Question: Find \( f'(8.23) \) where \( f(x) = 23|x| - 37\lfloor x \rfloor + 58\{x\} + 88\arccos(\sin x) - 40\max(x,0) \). Note: For a real number \( x \), \( \{x\} = x - \lfloor x \rfloor \) denotes the fractional part of \( x \).

Answer: 129


Question: Calculate the value of the following expression: \[ \frac{\int_0^\pi x^3\ln(\sin x)\,dx}{\int_0^\pi x^2\ln(\sqrt{2}(\sin x))\,dx} \]

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


Question: Find a number in the ring \(\mathbb{Z}[\sqrt{14}, \frac{1}{2}]\) that is not in \(\mathbb{Z}[\sqrt{14}]\). Provide an example of such a number.

Answer: \dfrac{1 + \sqrt{14}}{2}


Question: Let $P\subset\mathbb{R}^n$ be a convex lattice polytope. Does there always exist a lattice simplex $\Delta\subset P$ and an affine hyperplane $H\subset\mathbb{R}^n$ that separates $\Delta$ from the convex hull of the integer points of $P\setminus \Delta$? Equivalently, is there a degree one polynomial $h:\mathbb{R}^n\rightarrow\mathbb{R}$ that is positive on all the integer points of $\Delta$ and negative on all the integer points of $P\setminus \Delta$?

Answer: No


Question: The sides of a triangle have the combined equation \(x^2 - 3y^2 - 2xy + 8y - 4 = 0.\) The third side, which is variable, always passes through the point \((-5, -1)\). If the range of values of the slope of the third line, such that the origin is an interior point of the triangle, lies in the interval \((a, b)\), find the value of \(a + \frac{1}{b^2}\).

Answer: 24


Question: Determine the range of values for \(a \in \mathbb{R}\) such that the integral \(\int_0^\infty \frac{\arctan(5x)\,dx}{x^a}\) converges.

Answer: (1, 2)


Question: Calculate the volume of the region $K \subset \mathbb{R}^3$ defined by the points $(x,y,z)$ satisfying the conditions: \[ z(1-z) \leq \sqrt{x^2+y^2} \leq 2z(1-z),\ x \geq y. \] Use cylindrical coordinates to set up and evaluate the triple integral for the volume of $K$. Express your answer as a single numerical value.

Answer: \dfrac{\pi}{20}


Question: Consider a power series \(\sum_{k=0}^\infty a_k x^k\) with a radius of convergence of 1. Assume that \(\sum_{k=0}^\infty a_k = \infty\). Determine whether \(\lim_{x\to 1^-} \sum_{k=0}^\infty a_k x^k = \infty\).

Answer: \infty


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

Answer: 1


Question: Provide an example of a set that is not a $G_{\delta}$ set.

Answer: \mathbb{Q}


Question: Let $D \to A$ be a finite type algebra with $D$ a domain. Suppose $V \subset \operatorname{Spec}A$ is open and dense. Determine if $\dim V = \dim A$.

Answer: \dim V = \dim A


Question: Let $X$ be a Lebesgue-measurable subset of the real line $\mathbf{R}$ such that for any non-zero rational number $q$, the set $qX = X$. Assume that the Lebesgue measure $\mu(X)$ is greater than zero. Does it follow that the Lebesgue measure of the complement of $X$, $\mu(\complement X)$, is zero?

Answer: Yes


Question: Let $L = \mathbb{Q}(\alpha, \beta)$ where $\alpha$ satisfies $\alpha^3 + \alpha + 1 = 0$ and $\beta$ satisfies $\beta^2 + \beta - 3 = 0$. Find the degree of the field extension $[L : \mathbb{Q}]$. \(\boxed{?}\)

Answer: 6


Question: Determine the fundamental group of a sphere with two points removed.

Answer: \mathbb{Z}


Question: Evaluate the contour integral \( \int_{C}\frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy \) where \( C \) is the triangle with vertices at \((5,5), (-5,5),\) and \((0,-5)\) traversed counterclockwise. Note: Consider the domain of the integrand and the applicability of any theorems used.

Answer: 2\pi


Question: Evaluate the contour integral \( \oint_C \frac{dz}{2-\sin z} \) where the contour \( C \) is a circle of radius 0.5 centered at \( z=i \).

Answer: 0


Question: In a simple linear regression model under the assumptions of the Gauss-Markov theorem, is it necessary for the sample average of the dependent variable to be zero for the least squares estimators of the intercept and slope to be uncorrelated?

Answer: No


Question: Find a degree 4 polynomial \( f(x) \in \mathbb{Q}[x] \) such that \( f(\sqrt{2} + \sqrt{3}) = 0 \).

Answer: x^4 - 10x^2 + 1


Question: Find the normal curvature of the helicoid \(\sigma(u, v) = (u \cos v, u \sin v, v)\) in the direction of a general unit tangent vector at the origin \(p = (0, 0, 0)\).

Answer: -2ab


Question: If $G$ is a group with order $364$, does it have a normal subgroup of order $13$? Justify your answer.

Answer: Yes


Question: Evaluate the limit: $$ \lim_{(x,y) \to (0,0)}  (x+y+4)e^{-1/(x^2+y^2)} $$

Answer: 0


Question: Find the degree of the field extension \(\mathbb{Q}(\epsilon)|\mathbb{Q}(\epsilon^3)\), where \(\epsilon\) is a 9th primitive root of unity.

Answer: 3


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

Answer: e^6


Question: Solve the equation \(2a^x = b^x + c^x\) for \(x\), given that \(a, b, c\) are strictly positive real numbers and \(c = \sqrt{ab}\).

Answer: 0


Question: Evaluate the integral \( I = \int_{-\infty}^{\infty} \frac{\cos{x}}{x^2+1} \, dx \) using complex analysis techniques.

Answer: \dfrac{\pi}{e}


Question: Determine whether the following statement is true or false: \[ \lim_{n\to\infty}\int_0^1\frac{\sin(2n\pi x)}{\ln x}\,dx=\frac\pi2. \]

Answer: True


Question: Evaluate the surface integral \( \iint_{S} G(x, y, z) \, dS \) where \( G(x, y, z) = (1-x^2-y^2)^{\frac{3}{2}} \) and \( S \) is the hemisphere defined by \( z = (1-x^2-y^2)^{\frac{1}{2}} \).

Answer: \dfrac{\pi}{2}


Question: Let \( A = \begin{pmatrix} 1 & 1 & 1 & \cdots & 1 \\ 1 & 1-x & 1 & \cdots & 1 \\ 1 & 1 & 2-x & \cdots & 1 \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 1 & 1 & 1 & \cdots & n-1-x \end{pmatrix} \) be an \( n \times n \) matrix with \( n \geq 3 \). Solve the equation \( \det(A) = 0 \) for \( x \in \mathbb{R} \) and find the largest root.

Answer: n-2


Question: Consider a game played with a fair coin where you start at 0. Each heads flip moves you forward one unit (+1), and each tails flip moves you backward one unit (-1). The game ends when you reach either -32 or +63. Calculate the expected number of coin flips required for the game to end.

Answer: 2016


Question: Determine the number of ways to arrange red balls in 6 places such that in any 3 consecutive places, there are at least 2 red balls. Use the minimum number of red balls possible.

Answer: 6


Question: Find the residue of the function \( f(z) = \frac{\cos(z)}{(z-1)^2} \) at \( z = 1 \).

Answer: -\sin(1)


Question:
Given the differential system with initial conditions:
\[ y''_1 = -10y_1 + 6y_2 \]
\[ y''_2 = 6y_1 - 10y_2 \]
\[ y_1(0) = 1, \quad y_2(0) = 0, \quad y_1'(0) = 0, \quad y_2'(0) = 0 \]
Find the value of \( y_2(\pi/2) \).

Answer: -1


Question:
Given a triangle ABC with sides a, b, and c, and half-perimeter p, such that \(\frac{p-a}{11} = \frac{p-b}{12} = \frac{p-c}{13}\), find the value of \((\tan\frac{A}{2})^2\).

(A) \(\frac{143}{432}\)  (B) \(\frac{13}{33}\)  (C) \(\frac{11}{39}\)  (D) \(\frac{12}{37}\)

Answer: B


Question: Let $x_{n}$ be a sequence of operators in a von Neumann algebra $M$, and let $\Omega$ be a cyclic vector for $M$. If $x_{n}\Omega$ converges in the Hilbert space $\mathcal{H}$, does there exist a subsequence $\{y_{n}\}$ of $\{x_{n}\}$ that is uniformly bounded in operator norm?

Answer: No


Question: Let $R$ be a commutative ring with unity that is not a field. Suppose $I \subset R$ is a proper ideal such that every element of $R$ not in $I$ is invertible in $R$. Determine the number of maximal ideals of $R$.

Answer: 1


Question: Find the remainder when \(20^{20}\) is divided by 2020.

Answer: 600


Question: Calculate the limit: \[ \lim\limits_{x\to-\infty}\operatorname{arccotg}\frac{x}{(x^2-4)^{\frac12}} \]

Answer: \dfrac{3\pi}{4}


Question: Evaluate the integral: \[ \int_{-1}^{1} \LegP[n, x] \frac{d}{dx} (\LegP[n+1,x]) \, dx. \]

Answer: 2


Question:
Find the value of \(a \in \mathbb{R}\) such that the function \(f(x)\) is differentiable at \(x = 0\), where:
\[
f(x) = \begin{cases}
  x + xe^{\frac{1}{x}}, & x < 0, \\
  0, & x = 0, \\
  \frac{a-2\cos x}{\sin x}, & x > 0.
\end{cases}
\]

Answer: 2


Question: Compute the sum of the series: \[ \sum_{n=1}^\infty n^2 \cdot \left(\frac{2}{3}\right)^n \]

Answer: 30


Question: Given the equations \(x+y+z = 7\), \(x^3+y^3+z^3 = 7\), and \(x^2+y^2+z^2 = 49\), find the value of \(xyz\).

Answer: -112


Question: Determine the values of $r$ for which there exists a finite $r$-dense subset in the unit ball of $C([0,1])$ with respect to the supremum norm $d_{	ext{inf}}$. A subset $Y \subset (X, d)$ is $r$-dense if for every $x \in X$, there exists a $y \in Y$ such that $d(x, y) < r$. Provide the set of values for $r$ where such a finite subset exists.

Answer: (1, \infty)


Question: Let $G$ be a finite group and $H$ a subgroup of $G$. Is there a subgroup $K$ of $G$ such that $G = HK$ and $|G| = |H||K|$?

Answer: No


Question: Evaluate the limit: \[ \lim_{x\to \infty} \left((x+5)\tan^{-1}(x+5) - (x+1)\tan^{-1}(x+1)\right) \]

Answer: 2\pi


Question: Given the cubic equation \(x^3 - 6x^2 + 5x - 7 = 0\), find the value of \(\frac{1}{r^2} + \frac{1}{s^2} + \frac{1}{t^2}\), where \(r, s, t\) are the roots of the equation.

Answer: -\dfrac{59}{49}


Question: Find the limit of \( \frac{x^2 - \log(1+x^2)}{x^2 \sin^2 x} \) as \( x \to 0 \).

Answer: \dfrac{1}{2}


Question: Find the analytic solution to the advection equation \( \frac{\partial{u(x,t)}}{\partial{t}} + c \frac{\partial{u(x,t)}}{\partial{x}}=0 \) with the initial condition \( u(x,0)=\sin(x) \).

Answer: u(x, t) = \sin(x - ct)


Question: The series \( \sum_{n=1}^{\infty} \left(\frac{2z}{z+1}\right)^n \) converges in some neighborhood of 0 to a function that admits an analytic continuation \( f(z) \) to a neighborhood of the point \(-1\). Determine the value of \( f(-1) \).

Answer: -1


Question: Find the minimum value of \( f(b) = \max\left|\sin x + \frac{2}{3 + \sin x} + b\right| \) for all \( b \in \mathbb{R} \).

Answer: \dfrac{3}{4}


Question: For which values of \( \lambda \) does the integral \( \int_{0}^{1} t^\lambda \, \mathrm{d}t \) exist as a Lebesgue integral?

Answer: \lambda > -1


Question: Determine the number of irreducible polynomials of degree 3 over the field \( \mathbb{Z}_3 = \{\overline{0}, \overline{1}, \overline{2}\} \).

Answer: 8


Question: Determine the value of \( r > 0 \) such that with probability one, for all sufficiently large \( n \), the inequality \( M_n \le r\sqrt{\log n} \) holds, where \( M_n = \max\{|B_t - B_{n-1}| : n - 1 \le t \le n\} \) and \( B_t \) is a standard Brownian motion.

Answer: \sqrt{2}


Question: Let $u \in C^{\infty}(\mathbb{R}^3)$ be a harmonic function. Suppose $u$ has no critical points outside the unit ball but has at least one critical point inside the unit ball. Does it follow that $u$ is a polynomial?

Answer: Yes


Question:
Let $U_1, U_2, \ldots, U_n$ be independent and identically distributed random variables from a uniform distribution on the interval (0, 1). Let $U_{(1)}, U_{(2)}, \ldots, U_{(n)}$ denote the order statistics of these random variables. For a given $n \geq 1$ and $0 \leq \alpha \leq 1$, find the probability:
\[ P \left(\bigcap_{j=1}^{n} \left \{U_{(j)} > \frac{\alpha j}{n} \right \} \right) \]

Answer: 1 - \alpha


Question: Evaluate the limit: \[ \lim_{n \to \infty} n \int_2^e (\ln x)^n \, dx. \]

Answer: e


Question: Given the function \( f \) such that \( f(f(x)) = 4x + 3 \) for all positive integers \( x \), and \( f(5^k) = 5^k \times 2^{k-2} + 2^{k-3} \) for exactly one positive integer \( k \), find the value of \( f(2015) \).

Answer: 4031


Question: Find an integer $0 \le x < 527$ such that $x^{37} \equiv 3 \pmod{527}$. What is the value of $x$?

Answer: 148


Question:
Let $A$ be a self-adjoint operator on a functional Hilbert space $\mathscr{H}$. Define the norm $N(A)$ as follows:
\[ N(A) := \sup\{ |\langle A\widehat{k}_{\lambda}, \widehat{k}_{\lambda} \rangle| : \lambda \in \Omega \}, \]
where $\widehat{k}_{\lambda}$ is the normalized reproducing kernel of $\mathscr{H}$. Is it true that for any integer $n$, the equality $N(A^n) = N^n(A)$ holds?

Answer: No


Question: Find the limit \( \lim_{n \to \infty} A_n \) where \( A_n = \sum_{k = 1}^n \frac{k^6}{2^k} \).

Answer: 9366


Question: Evaluate the limit \( \lim_{x\rightarrow 0}\frac{e^{x^2}-\ln(1+x^2)-1}{\cos2x+2x\sin x-1} \) using Maclaurin's series expansion.

Answer: 3


Question: Determine whether the Galois group of the extension \( \mathbb{Q}(\alpha)/\mathbb{Q} \), where \( \alpha \) is a root of the polynomial \( h(X) = X^3 + 3X + \sqrt{3} \in \mathbb{Q}(\sqrt{3})[X] \), is isomorphic to \( C_6 \) or \( S_3 \). Justify your answer.

Answer: S_3


Question: Determine the number of ordered pairs \((x, y)\) that satisfy the congruence \(x^2 - y^2 \equiv a \pmod{p}\), where \(p\) is an odd prime and \(p\) does not divide \(a\).

Answer: p - 1


Question: Calculate the conditional expectation \( E(2X | Y) \) given the joint probability density function \( f(x,y) = 4e^{-2y} \) for \( 0 < x < y \) and \( y > 0 \).

Answer: Y


Question: Determine the number of elements of order 10 in the symmetric group $S_{10}$. Consider permutations written in disjoint cycles, where the order of the permutation is the least common multiple of the cycle lengths, and the sum of the cycle lengths is 10.

Answer: 514080


Question: Solve the differential equation $f' = f$ using the Fourier transform method. Verify that the solution is $f(x) = e^x$.

Answer: e^x


Question: Consider three independent Bernoulli random vectors $x, y, w$ of length $n$, where each entry follows the Bernoulli distribution $B$ with $P(B=0)=P(B=1)=\frac{1}{2}$. Let $X = \langle x, w \rangle$ and $Y = \langle y, w \rangle$, where $\langle \cdot, \cdot \rangle$ denotes the standard scalar product. Determine the expectation $\mathbb{E}((X-Y)^2)$ as a function of $n$.

Answer: \dfrac{n}{4}


Question: Determine the smallest value of $t$ for which the solution to the partial differential equation \( u_t + u_x = u^2 \) with initial condition \( u(0,x) = \cos(x) \) becomes undefined. The solution is given by \( u(t,x) = \frac{\cos(x-t)}{1-t\cos(x-t)} \). Identify the point \((t, x)\) where this occurs.

Answer: (1, 1)


Question: Does there exist an Euclidean domain that does not have a multiplicative Euclidean function?

Answer: Yes


Question:
Consider a positively oriented circle $S$ of radius 1 centered at the origin. Let $\phi(x)$ and $\psi(y)$ be smooth functions defined on $\mathbb{R}$. Determine which of the following integrals is zero:

I. $\int_S (\phi(y) + \psi(x)) \, dy$

II. $\int_S \phi(xy) \, (y \, dx + x \, dy)$

III. $\int_S \phi(x) \psi(y) \, dx$

Answer: II


Question: Let $G$ be a Lie group and $H$ a normal subgroup of $G$. Let $\mathfrak{g}$ and $\mathfrak{h}$ be the corresponding Lie algebras. Given that for all $X \in \mathfrak{g}$ and $h \in H$, $e^{tX} h e^{-tX} \in H$, does it follow that for all $X \in \mathfrak{g}$ and $Y \in \mathfrak{h}$, $e^{tX} Y e^{-tX} \in \mathfrak{h}$? In other words, can the adjoint map $\text{Ad}_{e^{tX}} : \mathfrak{g} \rightarrow \mathfrak{g}$ be restricted to $\mathfrak{h}$?

Answer: Yes


Question: Does there exist a non-injective homomorphism from a finite group $G$ to $GL(V)$, where $V$ is a field?

Answer: Yes


Question: Evaluate the limit without using L'Hôpital's rule: \[ \lim_{x \to 0}\frac{\sin^2{x}+2\ln\left(\cos{x}\right)}{x^4} \]

Answer: -\dfrac{1}{2}


Question: Determine whether the limit \( \lim_{n \rightarrow \infty} \frac{p(n)}{p(n-5)} \) exists, where \( p(n) \) denotes the partition function. If the limit exists, is it equal to 1?

Answer: 1


Question: Let $A$ be a $3 \times 3$ matrix with real entries. If $A$ commutes with all $3 \times 3$ matrices with real entries, how many distinct real eigenvalues does $A$ have?

Answer: 1


Question: Evaluate the integral \( \int_{\mathbb{R}^3} e^{-\left \| x \right \|}d^3x \).

Answer: 8\pi


Question: Calculate \((308!)^2 \mod 617\).

Answer: 616


Question: Suppose a polynomial with rational coefficients takes an integer value for all non-decreasing integers. Must it be true that the polynomial takes an integer value for all integer inputs? Provide a justification for your answer.

Answer: Yes


Question: Given the function \( y = f(x) \) with the property that its derivative's domain is \( \mathbb{R} \) and \( xf'(x) = e^x - 1 \) for all \( x \in \mathbb{R} \), and knowing that \( f(1) = 0 \), determine the value of \( \int_0^1 xf(x) \, \mathrm{d}x \).

Answer: -\dfrac{1}{4}


Question: Given matrices $A$ and $B$ in $M_n(\mathbb{R})$ such that $AB=BA$, where the minimal polynomial of $A$ is $x^2-1$ and the minimal polynomial of $B$ is $x^2+1$, find the minimal polynomial of the matrix product $AB$. Provide your answer as a polynomial.

Answer: x^2 + 1


Question: For what values of \( p \) is the limit comparison test invalid for the sequences \( a_n = 1 - 2^{-1/n} \) and \( b_n = \frac{1}{n^p} \)?

Answer: p \neq 1


Question: Evaluate the integral \( \int_{\ln 2}^{\ln 3} f(x) \, dx \), where \( f(x) = e^{-x} + 2e^{-2x} + 3e^{-3x} + \cdots \).

Answer: \dfrac{1}{2}


Question: Determine whether every binary number can be found at the start of some power of 3. Provide a justification for your answer.

Answer: Yes


Question:
Determine the number of integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 25\) subject to the constraints:
\(-2 \le x_1 \le 5\), \(1 \le x_2 \le 4\), \(-2 \le x_3 \le 6\), \(3 \le x_4 \le 7\), \(2 \le x_5 \le 8\), and \(3 \le x_6 \le 7\).

Answer: 3025


Question: Find the integer $x$ such that $x^{17} \equiv 20 \pmod{1001}$. Provide your answer as the smallest non-negative integer satisfying the equation.

Answer: 531


Question: Evaluate the limit \( \lim_{m\rightarrow \infty}\bigg[m^3\int^{2m}_{m}\frac{xdx}{x^5+1}\bigg] \) for \( m \in \mathbb{N} \).

Answer: \dfrac{7}{24}


Question: Let $H$ be a Hilbert space, and let $H_n$ be an increasing sequence of finite-dimensional subspaces such that $\bigcup _{n=1}^{\infty} H_n$ is dense in $H$. Suppose $P_n$ is the projection from $H$ to $H_n$ and $T:H\to H$ is a bounded linear operator such that $||TP_n-P_nT||\leq \frac{1}{2^{n+1}}$. Prove that there exists a linear operator $S$ that commutes with all $P_n$, and a compact operator $K$ such that $||K||<1$ and $T=S+K$. Provide the construction of $S$ and $K$.

Answer: T = S + K


Question: Find an upper bound for the recurrence relation \( T(x) \leq 4 T(\left \lfloor{\frac{x}{2}} \right \rfloor) + x \) using the master theorem, where \( x \in \mathbb{N} \).

Answer: O(x^2)


Question: Calculate the surface integral \( \iint_D \alpha \wedge \gamma \) where \( \alpha = x\,dx + y\,dy + z\,dz \) and \( \gamma = xy\,dz \). The surface \( D \) is the square defined by \( 0 \leq x \leq 1, 0 \leq y \leq 1, z = 1 \) and is oriented with the upward normal. Provide your answer as a single number.

Answer: 0


Question: Count the number of 10-digit numbers using the digits from \( \{1, 2, 3, 4\} \) such that no two adjacent digits differ by 1.

Answer: 21892


Question: Does there exist an injective function \(f: \mathbb{R} \to \mathbb{R}\) such that for every \(c \in \mathbb{R}\), there is a real sequence \((x_n)\) with \(\lim(f(x_n)) = c\), but \(f\) is neither continuous nor surjective?

Answer: Yes


Question: Given the expansion \((2x^2 + 3x + 4)^{10} = \sum_{r=0}^{20} a_r x^r\), find the ratio \(\frac{a_7}{a_{13}}\).

Answer: 8


Question: Let \( f:[0,1] \to \mathbb{R} \) be a continuous function such that \( \int^{x}_{0} f = \int^{1}_x f \) for all \( x \in [0,1] \). Prove that \( f(x) = 0 \) for all \( x \in [0,1] \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\rightarrow{\frac\pi2 }} (\sec(x) \tan(x))^{\cos(x)} \] without using L'Hôpital's rule.

Answer: 1


Question: Evaluate the integral \( \int_0^\infty \frac{x^2 \, dx}{(x^4+4)^2} \) using the residue theorem.

Answer: \dfrac{\pi}{64}


Question: Determine whether the integer part $I$ of the expression $(8+3\sqrt{7})^n = I + F$, where $F$ is a proper fraction, is even or odd.

Answer: odd


Question: Find the circulation of the vector field \( \vec{F}(x,y) = \langle x^{2018} - 233x + y\cos x, 5x + \sin x + e^{2018y - 233} \rangle \) along the circle traced by \( \vec{r}(\theta) = \langle 3\cos\theta + 1, 3\sin\theta - 1 \rangle \) from \( \theta = 0 \) to \( \theta = 2\pi \) using Green's theorem.

Answer: 45\pi


Question: Compute the limit: \[ \lim_{n \to \infty} \frac{n + \lfloor \sqrt[3]{n} \rfloor^3}{n - \lfloor \sqrt{n+9} \rfloor} \]

Answer: 2


Question: Does there exist an orthonormal basis for $L^2(\mathbb{R}^3)$?

Answer: Yes


Question: Evaluate the integral: \[ \int_0^2 \left((1+x^3)^{1/2} + (x^2+2x)^{1/3}\right) \, dx \]

Answer: 6


Question: Let $X, Y, Z, T$ be independent random variables each following a standard normal distribution $N(0,1)$. Consider two circles centered at the origin with radii $\sqrt{X^2 + Y^2}$ and $\sqrt{Z^2 + T^2}$, respectively. Let $A$ be the area of the annulus formed by these two circles. Calculate the expected value $\mathbb{E}(A)$ of the area of the annulus.

Answer: 2\pi


Question: Is it possible to define a binary operation on the set of prime numbers \( \{2, 3, 5, 7, \ldots\} \) such that it forms a group with the identity element being 2?

Answer: Yes


Question:
Let $R$ be a commutative ring and $J_1, J_2$ be two non-zero proper ideals of $R$. Consider the following two statements:

$P$: For any $r_1, r_2 \in R$, there exists a unique $r$ in $R$ such that $r - r_1 \in J_1$ and $r - r_2 \in J_2$.

$Q$: $J_1 + J_2 = R$.

Which of the following options are correct?
1. Statement $P$ implies $Q$, but $Q$ does not imply $P$.
2. Statement $Q$ implies $P$, but $P$ does not imply $Q$.
3. Neither $P$ implies $Q$ nor $Q$ implies $P$.
4. Statement $P$ implies $Q$ and $Q$ implies $P$.

Choose the correct option.

Answer: 1


Question: Evaluate the double integral: \[ \int_0^{\pi/2}\int_{x}^{\pi/2} \frac{\cos{y}}{y}\, dy \, dx \]

Answer: 1


Question: Compute \( \lim\limits_{x \to \infty} \frac{\ln (x!)}{x \ln (x)} \).

Answer: 1


Question: Determine the cardinality of Sym(X), the group of permutations of an infinite set X with cardinality k.

Answer: 2^k


Question: Let $p$ and $\ell$ be distinct prime numbers. Consider a pro-$p$-group $G$ acting continuously on a finite-dimensional $\mathbb{Q}_\ell$-vector space $V$. Assume the action of $G$ on $V$ is unipotent, meaning there exists an integer $n$ such that $(\sigma - 1)^n = 0$ for all $\sigma \in G$. Does it follow that the action of $G$ on $V$ is trivial?

Answer: Yes


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

Answer: 1


Question: Find the largest integer $m$ such that \( \sum_{n=0}^m 9(n+1)10^n \le 10^{100000} \).

Answer: 99994


Question: Find the multiplicative inverse of the polynomial $a(x) = x+1$ in the Galois Field $GF(8)$, where $GF(8)$ is defined by the polynomial $G(x) = x^3 + x + 1$. Provide your answer as a polynomial in $GF(8)$.

Answer: x^2 + x


Question: Evaluate the integral \( \int _{0} ^ {\infty} f(x+\frac{1}{x})\frac{\ln x}{x} \, dx \).

Answer: 0


Question: Evaluate the limit \( \lim_{x \to 0} \left( \frac{a^x + b^x}{2} \right)^{1/x} \) where \( a, b > 0 \).

Answer: \sqrt{ab}


Question: Compute the limit: $$\lim_{\Delta t\to 0}\frac{P(X(\Delta t)=1)}{P(X(\Delta t)\ge1)}$$ where $X(t)$ is a Poisson process with intensity $\lambda = 3$.

Answer: 1


Question: Evaluate the contour integral \( \oint_C \frac{1}{2z+1} \mathrm{d}z \), where \( C \) is the unit square with diagonal corners at \(-1 - i\) and \(1 + i\).

Answer: \pi i


Question: Let $N$ be a Poisson random variable with parameter $z$. Find the expected value of the product $E[N(N-1)(N-2)\cdots(N-k+1)]$. Express your answer in terms of $z$ and $k$.

Answer: z^k


Question: Can a compact Lie group have a non-compact Lie subgroup? Provide a justification for your answer.

Answer: Yes


Question:
Show that Kendall's tau for the Clayton copula, given by \( C(x,y)=(\sqrt{x}+\sqrt{y}-1)^2 \) when \( \theta =-\frac{1}{2} \), is \( -\frac{1}{3} \). Use the formula:
\[ \rho_{\tau}=4 \int_0^1 \int_0^1 C(x,y)\,dC(x,y)-1=4 \int_0^1 \int_0^1(\sqrt{x}+\sqrt{y}-1)^2\frac{1}{2 \sqrt{xy}}\,dx\,dy-1 \]

Answer: -\dfrac{1}{3}


Question: Determine the smallest number of data points, \( n \), such that the kernel matrix \( K \) is guaranteed to be singular for any matrix \( X \in \{1, 2, \ldots, m\}^{d \times n} \) and any kernel function \( k : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R} \).

Answer: m^d + 1


Question: Evaluate the integral \( \int_{0}^{\pi} D_{n}(y) \, dy \) where \( D_{n}(y) = \frac{1}{2\pi} \frac{\sin((n+\frac{1}{2})y)}{\sin(\frac{y}{2})} \). Show that the result is \( \frac{1}{2} \).

Answer: \dfrac{1}{2}


Question:
Prove that for any integer \( n \geq 0 \), the following identity holds:
\[ 3^n = \sum_{k=0}^n{(-1)^k\binom{n}{k}4^{n-k}}. \]

Answer: 3^n


Question: Let $(X, \tau)$ be a Hausdorff space. Define $[X]^2 = \{\{x,y\}: x,y\in X \text{ and } x\neq y\}$. For disjoint open sets $U, V \in \tau$, let $[U,V] = \{\{x,y\} \in [X]^2: x\in U \text{ and } y\in V\}$. Consider the topology $[\tau]^2$ on $[X]^2$ generated by the basis $\{[U,V]: U,V\in \tau \text{ and } U\cap V =\emptyset\}$. Is the space $([X]^2, [\tau]^2)$ Hausdorff?

Answer: Yes


Question: Let \( A \) and \( B \) be two square matrices of the same order such that \( A^2 = I \) and \( B^2 = I \), where \( I \) is the identity matrix. If \( \vert A \vert + \vert B \vert = 0 \), find the value of \( \vert A + B \vert \). Here, \( \vert A \vert \) denotes the determinant of matrix \( A \).

Answer: 0


Question: Determine whether the analytic function \( f(z) = \frac{2z-1}{z-2} \) defined on the domain \( D = \{ z \in \mathbb{C} : |z| < 1 \} \) satisfies the inequality \( |f(z)| \leq 1 \) for all \( z \in D \).

Answer: Yes


Question: Determine the value of the Eisenstein series of weight 2 for the lattice of Gaussian integers, given by \( G_{2} = \sum_{w \in \mathbb{Z}[i] - \{0\}} w^{-2} \). Is there a simplified or 'nice' value for this series?

Answer: 0


Question: Let \( \mathbf{A} \) be an \( m \times n \) matrix and \( \mathbf{B} \) be a symmetric \( n \times n \) matrix, where \( m < n \). Given that \( \operatorname{Rank}(\mathbf{A}) = \operatorname{Rank}(\mathbf{B}) = m \), determine the rank of the matrix product \( \mathbf{ABA}^\top \). Is it true that \( \operatorname{Rank}(\mathbf{ABA}^\top) = m \)?

Answer: m


Question: Given a Brownian motion $W_t$, does there exist another Brownian motion $W^*_t$ such that $W_t = W^*_{t+t_0} - W^*_{t_0}$ for some $t_0$? Provide a justification for your answer.

Answer: Yes


Question: If \( \theta: R \to S \) is an isomorphism of topological rings, does the restriction \( \theta|_{R^{\times}}: R^{\times} \to S^{\times} \) result in a topological group isomorphism when restricted to their groups of units?

Answer: Yes


Question: Determine the number of unique permutations of a list containing 5 A's and 6 B's, considering permutations equivalent under a cyclic shift. For example, the permutation [A, A, B, A, B, B, B, A, B, B, A] is considered equivalent to [A, B, A, B, B, B, A, B, B, A, A].

Answer: 42


Question: Determine the value of \( \lim\sup_{k\to\infty}\frac{c_k}{k} \), where \( c_k \) is the maximum chromatic number of a \( k \)-regular graph.

Answer: 1


Question: Let \( G \) be a group with order \( |G| = a \). Suppose \( H_1 \) and \( H_2 \) are subgroups of \( G \) with orders \( |H_1| = b \) and \( |H_2| = c \), respectively, such that \( bc = a \) and \( H_1 \cap H_2 = \{ e \} \). Is it true that \( H_1 H_2 = G \)?

Answer: Yes


Question:
Given a mixed random variable $X$ with the cumulative distribution function:
\[ F(x) = \begin{cases} 0, & x < 0 \\ x^2, & 0 \leq x < 1/2 \\ x, & 1/2 \leq x < 1 \\ 1, & x > 1 \end{cases} \]
Calculate the variance of $X$, $\operatorname{Var}(X)$. Include the steps to find $\mathbb{E}[X]$ and $\mathbb{E}[X^2]$ in your solution.

Answer: \dfrac{13}{288}


Question: Express the integral \( I_{n} = \int_{\mathbb{R}^{n}} \exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x \) in terms of \( I_1 \), where \( \left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots x_{n}^{2}\right)^{\frac{1}{2}} \) is the Euclidean norm.

Answer: I_n = (I_1)^n


Question: Find the positive integer $k$ such that the function \( f(19, k) = 20 \), where \( a_{n} = 4 - 3n \) for all integers \( n \geq 1 \) and \( f(x, y) = x + \sum_{i=1}^{\infty}\left [\left(\frac{\prod_{j=1}^{i}a_{j}}{3^i\cdot i!} \right )x^{a_{i+1}}y^{i}\right] \) for all real \( x \) and \( y \).

Answer: 1141


Question: Determine the number of group homomorphisms \( \phi: S_5 \to C_5 \), where \( S_5 \) is the symmetric group on 5 elements and \( C_5 \) is the cyclic group of order 5.

Answer: 1


Question: Determine the maximum value of \( C \) such that for any continuously differentiable function \( f \) on the interval \([0,1]\) with \( f(0) = f(1) = 0 \), the inequality \( \max_{x \in [0,1]} |f'(x)| \geq C \int_0^1 |f(x)| \, dx \) holds.

Answer: 4


Question: Given the function \( H(s) = \frac{\zeta(s)}{\phi(q)} \sum_{\chi \mod{q}} L(s,\chi) = \sum_{n=1}^{\infty} \frac{h(n)}{n^s} \), determine the smallest integer \( n \) as a function of \( q \) such that \( h(n) \neq 1 \).

Answer: q + 1


Question: A bakery makes doughnuts and sprinkles half of the surface area with sugar. Given that the doughnut is a torus with minor radius $r$ and major radius $R$, where $R=ar^{-3/2}$, the surface area is $A = 4 R r \pi^2$ and the volume is $V= 2R\pi^2r^2$. If $k$ is the price of dough and $l$ is the price of sugar, minimize the total cost $K = kV + \frac{lA}{2}$. Find the value of $r$ that minimizes the total cost $K$. (0<r<R)

Answer: \dfrac{l}{k}


Question: Consider the function \((a_{1},a_{2},\dots) \mapsto \liminf_{n}\sum_{t=1}^{n}a_{t}\)\ where \(a_{t} \geq 0\) for all \(t\). Determine if \(\frac{\partial}{\partial a_{t}} \left( \liminf_{n}\sum_{t=1}^{n}a_{t} \right) = 1\) for every \(t \in \mathbb{N}\).

Answer: No


Question:
Let \(f: (1,4) \to \mathbb{R}\) be a uniformly continuous function and let \(\{a_n\}\) be a Cauchy sequence in \((1,2)\). Define \(x_n = a_n^2 f(a_n^2)\) and \(y_n = \frac{1}{1+a_n^2} f(a_n^2)\) for all \(n \in \mathbb{N}\). Determine which of the following statements is true:

(A) Both \(\{x_n\}\) and \(\{y_n\}\) must be Cauchy sequences in \(\mathbb{R}\).
(B) \(\{x_n\}\) must be a Cauchy sequence in \(\mathbb{R}\) but \(\{y_n\}\) need not be a Cauchy sequence in \(\mathbb{R}\).
(C) \(\{y_n\}\) must be a Cauchy sequence in \(\mathbb{R}\) but \(\{x_n\}\) need not be a Cauchy sequence in \(\mathbb{R}\).
(D) Neither \(\{x_n\}\) nor \(\{y_n\}\) needs to be a Cauchy sequence in \(\mathbb{R}\).

Answer: A


Question: Determine whether the function \( f(x) = \sum_{n=1}^{\infty} \sin\left(\frac{x}{n^2}\right) \) is continuous on \( \mathbb{R} \).

Answer: Yes


Question: Is the set of all real numbers whose decimal expansions can be computed by a machine countable, assuming the machine cannot compute infinite decimal places?

Answer: Yes


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

Answer: e^2


Question: Let $X$ and $Y$ be topological spaces where $X$ is a $T_0$ space. Suppose $f: X \to Y$ is a continuous closed surjection. Is $Y$ necessarily a $T_0$ space?

Answer: Yes


Question:
Determine the values of \( a \) for which the infinite series \( \sum_{n=1}^{\infty} \frac{a^n \log_e n}{n^2} \) converges. Choose the correct option:

A. \( a \in [-1,1) \)
B. \( a \in (-1,1] \)
C. \( a \in [-1,1] \)
D. \( a \in (-\infty, \infty) \)

Answer: C


Question:
Evaluate the integral \( \int_{0}^{+\infty} \frac{x \ln(x)}{(1+x^2)^2} \, dx \) by finding the limits of the antiderivative \( F(x) = \frac{1}{4} \left( \frac{-2\ln(x)}{1+x^2} + \ln\frac{x^2}{1+x^2} \right) \) at the bounds of integration. Compute:

\[ \lim_{x \to \infty} F(x) - \lim_{x \to 0} F(x) \]

Answer: 0


Question: Consider the closed $n$-dimensional unit ball $\mathbb{D}^n \subseteq \mathbb{R}^n$ and a smooth map $A:\mathbb{D}^n \to \mathbb{R}^k \otimes \mathbb{R}^k$. Suppose that for every smooth map $V:\mathbb{D}^n \to \mathbb{R}^k$, the inequality \( \langle A , V \otimes V \rangle_{L^2} \ge 0 \) holds, where \( \langle , \rangle_{\mathbb{R}^k \otimes \mathbb{R}^k} \) is the tensor product metric. Is it true that \( \langle A(x) , v \otimes v \rangle_{\mathbb{R}^k \otimes \mathbb{R}^k} \ge 0 \) for every \( x \in \mathbb{D}^n \) and every \( v \in \mathbb{R}^k \)?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \infty} x \left( \sqrt[3]{5+8x^3} - 2x \right) \]

Answer: 0


Question: Determine the values of $p$ for which the series \( \sum_{n=2}^\infty \frac{1}{n(\ln(n))^p} \) is convergent.

Answer: p > 1


Question: Compute the limit: \( \lim_{x\to\infty}\ln(x)\cdot \ln(1-e^{-x}) \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{x - \sin(\sin(\cdots(\sin x)\cdots))}{x^3} \] where the sine function is applied \( n \) times.

Answer: \dfrac{n}{6}


Question: Determine the minimum number of coin tosses required so that the proportion of heads is within 0.05 of the true probability \( p \) with at least 90% confidence. Assume the coin produces heads with an unknown probability \( p \) and use the binomial normal approximation without continuity correction.

Answer: 271


Question: Determine the number of distinct homomorphisms from a group of order 70 to a group of order 91.

Answer: 7


Question: Evaluate the limit: \( \lim_{n \to \infty} f(a_n) \), where \( a_n = \sqrt{n+1} - \sqrt{n} \) and \( f(x) = \frac{\sin x + x \lfloor x \rfloor}{x \cos x - x} \).

Answer: -\infty


Question: Find the value of \( m \) such that the chord \( y = mx + 1 \) of the circle \( x^2 + y^2 = 1 \) subtends an angle of \( 45^\circ \) at the major segment of the circle.

Answer: \pm 1


Question: Find the smallest integer \( n_0 \) such that \( f(n) = 3n^3 - 39n^2 + 360n + 20 \geq 2.25n^3 \) for all \( n \geq n_0 \).

Answer: 40


Question: Determine if there exist integers $k < n$ with $k \geq 3$ and maximal linear hypergraphs $H_1 = (n, E_1)$ and $H_2 = (n, E_2)$ such that $|E_1| \neq |E_2|$. Provide a yes or no answer.

Answer: Yes


Question: Let \( A \) be a set and \( \kappa \) a cardinal such that \( \omega \leq \kappa \leq |A| \leq 2^{\kappa} \). Determine the cardinality of the set \( C = \{ B \subseteq A : |B| \leq \kappa \} \).

Answer: 2^{\kappa}


Question: Let S be a Stonean space, which is a compact Hausdorff space where the closure of any open set is open. If C is a clopen (both open and closed) subset of S, is C, with the subspace topology, also a Stonean space?

Answer: Yes


Question: Determine the minimal polynomial of $\mathbb{Q}(\sqrt{5}+\sqrt{3})$ over $\mathbb{Q}(\sqrt{10})$. Is the polynomial $x^4-16x^2+4$ the correct minimal polynomial for this extension?

Answer: x^4 - 16x^2 + 4


Question: Determine the number of degree \(d\) hypersurfaces in \(\mathbb{P}^n\) that contain a generic set of \(\binom{d+n}{d}-1\) points.

Answer: 1


Question: Determine whether the equation $2t = -\tan(1/t)$ has a finite or infinite number of solutions for $t \in (0,1)$. Provide your answer as either "finite" or "infinite."

Answer: infinite


Question: Find the value of \( a \) such that the region bounded by the curve \( y = x \sqrt{4-x^2} \) between \( x = 2 \) and \( x = a \), where \( 0 < a < 2 \), when rotated through \( 2\pi \) radians about the \( x \)-axis, generates a solid of revolution with volume \( \frac{657\pi}{160} \).

Answer: \dfrac{1}{2}


Question: Let $f(x)$ be a polynomial with complex coefficients such that there exists a positive integer $n_0$ for which $f(n)$ is an integer for all integers $n \ge n_0$. Prove or disprove that $f(n)$ is an integer for all integers $n$. \( \text{True or False?} \)

Answer: True


Question: Is it possible for an unbounded, non-negative function $f(x)$ to have roots $x_n$ such that $x_n \to \infty$ as $n \to \infty$? Provide a justification for your answer.

Answer: Yes


Question: For which positive integers \(n\) do there always exist positive integers \(a, b, c, d\) such that \(x < \frac{a}{b} < \frac{c}{d} < y\) and \(a^n + b^n = c^n + d^n\)?

Answer: 1


Question:
Given the complex number \( z = \left( \frac{\sqrt{3} - i}{1 + i} \right)^{12} \), determine which of the following statements is true:

A. \( z = 2^6 \)
B. \( \arg(z) = \pi \)
C. \( |z| = 2^{12} \)
D. \( z = 64i \)
E. \( \arg(z) = 2\pi \)

Answer: B


Question: Determine the cardinality of the set \(D_n = \{\langle d_k\rangle \in \mathbb{Q}^\mathbb{N} : (\exists q \in \mathbb{Q})(\forall k \geq n) \,d_k=q\}\). Is it true that \(|D_n| = |\mathbb{Q}^\mathbb{N}| = |\mathbb{Q}|^{|\mathbb{N}|} = \aleph_0^{\aleph_0} = \aleph_0\)?

Answer: \aleph_0


Question: Find the limit of the sequence \(a_n = \frac{\sqrt[n]{(n+1)(n+2)...(2n)}}{n}\).

Answer: \dfrac{4}{e}


Question: Maximize the expression \(x_2 - x_1 + y_1 - y_2\) subject to the constraints \(x_1^2 + y_1^2 = 1\) and \(x_2^2 + y_2^2 = 1\).

Answer: 2\sqrt{2}


Question: Evaluate the sum \(\sum_{d\mid2020}{\sigma(d)}\), where \(\sigma(n)\) is the sum of all divisors of \(n\).

Answer: 7931


Question: A particle moves on 12 points situated on a circle. At each step, it is equally likely to move one step in the clockwise or counterclockwise direction. Find the mean number of steps for the particle to return to its starting position.

Answer: 12


Question: Find the natural number $n$ such that $(2n)! = a^{2n}$, where $a$ is a real number. Use Stirling's approximation if necessary.

Answer: 1


Question:
Solve for \( x \) in the equation:
\[ 3^{\frac{x-1}{2}} - 2^{\frac{x+1}{3}} = 2^{\frac{x-2}{3}} + 3^{\frac{x-3}{2}} \]

Answer: 5


Question: For any distinct integers $a$ and $b$, does there exist an integer $n$ such that the sum of the decimal digits of $na$ is not equal to the sum of the decimal digits of $nb$?

Answer: No


Question:
Find all solutions for $x$ that satisfy the equation:
\[ \frac{1}{2(x-1)} = \int_0^\infty \frac{\sin (x\arctan y)}{(1+y^2)^{x/2}(e^{y\pi}+1)}\,dy \]

Answer: 2


Question: Does the probability \( \mathbb{P}(X = a) \) converge to 0 as \( X \) converges in distribution to \( N(0,1) \), where \( a \in \mathbb{R} \)?

Answer: Yes


Question: Find a simplified expression for the sum \( \sum_{i=0}^{N/2} \binom{N-i}{i} \).

Answer: F_{N+1}


Question: Calculate the Laplace transform of the function \( f(t) = \delta''(t-1)\theta(t) \), where \( \delta \) is the Dirac delta function and \( \theta \) is the Heaviside function.

Answer: s^2 e^{-s}


Question: Determine whether there exist uncountable sets $A \subset \mathbb{R}$ such that the Hausdorff dimension $\dim_H(A) = 0$. Provide a justification for your answer.

Answer: Yes


Question: Consider the standard probability space $(I, B, \mu)$ where $I = [0,1]$, $B$ is the Borel sigma-algebra, and $\mu$ is the Lebesgue measure. For each $n \in \mathbb{N}$, define the rotation $F_n(x) = x + \frac{1}{n} \pmod{1}$. Determine if it is true that for every Borel set $E$, $\mu(F_n E \; \triangle \; E) \to 0$ as $n \to \infty$. Provide a justification for your answer.

Answer: Yes


Question: Let \(\alpha \neq 0\) be such that for every \(\beta < \alpha\), there exists \(\beta < \gamma < \alpha\) and an elementary embedding from \(V_\gamma\) to \(V_\alpha\). Does it follow that \(V_\alpha\) models the replacement axiom?

Answer: Yes


Question: Calculate the surface integral \( \iint (2-y) \, dS \) over the surface defined by \( y = 1 \) and \( 0 \leq z \leq 1 \).

Answer: 1


Question: Does there exist a function \( m : P(\Bbb{R}) \to [0, \infty] \) such that for all \( a, b \in \Bbb{R} \) with \( a < b \), \( m([a, b]) = b - a \), and for any two disjoint subsets \( S, T \) of \( \Bbb{R} \), \( m(S) + m(T) = m(S \cup T) \)?

Answer: Yes


Question: Let $G$ be a finite group with a unique minimal subgroup, and suppose $G$ is a $p$-group with $p \neq 2$. Is it true that every subgroup $H \le G$ and every quotient $G/N$ have a unique minimal subgroup?

Answer: Yes


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

Answer: 0


Question: Consider a pentagon where each vertex is assigned an integer $x_i$ such that the sum $s = \sum x_i > 0$. If $x, y, z$ are numbers assigned to three successive vertices and $y < 0$, replace $(x, y, z)$ with $(x+y, -y, y+z)$. Repeat this step as long as there is a $y < 0$. Determine if this algorithm always terminates.

Answer: Yes


Question: One hundred indistinguishable ants are placed on a hoop with a diameter of 1 meter. Each ant moves at a constant speed of 1 meter per minute, either clockwise or counterclockwise. When two ants meet, they reverse directions. Determine if the ants will ever return to their original configuration, and if so, after how many minutes.

Answer: \pi


Question: Does there exist a topology that can be induced simultaneously by both a complete and an incomplete uniformity?

Answer: Yes


Question:
Find the general solution to the first-order partial differential equation given by:
\[
\begin{cases}
\displaystyle u(x+u)\frac {\partial }{\partial } - y(y+u)\frac {\partial }{\partial }  = 0 \\
u=\sqrt y ,x =1
\end{cases}
\]

Answer: u = \sqrt{xy}


Question: Consider the annular region \(\mathcal{A} := \{(r,\theta) \in \mathbb{R}^2 :  1 \leq r \leq 2\}\). Suppose that \(f: \mathcal{A} \rightarrow \mathbb{R}\) is a smooth function satisfying \(f|_{r=1} <0\) and \(f|_{r=2} > 0\). Also assume that the zero set of \(f\) is a collection of smooth curves. Is there a component of \(\{f=0\}\) which winds around the origin? Justify your answer.

Answer: Yes


Question: For a fixed integer $n \geq 1$, determine the largest integer $k$ such that there exists a finite collection of closed convex subsets of $\mathbb{R}^n$ that has the $k$-intersection property but does not have the $k+1$-intersection property. The $k$-intersection property is defined such that for any $k$ subsets $E_1, \ldots, E_k$ in the collection, their intersection is non-empty.

Answer: n


Question: Consider a function $h(t)$ such that $h(t) = o(t^2)$, meaning \( \lim_{t \rightarrow \infty} \frac{h(t)}{t^2} = 0 \). Define \( g(n) = h\left(\frac{t}{\sigma\sqrt{n}}\right) \). Prove that \( g(n) = o\left(\frac{1}{n}\right) \), i.e., \( \lim_{n \rightarrow \infty} h\left(\frac{t}{\sigma\sqrt{n}}\right) n = 0 \).

Answer: 0


Question:
Let \( f:[0,1]\to\mathbb{R} \) be a continuous function. Define \( g:[0,1] \to \mathbb{R} \) as follows:
\[
g(x)=\int_0^{x}{(x-t)^2\cdot f(t)}\,dt
\]
Find the third derivative \( g(x)''' \).

Answer: 2f(x)


Question: Let $f \in \mathbb{Z}[x_1, \dotsc, x_k]$ be a polynomial function such that for any roots $r_1, \dotsc, r_k$ of a monic polynomial of degree $k$ with integer coefficients, $f(r_1, \dotsc, r_k) \in \mathbb{Z}$. Is it true that $f$ must be a symmetric function of its arguments?

Answer: Yes


Question: Does there exist a Banach space $B$ with no complemented closed subspaces, meaning there do not exist closed subspaces $U,V \subset B$ such that $B = U \oplus V$?

Answer: No


Question: Given that the Fourier transform of the function \( f(x) \) is \( F(k) \), find the Fourier transform of the function \( f'(x) = [f(-x)]^* \).

Answer: [F(k)]^*


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{dx}{(x^2+1)(x^2+9)} \) using complex integration techniques.

Answer: \dfrac{\pi}{12}


Question: Find the smallest positive integer $X$ such that $478^{870} \equiv X \ (\text{mod} \ 273)$. Provide your answer as a single integer.

Answer: 1


Question: If $X + 1$ follows a chi-square distribution with 1 degree of freedom, what is the mean of $X^3$?

Answer: 8


Question: Given the set \( S = \{1, 2, \dots, 7\} \), determine the number of distinct values that can be obtained by multiplying the elements of each subset of \( S \).

Answer: 52


Question: In triangle $ABC$, where $AB = 30$, $BC = 51$, and $CA = 63$, points $P$ and $Q$ lie on line segment $BC$, point $R$ lies on line segment $CA$, and point $S$ lies on line segment $AB$. These points form a parallelogram $PQRS$ such that the center of $PQRS$ coincides with the centroid of triangle $ABC$. Determine the area of parallelogram $PQRS$. (Source: CMC)

Answer: 336


Question: Determine all real numbers \(x\) such that the series \(\sum_{n=1}^\infty \frac{n x^n}{2n^2+1}\) is absolutely convergent.

Answer: (-1, 1)


Question:
Let \( f : \mathbb{R} \to \mathbb{R} \) satisfy the following conditions:

- \( f(1) = 1 \);
- \( f(x + y) = f(x) + f(y) \) for all \( x, y \in \mathbb{R} \);
- \( f\left( \frac{1}{x} \right) = \frac{1}{x^2} f(x) \) for all \( x \in \mathbb{R}^* \).

Find \( f\left( \sqrt{2019} \right) \).

Answer: \sqrt{2019}


Question:
Use the Laplace transform to solve the differential equation \(xy''+(1-x)y'+my=0\). Choose the correct solution from the following options:

(a) \(y=\frac{e^t}{k!} \frac{d^k}{dt^k}(t^{-k}e^{-t})\)

(b) \(y=\frac{e^t}{k} \frac{d^k}{dt^k}(t^k e^{-t})\)

(c) \(y=\frac{e^t}{k!} \frac{d^k}{dt^k}(t^k e^{-t})\)

Answer: C


Question:
Determine if the following solution is correct:

Given that $f = 0$ almost everywhere, prove that $\int_{\mathbb{R}} f = 0$. The solution provided is:

Suppose $\phi$ is a simple function such that $0 \leq \phi \leq f$. Since $f = 0$ almost everywhere, $\phi = 0$ almost everywhere. Let $A = \{ x : \phi(x) = 0\}$. Therefore, $A^c = \{x : \phi(x) > 0 \}$ is a null set. Hence
$$ \int_{\mathbb{R}} \phi = \int_A \phi + \int_{A^c} \phi = 0 + 0  = 0 $$
Since $\mathbb{R} = A \cup A^c$, therefore $\int_\mathbb{R} f = 0$.

Is this solution correct?

Answer: Yes


Question: Find the limiting value of the p-norm \( \|f(x)\|_p \) as \( p \rightarrow \infty \) for the function \( f(x) = x \) over the interval \( a \leq x \leq b \).

Answer: \max\{|a|, |b|\}


Question: Given that $x_1 < x_2 < x_3$ are real numbers and solutions to the equation $x^3 - 3x^2 + (a+2)x - a = 0$, where $a$ is a real number, find the value of the expression $4x_1 - x_1^2 + x_3^2$. The equation can be factorized as $(x-1)(x(x-2)+a)=0$. Determine the value of $4x_1 - x_1^2 + x_3^2$.

Answer: 4


Question: If \( a, b, c \) are real positive numbers, and \( \theta = \tan^{-1}\left[\frac{a(a+b+c)}{bc}\right]^{\frac{1}{2}} + \tan^{-1}\left[\frac{b(a+b+c)}{ca}\right]^{\frac{1}{2}} + \tan^{-1}\left[\frac{c(a+b+c)}{ab}\right]^{\frac{1}{2}} \), find the value of \( \tan \theta \).

Answer: 0


Question: Let $A$ and $B$ be commutative rings. If $A[x] \subseteq B[x]$, is it true that $A \subseteq B$? Provide a justification for your answer.

Answer: Yes


Question: Given two reinforced concrete buildings A and B in a seismic region, an impending earthquake might be strong (S), moderate (M), or weak (W) with probabilities P(S) = 0.02, P(M) = 0.2, and P(W) = 0.78. The probabilities of failure for each building if these earthquakes occur are 0.20, 0.05, and 0.01, respectively. If building A fails, the probability that building B will also fail is 0.50, 0.15, and 0.02 for the three types of earthquakes. Calculate the probability that the earthquake was not strong given that building A has failed and building B has survived.

Answer: 0.890


Question:
Find a function \( f(x) \) that satisfies the following limit conditions:

1) For all \( b > 1 \), \( \lim\limits_{x \to \infty} \frac{f(x)}{x^b} = 0 \).
2) For all \( b \leq 1 \), \( \lim\limits_{x \to \infty} \frac{f(x)}{x^b} = \infty \).

Answer: x \ln x


Question: Is it possible for a chromatic polynomial to have a factor over \(\mathbb{Q}\) with coefficients that do not alternate in sign?

Answer: No


Question: Determine the value that \(\frac{B}{x}\) approaches, where \(B=\sqrt[3]{x\sqrt[5]{x^4\sqrt[9]{x^{24}\sqrt[17]{x^{240}\ldots}}}}\).

Answer: 1


Question: Find the limit of the sequence \( x_n = \left(\frac{e\sqrt[3]{e}\cdots\sqrt[n]{e}}{n}\right) \) as \( n \to \infty \).

Answer: e^{\gamma - \frac{1}{2}}


Question: Find the integer value of $b$ such that $44 \equiv 7^b \pmod{71}$.

Answer: 43


Question: Determine the number of prime numbers \(p\) less than 100 such that \(p\) divides \(x^2 + x + 1\) for some positive integer \(x\).

Answer: 12


Question: Evaluate the contour integral \( \oint_{|z|=1} \frac{1}{\cos\left(\frac{1}{z}\right)} \, dz \).

Answer: 0


Question: Determine whether there exist two distinct prime numbers \( p \) and \( q \) such that the equation \( 2^a p^b = q^c - 1 \) has no solution in natural numbers \( a, b, \) and \( c \).

Answer: Yes


Question: Find \( G'(2) \), where \( G(x) = \int_{\pi/x}^{\pi x} \cos^{15} t \, dt \).

Answer: \pi


Question: Evaluate the limit: \[ \lim_{z \to 0}{(1+z)^{\frac{1}{z}}} \]

Answer: e


Question: Determine the values of \(a > 0\) and \(b > 0\) for which the integral \(\int_{1}^{\infty} \frac{1}{x^a(1+x^b)} \,dx\) is convergent.

Answer: a + b > 1


Question: Given a random sample \(X_1\) and \(X_2\) of size \(n=2\) from an exponential distribution with parameter \(\theta\), find the efficiency of the estimator \(2Y_1\) relative to \(\bar{X}\), where \(Y_1\) is the first order statistic. Both \(2Y_1\) and \(\bar{X}\) are unbiased estimators of \(\theta\). Calculate the variance of \(2Y_1\) and use it to determine the relative efficiency \(\frac{V(\bar{X})}{V(2Y_1)}\), given that \(V(\bar{X})=\frac{\theta^2}{2}\).

Answer: \dfrac{1}{2}


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

Answer: e^{-2}


Question: Find a natural number $m$, which is the product of 3 prime numbers, such that the equation $x^2 + 1 \equiv 0 \pmod{m}$ has exactly 4 solutions. What is the value of $m$?

Answer: 130


Question: Show that the nilradical of the ring $F_p[x] \otimes_{F_p[x^p]} F_p[x]$ is a principal ideal.

Answer: (x \otimes 1 - 1 \otimes x)


Question: Is the monoid "group of units" functor, which is right adjoint to the forgetful functor, full? Specifically, is there a monoid $M$ with group of units $G \subseteq M$ such that there is no monoid homomorphism $\varphi: M \rightarrow G$ that restricts to the identity on $G$?

Answer: Yes


Question: Find the value of \(4L\) if \(L = \left|\lim_{x\to\frac{\pi}{2}^-}(1+\tan x)\left\{(1+\tan x)\ln\left(\frac{1+\tan x}{2+\tan x}\right)+1\right\}\right|\).

Answer: 2


Question: Determine whether the following statement is true: For a maximal ideal $I=\langle x-a,\,y-b\rangle$, the vector space $\mathbb{C}[x,y]/I$ always has dimension one.

Answer: True


Question:
Find the number of ordered pairs of integral solutions \((x, y)\) to the equation:
\[(1 + 3x)(1 - 3y) = -10xy\]

Answer: 8


Question: Find the smallest integer \( n \geq 2 \) such that the equation \( x^2 - x + \hat{2} = \hat{0} \) has a unique solution in \( \mathbb{Z}_n \).

Answer: 7


Question: Determine for which values of \( p \) the sequence \( \{f_n\}_n \), where \( f_n = n1_{[0,1/n]} \), is a Cauchy sequence in \( L^p(\mathbb{R}) \). Justify your answer.

Answer: (0, 1)


Question: Let $f$ and $g$ be nilpotent endomorphisms on a finite-dimensional vector space $V$ such that $\ker(f) = \ker(f \circ g)$. Prove or disprove that $\ker(f) = V$. \(\text{(Hint: Consider the implications of nilpotency and the given condition on the kernels.)}\)

Answer: \ker(f) = V


Question: Determine the number of infinite planar graphs with the degree sequence \(D=(4^4,6^\infty)\), where the superscripts denote the number of vertices with that degree.

Answer: 1


Question: Find the extremal function of the variational problem \( J[y] = \int_{0}^1 (y'(x))^2 \, dx \) subject to the conditions \( y(0) = 0 \), \( y(1) = 1 \), and \( \int_{0}^{1} y(x) \, dx = 0 \).

Answer: 3x^2 - 2x


Question: Given the joint distribution \( f(x,y) = x+y \) for \( 0 < x < 1 \) and \( 0 < y < 1 \), find the probability \( P(X+Y>\frac{1}{2}) \).

Answer: \dfrac{23}{24}


Question: Calculate \( \lim_{x \to 1^{-}} \frac{\arccos{x}}{\sqrt{1-x}} \) without using L'Hôpital's rule.

Answer: \sqrt{2}


Question: Given that $X, Y, Z$ are independent and identically distributed random variables following a uniform distribution on the interval (0, 1), compute the probability $P(X^2 < YZ)$. Provide your answer as a single numerical value.

Answer: \dfrac{4}{9}


Question: Find the function \( f(x,y) \) such that \( \int f(x,y) \, dx = (1-y) \times f(x,y) \) and \( \int f(x,y) \, dy = x \times f(x,y) \).

Answer: 0


Question: Evaluate the limit \( \lim_{z \to 1} \frac{\text{Log}(z^2)}{z-1} \), where \( z \) is a complex number given by \( z = x + iy \) and \( \text{Log}(z) = \ln(|z|) + i\theta \), with \( -\pi < \theta \leq \pi \).

Answer: 2


Question: Compute \( \sup_{f \in \mathcal{F}} |f(0)| \), where \( \mathcal{F} = \{ f \text{ holomorphic in } \mathbb{D} \text{ with } f(\mathbb{D}) \subset \mathbb{D}, f\left(\frac{1}{2}\right) = f'\left(\frac{1}{2}\right) = 0 \} \) and \( \mathbb{D} \) is the unit disc.

Answer: \dfrac{1}{4}


Question: Calculate the area under the curve of the function \( f(x) = \sqrt{x} \) on the interval \([0, 4]\) by computing the lower Riemann sum for \( f \) with the partition defined by \( x_i = \frac{4i^2}{n^2} \) and taking the limit as \( n \rightarrow \infty \).

Answer: \dfrac{16}{3}


Question: Let \( f_n : \mathbb{R}_{+} \to \mathbb{R} \) be defined by \( f_n(x) = 3x^n e^{-x^2} - 1 \). For each \( n > 1 \), let \( v_n \) be a real number such that \( v_n > 1 \) and \( f_n(v_n) = 0 \). Determine the limit \( \ell \) such that \( \lim_{n \to \infty} v_n = \ell \).

Answer: 1


Question: Let \( \omega = ze^{xy} dx \wedge dy \) be a 2-form on \( \mathbb{R}^3 \). Consider the map \( \varphi: \mathbb{R}^2_{\ne 0} \to \mathbb{R}^3 \) defined by \( \varphi(u,v) = \left( \frac{u}{v}, \frac{v}{u}, uv \right) \). Compute the pullback \( \varphi^*\omega \).

Answer: 0


Question: Solve the recurrence relation \( A(n) = 2A(n/2) + O(1) \).

Answer: O(n)


Question: Consider the series \( \sum_{n=-\infty}^{\infty} (-1)^{n} \frac{P(n)}{Q(n)} \), where \( P(z) \) and \( Q(z) \) are polynomials such that the degree of \( Q(z) \) is exactly one degree more than the degree of \( P(z) \). Assume this series converges. Evaluate whether the integral of the function \( \pi \csc (\pi z) \frac{P(z)}{Q(z)} \) around a square contour with vertices at \( \pm(N + \frac{1}{2}) \pm i (N+ \frac{1}{2}) \) vanishes as \( N \to \infty \) through the positive integers.

Answer: Yes


Question: In triangle \(ABC\), \(\measuredangle BAC = 45^\circ\) and \(\measuredangle ABC = 30^\circ\). A point \(M\) lies inside the triangle such that \(\measuredangle MAB = \measuredangle MBA = 15^\circ\). Determine \(\measuredangle BMC\).

Answer: 135^\circ


Question: Consider the operator \( T: (C[-1, 1], \|\cdot\|_\infty) \rightarrow \mathbb{R} \) defined by \( Tf = \int_{-1}^0 f(t)\, dt - \int_{0}^1 f(t)\, dt \). Prove that the norm of this operator is \( \|T\| = 2 \).

Answer: 2


Question: Assume $f(x)$ is a decreasing positive function such that $f(0)$ is infinite and $f(x)$ is comparable to $1/x$ near $x=0$. Consider only positive $x$. If the limit \( \lim_{a\to 0^+}\int_{a}^{2a} f(t)dt \) exists as a positive finite number, does it imply that \( \lim_{n\to\infty} f(2^{-n})\cdot 2^{-n} \) exists?

Answer: Yes


Question: A plane has 144 seats, and it is known that 15% of ticket holders do not show up for the flight, with each person's decision being independent. How many tickets can you sell to be 99% sure that not more than 144 people will show up for the flight?

Answer: 157


Question:
Find the value of $X$ that satisfies the equation involving recursive $n$-th roots:
\[ \sqrt[n]{X+\sqrt[n]{X+\sqrt[n]{X+ \dots}}}  =\sqrt[n]{X\sqrt[n]{X\sqrt[n]{X \dots}}} \]

Answer: 0


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed standard normal random variables. Find the probability $P(X_n \ge \max(X_1, X_2, \ldots, X_{n-1}))$. Express your answer in terms of $n$.

Answer: \dfrac{1}{n}


Question: Is every field of characteristic zero, where each nonzero element is either a square or the additive inverse of a square but not both, a Euclidean ordered field? A Euclidean ordered field is an ordered field where every positive element is a square. Determine if every such 'quasi-Euclidean' field of characteristic zero is indeed a Euclidean ordered field.

Answer: Yes


Question: Evaluate the integral of the function \( \tan(z) \) around the circle \( |z|=2 \).

Answer: -4\pi i


Question: Determine the condition on \( k \) such that the polynomial \( f(x) = x^4 + 4x^3 - 2x^2 - 12x + k \) has four real roots.

Answer: [-7, 9]


Question: Evaluate the series: \[ \sum_{k=1}^{\infty}\frac{1}{k(k+1)^2k!} \]

Answer: 3 - e


Question: Evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \), where \( \mathbf{F}(x,y,z) = 3x^2\mathbf{i} - 2xy\mathbf{j} + 8\mathbf{k} \), and \( S \) is the graph of the function \( z = f(x,y) = 2x - y \) for \( 0 \leq x \leq 2 \) and \( 0 \leq y \leq 2 \).

Answer: -8


Question: Given a function \( f(x) = \frac{g(x)}{h(x)} \) defined on the interval \((\alpha, 2]\), where \( h(\alpha) = 0 \) and \( \lim_{x \to \alpha} f(x) = +\infty \), and assuming that \( f \) is continuous and strictly decreasing on \((\alpha, 2]\), can we deduce that \( f \) is a bijection from the open interval \((\alpha, 2]\) to the open interval \([f(2), +\infty)\)?

Answer: Yes


Question: Are all finite extensions of the form \(\frac{\mathbb{Z}_p[x]}{\langle p(x)\rangle}:\mathbb{Z}_p\), where \(p\) is a prime and \(p(x)\) is irreducible, Galois extensions?

Answer: Yes


Question: Consider a group $G$ and a field $F$. It is known that there exists a group $H$ containing $G$ such that the group ring $F[H]$ is primitive. Construct the group and faithful, simple module as follows: let $G_1 = G$, $M_1 = F[G_1]$, $G_2 = \text{Aut}_F(M_1)$, $M_2 = F[G_2] \oplus M_1$, and continue this process with $H = \bigcup G_i$ and $M = \bigcup M_i$. Assume $M$ is both a left- and right-module. Is $M$ simple and faithful on both sides? If so, consider $\text{End}_{F[H]} M$ where $M$ is a left-module. Since it is left-simple, $\text{End}_{F[H]} M$ is a division ring by Schur's lemma. There is a ring homomorphism from $F[G]$ into $\text{End}_{F[H]} M$ by the action of right scalar multiplication. As $M$ is right-faithful, it has no non-zero annihilators, making the homomorphism injective. Does this imply that $F[G]$ is embedded into a division ring?

Answer: Yes


Question: Let $M$ and $N$ be 2-dimensional surfaces in $\mathbf{R}^3$. Suppose $F: M \to N$ is a function such that for any coordinate patch $\mathbf{x}$ in $M$ and any coordinate patch $\mathbf{y}$ in $N$, the composite map $\mathbf{y}^{-1} \circ F \circ \mathbf{x}$ is Euclidean differentiable as a function from $\mathbf{R}^2$ to $\mathbf{R}^2$. Can it be deduced that $F \circ \mathbf{x}$, as a function from $\mathbf{R}^2$ to $\mathbf{R}^3$, is Euclidean differentiable?

Answer: Yes


Question:
Given the cumulative distribution function (CDF) of a random variable X:

\[ F(x) = \begin{cases} 0, & \text{if } x < 0 \\ \frac{1}{4}, & \text{if } 0 \le x < 1 \\ \frac{2}{5}, & \text{if } 1 \le x < 2 \\ \frac{2x-3}{2}, & \text{if } 2 \le x < 2.5 \\ 1, & \text{if } x \ge 2.5 \end{cases} \]

Calculate the expected value E(X).

Answer: \dfrac{59}{40}


Question: Let $k$ be a field and $\mathrm{Br}(k)$ the Brauer group of $k$. Consider a field extension $k \subset L$ and an element $b \in \mathrm{Br}(k)$. If the base-change $b \otimes L$ is zero in $\mathrm{Br}(L)$, does there exist a subextension $k \subset K \subset L$ such that $K/k$ has finite degree and $b \otimes K = 0$? Provide a justification for your answer.

Answer: Yes


Question: Calculate the value of the expression \(\arccos(1/\sqrt{2}) + \arcsin(1/3) + 2 \arctan(1/\sqrt{2})\). Express your answer as a multiple of \(\pi\).

Answer: \dfrac{3\pi}{4}


Question: Let the function $f(x)$ be thrice differentiable and satisfy $f(f(x)) = 1 - x$ for all $x \in [0, 1]$. Given that $f''\left(\frac{4}{5}\right) = 0$, find the value of \( \int_0^1 f(x) \, dx \).

Answer: \dfrac{1}{2}


Question: Calculate \( \lim _{n\to \infty} \sum _{ k=1 }^{ n }{ \left| { e }^{ \frac { 2\pi ik }{ n } }-{ e }^{ \frac{ 2 \pi i(k-1) }{ n }} \right|} \).

Answer: 2\pi


Question: Evaluate the integral of the function \( f(x,y,z) = \frac{y}{\sqrt{z}} \) over the region defined by the conditions \( y \geq 0 \), \( 0 \leq z \leq x^2 \), and \( (x-2)^2+y^2 \leq 4 \).

Answer: \dfrac{64}{3}


Question: Does there exist a function \( f: \mathbb{R} \to \mathbb{R} \) such that for every disc in \( \mathbb{R}^2 \), the graph of \( f \) has at least one point inside that disc?

Answer: Yes


Question: Find the 100th derivative of the function \( f(x) = e^{x} \cos(x) \) evaluated at \( x = \pi \).

Answer: 2^{50} e^{\pi}


Question: Is there a field with a characteristic that is a composite number, such as 4?

Answer: No


Question: Evaluate the expression \( i\log(x-i) + i^2\pi + i^3\log(x+i) + i^4(2\arctan x) \) for \( x > 0 \).

Answer: 0


Question: Let H be a Hilbert space and A and B be closed subspaces of H such that \(A^{\perp} = B^{\perp}\). Determine whether A equals B.

Answer: A = B


Question: Provide an example of a function \( f: \mathbb{R} \to \mathbb{R} \) that is differentiable and bijective, but whose inverse is not differentiable.

Answer: f(x) = x^3


Question: Evaluate the integral \( \oint_{|z| = 2} \frac{e^{-z}}{z^2} \, dz \).

Answer: -2\pi i


Question: Let $M$ be an oriented compact $3$-manifold, either closed or with boundary. For any incompressible surface $F$, define a function $i_F$ on the set of homotopy classes of closed curves in $M$ by \(i_F (\alpha) = \alpha \cap F\), the geometric intersection number of $\alpha$ with $F$. Determine if it is true that two incompressible, $\partial$-incompressible surfaces $F$ and $F'$ are isotopic if $i_F = i_{F'}$. Provide a justification for your answer.

Answer: Yes


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

Answer: 0


Question: Find a finite extension of \( \mathbb{Q} \) in which all primes split. Use decomposition and inertia groups to justify your answer.

Answer: \mathbb{Q}


Question: Does there exist an increasing function $f$ that is continuous, vanishes at $0$, and satisfies \(f\left(\frac{1}{k}\right)e^{2k} \to 0\) as $k \to \infty$?

Answer: Yes


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

Answer: 1


Question:
Find the smallest value of the expression:
\[ S = \sqrt[5]{\frac{abc}{b+c}} + \sqrt[5]{\frac{b}{c(1+ab)}} + \sqrt[5]{\frac{c}{b(1+ac)}} \]
where \( a \ge 0 \) and \( b, c > 0 \).

Answer: 2


Question: Consider the differential equation \( x' = f(t, x) \) where \( f: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) satisfies \( x t f(t, x) < 0 \) for \( x t \neq 0 \). Determine if there exists at most one solution to this differential equation with the initial condition \( x(0) = 0 \), assuming \( f \) is continuous.

Answer: Yes


Question: Find a group $G$ and an element $a \in G$ such that the order of $a$ is 6, but the centralizer of $a$ in $G$ is not equal to the centralizer of $a^3$ in $G$. Provide an example of such a group and element.

Answer: D_6


Question: Calculate the volume in the first octant inside the cylinder \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and under the plane \( z = 3x \). Use the method of slicing to find the volume.

Answer: a^2 b


Question: Given a continuous function $f:[-2,8]\rightarrow\mathbb{R}$ such that $f(6-x)=f(x)$ for all $x\in[-2,8]$, and knowing that $\int_{-2}^8f(x)dx=10$, find the value of $\int_{-2}^8xf(x)dx$. Use any appropriate method to solve the problem.

Answer: 30


Question: Find \( \lim_{x \to 0^-} \frac{\sqrt{x+1}}{x} \).

Answer: -\infty


Question: Evaluate the integral \( \int_0^{2 \pi} \frac{1}{5-3 \sin z} \, dz \).

Answer: \dfrac{\pi}{2}


Question: Find the maximum value of \( \frac{a^3 + b^3 + c^3 }{\sqrt{a^4 + b^4 + c^4 }} \) given that \( a^2 + b^2 + c^2 = 1 \) and \( a, b, c \) are non-negative real numbers.

Answer: 1


Question: Let \( a_1 = 1 \) and \( a_n = n(a_{n-1} + 1) \) for \( n \geq 2 \). Define the product \( P_n = \prod_{i=1}^n \left(1 + \frac{1}{a_i}\right) \). Determine the limit of \( P_n \) as \( n \to \infty \).

Answer: e


Question:
Calculate the Jacobian determinant of the Fourier transform given by:
\[ x_n = \sum_k \frac{1}{\sqrt{N}} e^{-i 2 \pi k n/N} \tilde{x}_k \]
Show that the determinant is equal to 1.

Answer: 1


Question: Suppose \( U \) and \( V \) are subsets of a topological space \( X \). If \( U \) and \( V \) are homeomorphic when considered with the subspace topology, and \( U \) is open in \( X \), does it follow that \( V \) is open in \( X \)? Answer yes or no and justify your answer.

Answer: No


Question:
Find the 6-digit number \( N \) such that the sum of its digits is 43, and exactly one of the following statements is false:

1. \( N \) is a perfect square.
2. \( N \) is a perfect cube.
3. \( N < 500000 \).

Determine the value of \( N \).

Answer: 499849


Question: Given the recurrence relation \(T(n) = 7T\left( \frac{n}{2} \right) + n^2\) for the execution time of algorithm \(A\), and the recurrence relation \(T'(n) = aT'\left( \frac{n}{4} \right) + n^2\) for the execution time of algorithm \(A'\), determine the greatest integer value of \(a\) such that \(A'\) is asymptotically faster than \(A\).

Answer: 48


Question: Suppose that a sequence of random variables $X_n$ converges in probability to a constant $c$. Does it follow that \( \lim_{n\to\infty}P(X_n<c)=0 \)? Justify your answer.

Answer: No


Question:
Let $X$ denote the diameter of an armored electric cable and $Y$ denote the diameter of the ceramic mold that makes the cable. Both $X$ and $Y$ are scaled so that they range between 0 and 1. Suppose that $X$ and $Y$ have the joint density function given by:

\[
f(x,y) = \begin{cases}
\frac{1}{y}, & 0 < x < y < 1 \\
0, & \text{elsewhere}
\end{cases}
\]

Find the probability $P(X+Y > 1/2)$. Provide your answer to four decimal places.

Answer: 0.6534


Question: Find the real part of the complex nested radical expression: \[ \Re\left(\sqrt{1+\frac{i}{2}\sqrt{1+\frac{i}{2^2}\sqrt{1+\frac{i}{2^3}\sqrt{1+\frac{i}{2^4}\sqrt{\cdots}}}}}\right) = 1 \]

Answer: 1


Question: Calculate the double integral \( \iint\limits_D y \, dA \) where \( D = \mathbf{\Phi}(\mathcal{R}) \) with the transformation \( \mathbf{\Phi}(u,v) = (u^2, u+v) \) and \( \mathcal{R} = [4,8] \times [0,4] \).

Answer: \dfrac{4736}{3}


Question: Consider the short exact sequence of $R$-modules: $0 \to M_1 \to M_2 \to M_3 \to 0$. If $M_2$ is finitely generated, must $M_1$ also be finitely generated?

Answer: No


Question:
Given the recursive system of equations:
\[\begin{cases}
a_{n+1} = -2a_n - 4b_n \\
b_{n+1} = 4a_n + 6b_n \\
a_0 = 1, b_0 = 0
\end{cases}\]
Calculate the value of \( \frac{a_{20}}{a_{20} + b_{20}} \).

Answer: -39


Question: Let \( f(z) = \sum_{}^{} a_{k} z^k \) be a power series. Determine whether it is true that if \( f(x) \) converges for every \( x \in \mathbb{R} \), then \( f(z) \) converges for every \( z \in \mathbb{C} \).

Answer: Yes


Question: Given that $(a+1)(b+1)(c+1)(d+1)=81$ and $a,b,c,d>0$, find the maximum value of $abcd$.

Answer: 16


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left| \cos \left( \frac{\pi}{4(n-1)} \right) \right|^{2n-1} \]

Answer: 1


Question: Determine all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the equation \( xf(x) - yf(y) = (x-y)f(x+y) \) for all real numbers \( x \) and \( y \).

Answer: f(x) = ax + b


Question:
Determine which of the following conditions is necessary and sufficient for the existence of a continuous real-valued function \( f \) on \( \mathbb{R}^2 \) such that \( f(x) = a_j \) for \( x \in A_j \), where \( j = 1, 2, 3 \), and \( A_1 = \{ \text{closed unit disk in the plane} \} \), \( A_2 = \{(1,y):y\in \mathbb{R}\} \), \( A_3 = \{(0,2)\} \):

1. If and only if at least two of these numbers are equal.
2. All are equal.
3. For all values of these 3 numbers.
4. If and only if \( a_1 = a_2 \).

Use Urysohn's lemma if applicable.

Answer: 4


Question: Let $k$ be a field of characteristic zero, $A$ a simplicial commutative $k$-algebra, and $M$ a simplicial $A$-module. Consider the trivial square-zero extension $A \oplus M$ as an $A$-algebra. Is the relative cotangent complex of $A \oplus M$ over $A$ (i.e., the cotangent complex of the map $A \rightarrow A \oplus M$) isomorphic to $M$ in the derived category of $A$-modules?

Answer: M


Question: Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann was. The combined ages of Mary and Ann is 44 years. How old is Mary?

Answer: 27.5


Question: Determine whether every hypergraph \(H=(V,E)\) has a good transversal basis. A good transversal basis \(B\) is defined such that for all transversal bases \(B_1\) with \(I_{B}\subseteq I_{B_1}\), we have \(I_B=I_{B_1}\), where \(I_B:=\{e\in E:B\cap e\neq \emptyset\}\).

Answer: Yes


Question:
Determine whether the following statement is true:

If \( \int_{0}^{x}f(t)\,dt \leq \int_{0}^{x} c \,dt = cx \) for all \(x > 0\), where \(x\) is a real number and \(c\) is a fixed constant, and given that \(f(t)\) is a positive function on \(t > 0\) and satisfies \(f(t_{1}+t_{2})\geq f(t_{1})+f(t_{2})\) for all \(t_{1},t_{2}>0\), does it follow that \(f(t) \leq c\) for all \(t > 0\)?

Answer: True


Question: Find the minimum value of the function \( f(x) = \frac{9x^2(\sin^2x) + 4}{x\sin x} \) for \( x \in (0, \pi) \), using the inequality \( \frac{a_1+\cdots+a_n}{n} \ge \sqrt[n]{a_1\cdots a_n} \) with equality if and only if \( a_1 = a_2 = \cdots = a_n \). Determine when this minimum value occurs.

Answer: 12


Question: Compute \( \lim_{x\rightarrow 0} \frac{\ln{\cos{x}}}{e^{2x^2}-\cos{x}} \) without using L'Hopital's rule and without using the \(\sec(x)\) and \(\csc{x}\) functions.

Answer: -\dfrac{1}{5}


Question: Find the value of \( n \) such that the area under the curve \( y = x^n \) from \( x = 1 \) to \( x = a \) is one third of the area under the curve from \( y = 1 \) to \( y = a^n \).

Answer: 3


Question: Given a sequence of real numbers \( \{a_n\mid n \ge 1\} \), the partial sum of \( a_n \) forms a convergent series, while the partial sum of the absolute values \( |a_n| \) forms a divergent series. Determine if the radius of convergence \( R \) of the power series \( \sum_{n=0}^\infty a_nx^n \) is infinite.

Answer: No


Question:
Calculate the inverse Fourier transform of \( \frac{1}{a+\mathrm{j}w} \) using the definition:
\[
f(t)=\frac{1}{2\pi}\int_{-\infty}^\infty\frac{1}{a+\mathrm{j}w}\mathrm{e}^{\mathrm{j}wt}\,\mathrm{d}w.
\]

Answer: e^{-at} u(t)


Question: Let $A$ be a $6 \times 6$ real symmetric matrix with rank 5. Determine the rank of the matrix $A^2 + A + I$, where $I$ is the $6 \times 6$ identity matrix.

Answer: 6


Question: Calculate \( \int_{D} ||\nabla f||^2 \, dA \), where \( f(x,y) = y - x^2 + 1 \) and \( D = \{(x,y) \mid f(x,y) \geq 0, \, y \leq 0\} \). Determine the correct limits of integration and evaluate the integral.

Answer: \dfrac{12}{5}


Question: Calculate the expected number of trials needed to obtain a single success, given that the probability of success on the first trial is 5%. If the first trial is a failure, the probability of success on the next trial increases by 5%, and this pattern continues for subsequent trials. If a trial is successful, the probability resets to 5% for the next trial. Assume this process continues for an arbitrary number of trials.

Answer: 5.29


Question: Determine the Galois group of the polynomial \(x^4 + 2\) over the field of rational numbers \(\mathbb{Q}\).

Answer: D_4


Question: Consider a continuous distribution over the interval $(0,1)$ with cumulative distribution function $F$. Let $k_1 \in (0,1)$ be the value that maximizes the function $g_1(k) = k(1-F(k))$, and let $k_2 \in (0,1)$ be the value that maximizes the function $g_2(k) = \frac{k(1-F(k))}{2-F(k)}$. Prove or disprove that $k_1 \leq k_2$ for any such distribution.

Answer: k_1 \leq k_2


Question: Let \( K \) be a compact space and \( X \) be a Banach subspace of \( C(K) \). Suppose \( E \subseteq K \) is closed such that for every \( g \in C(E) \), there exists \( f \in X \) with \( f|_E = g \). Prove that there exists a constant \( c > 0 \) such that \( f \) can be chosen to satisfy \( ||f|| \le c||g|| \).

Answer: c


Question: A coffee shop roasts a random amount of coffee beans each morning, denoted by the random variable R ('00 kg), with a probability density function given by \( f(r) = -6(r-1)^2+1.5 \) for \( 0.5 < r < 1.5 \). On any given day, the amount of beans sold, denoted by S ('00 kg), is uniformly distributed such that \( S < r \). Calculate the expected value \( E[S] \).

Answer: 0.5


Question:
Find a closed form representation of the sum:
\[ \frac{1}{2k+1}+\frac{2k}{(2k+1)(2k-1)}+\frac{2k(2k-2)}{(2k+1)(2k-1)(2k-3)}+\cdots+\frac{2k(2k-2)\cdots 6\cdot 4}{(2k+1)!!}. \]
Here, \(k\) is a positive integer and \(!!\) denotes the double factorial.

Answer: 1


Question:
Given that $x_1=1$ and the sequence $x_1, x_2, \ldots, x_{100}$ satisfies the following inequalities:
\[(x_1 - 4x_2 + 3x_3) \geq 0,\]
\[(x_2 - 4x_3 + 3x_4) \geq 0,\]
\[\vdots\]
\[(x_{100} - 4x_1 + 3x_2) \geq 0,\]
find the value of $x_1 + x_2 + \ldots + x_{100}$. \(\)

Answer: 100


Question: Let \( f: [0,\infty) \rightarrow \mathbb{R} \) be a continuous function such that \( f(0) = 0 \). Determine whether the following statement is true or false: If the integral \( \int_0^{\pi/2} \sin(\theta) f(\lambda \sin(\theta)) \, d\theta \) is zero for every \( \lambda > 0 \), then \( f \) is identically zero.

Answer: True


Question: Let $I=\{1,...,n\}$ and consider a collection of functions $\{f_i\}_{i\in I}$ where each $f_i:\mathbb{R}\rightarrow\mathbb{R}$ is a Lipschitz function with constant $a > 0$. That is, for all $x, y \in \mathbb{R}$ and for each $i \in I$, the following holds: $$|f_i(x)-f_i(y)|\leq a|x-y|.$$ Define the function $f(z) = \max_{i\in I} f_i(z)$ for all $z \in \mathbb{R}$. Determine whether the function $f$ satisfies the Lipschitz condition: $$|f(x)-f(y)|\leq a|x-y|, \forall x,y\in \mathbb{R}.$$

Answer: Yes


Question: Find the greatest value of the function \( f(x) = 2(a-x)\left(x+\sqrt{x^2+b^2}\right) \) for real numbers \( x \).

Answer: a^2 + b^2


Question: Let \( A \in \mathbb{R}^{2\times 2} \) be a matrix such that \( \det(A) = d \neq 0 \) and \( \det(A + d \cdot \text{Adj}(A)) = 0 \). Evaluate \( \det(A - d \cdot \text{Adj}(A)) \).

Answer: 4


Question: Consider the hyperbolic plane $\mathbb{H}$ and a Fuchsian group $\Gamma < \text{Isom}(\mathbb{H})$. Let $S = \mathbb{H}/\Gamma$ be the quotient space. If $\gamma : [0,1] \rightarrow S$ is a geodesic on $S$, can it be lifted to a geodesic $\tilde{\gamma} : [0,1] \rightarrow \mathbb{H}$ on the hyperbolic plane? Provide a justification for your answer.

Answer: Yes


Question: Consider a set of balls numbered from 1 to \( m \). Balls are drawn with replacement, and each draw is equally likely to be any of the \( m \) balls. What is the expected number of draws required to first draw the ball numbered 1 twice in a row?

Answer: m(m + 1)


Question: Given the function \( w = z \tan^{-1}\left(\frac{x}{y}\right) \), calculate the value of \( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \).

Answer: 0


Question: Let \( \{X(t); t \geq 0\} \) be a Poisson process with parameter \( \lambda = 2 \). Find \( E[X(s) | X(t) = n] \), where \( 0 < s < t \).

Answer: \dfrac{n s}{t}


Question: Determine the number of non-similar matrices with the characteristic polynomial \((\lambda-1)^4(\lambda-2)^3\).

Answer: 15


Question: A casino patron will continue to make $5 bets on red in roulette until she has won 4 of these bets. Calculate her expected winnings when she stops.

Answer: -\dfrac{20}{9}


Question:
Compute the closed-form expression for the integral:
\[ R^* = \int\int \max_i\{\theta_i+\epsilon_i-1,0\}dF(\theta_i)dG(\epsilon_i) \]
where $\theta_i$ is uniformly distributed in $[1,2]$ and $\epsilon_i$ is uniformly distributed in $[-1/2,1/2]$, with both random variables being independent of each other and across $i$. Determine $R^*$ over the supports of each random variable.

Answer: \dfrac{25}{48}


Question: Calculate the limit using Taylor series: \[ \lim_{x \searrow 0} \frac{\sqrt{1 + 2x + 5x^2} - e^{3x}\cos(2\sqrt x)}{\sin(x^2)}. \]

Answer: \dfrac{17}{6}


Question: Consider an urn containing $2n$ balls numbered from $1$ to $2n$. You draw $n$ balls from the urn, where $n$ is an odd number. Let $S$ be the sum of the numbers on the $n$ balls you draw, and $R$ be the sum of the numbers on the remaining $n$ balls. Calculate the probability that $S > R$. Express your answer as a fraction or a decimal.

Answer: \dfrac{1}{2}


Question:
Suppose $S \subseteq \mathbb{Z}[i]$ satisfies the condition:

\[ \limsup_{n\rightarrow\infty} \frac{|S\cap A_n|}{|A_n|} > 0 \]

where \( A_n = \{z : z\in\mathbb{Z}[i], \text{Nm}(z)\le n\} \). Determine if $S$ contains arbitrarily long arithmetic progressions.

Answer: Yes


Question:
Given the system of equations:
\[ x^3 - 3xy^2 = 2010, \]
\[ y^3 - 3x^2y = 2009, \]
where \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) are the solutions, find the value of:
\[ \left(1 - \frac{x_1}{y_1}\right) \left(1 - \frac{x_2}{y_2}\right) \left(1 - \frac{x_3}{y_3}\right). \]

Answer: \dfrac{2}{2009}


Question: Calculate the limit \( \lim_{n\to\infty}\frac{\log n!}{n\log n} \).

Answer: 1


Question: In a $T_0$ topological space, if the space is totally disconnected, does it necessarily have a base consisting of closed sets?

Answer: No


Question: Evaluate the integral: \[ \int_0^{\infty} \frac{\sin^3{x}}{x} \, dx \]

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


Question: Suppose we have information about the supermarket purchases of 100 million people. Each person goes to the supermarket 100 times in a year and buys 10 of the 1000 items that the supermarket sells. Assume there are no criminals and that everyone behaves at random. Calculate the expected number of pairs of people who buy the same set of 10 items at least once during the year. Use this to determine if finding such pairs would likely indicate criminal activity.

Answer: 0


Question:
Find the kernel of the adjoint representation \( P: SU_2 \to \text{Aut}_\mathbb{R}(E) \), where \( P(A) \) maps \( X \) to \( AXA^{-1} \). Here, \( E \) is the space of traceless Hermitian matrices of the form:
\[
\begin{bmatrix}x_1 & x_2 + ix_3 \\ x_2 - ix_3 & -x_1\end{bmatrix}
\]
with \( x_1, x_2, x_3 \in \mathbb{R} \). Determine the matrices \( A \) such that \( AXA^{-1} = X \) for all \( X \) in \( E \).

Answer: \{\pm I\}


Question: If $N \unlhd G$, and $M,K \leq G$ such that $M \unlhd K$, does it imply that $MN \unlhd KN$? Justify your answer.

Answer: Yes


Question: Let $f$ be a continuous function on the interval $[0,1]$ mapping to $\mathbb{R}$. Define the linear operator $T$ by \[ T(f)(x) = \begin{cases} f(0), & \text{if } x = 0, \\ \frac{1}{x}\int_0^x f(t) \, dt, & \text{if } 0 < x \leq 1. \end{cases} \] Determine whether \[ \lim_{n \to \infty} \| T^n(f) - f(0) \|_{\infty} = 0. \]

Answer: 0


Question: Find the maximum value of \( P(X = Y) \) for random variables \( X \) and \( Y \) with Bernoulli distributions having parameters \( p \) and \( q \), respectively. The random variables \( X \) and \( Y \) are not necessarily independent.

Answer: 1 - |p - q|


Question: Determine the asymptotic complexity of the recurrence relation \( T(n) = T(\lfloor n/2 \rfloor) + T(\lfloor n/3 \rfloor) + \lfloor n \log_2 n \rfloor \) with initial conditions \( T(0) = T(1) = T(2) = 1 \). Express your answer using Big O notation.

Answer: O(n \log n)


Question:
Find the base $b$ such that for any $\epsilon > 0$, the following asymptotic behavior holds for the function \( f_n(2,2) = \sum_{k=0}^n \binom{2n}{k} 2^k \):
\[ \frac{f_n(2,2)}{(b+\epsilon)^n} \to 0 \quad \text{and} \quad \frac{f_n(2,2)}{(b-\epsilon)^n} \to \infty \quad \text{as} \quad n \to \infty. \]

Answer: 8


Question: Find the natural number \( n \) such that \( n^5 = 27^5 + 84^5 + 110^5 + 133^5 \). Use modular arithmetic to simplify the calculation.

Answer: 144


Question: Calculate the limit \( \lim_{r \rightarrow \infty} \frac{\Gamma(r\alpha)}{\Gamma((r+1)\alpha)} \) where \( 0 < \alpha < 1 \) and \( \Gamma(.) \) is the Gamma function.

Answer: 0


Question: Let $G$ be an abelian subgroup of the group of homeomorphisms on the circle. If every element of $G$ has a fixed point on the circle, does it imply that $G$ has a common fixed point?

Answer: Yes


Question:
Given that \( \phi_1(x) \) and \( \phi_2(x) \) are two linearly independent solutions of the differential equation:
\[ 2x^2y'' - (x + x^2)y' + (x^2 - 2)y = 0 \]
with the condition \( \phi_1(0) = 0 \), find the smallest positive integer \( n \) such that:
\[ \lim_{x \to 0} x^n \frac{\phi_2(x)}{\phi_1(x)} = 0 \]

Answer: 3


Question: Given that \( \frac{\partial CE}{\partial Z}=\hat{y}-y \) and \( Z=U^Tv_c \), where \( \hat{y}, y, v_c \) are vectors and \( U \) is a matrix, find the derivative \( \frac{\partial CE}{\partial v_c} \).

Answer: U(\hat{y} - y)


Question:
Calculate the length of the tangent vector \( \frac{\partial}{\partial \theta} \) at the north pole \( N \) of the sphere \( \mathbb{S}^2 \) with radius \( r \), given the metric:
\[ g = r^2 \, d\varphi \otimes d\varphi + r^2 \sin^2 \varphi \, d\theta \otimes d\theta \]
Verify if the length is zero at the north pole where \( \varphi = 0 \).

Answer: 0


Question: Let \( \lambda \) denote Lebesgue measure on \((0,\infty)\) and let \( \mu \) be a \( \sigma \)-finite Borel measure such that \( \mu \ll \lambda \) and \( \mu(B) = \alpha \mu(\alpha B) \) for any \( \alpha \in (0,\infty) \) and any Borel set \( B \subset (0,\infty) \). If \( \frac{d\mu}{d\lambda} \) is continuous, prove that there exists \( c \in (0,\infty) \) such that \( \frac{d\mu}{d\lambda}(x) = \frac{c}{x^2} \).

Answer: \dfrac{c}{x^2}


Question: Verify whether the function \( f(t,x) = t^3 e^{-tx^2} \) satisfies the Lipschitz condition on the domain \( (t,x) \in [0,1] \times \mathbb{R} \).

Answer: Yes


Question: Calculate the expected value of the random variable \( Z = \min\{X, Y\} \), where \( X \) and \( Y \) are independent and identically distributed with the probability mass function \( P(X = k) = P(Y = k) = \frac{1}{2^k} \) for \( k = 1, 2, \ldots \).

Answer: \dfrac{4}{3}


Question: Let \( X \) be a normally distributed random variable with mean \( \mu \) and standard deviation \( \sigma = \mu \). Given that \( P(-2 < X < 17) = 0.4375 \), find the value of \( \mu \).

Answer: 14.6


Question: Find the cosine of the angle at the vertex of an isosceles triangle that has the greatest area, given that the length of the median drawn to its lateral side is a constant value \( l \).

Answer: \dfrac{4}{5}


Question: Determine the number of real solutions for \(x\) in the equation \[ \left[\frac{2x+1}{3}\right] + \left[\frac{4x+5}{6}\right] = \frac{3x-1}{2} \] where \([x]\) denotes the greatest integer less than or equal to \(x\).

Answer: 9


Question: Let \( \{a_{n,k}\}_{n\in \mathbb{N}} \) be a sequence for some \( k\in \mathbb{N} \). Suppose that \( \lim_{n\rightarrow \infty} a_{n,k}=0 \) for all \( k=1,...,2n \). Does this imply \( \lim_{n\rightarrow \infty} \sum_{k=1}^{2n} a_{n,k}=0 \)? Justify your answer.

Answer: No


Question: The sequence $\{a_n\}$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n \geq 2$. What is the smallest positive integer $k$ such that the product $a_1 a_2 \cdots a_k$ is an integer?\n\nA. $17$\nB. $18$\nC. $19$\nD. $20$

Answer: A


Question: Calculate the conditional expectation \( E(B_t | B_1) \) for a Brownian motion \( B \) where \( t \in (0,1) \).

Answer: t B_1


Question: Evaluate the integral: \[ \int_{-\infty}^{+\infty} e^{(ix+b)^2} \, dx \] where \( x \) is a real variable and \( b \) is a constant.

Answer: \sqrt{\pi}


Question: Arrange four black balls, four white balls, and two red balls in a row such that for each color, not all the balls of that color are consecutive. Find the remainder when the number of such arrangements is divided by 1000.

Answer: 376


Question: Given that \((1+(2/3)x)^n \cdot (3+nx)^2 = 9+84x+\dotsb\), find the value of \(n\).

Answer: 7


Question: Find the Cauchy principal value of the integral \( \int_{-\infty}^{\infty} \frac{\sin(x)}{x-a} \, dx \).

Answer: \pi \cos a


Question: On a board, there are distinct natural numbers such that the fractional part of their arithmetic mean is 0.2016. Find the smallest possible value of the integer part of the arithmetic mean of these numbers.

Answer: 313


Question: Determine, as a formula in terms of \( n \), the maximum number of distinct \( n \)-letter words that can be read from a single convex \( n \)-gon, where each corner is labeled with a distinct letter. Assume the observer is outside the polygon and can see all letters without any being obscured.

Answer: 2n


Question: Calculate the sum of the following limits, rounded to four decimal places: \( I(2019,2020) + I(2018,2020) + I(2019,2021) \), where \( I(n,m) = \lim\limits_{x \to \pi} \frac{\sin (nx)}{\sin (mx)} \).

Answer: 0.9985


Question: Evaluate the infinite series \( \sum\limits_{n=1}^{\infty}\frac{H_n}{(n+1)(n+2)} \).

Answer: 1


Question: Evaluate the line integral \( \int_C \vec{F} \cdot d\vec{s} \), where \( \vec{F}(x,y) = xy^2 \hat{i} + x^2y \hat{j} \) and \( C \) is the upper half of the circle \( x^2 + y^2 = 1 \).

Answer: 0


Question: Given an infinite set $A$, determine whether the cardinality of $A$ is equal to the cardinality of the Cartesian square $A^2$. Provide a justification for your answer.

Answer: Yes


Question: Consider a function \( f: \mathbb{R} \longrightarrow \mathbb{R} \) that is continuous on the open interval \((a, b)\), differentiable on \((a, b)\), and satisfies \( \lim\limits_{x\to a^+} f(x) = \lim\limits_{x\to b^-} f(x) \). Determine whether there exists a point \( c \in (a, b) \) such that \( f'(c) = 0 \).

Answer: Yes


Question: Find all real solutions to the equation \((2^x + 3^x + 5^x)^3 = 160^x\).

Answer: 3


Question:
Let $N$ be a random variable uniformly distributed over the set \( \{0, 1, 2, \ldots, 19\} \). Compute the expected value:
\[ \mathbb{E}\left(\sum_{k=0}^{N} {N-k \choose k} (-1)^k \right) \]

Answer: \dfrac{1}{10}


Question: Determine the radius of convergence of the power series \( \sum_{n \geq 1} n^{\log(n)} x^n \).

Answer: 1


Question: Evaluate the limit \( \lim_{n \to \infty} n \cdot \left(1 - \frac{1}{n}\right)^{n \cdot \ln(n)} \).

Answer: 1


Question: Let $H$ be a Hilbert space and let $\mathcal{A}$ be a subalgebra of the algebra $\mathcal{B}(H)$ of all bounded operators on $H$. Suppose $\mathcal{A}$ consists of compact operators only, and each vector $v \in H$ lies in the closure of $\mathcal{A}v$. Determine whether there must exist an irreducible subspace for $\mathcal{A}$. Provide a justification for your answer.

Answer: Yes


Question: Let $K$ be an algebraically closed field of characteristic $0$. Determine the polynomial condition for $a \in K$ such that the polynomial $f(x) = x^3 - 3ax + 1$ has distinct roots.

Answer: 4a^3 - 1


Question:
Let $X$ be a standard normal random variable. Calculate $P(X < 0 \mid |\lfloor X \rfloor| = 1)$. Choose the correct answer from the following options:

- $\frac{\Phi(1) - \frac{1}{2}}{\Phi(2) - \frac{1}{2}}$
- $\frac{\Phi(1) + \frac{1}{2}}{\Phi(2) + \frac{1}{2}}$
- $\frac{\Phi(1) - \frac{1}{2}}{\Phi(2) + \frac{1}{2}}$
- $\frac{\Phi(1) - 1}{\Phi(2) + 1}$

Answer: A


Question: Calculate the area of the intersection between the sphere defined by $x^2 + y^2 + z^2 = 4$ and the region where $z \ge 1$. Express your answer as a single number.

Answer: 4\pi


Question:
Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function such that \( f(i) = 0 \) for all \( i \in \mathbb{Z} \). Which of the following statements is always true?

A. The image of \( f \) is closed in \( \mathbb{R} \).
B. The image of \( f \) is open in \( \mathbb{R} \).
C. \( f \) is uniformly continuous.
D. None of the above.

Select the correct option.

Answer: D


Question: Is a bijective entire function uniformly continuous? Specifically, if $f$ is an entire function that is bijective, is $f$ uniformly continuous?

Answer: Yes


Question: Determine the minimal polynomial of \(\sqrt{2} + \sqrt{3}\) over \(\mathbb{Q}\). Verify whether the polynomial \(x^4 - 10x^2 + 1\) is irreducible over \(\mathbb{Q}\).

Answer: x^4 - 10x^2 + 1


Question: Evaluate the limit: \[ I = \lim_{t\to0^{+}}\lim_{x\to+\infty}\dfrac{\displaystyle\int_{0}^{\sqrt{t}}dx\int_{x^2}^{t}\sin{y^2}dy}{\left[\left(\dfrac{2}{\pi}\arctan{\dfrac{x}{t^2}}\right)^x-1\right]\arctan{t^{\frac{3}{2}}}} \]

Answer: -\dfrac{\pi}{7}


Question: Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix such that $A$ is not of the form $A = c I_n$ for any $c \in \mathbb{R}$, and $(A-2I_n)^3 (A-3I_n)^4 = 0$. Determine the minimal polynomial $m_A(x)$ of $A$.

Answer: (x - 2)(x - 3)


Question: Evaluate the limit: $$\lim_{n \to \infty} \int _{n}^{2n}\frac{dx}{(f^{-1}(x))^3+f^{-1}(x)}$$ where $f(x)=x^3-x^2+x$. Provide your answer as a single value.

Answer: \ln 2


Question: In right triangle $\triangle ABC$ with right angle at $A$, the angle bisectors $BD$ and $CE$ divide the triangle such that $AE = 8$ and $AD = 9$. Find the perimeter of $\triangle ABC$. Use the angle bisector theorem and the Pythagorean theorem to assist in your calculations.

Answer: 72


Question: A ball has an initial volume of 3.0 cm³. The volume decreases over time, with the rate of change proportional to the surface area of the ball. After 1 month, the volume is 2.0 cm³. What is the volume of the ball after 4 months?

Answer: 0.36


Question: Evaluate the expression \( \sqrt{2013 + 2012 \sqrt{2013 + 2012 \sqrt{2013 + 2012 \sqrt{\ldots}}}} \) in its simplest form.

Answer: 2013


Question: Let $X$ be a Banach space and $T: \ell^2 \rightarrow \ell^2$ be a bounded linear map. Suppose the linear map $T \otimes \text{Id}_X: \ell^2 \otimes X \rightarrow \ell^2 \otimes X$, which maps $e_i \otimes x$ to $e_i \otimes T(x)$, is not bounded when using the norm on $\ell^2 \otimes X$ induced by the Banach space $\ell^2(X)$. Does there exist a sequence $(X_n)$ of finite-dimensional subspaces of $X$ such that \[ \|T \otimes \text{Id}_{X_n}\|_{\ell^2(X_n) \to \ell^2(X_n)} \xrightarrow[n \to +\infty]{} +\infty? \]

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \to \infty} \int_0^1 \int_0^x (x-y)^n f(y) \, dy \, dx,$$ where $f(x)$ is a continuous function over the interval $[0,1]$. Determine the value of this limit.

Answer: 0


Question:
Let \(\left( x_{n} \right)_{n \geq 1}\) be a sequence defined by:
\[ x_{1} = \frac{1}{2014} \quad \text{and} \quad x_{n+1} = x_{n} + x_{n}^{2}, \quad \forall n \geq 1 \]
Compute the integer part of the sum:
\[ S = \frac{x_1}{x_2} + \frac{x_2}{x_3} + \cdots + \frac{x_{2014}}{x_{2015}} \]
Find \( \left\lfloor S \right\rfloor \).

Answer: 2013


Question: Find the holomorphic function \( f = u + iv \) given that the real part \( u(x, y) = e^x \sin y \) and \( f(0) = 0 \). Verify that \( u \) is harmonic and determine the corresponding imaginary part \( v(x, y) \) to construct \( f \).

Answer: i(1 - e^z)


Question: Determine the value of \( r \) such that \( \int_{\mathbb{R}^d} |\nabla f_{\lambda}(x)|^2 dx = \lambda^{r}\int_{\mathbb{R}^d} |\nabla f(x)|^2 dx \), where \( f_{\lambda}(x) = f(\lambda x) \) and \( f \in \mathcal{S}(\mathbb{R}^d) \).

Answer: 2 - d


Question:
Determine the asymptotic behavior of the following sum as \( n \to \infty \):
\[
\sum_{i=1}^{n-1} \binom{n}{i}3^{\binom{n-i}{2}-\binom{n}{2}}2^{\binom{i}{2}}
\]
Does the sum approach infinity, zero, or some other value?

Answer: 0


Question: Given an \( n \times n \) matrix \( A \) such that \( \text{det}(A) = 9 \), and \( A^3 A^T = 3A^{-1} \), find the value of \( n \).

Answer: 10


Question:
Consider the function defined by
\[ f(x,y) = \begin{cases} \frac{x^{5}}{\sqrt{9 x^{8}+9(y + 2)^{8}}}, & \text{if } (x,y) \neq (0,-2), \\ 0, & \text{if } (x,y) = (0,-2). \end{cases} \]

1. Calculate the partial derivatives of \( f \) with respect to \( x \) and \( y \) assuming \((x,y) \neq (0,-2)\).

2. Calculate \( \frac{\partial f}{\partial x}(0,-2) \) and \( \frac{\partial f}{\partial y}(0,-2) \).

3. Calculate the directional derivative of \( f \) at \((0,-2)\) along a general vector \( \vec{v} = (a,b) \).

4. Calculate \( \nabla f(0,-2) \cdot (a,b) \).

5. By comparing your results from parts (3) and (4), what can you conclude about the function \( f \)?
   a) \( f \) is \( C^1 \) and differentiable.
   b) \( f \) is \( C^1 \) but not differentiable.
   c) \( f \) is not \( C^1 \) but is differentiable.
   d) \( f \) is not \( C^1 \) and is not differentiable.

Choose the correct option.

Answer: d


Question: Evaluate the integral \( \int _{|z|=2} \frac{z \, dz}{(z^2-9)(z+i)} \).

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


Question: Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \( f(f(x)+3y)=12x + f(f(y)-x) \).

Answer: f(x) = 3x + c


Question: Maximize the function \( g(x, y) = x^4 + y^4 \) subject to the constraint \( x^2 + y^2 = 9 \).

Answer: 81


Question: Let \( N \in \textrm{Mat}_n(\mathbb{C}) \) be a nilpotent matrix. Consider an invertible matrix \( g \in \operatorname{GL}_n(\mathbb{C}) \) with its multiplicative Jordan decomposition \( g = g_sg_u \), where \( g_s \) is diagonalizable, \( g_u \) is unipotent, and \( g_sg_u = g_ug_s \). Suppose \( N \) is an eigenvalue for the linear operator \( \operatorname{Ad}g \) on \( \textrm{Mat}_n(\mathbb{C}) \), meaning \( gNg^{-1} = \lambda N \) for some \( 0 \neq \lambda \). Does \( g_u \) necessarily commute with \( N \)?

Answer: Yes


Question:
Let S be a set with N elements, and let \( A_1, A_2, \ldots, A_{101} \) be 101 (possibly not disjoint) subsets of S with the following properties:

1. Each element of S belongs to at least one subset \( A_i \) for \( i \in \{1, 2, \ldots, 101\} \).
2. Each subset \( A_i \) contains exactly 1000 elements of S.
3. The intersection of any pair \( \{A_i, A_j\} \) of distinct subsets contains exactly 200 elements.
4. The intersection of any three distinct subsets \( \{A_i, A_j, A_k\} \) contains exactly 6 elements.
5. The intersection of any four or more distinct subsets is empty.

Using the inclusion-exclusion principle, compute the number of elements \( N \) in the set S.

Answer: 90900


Question: Evaluate the integral \( \iint_{T_a} (x+y) \, dx \, dy \) using the change of variables \( u = x+y \) and \( v = x \), where \( T_a \) is the region in the \( xy \) plane bounded by the \( x \) and \( y \) axes and the line \( x+y = a \).

Answer: \dfrac{a^3}{3}


Question: In a Noetherian ring $R$, every ideal is finitely generated. Suppose an ideal $I$ in $R$ is generated by an infinite set of generators $S$. Does there exist a finite subset of $S$ that also generates $I$?

Answer: Yes


Question: Let $f:B^{m}\to\mathbb{C}$ be a multivariate function defined by a power series $f(x)=\sum_{\alpha}c_{\alpha}x^{\alpha},\,\,x\in B^{m}$, where $\alpha\in\mathbb{N}^{m+1}$ and $B^{m}=\{x\in\mathbb{R}^{m+1},\,\,||x||<1\}$. If $\sum_{\alpha}c_{\alpha}x^{\alpha}<\infty,\,\forall\,\,x\in B^{m}$, is it true that for each $p\in B^{m}$ there exist $\{d_{\alpha}\}\subset \mathbb{C}$ and a neighborhood $p\in B^{m}$ such that $$f(x)=\sum_{\alpha}d_{\alpha}(x-p)^{\alpha}$$?

Answer: Yes


Question: Let $A$ be an $n \times n$ matrix with integer entries, each with an absolute value bounded by a constant $C$. Determine whether the entries of the inverse matrix $A^{-1}$ can grow more rapidly than exponentially with respect to $n$, or if they are bounded by $(C + O(1))^n$.

Answer: (C + O(1))^n


Question: Does there exist a smooth orthonormal basis for \(L^2(T)\) with respect to the Lebesgue measure on the unit circle \(T\)?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\rightarrow\pi/4}(2\sin(x)-\sqrt{2})\tan(x-3\pi/4) \]

Answer: -\sqrt{2}


Question: Given non-zero integers $a, b, c$ such that \( \frac{a}{b} + \frac{b}{c} + \frac{c}{a} = m \in \mathbb{Z} \) and \( \frac{a}{c} + \frac{c}{b} + \frac{b}{a} = n \in \mathbb{Z} \), calculate all possible values for $m + n$. Provide your answer as a set of integers.

Answer: \{-2, 6\}


Question: Find the limit of \( \frac{\log(n!)}{\log(n+1)} \) as \( n \to \infty \).

Answer: \infty


Question: A miner is trapped in a mine with three identical doors. The first door leads to safety in 2 hours. The second door leads back to the starting point in 3 hours. The third door leads to a maze, returning to the starting point in 5 hours on the first entry, and 1 hour on subsequent entries. The miner chooses each door with equal probability, independent of previous choices. What is the expected time until the miner reaches safety?

Answer: 8


Question: Consider a non-zero vector \( a \) such that \( a \in \{0,1\}^n \), and a vector \( b \) chosen uniformly and randomly from \( \{0,1\}^n \). What is the probability that the dot product \( \sum_{i=1}^{n} a_i b_i \) is odd?

Answer: \dfrac{1}{2}


Question: Does there exist a dense subset of functions $f \in L^{2}([0, 1])$ such that $x^{-1/2}f(x) \in L^{1}([0, 1])$ and $\int_{0}^{1}x^{-1/2}f(x)\, dx = 0$?

Answer: Yes


Question: Determine whether there exists a constant \( C \) such that for the sigmoid function \( \sigma(x) = \frac{1}{1+e^{-x}} \), the inequality \( \left| \frac{\sigma^{(n)}(x)}{n!} \right| \leq C \) holds for any \( x \in [-1/2, 1/2] \) and any \( n \geq 1 \).

Answer: Yes


Question: Evaluate the surface integral \( \iint_Y \mathbf{F} \cdot \mathbf{N} \, \mathrm{d}S \) where \( \mathbf{F}= (x^2 yz + x \sin z , x^2 + y(1 - \sin z ), x + y - xy z^2) \) and \( Y \) is the part of the conical surface \( 4x^2 + (y-1)^2 = z^2 \) which lies between \( z = 1 \) and \( z = 2 \). The normal vector points away from the \( z \)-axis. Use Gauss's theorem to assist in the calculation.

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


Question: Find the limit: \(\lim\limits_{x\to\frac{\pi}{2}}(\sin x)^{\tan x}\).

Answer: 1


Question: Determine whether the class of unitary matrices is the same as the class of diagonalizable matrices with all eigenvalues on the unit circle.

Answer: No


Question:
Find the surface area generated by revolving the curve $x = y^3$ about the $y$-axis for $0 \leq y \leq \sqrt[4]{11}$. Choose the correct answer from the following options:
(a) $23\pi$  (b) $37\pi$  (c) $46\pi$  (d) $62\pi$  (e) $73\pi$  (f) None of these.

Answer: B


Question: Evaluate the limit \( \lim_{N \to \infty} \int_{N}^{e^N} xe^{-x^{2017}} \, dx \).

Answer: 0


Question: What is the minimum number of times you need to roll a six-sided die so that the probability of rolling each number at least once is greater than 0.5?

Answer: 13