Question: Consider the sets $A=\{a_{1},a_{2},\ldots,a_{n}\}$, $B=\{b_{1},b_{2},\ldots,b_{n}\}$, and $C=\{c_{1},c_{2},\ldots,c_{n-1}\}$, which are discrete finite sets embedded in the unit sphere $S^2=x^2+y^2+z^2=1$. The topologies of $A$, $B$, and $C$ are induced from the topology of the sphere. Assume the points of $B$ are embedded in a great circle on $S^2$ with equal arc lengths between consecutive points, and similarly for the points of $C$. Define the function $f_{0}: A \rightarrow B$ by $f_{0}(a_{i})=b_{i}$ for $i=1,2,\ldots,n$, and the function $f_{1}: A \rightarrow C$ by $f_{1}(a_{1})=f_{1}(a_{n})=c_{1}$ and $f_{1}(a_i)=c_{i}$ for $i=2,3,\ldots,n-1$. Is it possible to define a homotopy $F: A \times [0,1] \rightarrow S^2$ between $f_{0}$ and $f_{1}$?

Answer: Yes


Question: In an abelian category \( \mathcal{A} \) that contains all colimits, let \( \mathcal{P} \) be a full subcategory of generating objects, which are assumed to be projective. Is it true that for any object \( X \in \mathcal{A} \), there exists an epimorphism \( \bigoplus_{i \in I} P_i \twoheadrightarrow X \) with all \( P_i \) in \( \mathcal{P} \)? Assume the definition of "generating" is that for any two morphisms \( f, g: X \to Y \) such that \( f \circ p = g \circ p \) for any morphism \( p: P \to X \) with \( P \in \mathcal{P} \), we have \( f = g \).

Answer: Yes


Question: A polyhedron is composed of P regular pentagons and H regular hexagons, where at each vertex, two hexagons and one pentagon meet. Using Euler's formula, determine the total number of faces of the polyhedron.

Answer: 32


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

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


Question: Let \( A = \{ 2013^{30n^2 + 4n + 2013} : n \in \mathbb{N} \} \). Determine the set \( X \) of remainders when dividing all numbers in \( A \) by 21.

Answer: \{9, 15, 18\}


Question: Calculate the residue of the function \( \frac{z^{2}}{(z+i)^{2}(z-i)^{2}} \) at the pole \( z = i \).

Answer: -\dfrac{i}{4}


Question: Let $\{f_n\}$ be a sequence of holomorphic functions defined in a domain $D \subset \mathbb{C}$. Suppose that $f_n$ converges uniformly to a function $f$. Is $f$ holomorphic in $D$?

Answer: Yes


Question: Is there a sequence \((f_{n})\) of \(L^{1}\) functions such that \(\lim_{n \to \infty}\int_{0}^{1}|f_{n}| = 0\) and \(\limsup_{n \to \infty} f_{n}(x) = 1\) for all \(x \in [0, 1]\)?

Answer: Yes


Question: Determine the number of closed subsets of \(\mathbb{R}\) up to homeomorphism.

Answer: 2^{\aleph_0}


Question: Evaluate the limit: \[ \lim_{x \to +\infty} (3x+1)(\log(2x)-\log(2x-1)) \] without using l'Hospital's rule.

Answer: \dfrac{3}{2}


Question: Convert the decimal number 44 to its representation in base -10.

Answer: 164


Question: Let $F$ be an orderable field, and let $\langle E, \leq \rangle$ be an ordered subfield of $F$. Can $F$ be ordered in a way that extends the ordering on $E$?

Answer: Yes


Question: In triangle $ABC$, point $P$ is located inside such that $\angle PBC = 30^\circ$, $\angle PBA = 8^\circ$, and $\angle PAB = \angle PAC = 22^\circ$. Determine the measure of $\angle APC$ in degrees.

Answer: 142


Question: Find a basis for the direct complement of the subspace \( L = \{(z_1, \ldots, z_n) \in \mathbb{C}^n : \text{Re}(z_i) - \text{Im}(z_i) = 0, \ i = 1, \ldots, n\} \) in \( \mathbb{C}^n \). Determine the dimension of this complement.

Answer: n


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

Answer: 0


Question: Find the supremum of $|f(2)|$ for all holomorphic functions $f: P \to B_1(0)$ such that $f(1) = 0$, where $P = \{ z \in \mathbb{C} : \text{Re} \, z > 0 \}$ is the right half-plane.

Answer: \dfrac{1}{3}


Question: Assume that $X_1, X_2, X_3, X_4$ are independent and identically distributed random variables such that $P(X_1=0)=0.3$, $P(X_1=1)=0.1$, and $X_1$ has a density $f(x)=0.6$ for $x$ in $(0,1)$. Calculate $P(X_1 + X_2 + X_3 + X_4 \leq 1)$. Provide your answer as a probability value.

Answer: 0.2295


Question:
Does there exist a positive constant $c_d$, depending only on the dimension $d$, such that for every convex body $K \subset \mathbb{R}^d$, there exists a parallelepiped $P \subset K$ satisfying the inequality:
\[
|P \cap \mathbb{Z}^d| \ge c_d |K \cap \mathbb{Z}^d|.
\]

Answer: Yes


Question: Is it true that for a holomorphic vector bundle $E$ on a compact Riemann surface $X$, there are no non-trivial sheaf cohomology groups in dimensions $\geq 2$?

Answer: Yes


Question: Let $p$ and $q$ be distinct primes, and let $G$ be a group of order $n = p^{\alpha}q^{\beta}$. If $H$ is a $p$-Sylow subgroup of $G$ and $K$ is a $q$-Sylow subgroup, is it true that $G = HK$? Justify your answer.

Answer: Yes


Question:
Determine the big-\(\Theta\) bound for the recurrence relation:
\[
T(n) = \begin{cases}
1, & \text{if } n = 0 \\
T(n/2) + T(2n/5) + 7n, & \text{if } n > 0
\end{cases}
\]
What is \(T(n)\) in big-\(\Theta\) notation?

Answer: \Theta(n)


Question:
Solve the following system of congruences for the integer $x$:

\[ x \equiv 6 \pmod{7} \]
\[ x \equiv 7 \pmod{13} \]
\[ x \equiv 4 \pmod{5} \]
\[ x \equiv 7 \pmod{11} \]

Answer: 1294


Question: Calculate the limit: \[ \lim_{(x,y) \to (0,1)} \frac{\ln\left(1+2x^3 y^2\right)}{x^3(1+y)}. \]

Answer: 1


Question: Find the sum of the series: \[ A = \frac{1}{1^3} + \frac{1}{2^3} + \frac{1}{3^3} + \cdots \]

Answer: \zeta(3)


Question: Evaluate the limit of the function \( f(x,y,z) = \frac{x^3y^3z^2}{x^6+y^8+z^{10}} \) as \((x, y, z)\) approaches the origin.

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=1}^{n}\frac{2015^{k}}{\sum_{i=0}^{k-1}2015^i\sum_{l=0}^k2015^l} \]

Answer: 1


Question:
Find the particular solution of the differential equation \(x^2y''+xy'-y=x^2e^x\) using the method of variation of parameters. Choose the correct option from the following:

(a) \( e^x+\frac{1}{x} \)
(b) \( e^x- \frac{e^x}{x} \)
(c) \( e^x+\frac{e^x}{x} \)
(d) \( e^x-\frac{1}{x} \)

Answer: B


Question: Let $f(x)$ be a concave function from $[0,1]$ to the reals such that $f(0) = f(1) = 0$. Consider the approximation of the integral $\int_0^1 f(x) \, dx$ by dividing $[0,1]$ into $n$ sub-intervals of equal length and using the maximum value of $f(x)$ on each sub-interval to form rectangles. Determine the maximum relative error of this approximation as a function of $n$.

Answer: \dfrac{2}{n}


Question: Find the maximum value of \( \frac{1}{(2a+b+c)^3}+\frac{1}{(2b+c+a)^3}+\frac{1}{(2c+a+b)^3} \) given that \( abc=1 \) and \( a, b, c \) are positive real numbers.

Answer: \dfrac{3}{64}


Question: Evaluate the integral \( \int_{l}(z^2+\bar{z}z)dz \) where \( l:|z|=1,0\leq\arg z\leq\pi \).

Answer: -\dfrac{8}{3}


Question: Determine the number of subgroups of order $p^2$ in the group $Z_{p^2} \times Z_p$, where $p$ is a prime number and $Z_k$ denotes the cyclic group of order $k$. Count isomorphic subgroups separately.

Answer: p + 1


Question: Consider the polynomial \( p(x) = x^{2021} - x + 1 \). Calculate the sum \( r_1^{2021} + r_2^{2021} + \ldots + r_{2021}^{2021} \), where \( r_i \) are the roots of \( p(x) \).

Answer: -2021


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^{n} \ln\left(\frac{k}{n} + \epsilon_n\right) \] where \( \epsilon_n > 0 \) and \( \epsilon_n \to 0 \).

Answer: -1


Question: Consider the plane $\mathbb{R}^2$ with a complete metric $g$ that has no conjugate points. Let $K$ denote the Gaussian curvature. Determine if there exists a constant $C > 0$ such that $K(p) \leq C$ for every point $p \in \mathbb{R}^2$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_0^1 [(t^2)^2 + i(t^2)^2][4t^3 + 4it^3] \, dt \).

Answer: i


Question: Let \( f(n_1,n_2) = \mathcal{O}\left(\frac{n_1n_2^2}{(n_1-n_2)^3}\right) \), where \( n_1 \) and \( n_2 \) are natural numbers. If \( n_1 \propto n_2 \), meaning the two variables grow proportionally, determine if \( f(n_1,n_2) = \mathcal{O}(1) \).

Answer: Yes


Question: Let \( f \) be a uniformly continuous function on \( \mathbb{R} \). Determine whether there exists a constant \( C \) such that \( \sup_{x, y \in \mathbb{R}} (f(y) - f(x) - C|x-y|) < \infty \).

Answer: Yes


Question: Let \( \mathbb{F} \) be a finite field of order \( q \). Consider polynomials \( p(x) \in \mathbb{F}[x] \) with degree at most \( \ell \) that satisfy \( p(a_1) = b_1 \) and \( p(a_2) = b_2 \) for two distinct \( a_1, a_2 \in \mathbb{F} \). Determine the maximum number of such polynomials. Can this number be bounded by \( O(q^{\ell-1}) \) as \( q \to \infty \) with \( \ell \) fixed?

Answer: q^{\ell - 1}


Question: Find the order of \(2^{36}\) modulo 107.

Answer: 53


Question: Determine the radius of convergence for the series \( \sum^{\infty}_{n=0} a_n z^{n!} \), where \( a_0 = 2i \) and \( a^2_n = a_{n+1} \).

Answer: 1


Question:
Given that two roots of the equation \((p-1)(x^2+x+1)^2-(p+1)(x^4+x^2+1)\) are real and distinct, and the function \(f(x)=\frac{1-x}{1+x}\), find the value of \(f(f(x))+f(f(\frac{1}{x}))\).

(a) \(p\)
(b) \(2p\)
(c) \(-p\)
(d) \(-2p\)

Answer: A


Question: Evaluate the limit: \[ \lim_{x \to \infty}{\frac{\int_{x^2}^{3x^2}{t\cdot \sin{\frac{2}{t}}dt}}{x^2}} \]

Answer: 4


Question: Given that $f(x)$ is a polynomial of degree 98 such that $f(k) = \frac{1}{k}$ for $k = 1, 2, 3, \ldots, 98, 99$, find the value of $f(100)$. A polynomial of this form exists.

Answer: \dfrac{1}{50}


Question: Calculate the mass of a bullet-shaped object formed by a cylinder with a curved top. The cylinder has a radius of \(a\) and is situated with its base on the \((x, y)\)-plane, with the origin at the center of the base. The curved top is described by the equation \(z = 4a - \frac{x^2}{a} + \frac{y^2}{a}\). The density of the object is given by \(D \cdot \left(\frac{\rho^2}{a^2} + 1 \right)\), where \(a\) and \(D\) are constants. Determine the mass of the object using appropriate integration techniques.

Answer: 6\pi D a^3


Question: Let $X$ be a Banach space. Determine whether the statement "if the dual norm of $X^*$ is Frechet differentiable, then $X$ is reflexive" is true or false.

Answer: True


Question: Find the partial derivative \( \frac{\partial}{\partial X} \text{tr}(XAX) \), where \( A \) and \( X \) are \( n \times n \) matrices, and \( A \) is a constant matrix. Provide a proof for your answer.

Answer: X^\top A^\top + A^\top X^\top


Question: Suppose that a matrix $A$ is diagonalizable. Can it be diagonalized using a matrix $P$ from the special linear group $SL_n(\mathbb{R})$, such that $\det(P) = 1$?

Answer: Yes


Question: Find the residue of the function \( \exp\left(\frac{2}{z-\pi}\right) \) at \( z=\pi \).

Answer: 2


Question: Determine the period of the function given by the equation \( f(x+\frac{1}{2}) + f(x-\frac{1}{2}) = f(x) \).

Answer: 3


Question: Evaluate the integral: \( \int_{-\infty}^{+\infty} e^{-x^2/2} \, dx \)

Answer: \sqrt{2\pi}


Question: Solve the equation for $z \in \mathbb{C}$: $$\text{Log}(z) - \text{Log}\left(\frac{1}{z}\right) = 1$$ where $\text{Log}{z} = \ln{r} + i\Theta$ for $-\pi < \Theta < \pi$ and $z \neq 0$. Find the value of $z$. \( \boxed{} \)

Answer: \sqrt{e}


Question: Let \( n \geqslant 3 \) and consider the linear map \( T : \mathbb{R}[x]_n \to \mathbb{R}[x]_n \) defined by \( T(p(x)) = 6p(x) + p'(x) - x^2p''(x) \). Determine the nullity of the linear map \( T \).

Answer: 1


Question: Calculate the area enclosed by the parametric curve \( \phi(t) = (a(2\cos(t)-\cos(2t)), a(2\sin(t)-\sin(2t))) \) using the appropriate method.

Answer: 6\pi a^2


Question: A trading card company produces packs containing 5 unique cards each, with a total of 50 unique cards in the set. Assuming perfect distribution and no card repeats within a pack, how many packs must you purchase to have a 95% probability of obtaining at least one of each card in the set?

Answer: 66


Question: Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a bijective function. Determine whether \( \lim_{n \rightarrow \infty} f(n) = \infty \).

Answer: \infty


Question: Evaluate the summation: \[ \sum_{n=1}^m \frac{n \cdot n! \cdot \binom{m}{n}}{m^n} \]

Answer: m


Question: Consider a sequence of integer-valued random variables $\{ X_{n}(\omega)\}$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that $X_{n}\overset{a.s.}{\to} c$, where $c$ is an integer constant. Define $N(\omega) = \min \{n_{2} \geq 1: X_{n}(\omega) = c, \forall n > n_{2}\}$. Is it true that $\mathbb{P}[N < \infty] = 1$?

Answer: Yes


Question: Is the product of a uniformly random permutation \(\sigma\) and an independent random permutation \(\tau\) also uniformly random?

Answer: Yes


Question: Evaluate the integral: $$\int_{-\infty}^{\infty}(x-2)^2e^{-\frac{(x-2)^2}{8}}\,dx$$. Hint: Consider constructing a probability density function.

Answer: 8\sqrt{2\pi}


Question: Let $H$ be a subgroup of a group $G$ such that the index $[G:H] = 2$. Prove that for all $a \in G$, $a^2 \in H$. Provide a rigorous justification for your proof.

Answer: a^2 \in H


Question: In right triangle $ABC$ with $AB=7$, $BC=24$, and $CA=25$, a circle $O$ is tangent to sides $AB$ and $BC$, and its center lies on side $CA$. Compute the radius of circle $O$. Express your answer as a common fraction.

Answer: \dfrac{168}{31}


Question: Find all pairs of natural numbers \((a, b)\) such that \(ab\) is a perfect square, \(4a + b - 3\) is a perfect square, and \(9a - 4b\) is a prime number.

Answer: (9, 16)


Question: Calculate the limit: $$\lim_{n\rightarrow\infty}\sqrt[n]{\prod_{i=1}^n\frac{1}{\cos\frac{1}{i}}}$$

Answer: 1


Question: Determine whether every compact space is compactly generated.

Answer: Yes


Question: Find the polynomial \( p(x) \) of degree at most 2 that minimizes the integral \( \int_{-1}^1 (p(x) - x^3)^2 \, dx \). Use the orthonormal basis of polynomials for the vector space of polynomials of degree at most 2 to compute the orthogonal projection of \( x^3 \) onto this space.

Answer: \dfrac{3}{5}x


Question: Consider a finite extension $K$ of $\mathbb{Q}_p$ with residue field $k$. There is a continuous surjective map $G_K \rightarrow G_k \cong \hat{\mathbb{Z}}$. Is this map an open map between the two topological groups?

Answer: Yes


Question: Does there exist a set $E \subseteq [0,1]^2$ such that for every $(x,y) \in [0,1]^2$, the sets $E \cap ([0,1]\times \{y\})$ and $E \cap (\{x\} \times [0,1])$ have measure zero in $\mathbb{R}$, but $E$ is not measurable in $\mathbb{R}^2$?

Answer: Yes


Question: Determine the smallest positive integer \( n \) such that \( \lim\limits_{x \to 0} f^{(n)}(x) \) is neither 0 nor \( \pm\infty \), where \( f(x) = \left(\frac{\sin x}{x}\right)^{\frac{x}{\sin x}} \) and \( f^{(n)}(x) \) denotes the \( n \)-th derivative of \( f(x) \).

Answer: 2


Question: Can the problem of choosing subsets \( \mathcal{A}_j \) from \( \mathcal{A}_i \subset \mathcal{G} \) with minimum cost, such that their union equals \( \mathcal{G} \), be formulated as a standard Integer Linear Programming (ILP) problem?

Answer: Yes


Question: Let \( f(x) \) be a continuously differentiable function on \([0,1]\) satisfying the following conditions: \( f(1) = 0 \), \( \int_0^1 [f'(x)]^2 \, dx = 7 \), and \( \int_0^1 x^2 f(x) \, dx = \frac{1}{3} \). Compute \( \int_0^1 f(x) \, dx \).

Answer: \dfrac{7}{5}


Question: If $R$ is a finite subring of a division ring $D$, is $R$ necessarily a division ring?

Answer: Yes


Question: Calculate the limit: \[ \lim_{x \to 0} \left(\frac{1+x2^x}{1+x3^x}\right)^{\frac{1}{x^2}}. \]

Answer: \dfrac{2}{3}


Question: Let \( F \) be a field and \( n, m \in \mathbb{N} \) be coprime numbers. Suppose \( \varepsilon \in \overline{F} \), where \( \overline{F} \) is an algebraic closure of \( F \), is both an \( n \)-th and \( m \)-th root of unity. Prove or disprove that \( \varepsilon = 1 \).

Answer: 1


Question: Calculate \( \int_C P \, dx + Q \, dy \) where \( P(x,y) = xe^{-y^2} \) and \( Q(x,y) = -x^2ye^{-y^2} + \frac{1}{x^2+y^2+1} \). The curve \( C \) is the boundary of the square determined by the inequalities \(-a \leq x \leq a\), \(-a \leq y \leq a\), oriented positively.

Answer: 0


Question: Consider a filtration $\mathcal F_t$ satisfying the usual conditions and a fixed real number $T > 0$. Define the filtration $\mathcal H_t := \mathcal F_{T + t}$. Let $X$ be a cadlag adapted process that is almost surely locally integrable, meaning for any compact set $C \subset \mathbb R_+$, we have $\int_C X_t \, dt < \infty$ almost surely. Define the moving average process $M$ by $M_t := \int_{[t, t + T]} X_s \, ds$. Suppose the following conditions hold: \n\n1. Almost surely, $X_0 = X_s$ for all $s \leq T$.\n2. $M_t$ is a $\mathcal H_t$-martingale.\n\nIs it true that $X$ is an $\mathcal F_t$-martingale?

Answer: Yes


Question: Find the area of the surface enclosed between the sphere \(x^2+y^2+z^2=4a^2\) and the cylinder \(x^2+(y-a)^2=a^2\). The answer should be \((8\pi-16)a^2\).

Answer: (8\pi - 16)a^2


Question: Evaluate the limit: \[ \lim_{x\to 3}((x+1)(x-3)\tan\left(\frac{x\pi}{2}\right)) \] using trigonometric identities, without using L'Hôpital's rule or Taylor series.

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


Question: Let $H$ and $K$ be subgroups of an infinite abelian group $G$ such that $K \subseteq H$ and $G \cong K$. Is it true that $G \cong H$?

Answer: Yes


Question:
Verify whether the following integral identity holds:
\[ \int_{\mathbb{R}} |x|^{3/2} e^{-x^2/t} \, dx = C t^{5/4} \]
where \( C \) is a constant involving a gamma function. If true, provide a method to evaluate the integral.

Answer: C t^{5/4}


Question: Evaluate the integral: \[ \int_{-\pi}^{\pi} \frac{x^3 \sin x}{1+e^{x \cos x}} \, dx \]

Answer: \pi^3 - 6\pi


Question: If $C_G(x) = G$ for a non-abelian group $G$, does it follow that $x = 1$? Justify your answer.

Answer: No


Question: Consider the interval \( I = [0,1] \). For each \( t \in I \), choose an open path-connected neighborhood \( V_t \). The collection \( \{V_t\} \) forms an open cover of \( I \). Determine whether there exists a finite subcover \( V_{t_1}, V_{t_2}, \ldots, V_{t_n} \) such that \( 0 = t_1 < t_2 < \cdots < t_{n-1} < t_n = 1 \) and \( V_{t_i} \cap V_{t_{i+1}} \neq \varnothing \) for \( i = 1, 2, \ldots, n-1 \).

Answer: Yes


Question: Suppose $V$ is an infinite dimensional vector space, and $W \subset V$ is another infinite dimensional subspace such that $V/W$ is finite dimensional. Let $U \subset V$ be a finite dimensional subspace with $U \cap W = \{ 0 \}$. Does the dimension formula \( \dim V/ ( W \oplus U) = \dim V/W - \dim U \) hold? Answer yes or no.

Answer: Yes


Question: Let $K$ be a number field and $G_K = \text{Gal}(\overline{K}/K)$ be its absolute Galois group. Consider a discrete $G_K$-module $M$ with a trivial $G_K$-action. For each place $v$ of $K$, let $K_v$ be the completion of $K$ at $v$. The restriction to a decomposition group $G_v$ at $v$ defines a homomorphism $H^1(K, M) \to H^1(K_v, M)$. Define $$F^1(K, M) := \text{Ker}\left(H^1(K, M) \to \prod_v H^1(K_v, M)\right).$$ Determine the structure of $F^1(K, M)$ when the $G_K$-action on $M$ is trivial.

Answer: 0


Question:
Given the following conditions:
\[ \lim_{x\to1} \cfrac{f(x)-f(1)}{x^2-1} = 4, \]
\[ g(x)+g(2)=2, \]
and
\[ (f \circ g)'(2) = 8, \]
find the value of \( g'(2) \).

Answer: 1


Question: Determine whether the limit \( \lim_{n\to\infty} M(n) \) exists, where \( M(n) \) is the median of the set \( \{L(m)/m : m \in \mathbb{N} \land 1 < m \leq n\} \) and \( L(n) \) is the largest prime divisor of \( n \). If the limit exists, what is its value?

Answer: 0


Question: Is it true that the Riemannian measure \( \mu \) is equal to the \( n \)-dimensional Hausdorff measure \( H^n \) for every Riemannian manifold \( M \)?

Answer: Yes


Question: Let X and Y be independent random variables, each following an exponential distribution with parameter \( \lambda = 1 \). Consider a rectangle with sides of length X and Y. What is the probability that a circle with radius \( r \) fits completely inside the rectangle?

Answer: e^{-4r}


Question: Suppose that \( f: \mathbb{R} \to \mathbb{R} \) is a continuous function such that \( f'(x) \) exists for all \( x > 0 \) and \( \lim_{x \to 0} f'(x) = 3 \). Does \( f'(0) \) exist?

Answer: No


Question: Suppose \( f\colon[0,\infty)\to[0,\infty) \) is a differentiable function with a continuous derivative such that its derivative \( f' \) is nonincreasing. Does this imply that \( f \) is non-decreasing? Answer yes or no.

Answer: Yes


Question: Find all functions \( f: \mathbb{N} \to \mathbb{N} \) such that for all natural numbers \( m \) and \( n \), the equation \( f(m^2 + f(n)) = f(m)^2 + n \) holds.

Answer: f(n) = n


Question: Given that $xf(x)=e^x-1$, calculate $\lim\limits_{n \to \infty}{nf^{(n)}}(x)$.

Answer: e^x


Question: Given that $x^{x^4} = 4$, find the value of $x^{x^2} + x^{x^6}$.

Answer: 18


Question: Use Lagrange multipliers to find the point on the surface \( \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1 \) with \( x, y, z > 0 \) that is closest to the origin.

Answer: (3, 3, 3)


Question: Find the limit of the sequence \( \sin^2 (\pi \sqrt{n^2 + n}) \) as \( n \to \infty \).

Answer: 1


Question: Find the equation of a circle that passes through the intersection of the circle \( S: x^2 + y^2 - 12x - 4y - 10 = 0 \) and the line \( L: 3x + y = 10 \), and has the same radius as the circle \( S \).

Answer: x^2 + y^2 = 50


Question: Determine the number of complex numbers \( z \) that satisfy the equation \( |z| = \max\{|z-1|, |z+1|\} \).

Answer: 0


Question: Determine whether there exists a subset $A \subset \mathbb{N}$ such that the upper density of its difference set $A-A$ is 1.

Answer: Yes


Question: Evaluate the limit: $$\lim _{x\to 0}\left(\frac{x\left(\sqrt{3e^x+e^{3x^2}}-2\right)}{4-\left(\cos x+1\right)^2}\right)$$

Answer: \dfrac{3}{8}


Question: Is there a number field $K$, and a smooth proper scheme $X \to \operatorname{Spec}(\mathfrak{o})$ over its ring of integers, such that $X(K_v) \neq \emptyset$ for every place $v$ of $K$, and yet $X(K) = \emptyset$?

Answer: Yes


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

Answer: 0


Question: Let \( n \) be a natural number that can be expressed as a sum of three non-zero squares, i.e., \( n = a^2 + b^2 + c^2 \) for some \( a, b, c \in \mathbb{N} \). Determine if it is possible for every natural power of \( n \), denoted as \( n^k \), to also be expressible as a sum of three non-zero squares, i.e., \( n^k = x^2 + y^2 + z^2 \) for some \( x, y, z, k \in \mathbb{N} \).

Answer: Yes


Question: Find all analytic functions $f(z)$ on the complex plane $\mathbb{C}$ such that $f(z) = f(-z)$ for all $z \in \mathbb{C}$.

Answer: f(z) = g(z^2)


Question:
In a Banach space $\langle V,+,\cdot, ||\cdot|| \rangle$ over the field $\mathbb{K}$ (where $\mathbb{K}$ is either $\mathbb{R}$ or $\mathbb{C}$), suppose there exists a subset $B$ of $V$ such that:

1. For all functions $f : B \to \mathbb{K}$, if $\sum_{b \in B} (f(b) \cdot b)$ exists and is the zero vector, then $f$ is identically zero.
2. For all members $v$ of $V$, there exists a function $f : B \to \mathbb{K}$ such that $\sum_{b \in B} (f(b) \cdot b) = v$.

Does it follow that all such subsets $B$ have the same cardinality?

Answer: Yes


Question: Determine whether the statistic \( T(\mathbf{X}) = (\min(\mathbf{X}), \max(\mathbf{X})) \) is a minimal sufficient statistic for \( \theta \) when \( \mathbf{X} \) is a random sample from a Uniform\((\theta, \theta+1)\) distribution.

Answer: Yes


Question: In the alternating group \(A_5\), let \(x\) be an element of order 2 and \(y\) be an element of order 3 such that \(\langle x, y \rangle = A_5\). Is it necessary that the order of the product \(xy\) is 5?

Answer: Yes


Question: Compute the relative cohomology group $H^0(\mathbb{R}^2, \mathbb{R}^2 - S^1)$. Use the long exact sequence of relative cohomology and the given information: $H^0(\mathbb{R}^2 - S^1) = \mathbb{Z}^2$, $H^1(\mathbb{R}^2 - S^1) = \mathbb{Z}$, and $H^2(\mathbb{R}^2 - S^1) = 0$. Also, $H^2(\mathbb{R}^2, \mathbb{R}^2 - S^1) = \mathbb{Z}$. Use the sequence $0 \to H^0(\mathbb{R}^2, \mathbb{R}^2 - S^1) \to \mathbb{Z} \to \mathbb{Z}^2 \to H^1(\mathbb{R}^2, \mathbb{R}^2 - S^1) \to 0$ to find $H^0(\mathbb{R}^2, \mathbb{R}^2 - S^1)$. Provide your answer in the form of a group.

Answer: 0


Question:
Determine which of the following formulations is correct for the likelihood of a log Gaussian Cox process (LGCP) with log intensity function \( \lambda(x)=S(x) \), where \( S \) follows a Gaussian process. The likelihood of an observed point process \((x_1,...,x_n)\) is given by:

1. \( p(X|S) \propto \prod_{i=1}^n \exp[S(x_i)] \exp[-\exp[S(x_i)]] \)

2. \( p(X|S) \propto \prod_{i=1}^n \exp[S(x_i)] \exp\left( -\int_\mathcal{D} \exp[S(x)]dx  \right) \)

Which formulation is correct?

Answer: 2


Question: Consider a polynomial $y = P(x)$ of the least degree that passes through the point $A(-1,1)$ and has two points of inflection: $B(1,2)$ and $C$ with an abscissa of 0. At point $C$, the curve is inclined to the positive x-axis at an angle of $\arcsec(\sqrt{2})$. Find the area of triangle ABC.

Answer: \dfrac{1}{4}


Question: Does there always exist an element \( \beta \in \overline{\mathbb{Q}} \) such that \( \beta^d = \alpha \) for any \( \alpha \in \overline{\mathbb{Q}} \) and any natural number \( d \)?

Answer: Yes


Question: Given the function \( f(t) = (t+1) \cos t \) for \(-\pi < t < \pi\), determine the sum of its Fourier series at \( t = 3\pi \).

Answer: -1


Question:
Given the matrices:
\[ P = \begin{pmatrix} a & 2d & 1 \\ b & 2e & -2 \\ c & 2f & -1 \end{pmatrix}, \quad U = \begin{pmatrix} a & b & c \\ 2 & 3 & 2 \\ d & e & f \end{pmatrix}, \quad V = \begin{pmatrix} a & b & c \\ d & e & f \\ 1 & 5 & 3 \end{pmatrix} \]
If \( \det(P) = 10 \) and \( \det(U) = -3 \), find the value of \( \det(V) \).

Answer: -2


Question: Let \((a_n)_{n\geq1}\) be a strictly increasing sequence of real positive numbers such that \(\lim\limits_{n\rightarrow\infty} \frac{a_{n+1}}{a_n}=1\). Find \(\lim\limits_{n\rightarrow\infty} \sum_{k=1}^{n} \frac{a_k}{a_k+a_1+a_2+...+a_n}\).

Answer: 1


Question: Provide an example of a non-commutative ring that has only the trivial ideals \( \{0\} \) and the ring itself, but is not a field.

Answer: M_2(\mathbb{R})


Question: Compute \(\lim_{n \rightarrow \infty} \left(1 + \frac{i}{n^2 + in}\right)^n\).

Answer: 1


Question: Determine whether the following statement is true or false: A continuous function \(f:\mathbb{R}\to \mathbb{R}\) is uniformly continuous if it maps Cauchy sequences into Cauchy sequences.

Answer: B


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

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{|z|=2} \frac{dz}{z^3(z^{10}-2)} \) using the residue theorem. Consider the substitution \( \zeta = 1/z \) to simplify the calculation.

Answer: 0


Question: Find the digital root of the number formed by a tower of 2013 twos: \( d(\underset{\text{The number of }2 \text{'s is }2013}{\underbrace{2^{2^{2^{.^{.^{.^{2}}}}}}}}) \). The digital root of a number \( n \) is given by \( d(n) = n - 9\left \lfloor \frac{n-1}{9} \right \rfloor \).

Answer: 7


Question: Consider a non-negative real sequence $(a_n)_n$. Define the set $S_\epsilon = \{n \mid a_n > \epsilon\}$ for $\epsilon > 0$. If the natural density of $S_\epsilon$ is $0$ for all $\epsilon > 0$, does the sequence $(a_n)_n$ have a subsequence that converges to zero and whose natural density is $1$? Provide a justification for your answer.

Answer: Yes


Question: Find the remainder when $99{,}999^{99}$ is divided by $999{,}999$.

Answer: 123579


Question: Find the equation of the line joining the points of contact of the tangents from the line \(2x-3y = c\) to the curve \(x^2+2y^2=2\). Given that \(c = \pm \sqrt{17}\), determine the equation of the line joining these points of contact.

Answer: 3x + 4y = 0


Question: Given a continuous random variable $X$ with cumulative distribution function $F_{X}$, find the expected value $E[F_{X}(X)]$. Provide your answer as a single value or expression.

Answer: \dfrac{1}{2}


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

Answer: p^n


Question: Find all positive integer pairs \((m, n)\) such that \(2^n + n = 3^m\).

Answer: (1, 1)


Question: Determine the number of values of \( n \) for which \( I_n = \int_0^1 \frac{1}{(1+x^2)^n} \, dx = \frac{1}{4} + \frac{\pi}{8} \).

Answer: 1


Question: Solve for \( x \) in the congruence equation: \[ \frac{38x + 23}{65} \equiv 22 \pmod{73} \]

Answer: 62


Question: Let \( \langle a_{n} \rangle \) be a sequence such that \( \lim_{n\rightarrow \infty}a_{n} = 0 \). Determine \( \lim_{n\rightarrow \infty}\frac{a_{1}+a_{2}+\cdots+a_{n}}{\sqrt{n(n+1)}} \).

Answer: 0


Question: Let $X$ be a compact topological space, and $d_1, d_2$ be two metrics that induce the topology on $X$. Is it true that for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $d_1(x, y) < \delta$, then $d_2(x, y) < \epsilon$ for all $x, y \in X$?

Answer: Yes


Question: Evaluate the integral \( \oint_{|z + 1| = 1} \frac{z^2}{z^2 - 1} \, dz \) using Cauchy's integral formula, where the contour is the circle \(|z + 1| = 1\) oriented counterclockwise.

Answer: -\pi i


Question: Given that $\Lambda \sim \text{Exp}(1)$ and $X$ has a probability mass function $P_\lambda(X=x)=\frac{\lambda^x e^{-\lambda}}{x!}$, where $\lambda$ is a realization of the random variable $\Lambda$, find the conditional expectation $E(e^{-\Lambda}|X=1)$. Assume $f(\lambda)=e^{-\lambda}$ for $\lambda>0$ and $0$ otherwise.

Answer: \dfrac{4}{9}


Question: Given a solution with a vertical asymptote at \(t = 8\), determine which of the following forms of ordinary differential equations (ODEs) this solution could match: \((1)\ y' + p(t)y = g(t)\) or \((3)\ y' = f(y)g(t)\). Justify your choice based on the behavior of the slope with respect to time.

Answer: 3


Question: Find the residue of \( \frac{e^z - 1 - z}{\sin z - z} \) at \( z = 0 \) using Taylor expansion.

Answer: -3


Question: Prove that there exists a constant \( C > 0 \) such that for the function \( R: \text{Mat}(n, \mathbb{R}) \to \mathbb{R} \) defined by \[ R(H) = \det(I+H) - 1 - \text{Tr}(H) \] the inequality \( |R(H)| \leq C ||H||^2 \) holds for all \( H \in \text{Mat}(n, \mathbb{R}) \) with \( ||H|| \leq 1 \).

Answer: C


Question:
Determine the order of convergence of the fixed point iteration given by:
\[
x^{(k+1)} = \begin{pmatrix}
0 & -1/2 & -1/4\\
-1/2 & 0 & 0\\
0 & -1/2 & 0
\end {pmatrix}x^{(k)}+\begin{pmatrix} 3/4 \\ -2 \\ -1/2 \end{pmatrix}
\]
for \(k \geq 0\), where \(x^{(k)} = \begin{pmatrix} x_1^{(k)} \\ x_2^{(k)} \\ x_3^{(k)} \end{pmatrix}\).

Answer: 1


Question: Let \(X_1, X_2, \ldots\) be independent and identically distributed random variables with mean 4 and variance 20. Define \(S_n = X_1 + X_2 + \cdots + X_n\) and \(V_n = X_1^2 + X_2^2 + \cdots + X_n^2\). Find a constant \(c\) such that \(P\left(\lim_{n \to \infty} \frac{S_n}{V_n} = c\right) = 1\).

Answer: \dfrac{1}{9}


Question: Given that $a, b, c, d$ are all prime numbers satisfying $a > 3b > 6c > 12d$ and $a^2 - b^2 + c^2 - d^2 = 1749$, determine the possible values of $a^2 + b^2 + c^2 + d^2$. Provide your answer as a list of possible values.

Answer: 1999


Question: Given \( z = \sqrt{2} + i\sqrt{2} \) and \( w = \frac{\pi}{4} + 2i \), find the real part of the expression \( z e^{iw} \).

Answer: 0


Question: Suppose $A = \{(x_n) \in \prod M_n(\mathbb{C}) : \lim_{n \to \infty} \text{tr}_n(x_n^*x_n) = 0\}$ is a $C^*$-algebra. If $B$ is any nonzero $C^*$-subalgebra of $A$, does there exist a finite-dimensional irreducible representation of $B$?

Answer: Yes


Question: Compute the line integral \( \oint_C \left( \frac{-y^3}{(x^2+y^2)^2} \right) dx + \left( \frac{xy^2}{(x^2+y^2)^2} \right) dy \) using Green's theorem, where \( C \) is the boundary of any simply connected region that contains the unit disk centered at the origin.

Answer: \pi


Question: In a Gaussian random walk, where $x_n$ is the sum of $n$ normal random variables, calculate the probability $P(x_1 > 0, x_2 > 0)$. Assume that the events $x_1 > 0$ and $x_2 > 0$ are not independent.

Answer: \dfrac{3}{8}


Question: Given a real sequence \( a_n \) defined by \( a_1 > 0 \) and \( a_{n+1} = \ln(a_n + 1) \) for \( n \geq 1 \), evaluate \( \lim_{n\to\infty} na_n \).

Answer: 2


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

Answer: 2e


Question: Find the intersection of the sets \(A_n = \{(n+1)k: k \in \mathbb{N}\}\) for all natural numbers \(n\).

Answer: \emptyset


Question: Find the value of \( y_0 \) such that the solution to the differential equation \( y' - \frac{3}{2}y = 3t + 2e^{t} \) with initial condition \( y(0) = y_0 \) separates solutions that grow positively as \( t \to \infty \) from those that grow negatively.

Answer: -\dfrac{16}{3}


Question: Determine whether the following statement is true or false: For every real matrix $X$ with $\det X = 1$, there exists a real symmetric traceless matrix $Y$ such that $X^TX = e^Y$. Provide a justification for your answer.

Answer: True


Question: Find the inverse of \( \alpha \) in the field \( \mathbb{Q}[\alpha] \), where \( \alpha \) is a root of the polynomial \( x^3 - 3x - 1 \). Express \( \alpha^{-1} \) as a linear combination of \( 1, \alpha, \) and \( \alpha^2 \).

Answer: \alpha^2 - 3


Question: Determine the number of right-angled triangles with integer sides that have an inradius of 2009.

Answer: 15


Question: Evaluate the sum \( \sum_{k=0}^n {2k\choose k}{2n-2k \choose n-k} \). Use the hint: \((1-4x)^{-1/2} = \sum_{n\ge0} {2n \choose n}x^n\).

Answer: 4^n


Question: How many ways are there to number the 8 vertices of a cube using the numbers 1 to 8 such that no two consecutive numbers are adjacent on the cube, considering 1 and 8 to be consecutive?

Answer: 480


Question: Find a function of a natural number \( M \) that yields an odd square, such as 9 or 25, only when \( M \) is a triangular number.

Answer: 8M + 1


Question: Compute the expected lifetime of an electronic tube with a probability density function given by \( f(x) = a^2 \cdot xe^{-ax} \) for \( x \geq 0 \).

Answer: \dfrac{2}{a}


Question:
Find all functions \( f : \mathbb{Q}^+ \to \mathbb{Q}^+ \) such that for all \( x \in \mathbb{Q}^+ \), the following conditions hold:
\[ f(x + 1) = f(x) + 1 \]
and
\[ f(x^2) = f(x)^2. \]

Answer: f(x) = x


Question: Find the extremal function for the fixed endpoint problem of the integral \( \int_1^2 \frac{\dot{x}^2}{t^3} \, dt \) subject to the boundary conditions \( x(1) = 2 \) and \( x(2) = 17 \).

Answer: t^4 + 1


Question: In a two-player game played on a 5×5 grid, two tokens, one red and one blue, start in random positions on the board. On each turn, a player can move either token one or two units to the right, or to the leftmost square of the above row. The last player who is able to move wins. Of the 625 possible starting positions for the two tokens, how many are winning positions for the first player if they play optimally?

Answer: 448


Question: Given the equation \(2y = \tan(x+y)\) where \(y = f(x)\), find the value of \(y''(0)\).

Answer: 0


Question:
Given a differentiable function $f$ satisfying the conditions:
\[
\begin{split}
f(2x)&=2\dfrac{f(x)}{e^x}\\
f'(0)&=1
\end{split}.
\]
Find the area bounded by $y=f(x)$ and the x-axis.

Answer: 1


Question: Evaluate the definite integral \( \alpha = \int_{\pi/6}^{\pi/3} \frac{\sin t + \cos t}{\sqrt{\sin 2t}} \, dt \). Then, find the value of \( f(\alpha) = (2\sin(\alpha/2) + 1)^2 \).

Answer: 3


Question: Compute \( \lim_{n\to\infty} nI_n \) where \( I_n = \int_1^e (\ln x)^n \, dx \).

Answer: e


Question: Given a countably infinite set, what is the minimum number of infinite subsets that must be added to a cofinite topology to transform it into a discrete topology?

Answer: \aleph_0


Question: Consider a sequence of independent and identically distributed discrete random variables \(X_n\) for \(n = 0, 1, \ldots\) with \(P(X_n = j) = a_j > 0\) for \(j = 0, 1, 2, \ldots\). Determine whether the process \(Y_n = \min(X_0, X_1, \ldots, X_n)\) is a Markov chain.

Answer: Yes


Question:
Given a positive integer $a$, find the maximum value of the expression:
\[ \sum_{k=1}^n (a-a_1)(a-a_2)\cdots(a-a_{k-1})a_k(a-a_{k+1})\cdots(a-a_n) \]
where $a_1, a_2, \ldots, a_n$ independently range over the interval $[0, a]$. Determine the maximum value of this expression.

Answer: a^n


Question: Two players each have $14. A fair coin is flipped repeatedly; if it lands heads, Player A pays Player B $1, and if it lands tails, Player B pays Player A $1. On average, how many coin flips will occur before one player runs out of money?

Answer: 196


Question: Identify the units in the polynomial ring $R[x]$, where $R$ is a domain.

Answer: R^\times


Question: Use the summation convention to simplify the expression \( g^{1}_{11} + g^{1}_{12} + g^{1}_{21} + g^{1}_{22} + g^{2}_{11} + g^{2}_{12} + g^{2}_{21} + g^{2}_{22} \) and determine the value of \( n \) necessary for this simplification.

Answer: 2


Question: Identify the unit element in the universal enveloping algebra of a Lie algebra \( \mathfrak{g} \) over a field \( k \) of characteristic zero. The universal enveloping algebra is formed by taking the tensor algebra \( T(\mathfrak{g}) \) and quotienting by the two-sided ideal \( I(\mathfrak{g}) \) generated by elements of the form \( X \otimes Y - Y \otimes X -[X,Y] \) for \( X,Y \in \mathfrak{g} \). What is the unit element in this associative algebra?

Answer: 1


Question: Determine the dimension of the quotient space \(c_0/W\), where \(c_0\) is the space of sequences converging to zero, and \(W\) is the subspace of \(c_0\) defined by \(W = \{ (y_n) \in c_0 : y_1 + y_2 + y_3 = 0 \}.\)

Answer: 1


Question:
Given the joint density function of two random variables $X$ and $Y$:
$$f(x,y) = \frac{1}{2\pi} e^{-(x^2+y^2)/2}$$
compute the probability $P(X < Y)$. Express your answer as a single value.

Answer: \dfrac{1}{2}


Question:
Find the exact value of the expression:
\[ 100! \times \left(1 + \sum_{n=1}^{100} \frac{(-1)^n (n^2+n+1)}{n!} \right) \]

Answer: 101


Question: Evaluate the limit: \[ \lim_{x\to 0} \left( \frac{e\cos(x)-e}{\ln(1+\ln(2-e^{\sin^2(x)}))} \right) \] without using L'Hôpital's rule or infinite series.

Answer: \dfrac{e}{2}


Question: Find the last two digits of the number represented by the power tower of 4s: $4^{4^{4^{\cdots}}}$, where there are 2013 instances of the number 4.

Answer: 96


Question: Evaluate the line integral \( \int_{C} F(r) \cdot dr \) where \( F = [x^3, e^{2y}, e^{-yz}] \) and the curve \( C \) is defined by \( x^2 + 9y^2 = 9 \) and \( z = x^2 \).

Answer: 0


Question: Determine the potential function for the vector field \( F = \langle 10xy + 3y^{2} + 2y, 5x^{2} + 6xy + 2x + 3 \rangle \). Then, calculate the line integral \( \int_C F \cdot dr \), where \( C \) is the path of the parabola \( y = x^{2} - 6x + 8 \) from the point \((0,8)\) to the point \((3,-1)\).

Answer: -69


Question: Consider a circulant $n \times n$ matrix $C$ over a finite field $F$ of order $q$, with the first row $c_0, c_1, \ldots, c_{n-1} \in F$. The determinant of $C$ can be expressed in terms of its representer polynomial $p_C(x) = c_0 + c_1 x + \cdots + c_{n-1}x^{n-1}$ and the roots of unity. A matrix $C$ is invertible if and only if $0$ is not an eigenvalue. Given that there are $q^n$ circulant matrices over $F$, determine how many of these matrices are invertible. Specifically, consider the case where $F=\mathbb{Z}_p$ and $C$ is a $(p-1) \times (p-1)$ matrix.

Answer: (p-1)^{p-1}


Question: Evaluate the integral \( I_{n} = \int_{-1}^{1} \frac{1}{x} P_{n}(x) P_{n-1}(x) \, dx \) using the recurrence relation \((n+1)I_{n+1} + nI_{n} = 2\).

Answer: \dfrac{1 - (-1)^n}{n}


Question: What is the topological dimension of Sierpiński's Triangle?

Answer: 1


Question: Find the remainder when \(2017^{2016^{2015}}\) is divided by 1000.

Answer: 281


Question: Compute the Frattini subgroup of \((\mathbb{Z}^n, +)\).

Answer: \{0\}


Question: In triangle $ABC$, the median $CM$ is perpendicular to the angle bisector $AL$, and their lengths are in the ratio $\sqrt{2} : 1$. Find $\cos A$.

Answer: \dfrac{1}{17}


Question: Determine the minimum number of times a die must be thrown so that the probability of the sum of the points being at least 4500 is at least 0.975. Use the central limit theorem to solve this problem.

Answer: 1321


Question: Use the Chinese Remainder Theorem to find the remainder of \(1030^{989}\) when divided by 3003.

Answer: 1933


Question:
Determine the value of \( k \) for which the functional \[ \int_0^1 (y^{\prime 2} + (y + 2y')y'' + kxyy' + y^2) ~dx, \] with boundary conditions \( y(0) = 0, \ y(1) = 1, \ y'(0) = 2, \ y'(1) = 3 \), is path independent. Choose the correct value of \( k \) from the following options:

(A) \( 1 \)
(B) \( 2 \)
(C) \( 3 \)
(D) \( 4 \)

Answer: B


Question: Find the first 10 digits after the decimal point in the number \((1+\sqrt{3})^{2015}\).

Answer: 0000000000


Question: Given the expression \(x-y+2x^2y+2xy^2-x^4y+xy^4\), where \(x = \sqrt{3-\sqrt{5}}\) and \(y = \sqrt{3+\sqrt{5}}\), express the value of the expression in the form \(\sqrt{p}+\sqrt{q}\) where \(p, q \in \mathbb{N}\). Find the value of \(p+q\).

Answer: 610


Question: Let \( \alpha \in \mathbb{C} \) be a root of the polynomial \( p(x) = x^7 - 4x^5 + 3x^2 + 2x + 1 - 2i \). What is the value of \( p(\overline{\alpha}) \)?

Answer: -4i


Question: Let \(n = 2^{13}3^{11}5^7\). Determine the number of positive integers \(d\) such that \(d \mid n^2\), \(d < n\), and \(d \nmid n\).

Answer: 3314


Question: Monic quadratic polynomials $P(x)$ and $Q(x)$ satisfy the following conditions: $P(Q(x))$ has zeros at $x = -23, -21, -17, -15$, and $Q(P(x))$ has zeros at $x = -59, -57, -51, -49$. Determine the sum of the minimum values of $P(x)$ and $Q(x)$. Provide your answer as a single number.

Answer: -100


Question: Find the value of $a$ such that \( \lim_{x\to 0^{+}}\ln (x\ln a)\ln \left(\dfrac{\ln ax}{\ln\frac xa}\right)=6 \).

Answer: e^3


Question: Assuming there are infinitely many twin primes, consider the Dirichlet series \( \sum_{n>0} a_{n} n^{-s} \) where the sequence of positive integers is replaced with the sequence of twin primes. If the original series has an abscissa of convergence of \( 1 \), what is the abscissa of convergence for this new series?

Answer: 1


Question: Let $f: D(0,1) \to \mathbb{C}$ be a holomorphic and bounded function. Do the antiderivatives of $f$ belong to the disc algebra, which consists of functions that are holomorphic on $D(0,1)$ and continuous on the closed disc?

Answer: Yes


Question: Consider a lattice \(\mathcal{L} = \mathbb{Z}v_1 \oplus \ldots \oplus \mathbb{Z}v_l\) in \(\mathbb{R}^n\) and let \(S_0\) be the set of edges of the fundamental unit of \(\mathcal{L}\). A region \(X\) is lattice path connected if for every two lattice points \(P\) and \(Q\), there exists a path of lattice points \(P = M_1, M_2, \ldots, M_{t - 1}, M_t = Q\) with \(M_i = M_{i-1} + u_i\) and \(u_i \in S_0\) and \(M_i \in X\). Let \(P \subset \langle v_1, \ldots, v_l \rangle\) be a convex \(k\)-dimensional polytope. Suppose the supporting linear space of \(P\) has a spanning set \(\mathbf{S}\) which is a subset of \(S_0\), and every codimension 1 face has a spanning set which is a subset of \(\mathbf{S}\). Is it true that \(P\) is lattice path connected?

Answer: Yes


Question: Evaluate the integral \( \int\limits_{S(1,1)} \left(\frac{z}{z-1} \right)^n dz \), where \( S(1,1) \) is a circle with center 1 and radius 1.

Answer: 2\pi i n


Question: In a right triangle \( \triangle ABC \) with hypotenuse \( AB = 6 \), a line \( BX \) is drawn from vertex \( B \) to leg \( AC \) such that \( AX = 3 \). Then, a line \( XY \) is drawn to the hypotenuse, forming a right triangle, with \( XY = 2 \). Find the length of \( BC \).

Answer: 4


Question: Compute the limit \( \lim_{n\to\infty}\frac{1}{\log n}\sum_{k=1}^n\left(1-\frac{1}{n}\right)^k\frac{1}{k} \). Note that \( \log x = \sum_{k=1}^{\infty}\left(1-\frac{1}{x}\right)^k\frac{1}{k},\quad x\ge \frac{1}{2}.\)

Answer: 1


Question: Find all positive integer solutions to the equation \( \sigma(n) = \frac{3n(n-1)}{2} \), where \( \sigma(n) \) is the sum of divisors function. Given that \( n = 2 \) is a solution, determine if there are any other solutions.

Answer: 2


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

Answer: e


Question: Evaluate the contour integral \( \oint_{C} \frac{e^z}{z-\pi i}\ \mathrm{d}z \), where \( C \) is the ellipse defined by \( |z - 2| + |z+2| = 6 \). Determine why the integral is zero.

Answer: 0


Question: Evaluate the triple integral: \[ \int_0^1 \int_0^{z\pi} \int_{\frac{y}{\pi}}^z \sin\frac{y}{x} \, dx \, dy \, dz \]

Answer: \dfrac{1}{3}


Question: In the cuboid $ABCD-A_1B_1C_1D_1$, points $M$ and $N$ are located on the diagonals $AB_1$ and $BC_1$, respectively. Given that $MN \parallel A_1C$, determine the ratio $\frac{MN}{A_1C}$.

Answer: \dfrac{1}{3}


Question: Find the norm of the linear functional \( f(x) = \sum_{j=1}^{\infty} \xi_j \frac{1}{\sqrt{j(j+1)}} \) defined by \( f: l^2 \rightarrow \mathbb{R} \) for all \( x = (\xi) \in l^2 \).

Answer: 1


Question: Calculate the expected value \( \mathbb{E}\big|X_P-X_Q\big|^2 \) for points \( P=(X_P,Y_P) \) and \( Q=(X_Q,Y_Q) \) chosen independently from the square with vertices \((-1,0), (0,-1), (1,0), (0,1)\) using geometric probability.

Answer: \dfrac{1}{3}


Question: Evaluate the integral \( \int_{\lambda}(n_1(x,y)x+n_2(x,y)y)ds \), where \( n = (n_1(x,y), n_2(x,y)) \) is the outward unit normal at the point \( P = (x,y) \) on the curve \( \lambda \) given by \( x^2 + 4y^2 = 4 \).

Answer: 4\pi


Question: Evaluate the integral: $$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} \, dx$$

Answer: \dfrac{\pi}{4}


Question: Evaluate the integral: \[ \int_0^{2\pi} \ln|i+2e^{ix}| \, dx \]

Answer: 2\pi \ln 2


Question: If $|\dot{x}(t)|$ converges to zero and $t|\dot{x}(t)|^4$ converges to zero, does it follow that $t|\dot{x}(t)|^2$ also converges to zero?

Answer: No


Question: Determine whether the function $X_t = W_t \sin(t)$ is a solution to the stochastic differential equation $$dX_t = \sin(t)dW_t.$$

Answer: No


Question: Evaluate the limit: \[ \lim_{x\to 1}\frac{x^{x^a}-x^{x^b}}{\ln^2 x} \]

Answer: a - b


Question:
Let \( p(x) \) be a polynomial such that \( p(x) - p'(x) = x^n \), where \( n \) is a positive integer. Find the value of \( p(0) \).

Options:
(A) \( n! \)
(B) \( (n-1)! \)
(C) \( \frac{1}{n!} \)
(D) \( \frac{1}{(n-1)!} \)

Answer: A


Question:
Consider the set $S = \{A=[a_{ij}]_{n\times n} : a_{ij} \in \{0,1,2,3,4,5\}\}$, where $A$ is an $n \times n$ matrix. Let $B = \{A \in S : |A| = 1\}$ and $C = \{A \in S : |A| = -1\}$. For $n \geq 2$, determine which of the following statements is true:

1) $B$ and $C$ are finite sets.
2) $B$ and $C$ are infinite sets.
3) $|B| = |C|$ if $n \geq 2$.
4) $|B| \neq |C|$.

Select the correct statement.

Answer: 3


Question: Find all solutions to the matrix equation \( AX = \frac{1}{2}X \), where \( A \) is a \( 5 \times 5 \) matrix with each element \( a_{ij} = \pm 1 \), and \( X \in \mathbb{R}^5 \).

Answer: 0


Question:
Consider a matrix $A \in M_3(\mathbb{R})$ that is not a diagonal matrix. Determine in which of the following cases $A$ is diagonalizable over $\mathbb{R}$:

(a) $A^2 = A$;
(b) $(A - 3I)^2 = 0$;
(c) $A^2 + I = 0$.

Provide your answer by identifying the cases where $A$ is diagonalizable.

Answer: a


Question: Evaluate the limit: $$\lim\limits_{x \to -3^+} \left((\ln(x+3))x^2 - 9\ln|2x+6|\right).$$

Answer: -9 \ln 2


Question: How many distinct necklaces can be formed using 12 beads, where each bead is either red or blue, and necklaces that differ only by rotation are considered identical?

Answer: 352


Question: Find the maximum value of the function \( f(x_{1},x_{2},...,x_{n})=x^{2}_{1}+x_{2}^{2}+...+x_{n}^{2} \) subject to the constraint \( x_{1}^{2}+2x_{2}^{2}+3x_{3}^{2}+...+nx_{n}^{2} = 1 \).

Answer: 1


Question: Evaluate the limit \( \lim_{n\to\infty}\dfrac{[1^2x]+[2^2x]+[3^2x]+\ldots+[n^2x]}{n^3} \), where \([.]\) denotes the greatest integer function.

Answer: \dfrac{x}{3}


Question: Determine whether the function \( f(x) = \int_0^x \cos \frac{1}{t} \cos \frac{3}{t} \cos \frac{5}{t} \cos \frac{7}{t} \, dt \) is differentiable at \( x = 0 \), given that \( f(0) = 0 \).

Answer: \frac{1}{8}


Question:
Let \( f:\mathbb{R}\rightarrow \mathbb{R} \) be a differentiable function at 0. Evaluate the limit:
\[
\lim_{x,y\rightarrow 0}\frac{f(x)-f(y)}{x-y}
\]
Is this limit equal to \( f'(0) \)?

Answer: f'(0)


Question: Consider a planar graph with 17 regions where each vertex has a degree of 5. Determine the number of vertices in the graph. Then, explain why such a planar graph cannot exist.

Answer: 10


Question: Determine the number of distinct types of functions \( f(x) \) that satisfy the equation \((f(x))^2 + (f'(x))^2 = \text{constant}\).

Answer: 2


Question:
Consider the following statements about functions in the complex plane:

(a) $|e^{\sin z}|$ does not assume its maximum value in $\mathbb{C}$.
(b) $|\sin (e^z)|$ does not assume its minimum value in $\mathbb{C}$.

Which of the following is true?

A. Only statement (a) is true.
B. Only statement (b) is true.
C. Both statements (a) and (b) are true.
D. Both statements (a) and (b) are false.

Answer: A


Question: Find the sum of the primitive roots of unity for the equation $z^{36} - 1 = 0$. Express your answer as a single number.

Answer: 0


Question: Determine the values of \( b > 0 \) for which the infinite series \( \sum_{n=0}^\infty \left(\frac{n}{b(n-3)}\right)^n \) converges.

Answer: (1, \infty)


Question: Let \( g: [0,1] \to \mathbb{R} \) be a function such that \( g \in C^1([0,1], \mathbb{R}) \). Prove that \( \lim_{n \to +\infty} \int_0^1 x^n \, dg(x) = 0 \).

Answer: 0


Question: Evaluate the limit: \( \lim_{x \to -\infty} \frac{x^2 - \sqrt{x^4 + 1}}{x^3 - \sqrt{x^6 + 1}} \) without using L'Hôpital's Rule.

Answer: 0


Question: Evaluate the integral \( \int_0^\pi e^{ix} \sqrt{1 - e^{2ix}} \, dx \).

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


Question: Calculate the dual norm \( \|T \|_{E^*} \) for the linear functional \( T: V \rightarrow \mathbb{R} \) defined by \( T(v) = \int_{0}^{1} v(t) \, dt \), where \( V = \{v:[0,1] \rightarrow \mathbb{R}  |\, v \; \text{cont.}, \, v(0) = 0  \} \) is a vector space with the supremum norm \( \|v\| = \sup_{x \in [0,1]} |v(x)| \).

Answer: 1


Question: Find the volume of the solid formed by the intersection of the surfaces \( z = x^2 + y^2 \) and \( z = 4y \) using triple integration.

Answer: 8\pi


Question: Evaluate the contour integral \( \oint_C \frac{3z-2}{z^2 - 2iz} \,dz \) where \( C \) is the circle of radius 2 centered at \( z = i \) with a counterclockwise orientation, using Cauchy's Integral Formula.

Answer: 6\pi i


Question: Let \( f \) be a continuous function on \([0, \infty)\) such that \( \lim_{x \to \infty} f(x) = L < \infty \). Compute \( \lim_{n \to \infty} \int_{0}^{2} f(nx) \, dx \).

Answer: 2L


Question: Evaluate the surface integral \( \int_S \boldsymbol{F} \cdot d\boldsymbol{S} \) where \( \boldsymbol{F} = x^2 \boldsymbol{i} + 2z \boldsymbol{j} - y \boldsymbol{k} \) over the curved surface \( S \) of the cylinder defined by \( x^2 + y^2 = 25 \), bounded by \( z = 0 \), \( z = 6 \), and \( y \geq 3 \).

Answer: 288


Question: Calculate the principal value of the integral \( P.V.\int_{-\infty}^{+\infty} \frac{1}{(2-x) (x^2+4)} \, dx \). Verify if the result is \( \frac{\pi}{8} \).

Answer: \dfrac{\pi}{8}


Question: Evaluate the improper integral \( \int_{\mathbb{R}} \frac{\sin(x)}{x} \, \mathrm{d}x \). Determine if it converges, and if so, find its value.

Answer: \pi


Question: Calculate the sum of the infinite series \( \sum_{i=0}^\infty \frac{i^2}{6^i} \).

Answer: \dfrac{42}{125}


Question: Consider a Lie group $G$ and its Lie algebra $\mathfrak{g}$. For an element $x \in \mathfrak{g}$, the coadjoint orbit $O_x$ is given by $O_x \simeq G/G_x$, where $G_x$ is the stabilizer of $x$. Given that there is a symplectic form on each orbit, determine whether the coadjoint orbit $O_x$ is a symplectic manifold.

Answer: Yes


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

Answer: -2


Question: Evaluate the limit: \[ \lim_{x\to\infty} \left( \frac{1}{x^2} \int_{t=1}^x \frac{t^2}{t + |\sin t|} \, dt \right) \]

Answer: \dfrac{1}{2}


Question: Calculate the volume of a torus using a double integral in Cartesian coordinates.

Answer: 2\pi^2 R r^2


Question: Given an $n \times n$ matrix $C$ over the complex numbers with trace 0, determine whether there always exist $n \times n$ matrices $A$ and $B$ such that $AB - BA = C$. Provide a proof or counterexample.

Answer: Yes


Question: Compute the limit: \[ \lim_{n\to+\infty}n\int_{0}^{\pi/2}x f(x) \cos^n x \, dx \] where \( f \in C([0,\pi/2]) \).

Answer: f(0)


Question: A factory uses 10 types of boxes to package its products, numbered from 1 to 10. Each box of type 10 can hold 6 boxes of type 9, each box of type 9 can hold 6 boxes of type 8, and so on, down to each box of type 2 holding 6 boxes of type 1, and each box of type 1 holding 6 products. Calculate the remainder when the total number of products that can be held in a box of type 10 is divided by 215.

Answer: 6


Question: Find the value of the infinite product \( \prod_{i=1}^\infty \frac{2i-1}{2i} \).

Answer: 0


Question: Let $x_1$ and $x_2$ be positive real numbers and define, for $n>2$: $$x_{n+1} = \sum_{k=1}^{n} \sqrt[n]{x_k}.$$ Evaluate $\lim_{n \to \infty} \frac{x_n - n}{\ln n}$. Provide your answer as a single value.

Answer: 1


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

Answer: \dfrac{4}{e}


Question:
Find all real solutions to the equation:
\[ \sqrt{\frac{x-7}{1989}} + \sqrt{\frac{x-6}{1990}} + \sqrt{\frac{x-5}{1991}} = \sqrt{\frac{x-1989}{7}} + \sqrt{\frac{x-1990}{6}} + \sqrt{\frac{x-1991}{5}} \]

Answer: 1996


Question: Compute the expectation \( \mathbb{E}\left(\left(1-\frac{2}{n}\right)^T\right) \) where \( T = \sum_{i=1}^n X_i \) and \( X_1, X_2, \ldots, X_n \sim \mathrm{Pois}(\lambda) \) are independent Poisson random variables.

Answer: e^{-2\lambda}


Question:
Which of the following functions is a solution to the differential equation \( y' = |y|^{\frac{1}{2}} \) with the initial condition \( y(0) = 0 \)?

- \( y(t) = \frac{t^2}{4} \)
- \( y(t) = -\frac{t^2}{4} \)
- \( y(t) = \frac{t|t|}{4} \)
- \( y(t) = -\frac{t|t|}{4} \)

Answer: C


Question: If some entries of a matrix are replaced by 0, does the resulting matrix necessarily have a lower operator norm, where the norm is defined as the maximum of the singular values of the matrix?

Answer: No


Question: Solve the differential equation \( \frac{dx}{dt} = x (1 - x) \cos (x) \) with the initial condition \( x(0) = \frac{1}{2} \). Find \( \lim_{t \to \infty} x(t) \).

Answer: 1


Question: Prove that \( \lim_{x\to\infty}\left(\frac{x}{x-1}\right)^x = e \).

Answer: e


Question: Given the joint probability distribution \( P(X = m, Y = n) = \frac{e^{-7}4^m3^{n-m}}{m!(n-m)!} \) for \( m \in 0, 1, 2, \ldots, n \) and \( n \in \mathbb{N} \), and zero otherwise, find the expected value \( E(X) \). It is shown that \( P(X = m) = \frac{e^{-4}4^m}{m!} \). Calculate \( E(X) \) using the formula \( E(X) = \sum_{m=0}^{\infty} m P(X = m) \).

Answer: 4


Question: Find the Taylor series of the function \( f(x) = e^{-1/x^2} \) about 0, where the function is defined as \( f(x) = e^{-1/x^2} \) for \( x \neq 0 \) and \( f(x) = 0 \) for \( x = 0 \). What can you conclude from the resulting series?

Answer: 0


Question: Minimize the function \( f = x^4 + 2y^4 + 3z^4 \) subject to the constraint \( x^2 + y^2 + z^2 = 1 \).

Answer: \dfrac{6}{11}


Question: Given two $3 \times 3$ matrices $A$ and $B$ such that $3A - B^2 = 0$ and $A^2 - 4B = 0$, find the possible values for the determinants of $A$ and $B$. Express your answer as a set of possible determinant pairs $(\det(A), \det(B))$.

Answer: \{(0, 0), (48, 36)\}


Question: Determine the fundamental group of the quotient space formed by identifying points on the circle \( \mathbb{S}^1 \) such that each point \( x \) is identified with \( r(x), r^2(x), \ldots, r^{n-1}(x) \), where \( r: \mathbb{S}^1 \to \mathbb{S}^1 \) is a quotient map defined by \( r(\cos \theta, \sin \theta) = (\cos(\theta + 2\pi/n), \sin(\theta + 2\pi/n)) \).

Answer: \mathbb{Z}


Question: Evaluate the limit: \( \lim_{n\rightarrow\infty}\frac{(3n)!}{(n!)^3} \) without using Stirling's approximation.

Answer: \infty


Question: Let $X$ be a metric space and $f: X \to \mathbb{R}$ be a continuous function. Suppose the support of $f$, denoted as $\textrm{supp}(f)$, is contained within a compact set $K$. Is it true that $f$ is zero on the boundary of $K$, i.e., $f|_{\partial K} = 0$?

Answer: Yes


Question: Determine the minimum value of \( \mathrm{trace}(AA^T) \) for an \( n \times n \) non-singular matrix \( A \) with integer entries.

Answer: n


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{\cos^2(1-\cos^2(1-\cos^2(1-\cos^2(\ldots 1\ldots \cos^2 x))\ldots))}{\sin\left(\pi\left(\frac{\sqrt{x+4}-2}{x}\right)\right)}$$

Answer: \sqrt{2}


Question: Evaluate the definite integral \( \int_0^{\pi} \frac{1}{a^2 \cos^2(x) + b^2 \sin^2(x)} \, dx \) using the given indefinite integral formula: \( \int \frac{1}{a^2 \cos^2(x) + b^2 \sin^2(x)} \,dx =\frac{1}{ab} \arctan\left(\frac{b}{a} \tan(x)\right) \).

Answer: \dfrac{\pi}{ab}


Question:
Let \( a, b, c \) be positive real numbers such that \( b^2 + c^2 < a < 1 \). Consider the matrix \( A = \begin{pmatrix} 1 & b & c \\ b & a & 0 \\ c & 0 & 1 \end{pmatrix} \). Determine which of the following statements is correct about the eigenvalues of \( A \):

1. All the eigenvalues of \( A \) are negative real numbers.
2. All the eigenvalues of \( A \) are positive real numbers.
3. \( A \) has a positive and a negative eigenvalue.
4. \( A \) has a non-real complex eigenvalue.

Note: Since \( A \) is a real symmetric matrix, all its eigenvalues are real.

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \to 0} (\sin(x))^{\cot^2(x)} \]

Answer: 0


Question: Find the value of \( \lambda \) such that \( \binom{\lambda}{n} = n+1 \).

Answer: n + 1


Question: Given three real numbers \(a, b, c\) such that \(a+b+c=5\) and \(a^2+b^2+c^2=11\), find the maximum value of \(abc\).

Answer: 3


Question: Find all entire functions $f$ such that for all $|z| \geq 1$, $|f(z)| \leq \frac{1}{|z|}$.

Answer: 0


Question: Solve the differential equation: \( f'(x) = f(f(x)) \)

Answer: 0


Question: Evaluate the integral of \( e^{1/z^2} \) around the contour \(|z|=1\) in the complex plane.

Answer: 0


Question: Determine the values of \( x \) for which the series \( \sum_{n=0}^{\infty} \left(\frac{n}{2n+1}\right)^{2n-1}x^n \) converges.

Answer: (-4, 4)


Question: Compute the sum \(\sum_{a+b+c=5} (6-a)(6-b)(6-c)\) where \(a, b, c\) are positive integers.

Answer: 465


Question:
In a quadratic programming problem, consider the following setup:

Minimize \( x^\top x + c^\top x \) subject to \( Ax = b \), where \( X \subset \mathbb{R}^n \) is a non-empty, convex, bounded polyhedron. Let \( x^* \) be the solution to this problem:

\[ x^* := \arg \min_{ x \in X } \ \{ x^\top x + c^\top x \} \text{ subject to } Ax=b \]

Define \( y \) as the solution to the unconstrained problem:

\[ y := \arg \min_{ x \in \mathbb{R}^n } \{ x^\top x + c^\top x \} \ \text{ subject to } Ax=b \]

Suppose that for all \( x \) such that \( Ax = b \), it holds \( A \left( P_X(x) \right) = b \). Is it true that \( x^* \) is the projection of \( y \) onto \( X \), i.e., \( x^* = P_X\left( y\right) \)?

The projection is defined as:

\[ P_X(y) := \arg\min_{x \in X} \left\|x-y\right\| \]

Answer: Yes


Question: If \( D \) is a finite division ring containing \( q \) elements, is it true that \( a^q = a \) for all \( a \in D \)? Justify your answer.

Answer: Yes


Question: Given a complex circular zero-mean Gaussian random process (or vector) \( \textbf{x} \) where \( \textbf{x} \sim \mathcal{CN}(0,\sigma^2) \), find the expectation \( \mathbb{E}\{\textbf{x}^n\} \) for \( n \in \mathbb{N}^{*+} \).

Answer: 0


Question: Let $M$ be a compact connected $3$-manifold with boundary $\partial M$. If $M$ is orientable, $\partial M$ is empty, and $H_1(M; \mathbb{Z}) = 0$, does it follow that $M$ has the same homology groups as a $3$-sphere? Answer yes or no.

Answer: Yes


Question: Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. Consider the set $S$ of points where $f$ has a local maximum. Determine whether $S$ is a closed set. Answer with "True" if it is closed, or "False" if it is not.

Answer: B


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

Answer: \dfrac{2}{3}


Question: Determine the number of connected components of the subset \( \mathcal{E} \) of \( GL_n(\mathbb{R}) \), where \( \mathcal{E} = \{V \in GL_n(\mathbb{R}): v_1, \dots, v_r \text{ is a basis for } \mathcal{S}\} \) and \( \mathcal{S} \subset \mathbb{R}^n \) is a subspace with dimension \( 1 \le r < n \). Here, \( v_1, \dots, v_r \) are the first \( r \) columns of \( V \).

Answer: 4


Question: Find the probability density function (PDF) of the random variable Y, where X is drawn uniformly from [0,1] and Y is drawn uniformly from [X, 1].

Answer: -\ln(1 - y)


Question:
Determine whether the function \( B_{n,k}(f'(x),f''(x),\cdots,f^{(n-k+1)}(x))_{(f \rightarrow g)^c} \) is symmetrical. Specifically, is the following equality true?
\[
B_{n,k}(f'(x),f''(x),\cdots,f^{(n-k+1)}(x))_{(f \rightarrow g)^c} = B_{n,k}(g'(x),g''(x),\cdots,g^{n-k+1}(x))_{(g \rightarrow f)^{n-c}}
\]

Answer: No


Question: Determine the interval \([a, b)\) within \([0, \infty)\) on which the sequence of functions \( f_n(x) = nx e^{-nx} \) converges uniformly to its pointwise limit \( f(x) = 0 \).

Answer: [a, \infty)


Question: Let $M$ be a finitely generated, non-zero, torsion-free module over a complete local 1-dimensional Noetherian domain $(R, \mathfrak{m})$. Let $n = \mu(M) = l_R(M/\mathfrak{m}M)$. Then $n \ge r := \dim_K M \otimes_R K$, where $K$ is the fraction field of $R$. The module $M$ embeds into $R^r$, making the canonical map $M \to M^{**}$ injective, allowing us to identify $M$ with a submodule of $M^{**}$. Does $M^{**}/M$ necessarily have finite length? Here, $(-)^* := \operatorname{Hom}_R(-, R)$. Answer with 'Yes' or 'No' and justify your answer.

Answer: Yes


Question: Find the Taylor polynomial \( p_2(x) \) for the function \( F(x) = \int_0^{x^2} e^{-\sqrt{t}} \, dt \) centered at \( a = 0 \).

Answer: x^2


Question: Determine whether there exists a natural number \( n \) such that \( 3^n \) begins with the digits 2019.

Answer: Yes


Question: Evaluate the triple integral \( \iiint_{D} z(x^{2}+y^{2}+z^{2}) \, \mathrm{d}x \, \mathrm{d}y \, \mathrm{d}z \) where the region \( D \) is defined by \( x^{2}+y^{2}+z^{2} \leq 2z \). Use spherical coordinates to find the correct value of the integral.

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


Question: Solve the Diophantine equation \( 12^x + y^4 = 56^z \) where \(x, y, z\) are non-negative integers.

Answer: (0, 0, 0)


Question: Determine the number of positive integer solutions to the equation \( \phi(x) = 2^{2013} \), where \( \phi \) is the Euler-totient function.

Answer: 32


Question: Determine whether there exists a sequence \(s_n\) of positive real numbers such that \(\sum \frac{1}{s_n \log s_n}\) converges but \(\sum \frac{1}{s_n}\) diverges.

Answer: Yes


Question: Determine whether the following statement is true or false: If $u$ and $v$ are any two vectors in $\mathbb{R}^n$ and $\mathcal{B}$ is any basis for $\mathbb{R}^n$, then $[u \cdot v]_{\mathcal{B}} = [u]_{\mathcal{B}} \cdot [v]_{\mathcal{B}}$. Justify your answer.

Answer: False


Question: Calculate the integral \( I = \int_{C} ze^{\frac{1}{z-1}} \, dz \) where \( C \) is the circle centered at 1 with radius \( \frac{1}{2} \).

Answer: 3\pi i


Question: Given the matrix \( A = \begin{pmatrix} 3x & 4x & 0 \\ 4x & 3x & 0 \\ 0 & 0 & 5x \end{pmatrix} \), where \( x \in \mathbb{R} \) is a constant, find the angle \( \alpha \) such that \( A \) can be diagonalized by a rotation about the \( z \)-axis.

Answer: \dfrac{\pi}{4}


Question: If a function \( f: [a,b] \to \mathbb{R} \) satisfies \( \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n f(k/n) = 1 \), does this imply that \( f \) is Riemann integrable on \([a,b]\)?

Answer: No


Question: Find the remainder when \(3^{2^{2016}} - 1\) is divided by \(2^{2019}\).

Answer: 2^{2018}


Question: Given that \( a = 7! \) and \( b = {}_{13}P_k \), and \( \frac{ab}{\operatorname{lcm}(a,b)} = 120 \), find the value of \( k \).

Answer: 4


Question: Find the limit of the sequence \( x_n = \left[ \left(\frac{2}{1}\right) \left(\frac{3}{2}\right)^2 \left(\frac{4}{3}\right)^3 \cdots \left(\frac{n+1}{n}\right)^n \right]^{\frac{1}{n}} \).

Answer: e


Question: Find the location and order of the zeros of the function \( f(z) = \sin^4{\frac{1}{2}z} \).

Answer: 4


Question: Consider a probability space $(\Omega,\mathcal{F},P)$ and the real line with the Borel sigma-algebra $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$. Let $(\Omega\times\mathbb{R},\mathcal{F}\otimes \mathcal{B}(\mathbb{R}),P\otimes \lambda)$ be the product measurable space. Given a set $A$ that is measurable in $\mathcal{F}\otimes \mathcal{B}(\mathbb{R})$ with $P\otimes\lambda (A)>0$, determine if there exists an $x\in\mathbb{R}$ and a set $B\in\mathcal{F}$ with $P(B)>0$ such that $B\times\{x\}\subset A$. Provide a justification for your answer.

Answer: Yes


Question: Let $P(x)$ be a polynomial of degree 1998 such that $P(k) = \frac{1}{k}$ for all integers $k$ from 1 to 1999. Determine the value of $P(2000)$. \( \boxed{} \)

Answer: \dfrac{1}{1000}


Question: Given the matrix \( A = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} \) and the expression for its powers \( A^n = \begin{pmatrix} x_n & -y_n \\ y_n & x_n \end{pmatrix} \), find the value of \( \frac{x_{10}^2 + y_{10}^2}{x_{8}^2 + y_{8}^2} \).

Answer: 4


Question:
Calculate the expected number of claims in a year for a firm where claims arrive according to a nonhomogeneous Poisson process with the following rate function \( \lambda(t) \):

\[
\lambda(t) = \begin{cases}
  t, & 0 \leq t \leq 5 \\
  5, & 5 < t \leq 7 \\
  12-t, & 7 < t \leq 12
\end{cases}
\]

where \( t \) is the time in months. Assume each month has 30 days. Provide your answer as the expected number of claims in a year.

Answer: 35


Question: Find the inverse Z-transform of the function \( X(z) = \log \left( \frac{z}{z-a} \right) \). Specifically, determine the value of \( x[n] \) when \( n = 0 \).

Answer: 0


Question: Calculate the limit: \[ \lim_{x\rightarrow \infty} e^{x \ln (\sin(1/x)+\cos(1/\sqrt{x}))}. \]

Answer: \sqrt{e}


Question: Given a complex polynomial \(P_n(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0\), where \(\alpha_i\) are its complex roots, calculate the product \((\alpha_1+1)(\alpha_2+1)\cdots(\alpha_n+1)\).

Answer: (-1)^n P_n(-1)


Question: Determine the value of the upper density \( \mu^+(I) \) of the set \( I \), where \( I \) is the set of positive integers \( n \) such that there exist positive integers \( a, b \) with \( n^2 = \frac{a^2 + b^2}{ab + 1} \).

Answer: 1


Question: Determine whether the following statement is true or false: Let $f$ be holomorphic with an essential singularity at $0$. Then for every $z_0 \in \mathbb{C}$, the Laurent expansion around $z_0$ has a nonvanishing principal part.

Answer: B


Question: Consider two distinct prime numbers \( p \) and \( q \). Are the numbers \( \frac{p^q-1}{p-1} \) and \( \frac{q^p-1}{q-1} \) relatively prime?

Answer: Yes


Question: Evaluate the integral \( \int \bar{F} \cdot d\bar{r} \) using Stokes' theorem, where \( \bar{F} = (x-y)\mathbf{i} + (y+z)\mathbf{j} + x\mathbf{k} \) and \( C \) is the boundary of the area of the triangle cut off from the plane \( 2x + y + z = 2 \) by the coordinate axes.

Answer: -2


Question: Calculate the group of continuous homomorphisms from the profinite completion of the integers, \( \hat{\mathbb{Z}} \), to the integers, \( \mathbb{Z} \), i.e., find \( \text{Hom}_{\text{cts}}(\hat{\mathbb{Z}}, \mathbb{Z}) \).

Answer: 0


Question: Determine the number of values of \(x\) in the interval \((0, 20)\) such that \(\{x\} + \{x^2\} = 1\), where \(\{\}\) denotes the fractional part of a number.

Answer: 400


Question:
Find the smallest positive integer \( c \) such that for all non-negative integers \( n \), the inequality \( f(n) \leq cn^2 \) holds, where \( f(n) \) is defined recursively by:

- Base case: \( f(0) = 4 \).
- Recursive case: \( f(n+1) = 7f\left(\lfloor \frac{n+1}{3} \rfloor \right) + 5(n+1)^2 \) for \( n > 0 \).

Answer: 33


Question:
Given the parametric equations for the Brachistochrone problem:
\[ x = \frac{k^2}{2}(\sin\theta + \theta + A), \quad y = \frac{k^2}{2}(1+\cos\theta) \]
Determine the constant $A$ such that the curve passes through the points $P_1 = (0,0)$ and $P_2 = (x_2, y_2)$. Assume $\theta$ is the parameter for the cycloid and $k$ is a constant.

Answer: -\pi


Question: Consider a min-degree graph process that starts with \( G_0 \), the complement of \( K_n \). Given \( G_t \), choose a vertex \( u \) of minimum degree in \( G_t \) uniformly at random, then choose a vertex \( v \) not adjacent to \( u \) in \( G_t \) uniformly at random. Add the edge \( A_{t+1} \) during period \( t+1 \). The process finishes with the complete graph. Determine if this graph process is Markovian.

Answer: Yes


Question: Evaluate the integral: $$\int_{0}^{2 \pi} \sqrt{(-5\sin t-10\cos t\sin t)^2+(-5\sin^2t+5\cos^2t+5\cos t)^2} \, dt$$

Answer: 40


Question: Is it possible for matrices $A$ and $B$ to have different ranks while satisfying the equation $A A^T = B B^T$?

Answer: No


Question: Find the general solution of the partial differential equation: \[ x\,\partial_x w - y\,\partial_y w = 0. \]

Answer: w(x, y) = f(xy)


Question: Evaluate the surface integral \( \iint\limits_S \vec{A} \cdot \hat{n}\,dS \) over the entire surface \( S \) of the region bounded by the cylinder \( x^2 + z^2 = 9 \), \( x = 0 \), \( y = 0 \), \( y = 8 \), \( z = 0 \), where \( \vec{A} = 6z\,\hat{i} + (2x+y)\,\hat{j} - x\,\hat{k} \).

Answer: 18\pi


Question:
Let $a, b, c$ be positive real numbers such that $abc = 1$. Determine the maximum value of the expression:
\[ \frac{a}{a^8+1} + \frac{b}{b^8+1} + \frac{c}{c^8+1}. \]

Answer: \dfrac{3}{2}


Question: Determine the number of solutions to the congruence equation \(x^{50} \equiv 1 \pmod{181}\).

Answer: 10


Question: Suppose matrices $A$ and $B$ have positive eigenvalues. Must the product $AB$ also have positive eigenvalues? Provide a justification for your answer.

Answer: No


Question: Let $L/K$ be an extension of number fields, with $\\mathcal{O}_L\\$ and $\\mathcal{O}_K\\$ as the rings of integers of $L$ and $K$, respectively. A prime ideal $\\mathfrak{p} \\subseteq \\mathcal{O}_K\\$ is said to be inert if $\\mathfrak{p}\\mathcal{O}_L\\$ remains a prime ideal in $\\mathcal{O}_L\\$. Does there exist at least one prime ideal in $\\mathcal{O}_K\\$ that is inert in $L$?

Answer: No


Question: Determine the minimum number of colors required to color the vertices of a graph such that no two adjacent vertices share the same color, given that each vertex lies on at most $k$ odd cycles.

Answer: 2k + 2


Question: Evaluate the integral \( \int_e \frac{1}{z^2+4} \, dz \) using the Cauchy Integral Formula, where the contour \( e \) is the ellipse given by \( 4x^2 + (y-2)^2 = 4 \).

Answer: \dfrac{\pi}{2}


Question: Let $A$ and $B$ be commutative rings with units, and let $f: A \to B$ be a ring homomorphism. Let $I \subset A$ be an ideal. Consider the kernel $J$ of the surjective map $B \to A/I \otimes_A B$, given by $b \mapsto 1 \otimes b$. Is $J$ equal to the ideal generated by $f(I)$ in $B$?

Answer: Yes


Question: Evaluate the sum \( \sum_{r=0}^{1006} \binom{2012}{2r} (-3)^r \).

Answer: -2^{2011}


Question: Is there a subset $D$ of the unit circle $S$ such that the only 1-iso map $\phi$ satisfying $\phi(D) = D$ and $\phi(S-D) = S-D$ is the identity map?

Answer: Yes


Question: Evaluate the integral \( \int_0^{\frac{\pi}{2}} \frac{dx}{{\left(\sqrt{\sin{x}}+\sqrt{\cos{x}}\right)}^4} \).

Answer: \dfrac{1}{3}


Question: Evaluate the line integral \( \int_C xy^2\mathrm{d}x + 2x^2y\mathrm{d}y \) using Green's Theorem, where \( C \) is the triangle with vertices \((0,0)\), \((2,2)\), and \((2,4)\).

Answer: 12


Question: Evaluate the integral \( \int_{-\pi/4}^{\pi/4} \frac{\sec^2x}{1+e^x} \, dx \).

Answer: 1


Question: In square $ABCD$ with side length 144, point $E$ is on side $BC$ such that $AE + EB = 216$. Point $F$ is on side $CD$ such that line $AF$ bisects angle $DAE$. Determine the length of $DF$. Express your answer as a single number.

Answer: 96


Question: Find the norm of the operator \( A:L_2[0,2] \rightarrow L_2[0,2] \) defined by \((Ax)(t) = t \, \operatorname{sgn}(t-1) x(t)\). Determine if the operator is bounded, and if so, find the operator norm.

Answer: 2


Question: Consider the region \( \Omega:\{ (x, y, z): x^2+y^2\leq 1, 0\leq z\leq 2 \} \) and the transformation \( T:(x, y, z)\to(x, y+\tan(\alpha z), z) \), where \( \alpha\in (0, \pi) \). Determine the volume of \( T(\Omega) \).

Answer: 2\pi


Question: Find the smallest integer \( n \) such that for a random sample \( X_1, \ldots, X_n \) from a normal distribution with mean \( \mu = 0 \) and variance \( \sigma^2 = 2 \), the probability \( P(\min(X_1^2, \ldots, X_n^2) \leq 0.002) \geq 0.8 \).

Answer: 63


Question: Evaluate the limit \( \lim_{x\to 3} \left(\dfrac{\sqrt{2x+3}-x}{\sqrt{x+1}-x+1} \right)^{ \left(\dfrac{x-1-\sqrt{x^2-5}}{x^2-5x+6}\right)} \) and express it in the form \(\dfrac{a\sqrt{b}}{c}\), where \(a,b,c \in \mathbb{N}\). Find the least value of \(a^2+b^2+c^2\).

Answer: 29


Question: Let $f$ be a continuous real-valued function on $[0,1]$ such that there exists a constant $K > 0$ for which $|f(x)| \le K \int_0^x |f(t)| \, dt$ for all $x \in [0,1]$. Prove or disprove that $f$ is identically zero on $[0,1]$.

Answer: 0


Question: Determine whether every $k_\omega$ space is also a $k$ space. A $k_\omega$ space is defined as the union of compact sets $K_n$ such that a set is closed if and only if its intersection with each $K_n$ is closed within that subspace. A $k$ space has the property that a set is closed if and only if its intersection with each compact set is closed within that subspace. Is it true that every $k_\omega$ space is also a $k$ space?

Answer: Yes


Question: Consider the transformation \( T: [0,1) \to [0,1) \) defined by \( T(x) = 2x \mod 1 \), which preserves the Lebesgue measure \( m \). Let \( \phi: [0,1) \to [0,1) \) be defined as \( \phi(x) = x^3 \). Determine the value of \( \lim_{n \to \infty} \frac{1}{n} \sum_{j=0}^{n-1} \phi(T^j x) \) for \( m \)-almost every point \( x \in [0,1) \).

Answer: \dfrac{1}{4}


Question: Find the Laplace Transform of the function \( \delta(t-2\pi)\cos(t) \), where \( \delta(t) \) is the Dirac Delta Function.

Answer: e^{-2\pi s}


Question: Let \( n \) be a positive integer and \( \mathbf{A} \) an \( n \)-by-\( n \) matrix with integer entries such that \( k = \big|\det(\mathbf{A})\big| \neq 0 \). Determine the number of \( n \)-by-\( 1 \) column vectors \( \mathbf{x} \in \mathbb{Q}^n \) with entries in the interval \([0,1)\) that satisfy \( \mathbf{A} \mathbf{x} = \mathbf{b} \) for some \( n \)-by-\( 1 \) column vector \( \mathbf{b} \in \mathbb{Z}^n \).

Answer: k


Question: Determine whether there exists a bijective function \( f: \mathbb{N}^+ \to \mathbb{Q} \setminus \{0\} \) such that \(|f(n)| > \frac{1}{n}\) for all positive integers \( n \).

Answer: Yes


Question: A 1-meter rope is cut at two uniformly random points, resulting in three pieces. What is the average length of the smallest piece?

Answer: \dfrac{1}{9}


Question: Evaluate the limit: $$\lim_{x \rightarrow +\infty} \frac{\sqrt x + x^3 -\sqrt{x^6-2x^4} -x}{\log(1+{e}^{3\sqrt x})}$$

Answer: \dfrac{1}{3}


Question: Calculate the integral \( I = \int_0^1 |f(x) - f^{-1}(x)| \, dx \), where \( f: [0, 1] \to [0, 1] \) is defined by \( f(x) = x e^{x^2 - 1} \).

Answer: \dfrac{1}{e}


Question: Evaluate the integral \( \int_0^2 \int_0^{\sqrt{1-(1-x)^2}} \frac{y}{y^2 + x^2} \, dy \, dx \) using polar coordinates.

Answer: 1


Question: Find the value of \( k \) such that there is no \( x^3 \) term in the expansion of \((k + 2x)(1 - \frac{3}{2}x)^6\).

Answer: 1


Question: Is it possible to choose ten different numbers from the set \{0, 1, 2, \ldots, 14\} and place them in ten circles such that all positive differences between numbers in adjacent circles are distinct?

Answer: Yes


Question:
Solve the following system of equations for \(x\) and \(y\):
\[
\begin{align*}
  & 2x^2 - 5xy - 5y^2 + x + 10y - 35 = 0, \\
  & x^3 + 5xy^2 + 42 = 0.
\end{align*}
\]

Answer: (-3, 1)


Question: Given the functions $f_0, f_1, \ldots, f_n$ defined by $f_0(x) = \frac{1}{x-2}$ and $f_{k+1}(x) = \frac{1}{1-f_k(x)}$ for every $k = 0, 1, \ldots, n-1$, evaluate $f_{2000}(2001)$. Provide your answer as a single value.

Answer: -1998


Question: Consider the space ${\cal P}(\omega)$, which is naturally bijective to $\{0,1\}^\omega$. This space is endowed with the product topology, where $\{0,1\}$ carries the discrete topology, making it a compact space with a Haar measure. Define the set $A_0 = \{a \in {\cal P}(\omega): 0 \in a\}$. Determine the Haar measure of the set $A_0$.

Answer: \dfrac{1}{2}


Question:
Given the polynomials $f(x)$ and $g(x)$ defined as follows:
\[f(x)=\prod_{i=0}^{2016}x^{i^{2017}}+\prod_{j=0}^{2017}(x^2+2016)^j\]
\[g(x)=\prod_{i=0}^{2016}x^{2017^i}+\left(\prod_{j=0}^{2017}x^{2j}\right)2016x\]
Find the positive difference between the sums of the roots of $f(x)$ and $g(x)$. Express your answer as a single number.

Answer: 0


Question: Let \( A = (a_{ij}) \) be a \( 10 \times 10 \) matrix such that \( a_{ij} = 1 \) for \( i \neq j \) and \( a_{ii} = \alpha + 1 \), where \( \alpha > 0 \). Let \( \lambda \) and \( \mu \) be the largest and smallest eigenvalues of \( A \), respectively. If \( \lambda + \mu = 24 \), find the value of \( \alpha \).

Answer: 7


Question: Let $x, y, z \in \mathbb{C}$. Given that $x+y+z = 0$ and $x^2+y^2+z^2 = 0$, determine if $x^2y^2 + y^2z^2 + z^2x^2 = 0$ is true.

Answer: 0


Question: Let $f(x)$ be a periodic function defined over $\mathbb{R}$, bounded near $x=0$, and satisfying $\lim\limits_{x \to +\infty}[2f(x)-f(2x)]=0$. Find $\lim\limits_{x \to +\infty}\frac{f(x)}{x}$. Assume $T$ is a positive period of $f(x)$. Show that $2f(x)=f(2x)$ for any $x \in \mathbb{R}$ and use this to determine the limit.

Answer: 0


Question: Let $d \geq 1$ be an integer. Consider a finite Galois extension $K/\mathbb{Q}$. Is it true that for all $1 \leq a \leq d$ with $\gcd(a,d)=1$, the arithmetic progression $a, a+d, a+2d, \ldots$ contains infinitely many primes that split completely in $K$?

Answer: No


Question:
Determine the set of all positive values of \(a\) for which the series \(\sum_{n=1}^\infty \left(\frac{1}{n} - \tan^{-1}\left(\frac{1}{n}\right)\right)^a\) converges. Choose the correct option:

(A) \((0,1/3]\)
(B) \((0,1/3)\)
(C) \([1/3,\infty)\)
(D) \((1/3,\infty)\)

Answer: D


Question: Determine whether the function \( f(\epsilon) = \lim_{n \rightarrow \infty}\frac{\sum_{i=1}^n\left|\{k\le i^\epsilon: k |i\} \right|}{\sum_{i=1}^n\left|\{k |i\} \right|} \) is well-defined for all \( \epsilon \in [0,1] \).

Answer: \epsilon


Question:
Given the polynomial:
\[10x^5+37x^4-83x^3-83x^2+37x+10\]
where \(a, b, c, d, e\) are the roots of the polynomial and \(a < b < c < d < e\), find the value of \((a + b)(b + c)(c + d)(d + e)(e + a)\).

Answer: -\dfrac{81}{5}


Question: Let $V$ be an infinite dimensional vector space. Given $n \in \mathbb{N}$, is there a subspace $U \subseteq V$ of dimension $n$?

Answer: Yes


Question:
Calculate the Fourier sine coefficient $b_1$ for the function $u(t)$, which is defined as 0 for $-\frac{\pi}{\omega} < t < 0$ and $E\sin(\omega t)$ for $0 < t < \frac{\pi}{\omega}$. The expression for the sine coefficient is given by:

\[ b_n = \frac{\omega E}{\pi} \int_0^\frac{\pi}{\omega} \sin(\omega t)\sin(n \omega t)\,dt \]

Evaluate this expression for $n=1$ and verify that $b_1 = \frac{E}{2}$. Provide a detailed explanation of your solution.

Answer: \dfrac{E}{2}


Question: Given a positive definite matrix $A$ with elements $a_{ij}$, a new matrix $A'$ is formed as $A'_{ij} = \frac{a_{ij}}{\sqrt{a_{ii}a_{jj}}}$. Is the matrix $A'$ also positive definite?

Answer: Yes


Question: Compute the value of the infinite series \( \sum_{n=1}^{\infty}\frac{1}{n2^n} \).

Answer: \ln 2


Question: Given a natural number \( n \geq 2 \), repeatedly subtract the largest proper divisor of \( n \) from \( n \) until the result is 1. For example, starting with \( n = 30 \), the sequence of operations is: 30 to 15, 15 to 10, 10 to 5, 5 to 4, 4 to 2, and finally 2 to 1, taking 6 steps in total. Determine the number of steps required to reduce \( n = 2016^{155} \) to 1.

Answer: 2015


Question: Let \(f = u + iv\) be an entire function such that \(u_x v_y - u_y v_x = 1\) throughout the complex plane. Prove that \(f(z) = az + b\), where \(a\) and \(b\) are constants and \(|a| = 1\).

Answer: f(z) = az + b


Question: A right circular cylindrical tank lies on its side with a length of 12 feet and a radius of 4 feet. The water level in the tank is 18 inches above the lowest part. Calculate the volume of water that must be drained from the tank. Round your answer to the nearest tenth of a cubic foot.

Answer: 78.3


Question: If a subgroup $H$ is characteristic in a group $G$, does it follow that for every automorphism $f \in \mathrm{Aut}(G)$, the image $f(H)$ is isomorphic to $H$?

Answer: Yes


Question: Determine whether the following claim is true or false: If there is a homomorphism \( \phi \) from a finite group \( G \) to \( S_n \) such that \( G/\ker(\phi) \cong A_n \), then the maximal subgroup of \( G \) is unique.

Answer: False


Question: Is the expression \( \frac{d}{dt}\phi\circ\gamma(t)=\frac{\partial}{\partial\xi^i}\phi(\gamma(t))\frac{d}{dt}\gamma(t) \) correct for the derivative of a curve \( \gamma: \mathbb{R} \to S \) on a manifold \( S \) with coordinate map \( \phi: S \to \phi(S) \subseteq \mathbb{R}^n \)?

Answer: Yes


Question: Let $f \in W^{1,p}(\mathbb{R}^d)$ be a Sobolev map with a weak derivative that is smooth, i.e., it has a representative $x \to df_x$ which is $C^{\infty}$ as a map $\mathbb{R}^d \to \mathbb{R}^d$. Determine if $f$ has a smooth representative.

Answer: Yes


Question: Let $N_1$ and $N_2$ be independent and identically distributed standard normal random variables. Determine if there exists a random variable $X$, dependent only on $N_1$ and satisfying $|X| < 2$, such that \( \mathbb{P}(N_1 + XN_2 > 0) \neq \frac{1}{2} \).

Answer: Yes


Question: Evaluate the integral: $$\int_{ - \pi /2}^{\pi /2} \frac{1}{2007^{x} + 1} \cdot \frac{\sin^{2008}x}{\sin^{2008}x + \cos^{2008}x} \, dx.$$

Answer: \dfrac{\pi}{4}


Question: Show that the function \( g(x) = \frac{4^x + x^2 - \ln(2) \cdot x - 1}{\tan(2x)} \) is continuous at \( x = 0 \), given that \( g(0) := \ln(\sqrt{2}) \). The function \( g(x) \) is defined on the intervals \( \left(\frac{-\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right) \).

Answer: \ln(\sqrt{2})


Question: Calculate \(11^{644} \mod 645\).

Answer: 1


Question: Calculate the inner product \( \langle xx^T, A^TA \rangle \) given the definition \( \langle A, B \rangle = \sum_{i=1}^n\sum_{j=1}^n a_{ij}b_{ij} \). Express your answer in terms of \( \|Ax\|^2_2 \).

Answer: \|Ax\|_2^2


Question: Compute the flux of the vector field \( F(x,y,z) = (2x-y^2) \mathbf{i} + (2x - 2yz) \mathbf{j} + z^2 \mathbf{k} \) through the surface consisting of the side and bottom of the cylinder of radius two and height two, i.e., \( \{(x,y,z) \mid x^2+y^2 = 4, 0 \le z \le 2\} \). Note that this surface does not include the top of the cylinder. Compute the surface integral \( \iint_{\mathbf{S}} \mathbf{F} \cdot \mathbf{n} \, dS \), where \( \mathbf{F} \) is the vector field above, \( \mathbf{S} \) is the bottom and side (but not the top) of the cylinder, and \( \mathbf{n} \) is the outward pointing unit normal vector to the surface.

Answer: 0


Question: Suppose $A$ and $B$ are square matrices such that $A^2 + B^2 = 2AB$. Determine whether $\det(AB - BA) = 0$.

Answer: 0


Question: Find the least possible sum of the digits of the function \( f(n) = p^4 - 5p^2 + 13 \), where \( p \) is an odd prime number.

Answer: 9


Question: Calculate \(\lim_{n\rightarrow \infty} \int g_n \, d\lambda\) where \(g_n(x) = g(x^n)\mathbb{1}_{[0,1]}(x)\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function. Express your answer in terms of \(g\).

Answer: g(0)


Question: Let \(f(x) = \sum_{k=0}^\infty a_kx^k\) be a power series with nonnegative coefficients and a radius of convergence of 1. Suppose \(f'(x_0)\) exists for \(x_0 \in (-1,1]\), considering the one-sided limit if \(x_0 = 1\). Is it always true that \(f'(x_0) = \sum_{k=0}^\infty ka_kx^{k-1}\)? Justify your answer for the case when \(x_0 = 1\).

Answer: Yes


Question: Find the value of $x$ in the equation $x^5 = 32$.

Answer: 2


Question: Evaluate the limit: \[ L = \lim_{x \rightarrow \infty} \left(\sqrt[11]{\prod_{n=1}^{11}\left(2x+\frac{n(2n+1)}{2}\right)}-2x\right) \]

Answer: 49


Question: Determine the period of the continuous-time signal \( y(t) = \sum_{n=-\infty}^{+\infty} e^{-|6t+n|} \).

Answer: \dfrac{1}{6}


Question: Find the limit: $$ \lim\limits_{x\to 0}\left( \frac{\sin x}{x} \right)^{\frac{1}{1-\cos x}} $$

Answer: e^{-\frac{1}{3}}


Question: Determine whether the following statement is true: \[ \lim_{h \to 0} \int_0^h |f(x+h)-f(x)| \, dx = 0. \]

Answer: True


Question: Determine the smallest integer \( k \) such that the approximation of the series \( S = \sum_{n=0}^{\infty}\frac{1}{e^n(n^2+1)} \) by the partial sum \( S_k = \sum_{n=0}^{k}\frac{1}{e^n(n^2+1)} \) satisfies the error bound \( R = \left| S - S_k \right| < 10^{-3} \).

Answer: 4


Question: Find the sum of the infinite series: \[ \frac{4}{1!} + \frac{8}{2!} + \frac{14}{3!} + \frac{22}{4!} + \cdots \]

Answer: 5e - 2


Question: Find the limit \( \lim_{n\to\infty} \int_0^n\left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n)) \, dx \).

Answer: \log 3


Question: Consider a symmetric positive definite matrix $X \in \mathbb{R}^{n \times n}$. Suppose there exists a matrix $Y \in \mathbb{R}^{n \times n}$ such that $X - Y^TXY$ is also symmetric positive definite. Prove that the spectral radius of $Y$, denoted as $\max{ |\lambda_i(Y)| }$, is less than 1, where $\lambda_i(Y)$ are the eigenvalues of $Y$.

Answer: \rho(Y) < 1


Question:
For an odd integer $k \geq 1$, let $F$ be the set of all entire functions $f$ such that \(f(x)= |x^k|\) for all \(x \in (-1,1)\). Determine the cardinality of the set $F$. Choose from the following options:

* $0$
* $1$
* $> 1$ but finite
* Infinite.

Answer: A


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

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


Question: If $m$ and $n$ are the smallest positive integers satisfying the relation \(\left(2\, \mathrm{cis} \frac{\pi}{6}\right)^m = \left(4\, \mathrm{cis} \frac{\pi}{4}\right)^n\), find the value of $m+n$. Here, $\mathrm{cis}\, \theta \equiv \cos \theta + i \sin\theta$. \(\mathrm{cis}\) represents the polar form of a complex number.

Answer: 72


Question: Let \( A \) be a real \( n \times n \) symmetric matrix with non-negative diagonal entries and non-positive off-diagonal entries. Additionally, assume that the sum of the entries in each column is strictly positive. Prove that \( \det(A) > 0 \).

Answer: \det(A) > 0


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

Answer: 2\sqrt{2} - 2


Question: Evaluate the integral \( \int _{\Gamma} \frac{\cos(2z)}{(z-\pi/4)^2}dz \), where \( \Gamma \) is the contour consisting of the sides of a triangle with vertices \( i \), \( -1-i \), and \( \pi -i \).

Answer: -4\pi i


Question: If $G$ is a nonabelian finite simple group, does $G$ contain a maximal subgroup that is not abelian?

Answer: Yes


Question: Does \(GL_5(\mathbb{R})\) have a subgroup of index 2?

Answer: Yes


Question: Let \( f \in L^1(\mathbb{D}) \) where \( \mathbb{D} \) is the unit disc in the complex plane. Suppose there exists \( r_0 \in (0,1) \) such that \( f(r_0e^{i\theta}) = \sum_{n=-\infty}^{\infty}\hat{f}(n){r_0}^ne^{in\theta} = 0 \) for every \( \theta \in (-\pi,\pi) \). Does this imply that \( \hat{f}(n) = 0 \) for every \( n \in \mathbb{N} \)?

Answer: Yes


Question: Determine the minimal degree $d$ such that there exists a map of degree $d$ from $S^2 \times S^2$ to $\mathbb{CP}^2$. Consider the cohomology ring of $\mathbb{CP}^2$ and the classes in $H^2(S^2 \times S^2)$ to find the minimal $d$.

Answer: 2


Question: Determine the Galois group $Gal(F/\mathbb{Q})$, where $F$ is the splitting field of the polynomial $X^4 - 2X^3 - 8X - 3$ over $\mathbb{Q}$.

Answer: S_3


Question: Evaluate the limit: \[ \lim_{x\to1} (\tan(\pi x/4))^{\tan(\pi x/2)} \]

Answer: \dfrac{1}{e}


Question: Determine the number of values of \( Z \) (real or complex) that simultaneously satisfy the following system of equations: \( 1 + Z^2 + Z^3 + \ldots + Z^{17} = 0 \) and \( 1 + Z^2 + Z^3 + \ldots + Z^{13} = 0 \).

Answer: 0


Question: Let $X$ and $Y$ be independent random variables with normal distributions $X \sim N(0, \sigma^2_X)$ and $Y \sim N(0, \sigma^2_Y)$. Calculate the probability $\mathrm{Pr}[X > Y]$. Express your answer in terms of $\sigma^2_X$ and $\sigma^2_Y$. \( \boxed{?} \)

Answer: \dfrac{1}{2}


Question: Evaluate the limit \( \lim_{n\to \infty} \left(\frac{\frac{a_n}{s_n}}{- \ln (1- \frac{a_n}{s_n})}\right) \) where \( a_n= \sqrt{n} \) for \( n \geq 1 \), and \( s_n= a_1+ a_2+ \cdots + a_n. \)

Answer: 1


Question: How many permutations of the set \( \{1, 2, \ldots, 8\} \) can be expressed as a product of a 1-cycle, two 2-cycles, and a 3-cycle?

Answer: 1680


Question: In the given configuration, circle ω touches circle Ω internally at point P. The center O of Ω is outside ω. Diameter XY of Ω is tangent to ω, with PY > PX. The line segment PY intersects ω at point Z, and it is given that YZ = 2PZ. Determine the magnitude of angle PYX in degrees.

Answer: 75


Question: In a sequence of independent flips of a fair coin, let \( N \) denote the number of flips until there is a run of three consecutive heads. Find \( P(N = 8) \), the probability that exactly 8 flips are needed to achieve this run of three consecutive heads.

Answer: \dfrac{13}{256}


Question: Does a rank 1 bounded operator from $\mathscr{K}:L^2([0,1])\to L^2([0,1])$ have at most 1 non-zero eigenvalue?

Answer: Yes


Question: Determine whether there exists a constant \( A \in \mathbb{R}^* \) such that for all functions \( f \in C^0([0, 1], \mathbb{R}) \), the inequality \( ||f||_1 \le A \cdot ||f||_2 \) holds, where \( ||f||_1 = \int_0^1 |f(x)|dx \) and \( ||f||_2 = \left(\int_0^1 (f(x))^2 dx\right)^{1/2} \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{y \rightarrow 0} 3y \int_{y^2}^{y} \frac{x^2 \sin(x^2 y^3)}{x^2 + \cos(x^2 y^3)} \, dx \] where \( x, y \in \mathbb{R} \) and \( |y| < 1 \).

Answer: 0


Question: Calculate the group of homomorphisms from \(\mathbb{Q}\) to \(\mathbb{Z}\), denoted as \(\text{Hom}_{\text{Grp}}(\mathbb{Q}, \mathbb{Z})\).

Answer: 0


Question: How many functions \( f: \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \to \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \) are there such that the 30th composition of \( f \) is the identity function, and 30 is the smallest such number for \( f \) to become an identity?

Answer: 120960


Question: Let $E$ be a separable Banach space and $A$ an uncountable subset of the unit sphere $S_E = \{x \in E \mid \|x\| = 1\}$. Must $A$ have an accumulation point?

Answer: Yes


Question: Consider the space of continuous functions on the interval \([0, 1]\), denoted as \(C[0,1]\). Do the supremum metric and the definite integral metric induce different topologies on this space?

Answer: Yes


Question: Consider the initial value problem (IVP): \( \frac{dy}{dt}=y^{2021}-y^{2019} \) with \( y(-2021)=\frac{1}{2019} \). Find \( \lim_{t \to \infty} y(t) \).

Answer: 0


Question: Given a constant symmetric tensor $V_{ij}$, calculate the product $V_{ij}V^{jk}$ by contracting the $j$ index. Express your answer as a matrix multiplication result.

Answer: V^2


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^\infty \left(\frac{n-1}{n}\right)^{n^2} z^n \).

Answer: e


Question: Given a quadratic Bezier curve with known endpoints and a specified length, along with the x-coordinate of the control point, determine the y-coordinate of the control point.

Answer: y_1


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

Answer: 0


Question: Find the exact area under the curve defined by the equation \([x+y] + [x-y] = 5\) with the constraints \(x \geq y\), \(x \geq 0\), and \(y \geq 0\), where \([\cdot]\) denotes the greatest integer function.

Answer: \dfrac{3}{2}


Question: Three circles have radii 1, 2, and 3, and each circle touches the other two. A smaller circle is constructed such that it touches all three of these circles. What is the radius of the smaller circle?

Answer: \dfrac{6}{23}


Question: Given the equations \(x = 3\sqrt{y\sqrt{3\sqrt{y\cdots}}}\) and \(y = 3\sqrt{x\sqrt{3\sqrt{x\cdots}}}\), find the value of \(x + y\).

Answer: 162


Question: Let $a_n = \frac{m}{n}$ for $m$ such that \( \frac{(m-1)!}{n^{m-1}} \le 1 < \frac{m!}{n^m} \). Find \( \lim_{n \to \infty} a_n \).

Answer: e


Question: Let \((X,Y)\) be a random variable distributed over \([0,1]\times[0,1]\) with the joint probability density function \(f(x,y)=6xy^2\). Calculate the probability \(P(XY^3 \leq \frac{1}{2})\).

Answer: \dfrac{3}{4}


Question: Let E be a subset of \( \mathbb{R} \) that has no isolated points, and let C be a countable subset of \( \mathbb{R} \). Does there exist a monotonic function on E that is continuous only at points in \( E \setminus C \)?

Answer: Yes


Question:
A machine consists of two components, whose lifetimes have the joint density function
\[f(x,y)=\begin{cases}
{1\over50}, & \text{for }x>0,y>0,x+y<10 \\
0, & \text{otherwise}
\end{cases}\]
The machine operates until both components fail. Calculate the expected operational time of the machine.

Answer: 5


Question: Let \(X, Y, Z\) be independent random variables, each uniformly distributed on the interval \([0, 1]\). Calculate the expected value of \((X + Y + Z)^2\).

Answer: \dfrac{5}{2}


Question: Two identical sticks, each with the same length and mass density, are used in an experiment. One stick is broken at a uniformly random point, and the other stick is supported at a uniformly random point. The pieces of the broken stick are placed on the ends of the supported stick. What is the probability that the shorter piece of the broken stick will tip downwards?

Answer: \dfrac{1}{4}


Question: Evaluate the surface integral \( \int_{S} (x^2 + y^2) \, dS \), where \( S \) is the part of the cone \( z = \sqrt{x^2 + y^2} \) with \( z \leq 2 \).

Answer: 8\sqrt{2}\pi


Question:
Let \( F: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} \) be the function defined by \( F(x,y) = \langle Ax, y \rangle \), where \( \langle , \rangle \) denotes the standard inner product on \( \mathbb{R}^n \) and \( A \) is an \( n \times n \) real matrix. Determine which of the following statements about the total derivative \( D \) of \( F \) is correct:

1) \((DF(x,y))(u,v) = \langle Au, y \rangle + \langle Ax, v \rangle\)
2) \((DF(x,y))(0,0) = (0,0)\)
3) \(DF(x,y)\) may not exist for some \((x,y) \in \mathbb{R}^n \times \mathbb{R}^n\)
4) \(DF(x,y)\) does not exist at \((x,y) = (0,0)\)

Answer: 1


Question: Determine the cardinality of the set formed by the union of all Cartesian products of the natural numbers with itself, i.e., the set of all sequences of natural numbers of any finite length. Is the cardinality of this set aleph-one or higher?

Answer: \aleph_0


Question: Determine the radius of convergence for the Taylor series expansion of the function \( f(x) = \frac{1}{1+x^2} \) about a point \( a \in \mathbb{R} \).

Answer: \sqrt{a^2 + 1}


Question: Consider forming a subset $X$ of $\mathbb{R}$ by the following process: for each $x \in \mathbb{R}$, toss a fair coin. If the result is heads, include $x$ in $X$. Assume that the coin tosses are independent. What is the probability that $X$ is a Lebesgue measurable subset of $\mathbb{R}$?

Answer: 0


Question: In triangle \( \triangle ABC \), \( \angle A = 30^\circ \) and \( BC = 13 \). Two circles \( \gamma_1 \) and \( \gamma_2 \) with radii \( r_1 \) and \( r_2 \), respectively, contain point \( A \) and touch side \( BC \) at points \( B \) and \( C \), respectively. Find the product \( r_1 r_2 \).

Answer: 169


Question: Evaluate the integral \( \int_{0}^{\infty}\frac{\sin^{2}(t)}{t^{2}}dt \) using the Laplace transform and the identity \( \int_{0}^{\infty}\frac{f(t)}{t}dt= \int_{0}^{\infty}\mathcal{L}\{ f(t) \}(u)du \).

Answer: \dfrac{\pi}{2}


Question: Let $X$ be a Banach space and $T: X \to X$ be a bounded and invertible operator. Determine whether the Fredholm index $\mathrm{ind}(T) = 0$. Provide a justification for your answer.

Answer: 0


Question: Let $f$ be a function satisfying the equation \( f\left(\frac{x}{2} + \frac{y}{2}\right) = \frac{f(x) + f(y)}{2} \) for all real numbers $x$ and $y$. Given that $f'(0) = -1$, find the value of $f(2)$. \( \boxed{?} \)

Answer: -2


Question: Determine whether the set of all polynomials \( P \in \mathbb{Q}[x] \) such that \( P(1) = P(3) = 0 \) forms an ideal. If it does, find the monic generator of this ideal.

Answer: x^2 - 4x + 3


Question: Calculate the cardinality of the product \( \prod_{0<\alpha<\omega_1}\alpha \).

Answer: 2^{\aleph_1}


Question: Let $G$ be a simple graph where each vertex has exactly 36 neighbors. For any two vertices $X$ and $Y$, if $X$ is a neighbor of $Y$, there are exactly 18 other vertices that are neighbors to both $X$ and $Y$. If $X$ is not a neighbor of $Y$, there are exactly 4 other vertices that are neighbors to both $X$ and $Y$. Determine the total number of vertices in $G$.

Answer: 190