question: Find the generating function in closed form for the sequence: \( \binom{5}{1}, 2\binom{5}{2}, 3\binom{5}{3}, \dots, 5\binom{5}{5} \).

Answer: 5x(1 + x)^4


Question:
Evaluate the limit:

$$\lim\limits_{a\to+\infty}\frac{a^n+\frac{1}{a^n}}{\frac{1}{3^n}(a+\frac{1}{a})^n}$$

where \(a = x + \sqrt{x^2 - 1}\) and \(x - \sqrt{x^2 - 1} = \frac{1}{a}\).

Answer: 3^n


Question: Is it possible for the alternating group $A_6$ to have a subgroup of order 18?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \to \infty} \int_{[2,4]}\left(1+\frac{\ln (x^3+1)}{n}\right)^{n} d\lambda$$

Answer: 62


Question: Determine the degree of the field extension \(\mathbb{Q}(\sqrt[3]{2}, i, \sqrt{2})\) over \(\mathbb{Q}\).

Answer: 12


Question: Calculate the area enclosed by the parametric curve defined by $x = t - \sin t$ and $y = 1 - \cos t$ for $t$ ranging from $0$ to $2\pi$, above the x-axis.

Answer: 3\pi


Question:
Find the value of \( k \) such that the following integral equation holds:

\[
\int_{6}^{16} \frac{1}{\sqrt{x^3 + 7x^2 + 8x - 16}} \, dx = \frac{\pi}{k}
\]

Given that \( x^3 + 7x^2 + 8x - 16 = (x - 1)(x + 4)^2 \), simplify the integral to express \( k \).

Answer: 6\sqrt{5}


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

Answer: (\ln 2)^2


Question: Given that $a^{n} \equiv 1 \pmod{m}$ and $\gcd(a,m)=1$, determine if there always exists an integer $n < m$ such that $a^{n-1}$ is the multiplicative inverse of $a$ modulo $m$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the probability that the order statistics of $n$ independent and identically distributed $U(0,1)$ random variables satisfy $U_{(j)} > \frac{\beta j}{n}$ for all $j = 1, \ldots, n$, where $0 \le \beta \le 1$. Provide your answer as a single probability value.

Answer: 1 - \beta


Question: Let $(X_n)_{n\ge 1}$ be a sequence of dependent nonnegative random variables. Each $X_n$ has a density with respect to the Lebesgue measure on $[0,n]$, and its expected value is finite, i.e., $\mathbb{E}(X_n) < \infty$. The sequence converges weakly to a random variable $X$, which also has a density with respect to the Lebesgue measure on $[0,\infty)$ and satisfies $\mathbb{E}(X) < \infty$. Determine whether $X_n/\log(n)$ converges almost surely to $0$. Provide a justification for your answer.

Answer: No


Question: Given two local martingales $M_1$ and $M_2$, each defined by stochastic integrals with respect to independent Brownian motions, find the covariation $[M]$ of the local martingale $M = M_1 + M_2$ in terms of $[M_1]$ and $[M_2]$. Provide the specific formula for $[M]$.

Answer: [M] = [M_1] + [M_2]


Question: Does every group of order $n$, where $p$ is a prime divisor of $n$, have a subgroup of index $p$?

Answer: No


Question: Let \( f(x) = \frac{1-2x^2}{1+x^2} \) for \( x \in \mathbb{R} \). Find the Chebyshev norm \( ||f||_\infty \).

Answer: 2


Question: Let \( A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \). Determine the maximum real part of the Rayleigh quotient \( \frac{x^* A x}{x^* x} \) for nonzero vectors \( x \) in \( \mathbb{C}^2 \).

Answer: 2


Question:
Given the differentiable function $z = f(x,y)$ with partial derivatives at points $(3,3)$ and $(2,5)$:

$$\begin{array}{ccc}
f(3,3) = 1, & f_x(3,3) = -2, & f_y(3,3) = 11, \\
f(2,5) = 1, & f_x(2,5) = 7, & f_y(2,5) = -3.
\end{array}$$

Suppose there is a differentiable function $w$ of $u$ and $v$ such that $f(w,w) = f(uv, u^2 + v^2)$ for all $(u,v)$. Find $\frac{\partial w}{\partial u}$ at the point $(u, v, w) = (1,2,3)$. \(\boxed{\text{Answer}}\)

Answer: \dfrac{8}{9}


Question: Evaluate the improper integral: $$ \int\limits_0^{\infty}\frac{\sin x}{x}\, dx$$

Answer: \dfrac{\pi}{2}


Question:
Determine if there exist positive integers A, B, C, x, y, z, p, q, and r, where A, B, and C are distinct, such that the following conditions are satisfied:
1. x + y + z = p + q + r
2. E1 = E2, where E1 = Ax + By + Cz and E2 = Ap + Bq + Cr
3. At least one of the following is true: x ≠ p, y ≠ q, or z ≠ r.

Answer: Yes


Question: Determine whether the following statement is true or false: Every nonzero integer can be represented in the form \( e_k3^k + e_{k-1}3^{k-1}+ \ldots + e_13+e_0 \) where \( e_j \in \{-1, 0, 1\} \) for \( j = 0,1,2,\ldots,k \) and \( e_k \neq 0 \).

Answer: True


Question:
Determine the rejection region for the most powerful statistical test using the Neyman-Pearson lemma for testing the null hypothesis \( H_0: \theta = \theta_1 \) against the alternative hypothesis \( H_1: \theta = \theta_2 \) at a significance level \( \alpha = 0.05 \). The discrete random variable \( X \) has possible values \{1, 2, 3, 4, 5\} with the following probability mass functions (PMFs):

- PMF for \( \theta_1 \):
  \[ p(x; \theta_1) = \begin{cases} 0.05 & \text{if } x = 1 \\ 0.1 & \text{if } x = 2 \\ 0.6 & \text{if } x = 3 \\ 0.02 & \text{if } x = 4 \\ 0.23 & \text{if } x = 5 \end{cases} \]

- PMF for \( \theta_2 \):
  \[ p(x; \theta_2) = \begin{cases} 0.45 & \text{if } x = 1 \\ 0.01 & \text{if } x = 2 \\ 0.19 & \text{if } x = 3 \\ 0.05 & \text{if } x = 4 \\ 0.3 & \text{if } x = 5 \end{cases} \]

Answer: \{1\}


Question: What is the average case complexity of the stable marriage problem?

Answer: O(n \log n)


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

Answer: 1


Question: Evaluate the integral \( \int_{0}^{1} (\log(x))^{2} x^{3/2} \, dx \) using the theorem of switching the differentiation and integration.

Answer: \dfrac{16}{125}


Question: Evaluate the line integral \(\int_C y^2 \, dx + (x-y) \, dy\) where \(C\) is the path of the square with vertices \((0,0), (1,0), (1,1), (0,1)\) traversed in a counterclockwise direction.

Answer: 0


Question: Determine whether the extension $\mathbb{Q}(\sqrt[4]{2}, \sqrt{7})/\mathbb{Q}(\sqrt{2})$ is normal.

Answer: Yes


Question: Calculate the integral \( \int_{S^{n-1}} x_1x_2 \, dS \), where \( S^{n-1} \) is the \( n-1 \) dimensional sphere in \( \mathbb{R}^n \).

Answer: 0


Question: Does a homeomorphism between two topological spaces map clopen sets to clopen sets?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \to \infty} \frac{1}{\sqrt{n}} \int_{-2}^{2} e^{\frac{-nx^{2}}{4}}g(x) \, \mathrm{d}x,$$ where $g(x)$ is a continuous function on the interval $[-2,2]$.

Answer: 0


Question: Find the smallest non-negative integer solution to the equation \( x^{2017^2} + x^{2017} + 1 \equiv 0 \pmod{2017} \), where 2017 is a prime number.

Answer: 1008


Question: Evaluate the limit \( \lim_{\Delta t \rightarrow 0^+} \frac{P(t < X < t + \Delta t)}{\Delta t} \) and show that it equals \( f(t) \), where \( f \) is the probability density function and \( F \) is the corresponding cumulative distribution function.

Answer: f(t)


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

Answer: 1


Question: Determine if the series \(\sum_{n=0}^\infty \frac{(2x)^n}{n!}\) converges uniformly on the interval \([-1, 1]\).

Answer: Yes


Question: Let $E$ be a measure zero set in $\mathbb{R}$. Does there exist a finite positive measure set $F$ such that for every $x\in E$, the symmetric density of $F$ at $x$ is equal to $0$ and $1$, i.e., \[\liminf_{r\to 0} \frac{m(F\cap B_{r}(x))}{2m(B_{r}(x))}=0 \quad \text{ and }\quad \limsup_{r\to 0} \frac{m(F\cap B_{r}(x))}{2m(B_{r}(x))}=1?\]

Answer: Yes


Question: Given the polynomial equation \(x^4 + x^3 + 5x^2 - x + M = 0\), find the value of \(M\) such that the sum of two of its complex roots equals the product of the other two roots. Assume \(X_1, X_2, X_3, X_4\) are the roots of the equation, and use Vieta's formulas for the relationships between the roots.

Answer: -6


Question: Determine the Baire function classes of the functions \( g_n(x)=e^{-(x-n)^2} \) for \( 0\leq{x}\leq{1} \) and \( G(x)=\lim_{n\to\infty}g_n(x) \).

Answer: 0


Question: Determine whether the sequence of functions \( f_n(x) = x^{n+1} - x^n \) converges in the space \( C([0,1]) \) with respect to the supremum norm \( ||f|| = \sup\{|f(x)| : x \in [0,1]\} \). If it converges, find the limit of the sequence.

Answer: 0


Question: Determine the quotient group $\mathbb{C}/P$, where $\mathbb{C}$ is the multiplicative group of non-zero complex numbers and $P$ is the subgroup of positive real numbers.

Answer: S^1


Question: Determine whether the prime decomposition of \(N = 10^{2n} + 10^n + 1\) for any positive integer \(n\) is always of the form \(N = pqrs...\) with distinct primes \(p < q < r < s < ...\).

Answer: Yes


Question: Given that $g$ is an integrable function on $[a, b]$ with $g(x) \geq 0$ for all $x \in [a, b]$, and $g(x) > 0$ at an infinite number of points within $[a, b]$, determine whether it must be true that $\int_a^b g > 0$. Answer "yes" or "no."

Answer: no


Question: Determine whether the statement is true or false: If two totally unimodular matrices are stacked, the resulting matrix is also totally unimodular.

Answer: False


Question: Determine the number of non-isomorphic semidirect products of the group \( \mathbb{Z}/p^2\mathbb{Z} \) by the group \( \mathbb{Z}/p\mathbb{Z} \), where \( p \) is an odd prime.

Answer: 2


Question: Determine whether the following statement is true or false: If a linear operator is bounded on an orthonormal basis of a separable Hilbert space, then the operator is itself bounded.

Answer: True


Question: Evaluate the integral \( \int_{0}^{\pi/2}\ln(4\sin^2\frac{u}{2})\,du \).

Answer: -2G


Question: Is the group of rotations of the unit disc with angles of the form \((m\alpha + n\beta)2\pi\), where \(\alpha\) and \(\beta\) are irrational and \(\mathbb{Q}\)-independent, isomorphic to \(\mathbb{Z}^2\)?

Answer: Yes


Question: Solve the differential equation \(xy'' + y' + 4xy = 0\) using the Laplace transform and converting it to a Bessel equation. The initial conditions are \(y(0) = 3\) and \(y'(0) = 0\).

Answer: 3 J_0(2x)


Question: Find the derivative of the function \( f(x) = (\sin x)^x \) at \( x = \frac{\pi}{2} \) without using differentiation rules.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to0}\left(\frac{1}{x^5}\int_0^x e^{-t^2}\,dt - \frac{1}{x^4} + \frac{1}{3x^2}\right).$$

Answer: \dfrac{1}{10}


Question: Let $G$ be a locally compact Hausdorff topological group, and $\Gamma$ be a discrete subgroup of $G$. Determine whether $G/\Gamma$ necessarily has a $G$-invariant measure.

Answer: No


Question: Consider the action of the symmetric group $S_6$ on the vector space $\mathbb{R}^6$ by permuting the coordinates. For the vector $\mathbf{v} = (1, 2, 2, 3, 3, 3)^T$, determine the order of the stabilizer subgroup of $\mathbf{v}$ under this action.

Answer: 12


Question: Let \( f(x) = \frac{1 - 2x^2}{1 + x^2} \) for all \( x \in \mathbb{R} \). Determine the Chebyshev norm, denoted as \( ||f||_\infty \), which is the supremum of \( |f(x)| \) over \( \mathbb{R} \).

Answer: 2


Question: Determine if the function \(f(x,y) = \begin{cases} \frac{x^2y^2}{x^2+y^2} & \text{if } x^2+y^2 \neq 0 \\ 0 & \text{if } x^2+y^2=0 \end{cases}\) is differentiable at the point \((0,0)\).

Answer: Yes


Question: Evaluate the integral \( \int _0^\infty \dfrac{\ln x}{(x^2+1)^2}dx. \)

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


Question: Given the power series \( P(z) = 1 + a_1 z^2 + a_2 z^4 + a_3 z^6 + \ldots \), which converges within the open unit disc centered at the origin, and knowing that for any complex number \( \omega \) with \( |\omega| = 1 \), \( \lim_{z \to \omega} |P(z)| = 1 \), determine if it can be concluded that \( a_i = 0 \) for all \( i \geq 1 \).

Answer: Yes


Question: Let $f$ be an entire function such that $f(0)=f'(0)=0$ and $|e^{f(z)}| \leq e^{|z|}$ for all $z\in \Bbb{C}$. Prove that $f \equiv 0$.

Answer: 0


Question: Determine the exact value of the norm \( \Vert g \Vert \) of the functional \( g \) defined on the normed space \( C[-1,1] \) by \[ g(x) = \int_{-1}^1 x(t)t \, \mathrm{d}t \] for all \( x \in C[-1,1] \).

Answer: 1


Question: Find the orthogonal projection of the vector $\underline{c} = (5, 7, 3, 2)$ onto the orthogonal complement of the subspace spanned by $\underline{a} = (1, 2, -2, 0)$ and $\underline{b} = (2, 1, 0, 4)$ in $\mathbb{R}^4$. Verify if this projection is $(2, 4, 5, -2)$. Provide your answer as a vector.

Answer: (2, 4, 5, -2)


Question: Evaluate the integral \(\int_{-\pi/4}^{\pi/4}(\cos(t)+\sqrt{1+t^2}\cos^3(t)\sin^3(t))dt\). Consider the parity of the integrand and any suitable substitution to find the result.

Answer: \sqrt{2}


Question: Let $g(x)$ be a strictly increasing function from $\mathbb{Z}^+$ to $\mathbb{Z}^+$ such that $g(g(x)) = 5x$. Given that $g(1) = 3$, find the value of $g(2) + g(10) + g(100)$. \( \boxed{} \)

Answer: 274


Question: Find the value of the constant \( K \) in the equation \( \vec{a} \times (\vec{b} \times \vec{c}) = K(\vec{b} (\vec{a} \cdot \vec{c}) - \vec{c} (\vec{a} \cdot \vec{b})) \) using the scalar triple product and the Binet-Cauchy identity.

Answer: 1


Question: Determine the length of the curve in the limit as the number of iterations approaches infinity, given that the measurement scale is equal to the height of the triangles.

Answer: \infty


Question: Let $f(x)$ be a cubic polynomial with rational coefficients. Determine the possible values of the degree of the field extension $[\mathbb{Q}(a_1,a_2,a_3):\mathbb{Q}]$, where $a_1, a_2, a_3$ are the roots of $f(x)$. Provide your answer as a set of possible values.

Answer: \{1, 2, 3, 6\}


Question: Determine the proportion of time the light bulb is on as time approaches infinity, given that it is turned on at $t=0$, off at $t=1$, on at $t=1+\frac{1}{2}$, off at $t=1+\frac{1}{2}+\frac{1}{3}$, and continues this pattern of alternation.

Answer: \dfrac{1}{2}


Question: Calculate the limit: $$\lim_{x\to 0 }\left( \frac{\sin(x)-x}{\tan^3(x)}\right)$$ without using L'Hopital's rule.

Answer: -\dfrac{1}{6}


Question:
Determine which of the following integral representations correctly describes the convolution of a function $f$ with its scaled version $f(\frac{x}{a})$:

1. \( f * f(\frac{x}{a}) = \int_\mathbb{R} f(u)f(\frac{x}{a}-u)du \)
2. \( f * f(\frac{x}{a}) = \int_\mathbb{R} f(\frac{u}{a})f(\frac{x}{a}-u)du \)

Answer: 1


Question: Find \( \lim_{n \to \infty}\int_{[0,n]}x\left(1-\frac{x}{n}\right)^{n} dm(x) \) using one of the Lebesgue convergence theorems.

Answer: 1


Question: Evaluate the integral \( \int_0^{2\pi} e^{e^{it}} \, dt \) by considering the unit circle \( e^{it} \), where \( t \) varies from \( 0 \) to \( 2\pi \), and the function \( f(z) = e^z \). Use the parameterization \( z(t) = e^{it} \), \( t \in [0,2\pi] \), to express the integral in terms of \( z \) and find its value.

Answer: 2\pi


Question: Determine whether the function \( f(x,y)=\dfrac{(xy+\sqrt{x^2+y^2})\sin^2 (x+y)}{x^2+y^2} \) for \( xy\neq0 \) and \( f(x,y)=0 \) for \( xy=0 \) is continuous at the origin.

Answer: Yes


Question: Find the maximum value of the function \( f(x,y,z) = xyz + xy \) subject to the constraints \( x, y, z \geq 0 \) and \( x^2 + y^2 + z^2 \leq 1 \).

Answer: \dfrac{16}{27}


Question: Given a continuous function $f(x)$ defined over the interval $[0, \\infty]$ with the properties: $f(0) = f(a) = 0$, $f(b) > 0$ for some $b$ such that $0 < b < a$, and $f'(x)$ has zeros only at $x_1$ and $x_2$, with $0 < x_1 < b < a$ and $x_2 > a$. Additionally, $f'(b) < 0$. Determine if the function $f(x)$ first increases from $0$ to $x_1$ and then decreases from $x_1$ to $a$ on the interval $[0, a]$, without computing the second derivative.

Answer: Yes


Question: Evaluate the limit: $$\lim _{x \rightarrow 0}\left(x^{3}\left(1+3+5+\dots+\left(2\left[\frac{1}{|x|}\right]-1\right)\right)\right)$$

Answer: 0


Question: Determine which is greater: $202^{303}$ or $303^{202}$.

Answer: 202^{303}


Question:
Consider an oscillating random walk defined by the sequence of partial sums \(S_n := X_1 + \dots + X_n\) with \(S_0 := 0\), where \(X_1, X_2, \dots\) are independent and identically distributed random variables. The walk oscillates if it satisfies:
\[
\liminf_{n \to \infty} S_n = -\infty \quad \text{and} \quad \limsup_{n\to\infty} S_n = +\infty.
\]
Does this imply that the expected value \(E(X_1) = 0\) for the random variables \(X_i\)? Assume no restrictions on the distribution of \(X_i\), which may have infinite variance.

Answer: No


Question: Evaluate the improper integral \( \int_0^{\infty} \frac{\sqrt{x}}{(x+1)^2}{\rm d}x \) using the residue theorem, assuming the principal branch of \(\sqrt{z}\) with a branch cut along the positive real axis.

Answer: \dfrac{\pi}{2}


Question:
Consider the sequence \( S_n = \frac{n^n}{(n+1)(n+2) \ldots (n+n)} \). Determine the limit to which the sequence \( \left(S_n\right)^{\frac{1}{n}} \) converges as \( n \to \infty \). Choose the correct answer from the following options:

(1) \( \frac{e}{2} \)
(2) \( \frac{e}{4} \)
(3) \( \frac{e}{8} \)
(4) \( 0 \)

Answer: B


Question: If $X_1, X_2, \ldots, X_n$ are independent and identically distributed standard normal random variables, does the vector $(\bar{X}, X_1 - \bar{X}, \ldots, X_n - \bar{X})$ follow a multivariate normal distribution, where $\bar{X}$ is the sample mean of $X_1, X_2, \ldots, X_n$?

Answer: Yes


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

Answer: \dfrac{\pi}{2}


Question:
Evaluate the limit as \( n \to \infty \) of the expression:
\[ \lim_{n \to \infty} n \left[ \left(1+\frac{c}{n}\right)^{n} - \left(1-\frac{c}{n}\right)^{-n} \right] \]
where \( c \) is a positive constant.

Answer: -c^2 e^c


Question: Determine whether there exists a number $n$ such that there are exactly 1 million abelian groups of order $n$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to\infty}\left(\sin{\frac 1x}+\cos{\frac 1x}\right)^x$$ using the limit formula $\lim_{x\to\infty}(1+\frac 1x)^x=e$.

Answer: e


Question: Let $f(z)$ be an analytic function defined on an open subset $U$ of $\mathbb{C}$ by the series $f(z)=\sum_{n=0}^\infty a_n e^{2\pi i n z}$. If $f(z)=0$ for all $z$ in a half-plane $H_s = \{z: \text{Im}(z) \ge s\}$, does it follow that $a_n=0$ for all $n$? Provide a justification for your answer.

Answer: Yes


Question: Determine whether every prime number greater than 2 has a primitive root that is also a prime number.

Answer: Yes


Question: Find the minimum value of $a$ such that the sum \((a+\frac{1}{8})^0 \cdot n + (a+\frac{1}{8})^1 \cdot n + (a+\frac{1}{8})^2 \cdot n + \dots + (a+\frac{1}{8})^{\log_8(n)} \cdot n\) attains its maximum value. Note that when $a=\frac{7}{8}$, the sum becomes $n \log(n)$. Provide a hint to solve the problem.

Answer: \dfrac{7}{8}


Question: Let $(X_n)_{n\geq 1}$ be an i.i.d sequence of random variables such that $EX_1=\mu$ and $\sigma^2=\text{Var}(X_1)<\infty$. Prove that $$ U_n=\binom{n}{2}^{-1}\sum_{1\leq i<j\leq n} X_iX_j\to\mu^2 $$ in probability as $n\to \infty$.

Answer: \mu^2


Question: Let $A \subseteq B(H)$ be a $*$-subalgebra of the bounded operators on the Hilbert space $H$. Let $B$ be the SOT-closure of $A$. Is $B$ abelian?

Answer: No


Question: Consider a sequence of vectors $V = \{v_1, v_2, \ldots\}$ in a linear space. If every consecutive triplet of vectors spans a subspace of dimension 2, determine whether the span of all vectors in $V$ can be finite-dimensional.

Answer: Yes


Question: If a vector space \( V \) is the direct sum of subspaces \( W \) and \( X \), and it is also the direct sum of subspaces \( W \) and \( Y \), does this necessarily imply that \( X = Y \)? Provide a justification for your answer.

Answer: No


Question: In triangle $ABC$, evaluate the expression $(b - c)\sin A + (c - a)\sin B + (a - b)\sin C$. Provide your answer as a single value.

Answer: 0


Question: Calculate the area enclosed by the cardioid given by the parametric curve \( \gamma(t) = \begin{pmatrix} (1+\cos t)\cos t \\ (1+ \cos t) \sin t \end{pmatrix} \), where \( t \in [0,2\pi] \), using Green's theorem.

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


Question:
Compute the arc length of the curve given by \( y = \sqrt{x-x^2} + \sin^{-1}(\sqrt{x}) \) for \( 0 \leq x \leq 1 \). Use the integral formula for arc length:
\[ L = \int_{0}^{1} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \ dx \]
where
\[ \frac{dy}{dx} = \frac{1-2x}{2\sqrt{x-x^2}} + \frac{1}{2\sqrt{1-x}\sqrt{x}} \]

Answer: 2


Question: Let $X = A \cup B$ and $F \subset A \cap B$. If $F$ is closed in $A$ and $F$ is closed in $B$, is $F$ necessarily closed in $X$? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the sequence \((y_n)\) defined by \(y_n=\frac{2^n}{\log^p (2^n)}\) converges or diverges for \(1< p<\infty\).

Answer: \infty


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

Answer: +\infty


Question: Calculate the limit: $$\lim_{x \to 0} (\cot^3(x)\cdot x\cdot(\ln(1+3x)-3x))$$

Answer: -\dfrac{9}{2}


Question: Does a function $f$ with continuous third-order partial derivatives in $\mathbb{R}^n$ have a symmetric Hessian matrix?

Answer: Yes


Question: In a laboratory, there is initially one amoeba. Each second, the amoeba either dies with probability \( \frac{1}{4} \) or splits into two with probability \( \frac{3}{4} \). Calculate the probability that at least one amoeba remains in the laboratory forever.

Answer: \dfrac{2}{3}


Question: Determine whether the following statement is true or false: For an open set $U \subset \mathbb{R}^n$ and a function $u:U \to \mathbb{R}$ that is differentiable of all orders, if $x_0 \in U$ is a local maximum point of $u$, then the first partial derivatives vanish, i.e., $u_{x_i}(x_0) = 0$, and the second partial derivative test holds, i.e., $u_{x_ix_i}(x_0) < 0$. Provide a justification for your answer.

Answer: B


Question: Evaluate \( \lim\limits_{x\to \frac{\pi}{2}} (\tan(x))^{\cos(x)} \).

Answer: 1


Question: Given a function $f(x)$ satisfying the equation \[ f(x) + 2f\left(\frac{1}{1-x}\right) = x, \] find the value of $f(2).$

Answer: \dfrac{2}{3}


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

Answer: \pi \ln 2


Question:
Evaluate the integral by reversing the order of integration:

$$\int_{0}^{\sqrt{3\pi}}\int_{y}^{\sqrt{3\pi}}\cos 7x^2\;dxdy.$$

Answer: 0


Question: Are the roots of the sequence of polynomials $p_d(x)$, each of degree $d$ with rational coefficients, dense in the real number line $\mathbb{R}$?

Answer: Yes


Question: Given a Poisson process $N(t)$ with parameter $\lambda > 0$ and a sequence of independent and identically distributed random variables $X_1, X_2, \ldots$ with a common mean $\mu$ and positive variance $\sigma^2$, let $$L(t) = \sum_{i=1}^{N(t)} X_i.$$ Find the expected value $E[L(t) \mid N(t) = n]$.

Answer: n\mu


Question: Consider a polynomial $f$ in $n$ variables. If the symmetrizing map $S(f)$ is non-zero, does there necessarily exist a permutation $\sigma$ such that $\sigma \cdot f = -f$? Answer yes or no.

Answer: No


Question: Determine whether it is possible to construct a function that is integrable over the entire real line, but whose values do not approach zero as the absolute value of the input approaches infinity.

Answer: Yes


Question: Calculate the conditional expectation $E(\xi | \xi^2)$ where $\xi$ is a random variable uniformly distributed on the interval $[-1, 1]$. Provide your answer as a function of $\xi^2$. \( \boxed{} \)

Answer: 0


Question: Determine whether the dominated convergence theorem can be applied to show that \( \lim_{a \to +\infty} \mathbb{E} (|X|\ 1_{|X| \geq a}) = 0 \) given that a random variable \( X \) is integrable, i.e., \( \mathbb{E}[|X|] < +\infty \).

Answer: 0


Question: Let $A$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$ and let $S$ be a closed subspace of $\mathcal{H}$. If the domain of the operator $\hat{A}$ (the restriction of $PAP$ to $S$) is dense in $S$, is it possible for $\hat{A}$ to be non-self-adjoint on $S$? Answer yes or no.

Answer: Yes


Question: Find the surface area of the solid obtained by revolving the curve $8y^2 = x^2 - x^4$ about the $x$-axis, where $x$ ranges from $-1$ to $1$. Use the formula for the surface area of revolution about the $x$-axis: $$S = 2\pi \int y \, dS.$$

Answer: \dfrac{\pi}{2}


Question: Consider a function of bounded variation on the interval $[a, b]$. If its values are altered at countably infinitely many non-dense points within the interval, does the function still remain of bounded variation?

Answer: No


Question: Determine whether there exists an unbounded function $f(x)$ such that $\int_{-\infty}^{\infty}|f(x)|dx < \infty$, and $f(x)$ approaches infinity as $x$ approaches zero.

Answer: Yes


Question: Calculate the derivative of the function \( f(x) = \begin{cases} x + 2x^2 \sin\left(\frac{1}{x}\right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \) at \( x = 0 \).

Answer: 1


Question: Evaluate the limit of the sequence \((S_n)_{n>0}\) as \(n \to \infty\), where \(S_n = \int_{0}^{1} \frac{nx^{n-1}}{1+x}dx\). Determine whether the sequence tends to 0, 1/2, 1, or \(+\infty\).

Answer: \dfrac{1}{2}


Question: Assuming \(a, b, c, d\) are integers greater than or equal to 2, and \(a \neq c\), can the equation \(n = a \times b^a = c \times d^c\) have more than one solution?

Answer: Yes


Question: A point moves along a circle defined by the equation \(x^2 + y^2 + 8x + 4y - 5 = 0\). The circle has a center \(C\), and the point breaks away at point \(A\) or \(B\) on the circle. It then continues along a tangent that passes through the point \(D(3, -3)\). Given that the area of triangle \(ACD\) is \(\frac{25}{2}\) square units when \(D\) lies on the tangent through \(A\), calculate the area of triangle \(BCD\).

Answer: \dfrac{25}{2}


Question: Find the norm of the operator $T: l^q \rightarrow l^q$ defined by $T((b_{n})) = b_{n+1} - 2b_{n} + b_{n-1}$, where $l^q$ is the space of sequences with the $q$-norm.

Answer: 4


Question: Given the functional equation \( f(z) = z + f(z^2) \), express it in terms of the coefficients of the power series \( f(z) = \sum_{j=0}^\infty a_j z^j \). Determine the radius of convergence of the resulting series.

Answer: 1


Question: Evaluate the improper integral \( \int_a^b \frac{dt}{\sqrt{(b-t)(t-a)}} \), where \( a, b \in \mathbb{R} \) with \( a < b \). Determine the most efficient approach to evaluate this integral.

Answer: \pi


Question: Determine the maximal interval of definition for solutions of the autonomous differential equation $\dot{x}=f(x)$, where $f:\mathbb{R}\rightarrow \mathbb{R}$ is locally Lipschitz continuous, $f(0)=0=f(1)$, and at time zero, $\dot{x}$ lies in the interval $[0,1]$. What is this interval?

Answer: (-\infty, \infty)


Question: Find the limit of the sequence defined by $b_1 = 5$ and $b_{n+1} = \frac{3^n + n^{(10^{18})}}{(1 + \frac{1}{n})^{(n^3)} - \frac{1}{n}} \cdot b_n$. Given that $\lim_{n \to \infty}{(1+\frac{1}{n})^{(n^3)} - \frac{1}{n}} = \exp(3)$, determine $\lim_{n \to \infty} b_n$. \( \boxed{} \)

Answer: \infty


Question:
Evaluate the limit as \(x\) approaches 0 of the expression:
\[ \left(\sum_{r=1}^{n}r^{\frac{1}{\sin^2x}}\right)^{\sin^2x} \]

Answer: n


Question: Determine whether it is possible for an equilateral triangle to have two vertices with integer coordinates and one vertex with irrational coordinates.

Answer: Yes


Question: Find the values of \( \beta \) for which the series \( \sum_{n=1}^{\infty} \left( \frac{n-1}{n+1} \right)^{n^\beta} \) converges.

Answer: \beta > 1


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

$$\left| 3 -x \right|^{ \log_7(x^2) - 7\log_x (49)} = (3-x)^3$$

Answer: 2


Question: Given that \( h(x) = \int_{x^3}^1 g(t)\, dt = \cos(x^3) \), find the value of \( g(1) \).

Answer: \sin(1)


Question: Calculate the integral \( \int_{0}^{\infty} (-1)^{\lfloor x \rfloor}\cdot \sin(x) \, \mathrm{d}x \).

Answer: 0


Question: For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. Find the remainder when $S(2015)$ is divided by 12.

Answer: 8


Question: Evaluate the limit \(\lim \limits _{x\to+0} ((x+9)^x-9^x)^x\) using logarithmic manipulation and L'Hôpital's rule.

Answer: 1


Question:
Evaluate the limit:

$$\lim_{x\to \infty} \frac{(f^{-1}(x))^2-(f^{-1}(5x))^2}{x^{1/10}}$$

where $f(x)=2023x^{2023}+x+1$. Determine the value of this limit.

Answer: 0


Question: Evaluate the integral \( \int_0^{2\pi} \frac{dx}{5-4\cos x} \).

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


Question: In a game, you start with 1 infinitely divisible coin. You flip a fair coin: if it lands on heads, your cash doubles; if it lands on tails, your balance halves. You play this game infinitely. What is your expected earnings?

Answer: \infty


Question: Find the coefficient of \(x^{12}\) in the expansion of \((x^5+x^6+x^7+\ldots )^2 \cdot (1+x+x^2+ \ldots)^8\).

Answer: 55


Question: Determine whether there exists a regular curve \( \gamma: \mathbb{R} \to \mathbb{R}^3 \) and a non-identity rigid motion \( f \) such that \( f \circ \gamma \) is a reparametrization of \( \gamma \). A regular curve is defined as a smooth function \( \gamma: I \to \mathbb{R}^3 \) with \(|\gamma'(t)| \neq 0\) for all \( t \in I \). A rigid motion \( f: \mathbb{R}^3 \to \mathbb{R}^3 \) is a function that preserves distances.

Answer: Yes


Question: In a finite \(p\)-group, if the number of maximal subgroups is less than \(p+1\), does this imply that the group is cyclic?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0^{+}} \frac{\arcsin(1-\{x\})\arccos(1-\{x\})}{\sqrt{2\{x\}}(1-\{x\})} \] where \(\{\cdot\}\) denotes the fractional part function.

Answer: \dfrac{\pi}{2}


Question: Calculate \(2 \cdot 4 \cdot 6 \cdots 56 \mod 29\) using Wilson's theorem, given that 29 is a prime number.

Answer: 28


Question: Given a function $F: \mathbb{R} \rightarrow \mathbb{R}$ such that $F(x-y) = F(x)F(y)$ and $F(1) = 3$, find all possible values of $F(2)$. \( \boxed{} \)

Answer: 9


Question: Find the operator norm \( \|T\| \) of the linear operator \( T: \ell^2 \rightarrow \ell^2 \) defined by \( (x_i) \mapsto \left(\frac{x_i}{i}\right) \), where \( \ell^2 \) is the space of square-summable sequences.

Answer: 1


Question: In a lattice $L$, determine whether the pseudocomplement of an element, if it exists, is unique. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Find the coefficient of \(x^{18}\) in the expansion of \((x^2+x^3+x^4)^3(1-x^5)^{-3}\) using generating functions.

Answer: 36


Question: Evaluate the limit: \[ \lim_{(x,y,z) \to (1,1,1)} \frac{(x-1)^2(y-1)^2(z-1)^2}{(x-1)^2+(y-1)^2+(z-1)^2} \]

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(\dfrac{n^2+5n+3}{n^2+n+2}\right)^n \]

Answer: e^4


Question: Given the set $N$ of positive integers and the set $S = \{(a,b,c,d) \in N^4 : a^2 + b^2 + c^2 = d^2\}$, find the largest positive integer $m$ such that $m$ divides $abcd$ for all $(a,b,c,d) \in S$. What is the value of $m$?

Answer: 12


Question: Find the spectrum of the operator $T = S + S^*$, where $S$ is the unilateral shift operator on $\mathcal{l}^2$ and $S^*$ is its adjoint.

Answer: [-2, 2]


Question: Find all points in the complex plane where the function $f(z) = \cot(z)$ is differentiable and compute the derivative of $f(z)$ at those points.

Answer: -\csc^2(z)


Question:
Consider a connected, finite, simple, undirected, and vertex-transitive graph $G$ with $N$ vertices. Let $
u_0 >
u_1 \geq \dots \geq \nu_{N-1}$ be the eigenvalues of the adjacency matrix of $G$. Is there a graph $G$ for which the eigenvalue $
u_1$ has a multiplicity greater than 1?

Answer: Yes


Question: Consider a sequence $b_i$ such that $b_i \geq 0$ for all $i \in {\mathbb Z_+}$. If the following condition holds: \[\liminf_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n} b_i > 0,\] does it imply that \(\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n} b_i\) exists? Answer yes or no and justify your answer.

Answer: No


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y, z \in \mathbb{R} \) with \( x + y + z = 0 \), the equation \( f(x^3) + f^3(y) + f^3(z) = 3xyz \) holds. Given that \( f(x) = x \) is a solution, determine if it is the only solution.

Answer: f(x) = x


Question: Given a sequence of functions $B_n \in C([0,1],\mathbb{R})$ and a function $B \in C([0,1],\mathbb{R})$, such that $\int_{[0,1]} B_n(x) f(x) \, dx \rightarrow \int_{[0,1]} B(x) f(x) \, dx$ for all bounded $f \in C([0,1],\mathbb{R})$, does this imply that $B_n(x) \rightarrow B(x)$ pointwise for all $x \in [0,1]$?

Answer: No


Question: Evaluate the integral \( \frac{1}{2i\pi} \oint_\gamma \frac{2i}{z^2 +1}dz \) where \( \gamma \) is a closed and continuously differentiable path in the upper half-plane \( \{z \in \mathbb{C} : z = x + iy, x, y \in \mathbb{R}, y > 0\} \) that does not pass through the point \( i \).

Answer: 1


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

Answer: \dfrac{15}{16}


Question:
Let $F(x,y): \mathbb{R}^2 \to [0,\infty]$ be a Lebesgue measurable function. If the set of values $(x,y) \in \mathbb{R}^2$ such that $F(x,y) = \infty$ has positive Lebesgue measure, determine whether the following integral is infinite:

$$\int_{-\infty}^\infty\int_{-\infty}^\infty F(x,y) \exp \{-|x| - |y|\} \, dx \, dy= \infty$$

Answer: \infty


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\frac{dx}{(x^2+9)(x^2+16)} \) using partial fractions.

Answer: \dfrac{\pi}{84}


Question: Find a generator of the ideal \((35-14i, 42+21i)\) in \(\mathbb{Z}[i]\) using the Euclidean Algorithm.

Answer: 7


Question: Determine whether there exists a function $g$ and a sequence $(b_n)_{n \in \mathbb{N}}$ such that for all $x \in (0, 2)$, the series $\sum_{n = 0}^\infty b_nx^n = g(x)$ and for all $k \in \mathbb{N}$, $k \geq 1$, $g\left(\frac{2}{2k+1}\right) = e^{\sqrt{2k+1}}$. Answer with 'yes' or 'no'.

Answer: no


Question:
Determine the correct expression for the mean of the distribution of \( \ln\left(\frac{X(t)}{Y(t)}\right) \) using Itô's lemma, given the dynamics:

$$dX(t) = X(t) (rdt + \sigma_X dW^X(t))$$
$$dY(t) = Y(t) (rdt + \sigma_Y dW^Y(t))$$

where \(W^X(t)\) and \(W^Y(t)\) are independent Brownian motions. Choose between the following expressions:

1. \(N\left(\ln\left(X(0)Y(0)\right) - \frac{1}{2} \left(\sigma_X^2 + \sigma_Y^2\right)t, (\sigma_X - \sigma_Y)^2t\right)\)
2. \(N\left(\ln\left(X(0)Y(0)\right) + \frac{1}{2} \sigma_Y^2 - \frac{1}{2} \sigma_X^2)t, (\sigma_X - \sigma_Y)^2t\right)\)

Answer: 2


Question: Evaluate the limit: $$\lim\limits_{n \to \infty}\prod_{k=2}^n\frac{k^3-1}{k^3+1}$$

Answer: \dfrac{2}{3}


Question: Let \( g(x), g: \mathbb{R} \to \mathbb{R} \) be a non-constant continuous function satisfying the equation \( \left(e^{2x}-1\right)g(x)=\left(e^x-1\right)g(2x) \). Given that \( g'(0)=2 \), find the value of \( \lim_{x\to \infty}\left(\frac{g(x)}{x}\right)^{2/x} \).

Answer: e^2


Question: Let X be a compact metric space, and let V and W be open subsets of X. Suppose there is a homeomorphism from V to W. Let B be a countable basis for X, and let B(V) and B(W) be the relativized bases for V and W, respectively. Are B(V) and B(W) order isomorphic under the subset order?

Answer: Yes


Question: Using Cauchy's integral formula, determine a tighter upper bound for \(|f^{(4)}(0)|\) for all functions \(f\) in the family \(\mathscr{F}\) as defined in the original context.

Answer: 24


Question: Determine whether the set of all ordinals \(\alpha < \Omega\) that are initial ordinals has a countable cardinality or a cardinality of \(\mathfrak{c}\), where \(\Omega\) is the first ordinal with cardinality \(2^\mathfrak{c}\).

Answer: \mathfrak{c}


Question: Given a directed multigraph $G$ of arbitrary cardinality, determine whether there exists a $G$-spacious topological space $T$ that is clean or admits an embedding of $G$. Consider known results such as: \(\mathbb{R}^2\) admits clean $G$-drawings for finite and some countable graphs, and embeddings for planar graphs; every finite graph $G$ can be embedded on a surface $S$; \(\mathbb{R}^3\) is clean $G$-spacious for finite and some countable graphs, and admits embeddings for all finite graphs; and some graphs, like $G=(V,E)$ with $V=2^\mathbb{R}$ and \(\{A,B\}\in E\) if \(A\triangle B\) is finite, cannot be \(\mathbb{R}^n\)-drawn for any \(n\in\mathbb{N}\).

Answer: Yes


Question: Determine the number of positive integer solutions to the equation \(a^{b^c}b^{c^a}c^{a^b}=5abc\).

Answer: 3


Question: Consider N couples randomly seated at a rectangular table, with men on one side and women on the other. Let X be the random variable representing the number of couples seated directly across from each other. Determine the variance of X.

Answer: 1


Question: Evaluate the limit: $$ \lim_{x\to 0^+} \left(\cfrac {\tan x}{x}\right)^{\cfrac{1}{x^3}}. $$

Answer: \infty


Question: Determine the values of $x$ in the interval $-\frac{\pi}{2} < x < \frac{\pi}{2}$ for which the series $1 - \tan^2 x + \tan^4 x - \cdots$ converges. Additionally, find the limiting sum of the series for those values of $x$.

Answer: \cos^2 x


Question: Determine the values of $p$ for which a function in the Sobolev space $W^{1,p}(\Omega;\mathbb{R}^n)$ has a continuous representative, given that $\Omega$ is a bounded and smooth domain in $\mathbb{R}^n$.

Answer: p > n


Question: Determine whether \( \lim\limits_{x \to \infty}f'(x)=0 \) as \( x \) approaches infinity implies that \( \lim\limits_{x \to \infty} f(x) \) exists in the extended real numbers \([-\infty,\infty]\).

Answer: No


Question: Determine whether the following statement is true or false: Let $(X, d)$ be a compact metric space and let $g_y:X\to X$ be a continuous function for every $y\in X$. If $f:X\to X$ is defined by $f(x)=g_x(x)$, then $f$ must be continuous.

Answer: False


Question: Let $x, y, z$ be positive real numbers such that $xy + yz + zx = 3$. Determine the minimum value of the expression \[ P = \frac{x^2}{\sqrt{x^3 + 8}} + \frac{y^2}{\sqrt{y^3 + 8}} + \frac{z^2}{\sqrt{z^3 + 8}}. \]

Answer: 1


Question: Find the limit of the sequence defined by \( a_1 = 1 \) and \( a_{n+1} = \frac{\sqrt{1+a_n^2}-1}{a_n} \) for \( n \in \mathbb{N} \).

Answer: 0


Question: Evaluate the limit: $$\lim_{n\to \infty}(\sin\sqrt{n+1} - \sin\sqrt{n}).$$

Answer: 0


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

Answer: -\dfrac{2}{3}


Question: Calculate the surface integral of the vector field $\underline{F} = 3x\underline{i} -5z\underline{k}$ over the cylindrical region $x^2 + y^2\leq 1$, where $0\leq z \leq 1$, using parameterizations for all three surfaces.

Answer: -2\pi


Question: Evaluate the improper integral \(\int_{-\infty}^{\infty}\frac{\sin(x)}{x+i}dx\) using the hint \(\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}\).

Answer: \dfrac{\pi}{e}


Question: Determine whether there exists a continuous function that maps every rational number to an algebraic number.

Answer: Yes


Question: Let $\{W_t\}_{t\ge 0}$ be a standard 2D Brownian motion starting at $(1,1)\in\Bbb R^2$. Determine the probability that $W_t$ hits the positive half of the $x$-axis before it hits the negative part.

Answer: \dfrac{3}{4}


Question:
Given the matrices:
\[ X = \begin{bmatrix} a & 1 & d \\ b & 1 & e \\ c & 1 & f \end{bmatrix} \quad \text{with} \quad \det(X) = 1 \]
and
\[ Y = \begin{bmatrix} a & 1 & d \\ b & 2 & e \\ c & 3 & f \end{bmatrix} \quad \text{with} \quad \det(Y) = 4 \]

Calculate the determinant of the matrix:
\[ Z = \begin{bmatrix} a & 5 & d \\ b & 7 & e \\ c & 9 & f \end{bmatrix} \]

Answer: 11


Question: Find the extreme values of the function \( f(x,y) = x^2 + y^2 \) subject to the constraint \( xy = 1 \).

Answer: 2


Question: Determine the size of the conjugacy class of a 2+2+1 cycle in the alternating group A5.

Answer: 15


Question: Let $X$ be a Banach space and consider a net $(x_\alpha^*)_\alpha$ in the continuous dual $(X \times X)^*$, where $x_\alpha^* = (x_{\alpha 1}^*, x_{\alpha 2}^*)$ with $x_{\alpha i}^* \in X^*$ for $i = 1, 2$. The norm of $x_\alpha^*$ is defined as $\|x_\alpha^*\|_1 = \|x_{\alpha 1}^*\|_{op} + \|x_{\alpha 2}^*\|_{op}$, where $\|x_{\alpha i}^*\|_{op}$ is the operator norm of $x_{\alpha i}^*$. Suppose that for all $(x_1, x_2) \in X \times X$, $\sup_\alpha |x_{\alpha 1}^*(x_1) + x_{\alpha 2}^*(x_2)| < \infty$. Does this imply $\sup_\alpha (\|x_{\alpha 1}^*\|_{op} + \|x_{\alpha 2}^*\|_{op}) < \infty$?

Answer: Yes


Question: Solve for integer values of \( x \) in the equation: \[ 3^{3x} - 3^x = (3x)! \]

Answer: 2


Question: Find the set of all 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 $X$ be a linear space and $T$ be a linear operator on $X$. Define a new linear operator $S$ by $S(x) = T(T(x))$. If $T^*$ is the adjoint operator of $T$, determine the adjoint operator of $S$.

Answer: (T^*)^2


Question: Calculate the residue of \((1+z+z^2)e^{1/z}\) at \(z=0\) using the Laurent series expansion.

Answer: \dfrac{5}{3}


Question: Determine the maximum value of the function \( f(x) = \frac{x - \ln(1 + x)}{x \ln(1 + x)} \) for \( x > 0 \).

Answer: \dfrac{1}{2}


Question: In a group $G$, let $a$ and $b$ be non-identity elements satisfying the relation $ab = b^2a$. If the subgroup generated by $a$ has order 3, determine the order of the subgroup generated by $b$. \( \boxed{?} \)

Answer: 7


Question: Evaluate the integral \( \int_0^1 \frac{x-1}{\ln(x)} \, dx \).

Answer: \ln 2


Question: Determine whether it is possible to find a basis in which a positive definite matrix acts as a diagonal matrix with all diagonal entries equal to 1.

Answer: Yes


Question: Given the open disk $D=\{ z \in \mathbb{C}: |z|<3\}$ and an analytic function $f:D\rightarrow\mathbb{C}$ satisfying $|f(z)|<1$ for all $z \in D$ and $f(\pm1)=f(\pm i)=0$, determine the maximum value of $|f(0)|$. Also, specify the conditions under which this maximum value is achieved.

Answer: \dfrac{1}{81}


Question: Is every finite subgroup of the multiplicative group of non-zero complex numbers, $C^*$, cyclic? In other words, can any finite subgroup of $C^*$ be expressed as $A_n = \{z \in \mathbb{C}^* : z^n = 1\}$ for some integer $n$?

Answer: Yes


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 3$ and $f$ satisfies the functional equation \[ f(x)f(y) = f(x+y) + f(x-y). \] Find the value of $f(7)$.

Answer: 843


Question: Let $f: [0, 1] \to \mathbb{R}$ be defined by $f(t) = t$ and $g: [0, 1] \to \mathbb{R}$ by $g(t) = 1 - t$. These paths satisfy $f(1) = g(0) = 1$. Define a continuous map $k: \mathbb{R} \to \mathbb{R}$ as $k(x) = x^2$. Determine if the concatenation property $k \circ (f * g) = (k \circ f) * (k \circ g)$ is satisfied for $f$, $g$, and $k$. Provide a justification for your answer.

Answer: Yes


Question:
Calculate the determinant of the matrix \( M \in M_{2n}(\mathbb{C}) \) defined as:

\[ M = \begin{pmatrix} I_n & iI_n \\ iI_n & I_n \end{pmatrix} \]

Determine if the determinant of \( M \) is equal to \( 2^n \). If not, provide the correct determinant.

Answer: 2^n


Question:
Evaluate the determinant of the matrix:
\[
\begin{vmatrix}
 p^2 & (p+2)^2 & (p+4)^2 & (p+6)^2 \\
 q^2 & (q+2)^2 & (q+4)^2 & (q+6)^2 \\
 r^2 & (r+2)^2 & (r+4)^2 & (r+6)^2 \\
 s^2 & (s+2)^2 & (s+4)^2 & (s+6)^2 \\
\end{vmatrix}
\]
without expanding it.

Answer: 0


Question: Evaluate the triple integral \( \iiint_Q y \, dV \), where \( Q \) is the solid region between the cylinders \( x^2+y^2=1 \) and \( x^2+y^2=4 \), above the xy-plane, and below the plane \( z=x+2 \).

Answer: 0


Question: Compute the limit: \[ \lim_{n\rightarrow \infty }\frac{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n}}{1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots+\frac{1}{2n+1}}. \]

Answer: 2


Question: Let $A_1 = \{ 5 \}$, and for each natural number $n$, define $A_{n+1} = \{ A_n \}$. Let $A = \bigcup_{n=1}^\infty \mathcal{P}(A_n)$. Determine whether $A \in A$. Provide a justification for your answer.

Answer: A \notin A


Question: Given a real or complex vector space $V$ with a finite basis $B$, determine whether there always exists an inner product on $V$ such that $B$ is an orthonormal basis with respect to it. Equivalently, is there always a positive definite symmetric matrix $A$ such that for $B=\{v_1,...,v_n\}$, the condition $v_i^tAv_j=\delta_{ij}$ holds for all $i, j$?

Answer: Yes


Question: Is it possible to define an alternative inner product \((\cdot, \cdot)_a\) on \(\mathbb{R}^2\) such that a given parallelogram becomes a square in the space \((\mathbb{R}^2, (\cdot, \cdot)_a)\)?

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to0}\int_{0}^{x}\frac{\cos(t^3)}{t+x}dt$$

Answer: \ln 2


Question: Evaluate the limit: $$\lim_{n\to \infty}\sum_{k=0}^n\frac{\binom{n}{k}}{(k+2)(n^k)}$$

Answer: 1


Question: Provide an example of a maximal ideal $P$ in the ring of integers $\mathbb{Z}$, where $P$ is of the form $p\mathbb{Z}$ for some prime $p$, such that when $P$ is extended to the polynomial ring $\mathbb{Z}[X]$, the ideal $P[X]$ is not maximal in $\mathbb{Z}[X]$. What is the prime $p$?

Answer: 2


Question: Find the equation of a plane that is perpendicular to the $xy$-plane and intersects the paraboloid $z = x^2 + 4y^2$ at the point $(2, 1, 8)$, such that the intersection is a parabola with a slope of 0 at this point.

Answer: x + 2y = 4


Question: Determine whether the Fourier series of a function $f \in L^2(0,1)$ is a linear map from $L^2(0,1)$ to $L^2(0,1)$. Provide a justification for your answer.

Answer: Yes


Question: Determine the values of $p > 0$ for which the function $g(x)$, defined as \( g(x) = \begin{cases} \cot(x) & x \in (0,\frac{\pi}{2}) \\ 0 & \text{else} \end{cases} \), is in the space $\mathcal{L}^p(\lambda)$.

Answer: (0, 1)


Question: Evaluate the limit \(\lim_{n\to\infty} \ln\left(\frac{n!e^n}{n^n}\right).\)

Answer: \infty


Question: Given the Taylor series representation of a function, \( f(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots \), determine the smallest non-negative integer \( n \) such that the \( n \)th derivative of \( f(x) \), denoted as \( f^{(n)}(x) \), is equal to the original function \( f(x) \) itself.

Answer: 4


Question: Determine whether the function \( y = e^{x^2} \int_{0}^{x} e^{-t^2} \, dt \) is a solution to the differential equation \( y' = 2xy + 1 \).

Answer: Yes


Question: Determine the number of ways to distribute $N$ distinct items among $N$ people such that each person receives at least one item and at most three items.

Answer: N!


Question: Evaluate the limit: \[ y = \lim_{n\to\infty}\left(\sin\frac{\pi}{2n}\times\sin\frac{2\pi}{2n}\times\cdots\times\sin\frac{(n-1)\pi}{2n}\right). \]

Answer: 0


Question: Determine the radius of convergence for the series \(\sum\limits_{n=0}^{\infty} \frac{(n!)^k \cdot x^n}{(kn)!}\) where \(k\) is a positive integer.

Answer: k^k


Question: Given the equations \(a+b+c=3\), \(a^2+b^2+c^2=5\), and \(a^3+b^3+c^3=7\), find the value of \(a^5 + b^5 + c^5\).

Answer: \dfrac{29}{3}


Question: If a function $g(z)$ is analytic on an open domain $U$, is the imaginary part of $g(z)$ harmonic on $U$?

Answer: Yes


Question: Determine the values of $r \in \mathbb{R}$ for which the series \[ S(r) = \sum_{x \in L^*_r} E_1(2 (x_1^2+x_2^2)) \exp((x_1^2+x_2^2) - (x_3^2+x_4^2)) \] is finite, where \( L^*_r = L_r \setminus W \), \( L_r = \left\{ \begin{pmatrix} a+cr \\ b+dr \\ c \\ d \end{pmatrix} : a,b,c,d \in \mathbb{Z} \right\} \), \( W = \left\{ \begin{pmatrix} 0 \\ 0 \\ x_3 \\ x_4 \end{pmatrix} : x_3,x_4 \in \mathbb{R} \right\} \), and \( E_1(s) = \int_s^\infty \exp(-t) \frac{dt}{t} \).

Answer: \mathbb{R}


Question: Suppose $f_n$ and $f$ are integrable functions on a measure space $(\Omega, \Sigma, \mu)$, $f_n \to f$ almost everywhere, and $\int |f_n| \to \int |f|$. Does it necessarily follow that $\int |f_n - f| \to 0$? Answer yes or no.

Answer: Yes


Question: Evaluate the integral: $$I=\int_{-\infty}^{\infty}\frac{x^2 \, dx}{1+x^6}$$

Answer: \dfrac{\pi}{3}


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

Answer: 1


Question: Evaluate the limit, if it exists, for integer values of \( n \): \[ \lim_{n\to\infty} \int_{-\infty}^{\infty} x^{2n}e^{(-\frac{1}{2})ax^2}dx \]

Answer: \infty


Question: Calculate the product of the trigonometric expression: \( \cot 1^\circ \cdot \cot 2^\circ \cdot \cot 3^\circ \cdots \cot 89^\circ. \)

Answer: 1


Question: Find all integer solutions $(a, b, c)$ to the equation $a \sqrt{2} - b = c \sqrt{5}$.

Answer: (0, 0, 0)


Question: Evaluate the limit using Taylor expansion: $$\lim_{x\to0} \frac{e^{x} - \ln(1+x) - 1}{x^2}$$

Answer: 1


Question: Determine the smallest integer $n$ such that in any red-blue edge coloring of the complete bipartite graph $K_{n,n}$, there is always a monochromatic 4-cycle.

Answer: 5


Question: In a normed vector space $(E, N)$, where the dimension of $E$ may be infinite, consider the set $E' = \left\{f: E \rightarrow \mathbb{K} \mid f \text{ is linear and continuous} \right\}$. Determine if it is true that $\bigcap_{f\in E'}\text{Ker}(f) = \left\{0 \right\}$.

Answer: \{0\}


Question: In the inner product space $V = P_3$, where $P_3$ is the set of polynomials of degree at most 3, consider the subspace $W$ with basis $\{1, t^2\}$. Find the basis for $W^\perp$, the orthogonal complement of $W$, with respect to the inner product defined by \(\langle p_1(t), p_2(t) \rangle = \int_{-1}^{1} p_1(x)p_2(x) \, dx\).

Answer: \{t, t^3\}


Question: Determine whether the following statement is true or false: A projection operator is not self-adjoint (Hermitian) if it is not an orthogonal projection operator.

Answer: True


Question:
Let $L^2([0, 1])$ be the Hilbert space of all real-valued square-integrable functions on $[0, 1]$ with the usual inner product. Let $\phi$ be the linear functional on $L^2([0, 1])$ defined by

$$\phi(f) = \int_{\frac{1}{4}}^{\frac{3}{4}}3\sqrt 2 f \, d\mu $$

where $\mu$ denotes the Lebesgue measure on $[0, 1]$. Find $\|\phi\|$. \(\boxed{\text{Answer}}\)

Answer: 3


Question: Determine the number of functions \( f: \mathbb{Z} \to \mathbb{Z} \) that satisfy the equation \( 3f(f(x)) - 7f(x) + 2x = 0 \).

Answer: 1


Question:
Evaluate the following limit as \( n \to \infty \):

\[ \lim_{n \to \infty}\dfrac{\left(\dfrac{n^2}{\ln \left(n^2\right)}+\dfrac{(n+3)^2}{\ln \left((n+3)^2\right)}\right)}{\left(\dfrac{n^2+(n+3)^2}{\ln \left(\dfrac{n^2+(n+3)^2}{2}\right)}\right)} \]

Does this limit approach 1?

Answer: 1


Question: Evaluate the integral \( I = \int_{-1}^2 \left\{|x-1| - \left\lfloor x \right\rfloor\right\}dx \), where \( \left\{x\right\} \) represents the fractional part function defined as \( \left\{x\right\} = x - \left\lfloor x \right\rfloor \).

Answer: \dfrac{3}{2}


Question: Evaluate the limit of the integral \( \lim_{n \to \infty} \int_0^n \frac{(1 - \frac{x}{n})^n}{ne^{-x}} \, dx \) for \( x > 0 \) using the Bounded Convergence Theorem or Dominated Convergence Theorem.

Answer: 0


Question: Find the minimum value of the expression \(4\sec^2(x) + 9\csc^2(x)\) for \(x\) in the first quadrant using algebraic methods.

Answer: 25


Question: Determine whether the following inequality holds for all points $x, y$ on the boundary of a bounded, convex domain $\Omega \subset \mathbb{R}^n$ (with $0 \in \Omega$): $$ C_2\left|x-y\right|^2\leq\left|\frac{x}{|x|}-\frac{y}{|y|}\right|^2\leq C_1\left|x-y\right|^2, \forall x,y\in b\Omega. $$ Here, $C_1$ and $C_2$ are positive constants that do not depend on $x$ or $y$. Is this inequality true?

Answer: Yes


Question: Find the value of $m$ that minimizes the expected squared deviation, $\mathbb{E}((X-m)^2)$, for a real random variable $X$ such that $X^2$ is integrable. Justify your calculation without assuming the integrability of $X$. Provide the value of $m$.

Answer: \mathbb{E}(X)


Question: Is the set of all possible outputs of a finite set of programs executed on a digital machine countable? Provide a justification for your answer.

Answer: Yes


Question: Find the greatest common divisor (GCD) of the expressions \(3^{3^{333}}+1\) and \(3^{3^{334}}+1\).

Answer: 3^{3^{333}} + 1


Question: Given a normed space $X$ and a subspace $M$, the quotient space $X/M$ is equipped with the seminorm $\left\|{x+M}\right\|=\inf\{\left\|{x+y}\right\|:y\in M\}$. If this seminorm is a norm, i.e., $\left\|{x+M}\right\| = 0$ implies $x+M = 0+M$, must $M$ be a closed subspace of $X$? Justify your answer with a proof or a counterexample.

Answer: Yes


Question: Evaluate the integral \( \int_0^{\pi/2}\frac{\cos(9x)\mathrm{d}x}{1+\sin^2(9x)} \) and find its exact value.

Answer: \dfrac{\pi}{36}


Question:
Compute the value of the integral \( \int_{\Bbb{R}} \int_{\Bbb{R}} |f(x,y)| \, dx \, dy \), where the function \( f: \Bbb{R}^2 \rightarrow \Bbb{R} \) is defined as follows:

\[
f(x,y) = \begin{cases}
\frac{1}{x^2}, & \text{if } 0 < y < x < 1, \\
-\frac{1}{y^2}, & \text{if } 0 < x < y < 1, \\
0, & \text{otherwise.}
\end{cases}
\]

Answer: \infty


Question: Evaluate the integral: $$\int_{\frac{1}{e}}^{\tan x}\frac{t}{1+t^2}dt+\int_{\frac{1}{e}}^{\cot x}\frac{1}{t(1+t^2)}dt$$

Answer: 1


Question: Let $G$ be a finitely generated group with a finite symmetric generating set $S$. Define the word length $||g||_S$ of an element $g \in G - \{e\}$ with respect to $S$ as \( ||g||_S = \inf\{k \geq 1: g = s_k s_{k-1} \ldots s_1, s_i \in S, 1 \leq i \leq k\} \). For $g, h \in G$, does there exist an element $k \in G$ such that $||g||_S \leq ||k||_S$ and $||h||_S \leq ||k||_S$?

Answer: Yes


Question: Determine whether there exists a periodic function that is infinitely differentiable but not analytic.

Answer: Yes


Question: Evaluate the line integral \( \oint_C \mathbf{G} \cdot d\mathbf{r} \), where \( C \) is the boundary of the surface \( T \) defined by \( z = 10x - 20y + 60 \) over the rectangle \( R = \{ (x,y) \mid -2 < x < 3, 0 < y < 2 \} \) and \( \mathbf{G}(x,y,z) = -y^2 \mathbf{i} + xy \mathbf{j} + zx \mathbf{k} \).

Answer: -8970


Question: Determine the radius of convergence of the series \(\sum a_n z^n\), where \(a_n = \frac{(n!)^3}{(3n)!}\). Use Stirling's formula, \(n! \sim \sqrt{2\pi n}(n/e)^n\), to find the limit superior of \(|a_n|^{1/n}\).

Answer: 27


Question: Find the volume of the region defined by the inequalities: \( \vert x - z \vert + \vert x - 2y \vert \leq \vert y \vert \leq 1 \).

Answer: \dfrac{4}{3}


Question: Find the maximum value of the function \( f(x) = \int_0^1 e^{|t-x|} \, dt \) for \( 0 \leq x \leq 1 \).

Answer: e - 1


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

Answer: \cos a


Question: Find the real solutions for $x$ in the equation $$2\cos\left(\frac{x^2+x}{6}\right)=2^x+2^{-x}.$$

Answer: 0


Question: Evaluate the integral \( I = \iint_R |xy| \, dx \, dy \), where \( R \) is the region bounded by \( x^2 + 4y^2 \ge 1 \) and \( x^2 + y^2 \le 1 \).

Answer: \dfrac{3}{8}


Question: Determine the value of $n$ such that the limit \( \lim_{x\rightarrow 0}\frac{x^n\sin^{n}(x)}{x^{n}-(\sin x)^{n}} \) is a non-zero finite number, given that \( n \neq 0 \).

Answer: 2


Question: Let $f$ be a continuous function on the interval $[a,b]$ such that for every function $g:[a,b]\rightarrow \mathbb{R}$, we have \(\int_{a}^{b}(f(x)\cdot g(x))dx =0\). Does this necessarily imply that $f(x)=0$ for all $x \in [a,b]$? Justify your answer.

Answer: Yes


Question: Let $(M,g)$ be a complete Riemannian manifold. Assume that for each $C > 0$, there exists a point $p \in M$ and unit vectors $X, Y \in T_pM$ such that the sectional curvature $K(X,Y) > C$. Does this imply that the diameter of $M$ is infinite?

Answer: Yes


Question: Let \( Q(x) = x^3 - bx + 1 \), where \( b \) is a positive integer. Given that the sum of the squares of the roots of \( Q \) is 73, find the sum of the fifth powers of the roots of \( Q \).

Answer: -\dfrac{365}{2}


Question: Given a nonsingular matrix $A \in M_n$ with ordered singular values $\sigma_1 \ge \sigma_2 \ge \dots \ge \sigma_n$, determine if the ordered singular values of its inverse $A^{-1}$ are $\sigma_1^{-1} \le \sigma_2^{-1} \le \dots \le \sigma_n^{-1}$. Answer with 'yes' or 'no'.

Answer: no


Question: Evaluate the limit: \[\lim_{x\to 0} \frac {1-(\cos 2x)^3(\cos 5x)^5(\cos 7x)^7(\sec 4x)^9(\sec 6x) ^{11}}{x^2}\] using a Taylor series expansion or L'Hôpital's Rule.

Answer: -30


Question: Determine whether the statement is true or false: The complement of every dense set is also dense in the same space.

Answer: B


Question: Evaluate the improper integral \( \int_D \frac{1}{x} \log(y) \, dA \) where \( D = \{(x, y) \in \mathbb{R}^2 : 0 < \sqrt{y} < x < 1\} \). Use the limit expression \( \lim\limits_{\epsilon \to 0} \int_{\epsilon}^{1} \int_{\epsilon}^{x^2} \frac{1}{x} \log(y) \, dy \, dx \) to find the value of the integral.

Answer: -1


Question: Suppose we have two independent real random variables $X$ and $Y$, and $X+Y$ has the same distribution as $X$. Does this imply that the distribution of $Y$ is degenerate at 0?

Answer: Yes


Question: Extend the function \( f(w) = \frac{w^3}{1-w^5} \) holomorphically to \( \mathbb{C} \cup \{ \infty \} \) and determine the order of the zero or pole of \( f \) at \( w = \infty \).

Answer: 2


Question: Given a real or complex vector space $V$ with a finite basis $B=\{v_1,\ldots,v_n\}$, determine if it is always possible to define an inner product on $V$ such that $B$ becomes an orthonormal basis with respect to this inner product. In other words, does there exist a positive definite, symmetric matrix $A$ such that $v_i^tAv_j=\delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta?

Answer: Yes


Question: Suppose in a certain basis $B$, the matrix representation of a linear operator $T$ and its adjoint $T^*$ are equal, i.e., $[T]_B = [T^*]_B$. Can you conclude that $T = T^*$?

Answer: Yes


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

Answer: e^{e - \pi}


Question: Evaluate the integral \( \int \int_D \frac{\arctan(x)}{1+x^2+y^2}dydx \) over the region \( D=\{(x,y)\in \mathbb{R}^2 : x^2+y^2\le1,\space y\ge0\} \).

Answer: 0


Question: Evaluate the limit: \[ L = \lim_{n \to \infty} \int_{0}^{\pi} \sqrt[n]{x} \sin x\, dx \]

Answer: 2


Question: Determine the number of abelian groups of order $10^5$, up to isomorphism.

Answer: 49


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{i=1}^n \frac{i^{1/i}}{n} \,. \]

Answer: 1


Question: Given the joint density function of continuous random variables $X$ and $Y$: $$f(x,y)=x+y, \quad 0<x,y<1$$ find the density function of $Z=XY$. Express your answer as a function of $z$.

Answer: 2(1 - z)


Question: Evaluate the integral \( \oint_C \frac{z^2-1}{z^2+2}\,dz \), where \( C \) is a circle of radius 2 centered at \( z=0 \).

Answer: 0


Question: Find the limit of the series as \( n \to \infty \): \[ \lim_{n\to \infty}\left(\frac{1}{2\log2}+\frac{1}{3\log3}+\cdots+\frac{1}{n\log n}\right) \]

Answer: \infty


Question: Prove that if $K$ is compact relative to $Y$ and $K \subset Y \subset X$, where $X$ is a metric space, then $K$ is compact relative to $X$.

Answer: K


Question: Determine whether the following statement is true: If $\lambda$ is an eigenvalue of a projection matrix $P$, then $\lambda \in \{0, 1, \exp\Bigl( i\frac{2\pi}{n-1}k\Bigr):k \in \{ 0,1,..,n-2\}\}$.

Answer: False


Question: Given a function \(f\colon \Bbb R^2\setminus\{(0,0)\}\rightarrow \Bbb R\) which satisfies the equation \(x \frac{\partial f}{\partial y} -y \frac{\partial f}{\partial x} = f\), prove that the function is zero on \(\mathbb{R}^2\setminus\{(0,0)\}\).

Answer: 0


Question: Evaluate the double integral \( \displaystyle\iint_{C} y \,dx\,dy \), where \( C \) is the region inside the ellipse \( \frac{(x+1)^2}{4}+\frac{(y-2)^2}{9} \leq 1 \).

Answer: 12\pi


Question: Find the multiplicative inverse of the polynomial $4x+3$ in the field $\mathbb{F}_{11}[x]/\langle x^2+1 \rangle$. Provide your answer as a polynomial in $\mathbb{F}_{11}[x]$.

Answer: 6x + 1


Question: Suppose \(X_1, X_2, \ldots, X_n\) are i.i.d. random variables following an exponential distribution with density function \(f_{\theta}(x) = \theta e^{-\theta x}\) for \(x > 0\). Given that the MLE of \(\theta\) is \(\hat \theta = \frac{1}{\bar X}\), find the MLE of the probability that a new observation from the underlying distribution is greater than 2.

Answer: e^{-2/\bar{X}}


Question: Given three vectors $v_1, v_2, v_3 \in \mathbb{R}^2$, if the minimum eigenvalue of $v_i v_i^T + v_j v_j^T$ is less than or equal to 1 for all distinct $i, j \in \{1, 2, 3\}$, determine an upper bound for the minimum eigenvalue of $v_1 v_1^T + v_2 v_2^T + v_3 v_3^T$ under the constraint $|z_1^2 + z_2^2 + z_3^2| = 0$, where $z_j = \alpha_j + i \beta_j$ and $v_j = [\alpha_j, \beta_j]^T$. Provide your answer as an expression or a number.

Answer: 3


Question: Let \( q(x) = 1 - (1-x) + (1-x)^2 - \cdots + (1-x)^{10} \). The roots of \( q(x) \) are \( \beta_i, i \in \{1, 2, \ldots, 10\} \). Find the value of \( \sum_{i=1}^{10} \frac{1}{\beta_i} \).

Answer: 5


Question: A cube $ABCD-A'B'C'D'$ has an edge length of $6$. Let $E$ be the midpoint of $A'B'$ and $F$ be a point on $A'D'$ such that $|A'F|=2|D'F|$. Calculate the volume of the tetrahedron $A'EF-ABD$.

Answer: 12


Question: Calculate the Lebesgue integral of the function \( f(x) \) defined on the interval \([0,1]\) as follows: \( f(x) = \begin{cases} 0, & \text{if } x \in \mathbb{Q} \\ x, & \text{if } x \notin \mathbb{Q} \end{cases} \). Construct an increasing sequence of simple functions \( \{\varphi_n\} \) that are less than or equal to \( f \) and converge to \( f \). Verify if the sequence \( \varphi_n(x) = \frac{1}{n}\lfloor nx \rfloor \) for \( x \notin \mathbb{Q} \) is appropriate, and calculate the integral.

Answer: \dfrac{1}{2}


Question: Find the direction of a light beam that should be projected onto the plane \(\pi: x-2y+4z=0\) such that its reflection has the direction \((-2,1,2)\).

Answer: (-50, 37, 10)


Question: Evaluate the integral \( \int_{0}^{+\infty} \mathrm{d}t \, f(t) \delta(t-a) \) where \( a > 0 \).

Answer: f(a)


Question: In the context of $\mathcal{V}$-enriched category theory, consider $\mathcal{C}$, $\mathcal{D}$, and $\mathcal{E}$ as $\mathcal{V}$-enriched categories. Let $F:\mathcal{C}\to\mathcal{D}$ and $G:\mathcal{D}\to\mathcal{E}$ be $\mathcal{V}$-functors such that $G\circ F$ and $G$ are fully faithful in the enriched sense. Is $F$ necessarily fully faithful as well?

Answer: Yes


Question:
Determine the limit of the expectation as $k$ approaches infinity for the sequence of random variables $W_k$ defined by:

$$W_k = \frac{1}{a + b \cdot \chi^2_k}$$

where $a, b \in \mathbb{R}_+$ and $\chi^2_k \sim \chi^2(k)$. Find the analytical expression for:

$$\lim_{k \to \infty} \mathbb{E}\left[ W_k \right]$$

Answer: 0


Question:
Consider a probability space $(\Omega,\mathcal{F},P)$ and a sequence of random vectors $(X_n)\subseteq (\mathbb{R}^k)^{\Omega}$ that converges in probability to a vector $c\in \mathbb{R}^k$. Assume $c$ is an interior point of a compact subset $K$ of $\mathbb{R}^k$. Given a sequence of measurable functions $\Psi_n:\mathbb{R}^k\times \Omega\to \mathbb{R}$ such that for all $n$ and all $x\in \mathbb{R}^k$, $\Psi_n(x,\cdot)$ is measurable, and a continuous function $\Psi:K\to \mathbb{R}$, determine if the following statement is true:

$$\forall \varepsilon>0,\ \lim\limits_nP(\sup\limits_{x\in K}|\Psi_n(x,\cdot)-\Psi(x)|>\varepsilon)=0 \Rightarrow \forall \varepsilon>0,\ \lim\limits_nP(|\Psi_n(X_n,\cdot)-\Psi(c)|>\varepsilon)=0.$$

Answer: True


Question: Find the first eigenvalue of the Laplacian operator $-\Delta = -\frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2}$ in the half-plane $\mathbb{R}_+^2 = \{(x, y): y > 0\}$ with the Dirichlet boundary condition $u = 0$ on $\partial\Omega$, where $\Omega = \mathbb{R}_+^2$. Assume the first eigenvalue of the same problem in a unit disk is known.

Answer: 0


Question: Evaluate the improper integral \( \int_{0}^{\infty} \frac{x^8 -1}{x^{10} + 1} \, dx \).

Answer: 0


Question: Determine if the function $g(x) = x^5$ can be invertible in a neighborhood of a point where its Jacobian determinant is zero.

Answer: Yes


Question: Determine whether the following statement is true or false: Given a Banach space $Y$ over $\mathbb{R}$, and a sequence of bounded functions $\{f_k\}_{k=1}^{\infty}$ from a set $X$ to $Y$, where each $f_k$ is bounded by a sequence of nonnegative real numbers $\{M_k\}_{k=1}^{\infty}$ with $\sum_{k=1}^{\infty} M_k < \infty$, does the sequence of partial sums $\sum_{k=1}^{n} f_k(x)$ converge uniformly to a bounded function $g: X \rightarrow Y$?

Answer: True


Question: Find the maximum value of the expression \(x^3+y^3+z^3-3xyz\) given that \(x^2+y^2+z^2=1\) for all real numbers \(x, y, z\).

Answer: 1


Question: Professor Liyung wants to form a math club with his 40 students. Each student is enemies with two other students, and no student will join the club if any of their enemies are members. Let M be the maximum number of members the club can have. Calculate the sum of all possible values of M.

Answer: 119


Question: Determine the number of homomorphisms from the group $\Bbb Z/3\Bbb Z \times \Bbb Z/3\Bbb Z$ to the symmetric group $S_4$. Provide your answer as a single integer.

Answer: 33


Question: Determine whether a linear transformation that preserves all angles must necessarily be unitary. Provide a justification for your answer.

Answer: No


Question: Evaluate the integral \( \int_C (z^3 - 2z) \, dz \) along the circle \(|z|=3\) from the point (3, 0) to the point (0, 3), moving counterclockwise. Use a suitable parameterization for the path.

Answer: 18


Question: Let $G$ be a finite group, and let $p$ be a prime number that divides the order of $G$. Determine whether the number of elements in $G$ with order coprime to $p$ is necessarily coprime to $p$. Provide a justification for your answer.

Answer: Yes


Question: Let $A$ and $B$ be matrices in $M_n(\mathbb{R})$ such that $x^2-2$ and $x^2+2$ are their minimal polynomials, respectively. Given that $AB=BA$, determine the minimal polynomial of the matrix $AB$. Provide your answer as a polynomial.

Answer: x^2 + 4


Question: Determine the period $T$ of the continuous time signal $y(t) = \sum_{n=-\infty}^{+\infty}{e^{-|6t+n|}}$. Provide your answer as a single value for $T$.

Answer: \dfrac{1}{6}


Question: Determine the degree of the canonical divisor of the curve $C = \mathbb{P}^2$ using the differential form $\omega = dx \wedge dy$ on $\mathbb{C}^2$. Provide your answer as a single integer.

Answer: -3


Question: Determine the number of distinct circular bracelets that can be created using $6n$ blue beads and $3$ red beads, where $n$ is a positive integer.

Answer: 3n^2 + 3n + 1


Question: Suppose $\Omega$ is open in $\mathbb{R}^N$ and $u \in BV(\Omega)$ such that $u = 0$ on an open subset $E \subseteq \Omega$. If $D_i u$ denotes the $i$-th partial derivative measure associated with $u$, does it hold that $D_i u(E) = 0$?

Answer: Yes


Question: Given a bijective decreasing function \(f:[2,4]\to[3,5]\), find the value of the expression \(\int_{2}^{4}f(t) \, dt - \int_{3}^{5}f^{-1}(t) \, dt\).

Answer: 2


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

Answer: \infty


Question: Determine whether a compact Lie group can have a non-compact Lie subgroup.

Answer: Yes


Question: Find the analytic extension of the function \( g(z)=\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}z^{2n+1} \) to the set \( \mathbb{C}\setminus \{z: \text{Im}(z)=0, |\text{Re}(z)|\geq 1\} \).

Answer: \arctan(z)


Question: If we write the 10 numerals in a non-repeating, non-terminating decimal sequence, does this sequence represent an irrational number?

Answer: Yes


Question: In triangle $ABC$, the angle bisectors $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ intersect at the incenter $I$. Given that $DI = 3$, $BD = 4$, and $CI = 5$, find the area of triangle $ABC$. \( \text{Express your answer in square units.} \)

Answer: \dfrac{384}{7}