Question: Determine whether the following statement is true or false: Let $A$ be a $\mathbb{Z}[\pi_1(X)]$-module. The homology of $X$ with local coefficients in $A$ is equal to $H_*(\tilde{X} \times_{\pi_1(X)} A,\mathbb{Z})$. Provide a justification for your answer.

Answer: False


Question: Determine the radius of convergence of the power series solution to the differential equation \((1-x^2)y''-2xy'+\lambda y = 0\) using the ratio test. The solution is given by \(y(x) = \sum_{n=0}^{\infty} C_n x^n\) with the recurrence relation \(C_{n+2} = \frac{n(n+1)-\lambda}{(n+2)(n+1)}C_n\).

Answer: 1


Question: Let $V$ be a finite-dimensional vector space and $F \subset V^*$ be a subspace. Determine the dimension of the subspace $Z(F) = \{v \in V : \forall \alpha \in F, \alpha(v) = 0\}$. Express your answer in terms of $\dim V$ and $\dim F$. \( \boxed{} \)

Answer: \dim V - \dim F


Question: Evaluate the definite integral: \[ \int_{-1}^{3} \left( \tan^{-1}\left(\frac{x}{x^2+1}\right) + \tan^{-1}\left(\frac{x^2+1}{x}\right) \right) \, dx \]

Answer: \pi


Question: Let \( \{A_n\}_{n \in \mathbb{N}} \) be a sequence of real numbers such that \( \lim_{n \rightarrow \infty}A_n=0 \). Does this imply that \( \operatorname{plim}_{n\rightarrow \infty}A_n=0 \), where \( \operatorname{plim} \) is the probability limit?

Answer: Yes


Question: Let $Y$ be an exponential random variable with mean $\frac{1}{\theta}$, where $\theta > 0$. The conditional distribution of $X$ given $Y$ is Poisson with mean $Y$. What is the variance of $X$? (A) $\frac{1}{\theta^2}$ (B) $\frac{\theta+1}{\theta}$ (C) $\frac{\theta^2+1}{\theta^2}$ (D) $\frac{\theta+1}{\theta^2}$

Answer: D


Question: Determine the number of homomorphisms \( f: (\mathbb{Z}, +) \to (\mathbb{Q}^*, \cdot) \) such that \( f(2) = \frac{1}{3} \).

Answer: 0


Question: Let \( \{a_n\} \) be a sequence of real numbers such that \[ \lim_{n \to \infty} \left|a_n + 3\left(\frac{n-2}{n}\right)^n\right|^{1/n} = \frac{3}{5}. \] Compute \( \lim_{n \to \infty} a_n. \)

Answer: -\dfrac{3}{e^2}


Question:
A machine starts operating at time $0$ and fails at a random time $T$. The distribution of $T$ has density $f(t) = \frac{1}{3}e^{-t/3}$ for $t > 0$. The machine will not be monitored until time $t = 2$. What is the expected time of discovery of the machine's failure? Choose the correct answer from the following options:

(a) $2 + \frac{e^{-6}}{3}$
(b) $2 - 2e^{-2/3} + 5e^{-4/3}$
(c) $2 + 3e^{-2/3}$
(d) $3$

Answer: C


Question: In a Galois Field \( GF(p^n) \), where \( p \) is a prime, the Frobenius automorphism \( \phi \) is defined by \( \phi(x) = x^p \) for each element \( x \) in the field. Determine the smallest positive integer \( m \) such that \( \phi^m \) is the identity automorphism on \( GF(p^n) \).

Answer: n


Question: Calculate the residue of \( \frac{1}{z^2 \sin z} \) at \( z = 0 \).

Answer: \dfrac{1}{6}


Question: Evaluate the integral \(I = \int_0^2 (3x^2 - 3x + 1) \cos(x^3 - 3x^2 + 4x - 2) \, dx\). If \(I\) can be expressed as \(p(\sin q)\), where \(p, q \in \mathbb{N}\), find the value of \(p + q - 1\).

Answer: 3


Question: Determine the number of connected components of the quotient space \( \mathbb{R}^2 / \sim \setminus \{(0,0)\} \), where \( \sim \) is the equivalence relation defined by \([0,0] = [0,1] = \{(0,0),(0,1)\}\) and \([x,y] = \{(x,y)\}\) if \(x \neq 0\).

Answer: 1


Question: Find the slope of the common tangent to the hyperbolas \(\frac{x^2}{9}-\frac{y^2}{16}=1\) and \(\frac{y^2}{9}-\frac{x^2}{16}=1\). Choose the correct option: (A) -2, (B) -1, (C) 2, (D) None of these.

Answer: D


Question: Does there exist a function $f$ such that for all $p > 0$, \( \lim_{n \to \infty}\frac {|f(n)|}{|n^p|}>1 \) and for all $a > 0$, \( \lim_{n \to \infty}\frac {|f(n)|}{|e^{an}|}<1 \)?

Answer: Yes


Question: Find the greatest common divisor (GCD) of the complex numbers \(29-3i\) and \(11+10i\) in the Gaussian integers \(\mathbb{Z}[i]\).

Answer: 1 + 4i


Question: Given two points randomly selected from a uniform distribution over the Euclidean plane, with a known Euclidean distance $d$ between them, find the expected value of the Manhattan distance $m$ between the two points, denoted as $E[m|d]$. Provide your answer in terms of $d$.

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


Question: Compute the expression of the derivative $DF_{(5,0,-4)}$ in the chosen coordinate charts of the manifolds $M$ and $S^1$. Here, $M = \{(x,y,z) \in \mathbb{R}^3 : x+y=5, x+z=\cos^2y\}$ is a 1-dimensional submanifold of $\mathbb{R}^3$, and $F: M \rightarrow S^1$ is given by $F(x,y,z) = \left(\frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}}\right)$. Use the chart $\varphi: M \rightarrow \mathbb{R}$ defined by $\varphi(x,y,z) = y$ for $M$, and the chart $\psi: U \subset S^1 \rightarrow (-1,1)$ defined by $\psi(x,y) = y$ for $S^1$, where $U = \{(x,y) \in S^1 : x > 0\}$. Determine $DF_{(5,0,-4)}$ in these coordinates.

Answer: \dfrac{1}{5}


Question: Evaluate the expression \( \frac{100^2}{100!} + \sum_{k=1}^{100} \left| (k^2 - 3k + 1) S_k \right| \), where \( S_k \) is the sum of the infinite geometric series with first term \( \frac{k-1}{k!} \) and common ratio \( \frac{1}{k} \).

Answer: 3


Question: Evaluate the limit: \[\lim_{x\to\infty} x\log(-e^x + 1)+\operatorname{Li}_2(e^x)-\frac{1}{2}x^2=\frac{\pi^2}{3}\] where \(\operatorname{Li}_n(x)\) is the polylogarithm function.

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


Question: Given two primes $p$ and $q$, determine an upper bound for the smallest natural number $K$ such that the set of natural number multiples $pN + qN$ spans all natural numbers greater than or equal to $K$. Provide your answer as an expression or a specific value for $K$.

Answer: pq - p - q + 1


Question: Consider the space \( (\mathbb{R}^2, T) = (\mathbb{R}, U) \times (\mathbb{R}, U) \), where \( U \) is the usual topology on \( \mathbb{R} \). Let \( Y = \mathbb{R}^2 - A^2 \), where \( A \) is the set of irrational numbers. Determine the connected component of the point \((0,0)\) in the space \((Y, T_Y)\), where \( T_Y \) is the relative topology on \( Y \).

Answer: Y


Question:
Simplify the expression:
\[
\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}
\]

Answer: 373


Question: Evaluate the limit \( \lim_{x\rightarrow +\infty} \left(\sqrt[n]{n + x} - \sqrt[n]{x}\right) \), where \( n \) is a positive integer greater than 1.

Answer: 0


Question: Evaluate the integral \( \int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}} \, dx \).

Answer: \pi^2


Question: Compute the limit \( \lim_{n \to \infty} P\left( \sigma^2 \in (A_n, B_n) \right) \), where \( A_n = \frac{(n-1)S_{n}^{2}}{\chi^{2}_{n-1,\alpha/2}} \) and \( B_n = \frac{(n-1)S_{n}^{2}}{\chi^{2}_{n-1,1-\alpha/2}} \). Assume \( X_1, X_2, \ldots, X_n, \ldots \) are i.i.d. random variables with \( \mathbb{E}(X_1^4) < \infty \).

Answer: 1 - \alpha


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

Answer: \dfrac{3e}{2}


Question: Determine the number of real values of \(x\) that satisfy the equation \((x^{2022}+1)(1+x^2+x^4+\ldots+x^{2020})=2022\cdot x^{2021}\).

Answer: 1


Question: Show that if $X_n$ is a Poisson random variable with parameter $n$, then the normalized random variable $\frac{X_n-n}{n^{1/2}}$ converges in distribution to a standard normal random variable $Z$ as $n \to \infty$, using the central limit theorem.

Answer: Z


Question: Calculate the limit \( \lim \limits_{x \to 0} \frac{\ln\left(\frac{\sin x}{x}\right)}{x^2} \) using L'Hôpital's rule.

Answer: -\dfrac{1}{6}


Question: Find the limit: \[ \lim_{n\rightarrow \infty} \frac{e^n}{(n+4)!} \]

Answer: 0


Question: Suppose $T$ is a bounded linear operator on a separable Hilbert space $H$ such that $0 \in \sigma_{e}(T)$, where $\sigma_{e}(T)$ represents the essential spectrum of $T$. Does this imply that the operator $T$ is compact?

Answer: No


Question: Determine whether the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$ is complete under the metric $d(f,g) = \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Provide a justification for your answer.

Answer: Yes


Question: Determine the values of \( \alpha \in \mathbb{R} \) such that the function \((x,y) \mapsto (1-xy)^{\alpha}\) is integrable on the region \((0,1)\times (0,1)\).

Answer: (-2, \infty)


Question: An alien arrives on Earth and each day, every alien independently chooses one of the following actions with equal probability of \(\frac{1}{4}\): destroy itself, split into 2 aliens, split into 3 aliens, or do nothing. Determine the probability that eventually there are no aliens left on Earth.

Answer: \sqrt{2} - 1


Question: There are 99 identical square tiles, each with a quarter-circle drawn on it. When the tiles are randomly arranged in a 9 by 11 rectangle, what is the expected value of the number of full circles formed?

Answer: \dfrac{5}{16}


Question: Calculate the work done by the force \( \vec{F}(x,y) = (x^2 - y^2) \ \vec{i} + 2xy \ \vec{j} \) in moving a particle counterclockwise around the square with corners \((0,0)\), \((a,0)\), \((a,a)\), \((0,a)\), where \(a > 0\).

Answer: 2a^3


Question: Determine if there exists a constant \( a \) such that the series \( \sum \left( \frac{1}{\sqrt{4n+1}} + \frac{1}{\sqrt{4n+3}} - \frac{a}{\sqrt{2n}} \right) \) converges.

Answer: \sqrt{2}


Question: Determine the number of subgroups isomorphic to \(\mathbb{Z}_{2} \times \mathbb{Z}_{2}\) in the dihedral group \(D_8 = \langle \sigma, \rho \; | \; \rho^8 = \sigma^2 = e \text{ and } \sigma\rho\sigma = \rho^{-1} \rangle\).

Answer: 4


Question:
Estimate the value of the expression \( \frac{1}{\Delta}\sqrt{abc(a+b+c)} \) for a triangle with sides \( a, b, c \) and area \( \Delta \). Use Heron's formula, where \( \Delta = \sqrt{s(s-a)(s-b)(s-c)} \) and \( s = \frac{a+b+c}{2} \). Choose the correct option:

(A) \( \leq 16 \)

(B) \( \geq 16 \)

(C) \( \leq 4 \)

(D) \( \geq 4 \)

Answer: D


Question: Evaluate the integral \[ \int_{(-\infty,\infty)^n} \frac{\prod_{k=1}^n \sin(a_k x_k)}{\prod_{k=1}^n x_k} \frac{\sin(\sum_{k=1}^n a_k x_k)}{\sum_{k=1}^n a_k x_k} \, dx_1 \cdots dx_n \] where \( a_1, \cdots, a_n > 0 \).

Answer: \pi^n


Question: Determine the number of distinct roots of the equation \( f^{1992}(x) = x \), where \( f(x) = 4x(1-x) \) and \( f : [0, 1] \to [0, 1] \).

Answer: 2^{1992}


Question:
Determine whether the following sum is correct:
\[ \sum_{k=1}^{\infty}\left(\frac{1}{kx}+\frac{1}{kx-1}+\frac{1}{kx-2}+\cdots+\frac{1}{kx-x+1}-\ln\frac{k+1}{k}\right)=\gamma+\ln{x} \]
where \( \gamma \) is Euler's constant and \( x \geq 1 \).

Answer: \gamma + \ln x


Question: Given the quadratic equation \(x^2 + (2-\lambda)x + (10-\lambda) = 0\), where \(\lambda \geq -2\), find the magnitude of the difference of the roots if the sum of the cubes of the roots is minimized.

Answer: 2\sqrt{5}


Question: Let \( f_n \) be a sequence of bounded measurable functions on a probability space. If \( f_n \) does not converge to 0 almost everywhere, does there exist a set of positive measure \( E \) such that \( \lim\sup_{n\to\infty}|f_n(x)|>2\epsilon \) for all \( x \in E \)? Answer yes or no.

Answer: Yes


Question: Consider the sequence \((a_n)\) with the property that for every \(\epsilon > 0\), there exists an \(N \in \mathbb{N}\) such that for all \(n \geq N\), \(|a_{n+1}| < \epsilon |a_n|\). Prove that \(\lim_{n \to \infty} a_n = 0\).

Answer: 0


Question: Determine for which values of \( p \geq 1 \) the functional \( F(f) = \int_{0}^{1}f(x^2)dx \) is bounded in \( L_p[0,1] \).

Answer: p > 2


Question:
Determine the values of \( t \in \mathbb{R} \) for which the function \( f_t: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) defined by the following mappings is a linear transformation:
\[
f_t\left(\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}\right)=\begin{pmatrix} -1 \\ 2 \\ -4-t \end{pmatrix}, \quad
f_t\left(\begin{pmatrix} 0 \\ 1 \\ t \end{pmatrix}\right)=\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}, \quad
f_t\left(\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}\right)=\begin{pmatrix} 7 \\ 2 \\ 2t+7 \end{pmatrix}
\]

Answer: t \neq 1


Question: Can a metric \(d\) be defined on the interval \((0,1)\) such that the topology induced by this metric is the same as the usual Euclidean topology on this set?

Answer: Yes


Question: Let $G$ be a connected linear algebraic group and $H$ be a closed subgroup of $G$ with codimension one. Is $H$ necessarily a normal subgroup of $G$?

Answer: No


Question: Given a ring $R$ and two distinct ideals $I$ and $J$ of $R$, is it possible for the quotient rings $R/I$ and $R/J$ to be isomorphic?

Answer: Yes


Question: Let \( F(x) \) be a polynomial of degree 2013 with integer coefficients such that \( F(0) = 1 \) and \( F(3) \) is a divisor of 2013. Determine the largest possible value of \( F(3) \).

Answer: 61


Question: Consider a random walk on the free group $F_2$ generated by $\{a, b\}$. Starting at the identity element $e$, at each step, you uniformly choose one of $\{a, a^{-1}, b, b^{-1}\}$ to append to the current word. What is the probability that the walk reaches the element $a$ before it reaches the element $b$?

Answer: \dfrac{1}{2}


Question: Let $a, b, c$ be three positive integers such that \(\text{lcm}(a,b) \cdot \text{lcm}(b,c) \cdot \text{lcm}(c,a) = a \cdot b \cdot c \cdot \gcd(a,b,c)\). Given that none of $a, b, c$ is an integer multiple of any other of $a, b, c$, find the minimum possible value of $a + b + c$. \(\text{lcm}\) denotes the least common multiple and \(\gcd\) denotes the greatest common divisor.

Answer: 300


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k=1}^{n^2} \frac{1}{\sqrt{n+k}} \]

Answer: \infty


Question: Let $X_1$ and $X_2$ be independent and identically distributed continuous random variables with probability density function $f(x) = 6x(1-x)$ for $0 < x < 1$, and $f(x) = 0$ otherwise. Using Chebyshev's inequality, determine the lower bound for the probability $P\left(|X_1 + X_2 - 1| \le \frac{1}{2}\right)$. Provide your answer as a fraction.

Answer: \dfrac{3}{5}


Question: Let $k$ be a characteristic $0$ field, and consider an étale morphism of $k$-schemes $f: X \rightarrow Y$. Let $K$ and $L$ be two extension fields of $k$ such that $K \subseteq L$ is a finite Galois extension. Suppose there is an $L$-point of $X$, $x \in X(L)$, such that its image under $f$ defines a point $y \in Y(K)$. Consider the absolute Galois group of $K$, $G = \text{Gal}(\bar{K}, K)$, and its action on $X(\bar{K})$. Is it true that for each element $\sigma$ of the Galois group $G$, the image of $\sigma^{\star} \circ \bar{x}$ is again $y$?

Answer: Yes


Question: Evaluate the limit \(\lim_{x\to0^+}\int_x^1\frac{f(t)}{t^3}\,\mathrm{d}t\), where \(f:[0,1]\) is a continuous function.

Answer: \infty


Question: Let $L$ be an infinite-dimensional subset of the linear space $\mathbb{R}^\omega$. Determine whether there exists an infinite set $I \subseteq \omega$ such that for every infinite set $J \subseteq I$, the projection of $L$ to $\mathbb{R}^J$ is infinite-dimensional.

Answer: Yes


Question: Find the holomorphic function \( f \) on the open unit disc \( |z|<1 \) that satisfies \( |f(z)| \leq \cos\left(\frac{\pi |z|}{2}\right) \).

Answer: 0


Question: Evaluate the line integral of the vector field \( F = \langle y^2, x, z^2 \rangle \) over the curve \( C \), which is the intersection of the plane \( x+y+z=1 \) and the cylinder \( x^2+y^2=1 \). The curve \( C \) is oriented clockwise when viewed from above.

Answer: -\pi


Question: Evaluate the expression \( \left\lfloor \sum_{n=1}^{\infty} \frac{n}{n^4+n^2+1} \right\rfloor - \left\lfloor \sum_{n=1}^{\infty} \frac{n^2}{n^8+n^4+1} \right\rfloor \).

Answer: 0


Question: Determine the values of \( p > 0 \) for which the integral \( \int_{-1}^{0} x^{-p} \, dx \) converges.

Answer: (0, 1)


Question: Determine the radius of convergence of the series \( \sum z^{2^{n}} \) and whether the series converges on the boundary of the unit circle, \( |z| = 1 \).

Answer: 1


Question: Let $X$ be a curve over a field $k$ that is geometrically connected, geometrically reduced, and stable. Let $Y \to X$ be the normalization of $X$. Determine whether $Y(k) = X(k)$. Provide a justification for your answer.

Answer: Y(k) \neq X(k)


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

Answer: \dfrac{1}{2}


Question: Find the sum of the series \( 1 - \ln 3 + \frac{(\ln 3)^2}{2!} - \frac{(\ln 3)^3}{3!} + \ldots \).

Answer: \dfrac{1}{3}


Question: Does the Hopf algebra $SU_q(2)$ contain any non-unital invertible elements?

Answer: No


Question: Suppose that $g$ is an integral form of discriminant $\mathbf{D}$ such that $g(u,v) = 1$. Can you deduce that $g$ is equivalent to the principal form?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty} \int_0^\infty \frac{n\sin x}{1+n^2x^2}\, dx \]

Answer: 0


Question: Determine the pointwise limit of the sequence of functions \( f_n: [0, 1/n] \to \mathbb{R} \) defined by \( f_n(0) = 0 \), \( f_n(1/n) = n \), and linear in between.

Answer: 0


Question: Find the coefficient of \(x^{18}\) in the expansion of \((x+1)(x+2)\cdots(x+10)(2x+1)(2x+3)\cdots(2x+19)\).

Answer: 5277440


Question: Find the sum of all possible values of \(\theta\) in the interval \((0, 2\pi)\) such that the quadratic equation \(x^2+(2-\tan\theta)x-(1+\tan\theta)=0\) has two integral roots.

Answer: 4\pi


Question: Given that $f(x) = g'(x)$ and $g(x) = -f'(x)$ for all real $x$, and that $f(2) = 4$ and $f'(2) = 4$, find the value of $f^2(4) + g^2(4)$. What can these functions be?

Answer: 32


Question: Given a point A outside a circle \( \Omega \), construct a secant line AB such that AC = CB, where C and B are points on the circle \( \Omega \). Use only a ruler and compass for the construction.

Answer: AB


Question: Find the extreme values of the function \( f(x,y,s,t) = xyst \) subject to the constraint \( x+y+s+t = 4c \), where \( c > 0 \) is fixed and \( x, y, s, t > 0 \). Use the method of Lagrange multipliers to solve this optimization problem.

Answer: c^4


Question: Evaluate the contour integral \( \oint_{\gamma} z^n dz \), where \( \gamma \) is a triangle with vertices at \( 0, 1, i \) oriented in the positive direction, and \( n \in \mathbb{Z} \).

Answer: 0


Question: Let $A, B \in \mathcal{M}_{n}(\mathbb{R})$ be matrices with integer entries. If $\det(A) = 0$, is $\det(A + 4B)$ always even? Provide a proof or counterexample.

Answer: Yes


Question: Find the area between the graph of the function \( f(x) = x \cdot e^{x^2 - 1} \) and its inverse function. Determine the points of intersection and calculate the integral to find the area.

Answer: \dfrac{2}{e}


Question: Find the radius of convergence for the series \( \sum_{k=0}^{\infty} \frac{(-1)^k x^{k+1}}{k+1} + \sum_{k=0}^{\infty} \frac{x^{k+1}}{k+1} \).

Answer: 1


Question: Evaluate the triple integral \( \iiint y^2 \, dV \) over the solid \( S \), which is a tetrahedron with vertices at \((0,0,0)\), \((0,0,1)\), \((1,1,0)\), and \((-1,1,0)\).

Answer: \dfrac{1}{10}


Question: Determine the values of \( p > 0 \) for which the series \( \sum_{n=1}^{\infty} p^{\sqrt{n}} \) converges.

Answer: (0, 1)


Question: Determine whether the limit \( \lim_{n\to\infty}\dfrac {C_n^{F_n}}{F_n^{C_n}} \) converges or diverges to infinity, where \( C_n=\dfrac {(2n)!}{n!(n+1)!} \) is the n-th Catalan number and \( F_n=2^{2^n}+1 \) is the n-th Fermat number.

Answer: \infty


Question: Find the volume of the solid bounded by the surfaces $y=0$, $z=0$, $y=x$, and $z=4-x^2-y^2$ in the first octant using a triple integral. Set up the integral with the correct bounds and evaluate it to find the volume.

Answer: \pi


Question: If a function satisfies that every pre-image of a recursively enumerable (r.e.) set is also a recursively enumerable set, can we conclude that the function is computable?

Answer: Yes


Question: Evaluate the surface integral \( \iint_{S} \vec{F} \cdot \vec{n} \, dS \) where \( S \) is the part of the cylinder \( x^2 + y^2 = 1 \) between the planes \( z = 0 \) and \( x + y + z = 2 \), and \( \vec{F} = x\vec{i} + y\vec{j} + z\vec{k} \).

Answer: 4\pi


Question: Evaluate the integral \( \int_K (x^2+y^2) \, dV \) where \( K \) is the solid cone defined by the inequalities \( \sqrt{x^2+y^2} \le z \le 1 \).

Answer: \dfrac{\pi}{10}


Question: A pond has a rare diatom present that is randomly distributed in its surface water with a rate of \( \lambda = 3 \) cells/ml. If three 1.5 ml samples of surface water are independently taken, what is the probability that exactly two of these three samples will contain four or more diatoms?

Answer: 0.444


Question: Let $X$ be an $n$-dimensional Alexandrov space with curvature $	ext{curvature} \geq k$. Determine if $X$ satisfies a reverse doubling condition. Specifically, does there exist a constant $C > 1$ such that for any $x \in X$ and $0 < r < \mathrm{diam}(X)/2$, the inequality $\mathrm{vol}(B_x(2r)) \geq C \cdot \mathrm{vol}(B_x(r))$ holds?

Answer: Yes


Question:
Let $u$ be a real-valued harmonic function on $
C$. Define $g:
R^2 \to
R$ by:
\[ g(x,y) = \int_0^{2\pi} u(e^{i\theta}(x+iy)) \sin \theta \, d\theta \]
Determine which of the following statements is TRUE about $g(x,y)$:
(A) $g$ is a harmonic polynomial
(B) $g$ is a polynomial but not harmonic
(C) $g$ is harmonic but not a polynomial
(D) $g$ is neither harmonic nor a polynomial

Answer: A


Question:
Let \( a > 1 \) be a real number that is not an integer, and let \( k \) be the smallest positive integer such that \([a^k] > [a]^k\). Determine which of the following statements is always true:

\(A)\,\, k\le 2([a]+1)^2\quad (B) \,\, k\le ([a]+1)^4\quad (C)\,\, k\le 2^{[a]+1}\quad (D)\,\, k\le\frac{1}{a-[a]}+1\)

Answer: D


Question: Consider a point \(x\) in the interval \([0,1]\). Construct a sequence \((y_n)\) where each term \(y_n\) is the supremum of the nth terms of all sequences that converge to \(x\) and are contained within \([0,1]\). Does the sequence \((y_n)\) converge to \(x\) or to 1?

Answer: 1


Question: Let $Z$ be a scheme, and $f, g : X \rightarrow Y$ be two morphisms of schemes over $Z$. Suppose the continuous function $i : \text{eq}(f,g) \rightarrow X$ is surjective. Does this imply that $f = g$? Provide a justification for your answer.

Answer: No


Question:
Solve for the real number \( x \) in the equation:
\[
\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{\cdots}}}}=x+\cfrac{x}{x+\cfrac{x}{x+\cfrac{x}{\cdots}}}
\]

Answer: 1


Question: Determine whether the function \( g(x) = \sum_{n=1}^\infty n(-x)^{n-1} \) is differentiable at \( x = 0 \). If it is, find \( g^{(n)}(0) \).

Answer: (-1)^n (n + 1)!


Question: Evaluate the integral \( \int_C \frac{e^{1/z}}{(z-i)^3} \, dz \) where \( C \) is the circle \( |z|=5 \).

Answer: 0


Question: Consider the open unit disk $D$ in the complex plane and its punctured version $D^* = D \setminus \{0\}$. Let $U_1, U_2, \ldots, U_n$ be an open cover of $D^*$. On each open set $U_j$, there exists an injective holomorphic function $f_j$ such that $df_j = df_k$ on every intersection $U_j \cap U_k$. Determine whether the differentials $df_j$ can be glued together to form a meromorphic 1-form on $D$. Justify your answer.

Answer: Yes


Question: Determine whether all topologies generated by uniform spaces are $T_1$. Suppose $X$ is a uniform space with uniformity $U$. A topology can be defined such that for any $x \in X$, the set $\{u[x]: u \in U\}$ forms a neighborhood base at $x$, where $u[x] = \{y \in X: (x,y) \in u\}$. Is the generated topology always $T_1$?

Answer: No


Question: Consider a party with ten people in a two-room apartment. Initially, it is equally likely that all ten people are in room A or that nine are in room A and one is in room B. Every minute, one person is chosen uniformly at random to move to the other room. What is the probability that the group is evenly split between rooms A and B (5 people in each) as the number of minutes goes to infinity?

Answer: \dfrac{63}{256}


Question: Given complex numbers \( c_1 = -i \) and \( c_2 = 3 \), and an arbitrary complex number \( z_0 \), we perform the following operations: Rotate \( z_0 \) around \( c_1 \) by \( \pi/4 \) counterclockwise to obtain \( z_1 \), and then rotate \( z_1 \) around \( c_2 \) by \( \pi/4 \) counterclockwise to obtain \( z_2 \). Determine the sum of the real and imaginary parts of a complex number \( c \) such that \( z_2 \) can be obtained from \( z_0 \) by rotating around \( c \) by \( \pi/2 \) counterclockwise.

Answer: 2 - \sqrt{2}


Question: Let \( G \) be a group of order 48 with a center consisting only of the identity element. Prove that \( G \) has a conjugacy class of order 3.

Answer: 3


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

Answer: e


Question: Evaluate the definite integral: \[ \int_0^\infty \frac{dx}{(x+\sqrt{1+x^2})^2}. \]

Answer: \dfrac{2}{3}


Question: Let the function \( f(z) = \frac{1}{z^2-2z+2} \). Compute \( \max_{z \in D(0,1)} |f(z)| \), where \( D(0,1) \) is the open disk centered at 0 with radius 1.

Answer: \sqrt{2}


Question: Determine all open sets on which the function \( f(z) = \frac{1}{x} + i\left(-\frac{1}{y}\right) \) is analytic.

Answer: \emptyset


Question: Let \((X_i)_{i=1}^{20}\sim N_6(\mu,\Sigma)\). Determine the distribution of the quadratic form \((X_1-\mu)^T\Sigma^{-1}(X_1-\mu)\).

Answer: \chi^2_6


Question: Let $Q$ be the subset of $\mathbb{R}^{n-1}$ consisting of vectors with strictly positive components, and let $R$ be the set of strictly positive real numbers. Consider $M = Q \times R$, the positive orthant of $\mathbb{R}^n$, and let $S$ be the graph of a smooth (e.g., $C^1$) strictly positive function $f$ on $Q$. Given that $S$ is an embedded submanifold of $M$ of codimension one, is $S$ closed as a subset of $M$?

Answer: Yes


Question: Does there exist a sequence of integers such that for every positive integer \( k \), it contains an arithmetic subsequence of length \( k \), but it does not contain an infinitely long arithmetic subsequence?

Answer: Yes


Question: Evaluate the limit \( \lim\limits_{n \rightarrow +\infty} \int_{0}^{1} nxe^{-\sqrt{n}x}e^{-x^2} \, dx \).

Answer: 1


Question: Find all positive integer solutions \((x, y, z)\) to the equation \[ x! + y! = z! \]

Answer: (1, 1, 2)


Question: Determine the value of \( a \) such that the equation \( \ln(x) = ax^4 \) has exactly one real root.

Answer: \dfrac{1}{4e}


Question:
Arrange the following products in increasing order from left to right:

\[ A = 1000! \]
\[ B = (400!)^2 \cdot (200!) \]
\[ C = (500!)^2 \]
\[ D = (600!) \cdot (300!) \cdot (1000!) \]
\[ E = (700!) \cdot (300!) \]

Answer: B < C < E < A < D


Question:
Given a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, let $\{X_n : n \ge 1\}$ be a sequence of identically distributed real-valued random variables with $E[|X_n|] < \infty$. Show that:
\[\lim_{n\to\infty} \frac{1}{n}\mathbb{E}\left[\max_{1 \le j \le n}|X_j|\right] = 0.\]

Answer: 0


Question:
Solve for real numbers \( x \):
\[ [2022+x(2+\sqrt{x})][111-x(2+\sqrt{x})]=-52^3 \]
Given that:
\[ \sqrt[3]{2022+x(2+\sqrt{x})}+\sqrt[3]{111-x(2+\sqrt{x})}=9 \]

Answer: 25


Question: Evaluate the limit \( \lim_{x\to 0} \frac{\sin|x|^2}{|x|^2+x_1x_2x_3} \) for \( x=(x_1,x_2,x_3) \).

Answer: 1


Question: Determine the dimension of the quotient space \( \frac{C_0}{M} \), where \( C_0 = \{(x_n): x_n \in \mathbb{R}, x_n \rightarrow 0 \} \) and \( M = \{(x_n): x_n \in C_0, x_1 + x_2 + \ldots + x_{10} = 0 \} \).

Answer: 1


Question: Find the last two digits of \(3^{7^{2016}}\).

Answer: 03


Question: Compute the net mass outflow through the cube formed by the planes \(x=0\), \(x=1\), \(y=0\), \(y=1\), \(z=0\), \(z=1\) for the mass flow vector \(\rho \vec{u} = (4x^2y, xyz, yz^2)\). Use the divergence theorem to simplify the calculation.

Answer: \dfrac{11}{4}


Question: Find the sum of the series: \[ \sum_{n=0}^{\infty} \frac{1}{(n+1)2^n} \]

Answer: 2 \ln 2


Question: Let \( f \) be an analytic function in \( \mathbb{C} \setminus (-\infty,0] \) such that \( f(x) = x^x \) for real \( x > 0 \). Find the value of \( f(i) \).

Answer: e^{-\pi/2}


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

Answer: p > 1


Question: Evaluate the integral \( \int_{0}^{\sqrt{\frac{\pi}{2}}}\int_{x}^{\sqrt{\frac{\pi}{2}}}\sin(y^2) \, \, \mathrm{d}y \, \, \mathrm{d}x \).

Answer: \dfrac{1}{2}


Question: Let $M$ be a maximal subgroup of a group $G$, and let $g \in G$ such that $g \notin M$. Determine whether it is possible for the subgroup $\langle M, gMg^{-1} \rangle$ to be equal to $M$.

Answer: Yes


Question: Evaluate the limit \( \lim_{R\to \infty}\int_{\gamma(0;R)}\frac{p(z)}{q(z)}\mathop{dz} \), where \( p \) and \( q \) are polynomials with \( \mathrm{deg}\ p < \mathrm{deg}\ q - 1 \).

Answer: 0


Question: Consider n independent tosses of a k-sided fair die. Let $X_1$ be the number of tosses that result in a 1, and $X_2$ be the number of tosses that result in a 2. Calculate the covariance between $X_1$ and $X_2$. Express your answer in terms of n and k.

Answer: -\dfrac{n}{k^2}


Question: Let $X$ be an $n$-manifold, $B$ an open ball in $\Bbb R^n$, and $f \colon B \to X$ an embedding. For a point $x \in f(B)$, is the map on homology \(H_n(X,X-f(B)) \to H_n(X,X-x)\) induced by the identity map of $X$ always an isomorphism?

Answer: Yes


Question: Let \( N \) be a natural number such that deleting its leftmost digit results in a number that is \( \frac{N}{57} \). Find the sum of the digits of \( N \).

Answer: 15


Question: Find the second-degree Taylor polynomial \( T_2(f, 0)(x) \) for the function \( f(x) = \log(\cos(x)) \) centered at \( x_0 = 0 \).

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


Question: Find the smallest multiple of 2019 that cannot be expressed as the sum of a perfect square and the product of two consecutive natural numbers.

Answer: 8076


Question:
Let $f(z)$ be a holomorphic function in the angle $A=\{0<\arg z<\frac{\pi}{2}\}$, continuous in $\bar{A}$, satisfying $|f(z)|\le M$ on $\partial A$ and the growth condition:
\[
|f(x+i y)|\le Ce^{y^2}\quad\text{in } A.
\]
Does it follow that $f$ is bounded in $A$?

Answer: Yes


Question: Compute the limit \( \lim_{k \to \infty} \int_0^k x^n \left(1 - \frac{x}{k} \right)^k \mathrm{d} x \) for a fixed \( n \in \mathbb{N} \).

Answer: n!


Question: Let $X$ be a vector space equipped with a seminorm $\|\cdot\|$. Is the kernel \( \text{Ker}(\|\cdot\|)=\{ x\in X : \|x\|=0 \} \) a closed subspace of $X$?

Answer: Yes


Question: Is the product of two measure-preserving transformations also measure-preserving?

Answer: Yes


Question: Using Stokes' theorem, compute the integral \( \int_{\partial S} \cos z \, dx + (xz + \tan y) \, dy + y^2z^3 \, dz \), where \( B = \{(x,y) \mid x^2+y^2 \le 1\} \) is a closed ball and \( S = \{(x,y,z) \mid z = x^2+y^2, (x,y) \in B\} \) is oriented such that the map \( f: B \to S \) defined by \( f(x,y) = (x,y,x^2+y^2) \) is orientation preserving.

Answer: \pi


Question:
Determine the number of solutions for the Cauchy problem given by the differential equation and initial condition:
\[ \begin{array}{l} x^6 + y^4 = y' \\
 y(2)=-1 \end{array} \]

Answer: 1


Question: Calculate the volume of the solid bounded by the plane \(z = 0\) at the bottom, the cylinder \(x^2 + y^2 = 4\) on the side, and the plane \(z = 3 - x - y\) at the top.

Answer: 12\pi


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

Answer: \dfrac{\pi}{2}


Question: Find the value of \( \alpha \) for the limit \( \lim_{n \to \infty}\left(\cos\left(\frac{x}{n}\right) + \sin\left(\frac{2x}{n}\right)\right)^{n} = \alpha \).

Answer: e^{2x}


Question: If the line \(y = kx + n\) is a common tangent to the circle \(x^2 + y^2 = 4\) and the ellipse \(2x^2 + 5y^2 = 10\), find the value of \(k^2 + n^2\).

Answer: 14


Question: Determine the number of zeros of the holomorphic function \( f(z) = e^{z + 3} + z^3 + z^2 + z + 1 \) within the unit disk.

Answer: 0


Question: Evaluate the double integral \( \int\int \sqrt{x^2+y^2} \, dA \) over the region \( D = \{(x,y) : x \leq x^2 + y^2 \leq 2x\} \).

Answer: \dfrac{28}{9}


Question: Find the fifth last digit of the number \(5^{5^{5^{5^5}}}\).

Answer: 0


Question: Find the roots of the equation: \( (a^2+1) \cosh (a (c -b)) - \cosh (c a) = 0 \).

Answer: 0


Question:
Let $F = \mathbb{Q}(\sqrt{2i})$. Determine which of the following statements is not true:

(A) $\sqrt{2} \in F$

(B) $i \in F$

(C) $x^8-16=0$ has a solution in $F$

(D) $\dim_{\mathbb{Q}}(F)=2$

Answer: A


Question: Determine whether the following statement is true or false: For a square matrix $X$, is it true that $\|X\|_2^2 = \|X^TX\|_2$, where $\|\cdot\|_2$ denotes the operator norm? Justify your answer.

Answer: True


Question: Find the surface area of the part of the sphere \(x^2+y^2+z^2=a^2\) that lies inside the cylinder \(x^2+y^2=ax\).

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


Question: Let $n < m$ be positive integers and $A \in \mathbb{R}^{m \times n}$ with $\text{rk}(A) = n$. Does there exist a matrix $B \in \mathbb{R}^{n \times (m-n)}$ such that $(I_n\;B) \cdot A$ is invertible? Equivalently, does $A$ have a left inverse $(C\;D)$ with $C \in \text{GL}(n, \mathbb{R})$ and $D \in \mathbb{R}^{n \times (m-n)}$?

Answer: Yes


Question: Given the asymptote of a hyperbola as the line $5x - 4y + 5 = 0$ and a tangent to the hyperbola as the line $4x + 5y - 7 = 0$, find the eccentricity of the hyperbola.

Answer: \dfrac{\sqrt{41}}{4}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(\dfrac{n^2+5n+3}{n^2+n+2}\right)^n \]

Answer: e^4


Question: Determine whether the group $E_6^3/\mathbb{Z}_3$, where the quotient is over the diagonal $\mathbb{Z}_3$, has an outer automorphism.

Answer: Yes


Question:
Given the differential equation for a driven harmonic oscillator:
\[ I\ddot{\theta} + \gamma\dot{\theta} + k\theta = \tau \]
where \( I \), \( k \), and \( \gamma \) are known constants, determine the angle \( \theta \) at which the system will be at equilibrium.

Answer: \dfrac{\tau}{k}


Question: Evaluate the limit: \[ \lim_{(x,y)\to (0,0)} \frac{x^3+y^2}{x^2+|y|}. \]

Answer: 0


Question: Determine the range of values for \( p \) such that the equation \( \left(\frac{1}{\pi}\right)^{|x|} = x^2 - p \) has at least one solution.

Answer: [-1, \infty)


Question: Find the function $y(x,t)$ that satisfies the partial differential equation \( \frac{\partial y}{\partial t} - 2\frac{\partial y}{\partial x} = 0 \) with the initial condition \( y(x, 0) = e^{-x^2} \).

Answer: e^{-(x + 2t)^2}


Question: Evaluate the integral \( \int_{0}^{\infty} u^{n}e^{-u} \, du \) for \( n \geq 0 \).

Answer: \Gamma(n+1)


Question: Let $(X, d)$ be a compact metric space and $h: (X, d) \to (Y, \rho)$ be a homeomorphism. Define $d_0: X \times X \to [0, \infty)$ by $d_0(x_1, x_2) = \rho(h(x_1), h(x_2))$. Consider the identity map $i_X: (X, d) \to (X, d_0)$. Is it true that the metrics $d$ and $d_0$ are equivalent?

Answer: Yes


Question: Given that \( u + v + w = 0 \) and \( \|u\| = \|v\| = \|w\| = 1 \), show that \( \langle u, v \rangle = -\frac{1}{2} \).

Answer: -\dfrac{1}{2}


Question: Is the statement "If $G_{\alpha}$ is a topological group for each $\alpha \in I$, then the direct product group $G = \prod G_{\alpha}$ is a topological group with the product topology" also true when the product topology is replaced with the box topology?

Answer: Yes


Question: Find the positive value of \( a \) that satisfies the equation \( \int^{a^{e}}_{e} \frac {dx}{x\int^{ax}_{a}\frac {dy}{y}}=1 \).

Answer: e


Question: Determine whether the following statement is true or false: In a topological space, if there exist uncountably many pairwise disjoint nonempty closed sets, then the space is not separable.

Answer: B


Question: What is \( \mathbb{E}[g(X)|X] \) for a continuous random variable \( X \) with distribution \( f_X \) and a continuous function \( g \)?

Answer: g(X)


Question: Let $R$ be a commutative ring (not necessarily with unity) such that $ab \ne 0, \forall a,b \in R \setminus \{0\}$. If every descending chain of non-finitely generated ideals of $R$ terminates, is it true that every ascending chain of ideals in $R$ terminates?

Answer: Yes


Question: Suppose you have a $k \times n$ matrix $A$ where $k \geq n$ and $\mathrm{rank}(A) = m$ with $m \leq n \leq k$. Let $r_1, \dots, r_k$ be the rows of $A$. Consider the $n \times n$ matrix $B = r_1^Tr_1 + \cdots + r_k^Tr_k$. Determine if $\mathrm{rank}(B) = m$.

Answer: m


Question: Determine the number of different groups of order 15.

Answer: 1


Question: Evaluate the contour integral \( \int_{|z|=1} \frac{e^z}{z^2} \, \mathrm{d}z \) using Cauchy's integral formula.

Answer: 2\pi i


Question: What is the probability that two independent univariate Gaussian random variables are equal? Let $X_1 \sim \mathcal{N}(m_1, \sigma_1^2)$ and $X_2 \sim \mathcal{N}(m_2, \sigma_2^2)$. Calculate $P(X_1 = X_2)$. Provide your answer as a single value.

Answer: 0


Question: Let \(f(x)\) be a cubic polynomial \(x^3 + ax^2 + bx + c\) such that \(f(x) = 0\) has three distinct integral roots and \(f(g(x)) = 0\) has no real roots, where \(g(x) = x^2 + 2x - 5\). Determine the minimum value of \(a + b + c\).

Answer: 719


Question: How many prime ideals in \(\mathbb{Z}[i]\) contain the integer 10?

Answer: 3


Question: Evaluate the improper integral \( \int_{0}^{1} \ln(e^{1/x}+1)\, dx \).

Answer: \infty


Question: Given the series \(a = 1 + \frac{w^2x^2}{2!} + \frac{w^4x^4}{4!} + \cdots\) and \(b = -x - \frac{w^2x^3}{3!} - \frac{w^4x^5}{5!} - \cdots\), where \(w\) is a non-real complex cube root of unity, find the value of \((a^2 - b^2w^2)\).

Answer: 1


Question: Determine whether there exists a uniformly continuous function \( f: \mathbb{R} \to \mathbb{R} \) such that for the sequence defined by \( a_1 = \frac{2}{3} \) and \( a_{n+1} = a_n - a_n^2 \) for \( n \geq 1 \), the relationship \( a_{n+1} = f(a_n) \) holds for all \( n \in \mathbb{N} \).

Answer: Yes


Question: Find the minimum value of \(|e^{z^{2}}|\) for all complex numbers \(z\) such that \(|z| \le 1\).

Answer: \dfrac{1}{e}


Question: Let $K = \mathbb{C}(x_1, \ldots, x_n)$ and let $G$ be a countable group of automorphisms of $K$, where $G \cong \mathbb{Z}$. The field of $G$-invariants, $K^G$, is an extension of $\mathbb{C}$ with transcendence degree $d$. For a generic $n$-tuple $x \in \mathbb{C}^n$, consider the Zariski closure of the $G$-orbit $\{ g(x) \}_{g \in G} \subset \mathbb{C}^n$. Can you conclude that the dimension of this Zariski closure is generically $n-d$?

Answer: n - d


Question: Find a set that is isomorphic to the quotient space $c/c_0$, where $c$ is the set of converging sequences and $c_0$ is the set of sequences that converge to 0.

Answer: \mathbb{R}


Question: Calculate the number of distinct strominoes, where each stromino is a $3 \times 1$ rectangle with dots ranging from 0 to 6, and has rotational symmetry of order 2. Consider the two methods provided and determine the correct number of distinct strominoes.

Answer: 49


Question: Determine the number of solutions for the complex equation \( z^3 + \frac{3(\bar{z})^2}{|z|} = 0 \), where \( z \) is a complex number.

Answer: 5


Question: Evaluate the line integral \( \int_{\gamma} \frac{\mathrm{d}z}{z} \) where \( \gamma \) is a closed curve centered at the origin in the complex plane. Determine the value of the integral.

Answer: 2\pi i


Question: For a given Lebesgue measure \( \mu \), determine if \( \left[\mu(\mathbb{R})\right]^{1/p} = \infty \) for \( 0<p<\infty \).

Answer: \infty


Question: What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles needed to divide a square?

Answer: 8


Question: Find the polynomial of minimal degree \(p(x)\) over \(\mathbb{Q}\) such that \(\beta = \cos(20^\circ)\) and \(p(\beta) = 0\).

Answer: 8x^3 - 6x - 1


Question: Does there exist a bounded metric space with a non-surjective self-isometry?

Answer: Yes


Question: Find the values of $x$ such that the recurrence relation defined by \( f_n(x) = f_1(f_{(n-1)}(x)) \) with \( f_1(x) = \frac{2}{3} - \frac{3}{3x+1} \) satisfies \( f_{1001}(x) = x - 3 \) for \( x \ge 2 \).

Answer: \dfrac{5}{3}


Question: Determine whether the metric space \(([1,2], d)\) is complete, where the metric \(d(x, y) = |x^4 - y^4|\).

Answer: Yes


Question:
Let $E$ be a differentiable function satisfying $E(u + v) = E(u)E(v)$ for all real numbers $u$ and $v$. Find the function $E(x)$. Hint: First find $E'(v)$.

Provide your answer in the form of an equation for $E(x)$. \( \boxed{} \)

Answer: E(x) = e^{kx}


Question: Given a graph where each edge has a capacity of 1, and the continuous edges have a flow of 0 while the dotted straight edges have a flow of 1, determine the maximum flow from source (s) to sink (t) and the minimum capacity of an s-t cut. Assume the graph is such that the total flow is initially 3, and after augmenting paths, the total flow becomes 5. What is the minimum capacity of an s-t cut in this graph?

Answer: 5


Question: Determine the asymptotic growth rate of the function \( f(x) = \int_e^\infty e^{-xt / \log t} \, dt \) as \( x \to 0 \). Specifically, find the value of \( p \) such that \( f(x) \approx \frac{\log^p(1/x)}{x} \) for some \( p > 0 \).

Answer: 1


Question: Does a maximal ideal of $\mathbb{Z}[X,Y]$ contain a prime number? Let $\mathfrak{m}$ be a maximal ideal of $\mathbb{Z}[X,Y]$. Determine if $\mathfrak{m}$ contains a prime number.

Answer: Yes


Question: Find the radius of convergence of the series \( \sum \frac{n! z^n}{n^n} \).

Answer: e


Question:
Find all solutions to the following system of congruences:
\[ x \equiv 2^{191889} \pmod{119} \]
\[ x \equiv 2^{119} \pmod{19} \]

Answer: 1345


Question: Compute the limit: \[ \lim_{n\to \infty} \int_1^\infty \frac{1+ (-1)^n \exp (-nx^2)}{x^2} \, dx. \]

Answer: 1


Question: Determine whether there exists an injection \( f: \omega_1 \times L \to L \) such that \( f(\alpha, \beta) > \beta \) for all \((\alpha, \beta)\) in the domain of \( f \), where \( L \) is the set of all non-zero countable limit ordinals.

Answer: Yes


Question: Minimize \(x^2 - y^2 - z^2\) subject to the constraint \(x^4 + y^4 + z^4 \leq 1\) using the Karush-Kuhn-Tucker (KKT) conditions. Determine the KKT points and find the minimum value of the objective function.

Answer: -\sqrt{2}


Question: Find a solution to the congruence equation $x^2 \equiv 7 \pmod{787}$, where the solution $x$ is between 1 and 786.

Answer: 105


Question: Find the exact value of the limit \( A_\infty \) for the sequence defined by \( A_1 = x \) and \( A_j = \cos(\sin A_{j - 1}) \) for all \( j \geq 2 \), where \( x \in \mathbb{R} \).

Answer: \cos(\sin A_\infty)


Question: Evaluate the integral \( \int_0^{2\pi} \left( \int_1^\infty \frac{\sin(s-t)}{t^3} \, dt \right) ds \).

Answer: 0


Question: Evaluate the surface integral \(\int_{\rm S}{\rm Curl}\vec{F}\cdot\vec{{\rm d}S}\) using Stokes' Theorem, where \(\vec{F}(x,y,z)=(z,x,y)\) and \(\rm S\) is the portion of the surface \(x^{2}+y^{2}+z=1\) lying above the \(\rm XY\)-plane, oriented upward.

Answer: \pi


Question: A point $(x, y, z)$ is picked uniformly at random inside the unit ball. Find the joint density function of $Z$ and $R$, where $R^2 = X^2 + Y^2 + Z^2$. Express your answer in terms of $r$ and $z$.

Answer: \frac{3}{2} r


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

Answer: (a, 0)


Question: Evaluate the limit: \[ \lim_{ x \to -\infty }\frac{\sqrt[n]{x+1}+\sqrt[n]{x+2}-2\sqrt[n]{x-3}}{\sqrt[n]{x-1}+\sqrt[n]{x-2}-2\sqrt[n]{x+3}} \] given that \( n \in \mathbb{N}, n>2, \text{ and } n \text{ is odd}."}

Answer: -1


Question: Determine the minimum value of \( s \) for which the series \( \sum_{n=1}^\infty \frac{1}{\lfloor n^s \rfloor} \) converges.

Answer: 1


Question: Let $K$ be a field extension of $F$, and let $S \subseteq K$ be a set such that each element $s \in S$ is algebraic over $F$. Determine whether $F[S] = F(S)$.

Answer: F[S] = F(S)


Question: Evaluate the definite double integral: \[ \int\int_D 6x\sqrt{y^2-x^2} \, dA, \quad D=\{(x,y) \mid 0 \leq y \leq 2, \; 0 \leq x \leq y\} \]

Answer: 8


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

Answer: 1


Question: Let $A_0$ be the set \( \{1, 2, 3, 4\} \). For each subsequent set \( A_{i+1} \), it is defined as the set of all possible sums obtained by adding any two numbers from \( A_i \), where the two numbers do not have to be different. Determine the number of unique numbers in the set \( A_8 \).

Answer: 769


Question: A 3D conical structure is described by the surface function \( z = f(x,y) = 8 - \sqrt{4x^2 +8x+4y^2 -24y + 40} \). During construction, the cone may have up to a 10% error in radius and a 5% error in height. Use linear approximation to determine the worst-case percentage error in the cone's volume.

Answer: 25\%


Question:
Evaluate the limit of the following integral:
\[ \lim_{n \rightarrow \infty} \int_{[0, \infty)} \frac{n \sin(\frac{x}{n})}{x(1+x^2)}\,\mathrm dm(x) \]

Answer: \dfrac{\pi}{2}


Question:
Given that $x^{x^x} = 2^{-
\sqrt{2}}$, find the value of $x^{-2}$.

Answer: 16


Question: Let $E$ be the splitting field of the polynomial $x^3 - 5$ over the field of rational numbers $\mathbb{Q}$. Determine the Galois group $\mathrm{Gal}(E:\mathbb{Q})$ and identify all subfields of $E$. Provide a justification for your answer.

Answer: S_3


Question: Given the partial differential equation \( u_{xt}(x,t) + u(x,t)u_{xx}(x,t) = h(t) \) for \( x \in [0, \pi] \) and \( t > 0 \), with boundary conditions \( u(\pi,t) = u_x(0,t) = u_x(\pi,t) = 0 \), find the function \( h(t) \). Assume all necessary regularity conditions are satisfied for the computations. Determine \( h(t) \).

Answer: 0


Question:
Classify the following partial differential equation in cylindrical coordinates as elliptic, parabolic, or hyperbolic:

\[ \frac{\partial^2 \phi}{\partial z^2} + \frac{\partial \phi}{\partial z} + \frac{\partial^2 \phi}{\partial r^2} + \left ( \frac{1}{r} \right )\frac{\partial \phi}{\partial \theta} = 0 \]

Answer: parabolic


Question: Given the joint probability density function \( f(x,y) = \frac{1}{2} e^{-x} \) for \( |y| \leq x \) and \( x > 0 \), find the conditional expectation \( E(X|Y=-1) \).

Answer: 2


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

Answer: 2


Question: A cylindrical drill with radius 5 is used to bore a hole through the center of a sphere with radius 7. Calculate the volume of the ring-shaped solid that remains after the hole is drilled.

Answer: 64\sqrt{6}\pi


Question: Let $X$ be a random variable with a Poisson distribution such that $P(X>2)=0.3$. Find $P(X<2)$. Express your answer as a decimal.

Answer: 0.43


Question: Let \( D^n \subseteq \mathbb{R}^n \) be the closed \( n \)-disk, and let \( p \in \text{int}(D^n) \). Does there exist a homeomorphism \( h: D^n \to D^n \) such that \( h(p) = 0 \) and \( h(\partial D^n) = \partial D^n \)?

Answer: Yes


Question: Consider an uncountable well-ordered set \(X\). For each element \(x \in X\), define the initial segment \(I(x) = \{y \in X \mid y < x\}\). Does there exist an element \(x \in X\) such that \(I(x)\) is countably infinite?

Answer: Yes


Question: Compute the least squares polynomial of degree three for the function \( f(x) = x^3 \) on the interval \((0, \infty)\) with respect to the weight function \( w(x) = e^{-x} \) using the Laguerre polynomials \( L_1(x) = x - 1 \), \( L_2(x) = x^2 - 4x + 2 \), and \( L_3(x) = x^3 - 9x^2 + 18x - 6 \).

Answer: x^3


Question: How many 4x3 matrices of rank 3 can be formed where the entries are from a field with 3 elements?

Answer: 449280


Question: Given that the roots of the quadratic equation \( ax^2 - 2bx + c = 0 \) are complex, determine the number of real roots of the equation \( 4e^x + (a+c)^2(x^3 + x) = 4b^2x \).

Answer: 1


Question: Let \( A \) be a \( 6 \times 6 \) real symmetric matrix with rank 5. Determine the rank of the matrix \( A^2 + A + I \).

Answer: 6


Question: Given that \( \nabla f(0, 1) = (3, 4) \), and the function \( h(x, y) = f(y - x - 1, y - x) \), find \( \nabla h(4, 3) \).

Answer: (-7, 7)


Question: Determine whether the following statement is true or false: Let $f$ and $g$ be two functions from $[a,b]$ to $[a,b]$ with $f$ strictly increasing. If $f$ and $f \circ g$ are continuous, then $g$ is continuous.

Answer: A


Question: Does there exist a $C^*$-algebra $A$ such that $A$ has a faithful tracial state, but there is a nontrivial ideal $I$ of $A$ such that the quotient $A/I$ has no tracial states?

Answer: Yes


Question: Evaluate the line integral \( \int_C y\,dx + z\,dy + x\,dz \) where \( C \) is the curve of intersection of the surfaces \( z = xy \) and \( x^2 + y^2 = 1 \), traversed once in a counterclockwise direction when viewed from above the \( xy \)-plane. Do not use Stokes' Theorem in your solution.

Answer: -\pi


Question: Determine the number of points on the curve $y = x^2$ that have the minimal Euclidean distance to a given point $P = (0, y_P)$, where $0 < y_P < 1$. Specifically, find the number $n$ of such points $x_1, x_2, \ldots, x_n$ such that for all $\tilde{x} \in \mathbb{R} \setminus \{x_1, x_2, \ldots, x_n\}$, the distance $d(P, (x_i, x_i^2))$ is less than $d(P, (\tilde{x}, \tilde{x}^2))$ for each $i$. Assume $n=2$ is possible with $P = (0, 5)$ and $x_1 \approx -2.179$, $x_2 \approx +2.179$. Is $n=3$ or higher possible?

Answer: 2


Question: Determine whether the subring \( R = \{a_{2n} x^{2n} + \ldots + a_2 x^2 + a_0 \mid n \in \mathbb{N}\} \) of \( \mathbb{Z}[x] \) is Noetherian.

Answer: Yes


Question: If an orbit $G \cdot x$ is closed in the standard topology on a finite-dimensional complex vector space $V$, where $G$ is a complex linear algebraic group acting linearly on $V$, is the orbit also Zariski-closed?

Answer: Yes


Question: Find the maximum value of the function \( f(t) = e^{-\frac{t^2}{2}}\frac{| e^{-ta}(1+at)+e^{ta}(1-at)|}{\sqrt{e^{-2ta}+e^{2ta}}} \) for some \( a > 0 \).

Answer: \sqrt{2}


Question:
Determine the rate of convergence in probability of \( \hat Q_{xx}^{-1} = \left(\frac{\mathbf{X}^T \mathbf{X}}{n}\right)^{-1} \) to its probability limit \( Q_{xx}^{-1} \) in the context of the simple linear regression model:
\[
y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \quad \quad \quad \quad i = 1,2,\dots,n.
\]
Given that \( \hat Q_{xx}^{-1} \stackrel{p}{\longrightarrow} Q_{xx}^{-1} \), verify if the convergence rate is \( O_p(n^{-1/2}) \).

Answer: O_p(n^{-1/2})


Question: For a group $G$, if $ba = a^6b$, simplify the expression $a^ib^ja^kb^lb^{-j}a^{-i}b^{-l}a^{-k}$. What is the simplified form?

Answer: a^{i(1 - 6^l) + k(6^j - 1)}


Question: Evaluate the limit: $$\lim_{x\to0} [\ln(1+\sin^2(x))\cdot\cot(\ln^2(1+x))]$$

Answer: 1


Question: Evaluate the integral: \[ \int^{2\pi}_0 (1+\cos(x))\cos(x)(-\sin^2(x)+\cos(x)+\cos^2(x))~dx \]

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


Question: Consider a finite Galois extension $L/K$ of number fields with group $G$. Let $\mathcal O_L$ be the ring of algebraic integers of $L$. Define $\mathcal A_{L/K}$ as the subring $\{x\in K[G]:x\mathcal O_L\subseteq \mathcal O_L\}$. If $\mathcal O_L$ is free as an $\mathcal A_{L/K}$-module, is it of rank 1?

Answer: Yes


Question:
Given the Laplace transform of a function \( f(t) \): \[ \mathcal{L}\{y\}(s)=\frac{9}{s(s^2 + 3s + 1)} \]
Determine the behavior of the function \( y \). Choose the correct option:

- The function oscillates but doesn't converge
- The function oscillates and converges to 3
- The function oscillates and converges to 9
- The function converges to 3 without any oscillations
- The function converges to 9 without any oscillations

Answer: E


Question: Calculate the quadratic covariance \( \langle t^{2}B^{1}_{t}, t^{3}B^{2}_{t} \rangle_{T} \) where \( B^{1} \) and \( B^{2} \) are independent Brownian motions.

Answer: 0


Question: In an equilateral triangle \( \triangle ABC \) with side length \( a \), point \( X \) is on segment \( \overline{AB} \) such that \( |AX| = \frac{a}{3} \), and point \( Y \) is on segment \( \overline{BC} \) such that \( |BY| = \frac{a}{3} \). Let \( T \) be the intersection point of lines \( AY \) and \( CX \). Find the measure of angle \( \angle CTB \).

Answer: 90^\circ


Question: Determine the order of convergence of the modified Newton-Raphson method given by the iteration formula: \[ x_{n+1} = x_n - m \frac{f(x_n)}{f'(x_n)} \] where \( m \) is the multiplicity of the root.

Answer: 2


Question: Can a sequence have infinitely many points of accumulation, meaning that we can extract infinitely many subsequences from it such that each subsequence converges to a distinct point of accumulation?

Answer: Yes


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \frac{\sin nx}{(1+2^x)\sin x} \, dx \) for \( n \in \mathbb{N} \).

Answer: \frac{\pi}{2} \left(1 - (-1)^n\right)


Question: Find a natural number $a$ such that for all natural numbers $n$, the expression $n^4 + a$ is not a prime number.

Answer: 64


Question: Find an integer $x \in \{0, \ldots, 196\}$ that satisfies the congruence $x^{131} \equiv 12 \pmod{197}$. Provide your answer as a single integer.

Answer: 152


Question: Evaluate the integral \( \int_{0}^{\infty} \left[ f(x^n + x^{-n}) \frac{\log x}{x} + \frac{1}{1+x^2} \right] dx \) for a continuous function \( f \).

Answer: \dfrac{\pi}{2}


Question: Does there exist a recursive set \( S \subseteq \mathbb{N}^2 \) such that for every recursive set \( T \subseteq \mathbb{N} \), there exists a constant \( c \in \mathbb{N} \) such that \( T = \{ n \in \mathbb{N} \mid (c, n) \in S \} \)?

Answer: No


Question:
Determine whether the following statement is true or false:

For each $p \in (1,\infty) \cup \{\infty\}$ and $k \in \mathbb{N}$, there exists a continuous function $f_k : [0,1] \to \mathbb{R}$ such that:

1. $\|f_k\|_1 = 1$,
2. $\displaystyle\lim_{k \to \infty} \|f_k\|_p = \infty$.

Here, $\|f_k\|_p$ denotes the $p$-norm of $f_k$. Provide a justification for your answer.

Answer: True


Question: Find the integer value of \( t \) such that the span of the set of vectors \( A = \left\{\begin{pmatrix}-5\\ 5\\ -2\\ -2\end{pmatrix},\begin{pmatrix}-5\\ -3\\ -3\\ -1\end{pmatrix},\begin{pmatrix}5\\ 4\\ -4\\ 5\end{pmatrix}\right\} \) is equal to the span of the set of vectors \( B = \left\{\begin{pmatrix} t \\ -6 \\ 3 \\ -13\end{pmatrix},\begin{pmatrix} 10 \\ 6 \\ 6 \\ 2\end{pmatrix},\begin{pmatrix} -10 \\ -17 \\ 14 \\ -13\end{pmatrix}\right\} \).

Answer: -20


Question: Determine the coefficient \(a\) in the function \(a\cos(x)\) such that the arc length of one complete cycle from \(0\) to \(2\pi\) is equal to \(2\pi\), given that \(b < 1\) or \(b = 1\).

Answer: 0


Question: The roots of a quadratic equation with a mistaken constant term are \(-17\) and \(15\), and with a mistaken leading coefficient, the roots are \(8\) and \(-3\). Find the actual roots of the quadratic equation.

Answer: -1 \pm \dfrac{\sqrt{215}}{5}i


Question: Compute the matrix exponential \( \exp(tE_{a,b}) \) where \( E_{a,b} \) is the elementary \( n \times n \) matrix with a 1 in the \((a,b)\)-entry and 0 elsewhere, and \( a \neq b \).

Answer: I + tE_{a,b}


Question: Consider a closed convex subset $A$ contained in the unit ball of a Banach space $X$, and bounded operators $T_n \in \mathcal L(X)$. Assume that for every $x \in A$, the sequence $\left\lVert T_n x \right\rVert$ is uniformly bounded in $n$. Does this imply that $\sup_{x \in A} \left\lVert T_n x \right\rVert$ is uniformly bounded in $n$?

Answer: Yes


Question: Find the number of paths of length 12 from the point \((0,0,0)\) to \((4,4,4)\) in a 3D lattice, where each step moves to an adjacent lattice point, and the path must avoid the points \((1,1,1)\), \((2,2,2)\), and \((3,3,3)\).

Answer: 14814


Question: Evaluate the limit: \[ \lim_{h\to 0} \frac{f(x+3h) - 3 f(x+h) + 3 f(x-h) - f(x-3h)}{h^3} \] where \( f'(x), f''(x), \) and \( f'''(x) \) are defined on the domain of the function.

Answer: 8 f'''(x)


Question: Find the minimum value of \( S = \frac{a}{\sqrt{1-a}} + \frac{b}{\sqrt{1-b}} \) given that \( a, b > 0 \) and \( a + b = 1 \).

Answer: \sqrt{2}


Question: Find a power series expression for the infinite sum \( \sum_{j=1}^\infty \left(\frac{1+z}{1-z}\right)^j \).

Answer: -\dfrac{1 + z}{2z}


Question: Find the smallest integer \( n > 1 \) such that the product of the factors of \( n \) is \( n^{15} \).

Answer: 720


Question: Determine for which values of \( \beta > 0 \) the integral \( \int_0^\infty \frac{\arctan(\ln^2x)}{\left|\beta-x\right|^e} \, dx \) converges.

Answer: 1


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

Answer: -1


Question:
Evaluate the limit of the following expression as \( n \to \infty \):
\[
\frac{\sum_{i=0}^{n/2-1}{\frac{(an^2)^{i}}{i!}}}{e^{an^2}}
\]
where \( a > 0 \). Determine if the limit is greater than 0.

Answer: 0


Question: Find the inverse Laplace transform of the constant function 1, i.e., compute \( \mathcal{L}^{-1}\{1\} \).

Answer: \delta(t)


Question: In a diagram, two circles with radii 1 and 2 are tangent to each other and both are internally tangent to a larger circle with radius 3. Determine the radius of a fourth circle that is internally tangent to the circle with radius 3 and externally tangent to the circles with radii 1 and 2.

Answer: \dfrac{6}{7}


Question: Show that the sequence of functions \( f_n(x) = n^2 x^n (1-x) \) converges pointwise to 0 on the interval \([0, 1]\).

Answer: 0


Question: Express the conjugate of the function \( g(x) = \inf_z f(x,z) \) in terms of the conjugate \( f^*(y,s) \) of \( f(x,z) \), where \( f(x,z) \) is convex in \((x,z)\).

Answer: g^*(y) = f^*(y, 0)


Question: Find the value of \( \left \lfloor \frac{2017!}{2016! + 2015! + \cdots + 2! + 1!} \right \rfloor \).

Answer: 2015


Question: Let $X$ and $Y$ be independent random variables, each uniformly distributed over the interval $(0,1)$. Define $U = \min \{X,Y\}$ and $V = \max\{X,Y\}$. Calculate the correlation coefficient of $U$ and $V$. \( \rho(U, V) \)

Answer: \dfrac{1}{2}


Question: Evaluate the sum: \[ \sum_{x=1}^{2012} \frac{9^{\frac{x}{2013}}}{9^{\frac{x}{2013}}+3} \]

Answer: 1006


Question: Determine whether \( \frac{\partial}{\partial x} f(x,y(x)) \) is equal to \( x \) or \( 2x \) given that \( f(x,y) = xy^2 \) and \( y^2 = x \).

Answer: x


Question: Evaluate \( \lim_{a \to \frac{\pi}{4}} \int_0^1 \frac{dx}{x^2 + 2x \tan a + 1} \).

Answer: \dfrac{1}{2}


Question: Find all real solutions for the system of equations: \[ \begin{cases} x^3 = y + y^5 \\ y^5 = z + z^7 \\ z^7 = x + x^3 \end{cases} \]

Answer: (0, 0, 0)


Question: Determine whether the following proposition is true or false: Given a function \( f \) such that \( f(0) = 0 \), \( f(1) = 1 \), \( f'(x) > 0 \), and \( f''(x) < 0 \) on \([0,1]\), there exists a \( c \) in the interval \( \left(\frac{1}{2}, 1\right) \) such that \[ 2 \int_{0}^{1} f(x) \, dx = \frac{f(c)}{c}. \]

Answer: True


Question: Evaluate the sum \( \sum_{z \in K} \frac{1+z}{1-z+z^2-z^3} \), where \( K = \{z : z^{1020} = 1 \text{ and } z^4 \neq 1\} \).

Answer: 508


Question: Find all functions \( f: \mathbb{Q} \rightarrow \mathbb{Q} \) that satisfy the equation \( f(x+y) + f(x-y) = 2f(x) + 2f(y) \) for all \( x, y \in \mathbb{Q} \).

Answer: f(x) = ax^2


Question: Is the space \(L^p(\mathbb{R}^n, B)\) complete for \(p \ge 1\) when \(B\) is a Banach space?

Answer: Yes


Question: Evaluate the complex integral \(\int_{C(0,2)} \frac{e^z}{i \pi - 2z} \, dz\) using the Cauchy Integral Formula.

Answer: \pi


Question: What is the expectation of a Gaussian random variable? For a random variable \( X \sim N(\mu, \sigma^2) \), find \( E[X] \).

Answer: \mu


Question:
Consider the ring $A = K[X,Y,Z]/(XZ-Y^2)$ and the ideal $P = (\bar{X},\bar{Y}) \subseteq A$. Let $M = A/P^2$ be an $A$-module, where $P^2 = (\bar{X}^2, \bar{Y}^2, \bar{X}\bar{Y}) = (\bar{X})\cap (\bar{X}, \bar{Y}, \bar{Z})^2$. Define the submodules of $M$ as follows:
\[N = \langle \bar{X} + P^{2}, \bar{Y} + P^{2}\rangle\]
\[L = \langle \bar{Y} + P^{2}\rangle\]
\[F = \langle \bar{X} + P^{2}\rangle\]
Determine if the isomorphism \(N/L \cong F\) holds.

Answer: N/L \cong F


Question: Determine whether the following statement is true or false: For Riemann-Stieltjes integrals, \( \int_a^b fd\phi \) exists if and only if \( \lim_{|\Gamma|\to0}R_\Gamma \) exists, where \( R_\Gamma=\sum_{i=1}^{m}f(\xi_i)[\phi(x_i)-\phi(x_{i-1})] \) and \( \xi_i\in[x_{i-1}, x_i] \).

Answer: A


Question: Determine the value of \(a + b\) if the equation \(\vert x^2 - 2ax + b\vert = 8\) has exactly three real roots, which are the sides of a right triangle.

Answer: 264


Question: Solve for \( x \) in the equation: \[ \frac{x+a^2}{a+b}+\frac{x+b^2}{b+c}+\frac{x+c^2}{c+a}=2(a+b+c) \] where \( a, b, \) and \( c \) are variables.

Answer: ab + bc + ca


Question: Calculate the number of paths from point (0,0) to point (9,9) on a 10x10 grid, where each move is either to the right or up, and the points (3,3), (6,3), and (6,6) must not be crossed.

Answer: 15564


Question: Suppose you have two sequences \((a_k)_{k=1}^N\) and \((b_k)_{k=1}^N\) such that \(|a_k-b_k| \leq \delta\) for all \(k\), and \(a_k, b_k \in [0,1]\) for all \(k\). Let \(A = \prod_{k=1}^N a_k\) and \(B = \prod_{k=1}^N b_k\). Find a bound on the difference \(|A-B|\) that approaches zero as \(\delta\) decreases, for a fixed \(N\).

Answer: N\delta


Question: Given an L-Lipschitz function \( f: X \subseteq \mathbb{R}^n \to Y \subseteq \mathbb{R}^n \), determine if it is true that \( \det(J_xf) \leq L^n \).

Answer: True


Question: Find the number of units in the quotient ring \( \Bbb Z_5[x]/(x^4-1) \), where \( \Bbb Z_5 \) is the finite field consisting of 5 elements.

Answer: 256


Question: Given that \(a\), \(b\), and \(c\) are real numbers such that \(a + b + c = 2\) and \(abc = 4\), find the minimum value of \(|a| + |b| + |c|\).

Answer: 6


Question: Determine the possible range of values for \( a \) such that the area under the curve \( y = e^x + \cos(x) \ln(x + \sqrt{1+x^2}) \) from \( x = -a \) to \( x = a \) is less than \( \frac{3}{2} \).

Answer: [0, \ln 2)


Question: Calculate the integral \( \int_{F(S^1)} \omega \) where \( F: S^1 \rightarrow S^1 \times S^1 \times S^1 \) is given by \( F(p) = (p,p,p) \) and \( \omega = dt_1 + dt_2 + dt_3 \) is a 1-form. Given that \( F^{*}\omega = 3\,dt \), find the value of the integral.

Answer: 6\pi


Question: Calculate the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) using Green's Theorem, where the vector field \( \mathbf{F}(x,y) = (x^3 - y, x + y^3) \) and \( C \) is the boundary of the region enclosed by \( y = x \) and \( y = x^2 \), oriented counterclockwise.

Answer: \dfrac{1}{3}


Question: Find all integer values of $n$ such that $1^n + 2^n + 3^n$ is a perfect square.

Answer: 3


Question: Is there a natural number \( k \) such that for every prime \( p \), there is a non-Abelian group of order \( p^k \)?

Answer: 3


Question: Evaluate the integral \( \int_0^1 f(x) \, dx \), where \( f(x) = \sum_{n=0}^{\infty} \frac{[2^n x] - 2[2^{n-1} x]}{3^n} \).

Answer: \dfrac{1}{4}


Question: Consider a separable Hilbert space $\mathcal{H}$. Define $T(\mathcal{H})$ as the trace class of $\mathcal{H}$, and $D(\mathcal{H})$ as the intersection of positive operators and trace class operators with trace no more than 1. For $A, B \in D(\mathcal{H})$, we say $A \leq B$ if $A - B$ is positive, satisfying the Lowner order. Given an increasing sequence $A_0, A_1, \ldots, A_n, \ldots$ where $(A_i, A_{i+1})$ satisfies the Lowner order and $A_i \in D(\mathcal{H})$, does this sequence have a least upper bound?

Answer: Yes


Question:
Given the equations \((1+2x)(1+x+x^2)^{n}=\sum_{r=0}^{2n+1} a_rx^r\) and \((1+x+x^2)^{s}=\sum_{r=0}^{2s} b_rx^r\), find the value of the expression \(\frac{\sum_{s=0}^{n}\sum_{r=0}^{2s} b_r}{\sum_{r=0}^{2n+1} \frac{a_r}{r+1}}\). Choose the correct option:

(A) \(2n+1\)
(B) \(\frac{2n+1}{2}\)
(C) \(\frac{n+1}{2}\)
(D) \(n+1\)

Answer: C


Question: Consider a sequence \(S = (a, b, c, d)\) of positive integers. Define the transformation \(T\) on \(S\) as follows: \(T(S) = (|a - b|, |b - c|, |c - d|, |d - a|)\). Starting with \(S\), repeatedly apply the transformation to generate the sequence \(S, S_1 = T(S), S_2 = T(S_1), S_3 = T(S_2), \ldots\). Does this sequence always eventually become \((0, 0, 0, 0)\)?

Answer: Yes


Question: Determine the supremum of the functional \( \mathcal{I}(f) = \int_0^1 f(x) \, dx - f(0) + f(1) \) over the set \( \mathcal{F} \) of continuous functions \( f: [0,1] \to \mathbb{R} \) such that \( \max_{0 \le x \le 1} |f(x)| = 1 \).

Answer: 3


Question: Determine all values of the parameter \(a\) for which the inequality \(3 - |x-a| > x^2\) is satisfied by at least one negative \(x\).

Answer: \left( -\frac{13}{4}, 3 \right)


Question: Find the number of positive integral solutions to the equation \(5x + 4y + 3z + 2m + k = 20\).

Answer: 7


Question: Let $S$ be a Cohen-Macaulay local ring, $I$ an ideal of $S$, and $R = S/I$. If $I$ is generated by $\dim S - \dim R$ elements, is $R$ necessarily Cohen-Macaulay?

Answer: Yes


Question: Evaluate the line integral \( \int_C (\ln y) e^{-x} \,dx - \dfrac{e^{-x}}{y}\,dy + z\,dz \) where \( C \) is the curve parametrized by \( r(t)=(t-1)i+e^{t^4}j+(t^2+1)k \) for \( 0\leq t\leq 1 \), using the potential function \( f(x, y, z) = -e^{-x}\ln y + \frac{z^2}{2} + C \).

Answer: \dfrac{1}{2}


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

Answer: e^{-2}


Question: Let \( A \subseteq \mathbb{R}^2 \) be a finite set such that the set \((-\delta, \delta) \times (-\delta, \delta)\) is generated by \( A \) in the sense of convex combinations. Is it true that the linear combination of elements of \( A \) is \( \mathbb{R}^2 \)? Here, \( \delta > 0 \) is some real number.

Answer: Yes


Question: Let $(B_t)_t$ be a standard Brownian motion with $B_0=0$. Define $T_{a,b} = \inf\{t>0 : B_t \not\in(a,b)\}$ where $a<0<b$. Find the expected first-passage time $\mathbf{E}[T_{a,b}]$.

Answer: -ab


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

Answer: e^{-2}


Question: The integral curve of the differential equation \((y + g(x))dx + (x - g(y))dy = 0\) passes through the point \((1, 1)\). Determine which of the following points it must also pass through: \((0, 0)\), \((2, 1/2)\), \((1/2, 2)\), \((-1, -1)\), or \((0, 1)\).

Answer: (-1, -1)


Question:
Minimize the function \((x-3)^2+(y-1)^2\) subject to the constraints:
\[ 2x+y \leq 2 \]
\[ x^2 + 2y = 3 \]
\[ x, y \geq 0 \]

Answer: 25 - 10\sqrt{3}


Question: Find a vector field \( \mathbf{F} = M(x,y,z)\mathbf{i} + N(x,y,z)\mathbf{j} \) such that \( \nabla \times \mathbf{F} = (-3xz^2, 0, z^3) \). Then, use Stokes' theorem to evaluate the surface integral \( \iint_{S} (-3xz^2, 0, z^3) \cdot \mathrm{d}\mathbf{S} \), where \( S \) is the surface defined by \( z = x^2 + y^2 \) for \( z \leq 4 \), oriented with an upward-pointing normal.

Answer: 256\pi


Question: Let $X$ be a random variable in $A = \{1,2,3,4\}$ with pmf $p(1) = \frac{1}{8}$, $p(2) = \frac{1}{8}$, $p(3) = \frac{1}{4}$, and $p(4) = \frac{1}{2}$. The binary Huffman code is given by $c(1) = 110$, $c(2) = 111$, $c(3) = 10$, and $c(4) = 0$. Generate a sequence in $A^n$ iid based on pmf $p$, then encode the sequence using $c$. What is the probability that a randomly chosen bit from this encoded sequence is $1$?

Answer: \dfrac{1}{2}


Question:
Consider the function \( f(x) \) defined by the integral equation:
\[
f(x) = x^2 + \int_0^x (x - t)f(t) \, dt.
\]
Derive an ordinary differential equation (ODE) and boundary conditions for \( f(x) \), and solve this to determine \( f(x) \).

Answer: e^x + e^{-x} - 2


Question: Given a 4th degree monic polynomial \( f(x) = x^4 + ax^3 + bx^2 + cx + d \) with the conditions \( f(1) = 10 \), \( f(2) = 20 \), and \( f(3) = 30 \), find the value of \( f(12) + f(-8) \).

Answer: 19840


Question: Kevin is taking an exam with $k > 1$ parts, numbered Part $1, 2, ..., k$. For $i = 1, 2, ..., k$, Part $i$ contains $i$ multiple choice questions, each with $i + 1$ answer choices. If he guesses randomly on every question, the probability of getting exactly one question correct is $10$ times the probability of getting no questions correct. Compute the total number of questions on the exam.

Answer: 55


Question: Evaluate the limit without using L'Hospital's rule: \[ \lim_{x \to 2} \frac{\sqrt{17 - 2x^{2}}\sqrt[3]{3x^{3} - 2x^{2} + 8x - 5} - 9}{(x - 2)^{2}} \]

Answer: -\dfrac{11}{3}


Question: Consider the linear system given by \(x'(t) = A(t)x(t)\), where \(A(t)\) is an \(n \times n\) matrix. Suppose \(A(t)\) is periodic with period \(T\). Let \(V\) be the vector space of all solutions, and \(W\) be the set of all \(T\)-periodic solutions. Given that \(\dim W \leq \dim V\), is it possible for \(0 < \dim W < n\)?

Answer: Yes


Question:
Let $ABC$ be a triangle and $M$ a point on side $BC$. Denote $\alpha=\angle BAM$ and $\beta=\angle CAM$. Determine whether the following inequality is true:
\[ \sin \alpha \cdot (AM-AC)+\sin \beta \cdot (AM-AB) \leq 0. \]

Answer: True


Question: Given the Markov chain with transition matrix \( \begin{pmatrix} 0 & 0 & 0.2 & 0.8 & 0 \\ 0 & 0 & 0 & 0.9 & 0.1 \\ 0.6 & 0 & 0 & 0 & 0.4 \\ 0.2 & 0.8 & 0 & 0 & 0 \\ 0 & 0.9 & 0.1 & 0 & 0 \end{pmatrix} \), calculate the expected number of visits to state B before reaching state A, starting from state C.

Answer: \dfrac{300}{149}


Question: Calculate the sum of the series \( \sum_{n,m=1}^{\infty}\frac{1}{(n+m)!} \).

Answer: 1


Question: Find an irreducible polynomial with integer coefficients that has \(\sqrt{2} + \sqrt{3}\) as a root.

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


Question: Let \( f(x) = (x-1)(x-2)(x-3)\cdots(x-n) \). Prove that for \( n \ge 1 \), the equation \( f'(x) = 0 \) has \( n-1 \) roots.

Answer: n-1


Question: Let $R$ be a finite von Neumann algebra and $S$ be a von Neumann subalgebra of $R$. Determine whether every normal tracial state on $S$ can be extended to a normal tracial state on $R$.

Answer: No


Question: Given the implicit function \( g(x,y,z) = x^2 + y^2 - z^2 - xyz = 12 \) and the solution point \((x,y,z) = (3,4,1)\), find the value of \( f(3,4) \) for the unique continuous implicit function \( z = f(x,y) \).

Answer: 1


Question: Solve the equation \( \sqrt[x]{\frac{2}{(x+1)}} = {(x+1)}^{x+2} \) for \( x \in \mathbb{R} \).

Answer: \sqrt{2} - 1


Question: Evaluate the limit of the expression \( \frac{x(1-q)+q^{-x}}{x(q-1)-1+q^{-x}} \) as \( x \to \infty \) for a constant \( 0 < q < 1 \). Determine if the limit is equal to 1.

Answer: 1


Question: Identify the group $G$ generated by the permutations $(1234)$ and $(3456)$ in the symmetric group $S_6$. The order of $G$ is 120. What is the group $G$?

Answer: S_5


Question: Let \( R \) be the set of primitive 42nd roots of unity, and let \( S \) be the set of primitive 70th roots of unity. Determine the number of elements that \( R \) and \( S \) have in common.

Answer: 0


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

Answer: 1


Question: Determine if 641 is the smallest factor of any composite Fermat number in the sequence \( a_n = 2^{2^n} + 1 \).

Answer: 641


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_0^{\infty} \arctan(nx) e^{- x^n} \, dx \]

Answer: \dfrac{\pi}{2}


Question: Find the unit digit of \((\sqrt{2} + \sqrt{3})^{100}\).

Answer: 7


Question: Compute the limit \( \lim_{n \to \infty} \frac{\sqrt{1} + \sqrt{2} + \cdots + \sqrt{n}}{n\sqrt{n}} \) using the integral of \( \sqrt{x} \) over the interval \([0, n]\).

Answer: \dfrac{2}{3}


Question: Determine the values of \(a\) for which the inequality \(ax^2 - (4-2a)x - 8 < 0\) holds for exactly three integral values of \(x\).

Answer: [2, 4)


Question: Find all positive integers \( M \) such that \( M^{n+1} + 12^{2n-1} \equiv 0 \pmod{133} \) for all positive integers \( n \).

Answer: 11


Question: Evaluate the integral \( \int_{\frac{1}{n}}^1 \cos(\{nx\} \cdot \pi) \, dx \), where \( n \) is a positive integer and \( \{x\} \) denotes the fractional part of \( x \).

Answer: 0


Question: Let \( \alpha, \beta, \gamma, \delta \) be the roots of the polynomial \( z^4 - 2z^3 + z^2 + z - 7 = 0 \). Find the value of \( (\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1) \).

Answer: 58


Question: Find all integer solutions \((x, y)\) to the equation \( y^2 = x^3 - 27 \).

Answer: (3, 0)


Question: If \( f: \mathbb{R} \to \mathbb{R} \) is a one-to-one and differentiable function, and the graph of \( y = f(x) \) is symmetrical about the point \((4,0)\), determine whether the statement is true or false: If \( f'(-100) > 0 \), then the roots of the equation \( x^2 - f'(10)x - f'(10) = 0 \) may be non-real.

Answer: B


Question: For any function \( f: B \to A \) with \( B \neq \emptyset \), determine whether a function \( h: A \to B \) can be constructed such that \( fhf = f \).

Answer: Yes


Question: Given the differential form $\omega = xydx + 3dy - yzdz$, verify by direct computation that $d(d\omega) = 0$. Compute the differential of $\omega$ and show that the differential of the result is zero.

Answer: d(d\omega) = 0


Question: Find the residue of the function \( \frac{z}{\log{z}} \) at \( z=1 \).

Answer: 1


Question: If \(X_1, \ldots, X_n \sim \mathcal{N}(\mu, 1)\) are independent and identically distributed (IID) random variables, compute \(\mathbb{E}(X_1 \mid T)\), where \(T = \sum_{i=1}^n X_i\).

Answer: \dfrac{T}{n}


Question: Let $V$ be a finite-dimensional vector space and $W$ a subspace of $V$. Given a linear functional $f \in W'$, prove that there exists a linear functional $g \in V'$ such that $g$ restricted to $W$ is equal to $f$. Provide a justification for your answer.

Answer: g


Question: Determine the values of the parameter \( \alpha \) for which the series \( \sum_{n=1}^{\infty} (n^{n^\alpha}-1) \) converges.

Answer: (-\infty, -1)


Question: Consider an affine scheme $X = \text{Spec}(A)$, which is smooth if for any closed embedding $X \subset \mathbf{A}^n$ with ideal $I$, the ideal $I$ can be locally generated by a sequence $f_{r+1}, \dots, f_n$ such that their Jacobian has maximal rank. Will the Jacobian of any set of $n-r$ generators of $I$ be of maximal rank?

Answer: Yes


Question:
Let \( V \) be a closed subspace of \( L^2[0,1] \) and let \( f, g \in L^2[0,1] \) be given by \( f(x) = x \) and \( g(x) = x^2 \). If \( V^{\perp} = \text{span}(f) \) and \( Pg \) is the orthogonal projection of \( g \) onto \( V \), determine \((g - Pg)(x)\) for \( x \in [0,1] \). Choose the correct option:

(A) \( \frac{3x}{4} \)
(B) \( \frac{x}{4} \)
(C) \( \frac{3x^2}{4} \)
(D) \( \frac{x^2}{4} \)

Answer: A


Question: Calculate the line integral \( \int_{dA} (x^2 \, dx + y^2 \, dy) \) over the region \( A = \{(x, y) \in \mathbb{R}^2 : -\frac{\pi}{2} \leq x \leq \frac{\pi}{2}, \ -1 \leq y \leq \cos(x)\} \).

Answer: 0


Question:
Let \( \Omega \subseteq \mathbb{R}^N \) be an open, bounded, and connected set, possibly with a smooth boundary. Consider a function \( \phi: \Omega \to \mathbb{R} \) such that \( \phi \in C^1(\overline{\Omega}) \), satisfying the following conditions:
- \( \Vert \phi \Vert_{C^1(\overline{\Omega})} \leq 1 \)
- \( \phi^{-1}(0) = \{ x \in \Omega \mid \phi(x) = 0 \} \neq \emptyset \)
- \( \nabla \phi(x) \neq 0 \) for all \( x \in \phi^{-1}(0) \).

Is it true that for each \( \varepsilon > 0 \), there exists a constant \( \delta = \delta(\varepsilon) > 0 \) such that:
\[ \lambda(\phi^{-1}(-\delta, \delta)) < \varepsilon \]?

Here, \( \lambda \) denotes the Lebesgue measure on \( \mathbb{R}^N \).

Answer: Yes


Question: Evaluate the limit of the double integral: \[ \lim_{b \to 0^+} \int_{b}^1 \int_b^1 \frac{y-x}{(y+x)^3} \, dy \, dx \]

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \left( \frac{\sin(x)}{x} \right)^2 \, dx \) using contour integration techniques.

Answer: \pi


Question: Given that the number of paying customers per day is a random variable \(X\) with mean 400 and variance 100, and the average amount a customer spends is a random variable \(Y\) with mean 5 and variance 4, find the variance of the total daily revenue, assuming that the number of customers and the amount spent are independent.

Answer: 642900


Question: Determine the number of ways to select 10 coupons from an infinite supply of coupons labeled A, B, C, and D such that the sequence "ABCD" does not appear consecutively in the selection.

Answer: 1020000


Question: Let \( \Omega \) be an open set in \( \mathbb{R}^n \) and \( K \subset \Omega \) be a compact subset. Assume \( \{f_i\} \) is a sequence of solutions of the equation \( \Delta f_i=0 \) on \( \Omega \), with \( \|f_i\|_{L^2(\Omega)} \leq C \) for some constant number \( C \) independent of \( i \). Does the inequality \( \|f_i\|_{L^{\infty}(K)} \leq \lambda \) hold for some constant number \( \lambda \)?

Answer: Yes


Question:
True or False: In the constrained optimization problem with integer parameters $N$ and $M$ where $1 \leq M \leq N$, consider the optimization of $P = (p_1, p_2, \ldots, p_{N+M})$ under the following constraints:

\[ \sum_{k=1}^{N+M} -p_k\log_2 p_k - (1-p_k)\log_2(1-p_k) \geq N, \]
\[ \forall 1 \leq k \leq N+M,\ 0 \leq p_k \leq 0.5 \]

Minimize:
\[ \sum_{k=1}^{N+M} p_k(1-p_k) \]

We define $0\log_2 0 = 0$. Determine whether the optimal $P^*$ that minimizes the objective function will have all coordinates equal, i.e., $P^* = (p, p, \ldots, p)$. Provide a true or false answer.

Answer: True


Question: What is the Euler characteristic of the hyperboloid of one sheet?

Answer: 0


Question: Evaluate the integral: \[ F(x) = \int_0^{\pi/2} \frac{1}{(\sqrt{\sin(x)} + \sqrt{\cos(x)})^4} \, dx \]

Answer: \dfrac{1}{3}


Question: Calculate the double integral of the function \( f(x,y) = |16xy| \) over the region defined by \( x^2 + y^2 \le 25 \) by converting to polar coordinates.

Answer: 5000


Question: Let \( T: M_3(F) \rightarrow M_3(F) \) be a linear transformation defined by \( T(B) = B^T \). Find the minimal polynomial of \( T \).

Answer: x^2 - 1


Question: Determine whether the following statement is true or false: If $f$ is a Taylor expansion about $x_{0} = 0$ with radius $R$, then $g(x) = f\left(\frac{x-1}{2}\right)$ has a Taylor expansion about $x_{0} = 1$ with radius $2R$. Justify your answer.

Answer: True


Question: Let $a_1 = 1$ and define the sequence $a_n = n(a_{n-1} + 1)$ for all $n \geq 2$. Determine the value of \( \lim_{n \to \infty} \prod_{r=1}^{n} \left(1 + \frac{1}{a_r}\right) \).

Answer: e


Question: Given the function \( g(t) = t^5 + t^3 \), find \( (g^{-1})'(2) \).

Answer: \dfrac{1}{8}


Question: Determine the value of \( c \) that leads to the fastest convergence of the sequence \( D_n(c) = \frac {4^n}{\binom {2n}{n} \sqrt{4n + c}} \) to \( \frac{\sqrt{\pi}}{2} \) as \( n \to \infty \).

Answer: 1


Question: Given a triangle $ABC$ with hypotenuse $AB$ lying in plane $Q$, sides $AC$ and $BC$ create angles $\alpha$ and $\beta$ with plane $Q$, respectively. If $\sin(\alpha) = \frac{1}{3}$ and $\sin(\beta) = \frac{\sqrt{5}}{6}$, find the angle between plane $Q$ and the plane of triangle $ABC$. Provide your answer in degrees.

Answer: 30


Question: Is it true that for a function $f$ in the disk algebra $A$, the spectrum $\sigma_A(f)$ is equal to $f(D)$, where $D$ is the unit disk? Answer yes or no.

Answer: Yes


Question: Find the value of $x$ such that $2^{133} \equiv x \pmod{133}$. Use Euler's theorem to simplify the calculation.

Answer: 128


Question:
Evaluate the following expression:
\[ \limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} \right) \]
If necessary, you may assume a suitable generalization of the Bunyakovsky conjecture.

Answer: 0


Question: Determine whether the following statement is true or false: If a closed, strictly convex curve has exactly four vertices (extrema of curvature), then any circle can intersect it at most four times.

Answer: True


Question: Find the residue of the function \( f(z) = \frac{z}{1 + \sin(z)} \) at the point \( z = -\frac{\pi}{2} \).

Answer: 2


Question: Evaluate the limit \( \lim_{n\to \infty}\int^1_0 f(x^n)\,dx \) for a continuous function \( f(x) \) on \([0,1]\).

Answer: f(0)


Question: Determine if it is possible to create new valid $50 notes by cutting full $50 notes, given that a note is considered valid if at least 70% of it is intact. If it is not possible, find the largest percentage of the full note that would allow the creation of new valid currency.

Answer: 50\%


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{a^b - b^a}{a - b} \] where $a(x)$ and $b(x)$ are continuous functions such that $a(0) = b(0) = 3$.

Answer: 27(1 - \ln 3)


Question: What is the smallest known constant \(c\) such that for any integer \(n \geq 2\), there are at least \(\frac{n}{\log_2{n}}\) primes in the interval \([n, cn]\]?

Answer: 2


Question: If $T: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}$ is a linear transformation, does there exist a matrix $C$ such that $T(A) = CA$ for every $A \in \mathbb{R}^{n \times n}$?

Answer: No


Question: Compute the limit: \( \lim \limits _{n \to \infty} \sqrt[n]{\frac{n!}{n^n}} \)

Answer: \dfrac{1}{e}


Question: On the pseudosphere, does the volume of metric balls and the area of metric spheres remain invariant under translation of the centerpoint?

Answer: No


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

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0}{\frac{1-\sqrt{\cos x}}{1-\cos \sqrt{x}}} \] without using l'Hospital's rule.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\rightarrow0}\frac{\ln\cos2x}{\left(2^{x}-1\right)\left(\left(x+1\right)^{5}-\left(x-1\right)^{5}\right)}$$

Answer: 0


Question: Determine whether there exists a pair of continuous surjective functions \( f_1 \) and \( f_2 \) from \( \mathbb{Q} \) to itself such that for every \( x, y \in \mathbb{Q} \), the intersection \( f_1^{-1}(x) \cap f_2^{-1}(y) \) is non-empty.

Answer: Yes


Question: A policeman is trying to catch a thief who is initially $a$ meters away. The thief moves with a constant velocity $v$ in a straight line to the right, starting at $(0, a)$. The policeman, starting at $(0, 0)$, moves with a constant velocity $k \cdot v$, where $k > 1$, and always runs directly towards the thief. The thief is caught at the point $(a, a)$. Determine the value of $k$.

Answer: 2


Question: Let $u$ be a harmonic function in the open simply connected set $U$. If $u$ is bounded in $U$, is its harmonic conjugate $v$ also bounded in $U$?

Answer: No


Question: Determine whether there exists an automorphism \( f : \mathcal{M}_{3} ( \mathbb{C} ) \to \mathcal{M}_{3} ( \mathbb{C} ) \) of \( \mathbb{C} \)-algebras such that \( f^3 ( M ) = M \) for all matrices \( M \). If such an automorphism exists, describe how to construct it.

Answer: Yes


Question: Let \( f : \mathbb{R} \to (0, \infty) \) be a continuous function. Prove that there exists a real number \( \alpha \) such that \( f(\alpha) = \frac{\alpha}{1-\alpha} \).

Answer: \alpha


Question: Let $(f_n)_{n\geq 1}$ be a sequence of continuous functions on $[0,1]$, converging pointwise to a function $f:[0,1]\to{\mathbb R}$ which is not assumed to be continuous. Fix $\varepsilon >0$, and define $F_n=\{ x\in [0,1] \mid |f_n(x)-f(x)|\leq\varepsilon\}$. Is it true that some $F_n$ has nonempty interior when $f$ is not assumed to be continuous?

Answer: Yes


Question: Evaluate the surface integral \(\int_{S} (z + x^{2}y)\,dS\), where \(S\) is the part of the cylinder \(y^{2} + z^{2} = 4\) that lies between the planes \(x = 0\) and \(x = 3\) in the first octant.

Answer: 48


Question: Calculate the volume of the solid that lies under the paraboloid $z = x^2 + y^2$ and above the region bounded by the curves $y = x^2$ and $x = y^2$.

Answer: \dfrac{6}{35}


Question: Evaluate the expression: \( \sqrt{1+ 2 \sqrt{1+3 \sqrt{1+\dots\infty}}} \)

Answer: 3


Question:
Let $A \in M_2(\mathbb{R})$ such that $AA^T = A^TA = I$ and $\det(A) = 1$. Which one of the following is necessarily true?

(a) $Av = v$ for some unit vector $v \in \mathbb{R}^2$.
(b) $Av \neq v$ for some unit vector $v \in \mathbb{R}^2$.
(c) $||Av|| \neq 1$ for some unit vector $v \in \mathbb{R}^2$.
(d) None of the above.

Answer: D


Question: Given the joint probability mass function \( P(X=x, Y=y) = e^{-2}\binom{x}{y} \left(\frac{3}{4}\right)^{y}\left(\frac{1}{4}\right)^{x-y}\frac{2^{x}}{x!} \) for \( y=0,1,...,x \) and \( x = 0,1,2,... \), find the expected value \( E(Y) \).

Answer: \dfrac{3}{2}


Question: Let $A$ and $B$ be different matrices such that $A^3 = B^3$ and $A^2B = B^2A$. Find the value of $\det(A^2 + B^2)$. \( \text{Express your answer as a single number.} \)

Answer: 0


Question: Determine whether the limit \( \lim_{x \to \infty} \sum_{n=1}^{x-1} \frac{1}{x-n} \) exists.

Answer: \infty


Question: Find all integer solutions for which the following holds: \(35x^{31} + 33x^{25} + 19x^{21} \equiv 1 \pmod{55}\).

Answer: 43


Question:
Evaluate the limit of the sequence given by:
\[ \lim_{n \rightarrow \infty} \left( \frac{n \pi}{4} - \left( \frac{n^2}{n^2+1^2} + \frac{n^2}{n^2+2^2} + \cdots + \frac{n^2}{n^2+n^2} \right) \right). \]

Answer: \dfrac{1}{4}


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left[\left(\left(\frac{x+1}{x-1}\right)^x - e^2\right)x^2\right] \] by substituting \( x \to \frac{1}{t} \) and considering \[ L = \lim_{t \to 0} \frac{\left(\frac{1+t}{1-t}\right)^{1/t} - e^2}{t^2} \]

Answer: \dfrac{2}{3}e^2


Question: Let $A$ be a unital C*-algebra and consider $PS(A)$, the set of all pure states on $A$ with the relative weak-star topology. Let $E$ be a relative weak-star open set in $PS(A)$ and let $\phi \in E$. Does there exist a positive element $a \leq 1$ in $A$ such that $\phi(a) = 1$ and $\psi(a) = 0$ for all pure states $\psi$ in the complement of $E$?

Answer: No


Question: Determine whether zero is a limit point of the sequence \((\sqrt{n} \sin n)_n\).

Answer: Yes


Question: Calculate the line integral \( \int_l (e^x \sin y - 7y) \, dx + (e^x \cos y - 7) \, dy \) where \( l \) is the upper section of the circle \( x^2 + y^2 = 16x \) from point \( A(16,0) \) to point \( B(0,0) \), using Green's theorem.

Answer: -224\pi


Question: Calculate the absolute value of the line integral \( \int_{C}\vec{F} \cdot \vec{dr} \) for the vector field \( \vec{F} \) with curl \( \vec{T} = \langle -x, -y, 0 \rangle \), where \( C \) is the closed curve forming the boundary of the surface of the cone \( z=\sqrt{x^2+y^2} \) bounded by the planes \( z=0 \) and \( z=3 \). Use Stokes' theorem to determine the correct value.

Answer: 18\pi


Question: Calculate the integral \( \int_{\gamma} \sqrt{z} \, dz \) where \( \gamma = C(2,1)^+ \), integrating over the principal value of \( \sqrt{z} \).

Answer: 0


Question: Given a convex set $C$ of points $x \in \mathbb{R}^d$ such that for all $x \in C$, $(-x) \in C$, determine if the centroid of $C$ can be the origin of $\mathbb{R}^d$ (the zero vector).

Answer: Yes


Question: Find the minimal polynomial of \( \sqrt{3} + \sqrt{5} \) over \( \mathbb{Q} \).

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


Question: Given the recursive sequence defined by \( a_{n+1} = \frac{a_n}{n} + \frac{n}{a_n} \) with the initial condition \( a_1 = 1 \), evaluate \( \left \lfloor a_{2015} \right \rfloor \).

Answer: 44


Question: Calculate the conditional expectation \( E[Y|T=5] \) where \( X_1, X_2, \ldots, X_5 \sim \text{exp}(1) \), \( Y=1_{[3,+\infty)}(X_1) \), and \( T=\sum_{i=1}^5X_i \).

Answer: \dfrac{16}{625}


Question: Find the value of the fourth derivative of the inverse function \((f^{-1}(y))^{(4)}\) evaluated at \(y=1\), where \(f(x) = \sin{x} + x^2 + 1\).

Answer: -140


Question: Evaluate the integral: $$\int_0^{2\pi}e^{-it}\prod_{k=1}^n(e^{2kit}-1)dt$$

Answer: 0


Question: Show using the definition that the function \( f(x + yi) = x^2 + y^2 + 2xyi \) is differentiable at the point \( x_0 + i0 \).

Answer: 2x_0


Question: In triangle $ABC$, the external angle bisector of $\angle B$ intersects the extension of $AC$ at point $P$, and the external angle bisector of $\angle A$ intersects the extension of $BC$ at point $Q$. Let $R$ be the intersection of lines $BP$ and $AQ$. Given that the circumcircle of $\triangle PQC$ passes through $R$ and $PQ = 1$, find the value of $PR + RQ$. Assume $PR = RQ$ due to the cyclic nature of $RPQC$. What is $PR + RQ$?

Answer: 2


Question: Find the natural number \( k \) that maximizes the expression \( \frac{k^2}{(1+10^{-3})^k} \).

Answer: 2001


Question: Given the joint probability distribution \( \mathbb{P}(X=k, Y=m) = \frac{1}{36} \left( \frac{5}{6} \right)^{k-2} \) for random variables \(X\) and \(Y\) with \(1 \leq m < k\), calculate the product of the expected values \(\mathbb{E}[X]\) and \(\mathbb{E}[Y]\).

Answer: 72


Question: Given the series \( y = \frac{2}{5} + \frac{1\cdot3}{2!} \left(\frac{2}{5}\right)^2 + \frac{1\cdot3\cdot5}{3!} \left(\frac{2}{5}\right)^3 + \cdots \), find the value of \( y^2 + 2y \).

Answer: 4


Question: Determine whether the following statement is true or false: Given an entire function $f(z)$, if there exists a polynomial $P \in \mathbb{C}[X,Y]$ such that $P(z,f(z)) = 0$ and $P$ is not identically zero, then $f(z)$ is a polynomial.

Answer: True


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

Answer: -\dfrac{1}{3}


Question: Calculate the limit without using L'Hôpital's rule: \[ \lim_{x\to 0} \left(\frac{(5x + 1)^{20} - (20x + 1)^{5}}{\sqrt[5]{1 + 20x^{2}}-1}\right) \] Use only Special Limits and simple transformations.

Answer: \dfrac{375}{2}


Question: Suppose the random variable X has a moment generating function given by \( M(t) = \frac{e^{at}}{1-bt^2} \) for \(-1 < t < 1\). Given that the mean and variance of X are 3 and 2, respectively, find the value of \(a + b\).

Answer: 4


Question: Let $U$ be the unit interval $[0,1]$ or $[0,1)$. Does there exist a bijective function $f: U \to U$ such that for every subinterval $I$ of $U$, the set $\{f(x)\ |\ x \in I\}$ is dense in $U$?

Answer: Yes


Question: A real number $U$ is chosen uniformly from the interval $(0,10)$. Next, an integer $X$ is chosen according to the Poisson distribution with parameter $U$. What is the expected value of $UX$?

Answer: \dfrac{100}{3}


Question:
Let $P$ and $Q$ be two non-empty subsets of a normed linear space $X$. Consider the set $P+Q=\{x+y \in X: x \in P \text{ and } y \in Q\}$. Determine which of the following statements is FALSE:

1. If $P$ or $Q$ is convex, then $P+Q$ is convex.
2. If $P$ or $Q$ is open, then $P+Q$ is open.
3. $P+Q$ must be closed if $P$ and $Q$ are closed.
4. If $P$ is closed and $Q$ is compact, then $P+Q$ is closed.

Answer: 3


Question: Determine whether the sequence \( e^{-(n^{\frac{1}{2}})}{(n+1)^{100}} \) converges. If it does, find the limit of the sequence.

Answer: 0


Question: Evaluate the integral \( I = \int_{e^{-2\pi n}}^1 \left|\left(\cos\left(\ln{1\over x}\right)\right)'\right|dx \) for \( n \in \mathbb{N} \).

Answer: 4n


Question: Calculate the work done by the vector field \( F = (y^2 + e^{x^2} + ye^{xy}, xe^{xy}) \) along the rectangle \( C \) with vertices at \((0,0)\), \((0,1)\), \((2,0)\), and \((2,1)\), oriented anticlockwise. Use Green's Theorem to evaluate the line integral.

Answer: -2


Question: Determine whether the following limit is true: \[ \lim_{x\rightarrow\infty}\frac{x}{\pi(x)}-\ln(x) =-1, \] where \( \pi(x) \) is the number of primes not exceeding \( x \).

Answer: -1


Question: Let $L/F$ be a Galois field extension with $	ext{char}F=0$. Consider an irreducible representation $V$ of a finite group $G$ over $F$. The representation $W = L \otimes V$ of $G$ over $L$ is Galois-invariant, meaning $W^{\sigma}$ is isomorphic to $W$ for each $\sigma \in \text{Gal}(L/F)$. Given that $W$ decomposes as $\bigoplus_{i=1}^{n} W_i$, where each $W_i$ is the direct sum of an entire $\text{Gal}(L/K)$-orbit of irreducible representations of $G$ over $L$, determine if $n=1$. In other words, is there always just a single orbit in the decomposition?

Answer: 1


Question: Find the value of \( f\left(f\left(4444^{4444^{4444}}\right)\right) \), where \( f(n) \) is the sum of the decimal digits of \( n \).

Answer: 7


Question: Evaluate the Legendre symbol \( \left(\frac{11}{71}\right) \).

Answer: -1


Question:
Let \(f: \mathbb{R} \to \mathbb{R}\) be a continuous function. Determine which of the following sets cannot be the image of \((0,1]\) under \(f\):

1. \(\{0\}\)
2. \((0,1)\)
3. \([0,1)\)
4. \([0,1]\)

Answer: B


Question: Determine the points at which the complex function \( w = z |z| \) is differentiable, where \( w = \sqrt{x^2+y^2} (x+i y) \). Use the Cauchy-Riemann equations to justify your answer.

Answer: 0


Question: Calculate the fundamental group \( \pi_1(\mathbb{R}^3 \setminus A) \), where \( A \) is a circle in \( \mathbb{R}^3 \). Use van Kampen's theorem to find the solution.

Answer: \mathbb{Z}


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

Answer: x^4 - 2x^2 + 9


Question: Given the matrix $A = (c_1,...,c_n)$ where $c_i$ are the column vectors of $A$ and $c_i \in \mathbb{R}^m$, with $c_1 = (1,...,1)$ and $c_2 = c_3 = ... = c_m = (0,...,0)$, compute the 2-norm of the matrix $A$, denoted as $||A||_2 = \sup_{x \in \mathbb{R}^m} \frac{||Ax||_2}{||x||_2}$.

Answer: \sqrt{m}


Question:
Evaluate the series:
$$(x+y)(x-y)+\frac{1}{2!}(x+y)(x-y)(x^2+y^2)+\frac{1}{3!}(x+y)(x-y)(x^4+y^4+x^2y^2)+\ldots$$
Which of the following is the value of the series?
$$(A)\quad e^{x^2}+e^{y^2}\quad\quad(B)\quad e^{x^2-y^2}\quad\quad(C)\quad e^{x^2}-e^{y^2}\quad\quad(D)\quad \text{none of these}$$

Answer: C


Question: Determine the measure of the set \(\Delta_n = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n : \sum_{k=1}^{n} x_k \leq 1, x_k \geq 0, \text{ for } k=1,\ldots,n \}\). Express \(\Delta_n\) in terms of \(\Delta_{n-1}\) and find a general formula for its measure.

Answer: \dfrac{1}{n!}


Question: Consider a sequence of independent Bernoulli random variables $X_1, X_2, \ldots, X_n$ with parameter $\theta$, where $0 < \theta < 1$ and $P(X_i = 1) = 1 - P(X_i = 0) = \theta$. Assume the prior distribution of $\theta$ follows a Beta(3,2) distribution. Given that the first 9 observations result in $\sum_{i=1}^9 X_i = 7$, calculate the Bayesian probability that $X_{10} = 1$.

Answer: \dfrac{5}{7}


Question: Determine the angle \(\theta\) between two radii of a circle, each of length \(r\), that maximizes the ratio of the area of a circle inscribed in the isosceles triangle formed by these radii and the chord connecting them, to the area of the shaded region (the difference between the triangle's area and the inscribed circle's area).

Answer: \dfrac{\pi}{3}


Question:
Find the value of \( x \) such that:
\[
(3+4)(3^2+4^2)(3^4+4^4)(3^8+4^8)(3^{16}+4^{16})(3^{32}+4^{32}) = 4^x - 3^x
\]

Answer: 64


Question: For which positive integer values of $n$ does the Diophantine equation \((3n-3)(x^2+y^2+z^2)=(9n+1)(xy+yz+zx)\) have no nontrivial solutions?

Answer: n \geq 3


Question: Determine whether the ring $\mathbb{M}[x]/I$ is a field for any of the following choices of $\mathbb{M}$: $\mathbb{R}, \mathbb{Q}, \mathbb{F_9}, \mathbb{C}$, where $I$ is the ideal generated by $x^4+2x-2$. Provide your answer with justification.

Answer: \mathbb{Q}


Question: What is the probability that a randomly selected number from the interval $[0,1]$ is algebraic?

Answer: 0


Question: Determine the minimum number of queries needed to identify the subset \( B \subseteq \{0,1\}^n \) such that if \((b_1,b_2,\ldots,b_n) \in B\) and \((c_1,c_2,\ldots,c_n) \in \{0,1\}^n\) with \(c_i \leq b_i\) for all \(i\), then \((c_1,c_2,\ldots,c_n) \in B\).

Answer: \binom{n}{\lfloor n/2 \rfloor}


Question: Does there exist a field $k$ and a subring $R$ of $S = M_2(k)$ such that $R$ is not finitely generated over its center, $S = kR$, and $1_R = 1_S$? Here, $S$ is the algebra of $2 \times 2$ matrices over $k$. Provide a yes or no answer.

Answer: Yes


Question: Let $K$ be a finite extension of $\mathbb{Q}_p$ with residue degree $f$ and ramification degree $e$. Let $R \subset K$ be its ring of integers, let $\mathfrak{p}$ be the prime of $R$ lying above $p$, and let $N_{R/\mathbb{Z}_p}$ be the norm map from $R$ to $\mathbb{Z}_p$. Is the mod-$\mathfrak{p}$ reduction of $N_{R/\mathbb{Z}_p}$ equal to the $e^{\text{th}}$-power of the norm map $N_{\mathbb{F}_{p^f}/\mathbb{F}_p}$ from $\mathbb{F}_{p^f}$ to $\mathbb{F}_p$?

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{2\pi} e^{\cos \theta} \cos( \sin \theta) \, d\theta \). To do this, define \( I(\alpha) = \int_{0}^{2\pi} e^{\alpha \cos \theta} \cos(\alpha \sin \theta) \, d\theta \) for \( \alpha \in \mathbb{R} \). Calculate \( I(0) \) and use it to evaluate \( I(1) = \int_{0}^{2\pi} e^{\cos \theta} \cos( \sin \theta) \, d\theta \). Consider the expression \( \frac{\partial}{\partial \theta}(e^{\alpha \cos \theta}\sin(\alpha \sin \theta)) \) to help evaluate the integral.

Answer: 2\pi


Question:
Given a parameter $a \in \mathbb{R}$ such that the sequence $(x_n)_{n \ge 1}$ is bounded, find the limit of the sequence $x_n$. The sequence is defined as:
\[ x_n = \left( 1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n-1} \right) - a \left( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) \]
Determine the value of the limit of $x_n$ as $n$ approaches infinity.

Answer: \ln 2


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{\sqrt{n}} \left( \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \right) \] without using integrals.

Answer: 2


Question:
Consider a circle with radius 1 and a fixed number of points, say 30, on its circumference. These points define a mean point inside the circle. Calculate the circular variance, which is the mean of the squares of the distances from each point to the mean point:
\[ \mathrm{msq} = \frac{1}{30} \sum_{k=1}^{30} ((x_k-x_m)^2+(y_k-y_m)^2) \]
Now, assume another set of 30 points on the circumference with the same mean point. Is the circular variance the same for this new set of points?

Answer: Yes


Question: Find all integer values of \( x \) that satisfy the equation: \[ 5^{2(x+1)} + 621 \times 10^x = 100 \times 4^x \]

Answer: -2


Question: Does there exist a signature $L$ and an $L$-theory $T$ such that $T$ is not the theory of finitely many of its models? In other words, can you find a theory $T$ for which there is no finite set of models $\{M_1,...,M_n\}$ such that $Th(\{M_1,...,M_n\})=T$?

Answer: Yes