Question: Evaluate the improper integral \( \int_2^\infty f(t) \,dt \) given that \( \int_2^x f(t) \,dt = \frac{6x}{\sqrt{9x^2-20}} - 3 \). Determine if the integral converges or diverges.

Answer: -1


Question: Find the surface area of an ideal tetrahedron in hyperbolic space $\mathbb{H}^3$ as a function of its vertices, which are parametrized by four complex numbers $z_1, z_2, z_3, z_4$ on the boundary $\mathbb{CP}^1$. Express the surface area in terms of $z_1, z_2, z_3, z_4$.

Answer: 4\pi


Question: Given the functions \( f(t) = \left( \int_0^{\sqrt{t}} e^{-x^2} \, dx \right)^2 \) and \( g(t) = \int_0^1 \frac{e^{-t(1+x^2)}}{1+x^2} \, dx \) for \( t > 0 \), find the value of \( f'(t) + g'(t) \).

Answer: 0


Question: In a right triangle \( \triangle ABC \) with \( \angle ABC = 90^\circ \), let \( \overline{AD} \) be the altitude from \( A \) to \( \overline{BC} \), with \( D \) on \( \overline{BC} \). Let \( \Gamma_1 \) and \( \Gamma_2 \) be the circumcircles of triangles \( \triangle ABD \) and \( \triangle ACD \), respectively. The circle \( \Gamma_1 \) intersects \( \overline{AC} \) at points \( A \) and \( P \), while \( \Gamma_2 \) intersects \( \overline{AB} \) at points \( B \) and \( Q \). Let \( X \) be the intersection of line \( BP \) with \( \Gamma_2 \) such that \( P \) is between \( B \) and \( X \). Similarly, let \( Y \) be the intersection of line \( QC \) with \( \Gamma_1 \) such that \( Q \) is between \( C \) and \( Y \). Given that \( A, X, \) and \( Y \) are collinear, find the smallest possible measure of angle \( \angle BAC \).

Answer: 45^\circ


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

Answer: e - 1


Question: Find the last two digits of the number \( \underbrace{7^{7^{7^{\cdots}}}}_{1+n} \) when there are \( n \) sevens in the ascending exponents.

Answer: 43


Question: Find the limit: \( \lim_{x \to 0^+}\Big(\ln\frac{1}{x}\Big)^x \)

Answer: 1


Question: Compute the remainder when \(67!\) is divided by \(71\).

Answer: 12


Question: Consider a graph G9 with 9 distinct vertices. Determine the number of triangles that can be formed in this graph, where a triangle is defined as a set of three vertices with edges between each pair of vertices. Provide your answer as a single number.

Answer: 84


Question: Evaluate the integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \log(\cos(\phi))\cos(\phi) \, d\phi \) and show that it equals \( \log(4) - 2 \).

Answer: \log(4) - 2


Question: Find a primitive root of 38.

Answer: 3


Question: Determine the number of ways to consider \(\Bbb Q\) as an \(\Bbb R\)-module.

Answer: 0


Question: Does there exist a bijective enumeration of the rationals in the interval $[0,1]$, denoted as $q_1, q_2, \ldots$, such that the series $\sum_{i=1}^{\infty} (q_i - q_{i-1})^2$ is finite?

Answer: Yes


Question: Solve for $x$ in the equation: \[ 0 = \frac{x^{36} - 1}{x - 1} - 10 \]

Answer: 0.9025


Question: Determine the Theta class for the recursion \( T(n) = T(3n/4) + T(n/6) + 5n \).

Answer: \Theta(n)


Question: Evaluate the infinite series \( \sum_{n=1}^\infty \frac{n^2}{3^n} \).

Answer: \dfrac{3}{2}


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f(0) = 0$ and $|f'(x)| \leq |f(x)|$ for all real numbers $x$. Does this imply that $f(x) = 0$ for all $x$? Justify your answer.

Answer: 0


Question: Is there a graph whose chromatic number exceeds twice its Hajós number?

Answer: Yes


Question: How many six-digit numbers, using digits from 0 to 9, have a digit sum of 27? Repetition of digits is allowed.

Answer: 50412


Question: Can the logical operators AND, OR, and NOT be used to represent any truth table for three inputs? Justify your answer.

Answer: Yes


Question: Given a convex polygon $P \subset \mathbb{R}^2$ with area 1 that contains the origin, and let $r$ be the maximum distance from the origin to any point in $P$. Let $S$ be a sector of a circle with radius $r$, centered at the origin, also with area 1. Is there an area-preserving map $f: P \to S$ such that for any point $x \in P$, $\|x\| \leq \|f(x)\|$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \pi/2}\frac{2x\sin(x) - \pi}{\cos x} \] without using L'Hôpital's rule.

Answer: -2


Question:
Consider the eigenvalue problem for the Dirichlet Laplacian:
\[
\begin{cases}
-\Delta u=\lambda u & \mbox{in }\Omega\\
u=0 & \mbox{on }\partial\Omega.
\end{cases}\]
Let \(\lambda(\Omega)\) be an eigenvalue of this problem. If \(A_k=\{x \in \mathbb{R}^n, \frac{1}{k} < |x|< 1 + \frac{1}{k}\}\) and \(B=\{x \in \mathbb{R}^n, |x|<1\}\), determine whether the following limit identity holds:
\[\lim_{k\to +\infty}\lambda(A_k)=\lambda(B).\]

Answer: \lim_{k\to +\infty}\lambda(A_k)=\lambda(B)


Question: Calculate \( \int_0^1 \int_x^1 \frac{f(t)}{t} \, dt \, dx \) given that \( f \) is a Lebesgue integrable function on \([0,1]\) with \( \int_0^1 f(t) \, dt = 1 \).

Answer: 1


Question: Evaluate the integral: $$ \int_{-\infty}^\infty e^{t^2} \ \mathrm{d}t $$

Answer: \infty


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

Answer: 1


Question:
Determine the number of real roots of the equation \(2013x^{2012} - 2012x^{2011} - 16x + 8 = 0\) within the interval \([0, 8^{\frac{1}{2011}}]\). Choose the correct option:

- Exactly one real root.
- No real root.
- At least one and at most two real roots.
- At least two real roots.

Answer: D


Question: Let $N$ be a minimal normal subgroup of a group $G$. Suppose both $N$ and $G/N$ are non-abelian simple groups. Is it true that $G$ can be expressed as $G = N \times A$, where $A$ is a non-abelian simple subgroup of $G$?

Answer: Yes


Question: Let $X$ be a disjoint union of sets $A_i$ for $1 \leq i \leq n$. Determine the number of functions $f: X \to X$ such that for any $a, b \in X$, $a$ and $b$ are in the same set $A_i$ if and only if $f(a)$ and $f(b)$ are in the same set $A_j$ for some $j$. Provide your answer as a single integer.

Answer: n!


Question: Determine whether the sequence \( \cos(\pi \sqrt{n^2 + n}) \) is convergent.

Answer: 0


Question: Evaluate the integral \( \int_{\partial D} \frac{z+4}{z-4}\frac{e^z}{\sin z} \, dz \), where \( D = \{ z \in \mathbb{C} ; |x| \leq 2, |y| \leq 1 \} \).

Answer: -2\pi i


Question: Suppose $X$ is a smooth projective variety over a field $k \subset \mathbb{C}$. Let $CH^r(X)_{hom}$ be the Chow group of codimension $r$ cycles defined over $k$ and homologous to zero. The usual Abel-Jacobi map is defined on $CH^r(X(\mathbb{C}))_{hom}$ taking values in the Griffiths intermediate Jacobian $IJ(H^*(X))$. If this intermediate Jacobian is an abelian variety, is it defined over the field $k$?

Answer: Yes


Question: Determine the values of the parameter \(a\) for which the inequality \(a \cdot 9^x + 4(a-1)3^x + a > 1\) holds for all real numbers \(x\).

Answer: [1, \infty)


Question: Given that the polynomial \( f(x) = x^3 - 3x + 2 \) has roots \( a, b, c \), find the value of \( ((a-b)(b-c)(c-a))^2 \).

Answer: 0


Question: Determine whether the expression \( \frac{\overline{X} - \mu}{S/\sqrt{n}} \) converges in distribution to \( N(0,1) \), given that \( X_1, X_2, \ldots, X_n, \ldots \) are i.i.d. random variables with expected value \( EX_i = \mu < \infty \) and variance \( Var(X_i) = \sigma^2 < \infty \).

Answer: Yes


Question: Determine whether the norm of the antipode $S$ on a Kac-type compact quantum group can be exactly 2. Specifically, is it possible that \( \|S\| = 2 \)?

Answer: No


Question: Find the remainder when the polynomial $p(x)$, which is the remainder of $x^{2019} - 1$ divided by $x^6 + 1$, is divided by $x - 3$. What is this remainder?

Answer: 26


Question: Find the line through the origin that is farthest from the points (1,1), (1,−1), and (−1,−1). Specifically, determine the line such that the sum of the square-distances between the points and their orthogonal projections onto the line is maximal.

Answer: y = -x


Question: Determine whether the statement is true or false: The group $\mathbb{Z} + \alpha\mathbb{Z}$ is not isomorphic to $\mathbb{Z} + \beta\mathbb{Z}$ as an ordered subgroup of the real numbers, where $\alpha$ and $\beta$ are different non-zero rational numbers.

Answer: B


Question: Let \( A \) be the subset of \([0, 1]\) consisting of all numbers that do not have the digit 4 in their decimal expansion. Find the measure \( m(A) \).

Answer: 0


Question: Determine the number of elements in the group \( \mathbb{Z}_{27} \times M \) that have an order not equal to a power of 3. The group \( M \) is defined as \( \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \mid ac \neq 0, \ a, b, c \in \mathbb{Z}_3 \right\} \) with respect to ordinary matrix multiplication. Assume the order of \( M \) is 12. Use Lagrange's Theorem to solve this problem.

Answer: 243


Question: If $X$ and $Y$ are random variables with the same characteristic function, are $X$ and $Y$ identically distributed?

Answer: Yes


Question: Prove or disprove the statement: If a field \(L\) contains \(\mathbb{C}\) and has a finite degree of extension over \(\mathbb{C}\), i.e., \([L:\mathbb{C}] < \infty\), then \(L = \mathbb{C}\).

Answer: \mathbb{C}


Question: Let \(P(x)\) be a quadratic polynomial with real coefficients such that \(P(x^3 + x) \ge P(x^2 + 1)\) for all real numbers \(x\). Find the sum of the roots of \(P(x)\).

Answer: 4


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \frac{\int_0^{\sin x} \sqrt{\tan t} \, dt}{\int_0^{\tan x} \sqrt{\sin t} \, dt} \]

Answer: 1


Question: Let $X$ and $Y$ be independent standard normal random variables, i.e., $X, Y \sim N(0,1)$. Define $Z = 3X + 4Y$. Find the conditional expectation $E(X \mid Z)$. Express your answer in terms of $Z$.

Answer: \dfrac{3}{25}Z


Question:
Let \( f : \Bbb{R} \to \Bbb{R} \) be a function such that \( f(x+y) = f(x)f(y) \) for all real numbers \( x, y \) and \( f(x) = 1 + xg(x) \) where \( \lim_{x \to 0} g(x) = 1 \). Determine which of the following functions \( f(x) \) satisfies these conditions:

(a) \( e^{x} \)
(b) \( 2^{x} \)
(c) A non-constant polynomial
(d) Equal to 1 for all real \( x \)

Answer: a


Question:
Evaluate the limit using Cauchy's Mean Value Theorem:
\[ \lim_{x\rightarrow 1} \left[\frac{\cos(\frac{1}{2}\pi x)}{\ln(1/x)}\right] \]

Answer: \dfrac{\pi}{2}


Question: Suppose that $x$ is a uniformly distributed random variable on the interval $(0,6)$ and $y = (x-3)^2$. What is the correlation between $x$ and $y$?

Answer: 0


Question: Let $X \subset C^1[0,1]$ be a subspace that is closed in $L^1(0,1)$. Prove that $\dim X < \infty$.

Answer: \dim X < \infty


Question:
Consider two Hermitian matrices \( \rho \) and \( A \), and a scalar parameter \( t \). Let \( f_i(\rho) \) denote the \( i \)-th eigenvalue of the matrix \( \rho \), and \( |x_i\rangle \) be the normalized eigenvector of \( \rho \) corresponding to \( f_i(\rho) \). Does the following expression hold in the limit as \( t \to 0 \)?
\[
f_i(\rho + tA) = f_i(\rho) + \text{tr} \{A |x_i\rangle\langle x_i|\}t + \mathcal{O}(t^2)
\]

Answer: Yes


Question: Compute \( \int_{S^1} i^* \omega \) where \( \omega \in \Omega^1(\mathbb{R}^2) \) is a 1-form such that \( d\omega = dx \wedge dy \). Let \( i: S^1 \to \mathbb{R}^2 \) be the inclusion map and \( S^1 \) be the unit circle. Use Stokes' theorem to evaluate the integral.

Answer: \pi


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

Answer: 0


Question:
A theme park has two zip wires modeled as straight lines \(L_1\) and \(L_2\) using coordinates in metres. The ends of one wire are located at \((0,0,0)\) and \((0,100,-20)\). The ends of the other wire are located at \((10,0,20)\) and \((-10,100, -5)\). Use this model to find the shortest distance between the zip wires:

\[L_1: \vec{r_1}(s) = s(100\hat{j} - 20\hat{k})\]
\[L_2: \vec{r_2}(t) = (10\hat{i}+ 20\hat{k}) + t(-20\hat{i} + 100\hat{j} - 25\hat{k})\]

Let \(P\) be a general point on line \(L_1\) and \(Q\) be a general point on line \(L_2\):

\[P = (100s\hat{j} - 20s\hat{k})\]
\[Q = (10-20t)\hat{i} + (100t)\hat{j} + (20 - 25t)\hat{k}\]

Vector \(\vec{PQ}\) is

\[\vec{PQ} = ((10-20t)\hat{i} + (100t - 100s)\hat{j} + (20 - 25t + 20s)\hat{k})\]

If \(\vec{PQ}\) is perpendicular to both lines, solve the following equations:

\[\vec{PQ} \cdot \vec{r_1}(s) = 0\]
\[\vec{PQ} \cdot \vec{r_2}(t) = 0\]

Find the values of \(s\) and \(t\) that satisfy these conditions and calculate the length of \(\vec{PQ}\) to determine the shortest distance between the zip wires.

Answer: \dfrac{50}{3}


Question: Evaluate the integral \( \int_0^4 g'(\sqrt{x}) \, dx \), where the function \( f(x) \) is defined as \( f(x) = \begin{cases} x^x \ln x + k & : \ (x \geq 1) \\ f(2-x) & : \ (x<1) \end{cases} \) and satisfies \( \int_0^1 f(x) \, dx = 5 \). The function \( g(x) \) is given by \( g(x) = \int_0^x f(t) \, dt \).

Answer: 20


Question: For how many integer values of $x$ is the expression \( N = \prod \limits_{k=1}^{40} \bigl(x-(2k-1) \bigr)^{2k-1} \) negative?

Answer: 20


Question: Is it consistent with ZFC that the only solution to the cardinal equation \(\kappa^{\aleph_0}=2^\kappa\) is \(\kappa=\aleph_0\)?

Answer: Yes


Question: Provide an example of a continuous function \( f: \mathbb{C} \rightarrow \mathbb{C} \) that does not have a primitive.

Answer: \overline{z}


Question: Compute the limit: \[ \lim_{n \to \infty} n \int_0^{\frac{\pi}{4}} \tan^n x \, dx \]

Answer: \dfrac{1}{2}


Question: Find the radius of convergence of the series \( \sum_{k=1}^\infty \frac{x^{2k-1}}{2k-1} \).

Answer: 1


Question: Given the function \( f(x) = x + \int_{0}^{1} t(x+t)f(t) \, dt \), find the value of \( \int_{0}^{1} f(x) \, dx \).

Answer: \dfrac{42}{23}


Question: Let X be an exponential random variable with parameter 9. Evaluate \( \ln \left( \frac{1}{P(X^2 + 10X \geq 24)} \right) \).

Answer: 18


Question: Is there a normal bitstream $s$ such that there are infinitely many finite $01$-strings that are frequent in $s$?

Answer: Yes


Question: Determine the number of pairs of natural numbers \((p, q)\) such that \(p^2q^2 - 4(p+q) = a^2\) for some natural number \(a\).

Answer: 4


Question: Evaluate the integral \( \int_D z^n \overline{z^m} \, dx \, dy \) over the domain \( D = \{ z \in \mathbb{C} : |z| < 1 \} \) and show that it equals zero when \( n \neq m \).

Answer: 0


Question: Let \(X\) and \(Y\) be two disjoint connected subsets of \(\mathbb{R}^2\). Can \(X \cup Y = \mathbb{R}^2\)? Justify your answer.

Answer: Yes


Question: Compute the limit: \[ \lim_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {\left\lvert \sin k\right\rvert} \]

Answer: \dfrac{2}{\pi}


Question: Determine the number of onto relations from the set \( A = \{1, 2, 3, \ldots, m\} \) to the set \( B = \{1, 2, 3, \ldots, n\} \), where the range of the relation is exactly \( B \).

Answer: (2^m - 1)^n


Question: Determine the degree of the line bundle over $\mathbb{P}^1$ with transition function $e^{1/z}$, where $z \neq 0, \infty$. What is the degree of this line bundle?

Answer: 0


Question: Let \( P(x) \) be a polynomial with real coefficients such that for all real numbers \( x, y, z \) satisfying \( xy + yz + zx = 1 \), the equation \( P(x) + P(y) + P(z) = P(x + y + z) \) holds. Additionally, it is given that \( P(0) = 1 \) and \( P(1) = 4 \). Determine the value of \( P(2017) \).

Answer: 4072324


Question: Determine the number of integer solutions \((a, n, p)\) to the equation \(a^p - a = n^p\), where \(p \ge 0\).

Answer: \infty


Question: Consider Banach spaces $X$ and $Y$, and let $T: X \to Y$ be a linear continuous map that is both injective and open. Does $T$ map closed sets in $X$ to closed sets in $Y$? Provide a proof or a counterexample.

Answer: Yes


Question: Let $f$ be a continuous function on the interval $[0, \pi]$. If \( \int_{0}^{\pi} f(x) \sin(nx)\,dx = 0 \) for all integers \( n \geq 1 \), is $f$ necessarily the zero function on $[0, \pi]$?

Answer: Yes


Question: If $f$ is $\mathbb{R}^2$-differentiable and the limit $\lim\limits_{h \to 0} \Re\left(\frac{f(z+h)-f(z)}{h}\right)$ exists, does this imply that $f$ is holomorphic on a domain $D \subset \mathbb{C}$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{(x,y,z)\to (0,0,0)}\frac{1}{x^2+y^2+z^2}e^{-\frac{1}{\sqrt{x^2+y^2+z^2}}} \]

Answer: 0


Question: Determine whether the following statement is true or false: If \( \{f_n\}_{n=1}^{\infty} \) is a sequence of real-valued functions on \( \mathbb{R} \) that converges uniformly to a function \( f \), then \( f \) must be continuous.

Answer: B


Question: What is the expected value of the geometric Brownian motion given by the equation \( dx_t = r x_t dt + \sigma x_t dB_t \)? Express your answer in terms of \( x_0, r, \sigma, \) and \( t \).

Answer: x_0 e^{r t}


Question: Calculate the spectrum of the operator \(T: \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})\) defined by \((T(x))_n = x_{n+1} + x_{n-1}\).

Answer: [-2, 2]


Question: Determine the angle between an antisymmetric matrix $A$ and a symmetric matrix $S$ in $M_n(\mathbb{R})$ using the inner product defined as $(A,B)=\text{tr}(AB^T)$. Express your answer in terms of the matrices' properties.

Answer: \dfrac{\pi}{2}


Question: Determine whether the following statement is true or false: If $k \rightarrow \infty$ and $\sqrt{\frac{1}{x_k}} = o(y_k)$, then $\frac{1}{x_k} = o(y_k)$, given that $x_k \rightarrow \infty$ and $y_k \rightarrow 0$. Provide a justification for your answer.

Answer: True


Question:
In a 3-D vector space, if a vector \( R \) is perpendicular to three vectors \( A, B, \) and \( C \), does this imply that \( A, B, \) and \( C \) are coplanar? Given the conditions:

\[ R \cdot A = 0 \]
\[ |R \times B| = |R||B| \]
\[ |R \times C| = |R||C| \]

Find the value of the scalar triple product \([A B C]\).

Answer: 0


Question: Find the angle between the parabolas \(y^2=2px-p^2\) and \(y^2=p^2-2px\) at their points of intersection.

Answer: 0


Question: Given 100 widgets, each with a probability of \( \frac{1}{2} \) of being defective, determine the minimum number of widgets \( k \) that must be sampled to ensure that the probability of all \( k \) sampled widgets being functioning properly and at least 6 of the 100 widgets being defective is less than 0.1.

Answer: 4


Question: Evaluate the limit: \[ \lim_{(x,y) \to (0,0)} \frac{x^{4}-x^{2}y^{2}+y^{4}}{x^{2}+y^{2}+x^{4}y^{4}} \] using polar coordinates.

Answer: 0


Question: Determine whether every $n \times n$ symmetric matrix over $\mathbb{F}_2$ can be decomposed into a sum of the form $\sum_{v=1}^k v_i v_i^T$, where $v_1, \ldots, v_k \in \mathbb{F}_2^n$ and $k$ is an integer.

Answer: Yes


Question: If a power $A^n$ of a matrix $A$ is symmetric, must the matrix $A$ itself be symmetric? Provide a justification for your answer.

Answer: No


Question: Determine the number of integer values of \( n \) for which the equation \( 3x^3 - 25x + n = 0 \) has three real roots.

Answer: 55


Question: Evaluate the limit: $$\lim_{n \to \infty}\frac{(\sum_{k=1}^{n} k^2 )*(\sum_{k=1}^{n} k^3 )}{(\sum_{k=1}^{n} k^6)}$$ by converting the sums into integrals.

Answer: \dfrac{7}{12}


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

Answer: \dfrac{2}{5}


Question: Compute the limit: $$\lim_{x\to 0}\left[(\sin x)^{\frac{1}{x}}+\left(\frac{1}{x}\right)^{\sin x}\right].$$

Answer: 1


Question: What is the minimum number of states in a deterministic finite automaton (DFA) that accepts the language consisting of strings over \( \{0,1\}^* \) such that the number of \(0\)'s in the string is divisible by \(k\) and the number of \(1\)'s is odd, where \(k \ge 2\)?

Answer: 2k


Question: Find the integer $x$ such that $x^{11} \equiv 10 \pmod{41}$. Use a calculator that can handle a maximum of 9 digits to find the solution.

Answer: 10


Question:
Calculate the integral \( \int_{[0,1]} x^2 \, d\mu_F \), where \( F \) is the Cantor function. Use the following properties of the Cantor function:

- \( F(1-x) = 1 - F(x) \)
- \( F\left(\frac{x}{3}\right) = \frac{F(x)}{2} \) for all \( x \in [0,1] \)
- \( F(0) = 0 \)

Answer: \dfrac{3}{8}


Question: Evaluate the line integral \(\oint_C |z|^2 \, dz\) around the square with vertices at \((0,0), (1,0), (1,1), (0,1)\).

Answer: -1 + i


Question: Let $\mu$ be a probability measure on the interval $I=[0,1]$, which is absolutely continuous with respect to the Lebesgue measure. Consider the "doubling angle map" $T$ on $I$, defined by $T(x)=2x \mod 1$. Determine if it is true, in general, that $\mu\left(T^{-n}\left[0,\frac{1}{2}\right)\right)$ converges to $\frac{1}{2}$ as $n$ approaches infinity.

Answer: \dfrac{1}{2}


Question:
The straight lines $L_1: x = 0$, $L_2: y = 0$, and $L_3: x + y = 1$ are mapped by the transformation $w = z^2$ into the curves $C_1$, $C_2$, and $C_3$, respectively. Determine the angle of intersection between these curves at $w = 0$. Choose the correct option:

(a) 0
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{2}\)
(d) \(\pi\)

Answer: D


Question:
Determine whether the following statement about partial derivatives is correct:

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) and \( g : \mathbb{R} \to \mathbb{R} \). Assume that all partial derivatives exist. Is the following equation true?

\[ \frac{\partial^2 (g \circ f)}{\partial x_1 \partial x_2} = \frac{\partial}{\partial x_1} \left( \frac{\partial g}{\partial f} \frac{\partial f}{\partial x_2} \right) = \frac{\partial g}{\partial f} \frac{\partial^2 f}{\partial x_1 \partial x_2} + \frac{\partial^2 g}{\partial f^2} \frac{\partial f}{\partial x_1} \frac{\partial f}{\partial x_2} \]

Answer: True


Question: Find \( \lim_{n \to \infty} n \sin(2\pi \sqrt{1+n^2}) \), where \( n \in \mathbb{N} \).

Answer: \pi


Question: Consider a self-adjoint bounded linear operator \( T: H \rightarrow H \) on an infinite-dimensional real separable Hilbert space \( H \), where \( T \) is not compact. Determine whether \( T \) can have a discrete spectrum.

Answer: No


Question:
Determine the radius of convergence of the series given by:
\[ C_n = (-1)^n\dfrac{m(m - 1)\cdots(m - n + 1)}{n!n!}C_0 \]

Answer: \infty


Question: Given a number $n > 1$, you repeatedly replace a number $t$ with $t - \sqrt{t}$. Determine asymptotically, in terms of $n$, how many times this operation must be performed until the number becomes less than 1.

Answer: 2\sqrt{n}


Question: Determine the smallest integer $n$ such that the $n$th Taylor polynomial $P_n(x)$ of $f(x) = \tan^{-1}(x)$ about $x_0 = 0$ approximates $f(x)$ within $10^{-6}$ on the interval $[-0.5, 0.5]$. Provide the value of $n$.

Answer: 15


Question: If $X$ is a separable infinite-dimensional Banach space, is it possible to construct another separable infinite-dimensional Banach space $Y$ that is not isomorphic to $X$?

Answer: Yes


Question:
Find the bias of the estimator \( \bar{\theta} = 2 \bar{X} - 1 \) for \( \theta \), where \( \bar{X} \) is the sample mean of \( n \) independent variables from the uniform distribution \( U[0, 1+\theta] \). The probability density function of \( \bar{X} \) is given by the Bates distribution:
\[ f(x) = \frac{n}{2(n-1)!} \sum_{k=0}^n (-1)^k \binom{n}{k}(nx - k)^{(n-1)} \sgn(nx -k) \]
where \( \sgn(nx-k) = \begin{cases} -1, & \text{if } nx<k, \\ 0, & \text{if } nx=k, \\ 1, & \text{if } nx > k. \end{cases} \]

Answer: 0


Question: Find \( \lim_{n\rightarrow \infty} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{n}}{n^{3/2}} \).

Answer: \dfrac{2}{3}


Question: Evaluate the limit of the infinite product: \[ 2 \times \frac{1}{2} \times 2^4 \times \frac{1}{2^8} \times 2^{16} \times \cdots \] Determine if the product converges to zero.

Answer: 0


Question: Consider the 3-sphere \( S^3 = \{(z_1, z_2) \in \mathbb{C}^2 \mid |z_1|^2 + |z_2|^2 = 1\} \) with a Riemannian metric induced from \( \mathbb{R}^4 \). There is an isometric \( S^1 \)-action given by \( g \cdot (z_1, z_2) = (gz_1, gz_2) \) for \( g \in S^1 \subset \mathbb{C} \). The quotient space is \( S^3/S^1 = S^2(r) \), where \( r = \frac{1}{2} \). Can a geodesic curve in \( S^3 \) be mapped onto a non-geodesic curve in \( S^2(r) \)?

Answer: Yes


Question: In triangle $\triangle ABC$, given that $\cos(2A-B) + \sin(A + B) = 2$ and $\overline{AB} = 4$, find the length of $\overline{BC}$.

Answer: 2


Question: Determine the radius of convergence of the function \( \frac{z-z^3}{\sin{\pi z}} \) at \( z=0 \).

Answer: 2


Question: Determine the cardinality of the set of all bijections from \( \mathbb{N} \to \mathbb{N} \).

Answer: 2^{\aleph_0}


Question: Find the rational number $x \in \mathbb{Q}$ such that $1 + \frac{2}{\lfloor x \rfloor} + \frac{3}{x - \lfloor x \rfloor} = 4x$. Determine the value of $x$.

Answer: \dfrac{7}{4}


Question: Evaluate the limit \( \lim_{x \to 0} \left(\frac{ \sin(x)}{x}\right)^\frac{1}{x} \) using Taylor's formula.

Answer: 1


Question: Determine whether there exist two finite complex measures \( \mu \) and \( \nu \) with compact supports on the real line, both singular with respect to the Lebesgue measure, such that their convolution \( \mu \ast \nu \) has a nonzero absolutely continuous component.

Answer: Yes


Question: Does there exist a continuous map \( f: S^1 \to \mathbb{R} \) such that for some odd number \( o \in \mathbb{N} \), there are infinitely many points in \( f(S^1) \) where the preimage of each point has exactly \( o \) points in \( S^1 \)?

Answer: Yes


Question: Calculate the number of set partitions of the set \( S = \{ a, b, c, d, e, f, g, h, i, j \} \) such that each subset in the partition has a size between 2 and 3.

Answer: 7245


Question: Prove that the conditional entropy of a random variable by itself, denoted as $H(X|X)$, is equal to 0.

Answer: 0


Question:
Determine the points in the domain of the function \( f: \mathbb{R} \to \mathbb{R} \) defined by
\[
f(x) =
\begin{cases}
x + 1,  & \text{if } x \in \mathbb{Q} \\
2, & \text{if } x \in \mathbb{R} - \mathbb{Q}
\end{cases}
\]
where the function is continuous and where it is discontinuous.

Answer: 1


Question: Find the coefficient of $x^{15}$ in the expansion of $\frac{x^3 - 5x}{(1-x)^3}$ using generating functions.

Answer: -509


Question: Suppose X follows the distribution N(0, $\sigma^2$) and Y|X=x follows N(\rho x, $\sigma^2$). Calculate the covariance Cov(X, Y).

Answer: \rho \sigma^2


Question: Compute the integral \( \int_C \frac{e^{3z} - z}{(z+1)^2 z^2} \, dz \), where \( C \) is a circle centered at the origin with radius \( \frac{1}{2} \).

Answer: 0


Question: Determine the minimum value of the function \( f(x, y) = x + y + \frac{1}{xy} \). Verify if the minimum value is indeed at \((1, 1)\) or if there is a lower value at another point.

Answer: 3


Question: Consider the power series \( \sum_{n,m = 0}^{\infty} a_{n,m} x^{n} y^{m} \) with real coefficients, which is absolutely convergent for every real \( x, y \) such that \( |x| < \alpha \) and \( |y| < \beta \), where \( \alpha > 0 \) and \( \beta > 0 \). Rearrange the series in ascending powers of \( x \) as \( \sum_{n = 0}^{\infty} f_n(y) x^{n} \), where \( f_n:(-\beta,\beta) \rightarrow \mathbb{R} \) are analytic functions. For every fixed \( y \in (-\beta,\beta) \), let \( R(y) \) be the radius of convergence of the series in \( x \). Is \( R(y) \) a continuous function of \( y \)?

Answer: Yes


Question: Compute \( \lim_{n\to\infty} \int_0^{\infty}\frac{e^{-x/n}}{1+(x-n)^2}dx \).

Answer: \dfrac{\pi}{e}


Question: Let $R$ be a commutative Noetherian ring with no nonzero nilpotents. Determine if every localization of $R$ at a maximal ideal is a field.

Answer: No


Question: Consider the functional equation \( f(r \cos \phi) + f(r \sin \phi) = f(r) \) for \( r \geq 0 \) and \( \phi \in [0, \pi/2] \). Assume that \( f: [0, \infty) \to \mathbb{R} \) is a monotone function. Given that \( f(x) = ax^2 \) is a solution, determine if there are any other solutions.

Answer: f(x) = ax^2


Question: Two teams, Team A and Team B, each consist of 10 people. For some integer \( n \) where \( 1 \leq n \leq 9 \), Team A chooses \( n \) people to remove from the team. Then, for some integer \( k \) where \( 0 \leq k \leq n \), Team B replaces \( k \) of its members with \( k \) of the removed members from Team A. After these changes, there are \( N \) distinct possible combinations of teams. Find the remainder when \( N \) is divided by 1000.

Answer: 696


Question: Let \( n \in \mathbb{N} \), and let \( a_i, b_i \in \mathbb{C} \). Consider the \( n \times n \) matrix \( A \) with \( a_i b_j \) as its \((i, j)\)-entry. Find \( \det(A) \).

Answer: 0


Question: Solve the inequality: \[ \frac{6x}{x-2} - \sqrt{\frac{12x}{x-2}} - 2\sqrt[4]{\frac{12x}{x-2}}>0 \]

Answer: (2, 8)


Question:
Given the following information:

1. The annual number of claims for a policyholder follows a binomial distribution with probability function: \( p(x \mid q) = {2\choose x}q^{x}(1-q)^{2-x} \), where \( x = 0, 1, 2 \).
2. The prior distribution is \( \pi(q) = 4q^{3} \), where \( 0 < q < 1 \).
3. The policyholder had one claim in each of Years 1 and 2.

Determine the Bayesian estimate of the number of claims in Year 3.

Answer: \dfrac{4}{3}


Question: A circle of radius 1 unit touches the positive x-axis and positive y-axis at points A and B, respectively. A variable line passing through the origin intersects the circle at points D and E. If the area of triangle DEB is maximized when the slope of the line is \( m \), find the value of \( \frac{1}{m^2} \).

Answer: 3


Question: Let \( \underline{X} \sim \text{Binomial}(n, p) \). Find the Uniformly Minimum Variance Unbiased Estimator (UMVUE) for the parameter \( p \).

Answer: \dfrac{X}{n}


Question: Calculate the distance from the complex number $1$ to $-2 i^{2/3}$. Express your answer in simplest form.

Answer: \sqrt{7}


Question: Determine for which dimensions \( n \in \mathbb{N} \) the Dirac-delta distribution is a bounded linear functional on the Sobolev space \( H_{0}^{1}(\Omega) \), meaning it is an element of \( H^{-1}(\Omega) \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{\sqrt{n}+ \sqrt[4]{n^2+n}}{\sqrt[6]{n^3-1}+\sqrt[3]{n+2}} \]

Answer: 2


Question: Determine the values of $a \in \mathbb{R}$ for which the function \( \frac{1}{n^2+x^{a}+2y^{a}+3z^{a}} \) is integrable over \([0,+\infty)^3\) for all \(n \in \mathbb{N} \cup \{0\}\).

Answer: a > 3


Question: Determine the support of the random variable \( Y = M\left( 1 - \frac{1 + N}{e^{a \times P \times X} + N} \right) \), where \( X \) is an exponentially distributed random variable with PDF \( f_X(x) = \frac{1}{\lambda} \exp\left( -\frac{x}{\lambda} \right) \). Given that \( N > 1, a > 0, P > 0, M > 0 \), find the interval over which the PDF \( f_Y(y) \) integrates to 1.

Answer: [0, M)


Question: Let $k: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $k(a) = k(e) = 0$, $k(b) = k(d) = 2$, and $k(c) = 1$, where $a < b < c < d < e$. Determine the minimum number of solutions to the equation $(k'(x))^2 + k(x)k''(x) = 0$ in the interval $[a, e]$. Provide your answer as a single integer.

Answer: 4


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{n^4} \sum_{k=1}^{n} \left\lfloor k^3 x \right\rfloor \]

Answer: \dfrac{x}{4}


Question: Find the minimum positive integer \( n \) such that the equation \( x + 11y + 11z = n \) has exactly 16653 solutions in positive integers \((x, y, z)\).

Answer: 2014


Question: Determine whether the Bernoulli measure of the infinite word \( x = (\ldots, y_1, y_1, y_1, \ldots) \), which consists entirely of the repeated entry \( y_1 \), is zero. Assume the Bernoulli measure \( \mu \) is defined on finite blocks as \( \mu([y_1,y_2,\ldots,y_n]m) = \prod_{j=1}^n p_j \), where \( p_j \) is the probability of \( y_j \).

Answer: 0


Question: Let $[\mathbf{a} \times \mathbf{b} \; \mathbf{b} \times \mathbf{c} \; \mathbf{c} \times \mathbf{a}] = k[\mathbf{a} \; \mathbf{b} \; \mathbf{c}]^2$. Find the value of $k$. Here, $[\mathbf{u} \; \mathbf{v} \; \mathbf{w}] = \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$.

Answer: 1


Question: Calculate the probability that the minimum of three independent random variables is greater than the maximum of two other independent random variables, where each random variable follows a continuous uniform distribution on the interval $[0,2]$. Specifically, find $$\Pr(\min(X_1, X_2, X_3) > \max(X_4, X_5)),$$ where $X_i \sim \operatorname{Uniform}[0,2]$ for $i = 1, 2, 3, 4, 5$. Provide your answer as a single probability value.

Answer: \dfrac{1}{10}


Question: Given a random sample of size 3, \(X_1, X_2, X_3\), from a population with probability density function \(f(x;\theta)=\dfrac{1}{\theta}e^{-\frac{x}{\theta}}\), where \(x>0\), test the null hypothesis \(H_0: \theta=3\) against the alternative hypothesis \(H_\alpha: \theta \neq 3\). The null hypothesis is rejected if the sample mean \(\overline{X} > 6.296\). What is the significance level of this test?

Answer: 0.05


Question: Find the integer $x$ such that $x^{311} \equiv 662 \pmod{713}$. Provide your answer as the smallest non-negative integer satisfying the congruence.

Answer: 303


Question: Determine the number of local maximum points of the function \( f(x) = e^{-|x|}|x^2-1| \). Choose from the following options: 0, 1, 2, or 3.

Answer: 3


Question: How many distinct rotations map a cube onto itself?

Answer: 24


Question: Evaluate the limit: \[ \lim_{x \rightarrow \pi/2}\left(\frac{\arctan\left(\frac{x}{\tan(x)-x}\right)}{x-\pi/2}\right) \] without using L'Hospital's Rule.

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


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

Answer: -1


Question: Determine whether the following statement is true or false: If $(x_n)$ is a positive real sequence such that $\sum x_n$ converges, then there exists $N \in \mathbb{N}$ such that $x_{\left\lceil\frac{1}{x_N}\right\rceil} < \frac{1}{N}$. Provide a justification for your answer.

Answer: True


Question: Determine the number of nilpotent elements in the quotient ring $R = \mathbb{Z}_3[x] / \langle x^2 - 1 \rangle$. A nilpotent element $a \in R$ satisfies $a^n = 0_R$ for some natural number $n$. How many such nilpotent elements exist in $R$?

Answer: 1


Question: A 3x3 square table can have each of its cells painted in one of three colors: black, white, or red. How many distinct colorings of the table are there if rotations of the table are considered equivalent?

Answer: 4995


Question: Express \( \lim_{h \to 0} \frac{f(x+2h) - f(x-h)}{h} \) in terms of \( f'(x) \), given that \( f \) is differentiable on \( \mathbb{R} \) and \( f' \) is continuous on \( \mathbb{R} \).

Answer: 3f'(x)


Question: Calculate the limit using Taylor series: $$\lim_{x\rightarrow0}\frac{\cos(x) \sin(x)-x}{\sin^3 x}$$

Answer: -\dfrac{2}{3}


Question: Determine whether the $n$-simplex with vertices $(0,0,\dots,r_i,r_{i+1},\dots,r_n)$, where $r_1,\dots,r_n$ are given natural numbers and $i=0,1,\dots,n+1$, admits a regular, unimodular triangulation.

Answer: Yes


Question: Find the volume of the largest rectangular box in the first octant with three faces on the coordinate planes and one vertex on the plane \(x + 3y + 6z = 18\).

Answer: 12


Question: Let \( f \) be a differentiable function such that the equation of the normal to the graph of \( y = f(x) \) at \( x = 3 \) is given by \( 3y = x + 18 \). Evaluate the limit \( L = \lim_{x \to 1} \frac{f(3 + (4\tan^{-1}x - \pi)^2) - f((3 + f(3) - x - 6)^2)}{\sin^2(x - 1)} \). Find the value of \( L \).

Answer: -12


Question: Determine whether every non-abelian finite group $G$ with the property that every subgroup of $G$ is normal is isomorphic to $G' \times Q_8$, where $G'$ is an abelian group that does not contain $C_4$ as a subgroup.

Answer: Yes


Question: Let \(H\) be a normal subgroup of \(G\) with finite index \([G:H] = n\). Suppose \(a\) is an element of \(G\) with finite order \(m\), and \(\gcd(n, m) = 1\). Prove that \(a \in H\).

Answer: a \in H


Question: Consider a sequence of measurable functions \( \{f_n\} \) that are uniformly bounded. If \( f_n \rightarrow f \) in measure, is the limit function \( f \) also bounded?

Answer: Yes


Question: Given $n \times n$ matrices $A$, $B$, and a projection matrix $C$ (which is singular and not the zero matrix), if $AC = BC$, does it necessarily follow that $A = B$? Provide a justification for your answer.

Answer: No


Question: Let $z$ be a complex number of maximum argument satisfying $|z-3|=\operatorname{Re}(z)$. Find $\operatorname{Re}(z)$. \(z\) is represented as \(x+iy\).

Answer: 3


Question: Calculate the limit: \[ \lim_{n \to \infty }n\int_{0}^{\pi}\left \{ x \right \}^{n}dx \] where \( \{ x \} \) denotes the fractional part of \( x \).

Answer: 3


Question: What is the degree of 0 in the context of differential forms?

Answer: 0


Question: Can a non-constant harmonic function of two variables, defined in the unit disk or on the half plane, have a set of zeroes with positive measure?

Answer: No


Question: Find a subgradient \( g \) of the function \( f(x) = ||x||_2^2 \).

Answer: 2x


Question: Find the value of \( p \) such that \( n^p X_{(2)} \) converges to a constant \( c > 0 \) in probability, where \( X_{(2)} \) is the second order statistic of a random sample from a standard exponential distribution.

Answer: 1


Question: Let $X$ be a locally compact Hausdorff space with a countable base, and let $f \ge 0$ be a lower semicontinuous function on $X$. Is it true that there exists a non-decreasing sequence $(f_n)$, where $f_n \in \mathcal{L}_+$, such that $f(x) = \lim f_n(x)$ for every $x \in X$?

Answer: Yes


Question: Evaluate the integral \( \int_C (2xy) \, dx + (xy^2) \, dy \) using Green's Theorem, where \( C \) is the closed, piecewise curve formed by traveling in straight lines between the points \((-2,1), (-2,-3), (1,-1), (1,5)\), and back to \((-2,1)\), in that order.

Answer: 61


Question: Let $R$ be a multiplicatively idempotent semiring with an additive identity, and a partial order relation $\leq$ is defined on $R$. For all $x$ in $R$, does the identity $x+2x=2x$ imply $x\leq 2x$? Justify your answer.

Answer: Yes


Question: Determine the time complexity of the recurrence relation \( T(n) = T(\sqrt{n}) + \sqrt{n} \).

Answer: O(\sqrt{n})


Question:
Let \( f(x) \) and \( g(x) \) be continuous, positive functions such that \( f(-x) = g(x) - 1 \) and \( f(x) = \frac{g(x)}{g(-x)} \). Given that \( \int_{-20}^{20} f(x) \, dx = 2020 \), find the value of \( \int_{-20}^{20} \frac{f(x)}{g(x)} \, dx \).\
\(\text{(A) } 1010 \quad \text{(B) } 1050 \quad \text{(C) } 2020 \quad \text{(D) } 2050\)

Answer: A


Question: Given that $x + y + z = 3$ and $xy + yz + zx = -9$ where $x, y, z \in \mathbb{R}$, determine the interval in which the value of $z$ lies.

Answer: [-3, 5]


Question: If there exists an elliptic curve $E$ and a morphism $f: X_0(p) \rightarrow E$ for a fixed prime $p$, does it imply that the conductor of $E$ is $p$?

Answer: Yes


Question: Find the Lebesgue measure of the set of numbers in the interval \([0, 1]\) whose decimal representation contains at least one digit equal to 9.

Answer: 1


Question: Let \( F(\alpha)/F \) be a Galois extension where \( \alpha \notin F \). Suppose there exists an element \( f \) in \( \operatorname{Gal}(F(\alpha)/F) \) such that \( f(\alpha) = \alpha^{-1} \). Prove that the degree \([F(\alpha):F]\) is even.

Answer: even


Question: Determine the condition under which an orthogonal projection \( P: \mathbb{C}^n \to \mathbb{C}^n \) is a unitary operator.

Answer: P = I


Question:
Evaluate the limit:
\[ \lim_{n \to \infty} \frac{\frac{\eta(1-\frac{1}{p_n})}{\zeta(1-\frac{1}{p_n})}}{\frac{\eta(1-\frac{1}{n})}{\zeta(1-\frac{1}{n})}} \]
where \( p_n \) is the sequence of prime numbers. Assume the known relationship between the Dirichlet eta function and the Riemann zeta function, \( \eta(s) = (1-2^{1-s})\zeta(s) \), holds for \( 0 < \Re s < 1 \).

Answer: 0


Question: Suppose $f$ is differentiable on $\mathbb{R}$ and that $\lim_{x \rightarrow 0} f'(x)=L$. Can we conclude that $f'(0)=L$? Provide a justification for your answer.

Answer: Yes


Question:
Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a continuous function such that $\int_{\mathbb{R}^n} |f(x)| \, dx < \infty$. Let $A$ be a real $n \times n$ invertible matrix and for $x, y \in \mathbb{R}^n$, let $\langle x, y \rangle$ denote the standard inner product in $\mathbb{R}^n$. Determine which of the following expressions is equal to \( \int_{\mathbb{R}^n} f(Ax) e^{i\langle y, x \rangle} \, dx \):

1. \( \int_{\mathbb{R}^n} f(x) e^{i\langle (A^{-1})^T y, x \rangle} \frac{dx}{|\det A|} \)
2. \( \int_{\mathbb{R}^n} f(x) e^{i\langle A^T y, x \rangle} \frac{dx}{|\det A|} \)
3. \( \int_{\mathbb{R}^n} f(x) e^{i\langle (A^T)^{-1} y, x \rangle} \, dx \)
4. \( \int_{\mathbb{R}^n} f(x) e^{i\langle A^{-1} y, x \rangle} \frac{dx}{|\det A|} \)

Answer: 1


Question:
A cat delivers $n$ kittens, each being male with a probability of $\frac{1}{2}$, independently. Let $1 \le k \le n$ be an integer, and denote $B$ as the event where at least $k$ of the kittens are male. If $k = \frac{n}{2} + \sqrt{n}$, determine the behavior of $P(B)$ as $n \to \infty$. Choose the correct statement:

- It converges to $a \le 0.05$
- It converges to $0.25 \le a \le 0.5$
- It converges to $a > 0.5$
- None

Answer: A


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_0^\infty n^2e^{-nx} \tan^{-1} x \, dx. \]

Answer: 1


Question: Find the integer \(a\) such that \(2^{1990} \equiv a \pmod{1990}\).

Answer: 1024


Question: Determine whether there exists a subset $A \subset \mathbb{F}_2^n$ such that $|A| \ge c 2^{n/k}$ for some absolutely positive constant $c$, and satisfying the condition that \( a_1 + a_2 + \dots + a_k \neq b_1 + b_2 + \dots + b_k \) for every pair of distinct $k$-element subsets \( \{a_1,...,a_k\} \neq \{b_1,...,b_k\} \) of $A$. Here, $k \in \mathbb{N}$ and $k \ge 2$. Provide a justification for your answer.

Answer: Yes


Question:
Consider the sequence \( a_n \) defined by:
\[ a_1 = \frac{1}{3} \]
\[ a_{n+1} = a_n^2 + a_n \]
Calculate \( S = \frac{1}{a_2} + \frac{1}{a_3} + \cdots + \frac{1}{a_{2008}} \).
Find the greatest integer less than or equal to \( S \), denoted by \([S]\).

Answer: 5


Question: Evaluate the integral \( \int_{-\pi}^{0} i e^{2it} \sin(2t) \, dt \).

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


Question: Evaluate the integral \( \int_\gamma \frac{e^\bar{z}}{z^2} \, \mathrm{d}z \), where \( \gamma \) is a circle centered at zero with a radius of 2.

Answer: 0


Question: Let $B$ be a random variable with a Binomial distribution $\text{Binomial}(n,p)$. Determine the value of \( \lim_{n \to \infty} \mathsf{P}\left[ \frac{B}{n} < p\right] \).

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true or false: Let $X$ be a Banach space. If $x_n \rightarrow x$ weakly and every subsequence of $\{x_n\}$ has a strongly convergent subsequence, then $x_n\rightarrow x$ strongly in $X$. Provide a proof or counterexample.

Answer: True


Question: Suppose \( f(x+1) = 2f(x) \) for all \( x \in \mathbb{R} \), and \( f(x) = x(x-1) \) for \( 0 < x \leq 1 \). Find the value of \( m \) such that \( f(x) \geq -\frac{8}{9} \) for all \( x \leq m \).

Answer: \dfrac{7}{3}


Question: Given two complex numbers $z_1$ and $z_2$ such that \( \left|\frac{z_1-2z_2}{2-z_1\bar{z_2}}\right|=1 \) and \( |z_2| \neq 1 \), find the magnitude \( |z_1| \).

Answer: 2


Question: Evaluate the double integral \( \int_{0}^{1}\int_{0}^{2-2x}\frac{\left(2x-y\right)^{2}}{2x+y}dydx \) using the substitution \( u=2x-y \) and \( v=2x+y \).

Answer: \dfrac{4}{9}


Question: Determine the degree of homogeneity of the function \( f(x_1,x_2) = \int_{0}^{\sqrt{{x_1}^2+{x_2}^2}} e^{-\frac{w^2}{{x_1}^2+{x_2}^2}} dw \).

Answer: 1


Question: Determine the Galois group of the field extension $F_{p^2}$ over $\mathbb{Z}_p$, where $F_{p^2}$ is the field with $p^2$ elements.

Answer: \mathbb{Z}/2\mathbb{Z}


Question: Find the equation of a plane that is parallel to the plane \(z = 2x + 3y\) and tangent to the surface defined by the function \(f(x, y) = x^2 + xy\).

Answer: z = 2x + 3y + 3


Question: If $f$ is a Darboux function and $g$ is a continuous function, is the function $f+g$ necessarily a Darboux function?

Answer: Yes


Question: Find the solution to the second-order ordinary differential equation \(2xx''-3(x')^2=4x^2\) with the initial conditions \(x(0) = 1\) and \(x'(0) = 0\).

Answer: \sec^2 t


Question: Determine the number of distinct values of the expression \( \sqrt{\sqrt{i}} \).

Answer: 4


Question: Let \( m \) be the number of integers \( n \), where \( 1 \le n \le 2005 \), such that the polynomial \( x^{2n} + 1 + (x + 1)^{2n} \) is divisible by \( x^2 + x + 1 \). Find the remainder when \( m \) is divided by 1000.

Answer: 337


Question: Evaluate the integral \( \int_0^1 \ln\left(\frac{1+x}{1-x}\right) dx \) by expanding the integrand into a power series.

Answer: 2 \ln 2


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\log_{\sin^2 x} \cos x}{\log_{\sin^2{\frac{x}{2}}} \cos \frac{x}{2}} \]

Answer: 4


Question: Evaluate the limit: \[ \lim_{(x,y)\to(0,0)} \frac{\sin(x^2y+y^2x)}{xy} \]

Answer: 0


Question: Calculate the last three digits of \(132^{1601}\).

Answer: 632


Question: In a group of eight friends, each of a different age, they decide to sit around an eight-person round table. The youngest friend sits at seat 1, and they alternate between older and younger friends around the table. How many ways can they arrange themselves? Assume the answer is 272.

Answer: 272


Question: Let $E$ be an elliptic curve over the finite field $F_p$, where $p > 7$ is a prime number. If $E(F_p)$ has a point of order $p$, is it true that the number of points on $E(F_p)$, denoted as #$E(F_p)$, is equal to $p$?

Answer: Yes


Question: Solve for $x$ in the equation \[ \frac{2ax}{ab+a+1}+\frac{2bx}{bc+b+1}+\frac{2cx}{ca+c+1}=1 \] given that $a$, $b$, and $c$ are positive numbers satisfying $abc=1$.

Answer: \dfrac{1}{2}


Question: Let $X, Y, Z$ be independent and identically distributed random variables with an exponential distribution with rate parameter $a > 0$. Find the probability $P(X < Y + Z)$.

Answer: \dfrac{3}{4}


Question: Let $\lambda$ be the Lebesgue measure and $\mu$ be a measure absolutely continuous with respect to $\lambda$. Suppose for Lebesgue measurable sets $X_n$, $n=1,2,...$, we have $\lambda(X_n)=1/n$. Determine whether $\mu(X_n)\to 0$ as $n\to \infty$. Justify your answer.

Answer: No


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a non-constant, three times differentiable function. If \( f\left(1 + \frac{1}{n}\right) = 1 \) for all integers \( n \), find \( f'(1) \).

Answer: 0


Question:
Given the joint density function for two random variables \(X\) and \(Y\):
\[ f(x,y) = \frac{e^{-\frac{x}{y}} e^{-y}}{y} \]
where \(x, y \in (0, \infty)^2\), find the conditional expectation \(E(X|Y=y)\) for all \(y > 0\).

Answer: y


Question: Evaluate the limit \( \lim_{n\rightarrow \infty} \int_{-\infty}^\infty \frac{1}{1+\frac{x^4}{n}} \, dx \) if it exists. If the limit does not exist, explain why.

Answer: \infty


Question: Find a curve passing through the points (0,0) and (1,1) that is an extremal of the functional \( J(x,y,y') = \int_{0}^{1} \left[ (y')^2 + 12x y \right] \, dx \).

Answer: y = x^3


Question: Evaluate the limit: $$\lim \limits_{x\to0}{\frac{\frac{\ln(1+x)}{\arcsin{(x)}}-1}{x}}$$

Answer: -\dfrac{1}{2}


Question: In the figure, ABCDE is a regular pentagon. Given that $AD = AF$ and $\angle DAF = 56^\circ$, find the measure of angle $x$.

Answer: 62


Question: Find the degree of the splitting field of the polynomial \(x^3 - 7\) over the rational numbers.

Answer: 6


Question: Evaluate the infinite series: \[ f(x) = \frac{1}{3} + \frac{1 \cdot 3}{3\cdot 6} + \frac{1\cdot 3\cdot 5}{3\cdot 6\cdot 9} + \frac{1\cdot 3\cdot 5\cdot 7}{3\cdot 6\cdot 9\cdot 12} + \ldots \] Find the value of the series.

Answer: \sqrt{3} - 1


Question: A firm is about to hire one of three candidates who take a test with scores that are continuous random variables distributed as \( \text{Unif}[0,100] \). Candidates apply sequentially, and the firm must decide to either select a candidate immediately or reject them permanently. If the first two candidates are rejected, the third candidate must be selected. What is the score threshold below which the first candidate will be rejected to maximize the expected score of the admitted candidate?

Answer: 62.5


Question: Determine if the region of interest on the XY plane, for finding the volume of the object bounded by the planes \(z = y + 4\), \(x^2 + y^2 = 4\), and \(z = 0\), is the entire circle defined by \(x^2 + y^2 = 4\).

Answer: Yes


Question: Find all distinct isomorphisms \(\sigma\) of \(\mathbb{Q}(\sqrt{1+\sqrt{2}})\) in a subfield of \(\mathbb{\bar{Q}}\) such that \(\sigma(a) = a\) for all \(a \in \mathbb{Q}\).

Answer: 4


Question:
Let \( f(x) \) be a function \( \mathbb{R} \to \mathbb{R} \) with an oblique asymptote at \( y = 2x - 1 \). Calculate the limit:
\[ \lim_{x\to\infty}\sin\left(\frac{2}{x}\right)f(x) \]

Answer: 4


Question:
Evaluate the triple integral by changing to polar coordinates:
\[ \int_0^1 \int_0^{\sqrt{1-x^2}} \int_0^{\sqrt{1-x^2-y^2}} xyz \, dz \, dy \, dx \]

Answer: \dfrac{1}{48}


Question: Determine the value of $a$ such that the existence and uniqueness theorem applies to the initial value problem: \( x' = \frac{3}{2}|x|^{1/3}, \ x(0) = a \).

Answer: a \neq 0


Question: Determine the radius of convergence of the series \( \sum_{k = 0}^{\infty} 3^{k^2}x^{k^2} \).

Answer: \dfrac{1}{3}


Question: Evaluate the limit: $$\lim_{n \to \infty} \frac{\big((n+1)(n+1)(n+3)\ldots3n\big)^{1/n}}{(n^{2n})^{1/n}}.$$

Answer: \dfrac{27}{e^2}


Question: For a twice differentiable function \(f(x)\), it is given that \(|f(x)| \leq 3\) for all \(x \in \mathbb{R}\). Additionally, for some \(\alpha\), we have \(f(\alpha) + (f'(\alpha))^2 = 80\). Determine the number of integral values that \((f'(x))^2\) can take in the interval \((0, 77)\).

Answer: 76


Question: Evaluate the integral: $$ \int_{-\infty}^{\infty} \frac{x^2}{(1+|x|^3)^4} \ dx.$$

Answer: \dfrac{2}{9}


Question: Compute \( \lim_{x \to \infty} \frac{P(x)}{P([x])} \), where \( P(x) \) is a polynomial with positive coefficients and \( [\cdot] \) denotes the greatest integer function.

Answer: 1


Question:
Determine whether the following inequality holds:
\[ \int_{l}^{w} \frac{t^{a-1}dt}{(1+ut)^{p+1}} \le  \int_{0}^{\infty} \frac{t^{a-1}dt}{(1+ut)^{p+1}} = \frac{1}{u^a}B(a,p+1-a) \]
where \(0 < l < w < \infty\), \(0<a\), and \(1<p\).

Answer: Yes


Question:
Consider \(n \times n\) matrices \(X\) and \(Y\). Determine the number of ordered pairs \((X, Y)\) such that \(XY - YX = I\), where \(I\) is the identity matrix. Choose the correct option:

- \(0\)
- \(1\)
- \(n\)
- \(\infty\)

Answer: A


Question: Let $f(x)=x^3+ax^2+bx+c$ and $g(x)=x^3+bx^2+cx+a$ where $a,b,c\in\mathbb{Z}$, $c\neq 0$. Suppose $f(1)=0$ and the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$. Find the value of $a^{2020}+b^{2020}+c^{2020}$. \(a, b, c \in \mathbb{Z}\).

Answer: 3


Question: Find the smallest positive integer \( m \geq 2 \) such that the equation \( 6x + 11y = m \) has no non-negative integer solutions, but for any integer \( k > m \), the equation \( 6x + 11y = k \) has at least one non-negative integer solution.

Answer: 49


Question: Consider the model \( Y_i = \beta + U_i \) for \( i = 1, 2, 3 \), where \( E(U_1) = E(U_2) = E(U_3) = 0 \) and \( \text{Var}(U_1) = 1, \text{Var}(U_2) = 2, \text{Var}(U_3) = 4 \). Determine the best linear unbiased estimate of the variance of \( \beta \).

Answer: \dfrac{4}{7}


Question: Evaluate the integral \( \int_{0}^{\infty}\frac{\sin^2 t}{t^2}dt \) using Laplace transforms.

Answer: \dfrac{\pi}{2}


Question: Determine the radius of convergence of the power series \( s(x) = \sum_{n=0}^\infty a_n x^n \), where the coefficients \( a_n \) are defined as follows: \( a_n = 1 \) if \( n \) is a perfect square, and \( a_n = 0 \) otherwise.

Answer: 1


Question: Two identical sets of 52 cards are shuffled separately. One set (set A) is shuffled, and both sets are placed face down on the table. Cards are drawn from the top of each set simultaneously until all cards have been drawn. What is the probability of drawing at least one pair of identical cards during this process?

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


Question: Evaluate the expression \( \log_3(1+2(3+1)(3^2+1)(3^4+1)(3^8+1)(3^{16}+1)(3^{32}+1)) \).

Answer: 64


Question: Evaluate the contour integral \( \oint_\gamma \frac{\sin^2(z)}{(z-1)^2} \,dz \), where \( \gamma \) is the circle of radius one centered at \( z=1 \).

Answer: 2\pi i \sin(2)


Question: Let $K$ and $L$ be two natural numbers such that $K$ has $L$ divisors and $L$ has $\frac{K}{2}$ divisors. Determine the number of divisors of $K + 2L$. \( \text{Express your answer as a single number.} \)

Answer: 4


Question:
Evaluate the limit using Taylor series:
\[ \lim_{x \to 0} \frac{(\ln(1+x) - x + \frac{x^2}{2})^4}{(\cos(x) - 1 + \frac{x^2}{2})^3} \]

Answer: \dfrac{512}{3}


Question: Find the relationship between $x$ and $a$ that minimizes the function $\epsilon = (1-e^{-x/a})^x$. It is claimed that this minimum occurs when $x \approx a\ln(2)$. Verify this claim analytically.

Answer: x = a \ln 2


Question: Find the sum of the infinite series: \[ \sum_{n=1}^\infty \frac{n}{3^n} \]

Answer: \dfrac{3}{4}


Question: Determine whether the identity \( \ln(AB) = \ln(A) + \ln(B) \) holds only if \( AB = BA \) for square matrices \( A \) and \( B \) where \( \ln(A) \), \( \ln(B) \), and \( \ln(AB) \) are all defined.

Answer: Yes


Question: Find the limit of the sequence \( a_n = \frac{3\ln(n^2+2n)}{\ln(\sqrt{n^4-1})+3\ln(5\ln(2n))} \).

Answer: 3


Question: Evaluate the limit \( \lim_{n \to \infty} (a_1 + a_2 + \cdots + a_n) \) where \( a_n = \left( \frac{n!}{1 \cdot 3 \cdot 5 \cdot 7 \cdots (2n+1)} \right)^2 \). Choose the correct option: \((a)\) Does not exist, \((b)\) Greater than \( \frac{4}{27} \), \((c)\) Less than \( \frac{4}{27} \), \((d)\) None of these.

Answer: C


Question: Is it necessary for the matrices $A$, $B$, and $C$ to be positive definite for the block matrix \( Y = \begin{bmatrix} A & F & G \\ F' & B & E \\ G' & E' & C \end{bmatrix} \) to be positive definite?

Answer: Yes


Question: Does the group \(\mathrm{GL}(n,\mathbb{Z})\) contain torsion-free abelian subgroups that are not finitely generated?

Answer: No


Question: Let \(K\subseteq\mathbb{R}^2\) be a convex compact set. Determine whether the boundary \(\partial K\) is a rectifiable curve, meaning it has a finite length.

Answer: Yes


Question: Determine whether it is true that if two finite simple graphs \((G, E)\) and \((G', E')\) are isomorphic, then all vertices in \(G\) being in a 4-cycle implies that all vertices in \(G'\) must also be in a 4-cycle.

Answer: True


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{x^2}{(x^2+1)^2} \, \mathrm{d}x \). You may use the substitution \( x = a \tan(\theta) \) to simplify the process.

Answer: \dfrac{\pi}{2}


Question: Suppose $f$ is a function that is continuous on the closed unit disk $\{z \in \mathbb{C} \mid |z| \leq 1\}$ and analytic on the open unit disk $\{z \in \mathbb{C} \mid |z| < 1\}$. If $f$ is constant on the open unit disk, must $f$ also be constant on the closed unit disk?

Answer: Yes


Question:
Let \( p_1, p_2, p_3 \) be the altitudes of \( \triangle ABC \) from vertices \( A, B, C \) respectively. Given that \( \Delta \) is the area of the triangle and \( R \) is the circumradius of the triangle, find the value of \( \frac{\cos A}{p_1} + \frac{\cos B}{p_2} + \frac{\cos C}{p_3} \). Choose the correct option:

(A) \( \frac{1}{R} \)

(B) \( \frac{a^2+b^2+c^2}{2R} \)

(C) \( \frac{\Delta}{2R} \)

(D) None of these.

Answer: A


Question: Verify if the following holds for a smooth function $f(x)$ with a zero at the origin and a smooth function $g(x)$ on a region $R$ around the origin: \[ \int_{-\epsilon}^\epsilon \delta(f(x))g(x)\,dx = \frac{g(0)}{f'(0)}. \] Specifically, does \[ \int_{-0.005}^{0.005} \delta(7x + x^2)(14x + 37\cos(x))\,dx = \frac{37}{7}? \]

Answer: \dfrac{37}{7}


Question:
Consider a function $f: [-1, 1] \to [-1, 1]$ with the following properties:

(a) $f(x) = 0$ at all points $x_n = 2^{1-n}$ and $y_n = -2^{1-n}$ for all natural numbers $n$.
(b) $f(x)$ is continuous at all these points.
(c) $f(x)$ is a bounded function.
(d) For all integers $n$, $f(x)$ restricted to $[-2^{-n}, 2^{-n}]$ satisfies $f(x) = \frac{1}{a^n}f(2^n x)$ for some fixed constant $a > 1$.
(e) $f(0) = 0$.

Is $f(x)$ necessarily continuous at $x = 0$?

Answer: Yes


Question: Find \( \lim_{x\rightarrow 0^{+}}x\cdot \left(\left\lfloor \frac{1}{x}\right\rfloor+\left\lfloor \frac{2}{x}\right\rfloor+\left\lfloor \frac{3}{x}\right\rfloor+\cdots +\left\lfloor \frac{15}{x}\right\rfloor\right) \), where \(\lfloor x\rfloor\) denotes the integer part of \(x\).

Answer: 120


Question: Determine if the function \( f(x) = \begin{cases} x^2 & \text{if } x \in \mathbb{Q}, \\ x^3 & \text{if } x \notin \mathbb{Q} \end{cases} \) is differentiable at \( x = 0 \).

Answer: 0


Question: Evaluate the integral \( \frac{(1-|a|^2)}{\pi} \int_{|z|=1} \frac{|dz|}{|z+a|^2} \), where \( a \in \mathbb{C} \) and \( |a| < 1 \).

Answer: 2


Question:
Let $B(x_0,R)$ be an open ball in $\mathbb{R}^n$ with $n \geq 2$. Suppose $u \in H^{1}(B(x_0,R))$ is a weak solution of $\Delta u = 0$ in $B(x_0,R)$. Let $u_{x_i}$, for $i \in \{1, \ldots, n\}$, be a weak derivative of $u$. Determine whether the following statement is true:
\[ u_{x_i}(x_0) = \frac{1}{|B(x_0,R)|} \int_{B(x_0,R)} u_{x_i}(y) \, dy. \]

Answer: True


Question: Evaluate the limit: $$\lim_{x \to 0^+} \left((x\cos(x))^x+(x\sin(x))\right)^{1/x}.$$

Answer: 0


Question: Evaluate the integral \( \iiint_{R} (xy+z^2) \, dV \), where the region \( R \) is defined as \( \{ (x,y,z) : 0 \leq z \leq 1 - |x| - |y| \} \).

Answer: \dfrac{1}{15}


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

Answer: n^a


Question:
Determine the radius of convergence, \( R \), for the power series:
\[ \sum_{n=2}^{\infty} \frac{(-1)^n z^{2n}}{(2n-1)!} \]

Answer: \infty


Question: Determine whether the set \( \{ \text{fractional part of } n\sqrt{2} \mid n \in \mathbb{N} \} \) is dense in the interval \([0,1]\).

Answer: Yes


Question: In triangle $ABC$, the altitude $AD$ is $18$, the median $BE$ is $9\sqrt{5}$, and the median $CF$ is $15$. Find the length of side $BC$. Express your answer as a single number.

Answer: 20


Question:
Evaluate the integral \( \int_C \text{Re}\;z\,dz \) along the contour \( C \) from \(-4\) to \(4\), which consists of the line segments from \(-4\) to \(-4-4i\) to \(4-4i\) to \(4\). Use the following parameterizations for each segment:

1. \( z(t) = -4 - 4it, \quad 0 \leq t \leq 1 \)
2. \( z(t) = -4 - 4i + 8t, \quad 0 \leq t \leq 1 \)
3. \( z(t) = 4 - 4i + 4it, \quad 0 \leq t \leq 1 \)

Calculate the integral for each segment and sum the results to find the total value of the integral.

Answer: 32i


Question: Given that $a + b + c = 2$ and $a, b, c > 0$, find the maximum value of $K$ such that the inequality \[ \sqrt{a^a b^b c^c} + \sqrt{a^b b^c c^a} + \sqrt{a^c b^a c^b} \leq K \] holds for all positive $a$, $b$, and $c$.

Answer: 2


Question:
Consider the initial value problem for the partial differential equation:

\[(x-y)\frac{\partial u}{\partial x} +(y-x-u)\frac{\partial u}{\partial y} =u\]

with the initial condition \(u(x,0)=1\). Determine which of the following expressions the solution satisfies:

1. \(u^2(x-y+u)+(y-x-u)=0\)
2. \(u^2(x+y+u)+(y-x-u)=0\)
3. \(u^2(x-y+u)-(x+y+u)=0\)
4. \(u^2(y-x+u)+(x+y-u)=0\)

Given that one solution is \(x+y+u=a\), identify the correct expression.

Answer: 2


Question: Evaluate the integral \( \int_0^{2\pi}\frac{\mathrm{d}x}{4\cos^2x+\sin^2x} \) using the residue theorem.

Answer: \pi


Question: Find the smallest natural number $b$ such that the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots is 20.

Answer: 1680


Question: Let $R$ be an integral domain and $I, J$ be ideals such that $IJ$ is a principal ideal. Is it true that $I$ is finitely generated?

Answer: Yes


Question:
Maximize \(|x|\) subject to the constraints:
\[-x + y \leq 1\]
\[x + y \leq 2\]
\[y \geq 0\]

Answer: 2


Question: Find the dimension of the quotient space \( c_0 / M \), where \( c_0 = \{(x_n) : x_n \in \mathbb{R},\ x_n \to 0 \} \) and \( M = \{(x_n) \in c_0 : x_1 + x_2 + \cdots + x_{10} = 0\} \).

Answer: 1


Question: Suppose you flip a coin where the probability of obtaining heads, denoted as $P$, is uniformly distributed over the interval (0, 1). Calculate the probability of obtaining the sequence HTH in the first three flips.

Answer: \dfrac{1}{12}


Question:
Let \(f(x)\) be a non-constant continuous function such that \((e^x-1)f(2x) = (e^{2x}-1)f(x)\). Given that \(f'(0) = 1\), calculate the limit:
\[ \lim_{x \to \infty} \left(\frac{f(x)}{x}\right)^{1/x} \]

Answer: e


Question: Is there always a prime number between $n$ and $2n$ for any natural number $n > 1$? Provide a justification for your answer.

Answer: Yes


Question: Is it possible to lay \( \aleph_1 \) many people on a railway track, assuming the axiom of choice? Provide a justification for your answer.

Answer: Yes


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

Answer: 2


Question: Does there exist a noncompact connected Lie group with a finite-dimensional, unitary, faithful, irreducible representation over \(\mathbb{C}\)?

Answer: No


Question: Can a faithful normal state always be defined on a C*-algebra acting on a non-separable Hilbert space?

Answer: No


Question: Is there a representation of the alternating group $A_4$ in $\operatorname{GL}_3(\mathbb{R})$ that preserves the unit 3-ball?

Answer: Yes


Question: Find the Radon-Nikodym derivative \( \frac{d\mu}{dm} \) where \( m \) is the Lebesgue measure on \([0,1]\), \( f(x)=x^2 \), and \( \mu(E)=m(f(E)) \).

Answer: 2x


Question: Compute the integral \( \int_0^{2\pi}\frac{1}{e^{i\theta}-r}d\theta \) for a positive number \( r < 1 \).

Answer: 0


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

Answer: -\infty


Question: Evaluate the integral \( \int_{|z| = 2} \overline{z} \, dz \), where \( \overline{z} \) is the complex conjugate of \( z \).

Answer: 8\pi i


Question: Find the largest integer $n$ such that $n! < 100^{n-1}$. Provide your answer as an integer.

Answer: 263


Question:
Given that \( f: [0,1] \to \mathbb{R} \) satisfies the following conditions:
\[ \int_{0}^{1} f(x) \, dx = 1, \]
\[ \int_{0}^{1} x f(x) \, dx = 1, \]
\[ \int_{0}^{1} f(x)^2 \, dx = 4, \]
find the value of:
\[ \int_{0}^{1} f(x)^3 \, dx. \]

Answer: 10


Question: Evaluate the limit: \[ \lim_{y\to 0^+} \int_0^1 \frac{x\cos{y}}{\sqrt[3]{1-x+y}}\,dx \] Justify your answer.

Answer: \dfrac{9}{10}


Question: Determine whether the following statement is true or false: If a function \( f \) is uniformly differentiable on \((a,b)\), then \( f' \) is continuous on \((a,b)\).

Answer: True


Question: Suppose $f$ is an entire function such that $f(k) = 0$ for all $k \in \mathbb{Z}$ and $|f(z^2)| \leq e^{|z|}$ for all $z \in \mathbb{C}$. Prove that $f(z) = 0$ for all $z \in \mathbb{C}$.

Answer: 0


Question: Calculate the limit \( \lim_{x\rightarrow +\infty} (1+3 \sin \frac{1}{x})^{\lfloor x \rfloor} \).

Answer: e^3


Question:
Let $V$ and $W$ be vector spaces over the field of real numbers $\mathbb{R}$. A function $f: V \rightarrow W$ is said to be linear if it satisfies the following conditions for any vectors $x, y \in V$ and any scalar $\alpha \in \mathbb{R}$:

1. $f(x + y) = f(x) + f(y)$
2. $f(\alpha x) = \alpha f(x)$

Is it possible to construct a function $f$ such that the first condition is satisfied but not the second one?

Answer: Yes


Question: Determine the maximum value of the function \( N(a - b) \) over all \( a, b \in U(V) \), where \( N: \text{End}_\mathbb{C} \to \mathbb{R}_{\ge 0} \) is defined as \( N(a) := \max_{\{v \in V\, :\, |v| = 1\}} |av| \). What is the diameter of the group \( U(V) \)?

Answer: 2


Question: Given a monic cubic polynomial $x^3 + ax^2 + bx + c$ with all roots being negative real numbers and $a < 3$, determine the range of the expression $b + c$. Here, $a$, $b$, and $c$ are real numbers.

Answer: (0, 4)


Question: Let \( Y_n \) be independent random variables uniformly distributed in \([0, 1]\). For \( N \geq 1 \), define \( X_n = Y_1 \cdot Y_2 \cdot \ldots \cdot Y_n \), the product of the \( Y_i \). Determine the limit as \( n \rightarrow \infty \) of \( \text{Prob}(X_n > (0.4)^n) \).

Answer: 0


Question: Identify the two additional symmetries of an octahedron that stabilize a given face, given that there are already four known symmetries. Use the Orbit-Stabilizer Theorem to determine the complete set of six symmetries.

Answer: 6


Question:
Determine whether the following statement about the Fermat primality test is correct:

For all integers $b$, if $n$ is a prime number and $(b,n)=1$, then $b^{n-1} \equiv 1 \pmod{n}$. The contrapositive is that if $b^{n-1} \not\equiv 1 \pmod{n}$, then either $n$ is not prime or $(b,n) \neq 1$. Is this statement correct?

Answer: Correct


Question:
Determine the number of distinct real roots of the equation given by the determinant:
\[\left| \begin{array}{cccc}
x & a & a & a \\
a & x & a & a \\
a & a & x & a \\
a & a & a & x \end{array} \right|=0.\]

Answer: 2


Question: Let \( M \) be an \( n \)-dimensional manifold and \( \pi : \tilde{M} \to M \) be the universal cover of \( M \). Consider a point \( \tilde{p} \in \tilde{M} \) that is a lift of \( p \in M \). Choose a measurable section \( j: B_1(p) \to B_1(\tilde{p}) \) such that \( \pi(j(x)) = x \) for any \( x \in B_1(p) \). Define \( T = j(B_1(p)) \). Let \( S \) be the union of \( g(T) \) over all \( g \in \pi_1(M) \) such that \( g(T) \cap B_1(\tilde{p}) \neq \emptyset \). Is it true that \( B_1(\tilde{p}) \subset S \)?

Answer: Yes


Question: In a Hausdorff compact space $X$, let $A$ and $B$ be two disjoint closed subsets. Is it possible to construct sequences of neighborhoods $\{ A_n \}$ and $\{ B_n \}$ for $A$ and $B$, respectively, such that $X = \bigcup\limits_{n\in\mathbb{N}} A_n \cup \bigcup\limits_{n\in\mathbb{N}} B_n$?

Answer: Yes


Question: Determine the number of distinct terms in the expansion of \((x + \frac{1}{x} + x^2 + \frac{1}{x^2})^{15}\), where distinct terms are defined by having different powers of \(x\).

Answer: 61


Question: Compute the limit: \[ \lim_{x\to\frac{\pi}{2}} \frac{(1-\sin x)(1-\sin^2x)\dots(1-\sin^nx)}{\cos^{2n}x} \]

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


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: Determine the value of \(n\) such that \(\lim\limits_{x \to 0}{\frac{\sqrt{x+\sqrt{x}}}{x^n}} \neq 0\).

Answer: \dfrac{1}{4}


Question: Consider the matrix $A \in \mathbb{R}^{(d+1)\times 2^d}$ whose columns are given by an enumeration $a: \{1,\dots,2^d\} \to \{1\}\times\{0,1\}^d$. For each $b \in \{1\} \times [0,1]^d$, determine if there exists a vector $x \in [0,+\infty)^{2^d}$ such that $Ax = b$. Is it true that such a solution always exists?

Answer: Yes


Question: Determine the largest possible radius of convergence for the Cauchy product of two power series, each with a radius of convergence of 1. Specifically, if $\sum_{n = 0}^{\infty} a_nz^n$ and $\sum_{n = 0}^{\infty} b_nz^n$ are two such power series, what is the upper bound on the radius of convergence of their Cauchy product $\sum_{n = 0}^{\infty} c_nz^n$, where $c_n = \sum_{k = 0}^{n} a_kb_{n - k}$?

Answer: 1


Question: Let $D$ be an integral domain with fraction field $K$. Let $V$ and $W$ be multiplicatively closed subsets of $D$. Consider the rings of fractions $V^{-1}D$ and $W^{-1}D$ as subrings of $K$. Is the intersection $(V^{-1}D) \cap (W^{-1}D)$ a ring of fractions of $D$? Specifically, does there exist a multiplicative subset $S$ of $D$ such that $(V^{-1}D) \cap (W^{-1}D) = S^{-1}D$?

Answer: Yes


Question: Evaluate the line integral \( \int_\ell \left(\frac{-y}{\sqrt{2x^2+2y^2}}\right) \, dx + \left(\frac{x}{\sqrt{2x^2+2y^2}}\right) \, dy \) over the curve \( \vec \ell(t)=(\sin(t), \sin(t)\cos(t)-1) \) for \( t \in [0,2\pi] \).

Answer: 0


Question: Find the value of $n$ that satisfies the equation: $$\frac{_{365}P_n}{365^n} \approx \frac{1}{5}.$$

Answer: 34


Question: Determine whether the elementary symmetric polynomials $e_2, e_3, \dots, e_{n-1}$ are irreducible elements in the polynomial ring $\mathbb{C}[x_1, x_2, \dots, x_n]$ for $n \geq 4$. Provide a justification for your answer.

Answer: Yes


Question: Find the area of the region $R$ enclosed by the curve parameterized by $g(t) = (t^4 - t^2, t^6 - t^2)$ for $0 \leq t \leq 1$ using Green's Theorem.

Answer: \dfrac{1}{60}


Question: In an acute triangle \( \triangle ABC \), inscribed in a circle with radius \( R \), the orthocenter \( H \) satisfies \( AH = R \). Find the measure of \( \angle BAC \).

Answer: 60^\circ


Question: Determine the number of ways for two people to each select \( k \) items from a set of \( n \) items such that each item is selected an even number of times. Specifically, find the number of solutions to the equations \( |a| = \sum_{i=1}^n a_i = k \) and \( |b| = \sum_{i=1}^n b_i = k \), where \( a_i + b_i \) is even for each \( i \).

Answer: \dbinom{n}{k}


Question: Let $D$ be a bounded, open, connected subset of $\mathbb{R}^n$. Suppose $u \in C^{1,\alpha}(D)$ for some $\alpha \in (0,1]$, meaning $u$ has a H\"older continuous derivative with exponent $\alpha$. Is it true that $u$ is Lipschitz on $D$? Justify your answer.

Answer: Yes


Question: Let \( \{x_k\}_{k\in\mathbb{N}} \) be a sequence of i.i.d. random variables with \( x_k\sim U_{[[ 1,n]]} \) for all \( k\in\mathbb{N} \), where \( U_{[[1,n]]} \) is the uniform distribution on the discrete set \( [[1,n]]:=\{1,2,\ldots,n\} \). Fix \( a\in[[1,n]] \). What is the probability that there exists a subsequence \( \{x_{k_j}\}_{j\in\mathbb{N}} \) such that \( \forall j\in\mathbb{N}, x_{k_j} = a \)?

Answer: 1


Question: Given a model \(X_t = X_{t-1} + Y_t\) where \(Y_t\) are iid random variables with \(\mathbb{E}(Y_t) = 0\) and \(\text{Var}(Y_t) = 4\), evaluate \(\text{Var}(X_1 + X_2 + X_3 + \ldots + X_8)\).

Answer: 816


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\left \{ \frac{\left ( n+1 \right )\left ( n+2 \right )...3n}{n^{2n}} \right \}^{\frac{1}{n}} \]

Answer: \dfrac{27}{e^2}


Question: Find the maximum integer $n$ such that there exists a sequence $a_1, a_2, \ldots, a_n$ where for any $s \in \{1, 2, \ldots, n\}$, the subsequence $a_1, a_2, \ldots, a_s$ has each element $a_i$ belonging to one of the $s$ intervals $(0, 1/s), (1/s, 2/s), \ldots, ((s-1)/s, 1)$. If no such maximum $n$ exists, determine if there is an infinite sequence $a_1, a_2, \ldots$ satisfying the same condition for all $s \in \mathbb{N}^+$. Provide an example for $n=6$ if possible.

Answer: 6


Question: Determine whether there exist infinitely many prime numbers of the form $x^3 - 1$. Justify your answer.

Answer: No


Question: Evaluate the improper integral: \[ I = \int_{0}^{\infty} \frac{x^3 \ln x}{(x^4+1)^3} \,dx \]

Answer: -\dfrac{1}{32}


Question: Determine whether the set $U$ of binarily universal members of $[0,1]$ is a Borel set. If it is, find its Borel measure.

Answer: 1


Question: Find the smallest integer value of the function \( f(x) = \left(\frac{1}{9} + \frac{32}{\sin(x)}\right)\left(\frac{1}{32} + \frac{9}{\cos(x)}\right) \) on the interval \((0, \pi/2)\).

Answer: 579


Question: Find the limit of the sequence \( \left\{ \left(\frac{n^2+2}{2n^2+1}\right)^{n^2}\right\}_{n\geq 1} \).

Answer: 0


Question: If $g$ is the inverse of the function $f$ and $f'(x)= \frac{1}{1+x^n}$, find $g'(x)$. Express your answer in terms of $g(x)$. \(g'(x) = 1 + [g(x)]^n\)

Answer: g'(x) = 1 + [g(x)]^n


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{(3n)! \, e^n}{(2n)!\,n^n\,8^n} \]

Answer: 0


Question: Evaluate the integral \( \int_0^\infty \! \!\int_x^\infty \frac{1}{2}e^{-x-\frac{y}{2}} \, \mathrm{d}y \, \mathrm{d}x \).

Answer: \dfrac{2}{3}


Question: Determine the number of ways to form hexagons by joining the vertices of a convex 20-gon such that no side of the hexagon is common with a side of the polygon.

Answer: 4290


Question: Suppose $M$ is an oriented closed manifold and $x \in H^*(M;\mathbb{Q})$ is a cohomology class. For every group $G$, every class $y \in H^*(BG,\mathbb{Q})$, and every continuous map $f: M \to BG$, it holds that $\langle x \cup f^*y,[M]\rangle=0$, where $[M]$ is the fundamental class in $H_*(M,\mathbb{Q})$. Does it follow that $x=0$?

Answer: Yes


Question:
Given that $a + b + c = 1$ and $a, b, c \geq 0$, find the maximum value of the expression:
\[ \frac{1}{a^2 - 4a + 9} + \frac{1}{b^2 - 4b + 9} + \frac{1}{c^2 - 4c + 9}. \]

Answer: \dfrac{7}{18}


Question: Compute the contour integral \( \int_{\Gamma} |z|^2 \, dz \) where \( \Gamma \) is the curve parameterized by \( \gamma(t) = a\cos(t) + ib\sin(t) \) for \( 0 \leq t \leq 2\pi \), and \( a, b \in \mathbb{R} \).

Answer: 0


Question: Find the value of \( a \) such that the vector \( \begin{pmatrix} a \\ -1 \\ -2 \end{pmatrix} \) is an eigenvector of the matrix \( \begin{pmatrix} a & 6 & 3 \\ -1 & a & 1 \\ 1 & 2 & -1 \end{pmatrix} \).

Answer: -\dfrac{4}{5}


Question: Find the remainder when $1978^{20}$ is divided by 125.

Answer: 26


Question: Evaluate the limit \( \lim_{n \to \infty} \sum_{k=1}^{n} \frac{(n+1)^k}{n^{k+1}} \) using the concept of Riemann sums.

Answer: e - 1


Question:
Find the equations of the orbits for the system of differential equations given by:
\[ \frac{dx}{dt} = y^2 \]
\[ \frac{dy}{dt} = \frac{-2}{3}x \]

Answer: y^3 + x^2 = C


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

Answer: e^{-8}


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

Answer: f(0)


Question: Find the Laurent series expansion of \( \left(\frac{z}{z-1}\right)^2 \) around \( z = 1 \) and determine the residue at this point.

Answer: 2


Question: Evaluate the limit \( \lim _{n\to \infty }\sum _{k=1}^n\frac{1}{n+k+\frac{k}{n^2}} \).

Answer: \ln 2


Question: In triangle \( \triangle ABC \), let \( I \) be the midpoint of side \( [AB] \). Consider point \( J \) on segment \( [BC] \) such that \( BJ = 2JC \) and \( \angle AIC = \angle BAJ \). Find the measure of angle \( \angle BAC \).

Answer: 90^\circ


Question: Evaluate the limit: $$\lim_{n\rightarrow\infty}\frac{1-2+3-4+5-\cdots+\left(-2n\right)}{\sqrt{n^{2}+1}+\sqrt{n^{2}-1}}.$$

Answer: -\dfrac{1}{2}


Question: Let $M$ be an n-dimensional differentiable manifold. Can we always find a chart $(U,f)$ such that $f(U) = \mathbb{R}^n$?

Answer: Yes


Question: Is a map of fibrant marked simplicial sets a fibration in \(\mathrm{sSet}^+\) if and only if its underlying map of simplicial sets is a fibration in \(\mathrm{sSet}\) with Joyal's model structure?

Answer: Yes


Question: Determine the values of $x_0 > 0$ for which the sequence defined by $x_n = \frac{x_{n-1}}{2} + \frac{3}{x_{n-1}}$ for $n \geq 1$ converges.

Answer: (0, \infty)


Question: Determine whether the series \( \sum_{i=1}^\infty \frac{r_i}{i} \) can converge to 0 for a non-trivial periodic sequence \( \{r_i\} \) of integers, where there exists some \( t \in \mathbb{N} \) such that \( r_i = r_{i+t} \) for all \( i \in \mathbb{N} \). Does \( \sum_{i=1}^\infty \frac{r_i}{i} = 0 \) imply that \( r_i = 0 \) for all \( i \in \mathbb{N} \)?

Answer: Yes


Question: Let $H$ and $H'$ be two Hopf algebras, and let $\phi: H \to H'$ be a bialgebra map. Is $\phi$ automatically a Hopf algebra map?

Answer: Yes


Question:
Find the function \( f(x) \) that satisfies the functional equation:
\[ f(x^2+x+3) + 2f(x^2-3x+5) = 6x^2 - 10x + 17 \]
for all \( x \in \mathbb{R} \).

Answer: 2x - 3


Question: Given a function sequence $f_n:[0,1]\rightarrow\mathbb{R}$, suppose that for all integers $n \geq 1$, the measure of the set $\{x \in [0,1] \mid |f_n(x)| > 1/n\}$ is at most $1/2^n$. Determine whether it is true that $\lim_{n\rightarrow\infty}f_n(x)=0$ for almost every $x \in [0,1]$. Justify your answer.

Answer: True


Question: Find the derivative of the function \( F = \|H - \mu_H\|_F^2 \) with respect to the matrix \( H \), where \( \mu_H = \frac{\Sigma H}{n_H} \) is the average of \( H \) in one dimension, repeated \( n \) times such that all columns are the same. Express your answer as \( \frac{\partial F}{\partial H} \).

Answer: 2(H - \mu_H)


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

Answer: e^{-e^2}


Question: How many ordered triples of rational numbers \((e, r, t)\) are there such that the polynomial \(g(x) = x^3 + ex^2 + rx + t\) has roots \(e, r,\) and \(t\)?

Answer: 3


Question: Prove that \( \sum_{cyc} \sqrt{\frac{a^3}{1+bc}} \geq 2 \) for positive real numbers \(a, b, c\) such that \(abc = 1\).

Answer: 2


Question: Consider the natural map \(f: \overline{\mathcal{M}}_{1,1} \to \overline{M}_{1,1} \cong \mathbb{P}^1\) from the stack of elliptic curves to the coarse space. Both spaces have \(\text{Pic} = \mathbb{Z}\). Determine the homomorphism \(f^*: \mathbb{Z} \to \mathbb{Z}\).

Answer: 12


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( \left( \dfrac{1-x}{e} \right) ^ { \dfrac{1}{x} } \right) ^ { \dfrac{1}{\sin x} } \]

Answer: 0


Question: Is it true that for any function of the first Baire class \(f:X\to\mathbb{R}\) on the Cantor cube \(X=2^\omega\), there exists a continuous function \(g:X\to[0,1]\) such that the image \((f+g)(X)\) is disjoint from the set \(\mathbb{Z}\) of integers?

Answer: Yes


Question: Given a convex quadrilateral $ABCD$, points $A_1$, $B_1$, $C_1$, and $D_1$ are located on the extensions of sides $AB$, $BC$, $CD$, and $DA$, respectively, such that $|BB_1|=|AB|$, $|CC_1|=|BC|$, $|DD_1|=|DC|$, and $|AA_1|=|DA|$. Express the area of quadrilateral $A_1B_1C_1D_1$ in terms of the area of quadrilateral $ABCD$.

Answer: 5


Question: Find the coefficient of $x^{32}$ in the expansion of $(x^3 + x^4 + x^5 + x^6 + x^7)^7$. Provide your answer as a single integer.

Answer: 6055


Question: Given a set of 40 elements, determine the maximum number of 4-element subsets that can be formed such that no triple of elements appears in more than one subset.

Answer: 2470


Question: Find the limit of \( \frac{1}{M} \int_1^M M^{\frac{1}{x}} \, \mathrm{d}x \) as \( M \to +\infty \).

Answer: 1


Question: In the Erdős–Rényi model on 4 vertices with probability p=0.5, calculate the probability that the graph is bipartite.

Answer: \dfrac{41}{64}


Question: Determine the cardinality of the quotient ring \( \mathbb{Z}[x]/(x^2-3,2x+4) \).

Answer: 4


Question: Assume $A$ is a nontrivial submodule of a free $R$-module, where $R$ is a ring. Is it true that $A$ is also a free module?

Answer: No


Question: Find the value of the constant \( k \) such that the cubic equation \( x^3 - 12x + k = 0 \) has distinct roots in the interval \([0, 2]\).

Answer: 16


Question: Find the constant C that satisfies the equation: \[ \int_0^x f(t)~dt = \int_x^1 t^2 f(t)~dt + \frac{x^2}{4} + \frac{x^4}{8} + C \]

Answer: -\dfrac{1}{8}


Question: Let \(a, b, c, d \ge 0\) satisfy \(a + b + c + d = 4\). Find the maximum value of \((ab + cd)(ac + bd)(ad + bc)\).

Answer: 8


Question: Prove that if $a$ and $b$ are positive integers such that $a \mid b^2$, $b^2 \mid a^3$, $a^3 \mid b^4$, and so on, then $a = b$.

Answer: a = b


Question: Given that $X$ and $Y$ are independent and identically distributed exponential random variables with rate 1, and their joint distribution is given by $f(x,y) = e^{-(x + y)}$ for $x > 0$ and $y > 0$, find the probability density function (PDF) of $U = e^{-(X + Y)}$. Express your answer in terms of $u$.

Answer: -\ln u


Question: Evaluate the limit: $$ \lim_{x\to-\infty} \sqrt{x^2-\sqrt{x^2+1}}+x $$

Answer: -\dfrac{1}{2}


Question: Determine the number of lines that are both normal and tangent to the curve \( y^2 = x^3 \).

Answer: 2


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

Answer: m


Question: Find the arc length of the curve defined by \((x-1)^{2/3}+(y-2)^{2/3}=1\).

Answer: 6


Question: Determine the number of elements of order 2 in the group \( \mathbb{Z}_{60} \times \mathbb{Z}_{45} \times \mathbb{Z}_{12} \times \mathbb{Z}_{36} \).

Answer: 7


Question: Given the smooth curve \( C: y^2z = x^3 - xz^2 \) in \( \mathbb{P}^2 \) and the point \( P = (0:0:1) \), consider the rational map \( \phi: \mathbb{P}^2 \dashrightarrow \mathbb{P}^1 \) defined by \( (x:y:z) \mapsto (x:y) \). The map \( \phi \) is undefined at \( P \). Define \( \psi = \phi|_C: C \dashrightarrow \mathbb{P}^1 \). Since \( C \) is smooth, \( \psi \) extends to a morphism \( \psi: C \to \mathbb{P}^1 \). Determine the value of \( \psi(P) \).

Answer: (0:1)


Question: Evaluate the contour integral \( \oint_{|z|=1} \frac{f(z)}{z^4} \, dz \), where \( f(z) = \sum_{k=0}^\infty \frac{k^3}{3^k}z^k \).

Answer: 2\pi i


Question: Assume that $A$ and $B$ are two $C^{*}$ algebras such that their minimal tensor product is a $Z^{*}$ algebra. Is it true that $A$ or $B$ is a $Z^{*}$ algebra?

Answer: Yes


Question: Determine the number of monic primitive quadratic polynomials in \( \mathbb{Z}_7[x] \).

Answer: 8


Question: Find the minimal area of a rhombus circumscribed around the region bounded by the curves $y = \cos(x)$ and $y = -\cos(x)$ for $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. The rhombus is formed by drawing a tangent line to the upper curve at a point $A = (x, \cos x)$, where $x \in \left(0, \frac{\pi}{2}\right)$, and reflecting the intersection points of this tangent line with the axes to form the vertices of the rhombus. What is the minimal area of such a rhombus?

Answer: 4


Question: Let $a = \{a_n\}_{n = 1}^{\infty}$ be a sequence in $l_{\infty}$, where $l_{\infty}$ denotes the space of sequences with a finite supremum norm. Define the linear operator $T : l_p \rightarrow l_p$ by $Tx = \{x_na_n\}_{n = 1}^{\infty}$. Determine the norm of the operator $T$. Is it $||a||_{\infty}$?

Answer: \|a\|_{\infty}


Question: Let $f:\mathbb{R} \to \mathbb{R}$ be a function such that \(\lim_{x \to 0} \dfrac{f(x) \sin x}{x^2 - x}= + \infty\). Determine \(\lim_{x \to 0} f(x)\).

Answer: -\infty


Question: Given the equation \(x^3 f(x) + f(x)^3 + f(x^3) = 3\) and the condition \(f(1) = 2\), find \(f'(1)\).

Answer: -\dfrac{3}{8}


Question:
Evaluate the determinant of the following matrix, where \(\alpha\) and \(\beta\) are non-real cube roots of unity:
\[
\begin{vmatrix}
\lambda +1 &\alpha &\beta\\
\alpha &\lambda + \beta &1\\
\beta &1 &\lambda + \alpha
\end{vmatrix}
\]

Answer: \lambda^3


Question: Find the limit of the expression \(2^{n+1}\sqrt{2-t_n}\) as \(n\to\infty\), where \(t_1=\sqrt{2}\), \(t_2=\sqrt{2+\sqrt{2}}\), \(t_3=\sqrt{2+\sqrt{2+\sqrt{2}}}\), and so on.

Answer: \pi


Question: Evaluate the double integral: \[ \frac{1}{2\pi} \iint_{\mathbb{R}^2} e^{\frac{-x^2}{2}} e^{\frac{-y^2}{2}} \, dA. \]

Answer: 1


Question: Consider performing a Lagrange interpolation for the points $x_0$, $x_0 + \varepsilon$, and $x_1$. As $\varepsilon \to 0$, what happens to the interpolation polynomial $P(x) = y_0L_0(x) + y_1L_1(x) + y_2L_2(x)$? Determine the degree of the resulting polynomial.

Answer: 2


Question: Find all positive integer solutions to the equation \((n+1)^n = 2n^k + 3n + 1\).

Answer: (3, 3)


Question: Differentiate the implicit function \( \sec\left(\frac{x^2-y^2}{x^2+y^2}\right) = e^a \) with respect to \( x \) and find \( \frac{dy}{dx} \).

Answer: \dfrac{y}{x}


Question: Consider a matrix of order $N$ with all distinct entries. Determine whether the sum of determinants of all possible matrices formed from these entries is zero.

Answer: 0


Question: Evaluate the double integral \( \int\int_R \sqrt{xy - y^2} \, dx \, dy \), where \( R \) is the triangular region with vertices at \((0,0)\), \((10,1)\), and \((1,1)\).

Answer: 6


Question: How many consecutive heads must you observe when tossing a coin to be 95% confident that the coin is not fair? Assume the null hypothesis is that the coin is fair, with a mean of \( \frac{1}{2}N \) and a variance of \( \frac{1}{4}N \). Use the z-score formula \( \frac{N-\frac{1}{2}N}{\frac{1}{2}\sqrt{N}} > 1.96 \) to determine the minimum number of tosses, \( N \), required.

Answer: 4


Question: Let $A$ be an $n \times n$ matrix with independent random entries $a_{ij}$, where $E(a_{ij}) = 0$ and $D(a_{ij}) = \sigma^2$. Calculate the variance $D(\det A)$ of the determinant of matrix $A$. Provide your answer in terms of $n$ and $\sigma^2$.

Answer: n! \sigma^{2n}


Question: Determine the values of \(p\) and \(q\) in \(\mathbb{R}\) for which the integral \(\int_{0}^{\pi/2} x^p (\sin x)^q \, dx\) converges.

Answer: p + q > -1


Question: Given the functions \( f(x) = x^3 - ax \) and \( g(x) = 2x^2 - bx \) where \( a \neq b \), the curves intersect at two points, one of which is a tangent. Find the area enclosed between the curves from \( x = 0 \) to \( x = 1 \).

Answer: \dfrac{1}{12}


Question: Determine the number of terms in the sum: \( q^{-N+1} + q^{-N+3} + \ldots + q^{N+1} \).

Answer: N + 1


Question:
Consider the sets \( A = \{x^2 : 0 < x < 1\} \) and \( B = \{x^3 : 1 < x < 2\} \). Determine which of the following statements is true:

1. There is a one-to-one, onto function from \( A \) to \( B \).
2. There is no one-to-one, onto function from \( A \) to \( B \) taking rationals to irrationals.
3. There is no one-to-one, onto function from \( A \) to \( B \) which is onto.
4. There is no onto function from \( A \) to \( B \) which is one-to-one.

Answer: 1


Question: Evaluate the limit: \[ \lim\limits_{x\rightarrow\infty} \frac{(x^2-1) \sqrt{x + 2}-x^2\sqrt{x+1}}{x\sqrt{x + 1}} \]

Answer: \dfrac{1}{2}


Question: Find the degree of the splitting field of the polynomial \( x^3 - 5 \) over \( \mathbb{Q} \).

Answer: 6


Question: Determine the number of distinct real roots of the polynomial equation: \[x^4 - 4x^3 + 12x^2 + x - 1 = 0\]

Answer: 2


Question: Evaluate the line integral \( \oint_C (x+2y) \, dx + (y-2x) \, dy \) using Green's Theorem, where \( C \) is the ellipse defined by \( x = 4\cos\theta, \; y = 3\sin\theta, \; 0 \leq \theta < 2\pi \), oriented counterclockwise.

Answer: -48\pi


Question: Calculate the integral \( \frac{1}{2\pi i}\oint_{|z|=1}\frac{2(1-\cos(z))e^z}{z^4}dz \).

Answer: 1


Question:
Prove that for any non-zero vector \(a\), the following limit holds:
\[ \lim_{h \to 0} \frac{\|a+h\| - \left(\|a\| + \frac{a\cdot h}{\|a\|}\right)}{\|h\|} = 0 \]

Answer: 0


Question: Evaluate the limit of the line integral \( \int_\mathrm{arc} \frac{\exp(iz)}{z+1} \, dz \) as the radius \( R \) of the semi-circular arc in the upper half complex plane goes to infinity. Does the integral converge to a fixed constant? If so, calculate this constant.

Answer: 0


Question: Evaluate the limit \( \lim_{z \to 0} z \log(z) \), where \( z \in \mathbb{C} \) and \( \log(z) \) is the principal branch of the logarithmic function.

Answer: 0


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

Answer: \dfrac{1}{2}


Question: Compute the integral \( \int_C i^*\eta \) where \( \eta=(x^2+y^2)dz \) and \( C=\{(x,y,-1): x^2+y^2=1\} \), with \( i \) being the inclusion map.

Answer: 0


Question: Evaluate \( \lim_{x\rightarrow \infty}x\cdot \ln\left(1+\frac{1}{x}\right) \) without using l'Hôpital's Rule.

Answer: 1


Question: Evaluate the integral \( \int_0^{2\pi} \frac{\sin^2x}{5+4\cos x} \ dx \) using complex integration techniques.

Answer: \dfrac{\pi}{4}


Question: Find the number of solutions to the equation \(x_1 + x_2 + x_3 + \ldots + x_6 = 20\) where each \(x_i\) is an integer satisfying \(0 \le x_i \le 8\).

Answer: 27237


Question: Using the Dirichlet convolution where \( f = \mu \ast \mu \) (Möbius function), find \( f(24) \).

Answer: 0


Question: Let $p$ be a prime number. Determine the minimum degree of a monic polynomial $q(x)$ with integer coefficients such that $q(0), q(1), \ldots, q(p-1)$ are all $(p-1)$-powers of distinct integers.

Answer: p-1


Question:
Simplify the following nested summation expression into a polynomial expression, if possible:
\[ \sum_{i = 1}^{z-2} (z - 1) - i + \left ( \sum_{j = i+1}^{z-2} (z - 1) - j \ + \left ( \sum_{k = j+1}^{z-2} (z - 1) - k \ + \ ... \right ) \right) \]
for \(z-1\) sigmas.

Answer: 2^{z-1} - z


Question: Verify the divergence theorem for a sphere by showing that the surface integral of a vector field over the sphere equals the volume integral of the divergence of the field over the volume enclosed by the sphere.

Answer: 4\pi R^3


Question: If \([M]\in H_n(M,\mathbb{Z})\) is a fundamental class for a manifold \(M\), is \(i(M)\in H_n(M,\mathbb{C})\) a fundamental class with respect to \(\mathbb{C}\)? In other words, does the image of \([M]\) in \(H_n(M,\mathbb{C})\) generate the complex homology?

Answer: Yes


Question: Determine whether the limit \( \lim_{(x,y) \to (0,0)} \frac{(1 + x^2 y^2)^{1/3} - 1}{x^2 + y^2} \) exists. If it does, find its value.

Answer: 0


Question: Find the infimum of the set \( A = \left\{ \left| \sqrt{m} - \sqrt{n} \right| : m, n \in \mathbb{N}, m \neq n \right\} \).

Answer: 0


Question: Given that $a, b, c \in \mathbb{R}$ and $a^2 + b^2 + c^2 = 2$, find the maximum value of $a + 2b + 3c$. Use any appropriate method to determine this maximum value.

Answer: 2\sqrt{7}


Question: Evaluate the limit: $$\lim_{n \to \infty} \left\{ \frac{n!}{\prod_{k=1}^{n} H_k} \right\}^{\frac{1}{n}}$$ where $$H_k = \sum_{i=1}^{k} \frac{1}{i}.$$

Answer: \infty


Question: Find the function $f$ that is continuous and differentiable for all $x$, satisfying the functional equation $$ f(x+y)f(x-y) = f^2(x) $$ with the conditions $f(0) \neq 0$ and $f(1) = 1$. What is $f$?

Answer: 1


Question: Suppose $f(x)$ and $g(x)$ are two continuous functions on $[0,1]$ with $0 \leq f(x) \leq 1$ and $0 \leq g(x) \leq 1$ for all $x$ in $[0,1]$, and they satisfy $f \circ g = g \circ f$. If $f(x)$ is not increasing, do $f(x)$ and $g(x)$ necessarily have a common fixed point?

Answer: Yes


Question:
Consider a scenario where you have n bottles arranged in a row, each with a unique cap. The caps are placed in a box, and you randomly select a cap from the box to place on each bottle, one by one. Let X be the random variable representing the number of bottles that end up with their matching caps. Determine the probability distribution of X from the following options:

1. Binomial
2. Geometric
3. Hypergeometric
4. X doesn't have a distribution with a name.
5. None of the above

Answer: 4


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

Answer: 1


Question:
Evaluate the limit of the expression as \(x\) approaches 0:
\[
\lim_{x \to 0^+} \left( \frac{e^x+e^{-x}}{-\cos x} \cdot \frac{\sin^2x}{e^x-e^{-x}} \right)
\]

Answer: 0


Question: Given the complex matrix \( z = \left| \begin{array}{ccc} 3 + 2i & 1 & i \\ 2 & 3 - 2i & 1 + i \\ 1 - i & -i & 3 \end{array} \right| \), and the condition \( |z + \overline{z}| = k|z| \), find the value of \( k \).

Answer: 2


Question: Determine the cardinality of the set of partitions of the set of rational numbers, \( \mathbb{Q} \). Let \( \mathcal{S} = \{ P \mid P \text{ is a partition of } \mathbb{Q} \} \). Find \( \operatorname{card}(\mathcal{S}) \).

Answer: 2^{\aleph_0}


Question:
Consider the series of natural numbers defined by:
\[
a_k=\begin{cases}
\frac{2(k+1)}{3} & \text{if }(k \mod3)=2\\
1 & \text{otherwise}
\end{cases}
\]
for every \(k\in\mathbb{Z}^+\). Determine the average of the reciprocals of the elements of this series, expressed as:
\[
\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{1}{a_k}
\]

Answer: \dfrac{2}{3}


Question: Consider a weighted graph $G$ with a minimum spanning tree $M$. If a new edge $e$ is added to $G$, determine whether it is sufficient to check if the minimum spanning tree of $M \cup e$ is still $M$ to conclude that $M$ remains the minimum spanning tree of the updated graph $G$. Is this statement true or false?

Answer: True


Question: Let \( f : \mathbb{Z}_{50}^* \to \mathbb{Z}_{50}^* \) be a group homomorphism defined by \( f(3) = 31 \), where \( \mathbb{Z}_{50}^* = \{ a \in \mathbb{Z}_{50} \mid \gcd(a, 50) = 1 \} \). Given that 3 is a primitive root modulo 50, find all elements of the set \( A = \{ x \in \mathbb{Z}_{50}^* \mid f(x) = 9 \} \).

Answer: \emptyset


Question: Suppose that $\{X_n\}$ is a sequence of nonnegative bounded random variables and $\{a_n\}$ is a sequence of positive constants such that \( \limsup_{n\to\infty}\sum_{i=1}^n a_iX_i <\infty \) almost surely. Determine if it is true that \( \frac{1}{n}\sum_{i=1}^n a_i\mathsf{E}X_i\to 0 \).

Answer: True


Question: Find the extrema of the function \( u = 2(x+y)z + xy \) subject to the constraint \( xyz = \frac{1}{2} \) with \( x, y, z > 0 \).

Answer: 3


Question: Suppose $E \subset \mathbb{R}^{n}$ with $m(E) = 0$, where $m$ denotes the Lebesgue measure. Determine if $m(\{(x,y) \in \mathbb{R}^{2n}: x-y \in E \}) = 0$. Provide a justification for your answer.

Answer: 0


Question: If \( f \) is an analytic function defined on the open unit disc such that \( f(z) = 0 \) for all \( |z| = \frac{1}{2} \), does it necessarily imply that \( f \) is a constant function? Justify your answer.

Answer: Yes


Question: Determine the limit of the series \( \sum_{k=0}^{\infty}(-1)^k\frac{1}{k+1} \).

Answer: \ln 2


Question: Determine whether every measurable subset of \( \mathbb{R} \) with non-zero Lebesgue measure contains arbitrarily long arithmetic progressions. A subset \( E \) of \( \mathbb{R} \) is said to contain arbitrarily long arithmetic progressions if, for every natural number \( n \), there exist \( a, d \in \mathbb{R} \), with \( d \neq 0 \), such that \( a + kd \in E \) for all natural numbers \( k \leq n \).

Answer: Yes


Question: Determine whether the following statement is true or false: Let $X$ be a $T_1$ topological space. There exists a cofinite topological space $Y$ such that $X$ can be embedded into $Y^J$.

Answer: False


Question: If $H$ is a subgroup of $G$ such that the index $[G:H] = 3$, is $H$ necessarily a normal subgroup of $G$?

Answer: No


Question: Let $A$ be a real matrix such that $AA^{\top} = I$ and $\det A < 0$. Calculate $\det(A + I)$. \(A\) is an orthogonal matrix with $\det(A) = -1$. Determine the determinant of the matrix $A + I$. \(\det(A + I) = ?\)

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty} \sin\left((2n\pi + \frac{1}{2n\pi}) \sin(2n\pi + \frac{1}{2n\pi})\right) \]

Answer: \sin 1


Question: Let $(x_n)$ be a convergent sequence of real numbers and $(y_n)$ be any sequence of positive real numbers. Is it true that there exists a subsequence $(x_{r_n})$ such that $|x_{r_{n+1}} - x_{r_n}| < y_n$ for all $n \in \mathbb{N}$?

Answer: Yes


Question: Find the minimum value of the function \( f(a,b,c) = \max\{|a+1|,|2a+b|,|b+2c|,|c-2|\} \) for all \( a, b, c \in \mathbb{R} \).

Answer: 1


Question: Find the maximum value of \( \left(\sum_{k=1}^{d} v_k\right)^2 \) subject to the constraint \( \|\boldsymbol{v}\|_2 = 1 \), where \( \boldsymbol{v} \in \mathbb{R}^d \) and \( d > 1 \).

Answer: d


Question: Calculate the dimension of the space of continuous functions that are piecewise polynomials of degree $k$ on the interval $[a,b]$, divided into $m$ subintervals. Is the dimension $m(k+1)$?

Answer: mk + 1


Question: Calculate the value of the Laplace operator \( \Delta\left(\frac{1}{r^2}\right) \) where \( r = |x| = \sqrt{x_1^2 + x_2^2 + x_3^2} \).

Answer: \dfrac{2}{r^4}


Question: Find the integer \( n \) such that \( 133^5 + 27^5 + 84^5 + 110^5 = n^2 \).

Answer: 248832


Question:
Given the power series $A(x)=\sum_{i\geq 0}a_ix^i$ which satisfies
\[A(x)=\frac{6-x+5x^2}{1-3x^2-2x^3},\]
and the fact that the denominator can be factored into $(1-2x)(1+x)^2$, express $A(x)$ in terms of partial fractions:
\[A(x)=\frac{C_1}{1-2x}+\frac{C_2}{1+x}+\frac{C_3}{(1+x)^2}.\]
Determine the constants $C_1$, $C_2$, and $C_3$, and then find the coefficient $a_5$ in the power series expansion of $A(x)$. Use the method of partial fractions to assist in your calculations.

Answer: 73


Question: In the figure, a square $ABCD$ is given with diagonal $BD$. A segment $AM$ is drawn such that $M$ is on side $BC$ and $MC = MP$, where $P$ is the intersection of $AM$ with $BD$. Find the measure of angle $\angle BMP$. \( \alpha \).

Answer: 60^\circ


Question: Let \( f: A \rightarrow \mathbb{R} \) be a Lebesgue integrable function, where \( A \) is an open, bounded subset of \( \mathbb{R}^n \) with a smooth boundary. Suppose there exists a measurable subset \( \Omega \subseteq A \) such that \( f(x) > 0 \) for all \( x \in \Omega \) and \( \int_{\Omega} f = 0 \). Does this imply that the measure of \( \Omega \), \( \mu(\Omega) \), is zero?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\to\infty} \sqrt[n]{\sum_{k =0}^n \binom{n}{k}^t} \) for \( t \geq 1 \).

Answer: 2^t


Question: Consider two probability density functions on the interval [0,1]: \( f_0(x) = 1 \) and \( f_1(x) = 2x \). Determine the maximum possible power of a test for the null hypothesis \( H_0 : X \sim f_0(x) \) versus the alternative \( X \sim f_1(x) \), given a significance level of \( \alpha = 0.10 \).

Answer: 0.19


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

Answer: 2


Question: Determine whether the limit \( \lim_{n \to \infty} \frac{\operatorname{exp}(H_n)}{n+1} \) exists. If it does, find its value.

Answer: e^{\gamma}


Question: Does the group of invertible 2x2 matrices over the real numbers, denoted as $GL(2,\mathbb{R})$, contain a cyclic subgroup of order 5?

Answer: Yes


Question: In a non-commutative ring $R$, let $p$ and $q$ be non-trivial idempotents such that $p = pq$ and $q = qp$. Does it follow that $p = q$?

Answer: No


Question: Calculate the work done by the vector field \( \vec{F} = \frac{1}{1+x^{2}} \; \vec{j} \) to move a particle along the parabolic arc \( y = x^{2} \) where \( x \) ranges from \( 0 \) to \( 1 \).

Answer: \ln 2