Question: Suppose $X \sim \mathsf{Bin}(18, \frac{1}{3})$ and $Y \sim \mathsf{Bin}(m, \frac{1}{3})$ are independent random variables. Compute the limit: $$ \lim_{m \to \infty} \mathbb{P}(X \leq \frac{m}{4} - Y). $$

Answer: 0


Question: Find the integer \(x\) such that \((9^4)^x \equiv 12 \pmod{23}\).

Answer: 6


Question:
Given a $4\times 4$ real matrix $T$ such that $T^4=0$, determine which of the following sequences $k_1,\ k_2,\ k_3,\ k_4$ is NOT a possible combination for the nullity of $T^i$:

1. $k_1 = 3,\ k_2 = 4,\ k_3 = 4,\ k_4 = 4$
2. $k_1 = 1,\ k_2 = 3,\ k_3 = 4,\ k_4 = 4$
3. $k_1 = 2,\ k_2 = 4,\ k_3 = 4,\ k_4 = 4$
4. $k_1 = 2,\ k_2 = 3,\ k_3 = 4,\ k_4 = 4$

Note: Since $T^4=0$, $T$ is nilpotent and its eigenvalues are all $0$. Consider the properties of nilpotent matrices to determine the answer.

Answer: 2


Question:
Evaluate the limit using Riemann sums:
\[ \lim_{n \to \infty} \sum_{i = 1}^{n}{1 \over n} \,\left\lfloor\,\sqrt{\,{4i \over n}\,}\right\rfloor \quad \text{where} \quad \left\lfloor\,x\,\right\rfloor \text{ is the greatest integer function.} \]

Answer: \dfrac{3}{4}


Question: Calculate the surface integral of the vector field \( \vec{F} = x\vec{i} + y^3\vec{j} + \vec{k} \) over the plane passing through the points \((0,0,2)\), \((0,1,0)\), and \((2,1,0)\). The equation of the plane is \( 2y + z = 2 \). Use the unit outward normal vector \( \hat{n} = \frac{2\vec{j} + \vec{k}}{\sqrt{5}} \). Compute the integral \( \int\int_{S} \vec{F} \cdot \hat{n} \, dS \).

Answer: \dfrac{9}{5}


Question: Determine whether the product of all primitive $n$th roots of unity is equal to 1 for $n > 2$. Provide a justification for your answer.

Answer: 1


Question: Determine whether there exist quadruplets \((x, y, z, p)\) of positive integers such that \(p\) is a prime number and \(x^3 + y^3 + z^3 = 3xyz + p\).

Answer: Yes


Question: How many different ways can the exponents of the multinomial \((\sqrt[3]{x}+\sqrt[5]{x}+2\sqrt{x})^5\) be chosen such that the resulting term is rational?

Answer: 2


Question: Determine the condition under which the vector equation \( A[x,y,z]^T=[y,2x,0]^T \) has a nonzero solution, where \( A \) is a 3x3 matrix. Express this condition in terms of the determinant of the matrix difference \( A - B \), where \( B = \begin{bmatrix} 0 & 1 & 0\\ 2 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} \). Is it true that the equation has a nonzero solution if and only if \( \det(A-B) = 0 \)?

Answer: \det(A - B) = 0


Question: Find the image of the region $|z+1| < 2$ under the Möbius transformation $w(z) = \frac{z-1}{iz+i}$.

Answer: |w + i| > 1


Question: Consider the function \(f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) defined by \(f(x,y) = (e^{x+y}, e^{x-y})\). Find the area of the image of the rectangle \(\{0 < x < 1, 0 < y < 1\}\).

Answer: e^2 - 1


Question: Let $\{a_n\} _{n\in\mathbb{N}} $ be a sequence of real numbers such that $\sum_{i=0}^{\infty} a_i \in\mathbb{R} ^{+} $. Determine if it is true that \( \lim_{k\rightarrow\infty} \sum_{i=k} ^{\infty} a_i = \lim_{n\rightarrow\infty} a_n =0, \) given that the series $\sum_i a_i$ is convergent.

Answer: True


Question: Find a continuous function \( g: [1, \infty) \to \mathbb{R} \) such that the improper integral \( \int_1^\infty g(x) \, dx \) exists, and \( \lim_{x \to \infty} g(x) \) does not exist.

Answer: \sin(x^2)


Question: Find the residue at $z=0$ of the function $$f(z)=\frac{\sqrt{1-z^2}}{z^4}$$ using its Laurent series expansion about $z=0$. Provide your answer as a single number.

Answer: 0


Question: Let $R$ be a commutative ring, $M$ a finitely generated $R$-module, and $E(N)$ the injective envelope of a non-zero submodule $N$ of $M$. Determine whether it is always possible to find a non-zero homomorphism from $M$ to $E(N)$. Provide a justification for your answer.

Answer: Yes


Question: Is the set $C$ of functions $f: \Bbb R \to \Bbb R$ that satisfy $f(x)=0$ for all but countably many $x$ dense in $X= \Bbb R^{\Bbb R}$ with the product topology?

Answer: Yes


Question: Evaluate the integral \( \int_{-a}^a \frac{1}{1+e^{h(x)}} \, dx \), given that \( h(x) \) is an odd function.

Answer: a


Question: Let \( \sum_{n=0}^{\infty} a_{n} z^{n} \) be a complex power series with a radius of convergence greater than 1, and assume that \( \left|\sum_{n=0}^{\infty} a_{n} z^{n}\right| \leq 1 \) for \(|z| \leq 1\). Prove that \( \sum_{n=0}^{\infty} |a_{n}|^{2} \leq 1 \).

Answer: \sum_{n=0}^{\infty} |a_{n}|^{2} \leq 1


Question: Calculate the volume of the solid generated by revolving the curve \( x=\sqrt{-W_{-1}\left(-e^{-4 y}\right)} \) with \( 0 \leq y \leq \frac{1}{4} \) about the x-axis using the disk method.

Answer: \dfrac{\pi}{8}


Question: Determine whether the surface of a 3-sphere embedded in Euclidean 4-space can be parameterized using complex numbers.

Answer: Yes


Question: Let $A$ be a local Artin $k$-algebra with residue field $k$, where $k$ is an algebraically closed field. Determine whether all vector bundles on $Spec(A)$ are trivial, i.e., whether all locally free modules over $A$ are globally free.

Answer: Yes


Question: Evaluate the limit using the squeeze theorem: $$ \lim_{x\rightarrow \infty} \frac{1}{x^2} \int_0^x \frac{t^3}{t^2 + |\cos t|} \, dt $$

Answer: \dfrac{1}{2}


Question: Determine the number of points on the curve defined by the equation \(x^3 + x + 1 = y^2\) over the finite field \(\mathbb{F}_{125}\). Given that the curve has 9 points over \(\mathbb{F}_5\), and using Hasse's theorem, the number of points over \(\mathbb{F}_{125}\) must be a multiple of 9. The possible orders are 108, 117, 126, 135, and 144. Find the correct number of points.

Answer: 108


Question: Evaluate the integral \( \int_{0}^{2\pi} \frac{x \sin^{100}x}{\sin^{100}x + \cos^{100}x} \, dx \).

Answer: \pi^2


Question: Find the value of \( \frac{m}{n} \), where \( m \) and \( n \) are positive integers, for the trigonometric series: \( T = \sin(5^\circ) + \sin(10^\circ) + \sin(15^\circ) + \cdots + \sin(175^\circ) = \tan\left(\frac{m}{n}\right) \).

Answer: \dfrac{175}{2}


Question: A fair die is thrown repeatedly. Let $X_n$ denote the sum of the first $n$ throws. Determine $\lim_{n\rightarrow \infty} P(X_n \text{ is a multiple of 13})$. \(\text{Express your answer as a single number.}\)

Answer: \dfrac{1}{13}


Question: Find all integers \( n \) such that the number of positive integers less than \( n \) and relatively prime to \( n \) is equal to half of \( n \).

Answer: 2^k


Question: Find the maximum possible area of triangle $OAB$, where $O$ is the origin, $A$ is the point where the tangent to the curve $y = e^{-x}$ (for $x \ge -1$) intersects the $x$-axis, and $B$ is the point of intersection with the $y$-axis.

Answer: \dfrac{2}{e}


Question: Suppose $G$ is a group defined by the presentation $G=\langle a,b\mid a^2=b^3=e\rangle$. Determine the order of the group $|G|$. \(\boxed{?}\)

Answer: \infty


Question: Evaluate the integral \( \iint \delta(\sqrt{x^2 + y^2} - R)\,\mathrm{d}x \,\mathrm{d}y \) using polar coordinates.

Answer: 2\pi R


Question: Find the limit of the function \( f(x, y) = \frac{4xy^2}{x^2+y^2} \) as \((x, y) \to (0, 0)\).

Answer: 0


Question: Evaluate the limit: $$\lim _{n \rightarrow \infty}\left(\frac{1}{n^{3}}+\frac{2}{n^{3}}+\cdots+\frac{(n-1)}{n^{3}}\right)\left(\frac{1}{2n+1}+\frac{1}{2n+3}+\cdots+\frac{1}{2n+(2n-1)}\right).$$

Answer: 0


Question: Let $V$ and $W$ be finite-dimensional vector spaces and $f: V \rightarrow W$ be a linear map. If $f$ is onto and $f^{-1}(U)$ has codimension 2 in $V$ for some subspace $U \subset W$, determine the codimension of $U$ in $W$.

Answer: 2


Question: Farmer Felix has 1 ton of rice and must pass inspections before selling it at the market. The inspector demands a portion of the rice for each inspection, with the cost of each subsequent inspection reduced by the amount given in the previous one. For example, if Felix gives 0.2, 0.3, and 0.5 tons in that order, the rice reaching the market is calculated as follows: \(0.2\times0 + 0.3\times0.2 + 0.5\times(0.2 + 0.3) = 0.31\). Determine the maximum amount of rice Felix can bring to the market, assuming the inspector allows infinite inspections.

Answer: \dfrac{1}{2}


Question: Given that $X_1, X_2, X_3 \sim_{i.i.d.} U(0,1)$, where $Y_1 = X_{(1)}, Y_2 = X_{(2)}, Y_3 = X_{(3)}$ are the order statistics such that $X_{(1)} \le X_{(2)} \le X_{(3)}$, find the expected value of $Z = X_{(3)} - X_{(1)}$. \( \text{Express your answer as a single value or expression.} \)

Answer: \dfrac{1}{2}


Question: Find the integer $x$ such that $x^{45} \equiv 7 \pmod{113}$. Provide your answer as the smallest non-negative integer satisfying the equation.

Answer: 83


Question: Compute the limit \( \lim_{n\to\infty}\frac{n^n}{(n!)^2} \) without using the gamma function or L'Hôpital's rule.

Answer: 0


Question: Determine the number of partial order relations for a set with 6 elements.

Answer: 130023


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^\infty n^{n^{1/3}}z^n \).

Answer: 1


Question:
Consider the polynomial \( f(x) \) of degree 11 defined such that \( f(x) = \frac{1}{x+1} \) for \( x = 0, 1, 2, \ldots, 11 \). What is the value of \( f(12) \)?

(A) \( \frac{1}{13} \)
(B) \( \frac{1}{12} \)
(C) \( 0 \)
(D) \( \frac{1}{7} \)

Answer: C


Question: Compute the first cohomology group $H^1(\mathbb{Z}, \mathbb{Z}_2)$, where $\mathbb{Z}$ is the group of integers.

Answer: \mathbb{Z}_2


Question: Evaluate the limit \( \lim_{x \to 0^+} x^{x^x} \).

Answer: 0


Question: Find all values of $t \geq 1$ such that the recursively defined sequence $(a_n)$, where $a_1 = t$ and $a_{n+1}a_n = 3a_n - 2$, is increasing.

Answer: (1, 2)


Question: Calculate the limit: $$\lim_{x \to 0} \left(2^x + \sin (3x)\right)^{\cot(3x)}.$$

Answer: e \sqrt[3]{2}


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

Answer: 0


Question: Verify whether the following expression is true: \[ e^x = \lim_{n \to \infty}\left(\frac{1+0.5\frac{x}{n}}{1-0.5\frac{x}{n}}\right)^n \]

Answer: e^x


Question: Determine the asymptotes of the function \( f(x) = \sqrt{x^2 + x\ln{(e^x+1)}} \) as \( x \to +\infty \).

Answer: y = \sqrt{2}x


Question: Determine whether the Riemann-Stieltjes integrability of $f^3$ on the closed interval $[a, b]$ implies the Riemann-Stieltjes integrability of $f$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$ \lim_{n \to \infty}\left( 1 - \frac{1}{3} \right)^{2} \left( 1 - \frac{1}{6} \right)^{2}\left( 1 - \frac{1}{10} \right)^{2}\ldots\left[ 1 - \frac{1}{n\left( n + 1 \right)/2} \right]^{2} $$

Answer: \dfrac{1}{9}


Question:
Find the arc length of the curve defined by the function \( y = \left(\frac{x^2}{8}\right) - \ln(x) \) over the interval \( 4 \le x \le 8 \). Use the formula for arc length:
\[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]

Answer: 6 + \ln 2


Question: Determine whether there exists a continuous function \( f: \mathbb{R} \setminus \mathbb{Q} \rightarrow \mathbb{R} \setminus \mathbb{Q} \) such that \( f \) does not have a fixed point.

Answer: Yes


Question: Determine the form of an entire function $f$ that takes every value no more than three times, considering the singularity at infinity.

Answer: f(z) = az^3 + bz^2 + cz + d


Question: If $X$ and $Y$ are independent random variables with characteristic functions $\phi_{X}(t)$ and $\phi_{Y}(t)$, respectively, is the product $\phi_{X}(t)\cdot\phi_{Y}(t)$ also a characteristic function?

Answer: Yes


Question: Evaluate the integral \( \int_1^\infty \frac{1}{x^4 \sqrt{x^2 + 3} } \, dx \) using a suitable substitution.

Answer: \dfrac{4}{27}


Question: A cylindrical container is constructed from a material with a fixed thickness \(T\). The container has an open top and a solid circular bottom, where the bottom's radius equals the outer radius of the container. Given that the inner volume of the container is \(V\) mm\(^3\) and the volume of the material used is minimized when the inner radius is 10 mm, determine the value of \(\frac{V}{\pi T}\).

Answer: 1000


Question: In a group, if the equation $abc = cba$ holds for all elements $a$, $b$, and $c$, does this imply that the group is Abelian?

Answer: Yes


Question: For any positive constant $c < 1$, determine whether there exists a positive integer $n$ that is coprime with 10, such that for all integers $k$, the number of zero digits in the decimal representation of $n^k$ is always greater than $c$ times the total number of digits in $n^k$. Provide a justification for your answer.

Answer: Yes


Question: Find the values of $c$ for which the $3 \times 3$ matrix \( A = \begin{bmatrix} i & c & 1 \\ 0 & i & 2i \\ 0 & 0 & 1 \end{bmatrix} \) is complex diagonalizable.

Answer: 0


Question: Let $x, y, z > 0$ such that $x + y + z = 1$. Find the minimum value of the expression \[ T = \sqrt[5]{\frac{x}{1 - x}} + \sqrt[5]{\frac{y}{1 - y}} + \sqrt[5]{\frac{z}{1 - z}}. \]

Answer: 2


Question: Determine whether the set of all functions $f(x)$ such that $f(q) = 0$ for all $q \in \mathbb{Q} \cap [0,1]$ is a subring of the continuous functions $C[0,1]$. Provide a justification for your answer.

Answer: Yes


Question: Is it feasible to decompose a 2D affine transformation matrix with non-uniform scale into two transformations, given the transformation matrix is composed as follows: $$ M(T,R,S) = \begin{bmatrix} s_1 \cos \alpha & -s_2 \sin \alpha & t_1 \\ s_1 \sin \alpha & s_2 \cos \alpha & t_2 \\ 0 & 0 & 1 \end{bmatrix} $$ where $T = \begin{bmatrix} t_1 \\ t_2 \end{bmatrix}$, $R = \alpha$, and $S = \begin{bmatrix} s_1 \\ s_2 \end{bmatrix}$ with $s_1 \neq 0$, $s_2 \neq 0$?

Answer: Yes


Question: Calculate the sum \( f \left(\frac{1}{2019}\right) + f\left(\frac{2}{2019}\right) + f\left(\frac{3}{2019}\right) + \cdots + f\left(\frac{2018}{2019}\right) \), where \( f(x) = \frac{3^{2x}}{3^{2x} + 3} \).

Answer: 1009


Question: Evaluate the Stieltjes integral: $$\int_{-\pi}^{\pi} (x+2) \, d(x \cdot \text{sign}(\sin x))$$

Answer: -\pi^2


Question:
Given three complex numbers \(a, b, c\) such that \(|a| = |b| = |c| = 1\) and \[\frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab} = -1,\] find the possible values of \(|a + b + c|\). Choose from the following options:

A) 0
B) 2
C) 1.5
D) 3

Answer: B


Question: For any natural number \(n \ge 3\), find the maximum number of odd entries in a matrix \(B\in\mathcal{M}_{n}\left(\mathbb{N}\right)\) such that \(\det(B)=1\).

Answer: n^2 - n + 1


Question: Let $A \in M^{\Bbb C}_{n \times n}$ be a square matrix with a minimal polynomial of degree $k$. Determine the dimension of $W=\operatorname{span} \{ I, A, A^2, A^3, \ldots, A^m, \ldots\}$.

Answer: k


Question: Determine whether there exists a function from \(\mathbb{R}\) to \(\mathbb{R}\) that is discontinuous at every rational point but continuous at uncountably many irrational points.

Answer: Yes


Question: Is it possible to inscribe a right-angled triangle within a rectangle such that when the triangle is rotated, it forms four right-angled triangles within the rectangle, each with sides that are Pythagorean triples?

Answer: Yes


Question: Is a real-valued function $f$ continuous on a topological space $X = \cup X_i$, where $X_i$ are its connected components, if $f$ is continuous when restricted to each $X_i$? Provide a justification for your answer.

Answer: No


Question: If the degree of the extension $F(a)$ over $F$ is $n$, does this imply that $a$ is algebraic over $F$? Provide a justification for your answer.

Answer: Yes


Question: Determine if the equality \( \arctan x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1} \) holds for all \( x \in [-1, 1] \) due to Abel's theorem.

Answer: Yes


Question:
Which of the following is the correct interpretation of the statement from Chernoff's paper regarding $S_n$ as the sum of $n$ independent observations $X_1,X_2,\ldots,X_n$ on a chance variable $X$?

(1) Let $(M,\mu)$ be a probability space and let $X:M\to\mathbb{R}$ be a measurable function. Let $X_1,\ldots,X_n$ be real-valued measurable functions on $M$ such that $(X_i)_\sharp\mu=X_\sharp\mu$ as measures on $\mathbb{R}$ for each $i$ and such that $(X_1,\ldots,X_n)_\sharp\mu=(X_1)_\sharp\mu\times\cdots\times(X_n)_\sharp\mu$ as measures on $\mathbb{R}^n$. Define $S_n:M\to\mathbb{R}$ by $S_n(p)=X_1(p)+\cdots+X_n(p).$

(2) Let $M$ be a measurable space and let $X:M\to\mathbb{R}$ be a measurable function. Then $S_n:M\times\cdots\times M\to\mathbb{R}$ is the function $S_n(p_1,\ldots,p_n)=X(p_1)+\cdots+X(p_n)$.

Choose the correct interpretation.

Answer: 1


Question: Determine whether the Kolmogorov Extension Theorem holds without any assumptions on the topology of the state space. Answer with True or False.

Answer: False


Question: Determine the minimum number of people required to attend 20 meetings, each with 8 attendees, such that no pair of attendees appears more than once across all meetings.

Answer: 36


Question: Determine the decay rate function $D(n)$ for the expression \( \frac{\Gamma (n)}{\Gamma \left(\frac{n}{a}\right)^a} \), where $a$ is a positive constant.

Answer: a^n n^{\frac{a - 1}{2}}


Question: Find all continuous functions \( f \) such that \( f(x) - 1999f\left(\frac{2x}{1-x^2}\right) = 18 \) for \(|x| \neq 1\).

Answer: -\dfrac{1}{111}


Question: Determine the number of ways to distribute $m$ distinct objects into $n$ distinct boxes such that each box can hold at most $k$ objects, given that $kn < m$.

Answer: 0


Question: Use Stokes' theorem to evaluate the surface integral \(\iint_{S} \text{curl}(\mathbf{F}) \cdot \vec{dS}\) where \(\mathbf{F} = (xz, -yz, 2)\) and \(S\) is the part of the sphere \(x^2 + y^2 + z^2 = 16\) above the plane \(z = 2\), oriented upward.

Answer: 0


Question: Suppose $X$ is a completely regular space. If $\mu$ and $\nu$ are regular Borel probability measures on $X$ such that \( \int _X f \, \mathrm{d} \mu = \int _X f \, \mathrm{d} \nu \) for all bounded measurable functions $f$, does it follow that $\mu(B) = \nu(B)$ for all Borel subsets $B$? Answer yes or no.

Answer: Yes


Question: Evaluate the integral \( \int_0^a\int_0^{2\pi}\sqrt{1-\sin^2(\phi)\cos^2(v)}\,d\phi\, dt \) to find the surface area of an oblique cylinder using the provided parametrization.

Answer: 4a E(\cos v)


Question: Determine whether the function spaces $C^0[a,b]$ and $C^0[0,1]$, both equipped with the $L^1$-Norm, are isometrically isomorphic.

Answer: Yes


Question: Determine whether a $p$-Sylow subgroup of a profinite group of order $lp^n$, where $l$ and $p$ are primes with $l<p$, is necessarily normal.

Answer: Yes


Question: In a sequence of independent flips of a fair coin, determine the minimum number of flips required to have at least a 90% probability of obtaining three consecutive heads.

Answer: 30


Question: Evaluate the integral \( \iint\limits_R r^2\,\, dA \) where \( R \) is the region inside the circle with polar equation \( r=2\sin\theta \).

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


Question: Given an L-shaped tile with outer side lengths \(c\) and inner side lengths \(a\), determine if there exists a procedure to partition the L-shaped tile into a finite number of pieces that can be reassembled to form a square with side length \(b = \sqrt{c^2 - a^2}\). Assume \(a, c \in \mathbb{R}\) with \(0 < a < b\).

Answer: Yes


Question: Find the remainder when \(652^{679}\) is divided by \(851\).

Answer: 717


Question: Determine if there is an exact solution for $x$ in the equation $x^3\cos(x^3) = 2\sin(x^3)$. Provide your reasoning.

Answer: 0


Question: Given a $4 \times 4$ matrix $A$ with eigenvalues $\lambda_A$, express the eigenvalues $\lambda_B$ of the block matrix $B = \left( \begin{array}{cc} A & I \\ I & A \end{array} \right)$ in terms of $\lambda_A$. Provide the relationship between $\lambda_B$ and $\lambda_A$.

Answer: \lambda_B = \lambda_A \pm 1


Question: Find all values of $n$ such that the inequality \( \sqrt{u + \sqrt{v^2 + w^2}} \ge x^{14/n} \) holds for all nonnegative real numbers $u$, $v$, and $w$, where $x = (u v^2 w^4)^{1/7}$.

Answer: 28


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1} \cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx \). You may use the series representations for sine and cosine: \( \sin x = \sum_{n=1}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} \) and \( \cos x = \sum_{n=1}^\infty \frac{x^{2n}}{(2n)!} \) for all \( x \).

Answer: \dfrac{\pi}{4}


Question: In the context of the Lebesgue measure on $\mathbb{R}$, consider two functions $f$ and $g$ where $f$ is left continuous everywhere and $g$ is continuous everywhere. If $f$ and $g$ agree almost everywhere, do they necessarily agree everywhere?

Answer: Yes


Question: Given two orthogonal projections $P_{1}$ and $P_{2}$ on a Hilbert space, where $P_{1} \neq P_{2}$, and $P_{1}P_{2} = P_{2}P_{1} = 0$, does it imply that the range of $P_{1}$ is orthogonal to the range of $P_{2}$?

Answer: Yes


Question: A fisherman continues fishing until the weight of the last fish caught is less than the one before. Calculate the probability that he catches exactly $k$ fish.

Answer: \dfrac{k - 1}{k!}


Question: Estimate from below the expected value of the absolute sum of independent Bernoulli random variables $b_1, \ldots, b_k$ with $P(b_i=1) = p$ and $P(b_i=0) = 1-p$: $$E \left|\sum_{i=1}^k b_i\right|.$$

Answer: kp


Question: Find the natural number \( n \) such that \( \frac{n^3+7}{2^n} \) is a perfect square.

Answer: 1


Question: In linear regression, given a noisy target $y = (w^{*})^T \textbf{x} + \epsilon$, where $\epsilon$ is a noise term with zero mean and $\sigma^2$ variance, independently generated for every example $(\textbf{x},y)$, determine the expected error of the best possible linear fit to this target.

Answer: \sigma^2


Question: Determine the number of solutions to the equation \( f(f(f(f(x)))) = 26 \), where \( f(x) = x^4 + 4x^3 + 26 \).

Answer: 0


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

Answer: 1


Question: Find the smallest integer \( n \) such that the remainder estimate \( |R_n| \leq \frac{M}{(n+1)!} (x-a)^{n+1} \) is satisfied, where \( M \) is the maximum value of \( |f^{(n+1)}(z)| \) on the interval \([-1,1]\) for the function \( f(x) = e^{-2x} \) expanded around \( a = 0 \). Determine \( M \) and ensure that \( |R_n| \leq \frac{1}{1000} \).

Answer: 10


Question: Let $M$ be an $(n-1)$-dimensional sub-manifold of $\mathbb{R}^n$ that is homeomorphic to the $(n-1)$-sphere $S^{n-1}$. According to the Jordan-Brouwer separation theorem, $M$ divides $\mathbb{R}^n$ into two connected components. Is the interior, bounded component guaranteed to be a topological $n$-ball, homeomorphic to the $n$-disc $D^n$, with $M$ as its boundary? Assume $M$ is smooth if necessary.

Answer: Yes


Question: Find the points of intersection between the functions \( f(x) = \frac{2}{1+x^2} \) and \( g(x) = |x| \). Then, calculate the area of the region enclosed by these curves.

Answer: \pi - 1


Question: In a regular tetrahedron $ABCD$ with edge length $x$, the points $P$, $Q$, $R$, and $S$ are the centroids of the triangles $ABD$, $BCD$, $CAD$, and $ABC$, respectively. If the tetrahedron $PQRS$ is also regular, determine the ratio of the side length of $PQRS$ to that of $ABCD$. Express your answer as a simplified fraction.

Answer: \dfrac{1}{3}


Question: Given a vector $x \in \mathbb{R}^m$ and a matrix $A \in \mathbb{R}^{m \times n}$, compute the Jacobian matrix of the product $x^T A$. Assume that the Jacobian of the transposed vector, $[x^T A]^T$, is $A^T$. What is the Jacobian matrix of $x^T A$?

Answer: A^T


Question: Evaluate the integral \( \int_0^\infty Ai(y) \, dy \), where the Airy function \( Ai(y) \) is given by the integral representation \( Ai(y) = \frac{1}{\pi} \int_0^\infty \cos(\alpha y + \frac{\alpha^3}{3}) \, d\alpha \).

Answer: \dfrac{1}{3}


Question: Find the indicial equation in terms of \( r \) for the differential equation \( xy'' - 4y' + 5xy = 0 \), given that \( x=0 \) is a regular singular point.

Answer: r(r - 5) = 0


Question: Given an unweighted graph and two vertices \( u \) and \( v \), determine if there exists a Hamiltonian cycle that visits every vertex exactly once and returns to the starting vertex. If such a cycle exists, find the length of the shortest cycle.

Answer: n


Question: Determine the intersection of all the kernels of irreducible characters of a finite group $G$ that contain a normal subgroup $H$.

Answer: H


Question: How many distinct pairs of coprime numbers \((a, b)\), where \(1 < a < b\), can be formed from the prime factorization \(2^2 \cdot 3^3 \cdot 5^5 \cdot 7^7\) such that their product equals the original factorization and the pairs are coprime?

Answer: 7


Question: In a biregular graph with 40 red vertices each of degree 7 and 30 blue vertices each of degree 9, determine the minimum number of additional blue vertices required so that the minimum distance between any two distinct red vertices is 2.

Answer: 2


Question: Let $C$ be the linear code over $\mathbb{F}_5$ with parity check matrix $H$ whose columns are vectors of the form $\begin{bmatrix}1\\ \alpha\\ \alpha^{2} \end{bmatrix}$, where $\alpha\in\mathbb{F}_5$. Determine the number of codewords in $C$ that have a Hamming weight of exactly 3.

Answer: 0


Question: Evaluate whether the limit \( \lim_{(x,y)\to(0,0)} \frac{xy}{|x|+|y|} \) exists, and if it does, determine its value.

Answer: 0


Question: Find the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfies the equation \[ f(f(x) - y) = f(x) + f(f(y) - f(-x)) + x. \]

Answer: -x


Question: Determine the value of \( \alpha \) such that the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \[ f(x) = \begin{cases} \frac{\ln(1 + \ln(2-x))}{(x-1)^\alpha}, & x \neq 1 \\ -1, & x = 1 \end{cases} \] is continuous at \( x = 1 \).

Answer: 1


Question: Compute the sum \((r_1+3)^{100}+(r_2+3)^{100}+\cdots+(r_{100}+3)^{100}\), where \(r_1,r_2,\ldots,r_{100}\) are the roots of the polynomial \(f(x) = (x^2+6x+9)^{50} - 4x + 3\).

Answer: -1500


Question: Determine the largest natural number \( n \) such that the group \( \mathbb{Z}_{180} \times \mathbb{Z}_{30} \) is isomorphic to the direct product of \( n \) cyclic groups.

Answer: 6


Question: Given a function $f: \mathbb{R} \to \mathbb{R}$ such that $\int_{-\infty}^{\infty} |f(z)| dz$ is finite, determine if it follows that $\lim_{h \to \infty} \int_{|f(z)| > h} f(z)dz = 0$. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Determine whether the intersection of two distinct maximal ideals in a Euclidean ring can contain a prime element.

Answer: No


Question: Let \( p \) be a non-singular matrix such that \( I + p + p^2 + \ldots + p^n = O \). Determine \( p^{-1} \).

Answer: p^n


Question: Evaluate \( \lim_{x \to \infty} \frac{\gamma(x+1, x)}{\Gamma(x+1)} \), where \( \gamma(x+1, x) \) is the lower incomplete gamma function and \( \Gamma(x+1) \) is the complete gamma function.

Answer: \dfrac{1}{2}


Question: Construct a holomorphic function \( f : D(0,1) \rightarrow \mathbb{C} \) such that \( \lim_{\zeta \rightarrow z}\text{Im}\ f(\zeta) = \text{Arg}(z) \) for all \( z \in \{\omega : |\omega| = 1\} \setminus \{-1\} \), where \( \text{Arg}(z) \in (-\pi, \pi) \).

Answer: 2 \log(1 + z)


Question: Suppose the domain of a function $f$ satisfies the following conditions: (1) if $f$ is defined for $x$, then it is also defined for $\frac{1}{x}$, and (2) the domain of $f$ is as large as possible. If $f(x) + f\left(\frac{1}{x}\right) = x^2$, find the minimum value of $f(x)$. \( \boxed{} \)

Answer: \dfrac{1}{2}


Question: Given the equation $a^2 = 2b^3 = 3c^5$, where $a, b, c$ are natural numbers, determine the smallest possible value of the product $abc$. Express your answer as a single integer.

Answer: 746496


Question: Consider an infinite countable set \( X = \{x_{1}, x_{2}, \dots \} \). Can the set \( X \) be written as a union of disjoint sets in countably many different ways?

Answer: Yes


Question: Determine whether the sequence \(v_n=\frac{1}{(n+1)!}\sum_{k=1}^{n+1}k!\) converges, using the Stolz-Cesaro theorem.

Answer: 1


Question: Calculate the covariance \( \operatorname{Cov}(X_1+X_2, \max(X_1,X_2)) \), where \( X_1 \) and \( X_2 \) are independent random variables uniformly distributed on \((0,1)\).

Answer: \dfrac{1}{12}


Question: Given the function \( f(x)=\frac{4x^3}{(x^2+1)} \), find the value of the derivative of its inverse function at \( x = 2 \), i.e., \( (f^{-1})'(2) \).

Answer: \dfrac{1}{4}


Question: Find the value of \(q + \frac{1}{p^2}\) if \((p, q)\) is the range of values of \(a\) for which the roots of the equation \(x^2 - 2x - a^2 + 1 = 0\) lie between the roots of the equation \(x^2 - 2(a+1)x + a(a-1) = 0\).

Answer: 17


Question: Find the largest positive integer $k$ such that the inequality $e^{|x|} - |x| + \cos x \geq k$ is satisfied for all $x \in \mathbb{R}$. \(\boxed{\text{Answer}}\)

Answer: 2


Question: Let $x$ and $y$ be positive integers such that $\frac{xy+2}{x+y}<\frac{5}{3}$. If $\frac{x^4y^4+2}{x^4+y^4}$ is maximized, find the sum of $x$ and $y$. \( \boxed{} \)

Answer: 5


Question: Let $f$ be a holomorphic function such that $f(0)=0$ and $|f(z)| \leq \exp[(\Re z)^2 - (\Im z)^2]$ for all $z$. Prove that $f(z) = 0$ for all $z$.

Answer: 0


Question: Let $X, Y,$ and $Z$ be independent random variables, and let $\mathscr{C} = \sigma(Y, Z)$. If $C \in \mathscr{C}$, are $X$ and $1_C$ always independent?

Answer: Yes


Question: Determine the number of factors of 2 in the factorial of \((2^{15} - 1)!\).

Answer: 32752


Question: Two points A and C lie on a line l. A point B is 4 units away from line l. The sum of the distances AB and BC is always equal to 10 units. Find the shortest possible distance for AC, given that A and C cannot be on the same side of the perpendicular to line l, although either A or C can be on the perpendicular.

Answer: 2\sqrt{5}


Question: Find all distributions $f \in D'(\mathbb{R})$ such that for the smooth function $c(x) = e^x$, the distribution $cf$ is zero, i.e., $(cf)(\phi) = f(c\phi) = 0$ for all test functions $\phi \in D(\mathbb{R})$. What are the possible values of $f$?

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to0} \frac{1}{x}\int_0^{3x} (e^t)^{t} dt$$

Answer: 3


Question: Suppose $X_1, \ldots, X_{100}$ are independent and identically distributed as $N(\mu,1)$, but only the values of $X_i$ that are less than 0 are recorded. If 40 observations are less than 0, find the maximum likelihood estimator (MLE) of $\mu$.

Answer: \Phi^{-1}(0.6)


Question: Find a polynomial \( f(x) \in \mathbb{Q}[x] \) of degree at most 2 such that \( f(x) \equiv 2 \pmod{x+1} \) and \( f(x) \equiv x+1 \pmod{x^2+1} \).

Answer: x^2 + x + 2


Question: Determine the point \((x, y)\) on the ellipse \(b^2x^2 + a^2y^2 = a^2b^2\) that minimizes the distance to the focus \((c, 0)\).

Answer: (a, 0)


Question: Find the spectral radius of the compact operator $T:C[0,1]\to C[0,1]$, defined by $Tf(x)= \int_{0}^{1-x} f(y)\,dy$. The operator has eigenvalues $\lambda_n = \frac{1}{\frac{\pi}{2} + 2n\pi}$ for $n \in \mathbb{Z}$, and it is known that $\|T\| \leq 1$ and $\sigma(T) \subset [-1,1]$. Determine the spectral radius $r(T)$. Assume $r(T) = \max\{|\lambda| : \lambda \in \sigma(T)\}$ or $r(T) = \lim_{n \to \infty} \|T^n\|^{1/n}$. Provide your answer as a single value.

Answer: \dfrac{2}{\pi}


Question: Find the values of $x$ that satisfy the equation $\Gamma(x - \sqrt{2x}) = (\sqrt{x + 1})!$.

Answer: 8


Question:
Evaluate the following limit of a multiple integral:

$$\lim _{ n\rightarrow \infty }{ \int _{ 0 }^{ 1 }{ \int _{ 0 }^{ 1 }{ \underbrace{\dots}_{n-3 \, times} \int _{ 0 }^{ 1 }{ \cos ^{2} { \left\{ \frac { \pi }{ 2n } \left({x}_{1}+{ x }_{ 2 }+\dots +{ x }_{ n }\right)\right\} \quad { dx }_{ 1 }{ dx }_{ 2 }\cdots { dx }_{ n } } } } } = \dfrac{1}{2} $$

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{-2}^{2} \sin(x^5)e^{x^8\sin(x^4)}\,dx \).

Answer: 0


Question: Evaluate the limit: $$\lim_{n \rightarrow \infty} \frac{n}{\log n} \int_{e^3}^n \left(\frac{1}{(\log x)^2}-\frac{2}{(\log x)^3}\right) dx$$

Answer: \infty


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} . \] Do not use Stirling's approximation in your solution.

Answer: e


Question: Determine the total derivative of the quadratic form \( q(x) = \langle x, Bx \rangle \) at a point \( a \in \mathbb{R}^{n} \), where \( B \) is a symmetrical \( n \times n \) matrix.

Answer: 2Ba


Question: Evaluate the limit \( \lim\limits_{x\to0}\frac{e^{-1/x^2}}{x} \) without using l'Hopital's rule.

Answer: 0


Question: Verify whether the last power of $2$ that only has $1$'s and $2$'s in its ternary expansion is $2^{15} = 1122221122_3$.

Answer: 2^{15}


Question: Calculate the surface integral of the vector field \( \vec{F}=y\hat{i}-x\hat{j} \) over the surface defined by the part of the paraboloid \( z=x^2+y^2 \) that lies below the plane \( z=1 \).

Answer: 0


Question: Given the complex numbers $z$ and $w$ that satisfy the conditions $|2z - w| = 25$, $|z + 2w| = 5$, and $|z + w| = 2$, determine the value of $|z|$.

Answer: 9


Question: Find the limit of the expression \( \lim_{x\to0}\frac{xe^{2x}+xe^{-2x}-4x^{3}-2x}{\sin^{2}x-x^{2}} \) using the Maclaurin series expansion.

Answer: 0


Question: Find the range of values for $p$ such that the expression $|\xi|^{2} = 1 - 4p^{2}(1-p^{2})\sin^4\left(\frac{\omega}{2} \right)$ satisfies the condition $|\xi| \leq 1$.

Answer: [-1, 1]


Question:
Evaluate the limit:

\[
\lim_{x \to 0} \frac{\left(1+x\right)\left(1+\frac{x}{2}\right)^2\cdots\left(1+\frac{x}{m}\right)^m - 1}{\left(1+x\right)\sqrt{1+2x}\cdots\sqrt[n]{1+nx} - 1}
\]

for positive integers $m$ and $n$.

Answer: \dfrac{m}{n}


Question: Consider a map \( f: \mathbb{C}_\infty \to \mathbb{C}_\infty \) that preserves all cross ratios. Determine whether this map must be a Möbius transformation.

Answer: Yes


Question: Evaluate the integral \( \left| \oint_{|z| = R} \frac{z^2}{\log z}\,dz \right| \) for \( R > 1 \).

Answer: 2\pi


Question: Evaluate the line integral \( \int_\gamma \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy + \ln{(z^4+z^2+1)}dz \) where \( \gamma \) is the curve formed by the intersection of the cone \( x^2+y^2=z^2 \) and the plane \( x+y+10z=10 \). The curve is oriented to make one clockwise revolution around the \( z \)-axis when viewed from above.

Answer: -2\pi


Question: Determine whether the following statement is true or false: If $A$ and $B$ are Hermitian matrices such that $v^*Av = v^*Bv$ for all vectors $v$, then $A = B$. Provide a justification for your answer.

Answer: True


Question: Evaluate the double integral \( \iint_R \frac{y^2}{x} \, dy \, dx \), where \( R \) is the region in the \( xy \)-plane bounded by the curves \( y = x^2 \), \( y = 4x^2 \), \( xy = 1 \), and \( xy = 5 \).

Answer: 12


Question: Find the limit: \[ \lim_{n\to \infty}\frac{n}{\sqrt[n]{n!}}. \]

Answer: e


Question: Evaluate the line integral \( \int_C (0, \frac{1}{x^2+y^2+1}) \cdot dr \), where \( C \) is the positively oriented boundary of the square defined by \( -a \leq x \leq a \) and \( -a \leq y \leq a \).

Answer: 0


Question:
Suppose $f(x)$ and $g(x)$ are nonconstant smooth functions satisfying the equations

$$\frac{f'(x)}{f(x)} + \frac{g'(x)}{g(x)} = 1$$

$$ \frac{f''(x)}{f(x)} + \frac{g''(x)}{g(x)} = \frac{f'''(x)}{f(x)} + \frac{g'''(x)}{g(x)}$$

(assume $f$ and $g$ are always nonzero). Compute the value of

$$ \frac{f''(2015)}{f'(2015)} + \frac{g''(2015)}{g'(2015)}.$$

Answer: \dfrac{2}{3}


Question:
Find the ordered pair \((a, b)\) such that the product \(f(x)g(x)\) is continuous on the interval \([0, 3)\), where:

\[ f(x) = \begin{cases}
  e^{x-1} & , 0 \leq x \leq 1 \\
  \lfloor x \rfloor + 1 & , 1 < x < 3
\end{cases} \]

\[ g(x) = x^2 - ax + b \]

Answer: (3, 2)


Question: Find the norm of the linear operator \( S: \ell^1 \rightarrow \ell^2 \) defined by \( Sx = \left( \frac{1}{2^k} x_k \right)_{k=1}^{\infty} \), where \( \ell^1 \) and \( \ell^2 \) are the sequence spaces of absolutely summable and square-summable sequences, respectively.

Answer: \dfrac{1}{2}


Question:
Consider the $15\times 15$ matrix

$$\begin{pmatrix}1&2&3&4&\cdots&15\\ 16&17&18&19&\cdots&30\\ 31&32&33&34&\cdots&45\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 211&212&213&214&\cdots&225 \end{pmatrix} $$

If you select one number from each row and one number from each column, will the sum of these numbers always be the same, regardless of the specific numbers chosen?

Answer: 1695


Question: Evaluate the integral: $$ \int_{0}^{\frac{\pi}{2}}\frac{{\cos}^{5}\left(x\right)\:dx}{{\cos}^{5}\left(x\right)+ {\sin}^{5}\left(x\right)} $$

Answer: \dfrac{\pi}{4}


Question: Let $K$ be a real quadratic number field. For a fractional ideal $\mathfrak{a}$, denote $[\mathfrak{a}]$ as its narrow ideal class. Recall that $[\mathfrak{a}]=[\mathfrak{b}]$ holds if and only if there is a $\lambda \in K^\times$ with $\lambda \mathfrak{a} = \mathfrak{b}$ and $N(\lambda)>0$. Let $\mathfrak{a}$ be a fixed ideal belonging to the principal genus, i.e., $\mathfrak{a}= \lambda \mathfrak{b}^2$ for $\lambda \in K^\times$ with $N(\lambda)>0$ and a fractional ideal $\mathfrak{b}$. Determine whether there are infinitely many prime ideals $\mathfrak{p} \subset \mathcal{O}_K$ such that $[\mathfrak{a}]=[\mathfrak{p}^2]$. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Let $a,b,c$ be continuous functions defined on $\Bbb R^2$. Let $V_1,V_2,V_3$ be non-empty subsets of $\Bbb R^2$ such that $V_1 \cup V_2 \cup V_3 =\Bbb R^2$ and the PDE: $$a(x,y)u_{xx}+b(x,y)u_{xy}+c(x,y)u_{yy}=0$$ is elliptic in $V_1$, parabolic in $V_2$, and hyperbolic in $V_3$. Which of the following statements is true?\n\n(A) $V_1,V_2,V_3$ are open sets in $\Bbb R^2$\n(B) $V_1,V_3$ are open sets in $\Bbb R^2$\n(C) $V_1,V_2$ are open sets in $\Bbb R^2$\n(D) $V_2,V_3$ are open sets in $\Bbb R^2$

Answer: B


Question: Simplify the integral: $$\int^{\pi/2}_0 \frac{\cos^4x + \sin x \cos^3 x + \sin^2x\cos^2x + \sin^3x\cos x}{\sin^4x + \cos^4x + 2\sin x\cos^3x + 2\sin^2x\cos^2x + 2\sin^3x\cos x} \, dx$$

Answer: \dfrac{\pi}{4}


Question:
The function \( f(x) \) is continuous and satisfies \( f(f(x)) = 1 - x \) for all \( x \in [0, 1] \). Evaluate the following combinations of \( f(x) \) evaluations and select the correct option:

(A) \( f\left(\frac{1}{8}\right) + f\left(\frac{7}{8}\right) \)
(B) \( f\left(\frac{2}{3}\right) + f\left(\frac{1}{3}\right) \)
(C) \( f\left(\frac{5}{6}\right) + f\left(\frac{1}{6}\right) \)
(D) None of These

Choose the correct option.

Answer: D


Question: Given a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) with a non-zero determinant (\(\operatorname{det}(A) \neq 0\)), determine if there exist two non-singular matrices \( B \) and \( C \) such that the product \( BAC \) equals \( \begin{pmatrix} a & -b \\ -c & d \end{pmatrix} \).

Answer: Yes


Question: Is it possible to map the point $(0,1)$ to infinity using a transformation, while keeping the points $(0,0)$ and $(1,0)$ regular (i.e., ensuring the determinant at these points is not zero)?

Answer: Yes


Question: In a commutative partial semigroup $(S,+)$, is the Stone-Čech compactification $(\beta S,+)$ also a partial semigroup, with the operation on $\beta S$ induced from the operation on $S$? Answer yes or no.

Answer: No


Question: Evaluate the integral \( \int_{-\infty}^{+\infty}\frac{\sin^3x}{x^2+4}\mathrm{d}x \).

Answer: 0


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

Answer: \dfrac{1}{2}


Question: Determine whether the group \( H = \{ a, b \ | \ a^4 = b^4 = (ab)^4 = 1\} \) is finite or infinite. If it is finite, find its order.

Answer: \infty


Question: Find the maximum speed of a particle whose position at time $t$ seconds is given by $$p(t)=(\ln(t^2+1), t^3-t).$$

Answer: \sqrt{5}


Question: Let $A$ be an operator on an infinite-dimensional complex Hilbert space $H$. Define $W(A)$ as the set of all $\langle Ax,x \rangle$ for unit vectors $x \in H$, and $w(A) = \sup_{z \in W(A)} |z|$. Given a sequence of unit vectors $(x_n)$ in $H$ such that $\lim_{n \to \infty} |\langle Ax_n, x_n \rangle| = w(A)$, determine if there exists a sequence of unit vectors $(y_n)$ in $H$ with $\langle x_n, y_n \rangle = 0$ for all $n$ and $\lim_{n \to \infty} |\langle Ay_n, y_n \rangle| = w(A)$. Provide a justification for your answer.

Answer: No


Question: Let $T: (C[0,1], ||.||_{\infty}) \to \mathbb{R}$ be a linear operator defined by $\displaystyle T(f) = \int_0^1 2xf(x)\,dx$ for all $f \in C[0,1]$. Determine $||T||$, where $||f||_{\infty} = \sup_{0 \le x \le 1} |f(x)|$. \(\boxed{\text{Answer}}\)

Answer: 1


Question: Is every subset of a measurable set also measurable?

Answer: No


Question: Calculate the product measure \((\mu\times\nu)(B)\), where \(B = \{(x,y) : x+y\leq1\}\), \(\mu\) is the Lebesgue measure on \([0, 1]\), and \(\nu\) is the counting measure on \([0, 1]\).

Answer: \infty


Question: A cube has 12 edges, each labeled with a unique number from 1 to 12. Each vertex of the cube is connected to three edges. The weight of a vertex is defined as the sum of the numbers on the three edges connected to it. Determine the number of ways to assign the numbers to the edges such that all vertices have the same weight.

Answer: 0


Question: Find the value of the Lambert W function at the point where the function \(f(x) = e^x\) intersects the line \(y = x\).

Answer: W(-1)


Question: In a Markov chain with six states, where state 6 is absorbing, each non-absorbing state has a 50% chance of moving one step up or down, except state 1, which always moves to state 2. Calculate the expected number of steps to reach the absorbing state if the starting state is chosen randomly with equal probability from all six states.

Answer: \dfrac{95}{6}


Question: Let $f, g \in L^{2}(\mathbb{R})$ and $\{x_{n}\} \subset (0, \infty)$ such that $x_{n} \to \infty$ as $n \to \infty.$ Assume that \[ \int_{\mathbb{R}} \left|\frac{1}{\sqrt{x_{n}}}f(y/x_{n})-g(y)\right|^2 dy \to 0 \] as $n \to \infty.$ Can we conclude that $f = g$?

Answer: f = g


Question: At 1 minute to midnight, 10 apples fall into a sack. The same happens at half a minute to midnight, then at a quarter minute to midnight, and so on. At each such event, you remove an apple randomly from the ones still present in the sack. What is the probability that the sack will be empty at midnight?

Answer: 1


Question: Find the value of the principal square root of the sum of a complex number and its square, given that the square of the complex number is equal to its principal square root but not equal to the number itself.

Answer: i


Question: For a non-prime integer $n$, does the congruence $a^n \equiv a \pmod{n}$ hold for all integers $a$ such that $\gcd(a,n)=1$? If so, provide an example of such an $n$.

Answer: 561


Question: Find the limit of the function \( f(t) = -t \sqrt{n} + n\log\left(1+ \frac{t}{\sqrt{n}}\right) \) as \( n \to \infty \).

Answer: -\dfrac{t^2}{2}


Question: Determine whether it is possible to draw $n$ circles ($n \ge 3$, $n \in \mathbb{N}$) such that they all intersect at a single point, but no three circles intersect at any other point.

Answer: Yes


Question: What is the ratio of the area of a triangle $ABC$ to the area of the triangle whose sides are equal in length to the medians of triangle $ABC$?

Answer: \dfrac{4}{3}


Question: Determine whether an entire function can map a line onto a circle. If it is possible, provide an example of such a function.

Answer: e^{iz}


Question: Let $\{A_n\}_{n \in \mathbb{N}}$ be a sequence of real numbers such that $\lim_{n \rightarrow \infty}A_n=0$. Determine if this implies that $plim_{n\rightarrow \infty}A_n=0$, where $plim$ denotes the probability limit.

Answer: Yes


Question: Determine the values of $m$ for which the quadratic equation $$(m+1)x^2 -2(m-1)x + m=0$$ has real roots $\alpha$ and $\beta$ such that $-1< \alpha < 1 < \beta < 2$. Provide your answer as a set of values for $m$.

Answer: (-\infty, -8)


Question: Determine whether there exists a right-angled triangle with integer sides such that its perimeter $P$ and area $A$ satisfy either $P = A \cdot k$ or $k \cdot P = A$, where $k$, $P$, and $A$ are positive integers.

Answer: Yes


Question: Evaluate the integral \( \int_0^\pi \int_0 ^\pi \vert \cos(x+y)\vert \, dx \, dy \) using the transformation: \(x+y=u \) and \( y=v\).

Answer: 2\pi


Question: Evaluate the integral \( \int_0^\infty \frac{dx}{(1+x^2)(1+x^c)} \) for arbitrary \(c \geq 0\).

Answer: \dfrac{\pi}{4}


Question: A square with side length $2n$ is divided into $4n^2$ smaller squares of equal size. A circle with diameter $2n-1$ is inscribed in the larger square. Determine the number of smaller squares that intersect the circumference of the circle.

Answer: 8n - 4


Question: Find the limit of the function \( \frac{x^2}{|x|+|y|}\cos(y^2) \) as \((x, y)\) approaches \((0, 0)\).

Answer: 0


Question: For a sequence generated by \(k^m, k^{2m}, k^{3m}, \ldots \pmod{m}\), where \(m\) is not prime, determine if there exists an element after \(k^m\) that is congruent to \(k^m \pmod{m}\).

Answer: Yes


Question: Find the average value of the function \( F(x, y, z) = z \) over the top half of a sphere with radius \( r = 2 \), where \( z \geq 0 \) and \( x^2 + y^2 + z^2 \leq 4 \). The volume of the region is \( \frac{16\pi}{3} \). Use the formula \( \overline{z}=\frac{1}{V}\int\int\int_{E} zdV \).

Answer: \dfrac{3}{4}


Question: What is the proportion of prime numbers \(p\) such that the Legendre symbols \(\left(\frac{2}{p}\right) = 1\) and \(\left(\frac{-5}{p}\right) = -1\)?

Answer: \dfrac{1}{4}


Question: Find all real constants \(a\) for which the existence and uniqueness theorem holds for the initial value problem (IVP) \(x' = \frac{3}{2}|x|^{1/3}\), \(x(0) = a\). Analyze the Lipschitz continuity of the function \(f(x) = \frac{3}{2}|x|^{1/3}\) at \(x = a\) to determine the correct values of \(a\).

Answer: a \neq 0


Question: Evaluate the limit: $$\lim_{n \to \infty} n \int_0^\pi \{ -x \}^n \, dx$$ where $\{ x \}$ denotes the fractional part of the real number $x$. Determine the value of this limit.

Answer: 4


Question: Calculate the sum \(\sum^{7}_{n=2} (n-1)P (X = n)\) for the random experiment of tossing a coin until the same number of heads and tails have been observed.

Answer: \dfrac{19}{16}


Question: Determine the number of real solutions to the equation \( f(f(f(f(x)))) = 0 \), where \( f(x) = x^2 + 10x + 20 \).

Answer: 2


Question: Evaluate the limit: $$\lim_{n \to \infty}\left(3 +\frac{1}{n}\right)^n\left(\frac{1}{3} + \frac{1}{n}\right)^{2n}$$ using the known limit $\lim_{n \to \infty}(1 + 1/n)^n = e$.

Answer: 0


Question: Consider an \(N \times N\) matrix \(\boldsymbol{M}\) composed of an upper submatrix and a lower submatrix, each with rank \(N/2\). What is the rank of the matrix \(\boldsymbol{M}\)?

Answer: N


Question: Evaluate the double integral \( \int_0^1\int_0^1 e^{\max(x^2,y^2)} \, dx \, dy \). Consider the cases where \( \max(x^2,y^2) = y^2 \) and \( \max(x^2,y^2) = x^2 \).

Answer: e - 1


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

Answer: -\dfrac{1}{2}


Question: Calculate the surface area of the portion of the cylinder \(x^2 + y^2 = 2ay\) that is contained within the sphere \(x^2 + y^2 + z^2 = 4a^2\), given that \(a > 0\).

Answer: 16a^2


Question: Find the value of $k$ if the angle between the planes given by the equation $6x^2+4y^2-10z^2+3yz+4zx-11xy=0$ is $\cos^{-1}(k)$. What is the value of $k$?

Answer: 0


Question: Determine whether there exists an ordinal \(\alpha\) such that \(\omega^{\omega^\alpha} = \alpha\).

Answer: Yes


Question:
Consider a sequence of real numbers $\{a_n\}$ such that the series $\sum _{n=1}^{\infty }\left|a_n-a_{n-1}\right|$ is convergent. Determine the convergence of the power series $\sum_{n=1}^\infty a_nx^n$ for $x \in \mathbb{R}$. Choose the correct option:

(a) Convergent nowhere on $\mathbb{R}$
(b) Convergent everywhere on $\mathbb{R}$
(c) Convergent on some set containing $(-1,1)$
(d) Convergent only on $(-1,1)$

Answer: C


Question: Assume that $\tau$ is a homeomorphism and $A$ is closed. Can the claim in Remark 2.1.1 be verified under these assumptions?

Answer: Yes


Question: Find the volume of the solid formed by revolving the region bounded by the circle \(x^2+y^2=1\) about the line \(y=\frac{\sqrt{3}}{2}\).

Answer: \pi^2 \sqrt{3}


Question:
Calculate the sum of the expression for integer \( n \ge 0 \):
\[
\frac{1}{n+2}\binom{2n+2}{n+1}-\sum_{k=1}^{n}\frac{1}{k+1}\binom{2k}{k}-\sum_{k=1}^{n-1}\frac{4}{k+3}\binom{2k+1}{k-1}
\]

Answer: 1


Question: Determine the number of linearly independent matrices in a commuting family of $4 \times 4$ complex matrices.

Answer: 5


Question: Determine the largest integer $k$ for which the equation \[ 25\lfloor x\rfloor - \lfloor x\lfloor x\rfloor \rfloor = 50k - 13\lceil x\rceil \] has a real solution for $x$. Provide your answer as an integer.

Answer: 7


Question: Evaluate the limit: $$\lim_{n \to \infty}\frac{\frac{\ln2}{2}+\frac{\ln3}{3}+\cdots+\frac{\ln n}{n}}{\ln^2 n}.$$

Answer: \dfrac{1}{2}


Question: Let $X$ be a metric space with metric $d$ such that $\text{diam}(X)=\infty$. Let $u:X\to(0,\infty)$ be a positive measurable function satisfying \( \int_{B_R(0)}u\,dx\leq\frac{1}{R} \) for every $R>0$, where $B_R(0):=\{x\in X:d(x,0)<R\}$. Does it follow that \( \int_{X}u\,dx=0 \)?

Answer: 0


Question:
Given a sequence of independent, identically distributed (i.i.d.) random variables $X_1, X_2, \ldots, X_i$ with $E[X_i] = 2$ and $\text{Var}(X_i) = 9$, determine the value to which the sequence converges:
\[ \lim_{i \to \infty} \frac{X_i}{2^i} \]

Answer: 0


Question: Let \( h(x) = \begin{cases} 1 - \frac{\sin x}{x} & \text{if } x \neq 0 \\ 0 & \text{otherwise} \end{cases} \) for \( x \in \mathbb{R} \). Define \( \alpha = \inf_{|x| \ge 1} h(x) \). Compute \( \alpha \) and justify why the infimum is attained at \( x = 1 \).

Answer: 1 - \sin 1


Question: Evaluate the expression: $$\frac{\sum\limits_{n=1}^{99} \sqrt{10 + \sqrt{n}}}{\sum\limits_{n=1}^{99} \sqrt{10 - \sqrt{n}}}$$

Answer: 1 + \sqrt{2}


Question: Let $X_1, X_2, X_3, \dots, X_n$ be $n$ independent and identically distributed random variables. Calculate the probability that these variables are in strictly increasing order: $$\Bbb P(X_1 < X_2 < X_3 < \dots < X_n).$$

Answer: \dfrac{1}{n!}


Question: Evaluate the improper integral: \[ \int_{0}^{+\infty} \frac{x - \sin{x}}{x^{3}}\, dx \] using the Dirichlet integral: \[ \int\limits_0^{+\infty} \frac{\sin \alpha x}{x}\,dx = \frac{\pi}{2}\text{sgn}\,\alpha \]

Answer: \dfrac{\pi}{4}


Question: Determine the values of \( \alpha > 0 \) for which the series \( \sum_{n=1}^\infty \frac{1}{n^\alpha \ln(1+n^\beta)} \) converges, given that \( \beta > 1 \).

Answer: \alpha > 1


Question: Determine a branch of the logarithmic function \(\log (2z - 1)\) that is analytic everywhere in the complex plane except on the ray \(\{x + iy : x \leq \frac{1}{2}, y = 0\}\).

Answer: \log(2z - 1)


Question: Determine the values of $x$ for which the matrix \( M = \begin{bmatrix} x & 0 \\ 0 & -\frac{1}{x} \end{bmatrix} \) cannot be expressed as a product of two involutory real matrices.

Answer: x \neq \pm 1


Question: Evaluate the limit: $$\lim_{n \to \infty} \dfrac{1}{n^4}\left(\left \lfloor{1^3 x}\right \rfloor + \left \lfloor{2^3 x}\right \rfloor + \cdots+ \left \lfloor{n^3 x}\right \rfloor\right)$$

Answer: \dfrac{x}{4}


Question: Determine whether there exists a logical structure in which the formula \( \forall x (A(x) \to B(x)) \to ( \exists x A(x) \vee \exists x B(x)) \) is true.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{\ln\left(\cos\left(\frac{x}{4}\right)\right)\ln\left(\sin^2\left(\frac{x}{8}\right)\right)}{\ln\left(\cos\left(\frac{x}{8}\right)\right)\ln\left(\sin^2\left(\frac{x}{4}\right)\right)}$$

Answer: 4


Question: Determine the cardinality of a 3-Sylow subgroup in the group of all invertible $4 \times 4$ matrices with entries in the field $\mathbb{F}_3$. Provide your answer as a single number.

Answer: 729


Question: For a field $k$ of characteristic $0$, consider the exact sequence $$1\to\pi_1(\mathbb A_{\overline k}^1\setminus\{0\})\cong\widehat{\mathbb Z}\to \pi_1(\mathbb A^1_k\setminus\{0\})\to\mathrm{Gal}(\overline k/k)\to 1,$$ which induces an action of $\mathrm{Gal}(\overline k/k)$ on $\widehat{\mathbb Z}$. This gives a group homomorphism $\mathrm{Gal}(\overline k/k)\to\widehat{\mathbb Z}^\times$. When $k=\mathbb Q$, is this homomorphism the same as the abelianization map $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\to \mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)^{\mathrm{ab}}\cong\widehat{\mathbb Z}^\times$?

Answer: Yes


Question: In a circle, points $A, B, C,$ and $D$ are arranged in order. The extensions of $AB$ and $DC$ intersect at point $E$, and the extensions of $AD$ and $BC$ intersect at point $F$. Tangents from $E$ and $F$ to the circle meet points $P$ and $Q$, respectively, with $EP = 60$ and $FQ = 63$. Find the length of segment $EF$. \( \boxed{} \)

Answer: 87


Question: Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of real numbers and $L$ be a real number. Determine whether the following statement is true or false: If for every neighborhood $U$ of $L$ there are infinitely many terms of $(a_n)$ in $U$, then $\lim_{n \to \infty} a_n = L$. Provide a justification for your answer.

Answer: False


Question: Let $f(x)$ be a rational non-constant polynomial such that $f(f(x)) = 3f(x)^4 - 1$. Determine the polynomial $f(x)$. \( \boxed{} \)

Answer: 3x^4 - 1


Question: Evaluate the integral \( \int_0^{\pi/2}\frac{\cos 3x+1}{\cos 2x-1}dx \).

Answer: -\infty


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

Answer: -1


Question: Evaluate the limit \( \displaystyle \lim_{x \to 0} \frac{\tan(x^2)-\sin(x^2)}{x^6} \) using L'Hopital's rule or Taylor series expansion.

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$\lim_{t\rightarrow\infty} \frac{1-\exp\left(-\frac{1}{1+tx}\right)}{1-\exp\left(\frac{1}{1+t}\right)}$$

Answer: -\dfrac{1}{x}


Question: Let $p_1, p_2 \in \mathbb{C}$ be linearly independent over $\mathbb{R}$. Consider an entire function $f: \mathbb{C} \to \mathbb{C}$ such that $f(0) = f'(0) = 1$, and $f(z + p_1) = a f(z)$ and $f(z + p_2) = b f(z)$ for all $z \in \mathbb{C}$, where $a, b \in \mathbb{C}$ are constants. Determine if it is true that $f(z) = e^z$ for all $z \in \mathbb{C}$, i.e., $f'(z) = f(z)$ for all $z \in \mathbb{C}$, and $f(z + w) = f(z)f(w)$ for all $z, w \in \mathbb{C}$. Provide a justification for your answer.

Answer: f(z) = e^z


Question: Find the area enclosed by the curve given by the equation $r^2 + \theta^2 = 1$ in polar coordinates.

Answer: \dfrac{2}{3}


Question: A ray of light is emitted from the point \((3,-2,-1)\) and reflects off the plane given by the equation \(x-2y-2z=0\). The reflected ray passes through the point \((4,-1,-6)\). Determine the point at which the ray hits the plane.

Answer: (2, 1, 0)


Question: Find the limit of the sequence \(y_n=\frac{a_1}{a_2}\cdot\frac{a_3}{a_4}\cdot...\cdot\frac{a_{2n-1}}{a_{2n}}\) where \((a_n)_{n\geq1}\) is an arithmetic progression such that \(0<a_1<a_2\).

Answer: 0


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

Answer: -2 \ln 2


Question: Evaluate the double integral: $$ \int_{-\infty}^\infty\int_{-\infty}^\infty \frac{xy}{1+(x^2+y^2)^2}\,dx\,dy $$

Answer: 0


Question: In an Artinian commutative ring, does the stabilization of every descending chain of ideals imply that every prime ideal is maximal?

Answer: Yes


Question: Consider the function \( f(x) \) defined as follows: \( f(x) = \begin{cases} x^2 + 1, & \text{if } x \text{ is irrational} \\ \tan(x), & \text{if } x \text{ is rational} \end{cases} \) in the interval \([0, 4\pi]\). Determine the number of points \( x \) within this interval where \( f(x) \) is continuous.

Answer: 4


Question: Given that $f(\pi)=\pi$ and \( \int_{0}^{\pi}\left(f(x)+f''(x)\right)\sin x\ dx=7\pi \), find $f(0)$, where $f(x)$ is continuous on $[0,\pi]$. \( \boxed{?} \)

Answer: 6\pi


Question: Evaluate the path integral \( \int_{\partial K(0,1)} \frac{dz}{(z-a)(z-b)} \) along the unit circle \( \partial K(0,1) \), where \(|a|,|b| < 1\) and \(a \neq b\).

Answer: 0


Question: Find the limit of the sequence \(\lim_{n\to\infty} \frac{(n!)^{1/n}}{n}.\)

Answer: \dfrac{1}{e}


Question: Determine if the polynomial $2x + 1$ has an inverse in the ring $\mathbb{Z}_3[x]/(x^2 + 2x + 2)$. If it does, find the inverse.

Answer: 2x


Question: Evaluate the integral of $|xy|$ over the region bounded by the ellipses $x^2+4y^2 \ge 1$ and $4x^2+y^2 \le 4$ using polar coordinates.

Answer: \dfrac{15}{8}


Question: Determine whether the sequence \(a_n= \frac{7^n + 6^n -n^{100}}{(7.1)^n-7^n+n^{101}}\) converges, and if so, find its limit.

Answer: 0


Question: Let $X$ be a topological space and $(A_{n})$ be a decreasing sequence of open sets, i.e., $A_{n+1} \subset A_{n}$. If the intersection of all $A_{n}$ is contained in a closed set $B$, is the closure of the intersection of all $A_{n}$ also contained in $B$?

Answer: Yes


Question: Evaluate the limit: $$\lim\limits_{x\to\ 0}\left(\frac{1}{\sin(x)\arctan(x)}-\frac{1}{\tan(x)\arcsin(x)}\right)$$ without using L'Hospital's rule or Taylor series.

Answer: 1


Question: Determine whether it is always possible to find a subinterval $[a, b] \subset [0,1]$ such that $\int_a^b f(x) \, dx = \int_a^b g(x) \, dx$ for any two functions $f$ and $g$ satisfying $\int_0^1 f(x) \, dx = \int_0^1 g(x) \, dx = 1$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to0}\left (\frac{4^{\tan(x)}+ \cos(x)}{2}\right)^{\cot(x)}$$ using logarithmic properties and standard limits.

Answer: 2


Question: Determine the quotient topology on the group \(\mathbb{R}\) given the equivalence relation \(x \sim y\) if and only if \(x-y \in \mathbb{Z}\) for all \(x, y \in \mathbb{R}\).

Answer: S^1


Question: Let $(\Omega, \mathcal{F}, P)$ be a probability space, where $X: \Omega \rightarrow \mathbb{R}$ is a random variable with $P(X = 0) = 1$. Define $\mathcal{N}$ as the collection of sets $A \subset \Omega$ for which there exists an $N \in \mathcal{F}$ with $P(N) = 0$ and $A \subset N$. The $\sigma$-algebra $\mathcal{G}$ is generated by $X$ and $\mathcal{N}$. If $Y: \Omega \rightarrow \mathbb{R}$ is a $\mathcal{G}$-measurable function, must it be the case that $P(Y = c) = 1$ for some $c \in \mathbb{R}$?

Answer: Yes


Question: Consider the heat equation \(\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}\) with the initial condition \(u(x,0) = \sin^2(x)\) and boundary conditions \(u(0,t) = u(\pi,t) = 0\). Determine \(\lim_{t\to \infty}u(\frac{\pi}{2},t)\).

Answer: 0


Question: Find the limit superior of the sequence \( \sin(x) + \sin(\pi x) \) as \( x \to \infty \).

Answer: 2


Question: Given the formula \( \nabla \bar{f}(y) = T^T \nabla f(Ty) \), derive the expression for the Hessian (second derivative) of \( \bar{f}(y) \).

Answer: T^\top \nabla^2 f(Ty) T


Question: In a Banach space \( \mathfrak{X} \), consider a weak-* convergent net in \( \mathfrak{X}^* \). Must this net be eventually bounded? Provide a justification for your answer.

Answer: Yes


Question: How many distinct homotopic curves are there in the complex plane connecting any two fixed points $a$ and $b$?

Answer: 1


Question: Determine the values of $c \in \mathbb{R}$ for which the sequence defined by $a_1 = c$ and $a_{n+1} = 1 + \frac{a_n^2}{4}$ converges.

Answer: [-2, 2]


Question: Determine the cardinality of a 3-Sylow subgroup in the group of all invertible $4 \times 4$ matrices with entries in the field of 3 elements.

Answer: 729


Question: Evaluate the integral \( \int_C \frac {3z^5 - 10z^3} {z^6-5z^4+10}\ dz \) where \( C = \{z \in \Bbb C : |z| < 2 \} \).

Answer: 4\pi i


Question: Suppose $V$ is an inner product space and $T \in \mathcal{L}(V)$. Determine whether $\langle Tv, w \rangle = \langle v, w \rangle$ for all $v, w \in V$ implies that $T = I$. Provide a justification for your answer.

Answer: T = I


Question: Does the continuity of a function $f: \mathbb{R}^2 \to \mathbb{R}$ imply the existence of its partial derivatives?

Answer: No


Question: Let $f(x)=x^4-2x^3-3x^2+4x-1$. If $f(x)$ has four distinct real roots $x_1,x_2,x_3,x_4$ such that $x_1<x_2<x_3<x_4$ and $x_1x_4=1$, find the value of $x_1^2+x_2^2+x_3^2+x_4^2$. Express your answer as a single number.

Answer: 10


Question: Determine all polynomial functions \( p(x) \) with complex coefficients such that \( p(x) \) maps real numbers to real numbers and non-real complex numbers to non-real complex numbers.

Answer: p(x) = ax + b


Question: Given that $x + y + z = xyz$ where $x, y, z \in \mathbb{Z}^{+}$, calculate the value of $xy + yz + xz$.

Answer: 11


Question: Let $X$ be a random variable with a Poisson distribution with parameter $n\lambda$. Find the expected value of $(X)_r$, where $(X)_r = \frac{X!}{(X-r)!}$. Express your answer in terms of $n$, $\lambda$, and $r$.

Answer: (n \lambda)^r


Question: Calculate the gradient of \( \text{Tr} \left( A^T A \right) \) with respect to the matrix \( A \), where \( A \) is a square matrix.

Answer: 2A


Question: If $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are differentiable functions and $h: \mathbb{R} \rightarrow \mathbb{R}$ is such that $f = g \circ h$, must $h$ also be differentiable? Answer 'yes' or 'no' and justify your answer.

Answer: no


Question: Let $(x_n)$ be the sequence defined by $x_1=2$ and the recursive formula $x_{n+1} = \frac{1}{2} + \sqrt{x_n}$. Find the value of $a+b+c$ where $a,b,c$ are the coefficients of the quadratic polynomial in the equation \( x_n - x_{n+1} = ax_{n+1}^2 + bx_{n+1} + c.\)

Answer: -\dfrac{3}{4}


Question: Let $M$ be the $(2n+1)\times (2n+1)$ skew symmetric matrix with entries given by \[a_{ij} = \begin{cases} \,\,\,\, 1 & \text{if } i-j\in \{-2n, -2n+1, \dots, -n-1\} \cup \{1, 2, \dots, n\}\\ -1 & \text{if } i-j\in \{-n, -n+1, \dots, -1\}\cup \{n+1,n+2, \dots, 2n\} \end{cases}\] Find the rank of the matrix $M$. \(\text{rank}(M)\).

Answer: 2n


Question: Consider a strictly increasing sequence of non-negative real numbers \((t_n)_{n\ge 0}\) with \(t_0 = 0\) and \(\lim_{n \to \infty} t_n = \infty\). The induced sequence \((S_n)_{n\ge 1}\) is defined by \(S_n = \sum_{i=0}^{n-1} \frac{t_{i+1} - t_i}{t_n - t_i}\). Determine whether there exists a sequence \((t_n)_{n\ge 0}\) such that the supremum of the sequence \((S_n)_{n\ge 1}\) is finite.

Answer: Yes


Question: Let \( f(x) = x^4 + ax^3 + bx^2 + cx + d \) be a polynomial with four real roots \( x_1, x_2, x_3, x_4 \). Given that \( b \geq d + 5 \), find the minimum possible value of the expression \((x_1^2 + 1)(x_2^2 + 1)(x_3^2 + 1)(x_4^2 + 1)\).

Answer: 16


Question: Determine the radius of convergence of the power series \(\sum_{n=1}^\infty \frac{(-1)^n}{4^n n^p} x^{2n}\), where \(p\) is a fixed parameter.

Answer: 2


Question: Evaluate the double integral \( \int_0^1 \int_0^{\sqrt{1-x^2}} e^{\sqrt{x^2 + y^2}} \, dy \, dx \) by converting to polar coordinates.

Answer: \dfrac{\pi}{2}


Question: Given two polynomials $f$ and $g$ in the polynomial ring $k[X_1, X_2]$, where $\frac{\partial f}{\partial X_2} = \frac{\partial g}{\partial X_1}$, determine whether there exists a polynomial $h$ in $k[X_1, X_2]$ such that $\frac{\partial h}{\partial X_1} = f$ and $\frac{\partial h}{\partial X_2} = g$. Provide a justification for your answer.

Answer: Yes


Question: Are all finite-dimensional algebras of the same dimension over a field $\mathbb{F}$ isomorphic to each other as $\mathbb{F}$-algebras? Answer yes or no.

Answer: No


Question: Evaluate the integral \( \int_{-0.5 \pi}^{0.5 \pi} \sin(x) F(\cos(x)) \, dx \), where \( F(x) = \int_{0}^{x} e^{\sin(t)} \, dt \).

Answer: 0


Question: Determine the number of ways to express $7^{13}$ as a product of three natural numbers.

Answer: 105


Question: Given a segment AB with only its endpoints painted, Matías can perform the following operations: 1) Choose two painted points X and Y, and paint the midpoint of XY. 2) Choose two painted points X and Y, a positive integer n, and paint the point Z on segment XY such that XZ/ZY = n/(n+1). Determine if Matías can paint a point C on segment AB such that AC/CB = p/q for any fraction p/q, where p and q are positive integers.

Answer: Yes


Question: Consider the function \( g(z) = (z^2 - 1)^\frac{1}{2} \). Determine the branch points of this function and find the value of \( g(-2) \) given a branch cut along the real axis from \( z = 1 \) to \( z = \infty \).

Answer: \sqrt{3}


Question: A triangular tower consists of 11 rows, with each square in the bottom row containing either a 0 or a 1. Each square above the bottom row is the sum of the two squares directly below it. Determine the number of distinct ways to fill the bottom row with 0s and 1s such that the number in the top square is divisible by 3.

Answer: 640


Question: Let \( n \) be an even natural number. The numbers \( 1, 2, \dots, n^2 \) are divided into two equal sets \( A \) and \( B \) such that \( |A| = |B| = \frac{n^2}{2} \). Let \( S_A \) and \( S_B \) be the sums of all elements in \( A \) and \( B \), respectively. Determine all even natural numbers \( n \) such that \( \frac{S_A}{S_B} = \frac{39}{64} \).

Answer: 206


Question:
Let $X$ be a topological space of infinite cardinality that is homeomorphic to $X \times X$. Determine which of the following statements is true:

(a) $X$ is not connected.
(b) $X$ is not compact.
(c) $X$ is not homeomorphic to a subset of $\mathbb{R}$.
(d) None of the above.

Answer: D


Question: Let \( f:[a,b]\to\mathbb{R} \) be a function that is continuous on \([a,b]\) and differentiable on \((a,b)\). Given that \( \frac{f(a)}{a} = \frac{f(b)}{b} \) and \( a, b \) have the same sign, prove that there exists a point \( c \) in \((a,b)\) such that \( f'(c) = \frac{f(c)}{c} \).

Answer: f'(c) = \dfrac{f(c)}{c}


Question: Calculate the operator norm of the linear operator $B: L^2[0,1] \to L^2[0,1]$ defined by $Bx(t) = \sin(t)x(t)$.

Answer: \sin(1)


Question: If $m$ is a non-zero vector and $Am=0$ has a solution for some $n \times n$ matrix $A$, can we be certain that there exists a non-zero vector $b$ such that $A^{T}b=0$?

Answer: Yes


Question:
Let \( f:\mathbb{R}\to\mathbb{R} \) be a continuous function such that \( f(x)\geq 0 \) for all \( x \). For \( r\geq 0 \) and \( n\in\mathbb{N} \), define
\[ J_n(r)=\int\cdots\int_{x_1^2+x_2^2+\cdots+x_n^2\leq r^2}f(x_1+x_2+\cdots+x_n)\,dx_1\,dx_2\cdots\,dx_n. \]
Find \( \lim_{n\to\infty} J_n(r) \) for a fixed \( r \).

Answer: 0


Question: Evaluate the convolution integral \( \int_{-\infty}^{+\infty} \delta''(t-b) h(t) \, dt \), where \( \delta''(t) \) is the second derivative of the Dirac delta function.

Answer: h''(b)


Question: Consider the paraboloid $P$ defined by $P = \{(x, y, z) \in \mathbb{R}^3: z = 4 - x^2 - y^2, x^2 + y^2 < 4\}$. Find the unit normal field of $P$ and compute the integral of the vector field $F(x, y, z) = (x, y, xy + z)$ over $(P, \nu)$. Provide your answer as a single number.

Answer: 24\pi


Question: Determine the largest number of consecutive fours that can appear at the end of a perfect square in decimal notation.

Answer: 3


Question: Find the constant value $c$ such that $y(t) = c$ is a solution to the differential equation: $$\frac{d^2y}{dt^2}(t) + (\sqrt{1+y^2}-2)\frac{dy}{dt}(t) + y(t) = 0.$$

Answer: 0


Question: Let $M$ be a W*-algebra with a non-empty set of minimal projections, and let $z(M)$ be the supremum of all minimal projections in $M$. It is known that $z(M)$ is a central projection. Consider the W*-algebra $M_0 = z(M)M$. Let $q$ be a projection in $M_0$. Is it true that $$q = \sup\{e: e \text{ is a minimal projection in } M_0\}?$$

Answer: Yes


Question: Evaluate the infinite series: $$\sum_{r=1}^{\infty} \frac{8r}{4r^4+1}$$

Answer: 2


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

Answer: 9


Question: Given that $x, y, z \ge 0$ and $(x^2+y^2)(y^2+z^2)(z^2+x^2)=2$, find the maximum value of $P = xy^2 + yz^2 + zx^2 + xyz$. Provide your answer as a single number.

Answer: 2


Question: Find all natural numbers $x$ such that there exists an integer $n$ for which $2^n + 15 = x^2$. Provide your answer as a list of all such $x$.

Answer: 4


Question: Find the limit: \[ \lim_{x \to \infty} \frac{\exp(2x)}{\exp(s\log(\cosh(\frac{2x}{s})))} \] for \( s > 0 \).

Answer: 2^s


Question: Compute the limit: \[ L= \lim\limits_{ x\to + \infty} \left[ (x+4)^{1+\frac{1}{x}} - x^{1+\frac{1}{x+\cos x}} \right] \]

Answer: 4


Question:
Given positive real numbers $x, y, z$ such that $xy + yz + zx = 1$, and two positive constants $k$ and $l$, find the minimum value of the expression:

$$E = kx^2 + ly^2 + z^2$$

The minimum value is given by $2t_0$, where $t_0$ is the unique root of the cubic equation:

$$2t^3 + (k + l + 1)t - kl = 0.$$

Answer: 2t_0


Question: Is the group of units \( \mathrm{Cl}^\times(V,Q) \) of a Clifford algebra \( \mathrm{Cl}(V,Q) \) a Lie group?

Answer: Yes


Question: Determine whether a non-continuous function $f:[0,3] \to \mathbb{R}$ can be bijective. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Determine whether the statement is true or false: A graph $G$ is $r$-partite if and only if its chromatic number $\chi(G)$ is $r$. Provide a justification for your answer.

Answer: False


Question: Find the residue of the function \( f(z) = e^{-\frac{3}{z^2}} \) at \( z = 0 \).

Answer: 0


Question: Determine the value of the constant $c$ in the differential equation $2(y')^2 = 2\cdot\sqrt{y} + c$ given the initial conditions $y(0) = 1$ and $y'(0) = 1$. Provide your answer as a single value for $c$.

Answer: 0


Question: Find the greatest common divisor (GCD) of the sequence \(A_0, A_1, \ldots, A_{2015}\), where \(A_n = 2^{3n} + 3^{6n+2} + 5^{6n+2}\) for \(n = 0, 1, \ldots, 2015\).

Answer: 7


Question: Calculate the Riemann-Stieltjes integral: $$\int_{0}^{1} x \, \mathrm{d}(4x)$$

Answer: 2


Question: Evaluate the limit: $$\lim_{x\to \infty}\frac{\sin(x^2)}{\sqrt{x}} $$

Answer: 0


Question: Determine the cardinality of the set of all sequences of rational numbers on the interval $[0,1]$. Consider sequences as functions from $\mathbb{N}$ to $\mathbb{Q}$. Express your answer in terms of cardinality notation.

Answer: 2^{\aleph_0}


Question: Determine the radius of convergence for the power series \(\sum_{n=1}^\infty \left(\frac{z-i}{z+2i}\right)^n\) using the Cauchy criterion.

Answer: \dfrac{3}{2}


Question: In a metric space with a bounded metric and the Heine-Borel property, is any closed set compact?

Answer: Yes


Question: If $f(n)$ is an injective function and the sequence $p_n$ converges to $L$, does the sequence $p_{f(n)}$ also converge to $L$?

Answer: Yes


Question: A polynomial $P(x)$ of degree 1998 satisfies $P(k) = \frac{1}{k}$ for each integer $k$ from 1 to 1999. Determine the value of $P(2000)$. \( \boxed{} \)

Answer: \dfrac{1}{1000}


Question: Calculate the area of the triangle formed by the points $A:(1,2,-3,3)$, $B:(3,-6,-4,2)$, and $C:(-3,-16,-4,0)$ in four-dimensional space $\mathbb{R}^4$. Provide your answer as a single numerical value.

Answer: 35


Question: Can the matrix \( \begin{matrix} 2 & 4 & 6 \\ 4 & 5 & 6 \\ 6 & 6 & 7 \\ \end{matrix} \) be transformed into a diagonal matrix using orthogonal matrices?

Answer: Yes


Question: Find the dimension of the symplectic group \(\operatorname{Sp}(2n, \mathbb{R})\), which is the group of invertible matrices \(A\) such that \(\omega = A^T \omega A\), where \(\omega = \begin{pmatrix} 0 & \mathit{Id}_n \\ -\mathit{Id}_n & 0 \end{pmatrix}\).

Answer: n(2n + 1)


Question: Find the maximum value of $a+b+c+d$ given that $a^2+b^2+c^2+d^2+abcd=5$.

Answer: 4


Question: Evaluate the integral \( \int_{-r}^r \sqrt{1 + \frac{x^2}{r^2 - x^2}} \, dx \) and verify that its value is \( \pi r \).

Answer: \pi r


Question: Determine the number of zeros of the function \( f(z) = 2z^5 + 8z - 1 \) within the annular region \( \{ 1 < |z| < 2 \} \).

Answer: 4


Question: Let $z = a + ib$ be a nonprime Gaussian integer such that $z = (c + id)(e + if)$. Determine whether it is true that either $|c + id| \le \sqrt{|a + ib|}$ or $|e + if| \le \sqrt{|a + ib|}$.

Answer: True


Question: Compute the real integral \( \int_{-\infty}^\infty \frac{dx}{(1+x^2)^3} \) using complex analysis techniques.

Answer: \dfrac{3\pi}{8}


Question: Determine the interval $I$ on which the function $f(x) = x\ln x$ is Lipschitz continuous. Find the interval $I$ such that there exists a constant $L$ satisfying the inequality $|f(x_1) - f(x_2)| \leq L|x_1 - x_2|$ for all $x_1, x_2 \in I$. Provide the interval $I$.

Answer: [a, b]


Question: Given the equation \(|z-\frac{4}{z}|=8\), find the maximum value of \(|z|\).

Answer: 4 + 2\sqrt{5}


Question: Determine whether every finitely presented group, such as $F_3$, can be expressed as a quotient of the free group $F_2$ on two generators.

Answer: No


Question:
Find the surface area of the part of the sphere defined by $x^2+y^2+z^2=1$ where $z \geq \frac{1}{2}$. Express your answer in terms of $
abla$.

Answer: \pi


Question:
Given that the vector functions \(\begin{pmatrix}1\\t-1\end{pmatrix}\) and \(\begin{pmatrix}t+1\\-1\end{pmatrix}\) are solutions to the homogeneous system \(\mathbf{x'}=\mathbf{Ax}\), where \(A\) is a 2x2 matrix with continuous functions as elements on the interval (0,1), determine which of the following functions is the second element of a solution to the system \(\mathbf{x'}=\mathbf{Ax}+\begin{pmatrix}t^2\\t^3\end{pmatrix}\):

(A) \(-\frac{1}{2}t^2+\frac{1}{3}t^3\)
(B) \(\frac{1}{2}t^2+\frac{1}{6}t^3+\frac{1}{3}t^4\)
(C) \(2t+4t^2\)
(D) \(\frac{1}{4}t^4\)

Answer: B


Question: Find the remainder when \(2^{14^{45231}}\) is divided by 31.

Answer: 16


Question:
Consider the function \( f(x,y) = \begin{cases} \frac{xy}{(x^2+y^2)^a}, & \text{if $(x,y)\neq (0,0)$ } \\[2ex\] 0, & \text{if $(x,y)= (0,0)$} \end{cases} \). Determine which of the following statements is true for \( f \) at the point \((0,0)\):

(A) For \(a=1\), \(f\) is continuous but not differentiable.

(B) For \(a=\frac{1}{2}\), \(f\) is continuous and differentiable.

(C) For \(a=\frac{1}{4}\), \(f\) is continuous and differentiable.

(D) For \(a=\frac{3}{4}\), \(f\) is neither continuous nor differentiable.

Answer: C


Question: In a topological vector space $X$, if $K$ is a compact set, is $K$ necessarily bounded? Answer true or false.

Answer: A


Question: Identify the appropriate set $F$ to represent a standard Wiener process $W = \{W_t, t \in [0,1]\}$ in the context of functional data analysis, where the regressor $X$ is a random variable taking values in an infinite-dimensional space $F$ equipped with a (semi/pseudo) metric $d$. \( \text{(Hint: Consider the space of continuous functions)} \)

Answer: C([0,1])


Question: Evaluate the integral \( \int_{\gamma}\frac{dz}{z} \) for a path \( \gamma \) from \( -i \) to \( i \) that avoids the negative real axis, using a parameterization of the path.

Answer: i\pi


Question: Determine whether the following statement is true or false: If a sequence of positive real functions converges in $L^1$-norm, then the supremum of the sequence is integrable.

Answer: B


Question: Determine whether the following statement is true or false: For any right-continuous function \( g \colon \mathbb{R} \rightarrow \mathbb{R} \), we have \( \inf_{x \in \mathbb{R}} |g(x)| = \inf_{x \in \mathbb{Q}} |g(x)| \).

Answer: True


Question: Determine the number of zeros of the function \( f(z) = z^2 + \cosh(iz) \) within the region \( |z| < \frac{1}{2} \).

Answer: 0


Question:
For any positive real number $\lambda$, define the set $A_\lambda$ as the collection of natural numbers $n$ such that $|\sin\sqrt{n+1}-\sin\sqrt n|<\lambda$. Let $A_\lambda^c$ denote the complement of $A_\lambda$ in the set of all natural numbers. Determine which of the following statements is true:

(A) $A_{1/2},A_{1/3},A_{2/5}$ are all finite sets.
(B) $A_{1/3}$ is a finite set, but $A_{1/2},A_{2/5}$ are infinite sets.
(C) $A_{1/2}^c,A_{1/3}^c,A_{2/5}^c$ are all finite sets.
(D) $A_{1/3},A_{2/5}$ are finite sets, and $A_{1/2}$ is an infinite set.

Answer: C


Question: Determine whether the following statement is true or false: If a function $f$ is uniformly continuous on any closed interval $[a,b]\subset \mathbb{R}$, then $f$ is uniformly continuous on the interval $(-\infty, \infty)$. Provide a justification for your answer.

Answer: False


Question: Does the function \(f(z) = \frac{z}{(z^2 - 1)(z^2 - 4)}\) admit a primitive in the set \(\{|z| < 1\}\)? Justify your answer.

Answer: Yes


Question: If you sum a certain number of rank-1 matrices: \[X = u_1 u_1^T + u_2 u_2^T + \cdots + u_N u_N^T\] where the individual vectors \(\{u_k\}_{k=1}^N\) are linearly independent, is the result guaranteed to be rank-\(N\)?

Answer: Yes


Question: Determine the convergence rate of the Newton iteration for the polynomial \( f(x) = 8x^4 - 12x^3 + 6x^2 - x \) when started close to the root \( r = 0 \).

Answer: 2


Question: Let \( \{X_n\}_{n=1}^\infty \) be a sequence of i.i.d. random variables with mean zero. Prove that there exists a constant \( K > 0 \) such that the probability that the partial sums \( |X_1 + \cdots + X_n| \) are bounded by \( K \) infinitely often is equal to 1.

Answer: 1


Question: For an RSA modulus \( n = pq \), where \( p \) and \( q \) are distinct primes, consider the map \( B_m: \mathbb{Z}_n \to \mathbb{Z}_n \) defined by \( B_m(r) = m + r \pmod{n} \). Does this map produce a uniform distribution over \( \mathbb{Z}_n \) when \( r \) is chosen uniformly at random from \( \mathbb{Z}_n \) and \( m \) is a fixed element of \( \mathbb{Z}_n \)? Provide a justification for your answer.

Answer: Yes


Question: In triangle ABC, O is the center of the incircle. Given that AB = 6 and angle ACB = 120 degrees, determine the radius of the circumscribed circle of triangle AOB.

Answer: 6


Question: Calculate the limit: $$\lim_{x\to\infty}\frac{\ln{2^{\frac{1}{x}}}-\ln{x^2}}{1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{x}}$$

Answer: -2


Question: Find the residue of the function \( f(z) \log \frac{z-b}{z-a} \) at \( z = a \), where \( f \) is an entire function and \( a, b \in \mathbb{C} \).

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^{+\infty} x e^{-|x|} \, dx \).

Answer: 0


Question: In probability theory, if the probability of the complement event $A^c$ occurring infinitely often is $0$, does it imply that the probability of event $A$ occurring infinitely often is $1$? (Here, "i.o." stands for "infinitely often")

Answer: Yes


Question: Determine the chromatic number of a graph $G$ where the vertex set consists of all non-empty subsets of $[n]$, and two vertices are adjacent if and only if their intersection is empty.

Answer: n


Question: A trough is 8 feet long and 1 foot high, with a vertical cross-section parallel to an end shaped like the graph of \( y = x^2 \) from \( x = -1 \) to \( x = 1 \). If the trough is full of water, calculate the amount of work required to pump all the water over the top of the trough.

Answer: 266.24


Question: If a subset of a sigma field is also a sigma field, must it be equal to the original sigma field? Answer yes or no and justify your answer.

Answer: No


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

Answer: 5


Question: Determine whether the limit \( \lim_{(x,y)\to (0,0)} \frac {x^3y^2}{x^4+y^6} \) exists.

Answer: 0


Question: Determine whether every nonsingular $n \times n$ matrix with real entries can be made singular by changing exactly one entry. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Let $P(x,y)$ be a point on the hyperbola $3x^2-4y^2=36$ that is nearest to the line $3x+2y=1$. Find the value of $\sqrt{2}(y-x)$.

Answer: -9


Question: Is it possible to construct an orthonormal basis for $L^2(\Omega)$ when $\Omega$ is a bounded domain in ${\bf R}^n$?

Answer: Yes


Question: Find the limit of the function as \( x \) approaches infinity: \[ \lim_{x \to \infty} x \left[ \frac{1}{e} - \left( \frac{x}{x+1} \right)^x \right] \]

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


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be an infinitely differentiable function such that \( f(0) = 0 \), \( f(x) > 0 \) for all \( x \neq 0 \), and \( f^{(n)}(0) = 0 \) for all \( n \in \mathbb{N} \). Determine whether the function \( g(x) = \sqrt{f(x)} \) is infinitely differentiable.

Answer: Yes


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

Answer: 0


Question: Find the possible values of the constant $S$ such that there exists a real number $x$ satisfying the equation $x^3+(S-x)^3+S^3+33x(S-x)=2662$. Provide all possible values of $S$.

Answer: 11


Question: Let \( b_n \) be the sequence of all positive integers such that the decimal expression for \( \frac{1}{b_n} \) terminates in an odd digit: 1, 2, 4, 8, 10, \ldots (For instance, 3 is not included because \( \frac{1}{3} = 0.33\ldots \) does not terminate, 4 is included because \( \frac{1}{4} = 0.25 \) terminates in 5, which is odd; 5 is not included because \( \frac{1}{5} = 0.2 \) terminates in 2, which is even.) Find the sum to infinity of \( \frac{1}{b_n} \).

Answer: \dfrac{20}{9}


Question: If $\lambda$ is an eigenvalue of matrix $A$, determine the eigenvalues of the matrix $A^TA$.

Answer: |\lambda|^2


Question: Let $u_n, u \in L^{\infty}(E)$ and suppose $u_n \rightarrow u$ almost everywhere. Determine whether the following equality holds: \[ \sup_{m \geq n} \, \esssup_E |u_n - u_m| = \esssup_E \, \sup_{m \geq n} |u_n - u_m|. \]

Answer: Yes


Question: Calculate the flux of the vector field \( F(x,y,z) = (3x-2y+z, 2x+3y-z, x-3y+z) \) through the surface defined by \( S: |3x-2y+z|+|2x+3y-z|+|x-3y+z|=1 \).

Answer: \dfrac{28}{9}


Question: Compute the integral of the 2-form $\omega=\mathrm{d}x\wedge\mathrm{d}y$ over the surface of the ellipsoid given by $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$.

Answer: 0


Question: Evaluate the limit \( \lim_{n\rightarrow\infty}\frac{1}{n}\sum^n_{k=2} (k+1)^{\frac{1}{k+1}} \).

Answer: 1


Question: In a plane, determine the minimum number of non-parallel straight lines required such that at least one angle formed between any two lines is less than 26 degrees.

Answer: 7


Question:
Consider the heat equation $u_t = u_{xx}$ on the domain $[0, \pi] \times [0, L]$ with boundary conditions $u(0,t) = u(\pi,t) = 0$ for $t \in [0, L]$ and initial condition $u(x,0) = \phi(x)$ for $x \in [0, \pi]$. If $u(x,L) = f(x)$, determine which of the following statements is true for a suitable kernel $k(x,y)$:

(A) $\int_{0}^{\pi} k(x,y) \phi(y) \, dy = f(x)$ for $0 \leq x \leq \pi$

(B) $\phi(x) + \int_{0}^{\pi} k(x,y) \phi(y) \, dy = f(x)$ for $0 \leq x \leq \pi$

(C) $\int_{0}^{x} k(x,y) \phi(y) \, dy = f(x)$ for $0 \leq x \leq \pi$

(D) $\phi(x) + \int_{0}^{x} k(x,y) \phi(y) \, dy = f(x)$ for $0 \leq x \leq \pi$.

Select the correct option.

Answer: A


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

Answer: -\dfrac{1}{3}


Question: Find the limit of the expression as \( n \) approaches infinity: $$\lim_{n\rightarrow\infty} \frac{(n+1)^{2n^2+2n+1}}{(n+2)^{n^2+2n+1}n^{n^2}}$$

Answer: \dfrac{1}{e}


Question: Determine the number of elements in the quotient ring $\mathbb{Z}[i]/\langle 5+2i\rangle$. Provide your answer as a single integer.

Answer: 29


Question: Determine the range of values for \(a\) such that the roots of the equation \(x^2 - 2x - a^2 + 1 = 0\) are between the roots of the equation \(x^2 - 2(a + 1)x + a(a - 1) = 0\). Express this range as \((p, q)\). Then, calculate the value of \(q + \frac{1}{p^2}\).

Answer: 17


Question: Calculate the line integral \( \int_C z\,dx + x\,dy + y\,dz \) where \( C \) is the curve defined by the parametric equations \( x = t \), \( y = t^2 \), and \( z = t^3 \) for \( 0 \leq t \leq 1 \).

Answer: \dfrac{91}{60}


Question: Determine whether the following statement is true or false: A metric space is totally bounded if and only if it contains a finite dense subset.

Answer: B


Question: Let $(f_n)$ be a sequence of differentiable real-valued functions on $[a,b]$. Suppose that $(f_n(c))$ converges for some $c \in [a,b]$ and $(f_n')$ converges uniformly on $[a,b]$. Does it follow that $(f_n)$ converges uniformly on $[a,b]$? Provide a justification for your answer.

Answer: Yes


Question: Find all positive real solutions to the equation \( \left(\dfrac{\ln (x)}{x}\right)^2=\pi^2 \).

Answer: \dfrac{W(\pi)}{\pi}


Question: Determine the number of natural numbers \( n \) such that \( 5^n - 1 \) is divisible by every prime greater than 3.

Answer: 0


Question: Determine the order of the automorphism group $\mathrm{Aut}(\mathbb{Z}_p \times \mathbb{Z}_q \times \mathbb{Z}_r)$, where $p$, $q$, and $r$ are distinct primes.

Answer: (p-1)(q-1)(r-1)


Question: If $f$ is a nonmeasurable function on a measurable set $E$, is it possible for $f^2$ to be measurable on $E$?

Answer: Yes


Question: Calculate the sum of the squares of the cosines of 180 angles in arithmetic progression, starting from the angle \( \alpha \).

Answer: 90


Question: Find the value of the coefficient \(a\) such that the limit \(\lim_{x\to0}\frac{e^{ax}-e^x-x}{x^2}\) is finite.

Answer: 2


Question: Suppose \((a, b, c) \in \mathbb{R}^3\), with \(a, b, c\) all nonzero, and we have the equation \(\sqrt{a+b}+\sqrt{b+c}=\sqrt{c+a}\). Determine the value of the expression \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\).

Answer: 0


Question:
Given a function $f$ defined for all positive integers $x$ and $y$, where:

1. $f(1, 1) = 2$
2. $f(x + 1, y) = 2(x + y) + f(x, y)$
3. $f(x, y + 1) = 2(x + y - 1) + f(x, y)$

Find the value of $x - y$ such that $f(x, y) = 2006$.

Answer: -20


Question: Let $(X,Y)$ be a random vector with a joint density function given by \(f(x,y) = \frac{6}{7}(x^2 + \frac{xy}{2})\) for \(0 < x < 1\) and \(0 < y < 2\). Given that the marginal density of \(X\) is \(f_X(x) = \frac{12}{7}x^2 + \frac{6}{7}x\) and \(\mathbb{P}(X > Y) = \frac{15}{56}\), find the conditional probability \(\mathbb{P}(Y > \frac{1}{2} | X < \frac{1}{2})\). Note that the domain for the conditional probability is \(0 < x < \frac{1}{2}\) and \(\frac{1}{2} < y < 2\).

Answer: \dfrac{69}{80}


Question:
Let \( f: [a, b] \to \mathbb{R} \) be a continuous function. Assume that the upper Dini derivative \( \overline Df(x) \) and the lower Dini derivative \( \underline Df(x) \) are bounded as follows: \(-M \leq \underline Df(x) \leq \overline Df(x) \leq M\) for all \( x \in [a, b] \). Using the notation from Royden & Fitzpatrick:
\[
\begin{split}
\overline Df(x) &= \lim_{h \to 0^+} \sup\left\{\frac {f(x+t)-f(x)} t:0<|t|<h\right\} \\
\underline Df(x) &= \lim_{h \to 0^+} \inf\left\{\frac {f(x+t)-f(x)} t:0<|t|<h\right\}
\end{split}
\]
Determine if there exists a positive integer \( n \) such that for each \( 1 \leq i \leq n \), the following inequality holds:
\[
3M \geq \left|\frac {f\left(x+ \frac {y-x} ni \right)-f\left(x+ \frac {y-x} n (i-1)\right)} {\frac {y-x} n}\right|
\]

Answer: Yes


Question: Evaluate the variance of the stochastic integral \(\mathbb{Var}\left(\int_{0}^t ((B_s)^2 + s) \mathrm{d}B_s \right)\) where \(B_s\) is a standard Brownian motion.

Answer: 2t^3


Question: Consider the sequence \((z_n)\) defined by \(z_n=(-1)^n+\frac{1}{n}\) for \(n\in\mathbb{N}_{+}=\{1,2,3,...\}\). Given a subset \(A\subset\mathbb{R}\) with \(\inf A=-2\) and \(\sup A = 1\), let \(B=\{az_n | a\in A, n\in\mathbb{N_{+}}\}\). Determine \(\sup B\) using the alternative characterization of supremum.

Answer: 2


Question: Determine the Krull dimension of the injective hull $E = E_R(R/\mathfrak{m})$, where $(R, \mathfrak{m})$ is a Noetherian local ring.

Answer: 0


Question: Given an affine variety $X$ with a known dimension, determine the maximum value of $m$ such that there exists a dominant morphism from $X$ to $\mathbb{A}_{k}^{m}$. Provide your answer as the maximum value of $m$. \( \boxed{} \)

Answer: \dim X


Question: In a metric space $X$, if every continuous function $f: X \to \mathbb{R}$ maps $X$ to an interval in $\mathbb{R}$, does it necessarily follow that $X$ is connected? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \(\lim_{n\to\infty}\sqrt[n]{\frac{(2n)!}{n!n^n}}\) and express your answer in terms of \(e\).

Answer: \dfrac{4}{e}


Question: For a set of chi-square random variables with degrees of freedom $k = 2$, calculate the probability that a specific variable $X_{\alpha}$ is the minimum among $n$ such variables as $n$ approaches infinity.

Answer: 0


Question: Determine whether the sequence \( y_{n} = n\ln(n) - n - (\ln(n))^{2} \) converges or diverges.

Answer: \infty


Question: Find the minimum distance from the origin to the surface defined by the equations: \( xyza = 1 \) and \( x + y + z + a = 4 \).

Answer: 2


Question: Find the greatest common divisor (gcd) of the polynomials \(x^4+1\) and \(x^{24}-1\) over the finite field \(\mathbb{F}_5\).

Answer: x^4 + 1


Question:
Suppose $C$ is the set of all one-variable computable functions. Let $C^{(n)}$ be the set of all one-variable computable functions that can be computed using a program with at most $n$ instructions, and $C_{(n)}$ be the set of all one-variable computable functions that can be computed using a program with variables $X, Y, Z_1, ..., Z_n$. Determine which of the following statements is correct:

(a) $ \exists n \ C^{(n)} = C , \exists n \ C_{(n)} = C$

(b) $ \exists n \ C^{(n)} = C , \forall n \ C_{(n)} \subsetneq C$

(c) $ \forall n \ C^{(n)} \subsetneq C , \forall n \ C_{(n)} \subsetneq C$

(d) $ \forall n \ C^{(n)} \subsetneq C , \exists n \ C_{(n)} = C$

Answer: D


Question: Find the minimum value of the expression \(|a+b|^2+|b+c|^2+|c+a|^2\) given that \(a, b, c\) are unit vectors.

Answer: 3


Question: Let $M$ be a smooth $n$-manifold, and consider an open set $U \subset M$. Define an atlas $\mathcal{A}_U$ on $U$ as the collection of all smooth charts $(V, \varphi)$ for $M$ such that $V \subset U$. Is $\mathcal{A}_U$ a maximal smooth atlas on $U$? Provide a justification for your answer.

Answer: Yes


Question: Determine the value of $g(0)$ given that $g: \mathbb{R} \to \mathbb{R}$ is a differentiable function satisfying \( \lim _{n \to \infty} n\int_{0}^{1/n}g(t)\,dt =1 \).

Answer: 1


Question: In a second price auction with common values, three players have values represented by random variables $x_1$, $x_2$, and $x_3$, each uniformly distributed on $[0,1]$ and independent. The valuations are given by $v_1 = v_2 = v_3 = x_1 + x_2 + x_3$, and the bidding strategy in the Bayesian Nash Equilibrium (BNE) is $b_i(x_i) = 3x_i$. Calculate the expected payment of the winner.

Answer: \dfrac{3}{2}


Question: Find the maximum value of the function \( f(x) = (1+x)^2(1-x) \) for \( x > 0 \).

Answer: \dfrac{32}{27}


Question: Suppose $u_1, u_2: \mathbb{R}^2 \rightarrow \mathbb{R}$ are two solutions to the wave equation \( \partial_t^2u_i = \partial_x^2u_i \) that satisfy the condition \( \partial_x \left(u_1^2 + u_2^2\right) = 0 \) and are globally bounded. Must $u_1$ and $u_2$ be of the form \( u_1(x,t)=C_1\sin(kz)\sin(kt+\phi_1) + C_2\cos(kz)\sin(kt+\phi_2) \) and \( u_2(x,t)=\pm \left( C_1\cos(kz)\sin(kt+\phi_1) - C_2\sin(kz)\sin(kt+\phi_2) \right) \)?

Answer: Yes


Question: Determine the positive values of $x$ for which the series \( \sum_{n=1}^{\infty}x^{\sqrt{n}} \) is convergent.

Answer: (0, 1)


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

Answer: \dfrac{1}{2}


Question: Calculate the index \([G:H]\) for the group \(G\), the group of all non-singular \(n \times n\) matrices over the finite field \(GF(11)\), and its subgroup \(H\), the set of matrices with determinant 1.

Answer: 10


Question: Evaluate the improper integral: $$ \int_{0}^{\infty} \frac{4}{(x+1)^2(x+3)} \, dx $$

Answer: 2 - \ln 3


Question: Find all positive integers $x$ such that both $2^x - 1$ and $2^x + 1$ are prime numbers.

Answer: 2


Question: Determine whether the expression \( \lim_{x \to \infty} \left(\ln x-\ln \left(x\left(1-\frac{1}{\ln^2 x}\right)\right)\right)\left(\ln \ln x+ B\right) \) converges, where \( B \) is the Meissel–Mertens constant.

Answer: 0


Question: Evaluate the expression \( \tan \left( \arcsin\left( \frac{\sqrt{1-x^2}+x}{\sqrt 2} \right) - \arcsin x \right) \) for \( 0 \le x \le \frac{1}{2} \).

Answer: 1


Question: Evaluate the limit: $$\lim_{x\rightarrow0^+}\frac{\sin^3x-x\arctan(x^2)+\beta x^5}{x^7}$$

Answer: \dfrac{53}{120}


Question:
Determine whether the following statement is true or false:
\[ \lim_{n \to \infty} \int_0^\infty f_n(x) \, dx \ne \int_0^\infty \lim_{n \to \infty} f_n(x) \, dx \]
where \( f_n(x) = ne^{-nx} \) for all \( x \in [0, \infty) \) and \( n \in \mathbb{N} \).

Answer: A


Question: Determine whether there exists a bijective differentiable function \( f: \mathbb{R^+} \rightarrow \mathbb{R^+} \) whose derivative is not a continuous function.

Answer: Yes


Question: Determine the Galois group of the polynomial \(x^5 - 9x + 3\) over \(\mathbb{Q}\).

Answer: S_5


Question: Consider a sequence of functions $(g_n)$ such that each function has infinitely many sign changes. Can the sequence $(g_n)$ uniformly converge to a function $g$ that has only a finite number of sign changes? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral: \[ \int^{\pi/2}_0 \frac{x+\sin x}{1+\cos x}\,\mathrm dx \]

Answer: \dfrac{\pi}{2}


Question: Suppose $S$ is an open set in $\mathbb{R}$ such that $\mathbb{Q} \subset S$. Is it possible for its complement, $T = \mathbb{R} \setminus S$, to be uncountable?

Answer: Yes


Question: Determine the radius of convergence of the Taylor series for the function \( \frac{2\tan x}{1+4x^2} \) centered at \( x=0 \).

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$\lim_{t\rightarrow 0}\left[-t^{-4} + t^{-5}\left(1+\frac{t^2}{3}\right)\arctan t\right]$$

Answer: \dfrac{4}{45}


Question: Evaluate the limit: \[ \lim_{n \to \infty} n^2\left[\left(1+\frac{1}{n}\right)^8 - \left(1+\frac{2}{n}\right)^4\right] \] without using L'Hôpital's rule.

Answer: 4


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \left( \sum_{k=1}^{10} k^2 \cos(kx) \right) \cos(2x) \, dx \).

Answer: 4\pi


Question: Evaluate the limit \( \lim_{x\to 0}\frac{f(2+\sin x)-f(2+x\cos x)}{x-\sin x} \) given that the function \( f(x) \) is differentiable, \( f'(x) \) is continuous for all real numbers, and \( f'(2)=14 \).

Answer: 28


Question: Calculate the degree of the field extension $[\mathbb{Q}(2^{1/4})(\sqrt{5}) : \mathbb{Q}(\sqrt{5})]$. This is equivalent to finding the degree of the minimal polynomial of $2^{1/4}$ over $\mathbb{Q}(\sqrt{5})$. Determine this degree.

Answer: 4


Question: Given real numbers $a, b, c, d, e$ such that $a+b+c+d+e=8$ and $a^2+b^2+c^2+d^2+e^2=16$, determine the minimum value of the expression $\sum_{cyc} (t-a)^2$ for $t \in \mathbb{R}$ and find the values of $a, b, c, d, e$ for which this minimum occurs.

Answer: \dfrac{16}{5}


Question: Let $A$ be an abelian group, and $B, C$ be subgroups of $A$. Suppose $B + C = A$ and $A$ is isomorphic to the direct product of $B$ and $C$. Does this imply that $B \cap C = 0$?

Answer: Yes


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

Answer: \infty


Question: Evaluate the integral \( I = \int_\gamma \left(\sqrt{x^2-y}+\frac{x^2}{\sqrt{x^2-y}}\right)dx-\frac{x}{2\sqrt{x^2-y}} \, dy \), where \( \gamma \) is a part of the curve \( x=y^2 \) from \((1,-1)\) to \((4,-2)\), using a different parametrization than the one provided in the original question.

Answer: 11\sqrt{2}


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2} f^{-1}(x) \, \mathrm{d}x \), where the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) is defined by \( f(x) = x + \sin x \).

Answer: 0


Question: For a positive integer \(n\), let \(d(n)\) denote the largest odd divisor of \(n\). Calculate the last three digits of the sum \(S = d(1) + d(2) + d(3) + \cdots + d(2^{99})\).

Answer: 782


Question: Determine the exact order of convergence of the sequence \( \{x_n\}_{n=0}^{\infty} \) defined by \( x_{n+1} = 2x_n - \pi x_n^2 \) with the initial value \( x_0 = \frac{1}{3} \).

Answer: 2


Question: Find the maximum value of the function \( f(x, y, z) = x^2 + y^2 + z^2 \) subject to the constraints given by the demand equation \( z = x + y \) and the ellipsoid equation \( \frac{x^2}{4} + \frac{y^2}{5} + \frac{z^2}{25} = 1 \).

Answer: 10


Question: Calculate the last two digits of \(9^{9^{9^{9}}}\) using modular arithmetic.

Answer: 89


Question: Determine whether there exist infinitely many integers \( n \) such that a given prime \( p \) divides \( a^n + n \) for any integer \( a \).

Answer: No


Question: Given a Lie group with its Lie algebra, determine if it is possible to compute the coefficients $c_i$ such that for two elements $A=e^{a^iE_i}$ and $B=e^{b^iE_i}$, the product $AB=C=e^{c^iE_i}$ can be expressed using the Lie algebra structure constants.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to 1} \frac{ \int_0^{\int _{1} ^ x (ue^{-u})du} \, (t^2e^t ) \, dt}{(x^3-1)} $$

Answer: 0


Question: Find the minimum value of the function \( f(a, b, c) = \int_0^1 (a + bx + cx^2)^2 \, dx \) subject to the constraint \( a + b + c = 1 \). The integral expands to \( a^2 + a(b + \frac{2c}{3}) + \frac{b^2}{3} + \frac{bc}{2} + \frac{c^2}{5} \).

Answer: \dfrac{1}{9}


Question: Find the moment generating function of the limit distribution of $X_n$ in the beta distribution problem, specifically for the case where $t=0$. Provide your answer as a simplified expression.

Answer: 1


Question: Calculate the sixth derivative with respect to $x$ and then the second derivative with respect to $y$ at the point $(0,0)$ for the function $f(x,y)=\frac{1}{1-x^2y}$. What is the value of this derivative?

Answer: 0


Question: Let $G = GL(2, GF(3))$ be the group of all $2 \times 2$ matrices with non-zero determinant, where the entries are from the finite field $GF(3) = \{0, 1, 2\}$. Define $H = \{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in G : ad - bc = 1\}$. Determine the order of the subgroup $H$, denoted as $o(H)$. \( \boxed{} \)

Answer: 24


Question: Evaluate the complex integral: $$\displaystyle\oint_{|z|=1}\sin\left (\displaystyle\frac{1}{z}\right ) dz$$

Answer: 2\pi i


Question: Find the residue of the function \( g(z) = \frac{1 - e^{2iz}}{\cosh^2(2z)} \) at \( z = \frac{\pi}{2} \).

Answer: 0