Question:
Let $x \neq 0$ be a vector in $\mathbb{R}^n$ and let $a \in \mathbb{R}$ with $a > -1$. Calculate the determinant of the matrix:
\[ |x|^2 \cdot \mathrm{id} + a \, x \otimes x, \] where $|x|$ denotes the Euclidean norm of $x$ and $x \otimes x$ is the outer product of $x$ with itself.

Answer: (1 + a) |x|^{2n}


Question: For an odd perfect number $L$ given in Eulerian form as $L = q^k n^2$, where $q$ is the Euler prime, does the inequality $k < q$ always hold?

Answer: Yes


Question: Let $X \subset \mathbb{P}^n$ be a projective variety over the field $\mathbb{C}$ of complex numbers. Consider a small open neighborhood $\tilde{X} \subset \mathbb{P}^n$ of $X$ in the complex topology. Assume that $X$ is locally a (set-theoretic) complete intersection. Does there exist a smooth projective subvariety of $\mathbb{P}^n$ contained in $\tilde{X}$ with dimension equal to the dimension of $X$?

Answer: Yes


Question: Determine the norm of the operator \(A\) in the space \(\ell^1\), where \(A\) is defined by \(A(x_n)_n = \left(\sum_{k=0}^\infty \frac{a^k}{k!} x_{n+k} \right)_n\) for a constant \(a > 0\).

Answer: e^a


Question: Solve for $x$ in the equation: \( (x^{2010}+1)(1+x^2+x^4+\ldots+x^{2008}) = 2010x^{2009} \)

Answer: 1


Question: Suppose $A$ and $B$ are two non-singular matrices such that $B \neq I$, $A^6 = I$, and $AB^2 = BA$. What is the least value of $k$ for which $B^k = I$?

Answer: 63


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

Answer: 0


Question: Calculate the mass of a solid body in the first octant bounded by the planes $z = 0$, $y = 0$, $z = y$, and the parabolic cylinder $y = 4 - x^2$. The body has a density given by $\rho = xz$. Determine the mass of the body.

Answer: \dfrac{16}{3}


Question: Find the coordinates of the point where all variable chords of the curve \(y^2 = 8x\) that subtend right angles at the origin are concurrent.

Answer: (8, 0)


Question: In the given figure, M, N, and L are points of tangency. Triangle ABD is a right triangle. Calculate the measure of \( \measuredangle PBN \).

Answer: 45^\circ


Question: A lamp is positioned at the point \((-2, h)\). A light ray from the lamp touches the circle defined by \(x^2 + y^2 = 1\) at a point in the first quadrant and then stops when it hits the x-axis at the point \((1.25, 0)\). Determine the height \(h\) of the lamp.

Answer: \dfrac{13}{3}


Question: Determine the values of \( p \) for which the series \( \sum_{n=N}^\infty \frac{1}{n(\log(n))(\log\log(n))\dotsm(\log_{j}(n))(\log_{j+1}(n))^p} \) converges.

Answer: p > 1


Question: Consider a bin containing 1047 distinct numbers. You extract 10 elements without replacement, note them, and then reinsert them into the bin. This process is repeated 50 times. What is the expected number of different elements you will have seen after these 50 extractions?

Answer: 399


Question: Determine the number of isomorphism types of groups of order $p^2q$ where the Sylow-$p$ subgroup is normal and isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$, the Sylow-$q$ subgroup is not normal, and no subgroup of order $p$ is normal in the group. Assume $p=2$ and $q=3$. It is known that there is such a group isomorphic to $A_4$. How many such isomorphism types exist?

Answer: 1


Question:
Determine the asymptotic order of the function defined by the recurrence relation:
\[ T(n) = n + \sum_{k=1}^n [T(n-k) + T(k)] \]
with the initial condition \( T(1) = 1 \).

Answer: \Theta(3^n)


Question: Evaluate the limit: \[ \lim_{n\to\infty} \int_{-\infty}^{\infty} e^{-x^{2n}} \, dx \]

Answer: 2


Question: Evaluate the limit: \[ \lim\limits_{x\to 0} \frac{\int_{0}^{x^2}\ln(1+t+t^2)dt}{\int_{0}^{x}\ln(1+t^3)dt} \]

Answer: 2


Question: Find the locus of a point $P$ such that the sum of the slopes of the normals from $P$ to the hyperbola $xy = c^2$ is a constant $k$ where $k > 0$. What is the equation of this locus?

Answer: x^2 = c^2 k


Question: A and B alternately throw a pair of dice until one of them wins. A wins if he rolls a total of 7, and B wins if he rolls a total of 5. A starts the game. What is the probability that one of the dice shows a face of 3 on the last throw of the game?

Answer: \dfrac{11}{28}


Question: A uniform lamina is formed by removing a quadrant of a circle of radius $l$ from a square of side $l$, with the center of the circle at one corner of the square. Calculate the distance of the center of mass of the remaining shape from the opposite corner of the square. Express your answer in terms of $l$. (The answer should be $0.316l$.)

Answer: 0.316l


Question: Let $G$ be a group of order $p^\alpha$, where $p$ is a prime number, and let $H \lhd G$. Determine if there exists a normal subgroup of $G/H$ with order $p$.

Answer: Yes


Question: Consider the sequence \( x_n \) defined by \( x_0 \) such that \( 0 < x_0 < 1 \) and the recursive relation \( x_{n+1} = 3x_n(1-x_n) \). Determine the limit of \( x_n \) as \( n \to \infty \).

Answer: \dfrac{2}{3}


Question: Calculate the probability that the sum of the top faces observed is 10 when rolling four dice with 2, 4, 4, and 6 faces, respectively. Express your answer as a fraction.

Answer: \dfrac{5}{32}


Question:
Consider the matrix \( A(x) = \begin{pmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{pmatrix} \) where \( x \in (0, 2\pi) \). Determine which of the following statements is true:

1. \( A \) has eigenvectors in \( \mathbb{R}^2 \) for all \( x \in (0, 2\pi) \).
2. \( A \) does not have eigenvectors in \( \mathbb{R}^2 \).
3. \( A \) has eigenvectors in \( \mathbb{R}^2 \) for exactly one value of \( x \).
4. \( A \) has eigenvectors in \( \mathbb{R}^2 \) for exactly two values of \( x \).

Answer: 3


Question: Given a random sample $X_1, X_2, \cdots, X_n$ from a distribution with the probability density function $$f(x;\theta)=\frac{1}{\theta}x^{(1-\theta)/\theta}\, , 0<x<\infty\, , 0<\theta<\infty,$$ find the asymptotic variance of the maximum likelihood estimator $\hat\theta$ using the Rao–Cramér lower bound.

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


Question: Let $M$ be a finitely generated $R$-module. Is it correct to express $M$ as $M = \sum_{i \in I} M_i$, where $I$ is a finite set of integers and $M_i \subseteq M$ for every $i$?

Answer: Yes


Question: Evaluate the surface integral \( \iint_S \langle F, \eta \rangle \,d\sigma \) where \( F(x,y,z)=(xz,yz,z^2) \) and \( S \) is the upper hemisphere of radius 1 centered at the origin. Here, \( \eta \) is the unit vector perpendicular to the unit tangent vector.

Answer: \pi


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = x^3 \) for \( x \ge 0 \) and \( f(x) = 0 \) for \( x < 0 \). Determine all \( n \in \mathbb{N} \) such that the \( n \)-th derivative \( f^{(n)} \) exists on all of \( \mathbb{R} \).

Answer: 2


Question:
Is the following statement true? For a random vector $X = (X_{t_1}, X_{t_2})$ defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, is it true that the expected value of the product $X_{t_1}X_{t_2}$ is given by:
\[ \mathbb{E}[X_{t_1}X_{t_2}] = \int_{\mathbb{R}}\int_{\mathbb{R}} x_1 x_2 \, \mathrm{d}F_{t_1,t_2}(x_1, x_2), \quad (x_1, x_2) \in \mathbb{R}^2? \]

Answer: True


Question: In an oriented simplicial complex \(K\), let \(\rho = [p_0, \cdots, p_q]\) and \(\sigma = [p'_0, \cdots, p'_q]\) be \(q\)-simplices. If \(<p_0, \cdots, p_q> = <p'_0, \cdots, p'_q>\), does it follow that \(|\rho| \subseteq |\sigma|\)?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 3^-} \frac{(e^{(x+3)\ln 27})^{\frac{x}{27}} -9}{3^x-27} \] Simplify the expression to find the limit.

Answer: \dfrac{1}{3}


Question: Let $M \in k[x]^{m \times n}$ be a polynomial matrix with entries having degrees bounded by $d$. Define $K = \{ \lambda \in k[x]^n \mid M \cdot \lambda = 0\}$ as the kernel of $M$. Is there a basis of $K$ over $k[x]$ such that the degree of each basis vector (maximum of degrees of the entries) is bounded by a constant multiple of $d$? Provide a justification for your answer.

Answer: Yes


Question: Find the minimum value of \( \tan x + \tan y + \tan z \) for an acute-angled triangle \( \Delta xyz \) where \( x + y + z = \pi \). Use the method of Lagrange multipliers to solve this problem.

Answer: 3\sqrt{3}


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{I_n}{I_{n-2}} \] where \( I_n = \int_0^1 x^n \sqrt{1-x^2} \, \mathrm{d}x \).

Answer: 1


Question: Let \( f: S^1 \rightarrow S^1 \) be the mapping defined by \( f(z) = z^k \) for some integer \( k \). Describe the induced homomorphism \( f_*: \pi(S^1, 1) \rightarrow \pi(S^1, 1) \) in terms of the isomorphism \( \pi(S^1, 1) \cong \mathbb{Z} \).

Answer: k


Question: Determine the number of positive integers \( n \) such that \( n^2 - 14n + 24 \) is a prime number.

Answer: 2


Question: Consider a Hilbert space $H$ and let $a \le b$ and $k \ge 0$ be fixed. Let $x$ and $y$ be two distinct elements of $H$. Suppose $u: \mathbb{R} \rightarrow H$ is a $C^k$-embedding connecting $x$ and $y$, such that the derivatives up to order $k$ vanish at infinity. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a $C^k$-map with support in $[a,b]$. It is possible to construct a $C^k$-map $\tilde{f}: H \rightarrow \mathbb{R}$ satisfying $\tilde{f}(u(s)) = f(s)$ for any $s \in \mathbb{R}$. Can the map $f \mapsto \tilde{f}$ be made continuous, where the domain is the space of $C^k$-embeddings and $\mathbb{R}$-valued $C^k$-functions supported in $[a,b]$, and the codomain is the space of $C^k$-maps on $H$?

Answer: Yes


Question: Determine the interval of \(c\) such that the rational function \(f(x) = \frac{x^2+2x+c}{x^2+4x+3c}\) takes all real values for all \(x\) in its domain.

Answer: (0, 1)


Question: Let \( \mathbb{P} \) be the set of real polynomials on \( \mathbb{R} \). For \( p \in \mathbb{P} \), define \( \| p \| = \sup_{t \in [0,1]} |p(t)| \). Define \( \psi: \mathbb{P} \rightarrow \mathbb{R} \) by setting \( \psi(p) = p\left(\frac{1}{2}\right) \). Determine if \( \psi \) is continuous, and if so, find its norm.

Answer: 1


Question:
Given the function \( K(x,y) \) defined by the limit:
\[
K_x = \lim_{h \to 0} \frac{y[(x+h)(x+1)]^{2019} - y[x(x+h+1)]^{2019}}{h(x+1)^{2019}(x+h+1)^{2019}}
\]
Find the fourth-order partial derivative \( K_{xxyy} \).

Answer: 0


Question: In a split reductive group $G$ over a field $k$, with maximal split torus $T$ and Borel subgroup $B = TU$, let $	ext{Φ}^+$ be a system of positive roots for $T$ in $G$ corresponding to $B$, and let $	ext{Δ}$ be the corresponding base. For each root $\alpha \in \text{Φ}$, let $U_{\alpha}$ be the root subgroup corresponding to $\alpha$. Consider the product mapping $\phi: \prod_{\alpha \in \text{Φ}^+} U_{\alpha} \rightarrow U$, which is an isomorphism of varieties. Let $x_{\alpha}: \mathbf{G}_a \rightarrow U_{\alpha}$ be a given isomorphism of algebraic groups. If $u_1 = \phi(x_{\alpha}(a_{\alpha})_{\alpha})$ and $u_2 = \phi(x_{\alpha}(b_{\alpha})_{\alpha})$, then $u_1u_2 = \phi(x_{\alpha}(c_{\alpha}))$ for $a_{\alpha}, b_{\alpha}, c_{\alpha} \in k$. Determine if $c_{\alpha} = a_{\alpha} + b_{\alpha}$ for all $\alpha \in \text{Δ}$. Is this statement true?

Answer: True


Question: Find the limit: \[ \lim_{x \rightarrow a} \left( \frac{\sin x}{\sin a} \right)^{\frac{1}{x-a}} \] where \(a \neq k\pi\) with \(k\) an integer.

Answer: e^{\cot a}


Question: Consider the function \( f: X \to Y \) defined by \( f(x) = 1 \) for \( x < 0 \) and \( f(x) = 2 \) for \( x \geq 0 \). If both \( X \) and \( Y \) are equipped with the Sorgenfrey topology, determine whether the function \( f \) is continuous.

Answer: Yes


Question: Consider the sequence of random variables $X_1, X_2, \ldots, X_n, \ldots$ where $X_n \sim \text{Uniform}(0,n)$. Define $Y_n = e^{-X_n}$. Determine whether the sequence $Y_n$ converges in probability, and if it does, find its limit.

Answer: 0


Question: Let \( f(x) = 2x\ln(x) \). Compute \( [f^{-1}]'(2e) \).

Answer: \dfrac{1}{4}


Question: Evaluate the integral \( \int_0^\infty \ln(x)\,x^2e^{-x}\,\mathrm{d}x \).

Answer: 3 - 2\gamma


Question: Suppose $f$ is a real function defined on $\mathbb{R}^1$ which satisfies \(\lim_{h\to 0}[f(x+h)-f(x-h)]=0\) for every $x\in \mathbb{R}^1$. Does this imply that $f$ is continuous? Provide a justification for your answer.

Answer: No


Question: Determine the values of \(x\) for which the series \(\sum_{n=2}^\infty \left|\frac{x^n}{n(\log n)^2}\right|\) converges.

Answer: [-1, 1]


Question: Evaluate the limit: \[ \lim_{x\to\infty}\left(\frac{(e^{1/x}-e^{\sin(1/x)})}{x^3}\right) \div \left((x!)^{(1/x)}\right) \]

Answer: 0


Question: Determine the number of zeros of \(\sin \pi x\) within the domain \(D = \{|z - 3 - 4i| < 6\}\).

Answer: 9


Question: Determine the degree of the splitting field of the polynomial \( f(x) = x^7 - 6 \) over \( \mathbb{Q} \).

Answer: 42


Question: Find the maximum value of \(|z - \sqrt{3} - i|^2\) for \(z\) in the set \(S = \{z \in \mathbb{C} : \overline{z} = iz^2\}\).

Answer: 9


Question: Evaluate the triple integral \( \iiint\limits_D \sin(xy) - \sin(xz) + \sin(yz) \, dx \, dy \, dz \), where \( D = \{ (x,y,z): 1 \leq x^2 + y^2 + z^2 \leq 4 \} \).

Answer: 0


Question: Determine whether the number of ordered 10-tuples of positive integers $(a_1, a_2, \ldots, a_{10})$ such that \( \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_{10}} = 1 \) is even or odd.

Answer: odd


Question: Determine the values of $z \in \mathbb{C}$ for which the infinite product \( \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right) \) converges.

Answer: 0


Question: Given that \(2^{\sqrt{200}} = a^a\), determine the value of \(a^2\).

Answer: 32


Question: Find the sum of the 5th powers of the roots of the equation \(x^3 + 3x + 9 = 0\).

Answer: 135


Question: Given the sequence \((x_n)_{n \ge 1}\) defined recursively by the condition \(x_1 > 0\) and \(x_1 + x_2 + \dots + x_n = \frac{1}{\sqrt{x_{n+1}}}\), compute the limit \(\lim_{n \to \infty} n^2 x_n^3.\)

Answer: \dfrac{1}{9}


Question: Is the null-space matrix for a matrix $H \in \mathbb{F}^{m \times n}$ always in the form $[I|A]$? Answer yes or no and explain why.

Answer: No


Question: Evaluate the double integral \( \iint_{S} \ln(1+x+y)\ dxdy \), where the region \( S \subset \mathbb{R}^2 \) is defined by \( S = \{(x,y) \mid x \geq 0, y > 0, x+y \leq 1\} \).

Answer: \dfrac{1}{4}


Question: Does there exist a connected topological space $T$ with cardinality $2^{\aleph_0}$ that is nowhere path connected? Equivalently, is there a connected space $T$ such that $|T| = 2^{\aleph_0}$ and any continuous function $f: [0,1] \rightarrow T$ is constant?

Answer: Yes


Question: Consider a sequence of random variables $X_n$ representing the number of heads obtained after tossing $n$ fair coins. Determine the limit of the probability $P(X_n < \frac{n}{2} + \sqrt{n})$ as $n \to \infty$.

Answer: \Phi(2)


Question: If $z = i^{2i}$, find the magnitude $|z|$.

Answer: e^{-\pi}


Question: In how many ways can 11 fish, consisting of 9 black fish, 1 yellow fish, and 1 blue fish, be distributed to 4 distinct persons such that each person receives at least one fish?

Answer: 1776


Question: Determine whether the converse of the generalized form of Fermat's Little Theorem is true. Specifically, if for every integer $a$, $a^m \equiv a^n \pmod{p}$ implies $m \equiv n \pmod{p-1}$, where $m$ and $n$ are positive integers and $p$ is a prime number, is this statement true?

Answer: True


Question: Consider a convex polygon that is not a triangle, has no pair of parallel sides, has vertices with integer coordinates, and has sides with integer lengths. Can the perimeter of such a polygon be an odd number?

Answer: No


Question: Let $T$ and $S$ be positive compact operators such that $\text{Tr}(TS) = 0$. Prove that $TS = ST = 0$. Use the spectral decomposition of $T$ and the properties of positive operators in your proof.

Answer: TS = ST = 0


Question: Find an element $g$ in $SO(3)$ such that the centralizer of $g$, $Z(g)$, is a disconnected subgroup of $SO(3)$, whose identity component is isomorphic to $SO(2)$. Provide the element $g$ and justify why its centralizer has the described properties.

Answer: \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}


Question:
In a linear programming problem given by:
\[ \min_x c^T x, \ \ \text{s.t.} \ Ax \le b \]
suppose it is known that at an optimal solution \(x^*\), one of the constraints \(a^T x \le 1\) is satisfied with strict inequality, i.e., \(a^T x^* < 1\). Can the constraint \(a^T x \le 1\) be removed from the problem while ensuring that \(x^*\) remains optimal for the modified problem? Ignore the possibility of divergence.

Answer: Yes


Question: Determine the number of solutions to the equation \(x_1 + x_2 + x_3 + x_4 = \frac{6(10^{100}-1)}{9}\), where each \(x_i\) is a hundred-digit positive integer composed only of the digits '1' and '2'.

Answer: 6^{100}


Question: Given that $Y \sim \text{Poisson}(10)$ represents the number of people entering an elevator on the ground floor, and there are $n$ floors above the ground floor, where each person is equally likely to exit on any floor independently, compute the expected number of stops $\mathbb{E}[X]$ the elevator makes. The expected number of stops is given by the expression $n - ne^{-10/n}$. Verify this result.

Answer: n - ne^{-10/n}


Question: Evaluate the limit of the expression \( \frac{-b+\sqrt{b^2-4ac}}{2a} \) as \( a \to 0 \), where \( b \) and \( c \) are constants.

Answer: -\dfrac{c}{b}


Question: Determine the condition under which a Möbius transformation of the form \( f(z) = \frac{az+b}{cz+d} \), where \( a, b, c, d \) are real numbers, maps the upper half-plane \( \Im(z) > 0 \) to itself. Show that this occurs if and only if \( ad - bc > 0 \).

Answer: ad - bc > 0


Question: In a complete graph with an odd number of vertices $n$, every vertex has $n-1$ edges. If we orient each edge to form a tournament, is it always possible to arrange the orientations such that every vertex has the same number of incoming and outgoing edges? Provide a justification for your answer.

Answer: Yes


Question: Has the Sieve of Eratosthenes been mathematically revamped as a recursive function?

Answer: Yes


Question: Find the equation of the locus of the other point of trisection of the segment PQ, where P and Q are points on the curve $y = x - x^3$ such that the tangent to the curve at P meets the curve again at Q, and one point of trisection lies on the y-axis.

Answer: y = x - 5x^3


Question: Find the polynomial $R(z)$ such that $z^{2021}+1 = (z^2+z+1)Q(z) + R(z)$, where the degree of $R(z)$ is less than 2.

Answer: -z


Question: Find the sum of the infinite series \( \sum_{n=1}^{\infty} \frac{F_n}{2^n} \), where \( F_n \) represents the Fibonacci numbers. Express your answer as a single value.

Answer: 2


Question:
Consider the polynomial $p(z,w) = a_0(z) + a_1(z) w + \dots + a_k(z) w^k$ in $\mathbb{C}^2$, where $k \ge 1$ and $a_i(z)$ are non-constant polynomials of $z$. Determine which of the following is correct for the set $A = \{(z,w) \in \mathbb{C} \times \mathbb{C} : p(z,w) = 0\}$:

A. Bounded with empty interior.
B. Unbounded with empty interior.
C. Bounded with nonempty interior.
D. Unbounded with nonempty interior.

Answer: B


Question: Calculate the integral \( \int_\sigma \omega \) directly, where \( \omega = x_3 \, dx_1 \wedge dx_2 \) and \( \sigma(\theta,\phi) = (\sin\phi\cos\theta,\sin\phi\sin\theta,\cos\phi) \) is defined on \([0,2\pi]\times[-\pi/2,\pi/2]\).

Answer: 0


Question: Find the number of real solutions to the equation \( g(x) = 0 \), where \( g(x) = \frac{1}{x-f(1)} + \frac{3}{x-f(2)} + \frac{4}{x-f(3)} \) and \( f(x) = x^3 + 6x^2 + 12x + 108 \).

Answer: 2


Question: Consider the function \( f(n) = \int_0^1 \ldots \int_0^1 \exp\left(-\left(\sum_{i=1}^{n} x_i\right)^{-1}\right) \, dx_1 \ldots dx_n. \) Determine whether \( f(n) \) is an increasing function of \( n \).

Answer: Yes


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function satisfying the functional equation \( f(x+y^3) = f(x) + f(y)^3 \) for all \( x, y \in \mathbb{R} \). Given that \( f'(0) \ge 0 \), find the value of \( f(10) \).

Answer: 10


Question:
Find the equation of the line that is parallel to \(\frac{x}{2} = \frac{y}{3} = \frac{z}{4}\) and intersects the lines \(9x + y + z + 4 = 0 = 5x + y + 3z\) and \(x + 2y - 3z - 3 = 0 = 2x - 5y + 3z + 3\). Choose the correct option:

(A) \(7x + y + 2z + 2 = 0 = x - 2y + z + 1\)
(B) \(7x - y + 2z + 2 = 0 = x + 2y + z + 1\)
(C) \(x + 7y + z + 1 = 0 = 2x - y + 2z + 2\)
(D) None of these

Answer: D


Question: Let $M$ be a smooth, compact manifold and let $\Delta$ be the Laplacian with eigenfunctions $f_0, f_1, \ldots$ ordered by increasing eigenvalues. Consider a function $f \in C^{\infty}(M)$ with $\int || \nabla f||^2 \leq K$ and $||f||_2 = 1$. Is there a function $n(K, \epsilon)$ such that there exist coefficients $a_1, \ldots, a_{n(K, \epsilon)}$ satisfying $|| \sum_{i = 0}^{n(K, \epsilon)} a_i f_i - f ||_2 < \epsilon$?

Answer: Yes


Question: Find the value of the infinite series: \[ \sum_{i=1}^{\infty} \frac{1+2+3+\dots+i}{i!} \]

Answer: \dfrac{3e}{2}


Question: Given a coin where the probability of landing heads is $p^2$, and $p$ is uniformly distributed over $[0,1]$, calculate the probability that $p < \frac{1}{2}$ given that a head is observed when the coin is flipped.

Answer: \dfrac{1}{8}


Question: Evaluate the limit \( \lim_{x \to 1} \sum_{n=1}^\infty \frac{x^n}{n(n+1)} \).

Answer: 1


Question: Calculate \( \lim \limits_{x \to 0^{+}} (\sin x)^{e^{x}-1} \) using Taylor series.

Answer: 1


Question: Find the exact value of the infinite sum \( \sum_{n=0}^{\infty}\frac{1}{L_{n}L_{n+2}} \), where \( L_{n} \) is the \( n \)-th Lucas number, defined by \( L_{0}=2, L_{1}=1 \), and \( L_{n}=L_{n-1}+L_{n-2} \).

Answer: \dfrac{1}{2}


Question: If $U$ is an irreducible $\mathbb{C}G$-module and $u \in U$, does there exist a basis for $U$ consisting of vectors of the form $u, ug_1, \dots, ug_n$, where $g_1, \dots, g_n \in G$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\rightarrow 0}\frac{1}{n}\int_{0}^{1}\ln(1+e^{nx})dx \]

Answer: \dfrac{1}{4}


Question: Let $f:X \to Y$ be a morphism of schemes of finite type over a field $k$. Suppose $X$ is smooth, $f$ is a closed immersion, and the Zariski tangent space of $X$ at a $k$-point $x$ is isomorphic to the Zariski tangent space of $Y$ at $f(x)$. Does there exist an open neighborhood $U$ such that the restriction of $f$ to $U$ is an open immersion of $U$ in $Y$?

Answer: Yes


Question: Determine the conflict-free chromatic number \(\chi_{\text{cf}}(\mathbb{R}, \tau)\) for the real numbers \(\mathbb{R}\) with the Euclidean topology \(\tau\).

Answer: 3


Question: Determine the values of \( n \) for which insertion sort, which runs in \( 8n^2 \) steps, is faster than merge sort, which runs in \( 64n\log_2n \) steps.

Answer: 2 \leq n \leq 43


Question: Find the remainder when \(5! \times 25!\) is divided by 31.

Answer: 1


Question: Identify the group with the presentation \( \langle x, y: xy=yx, x^5=y^3 \rangle \).

Answer: \mathbb{Z}


Question: Evaluate the contour integral \( \int_\gamma \frac{e^{-z^2}}{z^2} \, dz \) where \( \gamma \) is the ellipse parameterized by \( z(t) = a \cos t + i b \sin t \) for \( 0 \leq t \leq 2\pi \), with \( a, b > 0 \).

Answer: 0


Question: Calculate the operator norm of the operator $A: E \to E$ defined by $P(x) \mapsto x P(x)$, where $E$ is the normed vector space of all polynomials with real coefficients on $[0, 1]$ with the $\ell^2$ norm.

Answer: 1


Question: Evaluate the limit: $$\lim_{y\rightarrow \infty}\int_0^1 yx^yf(x)\,dx$$ where $f \colon [0,1]\rightarrow \mathbb{R}$ is a function with a continuous derivative and $f(1)=1$.

Answer: 1


Question: Johnny buys a pack of gum each day, aiming to collect all 6 different prizes, with each pack having an equal probability of containing any of the prizes. Calculate the standard deviation of the number of days Johnny needs to buy gum packs to collect all the different prizes.

Answer: \dfrac{\sqrt{3899}}{10}


Question: Find the sum of the series: \[ \sum_{n=1}^{\infty} \left( \sin\left(\frac{(n+1)\pi}{2n+1}\right) - \sin\left(\frac{n\pi}{2n-1}\right) \right) \]

Answer: 1


Question: Determine whether the following statement is true or false: Every continuous function $f: S^1 \to S^1$, where $S^1 = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$, has a fixed point. That is, there exists a point $p \in S^1$ such that $f(p) = p$. Provide a justification for your answer.

Answer: False


Question: Let $V$ be a finite-dimensional vector space with basis $e_1, e_2, \ldots, e_n$. Consider the function $f: V \rightarrow \mathbb{R}^n$ defined such that, for each $v \in V$, $f(v) = (a_1, a_2, \ldots, a_n)$ where $v = \sum_{i=1}^n a_i e_i$. Is the function $f$ continuous?

Answer: Yes


Question:
Find the limit of the sum as \( n \to \infty \):
\[ \dfrac{1}{7n+1} + \dfrac{2^{1/2}}{7n+2} + \dfrac{3^{1/3}}{7n+3} + \cdots + \dfrac{(70n)^{1/70n}}{77n} \]

Answer: \ln 11


Question: Consider four positive integers \(a_1, a_2, a_3, a_4\), each less than 45. Determine the number of different permutations \((a_1, a_2, a_3, a_4)\) such that \(a_1 + a_2 + a_3 + a_4 < 90\).

Answer: 1845646


Question: Evaluate the integral \( \int_{0}^{1} \int_{0}^{1} \frac{1}{(1+x y) \ln (x y)} \, dx \, dy \).

Answer: -\ln 2


Question: Let $k$ be a field, $f(x) \in k[x]$ be an irreducible polynomial over $k$, and $\alpha$ be a root of $f$. If $L$ is a field extension of $k$, determine what $k(\alpha) \otimes_k L$ is isomorphic to.

Answer: L[x]/(f(x))


Question:
Evaluate the ratio of the following two definite integrals:
\[ \frac{\int _0 ^\pi x^3 \ln (\sin x) \, \mathrm{d} x}{\int _0 ^\pi x^2 \ln(\sqrt{2} \sin x) \, \mathrm{d} x} \]

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


Question: Evaluate the integral \( \int_{C}{(xyz)dx+(xy)dy+(x)dz} \) using Stokes' Theorem, where \( C \) is the closed, piecewise smooth curve formed by traveling in straight lines between the points \((0,0,0), (2,1,5), (1,1,3)\), and back to the origin, in that order.

Answer: -\dfrac{13}{12}


Question: Let $T$ be a linear transformation from a finite-dimensional real vector space $V$ to itself, such that $T^2 = I$, where $I$ is the identity transformation. Does this imply that $T$ has a real eigenvalue? Justify your answer.

Answer: Yes


Question: Let $H$ be an infinite-dimensional Hilbert space over the complex numbers $\mathbb{C}$. Consider a continuous linear functional $f: H \to \mathbb{C}$ with $f \neq 0$. Suppose $\{v_n\}_{n \in \mathbb{N}}$ is a sequence in $H$ such that $\|v_n\| \to \infty$. Is it possible that $f(v_n) \to 0$? Provide a justification for your answer.

Answer: Yes


Question: Determine the number of degrees of freedom of a rank-1 matrix $X = ab^T$, where $a \in \mathbf{R}^m$ and $b \in \mathbf{R}^n$. Express your answer as a single integer.

Answer: m + n - 1


Question:
Calculate the limit, if it exists, as \( x = (x_1, x_2, x_3) \) approaches \( (0, 0, 0) \):
\[
\lim_{x \to 0} \frac{e^{|x|^2} - 1}{|x|^2 + x_1^2 x_2 + x_2^2 x_3 + x_3^2 x_1}
\]

Answer: 1


Question: Calculate the integral \( \int_C \frac{z^2 - 2z}{(z+1)^2(z^2+4)} \, dz \), where \( C = \{ z : |z| = 4 \} \), using the residue theorem. Identify the relevant residues inside the contour \( C \) and compute the integral.

Answer: 0


Question: Given positive integers \(n\) and \(x_1, x_2, \ldots, x_n \in (0, 1)\) such that for all \(i = 1, 2, \ldots, n\), the equation \(\sum_{i \ne j}\frac{1}{x_i - x_j} = \frac{1}{1 - x_i} - \frac{1}{x_i}\) holds, calculate the value of \(\sum_{i = 1}^n x_i\).

Answer: \dfrac{n}{2}


Question: Does the quotient of a complex algebraic group by its neutral component always split? Specifically, let $G$ be a complex algebraic group and $G^\circ \subseteq G$ be the connected component containing the neutral element. Given that $G^\circ$ is a normal subgroup and $G/G^\circ$ is a finite group, does there exist a subgroup $Q \subseteq G$ such that $Q \simeq G/G^\circ$ and $G = G^\circ Q$?

Answer: Yes


Question:
Determine which of the following sets is homeomorphic to \(\mathbb{R}^2 - \{(0,0)\}\) with the usual topology:

(A) The open unit disc in \(\mathbb{R}^2\)
(B) The cylinder \(\{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2=1 \}\)
(C) The paraboloid \(\{(x,y,z) \in \mathbb{R}^3 \mid z=x^2+y^2 \}\)
(D) The paraboloid \(\{(x,y,z) \in \mathbb{R}^3 \mid z=xy \}\)

Answer: B


Question: Let $(X, Y)$ have a joint mass function given by $P(k, n) = \frac{C \cdot 2^{-k}}{n}$, for $k = 1, 2, \ldots$ and $n = 1, 2, \ldots, k$, with a suitable constant $C$. Compute the conditional expectation $E(X|Y = y)$. Determine the range of $X$ for a given $Y = y$.

Answer: y + 1


Question: Given non-zero real numbers $p, q, r, x, y, z$ such that \( px + qy + rz + \sqrt{(p^2 + q^2 + r^2)(x^2 + y^2 + z^2)} = 0 \), find the value of the expression \( \frac{py}{qx} + \frac{qz}{ry} + \frac{rx}{pz} \).

Answer: 3


Question: Determine the fundamental group of the space $P_n$ for $n > 2$, where $P_n$ is constructed by taking $P_{n-1}$ and adjoining a circle to it such that it intersects exactly one of the already present circles in exactly one point, with no point being the intersection of more than two circles.

Answer: F_n


Question:
A differentiable function \( f : \mathbb{R} \to \mathbb{R} \) satisfies the following conditions:

- \( f(2) = 5 \)
- \( f(3) = 8 \)
- \( f'(x) \le 4 \sin^2(\pi x) \) for all \( x \in \mathbb{R} \).

Determine how many values of \( s \ge 2 \) satisfy \( f(s) = s^2 \).

Answer: 1


Question: Evaluate \( \lim_{n\to\infty}\int^2_0\frac{dx}{1+x^{2n}} \).

Answer: 1


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

Answer: 0


Question: Use the matrix tree theorem to show that the number of spanning trees of the complete graph $K_n$ is $n^{n-2}$. Consider the Laplacian matrix of $K_n$ and compute the determinant of any of its cofactor matrices to verify this result.

Answer: n^{n-2}


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

Answer: |z| = 1


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

Answer: \infty


Question: Evaluate the contour integral \( \int_{|z|=2} \sqrt{z^2-1} \, dz \).

Answer: -\pi i


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

Answer: \dfrac{\pi}{5}


Question: Find the coefficient of $x^{11}$ in the expansion of $(1+x^2)^4(1+x^3)^7(1+x^4)^{12}$.

Answer: 1113


Question: Is it known that for infinitely many integers \(d\), the curve given by \(d z^2 = F_{a,b}(u,v)\) has a rational point and a Jacobian with rank at least one?

Answer: Yes


Question: Determine whether the Cartesian product of two complete graphs, $K_n \times K_n$, is a Cayley graph. Justify your answer.

Answer: Yes


Question: For what basis $b > 6$ is the number 5654 a power of a prime number?

Answer: 7


Question: Find the sum of all integer values of $m$ that satisfy the equation: \[ \left(2^m3^5\right)^m 9^7=\dfrac{\left(256\cdot3^m\right)^m}{\left(\sqrt2\right)^{14}}. \]

Answer: 7


Question: Evaluate the integral \(\displaystyle\oint_C\left(\frac{1}{\cos(z)-1}\right)dz\) where \(C\) is a circle with radius 7, oriented counterclockwise.

Answer: 0


Question: Determine the values of \(p\) and \(q\) for which the integral \(\int_0^\infty \frac{x^p}{1+x^q}\,dx\) converges, given that \(q > 0\).

Answer: -1 < p < q - 1


Question: Determine the number of non-isomorphic non-abelian groups of order 10.

Answer: 1


Question: Determine whether the following statement is true or false: If $X$ and $Y$ are Gaussian random variables and $\mathbb{E}[XY] = 0$, then $X$ and $Y$ are independent.

Answer: B


Question: Given that the characteristic polynomial of a matrix \(A\) is \(P_A(t) = t^4 - t\), where \(A \in M_{4\times4}(\mathbb{R})\), find the characteristic polynomial of \(A^2\).

Answer: t^4 - t


Question:
Evaluate the limit of the product as \( n \to \infty \):
\[
P = \lim_{n\to \infty} \prod_{i=1}^n \frac{1 - (2i + 1)a/(2n)}{1 - ia/n}
\]
where \( 0 < a < 1 \).

Answer: \sqrt{1 - a}


Question: Find the minimum positive odd integer $m$ such that $(m+4\cdot41)(m^2+4^2\cdot41^2)$ is a perfect square.

Answer: 205


Question: Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent and identically distributed random variables with finite expected value. Find the expected value $\mathbb{E}[X_{1}|(X_{1}+X_{2}+ \ldots + X_{n})=x]$, assuming that $f_{(X_{1}+X_{2}+ \ldots + X_{n})}(x) \neq 0$. Express your answer in terms of $x$ and $n$.

Answer: \dfrac{x}{n}


Question: Evaluate the limit: $$\lim_{n\to\infty}\int_{0}^{1}\left(\frac{x^2+x+1}{4}\right)^n\sin(nx)dx$$

Answer: 0


Question: Let $P(x)$ be a polynomial of degree 1996 such that $P(n) = \frac{1}{n}$ for $n = 1, 2, 3, \ldots, 1997$. Find the value of $P(1998)$. \( \text{Express your answer as a fraction.} \)

Answer: \dfrac{1}{999}


Question: Given the points on a plane: $A = (1, 1, 1)$, $B = (1, 0, 1)$, and $C = (0, 1, 1)$, and their affine transformations $f(A) = (0,0)$, $f(B) = (1, 0)$, $f(C) = (0, 1)$, find all points $x$ on the plane such that $f(x) = (1, 1)$. Provide the set of all such points.

Answer: (0, 0, 1)


Question: Find the volume of the region bounded by the paraboloid \( z = x^2 + y^2 \) and the plane \( z = 1 - x - y \).

Answer: \dfrac{9\pi}{8}


Question: Determine the cardinality of the degree of the field extension $[F(x) : F]$, where $F$ is a field and $F(x)$ is the field of fractions of the polynomial ring $F[x]$. Is it $\aleph_0$? Does it depend on the field $F$?

Answer: \aleph_0


Question: Given that \( f(x) = -9 \) at \( x = 0 \) and the \( n \)-th derivative of the function at \( x=0 \) is \( f^{(n)}(0) = 4\frac{n!}{5^n} \) for \( n \geq 1 \), find \( f(0.5) \). The function \( f(x) \) is analytical and equal to its Maclaurin series on the interval \((-5, 5)\).

Answer: -\dfrac{77}{9}


Question: Find the sum of all positive integers less than \(2n\) that are relatively prime to \(n\), where \(n\) is a positive integer.

Answer: 2n\phi(n)


Question: What is the exact distinguishing number of the Fano plane?

Answer: 3


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(1- \frac{1-e^{-\frac{x(e-1)}{n}}}{1-e}\right)^n, \] where \( x \) is a negative real number.

Answer: e^x


Question: Let $f \in C^1(\mathbb{R}, \mathbb{R})$ such that \(|f(x)| + |x||f'(x)| \to 0\) as \(|x| \to \infty\). Is it necessary that $f'(x) \in L^1(\mathbb{R})$ with respect to Lebesgue measure?

Answer: Yes


Question: Consider a metric space \(X\) and a point \(a \in X\). Suppose that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that for every homeomorphism \(f: X \to X\) with \(d_{C^0}(f, \text{id}) < \delta\), we have \(d(a, f^n(a)) < \epsilon\) for all \(n \in \mathbb{Z}\). Is it true that \(a\) is an isolated point of \(X\)?

Answer: No


Question: Find the value of \( z \) such that \( P(Z < z) = 0.01 \), where \( Z = X + Y \), and \( X, Y \sim N(0, 1) + \eta \). The random variable \( \eta \) has a discrete distribution with values 0 and -10, with probabilities 0.991 and 0.009, respectively. Is there a closed form solution for \( z \)?

Answer: -9.80


Question: Evaluate the integral \( \int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi{\det}^2\begin{Vmatrix}\sin \alpha x&\sin \alpha y&\sin \alpha z\\\sin \beta x&\sin \beta y&\sin \beta z\\\sin \gamma x&\sin \gamma y&\sin \gamma z\end{Vmatrix} \text{d}x\,\text{d}y\,\text{d}z \) where \(\alpha,\beta,\gamma\) are integers.

Answer: 6\pi^3


Question: Determine the location of the set of all numbers \( e \) such that \( \phi_e \) has a primitive recursive extension in the arithmetical hierarchy, where \( \phi_e \) is the \( e \)th partial recursive function.

Answer: \Sigma^0_3


Question: Calculate \( \iint F \cdot n \, d\sigma \) over the part of the surface \( z = 4-x^2-y^2 \) that is above the xy-plane, where \( F = x\mathbf{i} + y\mathbf{j} \). Use the Divergence Theorem to evaluate the integral over the volume bounded by the surface and the region it cuts out of the xy-plane.

Answer: 16\pi


Question: Let \( f: \mathbb{C}[x,y] \rightarrow \mathbb{C}[t] \) be a non-zero homomorphism such that \( \ker f \) strictly contains a prime ideal \( P \neq 0 \). Determine whether \( \text{im} \ f = \mathbb{C} \).

Answer: \mathbb{C}


Question: Evaluate the surface integral \( \iint_S \mathbf{A} \cdot \mathbf{n} \, \mathrm{d}S \) using the divergence theorem, where \( \mathbf{A} = y\mathbf{i} + 2x\mathbf{j} - z\mathbf{k} \) and \( S \) is the surface of the plane \( 2x + y = 6 \) in the first octant, cut off by the plane \( z = 4 \).

Answer: 108


Question: Let \( f_{n}(x) = \dfrac{1}{n} \sum_{k=0}^{n} \sqrt{k(n-k)}{n\choose k}x^{k}(1-x)^{n-k} \) for \( x \in [0,1] \) and \( n = 0, 1, 2, \ldots \). If \( \lim_{n\to \infty} f_{n}(x) = f(x) \) for \( x \in [0,1] \), determine the maximum value of \( f(x) \) on \([0,1]\). Choose from the following options: \( \text{(A) 1} \), \( \text{(B) } \frac{1}{2} \), \( \text{(C) } \frac{1}{3} \), \( \text{(D) } \frac{1}{4} \).

Answer: B


Question: Solve the recurrence relation: \[ T(n) = T(n-1) + T(n/2) + n \]

Answer: O(2^n)


Question: Evaluate the integral \( \int_{|z|=1}(z^2+2z)^{-1}dz \) and determine if the result is \( \pi i \).

Answer: \pi i


Question: Find the integer $i$ such that $0 \leq i < 15$ and $1 + a + a^2 = a^i$ in the finite field $\mathbb{F}_{16}$, where $a$ is a primitive element satisfying $a^4 = 1 + a$. What is the logarithm of $1 + a + a^2$ with base $a$?

Answer: 10


Question: Find the area contained within the curve \( \left(\frac{x}{5}\right)^2 + \left(\frac{y}{4}\right)^{2/3} = 1 \).

Answer: 15\pi


Question: Let $X_1$ be an exponential random variable with mean $1$, and $X_2$ be a gamma random variable with mean $2$ and variance $2$. If $X_1$ and $X_2$ are independent random variables, find $P(X_1 < X_2)$. Express your answer as a probability.

Answer: \dfrac{3}{4}


Question: Let $(Y,d)$ be a non-degenerate compact metric space, and let $d_H$ be the Hausdorff metric on $K(Y)$, the set of non-empty compact subsets of $Y$. Let $M$ be the maximum value of $d_H$. If $A \in K(Y)$ and $d_H(A,Y) = M$, is $A$ necessarily nowhere dense in $Y$? Provide a justification for your answer.

Answer: No


Question: Prove that for any positive integer \( n \), the inequality \( \sum_{k=1}^{n^3} \frac{1}{k^{\frac{2}{3}}} < 3n \) holds.

Answer: 3n


Question: Suppose you have 5 consecutive positive integers \( r, o, h, a, n \). The sum of \( o + h + a \) is a perfect square, and the sum of \( r + o + h + a + n \) is a perfect cube. What is the lowest possible value of \( h \)?

Answer: 675


Question: Compute the limit: \[ \lim_{x \to 0^{+}} \frac{4^{-1/x}+4^{1/x}}{4^{-1/x}-4^{1/x}} \] without using L'Hopital's rule.

Answer: -1


Question: Given a periodic function $f: \mathbb{R}^n \to \mathbb{C}$ that is locally in $L^p$, can you find a sequence of smooth periodic functions $f_n$ such that $\|f - f_n\|_{L^p((0,1)^n)} \to 0$? Assume the periodicity condition $f(x+y) = f(x)$ for any $y \in \mathbb{Z}^n$. Provide a justification for your answer.

Answer: Yes


Question: Is it possible to replace the condition \( \exists \xi \in [a,b] \) with \( \exists \xi \in (a,b) \) in the first mean value theorem of integration, given that \( f \in C[a,b] \) and \( g \in \mathcal{R}[a,b] \) with \( g \) being nonnegative?

Answer: Yes


Question: Evaluate the limit: $$\lim \limits_{x \rightarrow 0}\left(\frac{1}{x^2}-\frac{1}{\sin^2(x)}\right)$$

Answer: -\dfrac{1}{3}


Question: Determine whether there exists a 3-sheeted covering map for the torus \( S^1 \times S^1 \).

Answer: Yes


Question: In the proof of the property of the Jacobson radical, if $a \in J(A)$, then $1-a$ is invertible. Consider the step where it is shown that $(1-a)A = A$. Can we directly conclude that there exists an $x \in A$ such that $(1-a)x = 1$, thereby proving that $1-a$ is invertible?

Answer: Yes


Question: Determine the size of the conjugacy class of the permutation $(123)(456)$ in the alternating group $A_7$.

Answer: 280


Question: Let \( \{a_k\} \) be a positive non-increasing sequence such that the series \( \sum_{k=1}^{\infty} a_k \) converges. Prove that \( b_n = n a_n \to 0 \) as \( n \to \infty \).

Answer: 0


Question: Let $P(x)$ and $Q(x)$ be two power series. If $P(x)$ converges if and only if $Q(x)$ converges, does this imply that both series have the same radius of convergence?

Answer: Yes


Question: Find the maximum value of the function \( f(x,y) = x^2 + y^2 + 2x + y \) on the closed disc of radius 2 centered at the origin.

Answer: 4 + 2\sqrt{5}


Question: In a quadratic field $K$, is every prime ideal in the ring of integers $\mathcal{O}_K$ of norm either $p$ or $p^2$, where $p$ is a prime number?

Answer: Yes


Question: Let $M$ be a well-ordered set with at least two elements, and let $L$ be a well-ordered set with the same cardinality as the natural numbers. Consider the set $[L, s.f, M]$ of all functions $f: L \to M$ with finite support, where the support of $f$ is the set of all $x \in L$ such that $f(x) \neq a$, with $a$ being the least element of $M$. Is there an injection from $L$ to $[L, s.f, M]$?

Answer: Yes


Question: Determine the number of solutions of the equation \(2x^5 - 6x^3 + 2x = 4x^4 - 6x^2 + 1\) within the interval \([-2, 2]\).

Answer: 4


Question: Find the value of \( S(S(S(S(2018^{2018})))) \), where \( S(n) \) denotes the digit sum of the integer \( n \).

Answer: 4


Question: Provide an example of a function that is infinitely differentiable (belongs to \( C^{\infty} \)) but is not holomorphic.

Answer: f(z) = \overline{z}


Question: Evaluate the integral \( g(z) = \int_{C} \frac{s^2 + s + 1}{s - z} \, ds \) using the Cauchy integral formula, where \( C \) is the ellipse \( 9x^2 + 4y^2 = 36 \) traversed once in the counterclockwise direction. Find the value of \( g(4i) \).

Answer: 0


Question: Let $f$ be a decreasing continuous function satisfying the functional equation \( f(x+y) = f(x) + f(y) - f(x)f(y) \) for all \( x, y \in \mathbb{R} \), and the derivative condition \( f'(0) = -1 \). Calculate \( \int_0^1 f(x) \, \mathrm{d}x \).

Answer: 2 - e


Question: Consider a homomorphism of compact Lie groups $i:H \to G$. Let $\iota_*:R(H) \to R.\!(G)$ be the homomorphism induced by the representation $\iota_*V := \mathrm{Map}^H(G,V)$ of an $H$-representation $V$. On the other hand, the pullback of representations via $i$ gives rise to a homomorphism $i^*:R(G) \to R(H)$, and taking its transpose, we obtain a homomorphism $i_*:R.\!(H) \to R.\!(G)$. Does $\iota_*:R(H) \to R.\!(G)$ agree with $i_*:R(H) \to R.\!(G)$?

Answer: Yes


Question: Given a coin with uneven sides, where side 1 is heavier than side 2, denote the weight of side 1 by $x$ and side 2 by $y$. You flip the coin 100 times, and side 2 faces downwards 30 times. Calculate the probability $Pr(y \leq 2x | \text{side 2 faces down 30 times})$. Assume $Pr(\text{side 1 down}) = \frac{x}{x+y}$ and $Pr(\text{side 2 down}) = \frac{y}{x+y}$. Use the binomial distribution to express $Pr(\text{side 2 faces down 30 times})$ and find the conditional probability.

Answer: 1


Question:
Determine the correct statement about the monotonicity of the function \( f(x) = \max\left\{\frac{|x-y|}{x+y+1}:0\le y\le1\right\} \) for \( x \in [0,1] \):

(A) \( f \) is strictly increasing on \( \left[0,\frac{1}{2}\right] \) and strictly decreasing on \( \left[\frac{1}{2},1\right] \)

(B) \( f \) is strictly decreasing on \( \left[0,\frac{1}{2}\right] \) and strictly increasing on \( \left[\frac{1}{2},1\right] \)

(C) \( f \) is strictly increasing on \( \left[0,\frac{\sqrt3-1}{2}\right] \) and strictly decreasing on \( \left[\frac{\sqrt3-1}{2},1\right] \)

(D) \( f \) is strictly decreasing on \( \left[0,\frac{\sqrt3-1}{2}\right] \) and strictly increasing on \( \left[\frac{\sqrt3-1}{2},1\right] \)

Answer: D


Question: Use Stokes' Theorem to evaluate the line integral \( I = \int\limits_C \textbf{F} \cdot \textbf{ds} \) where \( \textbf{F} = 3zx\textbf{i} + 3xy\textbf{j} + yz\textbf{k} \) and \( C \) is the path consisting of the three edges of the triangle formed by the portion of the plane \( x + y + z = 1 \) in the first octant of 3-space. The path \( C \) is oriented counterclockwise when viewed from above. Calculate the value of the integral.

Answer: \dfrac{7}{6}


Question: Evaluate the line integral \( \oint_C F \cdot dr \) using Green's Theorem, where \( C \) is a closed \( C^1 \) path oriented counterclockwise, consisting of a piece of the curve \( y^2 = 2(x+2) \) and the vertical segment \( x = 2 \). The vector field \( F \) is given by \( F(x,y) = \left(\frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2}\right) \).

Answer: 2\pi


Question: Compute the integral \( \int_{|z|=1} |z-1| \cdot |dz| \).

Answer: 8


Question: Given a polynomial function \(f(x)\) of degree 10, which is symmetric about \(x=3\) and defined on the interval \([1,5]\), determine the number of solutions to the equation \(f(x) = f(f(x))\).

Answer: 10


Question: Find the minimum value of the integral \( \int_{-1}^{1} |x^3-a-bx-cx^2|^2 \, dx \) over all real numbers \( a, b, c \).

Answer: \dfrac{8}{175}


Question: Suppose $f(x) \in C^{\infty}$ and $g(x) = f(x) + f(2x)$ is quasianalytic, belonging to the Denjoy-Carleman quasianalytic class $C_M(U)$ for a compact neighborhood $U$ of the origin in $\mathbb{R}$. Given that the sequence $M_n$ satisfies \(\sum_{n=0}^{\infty}\frac{M_n}{(n+1)M_{n+1}}=\infty\), and there exists a constant $A$ such that for every $x \in U$ and $n \in \mathbb{N}$, \(|g^{(n)}(x)| \leq n!A^nM_n\), determine whether $f(x)$ is necessarily quasianalytic in some compact neighborhood $V$ of the origin, i.e., \(|f^{(n)}(x)| \leq n!B^nM_n\) for some constant $B$ and any $x \in V$, and $n \in \mathbb{N}$. Provide a justification for your answer.

Answer: Yes


Question: A spring with a damping constant of 9 can be held stretched 2.5 meters beyond its natural length by a force of 7.5 Newtons. If the spring is stretched 5 meters beyond its natural length and then released with zero velocity, find the mass that would produce critical damping.

Answer: \dfrac{27}{4}


Question: Determine the degree of the field extension $[\Bbb Q(\sqrt{42}, \sqrt{-42}):\Bbb Q]$. Provide your answer as a single integer.

Answer: 4


Question: Evaluate the limit \( \lim_{x\rightarrow 0}{\frac{(2+x)^x-2^x}{x^2}} \) using the concept of equivalent functions.

Answer: \dfrac{1}{2}


Question: Two perpendicular normals to a variable circle are tangent to a fixed circle \( C_1 \) with radius 2. The locus of the center of the variable circle is the curve \( C_2 \). Find the product of the maximum and minimum distances between the curves \( C_1 \) and \( C_2 \).

Answer: 4


Question: Is the group \( \mathbb{Z}_p^* = \{ 1, 2, 3, \ldots, p-1 \} \) cyclic when \( p \) is a prime number?

Answer: Yes


Question: Let $G$ be a finite group. Consider its embedding into the symmetric group $S_{|G|}$ via the Cayley embedding. Is it true that any automorphism of $G$ arises from conjugation by an element in the normalizer $N_{S_{|G|}}(G)$?

Answer: Yes


Question: Determine whether the limit \( \lim_{n \rightarrow \infty} \sum_{k = 1}^\infty \frac{(1-\epsilon(n))^k}{k} \) exists and is finite, given that \( \epsilon(n) \rightarrow 0 \) as \( n \rightarrow \infty \).

Answer: \infty


Question: Determine the class number $h(G)$ for the group $G = \textbf{PGL}_{2,K}$, where $K = F_q(x)$ is the rational function field.

Answer: 1


Question: What is the best known upper bound on the index of an abelian subgroup in a discrete subgroup of the Euclidean group of isometries, \( \isom(\mathbb{R}^n) \)?

Answer: 2^n n!


Question: Evaluate the limit \( \lim\limits_{n \to \infty}\frac{|\sin 1|+2|\sin 2|+\cdots+n|\sin n|}{n^2}. \)

Answer: \dfrac{1}{\pi}


Question: Evaluate the integral \( \int_{-\infty}^{+\infty} \left[ f(t+1) - f(t) \right] dt \) for a continuous function \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( \lim_{t \rightarrow -\infty} f(t) = l_1 \) and \( \lim_{t \rightarrow +\infty} f(t) = l_2 \).

Answer: l_2 - l_1


Question: Determine the values of \( \alpha > 0 \) for which the improper integral \( \int_{2}^{\infty} \frac{dx}{x^2 \cdot (\ln(x))^{\alpha}} \) is convergent.

Answer: \alpha > 0


Question: Evaluate the surface integral \( \iint \vec{F} \cdot d\vec{S} \) for the vector field \( \vec{F}(x,y,z) = x\hat{i} + y\hat{j} + z^4\hat{k} \), where \( S \) is the part of the cone \( z = \sqrt{x^2 + y^2} \) below the plane \( z = 1 \) and is oriented downward. Do not use Stokes' theorem in your solution.

Answer: \dfrac{\pi}{3}


Question: Evaluate the line integral \( \int_C x \, dy - y \, dx \), where \( C \) is the curve composed of a straight line segment from \((-2, 0)\) to \((0, 0)\), a straight line segment from \((0, 0)\) to \((0, -2)\), and the part of the circle of radius 2, centered at the origin, traversed counterclockwise starting from \((0,-2)\) and ending at \((-2, 0)\).

Answer: 6\pi


Question: Find the limiting value of \( \frac{\binom{n}{n/2}}{2^n} \) as \( n \to \infty \), assuming \( n \) is even.

Answer: 0


Question: Evaluate the double summation: $$\sum_{m=1}^{\infty}{\sum_{n=1}^{\infty}{\frac{m^2\cdot n}{3^m \cdot (n\cdot 3^m+m\cdot3^n)} } }$$

Answer: \dfrac{9}{32}


Question: Find the coordinate ring of the intersection of the affine curves defined by $Z(y^n-x^{n-1}+1)$ and $Z(v^n-u+u^n)$ in the projective plane $\mathbb{P}^2(k)$, where $x_0 \neq 0$ and $x_2 \neq 0$. Assume the curves are given by the coordinate rings $\mathbb{C}[x,y]/(y^n-x^{n-1}+1)$ and $\mathbb{C}[u,v]/(v^n-u+u^n)$ respectively.

Answer: \mathbb{C}[x, y]/(y^n - x^{n-1} + 1, x^n - y^{n-1} + 1)


Question:
Calculate the operator norm of the operator $A$ on the Hilbert space $L^2([0, \pi])$, where $A$ is defined by
\[ A(f)(x) = \int_0^\pi \sin(x-y) f(y) \, dy, \quad 0 \le x \le \pi. \]
Determine \( \|A\| \).

Answer: \dfrac{\pi}{2}


Question: Is there an injective function \( f: \mathcal{E} \setminus \{\emptyset\} \to \mathbb{R} \) such that \( f(U) \in U \) for all \( U \in \mathcal{E} \setminus \{\emptyset\} \), where \( \mathcal{E} \) is the collection of open sets in \( \mathbb{R} \) with respect to the Euclidean topology?

Answer: Yes


Question: Consider all the positive integers up to $10,000,000,000$ in which every digit is either $0$ or $2$. What is the total number of $0$s among their digits?

Answer: 4097


Question: Let $C$ be the Cantor ternary set. Prove that the set $C + C = \{x + y : x, y \in C\}$ is equal to the interval $[0, 2]$. Use the standard base-3 representation for elements in $C$ to show this inclusion.

Answer: [0, 2]


Question: Consider a locally compact Hausdorff space $X$ and let $C_0(X)$ be the space of continuous functions vanishing at infinity. Suppose $\{\mu_\alpha\}$ is a net of positive Radon measures on $X$ such that $\mu_\alpha \to \mu$ in the weak$^*$ topology $\sigma(M(X), C_0(X))$. Is the net $\{\mu_\alpha\}$ eventually norm-bounded, i.e., does there exist an index $\alpha_0$ such that $\sup_{\alpha \geq \alpha_0} \|\mu_\alpha\| = \sup_{\alpha \geq \alpha_0} \mu_\alpha(X) < \infty$?

Answer: Yes


Question: Find the smallest integer $N$ such that for all $n > N$, the inequality $|n^{1/n} - 1| < 0.1$ holds.

Answer: 38


Question: Does there exist a non-identically zero sequence \( A = \{a_i\}_{i=0}^\infty \) such that all the sequences \( A, SA, SSA, SSSA, \ldots \) are convergent, where \( SA \) is the sequence of partial sums of \( A \)?

Answer: No


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{(4n^7+3^n)^n}{3^{n^2}+(-9)^n\log(n)} = 1 \]

Answer: 1


Question: Let $A$ be an artinian ring and $f : X \rightarrow \bigoplus_{j=1}^{n}I_{j}$ be a monomorphism of $A$-modules, where each $I_{j}$ is injective and indecomposable. Determine if there exists an injective envelope $g : X \rightarrow \bigoplus_{t=1}^{m}I_{j_{t}}$, where $\{ j_1, \ldots, j_m \} \subset \{1, \ldots, n \}$. Provide a justification for your answer.

Answer: Yes


Question: Determine the order of the pole of the function \( \frac{\mathrm{Log}(z)}{(z-1)^3} \) at \( z=1 \).

Answer: 2


Question: Evaluate the integral \( \int_{0}^{2\pi} \cos(\cos t) \, dt \).

Answer: 2\pi J_0(1)


Question: Is there a Hausdorff compactification of the irrational numbers with its usual topology such that the remainder is a discrete space?

Answer: No


Question: Given the function \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \), let \( x_1 \) and \( x_2 \) be the points of maximum and minimum, respectively. If \( x_1^2 = x_2 \), determine the value of \( a \). Choose from the following options: \((A)\ 0, (B)\ 2, (C)\ 1, (D)\ \frac{1}{4}\).

Answer: B


Question: Let $M$ be a Riemannian manifold and $N$ be a submanifold of $M$. Let $v$ be a vector field on $N$. Consider the Levi-Civita connection on $N$ and differentiate $v$ along a curve $\gamma$ in $N$, resulting in a vector field $u$. Since $N$ is embedded in $M$, the curve $\gamma$ and the vector field $v$ can be extended to $M$. Using the Levi-Civita connection on $M$, differentiate $v$ along $\gamma$ to obtain a vector field $w$. Is the orthogonal projection of $w$ onto $T_p N$ equal to $u$ for each point $p$ on $N$?

Answer: Yes


Question: Determine the coefficient of \(x^{15}\) in the expansion of the generating function \((1+x^3+x^6+x^9+x^{12}+x^{15})(1+x^6+x^{12})(1+x^9)\). Use the fact that this coefficient represents the number of partitions of 15 into parts from the set \{3, 6, 9\}.

Answer: 5


Question:
Given the set \( N \) defined by the Axiom of Infinity as follows:
\[ N = \{ \emptyset, \{ \emptyset \}, \{ \{ \emptyset \}, \{ \{ \emptyset \} \} \}, \{ \{ \{ \emptyset \}, \{ \{ \emptyset \} \} \}, \{ \{ \{ \emptyset \}, \{ \{ \emptyset \} \} \} \}, \ldots \} \]
which represents the natural numbers, and the Axiom of Regularity which states that for every non-empty set \( x \), there exists a member \( y \) that is disjoint from \( x \), identify the set \( y \) for \( N \).

Answer: \emptyset


Question: Let $S = \{1, 2, 3, \ldots, n\}$. Consider the set $T$ of all ordered triples of subsets $(A_1, A_2, A_3)$ such that $A_1 \cup A_2 \cup A_3 = S$. Determine, in terms of $n$, the value of \( \sum_{(A_1, A_2, A_3) \in T} |A_1 \cap A_2 \cap A_3| \), where $|X|$ denotes the number of elements in the set $X$. Provide your answer as a formula in terms of $n$. \( \boxed{} \)

Answer: n \cdot 7^{n-1}


Question: Evaluate the line integral \( \int 3yz \, dx + z^3 \, dy + 3yz^2 \, dz \) along the curve parameterized by \( r(t) = (\sin^2 t - \cos^2 t) \cdot \hat{i} + \sin t \cdot \hat{j} + \cos t \cdot \hat{k} \) for \( 0 \leq t \leq 2\pi \), which lies on the surface \( x = y^2 - z^2 \).

Answer: 3\pi


Question: Determine whether the function \( f(x,y) = x^2 + y^2 - \alpha \log(x^2y^2) \) is convex in its domain.

Answer: \alpha \geq 0


Question: Let $f(x)$ be a four-times differentiable function such that $f(2x^2-1)=2xf(x)$. Find the value of $f''''(0)$.

Answer: 0


Question: Let $V$ be a finite dimensional vector space over the finite field $\mathbb{F}_q$ with $q$ elements, and let $U$ be a subspace of $V$. Determine the number of subspaces $W \subset V$ such that $W \cap U = \{0\}$ and $V = W + U$. Provide your answer as a single integer.

Answer: q^{k(n - k)}


Question: Determine the convergence order of the Newton-Raphson method when the derivative at the root, \( f'(x_r) \), is zero.

Answer: 1


Question: Find the product of all quadratic nonresidues in $\mathbb{Z}_{103}$.

Answer: 102


Question: Calculate the limit: \[ \lim_{n \to \infty} \sqrt[n]{\frac{(3n)!}{n!(2n+1)!}} \]

Answer: \dfrac{27}{4}


Question: Determine the maximum number of vertices in a 5-regular planar graph.

Answer: 12


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

Answer: -\dfrac{1}{3}


Question:
Given the joint density function of continuous random variables $X$ and $Y$:
\[ f(x,y) = \frac{3}{4}(2-x-y) \cdot \mathsf{1}_{(0,2)^3}(x,y,x+y) \]
find the expected value of the total $X+Y$. Express your answer as a single number.

Answer: 1


Question: Find the area of the region inside the cardioid \( r = 1 + \cos \theta \) and outside the circle \( r = 3 \cos \theta \) using a double integral. The points of intersection are at \( \theta = \pi/3 \) and \( \theta = -\pi/3 \).

Answer: \dfrac{\pi}{4}


Question: Compute the limit \( \lim_{h\to 0} \frac{f(a+h)+f(a-h)-2f(a)}{h^2} \) given that the function \( f \) is twice differentiable at \( a \).

Answer: f''(a)


Question: Consider a carnival game that starts with two dollars in a bowl. Each turn, a coin is flipped. If the result is heads, the carnival doubles the money in the bowl. If the result is tails, you take the money out of the bowl. What is the expected value of the money you would receive from this game?

Answer: \infty


Question: Is the sequence \((\{10^n\pi\})_{n=1}^\infty\) dense in the interval \([0,1]\)? Here, \(\{x\} = x \mod 1 = x - \lfloor x \rfloor\) represents the fractional part of \(x\).

Answer: Yes


Question: Find the norm of the operator \( T: \ell^2 \to \ell^2 \) defined by \( T(x_1, x_2, x_3, \ldots) = (0, 0, 3x_1, 2x_2, 4x_3, 3x_4, 2x_5, 4x_6, \ldots) \).

Answer: 4


Question: Determine the number of solutions of the equation \(3z^5 + z^2 + 1 = 0\) that lie in the annular region where \(1 < |z| < 2\).

Answer: 0


Question: Find the expectation of a Gamma distribution with shape parameter \( \alpha \) and scale parameter \( \beta \).

Answer: \alpha \beta


Question: Find the area that is inside the curve \( r = 3\cos(\theta) \) and outside the cardioid \( r = 1 + \cos(\theta) \).

Answer: \pi


Question: Determine the number of integer values of $p$ for which the equation \( 3p \sin x - (p + \sin x)(p^2 - p \sin x + \sin^2 x) = 1 \) has a solution for $x$.

Answer: 4


Question: Evaluate the double integral \( \int^{10}_0 \int^0_{-\sqrt{10y-y^2}} \sqrt{x^2+y^2} \,dx\,dy \) by changing to polar coordinates.

Answer: \dfrac{2000}{9}


Question: Find the smallest integer \( k \) such that \( \frac{k}{10^5} \leq \arctan(0.1) < \frac{k+1}{10^5} \).

Answer: 9966


Question: Find the last three digits of the number \(5^{5^{5^{5^5}}}\).

Answer: 125


Question: Evaluate the limit \( \lim_{x\to 0}\frac{(\tan x)^{2008}-(\arctan x)^{2008}}{x^{2009}} \) without using Taylor series.

Answer: 0


Question: Find the Taylor series expansion of \( \ln(x) \) around \( x = 3 \). Determine the interval of \( x \) for which this series is valid.

Answer: (0, 6]


Question: What is the probability that four points placed uniformly at random in the unit square of \(\mathbb{R}^2\) form a convex quadrilateral?

Answer: \dfrac{25}{36}


Question: Let \( f(x) = 2\arccos x + 4\operatorname{arccot} x - 3x^2 - 2x + 10 \) for \( x \in [-1, 1] \). If \([a, b]\) is the range of \( f(x) \), find \( 4a - b \).

Answer: 11 - \pi


Question: Find an integer $x$ such that $x^3 \equiv 3 \pmod{257}$, given that 3 is a primitive root modulo 257.

Answer: 147


Question: Determine the value to which the series \(\sum_{n=0}^{\infty} \frac{n^{x}}{n!}\) converges.

Answer: e B_x


Question:
Let \( A = (a_{ij}) \) be an \( n \times n \) matrix. Show that for \( i \neq j \), the following holds:
\[ a_{i1}C_{j1} + a_{i2}C_{j2} + \cdots + a_{in}C_{jn} = 0 \] where \( C_{ij} \) is the cofactor of \( a_{ij} \).

Answer: 0


Question: Find the residue at $z=0$ of the function $f(z)=\frac{\cot z}{z^4}$. Express your answer as a fraction.

Answer: -\dfrac{1}{45}


Question: Let $(X, \mu)$ be a finite measure space and $f : X \to \mathbb{R}$ be a measurable function. Assume $\mu(f > 0) > 0$. Determine whether it is true that for every $\epsilon > 0$, there exists a $\delta > 0$ such that $\mu(0 < f < \delta) < \epsilon$. Provide a proof or counterexample.

Answer: True


Question: Determine whether the tensor product \( \mathbb{Z}[G] \otimes_{\mathbb{Z}} \mathbb{Q} \) is isomorphic to \( \mathbb{Q}[G] \) as a \( \mathbb{Q} \)-algebra.

Answer: Yes


Question: Does there exist a power series \( s(z)=\sum\limits_{k=0}^{\infty}a_{k}z^{k} \) such that the set \( D=\{z\in\mathbb{C}:|z|=1, \;s(z) \text{ diverges} \} \) is not a countable union of arcs (closed, open, or otherwise)?

Answer: Yes


Question: Let \( D \) be an indefinite quaternion algebra over \( \mathbb{Q} \). Given an isomorphism \( \iota : D \otimes_{\mathbb{Q}} \mathbb{R} \cong \mathrm{M}_2(\mathbb{R}) \), if we choose another isomorphism \( \iota' : D \otimes_{\mathbb{Q}} \mathbb{R} \cong \mathrm{M}_2(\mathbb{R}) \), is it true that \( \iota(D) = a\,\iota'(D)\,a^{-1} \) for some element \( a \in \mathrm{GL}_2(\mathbb{R}) \)? Justify your answer using the Skolem-Noether theorem.

Answer: Yes


Question: Given a non-invertible $n \times n$ matrix $A$, is it possible to make $A$ invertible by changing exactly $n$ entries? Provide a justification for your answer.

Answer: Yes


Question: Given that \( 2\sin a \sin b + 3\cos b + 6\cos a \sin b = 7 \), find the value of \( \sec^2 a + 2\sec^2 b \).

Answer: 12


Question: Determine whether the integral \( \int_0^1 (1-x)\ln(x) \, dx \) converges. If it does, find its value.

Answer: -\dfrac{3}{4}


Question:
In a chess tournament with \( n \) players participating in a single round-robin format, each player earns 1 point for a win, 0.5 points for a draw, and 0 points for a loss. The total number of points distributed in the tournament is \( \frac{1}{2}n(n-1) \). At the end of the tournament, the following conditions are met:

- The winner has \( w \) points.
- The second-placed player has \( s \) points.
- The remaining \( n-2 \) players each have \( d \) points.
- The sum of the winner's and second-placed player's points is \( w + s = 8 \).

Determine the possible values of \( n \).

Answer: 9


Question: Find the explicit sum of the series \( \sum_{n=1}^\infty n(n+1)\left(\frac{1}{2}\right)^n \).

Answer: 8


Question: Let $Y$ be a real normed linear space. If there exists a Banach space $X$ such that there is a continuous linear open mapping from $X$ to $Y$, is $Y$ necessarily a Banach space?

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\infty} e^{-x} f(x) \, dx \), where the function \( f(x) \) is defined by the series \( f(x) = \sum_{n=0}^{\infty} (-1)^n \frac{1}{(2n!)^2} (2015x)^{2n} \).

Answer: \cos(2015)


Question: Calculate the sum of Legendre symbols: \( \left(\frac{1}{73}\right) + \left(\frac{2}{73}\right) + \cdots + \left(\frac{72}{73}\right) \).

Answer: 0


Question: Let $q$ be a prime power and let $\mathbb{F}_q$ be the finite field with $q$ elements. Consider the field extension $\mathbb{F}_{q^n}$, which can be viewed as an $n$-dimensional vector space $V$ over $\mathbb{F}_q$. The general linear group $GL(V)$ acts on $V$. For any element $x$ in $\mathbb{F}_{q^n}$, multiplication by $x$ defines an element $g_x$ in $GL(V)$. Suppose an element $h \in GL(V)$ commutes with $g_x$ for some $x \neq 0, 1 \in \mathbb{F}_{q^n}$. Does this imply that $h = g_y$ for some $y \in \mathbb{F}_{q^n}$?

Answer: No


Question: Evaluate the limit \( \lim_{z\rightarrow 0}\left(\frac{\sin z}{z}\right)^{1/z^{2}} \) where \( z \) is a complex number.

Answer: e^{-1/6}


Question:
Let $(M^n,g)$ be a complete Riemannian manifold with $|Rm| \le 1$. Determine whether there exist two positive constants $C$ and $\epsilon$, depending only on $n$, such that in the normal coordinates $(g_{ij})$ with respect to any point $p \in M$, the inequality
\[ |
abla_k g_{ij}(x)| \le C \]
holds for any $|x| \le \epsilon$. Consider the case where the injectivity radius at $p$ is small, and the estimates should be understood for the pull-back of $g$ to the tangent space.

Answer: Yes


Question: Compute the equation that defines the quotient variety $V/(\mathbb{C}^*)^2$, where $V = \{ (x,y,z) \in \mathbb{A}^3\ |\ x = yz \}$ and the action of $(\mathbb{C}^*)^2$ on $V$ is given by $(\lambda, \mu) \star (x,y,z) = (\lambda \mu\ x , \lambda\ y,\mu\ z)$.

Answer: 0


Question: If \( z = 3+i \), find all integer values of \( n \) for which \( \text{Im}(z^n) = 0 \).

Answer: 0


Question: Find the remainder when \(8^{13^{48}}\) is divided by 1000.

Answer: 808


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{(\tan(x) - x)^2}{(\ln(x+1) - x)^3} \]

Answer: -\dfrac{8}{9}


Question: Determine the radius of convergence of the series \( \sum a_n z^n \) where \( a_n = \frac{(n!)^3}{(3n)!} \).

Answer: 27


Question: Evaluate the limit: \[ \lim_{x \to 0^+}(1+3x)^\frac{-2}{3x^3} \]

Answer: 0


Question: Given the polynomial \(x^2 + Ux + V\), where \(U\) and \(V\) are independent and uniformly distributed on \([0, 1]\), find the probability that the polynomial has two different real roots.

Answer: \dfrac{1}{12}


Question: Evaluate the integral \( \int_{0}^{2\pi} \log|1-ae^{i\theta}| \,\mathrm{d}\theta \) for \( a \leq 1 \).

Answer: 0


Question: Find all prime numbers \( p \) such that there exist natural numbers \( a \) and \( b \) for which \( a^3 + b^3 = p^2 \).

Answer: 3


Question: Compute the integral \( \int_{\mathbb{R}} \int_{\mathbb{R}} \chi_S(x,y)\, d c(x)\, d \lambda(x) \) where \( S = \{ (t,t) \in \mathbb{R}^2, t \in [0,1]\} \), \( c \) is the counting measure, and \( \lambda \) is the Lebesgue measure.

Answer: 1


Question: Consider a function \(N:[0,1]\to [0,1]\) called a negation, which satisfies the conditions \(N(0)=1\), \(N(1)=0\), and is non-increasing \((x \leq y \Rightarrow N(x) \geq N(y))\). If \(N\) is continuous, is there a unique balance point where \(N(a)=a\)? Justify your answer.

Answer: Yes


Question: Is the fourth cohomology group $H^4(PSL(2,\mathbb{Z}),\mathbb{Z})$ known?

Answer: \mathbb{Z}/6\mathbb{Z}


Question:
Determine the number of solutions to the equation $f(x) = g(x)$, where $f(x) = x^2 - \frac{\cos x}{2}$ and $g(x) = \frac{x\sin x}{2}$. Choose the correct option:

(A) $f(x) = g(x)$ for more than two values of $x$
(B) $f(x) \neq g(x)$, for all $x \in \mathbb R$
(C) $f(x) = g(x)$ for exactly one value of $x$
(D) $f(x) = g(x)$ for exactly two values of $x$

Answer: D


Question: Suppose $X$ and $Y$ are two independent non-negative random variables such that $E[X] = E[Y] = 1$. Determine the value of $\sup_X \{ \inf_Y \{P(X > Y)\}\}$.

Answer: \dfrac{1}{2}


Question: Determine for which values of $k$ the number $M(k) = 100^{k-1} + 100^{k-2} + \cdots + 100 + 1$ is a prime number. Here, $M(k)$ consists of exactly $k$ ones in its decimal representation.

Answer: 2


Question: Compute the value of \(5^{288} \bmod 577\).

Answer: 576


Question: Given a reduced finite type \( \bar{k} \)-algebra \( R \), a projective morphism \( \pi : V \rightarrow \mathrm{Spec}(R) \), and ideals \( I, J \subseteq R \), assume there is a split \( s_{IJ} : \mathrm{Spec}(R/IJ) \rightarrow V \times_{\mathrm{Spec}(R)} \mathrm{Spec}(R/IJ) \) of \( \pi \). The sections \( s_{J} : \mathrm{Spec}(R/J) \rightarrow V \times_{\mathrm{Spec}(R)} \mathrm{Spec}(R/J) \), \( s_{I} : \mathrm{Spec}(R/I) \rightarrow V \times_{\mathrm{Spec}(R)} \mathrm{Spec}(R/I) \), and \( s_{I+J} : \mathrm{Spec}(R/I+J) \rightarrow V \times_{\mathrm{Spec}(R)} \mathrm{Spec}(R/I+J) \) lift to sections \( \hat{s}_{J} : \mathrm{Spec}(\hat{R}_J) \rightarrow V \times_{\mathrm{Spec}(R)} \mathrm{Spec}(\hat{R}_J) \), \( \hat{s}_{I} : \mathrm{Spec}(\hat{R}_I) \rightarrow V \times_{\mathrm{Spec}(R)} \mathrm{Spec}(\hat{R}_I) \), and \( \hat{s}_{I+J} : \mathrm{Spec}(\hat{R}_{I+J}) \rightarrow V \times_{\mathrm{Spec}(R)} \mathrm{Spec}(\hat{R}_{I+J}) \) on the completions. Does \( s_{IJ} \) lift to a morphism \( \hat{s}_{IJ} \)?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\rightarrow\infty}n^k\int_0^n(\sin x\sin\frac x{2^2}...\sin\frac x{n^2})dx \) for any \( k \in \mathbb{N}^\ast \).

Answer: 0


Question: Evaluate the expression \(10\sqrt{5}\,\alpha\), where \(\alpha = \lim_{n\to\infty}\sum_{m = n^2}^{2n^2}\dfrac{1}{\sqrt{5n^4 + n^3 + m}}.\)

Answer: 10


Question: Determine the number of elements in the quotient module \( \mathbb{Z}[1/6]/(2020)\mathbb{Z}[1/6] \).

Answer: 505


Question: Express the function \( g(x) \) in terms of \( f(x) \) given the equation \( f(x) = g(x) \cdot g(-x) \).

Answer: g(x) = \sqrt{f(x)}


Question: Let $M$ be a left $R$-module, where $R$ is a ring with unity $1_R$. Suppose $N \subseteq M$ is a submodule and $x \in M - N$ for some $x \in M$. Prove that there exists a maximal submodule $L$ of $M$ such that $L \supseteq N$ and $x \not\in L$.

Answer: L


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{\ln\left(1+\sin^2(2x)\right)}{1-\cos^2(x)}. \]

Answer: 4


Question: Determine whether the following statement is true or false: A subset $A$ of a metric space $X$ is precompact if and only if every sequence in $A$ has a Cauchy subsequence.

Answer: True


Question: Calculate \( \lim_{m \rightarrow \infty} \int_{[0, 1]} f_m(x) \, d\lambda(x) \), where \( f_m(x) = ((1 - x^p)^{1/p})^m \) and \( p \in (0, \infty) \).

Answer: 0


Question: Compute the limit: $$\lim_{n\to\infty} \frac{1}{\log(n)}\sum _{k=1}^n \frac{\cos (\sin (2 \pi \log (k)))}{k}$$

Answer: J_0(1)


Question:
Show that the circle $r = 0$ is a periodic orbit with period $2\pi$ for the differential equations given by:
\[ \dfrac{dr}{dt} = r(1 + a\cos \theta - r^2) \\
\dfrac{d \theta}{dt} = 1 \]
where $a$ is a positive number less than $1$. Assume $r = 0$ and determine if it is a periodic orbit.

Answer: 2\pi


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

Answer: 2x^2 - 1


Question: Consider a truly random binary string of 64 bits, where all the ones and zeros are independent and identically distributed (IID) variables. If the Fisher-Yates shuffle is applied to this string to rearrange the bits, do the bits remain IID variables after shuffling?

Answer: Yes


Question:
Let $f:[0,1]\to[0, 1]$ be a twice differentiable function satisfying $f(ax+(1-a) y) \leq af(x)+(1-a)f(y)$ for all $x, y \in [0, 1]$ and $a \in [0, 1]$. Determine which one of the following is true for all $x \in (0, 1)$:

A. $f'(x) \geq 0$.
B. $f''(x) \geq 0$.
C. $f'(x) \leq 0$.
D. $f'''(x) \leq 0$.

Choose the correct option.

Answer: B


Question: Determine whether the function \( f(\tau, x) = \int_{-\infty}^{+\infty} \frac{dp}{\sqrt{p^2+1}} e^{-\sqrt{p^2+1}\tau + ipx} \), where \( \tau > 0 \) and \( x \in \mathbb{R} \), is invariant under the group of rotations \( SO(2) \), i.e., whether it depends only on \( \tau^2 + x^2 \).

Answer: Yes


Question: In an exchange economy with two types of goods and n consumers, half of the consumers have a utility function given by \(U(x)= 5\ln{x} +m \) and the other half have \(U(x) = 3\ln{x} + m\). Each consumer has an initial endowment of 20 units of good \(x\) and 10 units of good \(m\). Determine the maximum amount of good \(x\) that the first type of consumers can obtain at a Pareto Optimal allocation, given the constraint \(m > 0\) for all consumers.

Answer: 25


Question: Suppose $A=(a_{jk})_{j,k=1}^n$ is a symmetric complex-valued matrix, meaning $a_{jk}=a_{kj}$ for all $j,k=1,\dotsc,n$. Given any two linearly independent vectors $\alpha=(\alpha^j)_{j=1}^n, \beta=(\beta^j)_{j=1}^n \in \mathbb{C}^n$ that satisfy $\sum_{j=1}^n(\alpha^j)^2=\sum_{j=1}^n(\beta^j)^2 =0$, it holds that $A(\alpha,\beta)=0$. Does it follow that $A=0$?

Answer: No


Question: Given that \( \lambda = \frac{41}{10}\left(\frac{1}{2^2-1}+\frac{1}{4^2-1}+\frac{1}{6^2-1}+\ldots+\frac{1}{40^2-1}\right) \), where \( w \) is a cube root of unity other than 1, find the value of \( w + w^\lambda \).

Answer: -1


Question: Evaluate the line integral \( \int_{C} \frac{y\,dx - (x+1)\,dy}{x^2 + y^2 + 2x + 1} \) where \( C \) is the curve \( |x| + |y| = 4 \) oriented counterclockwise for a full revolution.

Answer: -2\pi


Question: Show that the sequence \( \left\{\frac{s_n}{n}\right\} \) converges, given that \( s_{m+n} \leq s_m + s_n \) and \( s_n \geq 0 \) for all \( m, n \in \mathbb{N} \). Use the hint: Let \( a = \lim \inf \frac{s_n}{n} \), then show \( \frac{s_n}{n} \rightarrow a \).

Answer: a


Question: Let $(R, \mathfrak m)$ be a complete local normal domain of dimension $2$ with an algebraically closed residue field $R/\mathfrak m$ of characteristic $0$. Assume that Spec$(R)$ has a rational singularity, and let $\pi: X \to \text{Spec}(R)$ be the minimal resolution of singularities with the exceptional divisor $E=\pi^{-1}(\mathfrak m)$. If $E$ has exactly one irreducible component, is it true that $R$ is a cyclic quotient singularity?

Answer: Yes


Question:
Find the greatest integer less than or equal to the sum of the reciprocals of the square roots of the integers from 1 to 100:
\[ \left\lfloor \sum_{k=1}^{100}\frac{1}{\sqrt{k}} \right\rfloor \]

Answer: 18


Question:
Given that \( \sum_{i=1}^n f(i) \sim n \log n \) and \( f \) is a monotonically increasing function that is always positive, does it follow that \( f(n) \sim \log n \)?

Note: \( A(n) \sim B(n) \) means:
\[ \lim_{n\to\infty } \frac{A(n)}{B(n)}=1 \]

Answer: Yes


Question: Let $X$ be a simply connected space with $H_2(X; \mathbb{Z}) \cong \mathbb{Z}/5$, $H_3(X; \mathbb{Z}) \cong \mathbb{Z}$, and all higher homology groups zero. Prove that there exists a CW-complex $Z$ with one $0$-cell, one $2$-cell, and two $3$-cells, and a weak homotopy equivalence $Z \to X$. Use the relative Hurewicz theorem to justify your answer.

Answer: Z \to X


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

Answer: \dfrac{\pi}{4}


Question:
Consider an almost Hermitian manifold $(M, \mathcal{J}, g)$, where $\mathcal{J}^2 = -I$ and $g(\mathcal{J} X, \mathcal{J} Y) = g(X, Y)$. Suppose for any local orthonormal $\mathcal{J}$-frame $\{X_i, \mathcal{J}X_i\}$, the following conditions hold for $i \neq j$:
\[ g(Q\mathcal{J}X_i, \mathcal{J}X_i) = g(QX_j, X_j), \quad K(X_i, X_j) = K(X_i, \mathcal{J}X_i), \]
where $Q$ is the Ricci operator and $K$ is the sectional curvature. Can it be deduced that $(M, \mathcal{J}, g)$ is of constant curvature?

Answer: Yes


Question: Determine if there exists a differentiable function \( f(x, y) \) for vectors \( x \) and \( y \), such that \( f(x, y) = -f(y, x) \), where the output of \( f \) is a single real number.

Answer: Yes


Question: How many triangles with integer altitudes 6, 8, and an unknown integer value X can be formed?

Answer: 20


Question:
Which of the following functions is a solution to the integral equation \( u(x) = x + \int_{0}^{x} (t-x)u(t)dt \)?

(A) \( \sin x \)

(B) \( x \cos x \)

(C) \( \ln (x+1) \)

(D) \( x e^{-x} \)

(E) \( xe^x \)

Answer: A


Question: Determine the smallest integer $n$ such that the group $Q_8 \times Q_8$ is isomorphic to a subgroup of the symmetric group $S_n$.

Answer: 16


Question: Let \( A \) be a bounded subset of \( \mathbb{C}^d \). If the closure \( \overline{A} \) is convex, is \( A \) necessarily convex? Provide a justification for your answer.

Answer: No


Question: Compute the surface integral \( \iint_Y F \cdot N\, dS \) using Gauss's theorem, where \( Y = (x-z)^2 + (y-z)^2 = 1 + z^2 \) for \( 0 \leq z \leq 1 \) and \( N \) is the outward-pointing normal vector. The vector field is given by \( F = (y, x, 1 + x^2z) \).

Answer: -3\pi


Question: Determine the maximal number of members, or an upper bound, in a family of ordered set partitions $C_1, \dots, C_m$ of $[n]$ with exactly $k$ blocks, such that for each $j=1, \dots, k$, the union $B_{1j} \cup \cdots \cup B_{mj}$ is a partition of $[n]$. Consider the case $k=2$ specifically.

Answer: 2


Question: Consider the complex function \( f(z) = z^2 \). If the domain is restricted such that \( f(re^{i\theta}) = r^2e^{2i\theta} \) with \( 0 \leq \theta < \pi \), does this restriction make the function bijective?

Answer: Yes


Question: Is the set of continuously differentiable functions $C^1([0,1])$ dense in the space of continuous functions $C([0,1])$ with respect to the supremum norm?

Answer: Yes


Question: Estimate the number of random integers you need to check before finding a prime number with 100 digits.

Answer: 230


Question: Determine whether any field automorphism of the $p$-adic numbers $\mathbb{Q}_p$ is continuous.

Answer: Yes


Question:
Determine whether there exists a $C^\infty$ monotonically non-decreasing function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f((-
fty, -2]) = \{-1\}$, $f([2, \infty)) = \{1\}$, and $f(x) = x$ near $0$. Provide a justification for your answer.

Answer: Yes


Question: Find the limit: \[ \lim_{(x,y) \to (0,0)} \frac{\sin^2(xy)}{3x^2+2y^2} \]

Answer: 0


Question: Find the minimum value of the function \( g(x) = \max_{0 \leq y \leq 1} |y^2 - xy| \).

Answer: 3 - 2\sqrt{2}


Question: Let \( f_n(x) = \sum_{r=1}^{n} \frac{\sin^2 x}{\cos^2 \frac{x}{2} - \cos^2 \frac{(2r+1)x}{2}} \) and \( g_n(x) = \prod_{r=1}^{n} f_r(x) \). Define \( T_n = \int_{0}^{\pi} \frac{f_n(x)}{g_n(x)} \, dx \). If \( \sum_{r=1}^{10} T_r = k\pi \), find the value of \( k \).

Answer: 5


Question: Determine the values of $a_1$ for which the power series \(-1 + a_1x + a_2x^2 + a_3x^3 + \ldots\) converges for all $x \in \mathbb{C}$, where $a_0 = -1$ and $a_n = (-1)^{(n+1)} \frac{n^{n-1}}{(n-1)!} a_1^n$ for $n \geq 1$. Consider the condition $|a_n| \leq \frac{c}{n!}$ for some constant $c$ to determine convergence.

Answer: 0


Question: Given that \(\lim_{n \to \infty} \sum^n_{k =0} \frac{nC_k}{n^k(k+3)} =ae+b\), find the value of \(a-b\).

Answer: 3


Question: Determine whether the sequence $x_n = (1+1/n)^n$ converges in the metric space $(X, d)$, where $d(x, y) = \frac{2|x-y|}{3+2|x-y|}$. If it converges, find its limit.

Answer: e


Question: Let $R$ be a commutative ring with $1$ and $I$ be an ideal in it. Consider $I^{j}/I^{j+1} \cong R/I$ for any ideal $I$ in ring $R$. Let $\overline{\alpha} \in I^j\setminus I^{j+1}$ and define $\theta\colon R \to I^j/I^{j+1}$ by $\theta(x)=\overline{\alpha x}$. Determine whether $I=\ker(\theta)$. Provide a proof or counterexample.

Answer: I = \ker(\theta)


Question: Calculate the present value of a 10-year deferred life annuity-due of one dollar per annum at Kimberly's issue age, given an interest rate of 3%. Use the formula for a deferred annuity and provide the final present value.

Answer: 25.55


Question:
Let $P(x)$ be a non-zero polynomial of degree $N$. Determine the radius of convergence of the power series $\sum_{n=0}^{\infty} P(n)x^n$. Choose the correct option:

(a) It depends on $N$.
(b) It is $1$ for all $N$.
(c) It is $0$ for all $N$.
(d) It is $\infty$ for all $N$.

Select the correct answer.

Answer: B


Question: Given a group homomorphism $\phi: G \to G_0$ where $|G| = 1013$ and $|G_0| = 55$, determine the possible orders of the image of $G$ under $\phi$. Justify your answer using group theory principles.

Answer: 1


Question: Find a simplified expression for the integral \( \int_a^\infty \Phi(a - x) f(x) \, dx \), where \( f \) is the probability density function of a normal distribution with mean \( a \) and variance \( 1 \), and \( \Phi \) is the standard normal cumulative distribution function.

Answer: \dfrac{1}{8}


Question: Find the sum of the series \( \sum_{n=0}^{\infty}\binom{2n}{n}\left(\frac{1}{5}\right)^n \).

Answer: \sqrt{5}


Question: Consider 1-dimensional CW-complexes $A$ and $B$, both having a fundamental group isomorphic to $\mathbb{Z}$. Does the inclusion map $A \hookrightarrow B$ induce an isomorphism on their fundamental groups?

Answer: Yes


Question: Evaluate the contour integral \( I = \oint_{\gamma} e^{\frac{1}{z}} z^{-3} \, dz \), where \( \gamma \) is the circle centered at \( z = 0 \) with radius 2.

Answer: 0


Question: Calculate \( 555^{555^{555}} \mod 1000 \).

Answer: 875


Question:
Calculate the value of \( \frac{dy}{dx} \) at \( x = -1 \) for the equation:
\[
(\sin y)^{\sin(\frac{\pi x}{2})}+ \frac{\sqrt{3}}{2}\sec^{-1}(2x)+ 2^x\tan (\log_e(x+2))=0
\]

Answer: \dfrac{3}{\pi \sqrt{\pi^2 - 3}}


Question: Given the function \( y = e^{2\cos^{-1}x} \), find the value of \( \lambda \) such that the differential equation \((1-x^2)y'' - xy' - \lambda y = 0\) is satisfied.

Answer: 4


Question: Find an irreducible polynomial of degree 7 over the field \( \mathbb{Z}_2 \) to construct a finite field with 128 elements.

Answer: x^7 + x + 1


Question: Let \( f(x) = x^3 - 4x^2 + 4x \) and its graph be denoted by \( C \). For any \( x_1 \neq \frac{4}{3} \), the tangent to \( f(x) \) at the point \( P_1(x_1, f(x_1)) \) intersects \( C \) at another point \( P_2(x_2, f(x_2)) \). Similarly, the tangent to \( f(x) \) at \( P_2 \) intersects \( C \) at another point \( P_3(x_3, f(x_3)) \). Let \( S_1 \) and \( S_2 \) be the areas between \( C \) and the line segments \( P_1P_2 \) and \( P_2P_3 \), respectively. Find the value of \( \frac{S_1}{S_2} \).

Answer: \dfrac{1}{16}


Question: Consider a probability space $(\Omega,\mathcal A,\operatorname P)$ and a sequence of random variables $(X_n)_{n\in\mathbb N}\subseteq\mathcal L^1(\operatorname P)$ with $X\in\mathcal L^1(\operatorname P)$. Suppose $\left|\operatorname E\left[X_n-X\right]\right|\xrightarrow{n\to\infty}0$. Let $Y:\Omega\to\mathbb R$ be a bounded and $\mathcal A$-measurable function. Does it follow that $\left|\operatorname E\left[(X_n-X)Y\right]\right|\xrightarrow{n\to\infty}0$?

Answer: No


Question: Is the Lebesgue measure of a closed set with empty interior in \(\mathbb{R}^{n}\) always zero?

Answer: No


Question: Determine the number of zeros of the function \( f(z) = z^5 + iz^3 - 4z + i \) in the annular region \( 1 < |z| < 2 \).

Answer: 4


Question: Determine the sets where the function \( f(x,y) = \begin{cases} e^{\frac{1}{x^2+y^2-1}}, & \text{if } x^2+y^2<1 \\ 0, & \text{if } x^2+y^2\ge 1 \end{cases} \) is continuous.

Answer: \mathbb{R}^2


Question: Find the number of integer solutions to the inequality \(x + y + z \leq 13\) subject to the constraints \(x \leq y \leq x+2\) and \(x \leq z\). Use generating functions to determine the number of possibilities.

Answer: 105


Question: Consider the self-adjoint matrix \( A_X = \begin{pmatrix} 0 & -i \\ i & X \end{pmatrix} \), where \( i \) is the imaginary unit and \( X \) is a uniformly distributed random variable on the interval \([-\varepsilon,\varepsilon]\). Define the product \( M_n = A_{X_n} \cdot \ldots \cdot A_{X_1} \), where \( X_1, \ldots, X_n \) are independent and identically distributed copies of \( X \). Given that \( \operatorname{det}(M_n)=(-1)^n \), \( M_n \) has an eigenvalue with modulus at least 1. Consider the Lyapunov exponent \( \mu = \lim_{n \to \infty}\frac{1}{n} \log \Vert M_n\Vert \). Is it true that \( \mu > 0 \)?

Answer: Yes


Question: Let \( f \in \mathcal{S}(\mathbb{R}) \) be a positive function. Find the point \( \xi \) where the function \( \xi \mapsto |\hat{f}(\xi)|^2 \) reaches its maximum value.

Answer: 0


Question: Given a signal $s(t)$ sampled at $1000$ observations, where $h(k) = s(k\Delta t + t_0)$ for $k = 0, 1, \dots, 999$, with $\Delta t = 1/200$ and $t_0 = -2$ seconds, the discrete Fourier transform $|\hat{h}(m)|$ has its largest values at $m = 193$ and $m = 807$. What is the corresponding positive frequency in hertz for the signal $s$, rounded up to the nearest integer?

Answer: 39


Question:
Consider the sequence \(a_n = n\big((ea)^{\frac {1}{n}}- a^{\frac {1}{n}}\big)\) for \(n \geq 1\) and positive \(a\). Determine \(\lim_{n \to \infty} a_n\). Choose the correct answer from the following options:

A. \(\lim_{n \to \infty} a_n\) does not exist
B. \(\lim_{n \to \infty} a_n = e\)
C. \(\lim_{n \to \infty} a_n = 0\)
D. None of the above

Answer: D


Question: Design an algorithm that allows you to find your room in an infinite corridor by walking at most \(O(n)\) steps, where \(n\) is the unknown number of steps between your initial position and your room. You will only know you have reached your room when you are directly in front of it.

Answer: O(n)


Question: In a clinical trial, only 0.1% of new treatments are actually effective. Given a significance level of 0.05 and a power of 0.8, calculate the proportion of tests that declare a new treatment effective which are actually false positives.

Answer: 0.984


Question:
Evaluate the Fourier series \( S_f(x) \) at \( x = \frac{\pi}{2} \) for the function \( f: [-\pi, \pi] \to \mathbb{R} \) defined by
\[
x \mapsto
\begin{cases}
0  & \text{if } -\pi \le x \le -\pi /2 \\
1 & \text{if } -\pi/2 < x \le \pi /2 \\
0 & \text{if } \pi/2 < x \le \pi
\end{cases}
\]
The function is extended periodically to the whole of \( \mathbb{R} \), and the Fourier series is given by:
\[
S_f(x) = \frac{1}{2} + \sum_{n=1}^\infty \frac{2}{\pi n} \sin\left(\frac{\pi n}{2}\right) \cdot \sin(nx)
\]
Find \( S_f\left(\frac{\pi}{2}\right) \).

Answer: \dfrac{1}{2}


Question: Let \( f(x) \) be a function that is continuous and differentiable everywhere, satisfying the functional equation \( f(x+y) = f(x)f(y) \) for all real numbers \( x \) and \( y \). Given that \( f'(0) = 2 \), find the value of \( f'''(0) \).

Answer: 8


Question: Determine whether the integral operator \( I: L^1([0,1]) \to L^1([0,1]) \), defined by \( f \mapsto \left(x \mapsto \int_0^x f \,\mathrm{d}\lambda\right) \), is compact.

Answer: Yes


Question:
Consider a sequence $(u_n)$ that is bounded in the Hilbert space $H^1(\mathbb{R}^N)$. Verify whether the following convergence results hold, up to a subsequence:
\begin{eqnarray}
&&u_n \rightharpoonup u\ \mbox{ weakly in }H^1(\mathbb{R}^N),\\
&&u_n \rightarrow u \mbox{ in }L_{loc}^s(\mathbb{R}^N)\mbox{ for all } 2\leq s< 2^*,\\
&&u_n(x) \rightarrow u(x) \mbox{ for a.e. }x \in  \mathbb{R}^N,\\
&& \nabla u_n(x) \rightarrow \nabla u(x)\mbox{ for  a.e. } x \in  \mathbb{R}^N,\\
&&\nabla u_n \rightarrow \nabla u \mbox{ in } \bigg(L_{loc}^s(\mathbb{R}^N)\bigg)^N\mbox{ for all }  2\leq s<2^*.\\
\end{eqnarray}

Answer: 1, 2, 3


Question: Given a set of ordered pairs \( S = \{(x_1, y_1), (x_2, y_2), \ldots, (x_k, y_k)\} \) where \( 0 < x_i < y_i \leq n \) for some upper bound \( n \), determine the length of the longest subset of \( S \) such that all numbers within the ordered pairs are unique. For example, if \( S = \{(1,2), (1,3), (1,4), (2,3), (3,4)\} \), the longest subset with unique numbers is \{(1,4), (2,3)\}, which has a length of 2.

Answer: 2


Question:
Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Determine which of the following statements implies that $f(0) = 0$:

(A) $\lim_{n \to \infty} \int_{0}^{1} f(x)^n \, dx = 0.$

(B) $\lim_{n \to \infty} \int_{0}^{1} f(x/n) \, dx = 0.$

(C) $\lim_{n \to \infty} \int_{0}^{1} f(nx) \, dx = 0.$

(D) None of the above.

Choose the correct option.

Answer: B


Question: Find the maximum value of the function \( f(x,y) = 5x^2 - 22xy + 5y^2 + 8 \) within the disk defined by \( x^2 + y^2 \leq 25 \).

Answer: 408


Question: Let $g$ be a real-valued function defined on the interval $(-1,1)$ such that \[ e^{-x}(g(x)-2e^x)=\int_0^x \sqrt{y^4+1}\,dy \] for all $x\in (-1,1)$. Let $f$ be another function such that \[ f(g(x))=g(f(x))=x. \] Find the value of $f'(2)$. \( \)

Answer: \dfrac{1}{3}


Question: Evaluate the integral \( \int_C \frac{1}{z^4+1} \, dz \), where \( C = \{ z \mid |z-1| = 2 \} \).

Answer: 0


Question: Calculate the integral \( \int\limits_{\overline{B^3(1)}} \left(\frac{\partial X_1}{\partial x_1} + \frac{\partial X_2}{\partial x_2} + \frac{\partial X_3}{\partial x_3}\right) \, dx_1 \, dx_2 \, dx_3 \) where \( X(x) = x \) for each \( x \in S^2 \) and \( \overline{B^3(1)} = \{x \in \mathbb{R}^3 \mid \|x\| \leq 1\} \).

Answer: 4\pi


Question: Find the smallest positive integer \( n \) such that the symmetric group \( S_n \) contains an element of order 60.

Answer: 12


Question: Find the limit of \( n!^{1/n^2} \) as \( n \to \infty \) using the Squeeze Theorem.

Answer: 1


Question: Let \( P(n) = \sum_{m=1}^\infty \left(\prod_{k=0}^n \frac{1}{m+k}\right) \) and \( Q(n) = \sum_{k=1}^n \frac{1}{P(k)} \). Find the last 4 digits of \( Q(2015) + 2015 \).

Answer: 2014


Question: Calculate \( \lim_{n \rightarrow \infty} \int_1^\infty \frac{\sqrt{x}}{1 + nx^3} \, d\lambda(x) \).

Answer: 0


Question: Determine the number of ways to distribute 5 distinguishable toys and 7 indistinguishable sweets among 4 distinguishable children, ensuring that each child receives at least one item.

Answer: 89420


Question:
Let \( A \) and \( B \) be \( n \times n \) matrices such that \( \text{trace}(A) < 0 < \text{trace}(B) \). Consider the function \( f(t) = 1 - \det(e^{tA + (1-t)B}) \). Determine which of the following statements is true:

1) \( f(t) \) has infinitely many zeros in \( 0 < t < 1 \).
2) \( f(t) \) has at least one zero in \( \mathbb{R} \).
3) \( f(t) \) has no zeros.
4) \( f(t) \) has either no zeros or infinitely many zeros in \( \mathbb{R} \).

Answer: 2


Question: Given that $Y_1$ and $Y_2$ have a bivariate normal distribution with $E(Y_1|Y_2) = 4.7 - 0.16Y_2$ and $E(Y_2|Y_1) = 0.8 - Y_1$, and a conditional variance of 3.64, find the correlation coefficient between $Y_1$ and $Y_2$. Provide your answer as a numerical value.

Answer: -0.4


Question: Let $U$ be an open set in $\mathbb{R}^n$. If $x \in U$ and $y \in U^c$, must some convex combination of $x$ and $y$ lie on the boundary of $U$? Justify your answer.

Answer: Yes


Question: Let $G$ be a finite group and $P$ a Sylow $p$-subgroup of $G$ contained in the center of $G$. Does there exist a normal subgroup $N$ of $G$ such that $P \cap N = \{e\}$ and $PN = G$?

Answer: Yes


Question: Given that $f$ and $g$ are homogeneous functions of degree $k$, determine if the composition $f(g)$ is homogeneous. If it is, find the degree of homogeneity of $f(g)$. Provide a justification for your answer.

Answer: k^2


Question: Let \( f : \mathbb{R}^+ \to \mathbb{R} \) be an increasing function such that \( x f(x) + 2 > 0 \) and \( f(x) f\left( \frac{x f(x) + 4}{x} \right) = 1 \). Find the value of \( \lfloor f'(1) \rfloor \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function.

Answer: 1


Question: For each natural number \(m\), determine whether there exists a prime number \(p > m\) such that the interval \([p-m, p+m]\) contains no other prime numbers.

Answer: Yes


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

Answer: 1


Question: Determine whether the Hessian matrix of the function \( f: \mathbb{R}^n \to \mathbb{R} \) defined by \( f(x) = \sqrt{\|x\|^2+\epsilon^2} \), where \( \epsilon = 0.00001 \), is positive definite for all \( x \in \mathbb{R}^n \).

Answer: Yes


Question:
Consider the sequence \(\{a_n\}_{n=1}^{\infty}\) defined recursively by
\[a_{n+1} = \sqrt{n^2 - a_n}\] with \(a_1 = 1\). Compute \(\lim_{n\to\infty} (n-a_n)\).

Answer: \dfrac{3}{2}


Question: Determine all functions \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that, for every positive integer \( n \), the inequality \( 2n+2001 \leq f(f(n)) + f(n) \leq 2n+2002 \) holds.

Answer: f(n) = n + 667


Question: Compute the partial derivative \( u_{xy} \) of the function \( u(x,y) = \frac{xy(x^2-y^2)}{x^2+y^2} \) at the point \((0,0)\), given that \( u(0,0) = 0 \).

Answer: -1


Question: What is the surface area of a 4-dimensional sphere (hypersphere) with radius \( r \)?

Answer: 2\pi^2 r^3


Question: Evaluate the integral \( \int_{\mathbb{R}^2} |f(x,y)| \, dx \, dy \) where \( f(x,y) = 2(x-y)e^{-(x-y)^2} \) if \( x > 0 \), and zero otherwise. You may use the fact that \( \int_{-\infty}^{\infty} e^{-z^2} \, dz = \sqrt{\pi} \).

Answer: \infty


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

Answer: \dfrac{2}{3}


Question: Find the sum of all integers \( n \) such that \( \frac{1}{4} < \sin\frac{\pi}{n} < \frac{1}{3} \).

Answer: 33


Question: If a subset $A$ of the natural numbers $\mathbb{N}$ has a lower density of $0$, does it follow that the lower density of its complement $\mathbb{N} \setminus A$ is $1$?

Answer: No


Question: The stock prices of two companies at the end of any given year are modeled with random variables $X$ and $Y$ that follow a distribution with joint density function $f(x, y) = 2x$ for $0<x<1, x<y<x+1$ and $0$ otherwise. Calculate the conditional variance of $Y$ given $X=x$. Provide your answer in terms of $x$.

Answer: \dfrac{1}{12}


Question: Given a $3 \times 3$ matrix $C$ with singular value decomposition $C = USV^T$, where $U = [u_1 \hspace{1mm} u_2 \hspace{1mm} u_3]$ and $V = [v_1 \hspace{1mm} v_2 \hspace{1mm} v_3]$ are orthogonal matrices, and $S$ is a diagonal matrix of singular values, determine if the matrix 2-norms of $u_iv_i^T$ are equal to 1 for each $i=1,2,3$. Provide a mathematical justification for your answer.

Answer: 1


Question: Let \( f \in C^1([0,1]) \) with \( f'(0) \neq 0 \). For \( x \in (0,1] \), define \( \theta(x) \) by the equation \( \int_0^x f(t) \, dt = f(\theta(x))x \). Determine \( \lim_{x \to 0^{+}} \frac{\theta(x)}{x} \).

Answer: \dfrac{1}{2}


Question: Determine the number of solutions for the equation \(\underbrace{x*x*\ldots*x}_{x\text{ 10 times}}=\frac{1}{10}\) using the binary operation \(x*y=\dfrac{2xy+3(x+y)+2}{3xy+2(x+y)+3}\) for \(x, y \in (-1, 1)\).

Answer: 1


Question: Evaluate the Lebesgue integral: $$ \int_{[0,\infty)}\dfrac{1}{\lfloor{x+1}\rfloor\cdot\lfloor{x+2}\rfloor}d\lambda(x). $$

Answer: 1


Question: In the permutation group $S_n$ with $7 \leq n \leq 10$, consider subgroups $H_1$ and $H_2$ where $H_1 \cong S_k$ and $H_2 \cong S_{n-k}$. If every element of $H_1$ commutes with every element of $H_2$, are $H_1$ and $H_2$ the stabilizers of $n-k$ and $k$ letters, respectively? Answer true or false.

Answer: True


Question: Determine the number of subgroups isomorphic to \( \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \) in the group \( G \) isomorphic to \( \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/9\mathbb{Z} \times \mathbb{Z}/81\mathbb{Z} \).

Answer: 13


Question: Is the space of compactly-supported orientation-preserving homeomorphisms on \(\mathbb{R}^n\), denoted as \(Homeo_c(\mathbb{R}^n)\), dense in the space of all orientation-preserving homeomorphisms on \(\mathbb{R}^n\) to itself, denoted as \(Homeo_0(\mathbb{R}^n)\), when equipped with the topology of compact convergence?

Answer: Yes


Question:
Let $f$ be a real-valued function defined on the interval $(-1,1)$ such that:
\[ e^{-x}\cdot f(x)= 2 + \int_{0}^{x} \sqrt{t^4+1} \, dt \quad \text{for} \; x \in (-1,1) \]
Let $f^{-1}$ be the inverse of $f$. What is the value of $(f^{-1})'(2)$?

Answer: \dfrac{1}{3}


Question: Determine the values of \(a \in \mathbb{R}\) for which the integral \(\int_0^{\infty} \frac{x^{a-1}}{1+x}\, dx\) converges.

Answer: (0, 1)


Question: Consider a graph with 8 vertices labeled $1, 2, \ldots, 8$. Each edge is independently present with probability $p$. The degree of vertex $i$, denoted as $D_i$, is the number of edges that include vertex $i$. Calculate $\text{Corr}(D_i, D_j)$, the correlation between $D_i$ and $D_j$.

Answer: \dfrac{1}{7}


Question: Given a sequence \( \{x_{n}\} \) of positive real numbers where \( 0 < x_{n} < 1 \) and \( \lim_{n\to\infty}x_{n}=0 \), determine if there exists a subsequence \( x'_{n} \) of \( x_{n} \) such that \( \lim_{n\to\infty}\frac{x'_{n}}{x_{n}}=c \) for some nonzero constant \( c \).

Answer: Yes


Question: Determine the values of \(a \in \mathbb{R}\) for which the integral \(\int_{0}^{\frac{\pi}{2}} \log(\cos(x)) \cdot (\tan(x))^a \, dx\) converges.

Answer: (-3, 1)


Question: Determine which integers between 00 and 99, inclusive, occur as the last two digits in the decimal expansion of infinitely many terms of the sequence \( \{a_n\} \) defined by \( a_1 = 3 \) and \( a_{i+1} = 3^{a_i} \) for \( i > 0 \).

Answer: 87


Question:
Compute the line integral over the smooth Jordan curve \( \gamma \) in \( \mathbb{R}^2 \setminus \{(0,0)\} \) from \((1,0)\) to \((1,0)\), which winds once around the origin in the clockwise direction:
\[ \int_{\gamma}\frac{y}{x^2+y^2}dx-\frac{x}{x^2+y^2}dy \]

Answer: 2\pi


Question: Evaluate the line integral of the vector field \( \vec{F} = -e^{-x}\sin y\,\vec{i} + e^{-x}\cos y\,\vec{j} \) over the closed path of a square in the \(xy\)-plane with vertices at \((0,0), (1,0), (1,1), (0,1)\).

Answer: 0


Question: What is the largest amount that cannot be paid using any combination of 4 paise, 7 paise, and 11 paise coins?

Answer: 17


Question: Consider a hyperkähler manifold $X$ and its associated twistor space $T = S^2 \times X$ with the tautological complex structure. For each point $x \in X$, there is a holomorphic map $u_x: S^2 \to T$ given by $u_x(\theta) = (\theta, x)$. Is every holomorphic map $u: S^2 \to T$ that satisfies ${\rm pr}_1 \circ u = {\rm id}_{S^2}$ of the form $u_x$ for some $x \in X$?

Answer: Yes


Question: Transform the complex function \(f(x, y) = (x^3 - 3xy^2) + i(3x^2y - y^3) + iC\) into a function of \(z\) only, where \(z = x + iy\). Express \(f(z)\) without involving \(\overline{z}\).

Answer: z^3 + iC


Question: Evaluate the line integral \(\int_C(2ye^{x^2-z}\cos(y^2)-9xy^2)dy+(12z-e^{x^2-z}\sin(y^2))dz+(2xe^{x^2-z}\sin(y^2)-3y^3)dx\), where \(C\) is the broken line path from \(A(0, \sqrt{\pi}, 3)\) to \(B(0, \sqrt{\frac{\pi}{2}}, -1)\) passing through the points \((0, \sqrt{\pi}, 3)\), \((1,3,5)\), and \((0,\sqrt{\frac{\pi}{2}},-1)\).

Answer: e - 48


Question: Suppose $k$ is an algebraically closed field, and let $f \in k(\mathbb{A}^1)$ be a rational function over the affine line $\mathbb{A}^1$. If $f^2 \in k[\mathbb{A}^1]$, does it follow that $f \in k[\mathbb{A}^1]$? Justify your answer.

Answer: Yes


Question:
Given the ordinary differential equation \( y'' - 2xy' + (\sin x^2)y = 0 \), let \( y_1 = \phi(x) \) and \( y_2 = \psi(x) \) be solutions such that \( \phi(0) = 1, \phi'(0) = 1 \) and \( \psi(0) = 1, \psi'(0) = 2 \). Determine the value of the Wronskian \( W(\phi, \psi) \) at \( x = 1 \). Choose the correct answer from the following options:

- \( 0 \)
- \( 1 \)
- \( e \)
- \( e^2 \)

Answer: C


Question: Find \( \lim_{n\to \infty} \sum_{r=1}^n \frac{6n}{9n^2-r^2} \).

Answer: \ln 2


Question: Find the sum of the infinite series: \[ \frac{1}{3!} + \frac{4}{4!} + \frac{9}{5!} + \ldots \]

Answer: 2e - 5


Question: Let $G$ and $H$ be groups such that gcd$(|G|,|H|)=1$. If $f: G \rightarrow H$ is a homomorphism, determine the kernel of $f$. Provide a justification for your answer.

Answer: G


Question: Find the first three 7-adic digits of a root of the polynomial \(x^3 - 1\) in \(\mathbb{Z}_7\) that is not equal to 1.

Answer: 246


Question: Given that radioactive decay of an element occurs according to a Poisson process with a rate of 10,000 decays per second, calculate the approximate probability that the millionth decay occurs within 100.2 seconds.

Answer: 0.9772


Question: Let \( G \) be a group with \(|G| = 51\). Determine the number of elements \( a \in G \) such that the order of \( a \), denoted \( o(a) \), is 3.

Answer: 2