Question: Let $g(x)$ be a quadratic polynomial with real coefficients such that $g(3) = 0$ and the equation $g(g(x)) = 0$ has only one real root at $x = 7$. Determine the value of $g(1)$. \( \boxed{} \)

Answer: -\dfrac{15}{4}


Question: Determine the number of distinct terms in the binomial expansion of \(\left(x+\frac{1}{x}+x^2+\frac{1}{x^2}\right)^{15}\).

Answer: 61


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

Answer: \dfrac{1}{e}


Question: Determine the number of points at which a line in $\mathbb{CP}^n$, defined as a copy of $\mathbb{CP}^1$, intersects with $\mathbb{CP}^2$. Provide your answer as a single integer.

Answer: 1


Question: Given the function \( g(x) = \int_0^x f(y) (y - x) \, dy \), where \( g(x) \) is exactly 3 times continuously differentiable, find the greatest integer \( n \) for which \( f \) must be \( n \) times continuously differentiable.

Answer: 1


Question: Evaluate the limit \( \lim\limits_{R\to\infty}\oint_{\partial C_R(0)}\frac{d z}{z^2+2z+2} \), where \( \partial C_R(0) \) is the circle centered at 0 with radius \( R>0 \).

Answer: 0


Question: Suppose that $I$ is an interval, and $g,f,h$ are functions defined on $I$, except possibly at $x_0$. Furthermore, for all $\varepsilon>0$, for all $x\in I$, and for some constant $L$, it holds that $$(L-\varepsilon)g(x) < f(x) \le Lh(x) .$$ Also, $$\lim_{x\to x_0} g(x) = \lim_{x\to x_0} h(x) = 1.$$ Prove that for all $\varepsilon>0$, $$L-\varepsilon < \lim_{x\to x_0} f(x) \le L,$$ and conclude that $$\lim_{x\to x_0} f(x)=L.$$

Answer: L


Question: Suppose \(\{x_n\}\) is a real sequence for which the limit \(\lim_{n\to\infty}(x_{n+1}-x_n)=c\) exists. Determine the convergence behavior of the sequence \(\left\{\frac{x_n}{n}\right\}\).

Answer: c


Question: Determine the values of \( \alpha \) for which the set \( S_{\alpha} = \{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 = z^2, z - y + \alpha = 0\} \) is a subspace of \( \mathbb{R}^3 \).

Answer: 0


Question: Let $\{x_1, x_2, \ldots, x_n\}$ and $\{y_1, y_2, \ldots, y_n\}$ be non-zero real vectors. Define $b_{ij} = x_i y_j^T$. Determine the rank of the matrix $B = [b_{ij}]$. \(\text{Express your answer as a single integer.}\)

Answer: 1


Question: Evaluate the limit \(\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)\) where \(p\) is a positive integer.

Answer: \dfrac{1}{2}


Question: Consider a zonotope $Z \subset \mathbb{R}^d$, defined by generators $G \in \mathbb{R}^{n\times d}$. Each facet of this zonotope is a $(d-1)$-dimensional polytope. Determine whether it is true that every edge of these facets can be represented as exactly one generator shifted by some positional offset.

Answer: True


Question: Determine if the Lebesgue Dominated Convergence Theorem can be applied to the sequence of functions \( g_n(x) = \frac{x^n}{(1+x)^n} \) defined on \([0, \infty)\). If applicable, find the limit \( \lim_{n\to \infty} \int_{0}^{\infty} g_n(x)\,dx \).

Answer: \infty


Question: Evaluate the limit: $$\lim_{n \to \infty} n \sum_{k=0}^{3} \int_{k}^{k+1} (x-k)^n dx$$

Answer: 4


Question: Evaluate the limit as \( n \) approaches infinity of the series: \[ \sum_{r =1}^{n}\frac{6n}{9n^2-r^2}. \]

Answer: \ln 2


Question:
Consider a collection of matrices with dimensions $N \times M$, where each element is an integer from $0$ to $K-1$ inclusive. For each such matrix $A$, define a sequence $L[1], L[2], \ldots, L[N+M]$ as follows:

1. For $1 \leq i \leq N$, $L[i]$ is the maximum element in the $i$-th row of $A$.
2. For $1 \leq i \leq M$, $L[N+i]$ is the maximum element in the $i$-th column of $A$.

Find the number of distinct sequences that can be formed using this method. For example, calculate the number of different sequences when $N = 2$, $M = 2$, and $K = 2$. \( \boxed{?} \)

Answer: 10


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

Answer: \infty


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2} \frac{\tan^5x}{\tan^5x+\cot^5x}\,\mathrm dx \).

Answer: \dfrac{\pi}{2}


Question: Find the real number \( x \) that satisfies the equation: \[ 5^{2x-1} - 6e^{x-1} + 1 = 0 \]

Answer: 1


Question: Determine whether the sequence \( \cos(\pi \sqrt{n^2 + n}) \) converges as \( n \to \infty \).

Answer: 0


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

Answer: (1, 2)


Question: Let $R$ be the ring $\mathbb{Z}[x] / ((x^2 + x + 1)(x^3 + x + 1))$ and $I$ be the ideal in $R$ generated by $2$. Find the cardinality of the quotient ring $R/I$. \( \boxed{} \)

Answer: 32


Question: Let $Y$ be a random variable with $E[Y^2]=3$. Assume that $Y_1$, $Y_2$, and $Y$ are independent and identically distributed, and that $Y$ has the same distribution as $\frac{Y_1 + Y_2}{\sqrt{2}}$. Find the variance of $Y$, denoted as $\text{Var}[Y]$. \( \boxed{} \)

Answer: 3


Question: Determine the Lebesgue measure of the set \(\{(a,b) \in \mathbb{R}^2 \mid ab \in \mathbb{Q}\}\) in \(\mathbb{R}^2\).

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to +\infty}\frac{1}{n^2}\sum_{i=1}^{n}\log\binom{n}{i} \] where \(\binom{n}{i}\) is the binomial coefficient.

Answer: \dfrac{1}{2}


Question: Given the equation \(\frac{a_0}{n+1}+\frac{a_1}{n}+\frac{a_2}{n-1}+\ldots+\frac{a_{n-1}}{2}+a_{n}=0\), determine the maximum possible number of roots of the polynomial \({a_0}{x^n}+{a_1}{x^{n-1}}+{a_2}{x^{n-2}}+\ldots+{a_{n-1}}{x}+{a_n}=0\) within the interval \((0,1)\).

Answer: n


Question: A polyhedron consists solely of triangular and quadrilateral faces. Each vertex in the polyhedron has a degree of 4, and there are exactly 4 quadrilateral faces. Determine the number of triangular faces in the polyhedron.

Answer: 8


Question: Given an integer $P$ and an odd positive integer $Q$, if $P^2 + 17^Q = 10^8$, determine how many integer values $P$ can take.

Answer: 0


Question: Find the formula for the length of the shortest string that contains all possible substrings of length $n$ over an alphabet of size $k$.

Answer: k^n + n - 1


Question: A light elastic string with stiffness \( k \) and natural length \( l \) has a bead of mass \( m \) attached to one end, while the other end is fixed at a point O. The bead is released from rest at O. Determine the maximum extension of the string below point O.

Answer: \dfrac{2mg}{k}


Question: Evaluate the limit \(\lim_{n\to\infty}\left(H_n^2 - 2\sum_{i=1}^n \frac{H_i}{i} + \sum_{k=1}^n \frac{1}{k^2}\right)\), where \(H_i\) is the \(i\)-th harmonic number.

Answer: 0


Question: Given the differential equation \( f'(x) = \frac{1}{2}f(x) \) with \( f(0) = 1 \) and \( |f(x)| \leq 6 \) for all \( x \in [-3, 3] \), determine the maximum error in approximating \( f(2) \) using the 6th degree MacLaurin polynomial \( T_6(2) \).

Answer: \dfrac{1}{840}


Question: Let $N$ be a positive integer, and let $p$ and $q$ be distinct positive integers less than $N$. If the number of distinct remainders when multiples of $p$ are divided by $N$ is equal to the number of distinct remainders when multiples of $q$ are divided by $N$, does it imply that the two sets of remainders are equal? Answer yes or no.

Answer: Yes


Question: Evaluate the integral \( \int_{\pi/2}^{3\pi/2}\frac{\sin x}{x}\,dx \) using the Fundamental Theorem of Calculus, given that \( f'(x) = \frac{\sin x}{x} \), \( f(\pi/2)=0 \), and \( f(3\pi/2)=1 \).

Answer: 1


Question: If $a, b, c$ are natural numbers such that the concatenated number $(\overline{ab})^3$ equals $\overline{1c8ab}$, find the value of $2b - a - c$. \(\overline{ab}\) denotes the number formed by concatenating the digits of $a$ and $b$, and \(\overline{1c8ab}\) denotes the number formed by concatenating the digits 1, $c$, 8, $a$, and $b$. \(a, b, c\) are natural numbers.

Answer: 3


Question: Find the residue of the function \( f(z) = \dfrac{z^2}{(z-1)(z-2)(z-3)} \) at infinity, given that the sum of the residues at its poles 1, 2, and 3 is 1.

Answer: -1


Question: In the context of integral domains, if $R$ is a free $\mathbb{Z}$-module and $I$ is a non-zero ideal of $R$, do $R$ and $I$ have the same rank as $\mathbb{Z}$-modules? Answer yes or no.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{4} \frac{1}{(2x+8)\, \sqrt{x(x+8)}}\, dx \) using the substitution \( 4\sinh(z) = x + 4 \).

Answer: \dfrac{\pi}{24}


Question: Consider four Bernoulli-distributed random variables, $X_1$, $Y_1$, $X_2$, and $Y_2$, with the following independence relationships: $X_1$ is independent of $Y_2$, and $X_2$ is independent of $Y_1$. Determine if it is possible to express the covariance of $X_1Y_1$ and $X_2Y_2$ in terms of the covariances of the individual random variables, such as Cov$(X_1, Y_1)$ and Cov$(X_2, Y_2)$. Provide your answer as "Yes" or "No".

Answer: No


Question: Evaluate the limit: $$\lim\limits_{n \to \infty} \frac{n^{1/n^2+x}−1}{\ln(n)\cdot \frac{1}{n^2}}$$

Answer: 1


Question: Determine the value of $c$ such that the function \( y = \frac{x^2 - 3x + c}{x^2 + 3x + c} \) achieves a minimum value of 7 and a maximum value of \( \frac{1}{7} \).

Answer: 4


Question: Evaluate the integral \( \iiint \limits _D z^3 \, \mathrm{d}V \) over the region \( D \) bounded by the surface of the hemisphere \( z = \sqrt{1-x^2-y^2} \) and the plane \( z=0 \) using spherical polar coordinates.

Answer: \dfrac{\pi}{12}


Question: Determine the maximal ideal space $\hat{A}$ of the $C^*$-subalgebra $A$ of $C_b(\mathbb{R})$, which is generated by $C_∞(\mathbb{R})$ together with the function $f(t) = e^{it}$.

Answer: \mathbb{R} \cup S^1


Question: Given a positive $L^1$ function $f$ on $\mathbb{R}^2$, determine if there exists a subset $A \subseteq \mathbb{R}^2$ such that \( \int_A f \, d\mu = \int_{\mathbb{R}^2 \setminus A} f \, d\mu \).

Answer: Yes


Question: Given a $\sigma$-algebra $\Sigma$ on a finite set $\Omega$ with $2^n$ elements, determine the cardinality of the set of equivalence classes on $\Omega$ induced by $\Sigma$. Provide your answer as a single number.

Answer: 2^n


Question: Let $A$ be a non-singular 2x2 matrix with elements from the set $S=\{-1,0,1\}$. If $A^2 + I = 0$, where $I$ is the identity matrix, determine the probability that $\det(A) = -1$. Express your answer as a fraction.

Answer: 0


Question: Evaluate the integral of a function where the integrand simplifies to 1 due to the properties of the binomial distribution.

Answer: 1


Question: Evaluate the triple integral \(\iiint (x+y+z) \, dx\,dy\,dz\) over the region \(D:=\left \{ (x,y,z) \in\mathbb{R}^3 :x^2+y^2+z^2 \leq 9 \wedge z > y\right \}\).

Answer: 0


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

Answer: e


Question: Determine the maximum area of a triangle in the first quadrant formed by the x-axis, y-axis, and a tangent line to the curve \( f(x) = (x + 2)^{-2} \). Given that the height \( h \) of the triangle is expressed as \( \frac{-2}{4(a+2)^3} \), where \( a \) is the x-coordinate of the tangent point, find the x- and y-intercepts of the tangent line and maximize the area of the triangle.

Answer: \dfrac{1}{4}


Question: Determine whether the function \( f(x,y,z) = \frac{x^2 y^2 z^2}{x^2 + y^2 + z^2} \) is continuous at the point (0,0,0) using the AM-GM inequality.

Answer: Yes


Question: Let $f(x) = x^2 + kx$, where $k$ is a real number. Determine the set of values of $k$ for which the equations $f(x) = 0$ and $f(f(x)) = 0$ have the same set of real solutions.

Answer: [0, 4)


Question: Let $\{f_n\}_{n=1}^\infty$ be a sequence of functions in $C^1[0, 1]$ such that $f_n(1) = 1$ for all $n \in \mathbb{N}$. If the sequence $\{f'_n\}$ converges uniformly on the interval $[0, 1]$, does it follow that the sequence $\{f_n\}$ also converges uniformly on the same interval? Provide a justification for your answer.

Answer: Yes


Question: Calculate the work done by the force \( \vec{F} = \sin(x^2)\hat{x} + (3x-y)\hat{y} \) in moving a particle along the path from the origin \( O \) to point \( A = (3,0) \) and then to point \( B = (0,4) \) in straight lines.

Answer: 10


Question: Evaluate the flow of the vector field \( F(x, y, z) = (0, 0, z) \) through the portion of the plane \( x + y + z = 1 \) that lies within the first octant. Determine the appropriate limits for \( x \), \( y \), and \( z \), and compute the integral.

Answer: \dfrac{1}{6}


Question: Find the maximum value of the product $abc$, given that $a$, $b$, and $c$ are real numbers satisfying $a+b+c=5$ and $a^2+b^2+c^2=11$.

Answer: 3


Question: Calculate the mass of a wire that lies at the intersection of the surfaces $x^2 + y^2 + z^2 = 1$ and $x + y + z = 0$, where the density of the wire, $\rho({\bf x})$, is given by $x^2$ per unit length.

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


Question: Determine the Lipschitz constant for a solution $x(t)$ to the differential equation $y'(t) = \sin(3\pi(y(t)+2t))$. Provide your answer as a single numerical value.

Answer: 3\pi


Question: Find the remainder when \(37^{19^{19}}\) is divided by 255.

Answer: 58


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

Answer: 0


Question: Determine the radius of convergence for the function \( f(x)= \begin{cases} e^{-\frac{1}{x^2}}\quad &\text{if $x\neq 0$}\\ 0\quad &\text{if $x=0$} \end{cases} \). Justify why the radius of convergence is zero.

Answer: 0


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{(\int_{0}^x e^{t^2}dt)^2 }{\int_{0}^{x} e^{2t^2}dt}$$

Answer: 0


Question: True or false: Every harmonic function on \(\Bbb C \setminus \{0\}\) has a harmonic conjugate.

Answer: B


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

Answer: 0


Question: Calculate the expectation of the cube of Brownian motion at time \( t \), given that \( E(W_t) = 0 \) and \( E(W_t^2) = t \).

Answer: 0


Question: Given a $4\times 4$ real matrix $A$ with eigenvalues $1, -1, 2, -2$, determine the determinant of the matrix $B$ defined as $$B = 2A + A^{-1} - I.$$ Using the Cayley-Hamilton theorem, $A$ satisfies its characteristic equation $A^4 - 5A^2 + 4I = 0$. Express $B$ in terms of $A$ and find $\det(B)$. Provide your answer as a single number.

Answer: 154


Question: Calculate the surface area of the portion of the cone defined by $x^2 + y^2 = z^2$ that is above the $xy$-plane and within the cylinder $y^2 + z^2 = 16$. Provide your answer as a single numerical value.

Answer: 16\pi


Question: Given a set of complex numbers $\{\lambda_1, \lambda_2, \dots, \lambda_k\}$ such that for all natural numbers $n$, the equation \(\lambda_1^n + \lambda_2^n + \dots + \lambda_k^n = 0\) holds, determine whether it must be true that $\lambda_1 = \lambda_2 = \dots = \lambda_k = 0$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to\infty}\left(\sin{\frac{\ln{2}}{2}}+\sin{\frac{\ln{3}}{3}}+\cdots+\sin{\frac{\ln{n}}{n}}\right)^{1/n}$$

Answer: 1


Question: Find the minimum value of the constant \( A > 0 \) such that the function \( f(x) = 3x^3 + \frac{A}{x^3} \) satisfies \( f(x) \geq 20 \) for all \( x > 0 \).

Answer: \dfrac{100}{3}


Question: Given vectors $a$ and $b$ such that $a \times b = 2i + j - k$ and $a + b = i - j + k$, find the minimum possible magnitude of vector $a$. Here, $i$, $j$, and $k$ are unit vectors along the $x$, $y$, and $z$ axes, respectively.

Answer: \sqrt{2}


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\sin (3x)}{\cosh^3(x)} \, dx \) using complex integration techniques.

Answer: 0


Question: Find the limit of the sequence \(x_n\) defined by \[x_n = \left(\frac {n}{n+\ln(n)}\right)^{\ln(n)}\].

Answer: 1


Question: Determine the number of ways to rearrange the letters in the word INDIVISIBILITY such that no two 'I's are adjacent.

Answer: 3386880


Question: Determine all natural numbers \(n\) such that the polynomial \((X^2+1)^{6n+2} + X^4 + 1\) divides \(X^2 + X + 1\).

Answer: \emptyset


Question: Evaluate the integral \( \oint_{C} u \, dy + v \, dx \), where \( u \) and \( v \) are the real and imaginary parts of the function \( f(z) = \frac{1}{z^2 - 6z + 8} \), and \( C \) is the simple closed curve \( |z| = 3 \) oriented counterclockwise.

Answer: -\pi


Question: Evaluate the limit: $$ \lim_{x \to 0^+} (x |\ln x|)^{\frac{1}{x}} $$

Answer: 0


Question: Find the equation of the osculating plane to the curve given by the parametric equations \( x = a\cos(\theta) \), \( y = a\sin(\theta) \), \( z = a(1+\sin(\theta)) \) at \( \theta = \frac{\pi}{4} \).

Answer: z - y = a


Question: If $AB = I$, where $A$ and $B$ are square matrices, is it necessarily true that $BA = I$? Provide a justification for your answer.

Answer: Yes


Question: Let $T$ be a Noetherian ring. Suppose $R$ is a subring of $T$ such that $R[x]$ is integrally closed in $T[x]$. Does it follow that $R$ is integrally closed in $T$? Answer yes or no and justify your answer.

Answer: Yes


Question: Consider the group \( S \) defined by the set of positive real numbers with the operation \( a \circ b = 4ab^2 \). Determine the identity element of this group.

Answer: \dfrac{1}{2}


Question:
Determine the number of roots of the equation \(F(x) = \frac{n}{2}\) within the interval \(I = \left(\frac{n}{2}, n\right)\), where \(n > 1\) and \(n \in \mathbb{N}^+\). The function \(F(x)\) is defined as:
\[ F(x) = \int_0^x e^{-t} \left(1 + t + \frac{t^2}{2!} + \cdots + \frac{t^n}{n!}\right) \, dt \]

Answer: 1


Question: Let \( X = (C[0, \pi], \| \cdot \|) \). For \( f \in C[0, \pi] \), define the operator \( T(f)(x) = \int_0^x \sin(x+y)f(y) \, dy \). Find the operator norm \( \| T \| \).

Answer: 2


Question: Let $(u_n)$ and $(v_n)$ be sequences of complex numbers. Suppose that $(u_n)$ converges in mean and that there exists a sequence $(\varepsilon_n)$ such that $v_n = u_n(1 + \varepsilon_n)$ and $\lim_{n\to\infty} \varepsilon_n = 0$. Does $(v_n)$ necessarily converge in mean? Provide a justification for your answer.

Answer: Yes


Question:
Find the coefficient of \(x^5\) in the expansion of the product of the following infinite series:
\[(1+x+x^2+x^3+\ldots)(1+x^2+x^4+x^6+\ldots)(1+x^3+x^6+x^9+\ldots)(1+x^4+x^8+\ldots)\]

Answer: 6


Question: Consider a Borel set $A \in \mathfrak{B}(\mathbb{R})$ with Lebesgue measure $\lambda(A) > 0$. Determine if it is possible to find two disjoint Borel sets $B, C \in \mathfrak{B}(\mathbb{R})$ such that $B \cup C = A$ and both $\lambda(B), \lambda(C) > 0$. Provide a justification for your answer.

Answer: Yes


Question: Suppose $g:\Bbb R\to \Bbb R$ is a continuous function such that $\int _{-\infty}^\infty g(x)\text{dx}$ exists. If $\lim _{x\to -\infty } g(x)$ exists, does it follow that $\lim_{x\to -\infty} g(x)=0$? Answer yes or no.

Answer: Yes


Question: Determine the parameters (length, dimension, and minimum distance) for the subfield subcode of the $[21,18,3]$ Hamming code $Ham(3,4)$ over the alphabet $\mathbb{F}_4$, when restricted to $\mathbb{F}_2$. That is, find the parameters for $Ham(3,4)\mid_{\mathbb{F}_2}=Ham(3,4)\cap \mathbb{F}_2^{21}$. Provide your answer in the form of $[n, k, d]$. \( \boxed{[n, k, d]} \)

Answer: [21, 15, 3]


Question: Evaluate the contour integral \( \int_C \frac{e^z}{z^2-4} \, dz \), where \( C \) is the circle defined by \( |z-1| = 1 \) with positive orientation.

Answer: \dfrac{\pi i e^2}{4}


Question: Determine the order of the element 250 in the group $U(641)$, which consists of positive integers coprime to 641 under multiplication modulo 641.

Answer: 16


Question: Find the Frobenius number for the arithmetic sequence of integers 100, 105, 110, ..., 200.

Answer: 95


Question: Evaluate the integral \( \int_{|z|=2} \frac{\cos^{4} z}{z^{3}}dz \) using the method of residues.

Answer: -4\pi i


Question: In a finite $p$-group $G$, determine if a proper subgroup $H$ can be equal to its own normalizer $N_G(H)$. Provide a justification for your answer.

Answer: No


Question: Evaluate the limit \( \lim_{x \to 0} \frac{\int_0^x e^t\cos t\, dt}{x} \) without using L'Hospital's rule.

Answer: 1


Question: Calculate the sum of the infinite series: \[ \frac{1.3}{2} + \frac{3.5}{2^2} + \frac{5.7}{2^3} + \cdots \]

Answer: 23


Question:
Calculate the value of the following double limit as both $v$ and $V$ approach the speed of light $c$:
\[
\lim_{v\to c}\lim_{V\to c}\frac{v - V}{1 - \frac{vV}{c^2}}.
\]

Answer: -c


Question: Let \( f(x) = \sin x \) for \( x \in [0, \frac{\pi}{2}] \), \( f(x) = 2 - \sin(\pi - x) \) for \( x \in (\frac{\pi}{2}, \pi] \), and \( f(x) = f(2\pi - x) \) for \( x \in (\pi, 2\pi] \). If the area enclosed by \( y = f(x) \) and the x-axis is \( a\pi + b \), find \( a^2 + b^2 \).

Answer: 4


Question: Extend the recursion formula for the number of involutions to include the case where $m = 7$. Provide the resulting expression or value.

Answer: 232


Question: Find all positive integer solutions to the Diophantine equation $x^3 + 3 = 4^n$.

Answer: (1, 1)


Question:
Let $(x_n)_{n \geq 0}$ be a sequence defined by the recurrence relations:

\[\begin{cases}
x_0=a \\
x_{n+1}=x_n+e^{-bx_n}
\end{cases}\]

where $a$ and $b$ are positive constants. Determine the limit:

$$\lim_{n\to \infty}\frac{\ln(c+n)}{x_n}$$

where $c$ is a positive constant.

Answer: b


Question: Let $A$ be a finite-dimensional $\mathrm{C}^*$-algebra with the absolute value defined as $|a|:=\sqrt{a^*a}$ and the cone of positive elements $A^+:=\{a\in A:\exists \,b\in A, a=b^*b\}$. An operator $T\in B(A)$ is said to be positive if $T(A^+)\subset A^+$. Determine whether it holds that $|Tf|=T|f|$ for a positive operator $T$.

Answer: No


Question: Evaluate the limit using the Mean Value Theorem: $$\lim _{x\to \infty \:}\frac{\int _{2x}^{3x}\frac{3t-1}{t+1}\,dt}{x}$$

Answer: 3


Question: Evaluate the integral \( \int_0^1 \frac{\ln x}{x^2 + 1} \, dx \).

Answer: -\mathbf{G}


Question: Determine the number of distinct homomorphisms from the group $\mathbb{Z} / 8 \mathbb{Z}$ to the dihedral group $D_4$, which has order 8.

Answer: 8


Question: Find the remainder when the number \(2^{2^{2^{\cdot^{\cdot^{2}}}}}\), where 2 appears 2016 times, is divided by 2016.

Answer: 1024


Question: Evaluate the definite integral \(\int_2^{12}\frac{\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}}{\sqrt{x\sqrt{x\sqrt{x}\ldots}}}\,\mathrm{d}x\) where the series in the numerator and denominator continue infinitely.

Answer: 4 + \ln 3


Question: Given two discrete random variables $X$ and $Y$, if their probability generating functions $E(t^X)$ and $E(t^Y)$ are equal for all $t$ in the interval $(0,1)$, can we conclude that $X$ and $Y$ have the same distribution? Provide a justification for your answer.

Answer: Yes


Question: Determine the value of \( t \) for which the integral \( I = \int_{0}^{\infty} \left( \frac{x+1}{3x^2-t} - \frac{t}{2x+1} \right) \, dx \) converges.

Answer: \dfrac{2}{3}


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

Answer: \dfrac{1}{6}


Question: Consider a solution \(\phi_t(x)\) of the differential equation \(x' = f(x)\), where \(x \in \mathbb{R}^n\), \(f: \mathbb{R}^n \to \mathbb{R}^n\), and \(f\) is continuously differentiable (\(f \in C^1\)). If the solution converges to a fixed point \(x_0\) as \(t\) approaches infinity, i.e., \(\lim_{t \to \infty} \phi_t(x) = x_0\), does it follow that \(f(x_0) = 0\)? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the sum: $$ \sum_{r=0}^n \left[\binom{n}{r}\cdot\sin rx \cdot \cos (n-r)x\right] $$

Answer: 2^{n-1} \sin(nx)


Question: Find the smallest integer $n \ge 5$ such that there exists an integer $x$ with $x \not\equiv 1 \pmod{n}$ and $x^3 \equiv 1 \pmod{n}$.

Answer: 7


Question: Evaluate whether the limit \( \lim_{x\to+0}x^{p}\int^{x}_1w^{-p}f\left(w\right)dw \) approaches zero, given that \( f \) is a continuous function and \( p > 0 \).

Answer: 0


Question: In Zermelo-Fraenkel (ZF) set theory, is it true that everything is a set and there is no primitive notion of class?

Answer: Yes


Question: Consider the function \(h(x):=\begin{cases}\displaystyle1-\frac{\sin x}{x}&\text{if }x\ne0\\0&\text{otherwise}\end{cases}\) for \(x\in\mathbb R\). Let \(\alpha:=\inf_{\substack{x\in\mathbb R\\|x|\ge1}}h(x)\). Determine \(\alpha\) and rigorously justify why the infimum is attained at \(x=1\).

Answer: 1 - \sin 1


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

Answer: \dfrac{1}{e}


Question: Determine if the bilinear form \( C(u,v) = \int_{0}^{2} u'v' + muv \) is coercive for \( u, v \in H^1_0((0,2)) \) and \( m > 0 \).

Answer: Yes


Question: Determine the number of different subgroups of order 9 that are isomorphic to $C_3 \times C_3$ in the symmetric group $S_6$. Provide your answer as a single integer.

Answer: 10


Question:
Calculate the determinant of the linear transformation \( T(X) = \frac{1}{2}(AX + XA) \), where \( V \) is the vector space of all \( 3 \times 3 \) matrices and \( A \) is the diagonal matrix given by:
\[ A = \begin{pmatrix} 1 & 0 & 0\\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

Answer: \dfrac{81}{8}


Question: Determine the condition on the value of 'a' such that the characteristic polynomial \(x^3 - 3x + a\) has three distinct real roots.

Answer: (-2, 2)


Question:
Evaluate the integral:

$$\int_{-2}^2 \int_{-\infty}^2 \left[u < x\right] f(u) \, du \, dx$$

where $f(u)$ is the probability density function of a random variable $X$ that is symmetric about 0, and $\left[u < x\right]$ is the Iverson bracket, which is equal to 1 if $u < x$ and 0 otherwise.

Answer: 2


Question: Assume that $F$ is a field such that $GL_n(F)$ is a divisible group. Is it true that the multiplicative group $F^*$ is also divisible?

Answer: Yes


Question:
Find the extreme values of the function \( f(x,y) = 2(x-y)^2 - x^4 - y^4 \) by solving the system of equations given by the partial derivatives:
\[ \frac{\partial f}{\partial x} = 4(x - x^3 - y) = 0 \]
\[ \frac{\partial f}{\partial y} = 4(y - y^3 - x) = 0 \]

Answer: 8


Question: Determine the function \( f(x) \) that transforms the differential equation \( y'' - 4xy' + (4x^2 - 2)y = 0 \) into \( u'' + f(x)u = 0 \) using the substitution \( y(x) = u(x)v(x) \).

Answer: 0


Question: Find the function $f(X)$ that minimizes the conditional expected value $E[(Y-f(X))^2|X]$, given that $X$ and $Y$ are random variables with $E(Y) = \mu$ and $E(Y^2) < \infty$. Provide your answer in terms of $X$ and $Y$.

Answer: E(Y|X)


Question: Evaluate the limit: $$\lim_{x\to 0} \Big((9+x)^x-9^x\Big)^x$$

Answer: 1


Question: Simplify the expression \((2\text{proj}_v-I)^T(2\text{proj}_v-I)\) to obtain \(I\).

Answer: I


Question: Consider a positive definite, symmetric $n \times n$ matrix $G$ with integer entries. The matrix $G$ can be diagonalized as $G = Q^T D Q$, where $Q$ is an orthogonal matrix and $D$ is a diagonal matrix with eigenvalues of $G$. Determine whether the matrix $Q$ must also have integer entries.

Answer: No


Question: Consider a Galois group $Gal(K/\mathbb{Q})$ that is isomorphic to the cyclic group $C_{16}$. Determine the number of subfields $L$ such that the degree $[K:L] = 4$. Provide your answer as a single integer.

Answer: 1


Question: Two teams, each with seven players numbered 1 to 7, compete in a series of games. In the first game, the players numbered 1 from each team play against each other. The loser of each game is eliminated and replaced by the next player from the same team. This process continues until all players from one team are eliminated. Let $N$ be the total number of possible sequences of games. Determine the remainder when $N$ is divided by 1000.

Answer: 432


Question: Does the identity of indiscernibles hold for Nei's genetic distance, i.e., is it true that $D[X,Y] = 0$ if and only if $X = Y$?

Answer: Yes


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

Answer: 1


Question: Determine the number of group homomorphisms from the dihedral group $D_4$ to the symmetric group $S_3$. Use the fact that the order of the image of an element must divide the order of the element.

Answer: 10


Question: Evaluate the limit of the function \( f(x,y) = \frac{x^3+y^3}{(|x|+|y|)\sqrt{x^2+y^2}} \) as \((x,y)\) approaches \((0,0)\).

Answer: 0


Question: Determine whether there exists a group $G$ such that for every proper subgroup $H$ of $G$, the index $[G:H]$ is infinite.

Answer: Yes


Question: Determine the number of group homomorphisms from $\mathbb{Z}_{15}$ to $\mathbb{Z}_{20}$ whose image has exactly 5 elements.

Answer: 4


Question: Evaluate the limit: \[\lim_{n\to +\infty}\left(\frac{f(a+\frac{1}{n})}{f(a)}\right)^{\frac{1}{n}}\] where \(f: \mathbb{R} \to \mathbb{R}\) is differentiable at \(a \in \mathbb{R}\) with \(f(a) > 0\).

Answer: 1


Question: Consider two real $n \times n$ matrices $A$ and $B$ with elements $\{a_{ij}\}$ and $\{b_{ij}\}$, respectively. If $a_{ij} = b_{ij}$ for $i = j$ and $a_{ij} \leq b_{ij}$ for $i \neq j$, and $B$ is positive-definite, does it imply that $A$ is also positive-definite? Answer with 'Yes' or 'No' and justify your answer.

Answer: No


Question: Find the sum of all real values of $x$ that satisfy the equation $(x^2-5x+5)^{x^2+4x-60}=1$. Provide your answer as a single number.

Answer: 3


Question: Find all positive integer solutions \((x, y, z)\) to the equation \(3^x - 5^y = z^2\).

Answer: (2, 1, 2)


Question: Evaluate the limit \( \lim_{x\to\infty} (x+\log x)\cdot \log\left(\frac{x^2+3x}{x^2-5x}\right) \) without using L'Hopital's rule.

Answer: 8


Question: Evaluate the limit: $$\lim_{x \to \pi}\frac{5e^{\sin 2x}-\frac{\sin 5x}{\pi-x}}{\ln(1+\tan x)}$$ without using l'Hôpital's rule or equivalents.

Answer: 10


Question: Determine whether the set $T = \{x \in \mathbb{R}: 2x^2 \cos\left(\frac{1}{x}\right) \le 1\} \cup \{0\}$ is a complete subset of $\mathbb{R}$ with respect to the standard metric.

Answer: Yes


Question: Calculate the residues of the function \( f(z) = \frac{e^z}{\sin^2 z} \) at its isolated singularities \( z = k\pi, \; k \in \mathbb{Z} \).

Answer: e^{k\pi}


Question: Evaluate the limit: $$\lim_{n\to \infty} \int_0^1 \sqrt[n]{x} e^x \,dx.$$

Answer: e - 1


Question: Let $D$ be a unique factorization domain (UFD), and let $F$ be the field of fractions of $D$. Suppose $a \in D$ is such that $x^2 \neq a$ for all $x \in D$. Determine whether it is true that $x^2 \neq a$ for all $x \in F$. Provide a justification for your answer.

Answer: Yes


Question: Consider the irreducible polynomial \( g = X^4 + X + 1 \) over the finite field \( \mathbb{F}_2 = \{0, 1\} \). Let \( E \) be the field extension of \( \mathbb{F}_2 \) that includes a root \( \alpha \) of \( g \). Determine the number of elements in the field \( E \).

Answer: 16


Question:
Determine whether the series \(1 + \frac{1}{2} -\frac{1}{3} + \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \dots\) converges. If it diverges, specify whether it diverges to \(\infty\) or has no limit. The generic term of the series is given by:
\[a_n = \begin{cases} -\frac{1}{n}, & \text{if } 3 \mid n \\ \frac{1}{n}, & \text{otherwise} \end{cases}\]

Answer: \infty


Question: Evaluate the integral \( \int_{0}^{\frac{\pi}{4}} \left( \tan^9 x + 5\tan^7 x + 5\tan^5 x + \tan^3 x \right) \, dx \) using the substitution \( u = \tan x \).

Answer: \dfrac{25}{24}


Question: Determine whether the following statement is true or false: If $M/N$ and $N/K$ are both normal field extensions, then all the $K$-embeddings of $N$ into an algebraic closure of $K$ can be prolonged to $M$.

Answer: True


Question: Let \( f \) and \( g \) be continuous functions on \([a,b]\) and differentiable on \((a,b)\). Suppose that \( g'(x)=0 \) for some \( x \) in \((a,b)\). Determine if there exists a \( c \) in \((a,b)\) such that \[ \frac{f(b)-f(a)}{b-a}-\frac{b-a}{g(b)-g(a)}=f'(c)-\frac{1}{g'(c)}. \]

Answer: Yes


Question:
Consider a finitely generated group $G = K \rtimes \mathbb{Z}$, where $K$ is not finitely generated. Let $T$ be a finite subset of $K$ such that $\langle (0,1), (k,0) \mid k \in T \rangle$ is a generating set of $G$. Define $\phi(n)$ as the automorphism on $K$ corresponding to $n \in \mathbb{Z}$, which defines the semi-direct product. Let $H = \langle T \rangle$, a finitely generated subgroup of $K$. Suppose the automorphisms satisfy the condition:

$$\ldots \phi(-1)(H) \supsetneqq \phi(0)(H) \supsetneqq \phi(1)(H) \supsetneqq \phi(2)(H) \ldots$$

Is it possible for $H$ to have a finite abelianization, i.e., for the quotient group $H \big/ [H,H]$ to be finite?

Answer: Yes


Question: Given two Banach spaces $X$ and $Y$, and a surjective continuous linear transformation $T: X \rightarrow Y$, determine if for every convergent sequence $\{y_n\}$ in $Y$ converging to $y \in Y$, there exists a sequence $\{x_n\}$ in $X$ converging to $x \in X$ such that $T(x_n) = y_n$ for all $n \in \mathbb{N}$ and $T(x) = y$. Provide a justification for your answer.

Answer: Yes


Question: Transform the basis of the ideal $I = (5x+7y, 3x+4y)$ in the ring $R = \mathbb{Z}[x,y]$ to a triangular form using basis transformations.

Answer: (x, y)


Question: Find the value of $n$ such that the cyclotomic polynomial $\Phi_n(x)$ is given by $f(x) = (x^4 - x^2 + 1)(x^2 + 1)$. What is $n$?

Answer: 24


Question: Does there exist a ring with 5 elements such that the characteristic of the ring is 2?

Answer: No


Question: Determine whether there exists a real-valued function \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(x) \cdot f(y) = f(x + y) \) for all \( x, y \in \mathbb{R} \), and the function is discontinuous at some points.

Answer: Yes


Question: Can a finite group $G$ be embedded in a finite group $G_1$ such that two distinct elements $x$ and $y$ of $G$ with the same order become conjugate in $G_1$?

Answer: Yes


Question: Evaluate the limit \(\lim_{x\to0}\frac{1}{x^2}\Big(\frac{3}{3+x}-\frac{\sin(x)}{x}\Big)\) using Taylor series expansion.

Answer: -\infty


Question:
Given a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f'(0) = 1$ and $f(x + y) = f(x)f(y)$ for all $x, y \in \mathbb{R}$, determine which of the following statements is true:

1. Both $f$ and $f'$ are decreasing functions.
2. Both $f$ and $f'$ are increasing functions.
3. $f$ is an increasing function but $f'$ is a decreasing function.
4. $f$ is a decreasing function and $f'$ is a decreasing function.

Answer: 2


Question: Let \( \{x_n\} \) be a real-valued sequence such that \( \lim_{n\to\infty} n \cdot x_n = 0 \). Determine all possible real numbers \( t \) for which \( \lim_{n\to\infty} x_n \cdot (\log n)^t = 0 \).

Answer: \mathbb{R}


Question: Solve the equation: \( (x^{2010}+1)(1+x^2+x^4+\ldots+x^{2008})=2010x^{2009} \). The second term on the left-hand side is a geometric series.

Answer: 1


Question: Determine whether the statement $|f(x) - f(y)| \leq \epsilon$ for all $\epsilon > 0$ implies that $f(x) = f(y)$. Provide a justification for your answer.

Answer: f(x) = f(y)


Question: Evaluate the integral \( \oint_{|z|=1} \frac{\cos z}{z^3} \, dz \) using Cauchy's Integral Formula.

Answer: -\pi i


Question:
In the inner product space $P([0,1])$, consisting of all real polynomials on $[0,1]$ with inner product $\langle f,g\rangle = \int_0^1 f(x)g(x)dx$, consider the subspace $V = \text{span}\{t^2\}$. Let $h(t) \in V$ be a polynomial such that $\|2t - 1 - h(t)\| \leq \|2t - 1 - x(t)\|$ for all $x(t) \in V$. Determine the correct form of $h(t)$ from the following options:

(a) $h(t) = \frac{5t^2}{6}$

(b) $h(t) = \frac{5t^2}{3}$

Verify which option is correct.

Answer: a


Question: Determine the cardinality of the set $S$ consisting of all infinite subsets of the set of rational numbers, $\mathbb{Q}$. \( \boxed{} \)

Answer: 2^{\aleph_0}


Question: Evaluate the integral \( \oint_{|z|=3} \frac{z^4}{(z^2+1)(z-2)} \, dz \) using the residue theorem.

Answer: 6\pi i


Question: How many unique pairs (m, n) of positive integers exist such that their least common multiple is \(3^3 \cdot 5^7 \cdot 11^{13}\), and the greatest common divisor of m and n is 1?

Answer: 8


Question: Calculate the integral \( \int_V \Delta \frac{1}{r} \, dV \) over a sphere of radius \( R = 1 \) centered at the origin, where \( r = \sqrt{x^2 + y^2 + z^2} \). Use Gauss's theorem to perform the calculation, given that the differential surface element in spherical coordinates is \( d\textbf{S} = \sin(\vartheta)\textbf{e}_r d\vartheta d\varphi \).

Answer: -4\pi


Question: Given a function $f \in L^p(\mathbb{R})$ with $1 \leq p < \infty$ and a positive tolerance $\epsilon$, determine if there exists a radius $R > 1$ such that the $L^p$ norm of $f$ over the set $B_R = \{x: |x| \geq R\}$ is less than $\epsilon$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{1}{x^2+1} \, dx \) using the complex contour integral \( \oint_{C(r)} \frac{1}{z^2+1} \,dz \), where \( C(r) \) is the closed semicircle in the upper half-plane with endpoints \((-r,0)\) and \((r,0)\). Use the hint: \( \frac{1}{z^2+1}=\frac{-1}{2i}\left(\frac{1}{z+i}-\frac{1}{z-i}\right)\), and show that the integral along the open semicircle vanishes as \( r \rightarrow \infty \).

Answer: \pi


Question: Let $V$ be the $\mathbb{R}$-vector space of polynomials with degree $\leq 2$. Define the bilinear form $\psi: V \times V \rightarrow \mathbb{R}$ by $\psi(p,q) = \int_{0}^{1} p(x)q(x) \, dx$. Let $\phi: V \rightarrow \mathbb{R}$ be the linear form given by $\phi(p) = p(0)$. Find a polynomial $q \in V$ such that $\phi(p) = \psi(p,q)$ for all $p \in V$. \( \text{Express your answer as a polynomial.} \)

Answer: 30x^2 - 36x + 9


Question: Evaluate the integral \( \int \frac{\exp(z) \sin(3z)}{(z^2-2)z^2} \, dz \) on the contour \(|z|=1\).

Answer: -3\pi i


Question: At a square table, 8 friends are seated such that each side of the table has 2 friends. Given that every pair of friends knows each other, what is the probability that no friend is seated next to another friend they know?

Answer: 0


Question: If a sequence \( \{p_n\} \) of real numbers diverges to infinity, does it imply that the limit superior of \( \{p_n\} \), denoted as \( \limsup p_n \), is also infinity? Justify your answer.

Answer: \infty


Question: Evaluate the limit: $$\lim\limits_{x\to{-2}}(2\sqrt{-1-x}-1)^{\frac{1}{4^{-x}-16}}$$ without using l'Hôpital's rule, derivatives, or Taylor series.

Answer: e^{\frac{1}{32 \ln 2}}


Question: Evaluate the integral \( \int_\gamma \frac{1}{1+z^2} \, dz \) using Cauchy's Theorem, where \( \gamma \) is the circular contour defined by \( t \mapsto 2+e^{it} \) for \( 0 < t < 2\pi \).

Answer: 0


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

Answer: (-\infty, 1)


Question: Let $Y = \sqrt{X}$, where $X$ has the cumulative distribution function \[ F_X(x) = \begin{cases} 0 & \text{if } x < 0, \\ \frac{1}{8} x^3 & \text{if } 0 \leq x \leq 2, \\ 1 & \text{if } x > 2. \end{cases} \] Find $\mathbb{P}[Y \leq X]$. \(\text{Express your answer as a single probability value.}\)

Answer: \dfrac{7}{8}


Question: Evaluate the integral of the 2-form \( x_2 \, dx_2 \wedge dx_3 + x_1 x_3 \, dx_1 \wedge dx_3 \) over the image set \( Y \) of the mapping \( \alpha: (0,1)^2 \to \mathbb{R}^3 \) given by \( \alpha(u,v) = (u,v,u^2+v^2+1) \).

Answer: \dfrac{1}{2}


Question: Determine the dual space of the sum $X=L^2(\mathbb{R})+L^3(\mathbb{R})$ using the norm defined in the problem.

Answer: L^2(\mathbb{R}) \cap L^{3/2}(\mathbb{R})


Question: Determine the value of \(x\) such that \(\sum_{i=1}^{2013}\left\lfloor\frac{x}{i!}\right\rfloor=1001\).

Answer: 584


Question: Find the multiplicative inverse of the element $x^2+1$ in the field $\mathbb{Z}[\alpha] := \mathbb{Z}_2[x]/(x^3+x+1)$. Express your answer as a polynomial in $\mathbb{Z}_2[x]$ of degree less than 3.

Answer: x


Question: Determine whether the statement is true or false: If $z_n$ is a complex sequence, then $1/z_n$ converges to 0 if and only if $z_n$ diverges.

Answer: B


Question: Simplify the expression \( \cos(2^k \cos^{-1}(x)) \).

Answer: T_{2^k}(x)


Question: Consider a topological space $X$ and a subspace $A$. Suppose there exists a continuous map $f:X \rightarrow A$ such that $f(a) = a$ for all $a \in A$. Define $g := (f, \operatorname{id}) : X \to X \times X$, where $\operatorname{id}$ is the identity map. Describe the inverse image of the diagonal $\Delta_X$ under $g$. What can be concluded about this inverse image?

Answer: A


Question: Let $p: T \to B$ be a vector bundle with a subbundle $p|_E: E \to B$. Consider the quotient bundle $T/E \to B$, defined by identifying vectors in the fibers of $T$ whose difference lies in the subspace $E_b \subseteq T_b$. Assume $T$ is equipped with an inner product, and let $E^{\perp} \to B$ be the orthogonal complement to $E$. Are the vector bundles $T/E$ and $E^{\perp}$ isomorphic?

Answer: Yes


Question: In a Poisson process with a mean of 5 people joining a tennis club per month, the probability that a person joining is a woman is 0.3. What is the expected waiting time until 3 women join the club from the beginning of the period, given that the probability of fewer than 3 people joining is $18.5 \cdot e^{-5}$?

Answer: 2


Question: Evaluate the integral \( \int_0^\infty \frac{\sin x}{x} \, \mathrm{d}x \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit: $$\lim_{x\to\infty}\bigg({\frac{e^x-x-1}{e^x-\frac{1}{2}x^2-x-1}}\bigg)$$

Answer: 1


Question: Let $X, Y, Z$ be independent and identically distributed random variables, each following a standard normal distribution, $\mathcal{N}(0,1)$. Given that $X > Z$, find the conditional probability $\mathbb{P}(Y > Z \mid X > Z)$. Express your answer as a numerical value.

Answer: \dfrac{2}{3}


Question: Find the integer $N$ such that the inequality \( \left(\dfrac{1}{2}\right)^{1000} \sum_{x=N+1}^{1000} {1000 \choose x} < 0.01 \) holds true.

Answer: 537


Question: In \(\mathbb{R}^3\), a sphere has a diameter \(d\). What is the most probable length of random secants (including tangents, where the length is 0) drawn through the sphere?

Answer: d


Question: An abelian group $G$ is generated by elements $x$ and $y$, where $|x| = 16$, $|y| = 24$, and $x^2 = y^3$. What is the order of $G$?

Answer: 48


Question: Determine if the following limit exists: $$\lim_{(x,y) \rightarrow (0,0)}\frac{\sin{y}-y}{x^2+y^2}$$

Answer: 0


Question: Determine the number of roots of the equation \(x^3 + x^2 + 2x + \sin x = 0\) within the interval \([-2\pi, 2\pi]\).

Answer: 1


Question: Determine if the derivative of the function \( g(x)=x^3\left|\sin\left(\dfrac{\pi}{x}\right)\right| \) exists at \( x=0 \), where \( g(0)=0 \). If it exists, find its value.

Answer: 0


Question:
Let $A$ be the set of all real numbers in the interval $[0,1]$ that can be represented in base 5 using only the digits $0, 1, 2, 3,$ and $4$. Determine which of the following statements about $A$ is true:

1. $A$ is a finite set.
2. $A$ is a countably infinite set.
3. $A$ is an uncountable set but does not contain an open interval.
4. $A$ contains an open interval.

Determine the correct statement about the set $A$. Is there any relationship between $A$ and the Cantor set?

Answer: 4


Question: Consider the set \( E = \{ f \in C[0,1] : f(1)=0, \int_0^1 tf(t)\,dt =1 \} \). Determine the value of \( \delta = \inf \{\lVert f \rVert_{\infty} : f \in E\} \).

Answer: 2


Question: Compute \(2^{2475} \bmod 9901\).

Answer: 1000


Question: Consider a group $G$ of order $p^nq^m$, where $p$ and $q$ are primes with $p < q$, and the order of $[p]$ in $\mathbb{Z}_q^{\times}$ is greater than $n$. Determine if there exists a chain of subgroups of $G$ such that each subgroup is normal in the next, and the order of each quotient group is a prime.

Answer: Yes


Question: Let $(x_n)$ be a sequence in $\mathbb{R}$. If every subsequence of $(x_n)$ that omits only finitely many terms converges, does this imply that $(x_n)$ converges? Provide a justification for your answer.

Answer: Yes


Question: Does a function being both Lipschitz continuous and strictly monotonic guarantee its differentiability? Provide a justification for your answer.

Answer: No


Question: Determine the radius of convergence of the Taylor series at zero for the holomorphic function $g(z)$, given that $g^4 + 2g^2 - 5 = z$ and $g(0) = 1$. Provide your answer as a single numerical value.

Answer: 5


Question: Can a ring with positive characteristic have an infinite number of distinct elements? Provide a justification for your answer.

Answer: Yes


Question: Compute the fundamental group of the space \( \mathbb{R}^4 \setminus \mathbb{R}^2 \).

Answer: \mathbb{Z}


Question: Let $A$ be an integral domain such that every element $a \in A$ generates a finite subring. Is $A$ necessarily commutative? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral: \[ I = \int_{0}^{2 \pi} \ln (\sin x +\sqrt{1+\sin^2 x}) \, dx \]

Answer: 0


Question: Given two matrices $x$ and $y$ of the same rank, does there exist a pair of matrices $(g_1, g_2) \in GL(m,k) \times GL(n,k)$ such that $g_1 x g_2^{-1} = y$?

Answer: Yes


Question: Given two similar matrices A and B, where A is symmetric and non-negative, determine if there exists a matrix P that is not orthogonal and a matrix B that is not symmetric such that A = PBP^{-1}.

Answer: Yes


Question: Solve the differential equation \( \frac{d}{dx}(2y y') = (y')^2 \) with the initial conditions \( y(0) = 0 \) and \( y(1) = 0 \).

Answer: 0


Question: Evaluate the limit \( \lim_{n \rightarrow \infty} \frac{\sum_{i=0}^k \binom{n}{4i}}{3^n} \) where \( n = 4k \).

Answer: 0


Question: If the subsequences $x_{2n}$ and $x_{3n}$ of a sequence $x_n$ both converge to the same limit, does the sequence $x_n$ necessarily converge? Justify your answer.

Answer: No


Question: Given the function \( f(x)=\left\{\begin{matrix} \frac{g(x)}{x},&\text{if }x\neq0\\ 0,&\text{if }x=0 \end{matrix}\right. \) with conditions \(g(0) = 0\), \(g'(0) = 0\), and \(g''(0) = 3\), determine the derivative \(f'(0)\).

Answer: \dfrac{3}{2}


Question: Calculate the limit: $$\lim_{x\to0}\left(\ln(1+x)+\ln(1+2x)+\cdots+\ln(1+px)\right)^x$$

Answer: 1


Question: Calculate the result of the convolution \( \delta_{im} \epsilon_{ijk} \) using Einstein summation notation, considering that two indices coincide.

Answer: 0


Question: Suppose $V$ is a complex finite-dimensional inner product space, and $TT^*-T^*T$ is a positive operator on $V$. Is $TT^*-T^*T$ necessarily the zero operator on $V$?

Answer: Yes


Question: Can a three-body system have a positive total energy ($E > 0$), suggesting the possibility of escape, while no object can actually escape due to the conservation of angular momentum? Assume the total energy is given by the sum of kinetic and potential energies: $$E = \sum_{i=1}^{3}\frac{1}{2}m_i v_i^2 - \sum_{i=1}^{3} \sum_{j>i}^{3} \frac{m_i m_j}{r_{ij}},$$ with the condition that the center of mass is stationary, $\sum_{i=1}^3m_iv_i=0$. Consider angular momentum with respect to an arbitrary point.

Answer: No


Question: A sphere centered at the origin rotates about the z-axis with angular velocity \( \omega \). Let \( P(r, \theta, \phi) \) be any point inside the sphere. Determine the direction of movement of \( P \) as a unit vector in spherical coordinates.

Answer: \hat{\boldsymbol{\phi}}


Question: Determine the function $f: \mathbb{R} \rightarrow \mathbb{R}$ given that $f(a + b\sqrt{2}) = (a + b\sqrt{2})^{2k+1}$ for all integers $a$ and $b$, where $k$ is a non-zero natural number and $f$ is monotonic.

Answer: f(x) = x^{2k+1}


Question: In a metric space $(X, d)$, a space is defined as weakly totally bounded if for every $\varepsilon > 0$, there exist countably many points $\{x_i\}_{i=1}^\infty$ in $X$ such that $X \subseteq \bigcup_{i=1}^\infty B_\varepsilon(x_i)$. Determine whether every weakly totally bounded space is separable.

Answer: Yes


Question: Evaluate the double limit \( \lim\limits_{n\to \infty}\lim\limits_{x\searrow 0}f^{(n)}(x) \) for the function \( f:[0,\infty)\to \mathbb{R},~f(x)=\exp(\sqrt{x})+\exp(-\sqrt{x}) \).

Answer: 0


Question: Calculate the covariance of two random variables $X$ and $Y$ given the joint probability density function: $$f(x,y)=\mathrm{Pr}[X{\approx}x; Y{\approx}y]=\begin{cases}2 & \text{on the triangle with vertices at the origin, $(1,1)$ and $(0,1)$} \\0 & \text{otherwise}.\end{cases}$$

Answer: \dfrac{1}{36}


Question: Evaluate the infinite series: $$\sum_{k=1}^{\infty} \sum_{j=1, j \neq k}^{\infty} \sum_{i=1, i \neq j, i \neq k}^{\infty} \frac{1}{4^i4^j4^k}$$

Answer: \dfrac{2}{945}


Question: Is $P^k$, where $P$ is a prime ideal in a Dedekind domain $R$, an irreducible ideal? An ideal $I$ is irreducible if it cannot be expressed as an intersection of two proper ideals, $I = J \cap K$, with $J, K \neq I$. Provide a justification for your answer.

Answer: Yes


Question: Suppose $P(x) = x^5 - 20x^4 + 160x^3 + cx^2 + dx + e$ has only real roots. Determine the maximum possible value of $c$. \( \text{(Assume } b = 160 \text{)} \)

Answer: -640


Question: Find all Fibonacci numbers \( f_n \) such that \( f_{n-1}^2 = \left(\frac{f_n}{2}\right)^2 + h^2 \) for some positive integer \( h \), and \( f_n \) is twice a perfect square.

Answer: 8


Question: Evaluate the double integral \( \iint_A (x^2-y^2) \, dA \) where \( A \) is the unit circular disk defined by \( r \le 1 \).

Answer: 0


Question: Evaluate the limit: $$\lim _{n \rightarrow \infty}\left(n \ln \left(1-\frac{1}{n}\right) \cos \left(\pi \sqrt{4 n^2+10}\right)\right)$$

Answer: -1


Question: Find all integer solutions to the Diophantine equation \(x^3 = 3^y - 10\).

Answer: (-1, 2)


Question: Is it true that for real random variables $X$ and $Y$ on a probability space $(\Omega,\mathcal A,P)$, a sub-$\sigma$-algebra $\mathcal F$ of $\mathcal A$, and a Borel measurable function $f:\mathbb R^2\to \mathbb R$, if $f(X,Y)$ is $P$-integrable and $Y$ is $\mathcal F$-measurable, then the following holds for $P$-almost all $\omega\in\Omega$? $$E[f(X,Y)|\mathcal F](\omega)=\int f(x,Y(\omega))\kappa_{X,\mathcal F}(\omega,dx)$$ where $\kappa_{X,\mathcal F}$ is a regular conditional distribution of $X$ given $\mathcal F$. Answer with True or False.

Answer: True


Question: Given that for a fixed integer $m$, the inequality $a_{n,m} < b_{n,m}$ holds for all $n \geq 1-m$, and $\limsup_{n\to\infty}\frac{1}{n}\ln b_{n,m} = c$, where $c$ is a constant, determine whether $\limsup_{n\to\infty}\frac{1}{n}\ln a_{n,m} < c$ or $\limsup_{n\to\infty}\frac{1}{n}\ln a_{n,m} \leq c$.

Answer: \limsup_{n\to\infty}\frac{1}{n}\ln a_{n,m} \leq c


Question: Evaluate the integral: $$\int_0^\frac{\pi}{2} \sin^7x \cos^5x\, dx $$

Answer: \dfrac{1}{120}


Question: Given a closed path \(c\) with a winding number \(k\), and a transformation \(\tilde{c} = \rho(t)c(t)\) where \(\rho:[0,1]\to(a,b)\) with \(0<a<b\), determine the winding number of \(\tilde{c}\).

Answer: k


Question: Determine whether the Lyapunov Central Limit Theorem (CLT) condition holds for any sequence of independent Bernoulli random variables $X_1, X_2, \dots, X_n$, regardless of their distribution.

Answer: No


Question: Determine whether the Galois group of the sextic polynomial $Q(x)$ over the field $\mathbb{Q}(a, b)$ is isomorphic to the group $C_2 \times S_4$. Assume $Q(x)$ is defined in the given context.

Answer: C_2 \times S_4


Question: Does there exist a set $X$ such that $X \in X$?

Answer: No


Question: Calculate the area enclosed by the curve defined by the equation \(\lfloor x + y\rfloor + \lfloor x - y\rfloor = 5\) for \(x \ge y\) and \(x, y \ge 0\), where \(\lfloor z\rfloor\) represents the greatest integer less than or equal to \(z\).

Answer: \dfrac{3}{2}


Question: Given that $2017$ is a prime number, find the remainder when the sum $S = \sum_{k=0}^{62} \binom{2014}{k}$ is divided by $2017$. Simplify the expression for $S$ using combinatorial identities and modulo arithmetic.

Answer: 1024


Question: Determine the points where the function $f(z) = z|z|^2$ is complex differentiable using the Wirtinger derivatives.

Answer: 0


Question: Calculate the volume of the solid bounded by the surfaces $y = 2 - x^2$, $y = x^2$, $z = 0$, and $z = y + 3$ using a triple integral, considering the symmetry of the solid.

Answer: \dfrac{32}{3}


Question: Let \(n \geq 1\) be an integer and let \(c_{0}, \dots, c_{n} \in \mathbb{R}\). If \(c_{n} > 0\), is there necessarily an \(x \in \mathbb{R}\) such that \(\sum_{k=0}^{n}c_{k}x^{k} > 0\)?

Answer: Yes


Question: Calculate the expected value \( E[\Phi(c\bar{X})] \) where \( X_i \sim N(0, \sigma^2) \) are independent and identically distributed random variables, \( \bar{X} \) is the sample mean, and \( c \) is a constant.

Answer: \dfrac{1}{2}


Question:
Evaluate the limit:

$$\lim_{n\to\infty} \left(\prod_{k=1}^n \frac{2k}{2k-1}\right) \int_{-1}^{\infty} \frac{(\cos x)^{2n}}{2^x} \, dx$$

Answer: \pi


Question: Calculate the probability of obtaining at least three consecutive heads in 20 coin tosses, without using the inclusion-exclusion principle.

Answer: \dfrac{825259}{1048576}


Question: Determine the rank of the matrix $A = \sum\limits_{i=1}^5 y_i y_i^T$, where $y_1 = (1,0,1)^T$, $y_2 = (0,1,1)^T$, $y_3 = (1,1,2)^T$, $y_4 = (1,1,0)^T$, and $y_5 = (1,-1,1)^T$.

Answer: 3


Question:
Find the limit of the following expression as $n$ approaches infinity, given that $b > 0$:
\[ \lim_{n \to \infty} \frac{\sum_{i=0}^{n/3-1}{\frac{(bn^3)^{i}}{i!}}}{e^{bn^3}} \]

Answer: 0


Question: Calculate the limit \( \lim_{x\rightarrow 0}\dfrac{1}{x}\int_x^{2x}e^{-t^2}dt \).

Answer: 1


Question: Given a function $\mathbf{f}(\mathbf{x})$ where $\mathbf{x}\in\mathbb{R}^n$ and $\mathbf{f}:\mathbb{R}^n\rightarrow\mathbb{R}^n$ with a Jacobian matrix $\mathbf{J} = \partial \mathbf{f}/\partial \mathbf{x}\in\mathbb{R}^{n\times n}$. If the Jacobian matrix $\mathbf{J}$ is real and symmetric everywhere, does there exist a function $y:\mathbb{R}^n\rightarrow\mathbb{R}$ such that $\mathbf{J}=\frac{\partial^2 y}{\partial \mathbf{x}\partial \mathbf{x}}$?

Answer: Yes


Question: Find the maximum area of a rectangle whose vertices lie on the ellipse \(x^2 + 4y^2 = 1\).

Answer: 1


Question: Calculate the limit: $$\lim_{h\rightarrow0}\sup_{n\in\mathbb{N}}\left\|\sqrt{n}\,\mathbb{1}_{(0,\frac{1}{n})}(x+h)-\sqrt{n}\,\mathbb{1}_{(0,\frac{1}{n})}(x)\right\|_{L^2(\mathbb{R})}$$

Answer: \sqrt{2}


Question: Consider a compact metric space \(Y\) and a continuous real function \(g\). Suppose there is a point \(\rho\) such that \(g(\rho)\) is the minimum value of \(g(y)\) for all \(y \in Y\). If \(\rho\) is not a limit point of \(Y\), can we still conclude that \(g\) is uniformly continuous? Provide a justification for your answer.

Answer: Yes


Question: Determine the number of functions $f: \mathbb{N} \rightarrow \mathbb{N}$ that satisfy the condition $f(a^2 - b^2) = f(a)^2 - f(b)^2$ for all $a, b \in \mathbb{N}$ with $a \geq b$, and also satisfy $f(2) = 2$. \( \mathbb{N} \) denotes the set of natural numbers.

Answer: 1


Question: Is the fundamental group of a graph always isomorphic to a free group?

Answer: Yes


Question: In the ring \(A\) of continuous real-valued functions on \(\mathbb{R}\), let \(I\) be the set of functions that vanish at infinity. Are there non-zero functions in \(A\) whose product is in \(I\), even though the functions themselves are not in \(I\)?

Answer: Yes


Question: Find the number of non-trivial ring homomorphisms from \( \mathbb{Z}_{12} \) to \( \mathbb{Z}_{28} \).

Answer: 1


Question: Calculate the limit: $$ \lim_{x\to 0}\frac{2-x}{x^3}e^{(x-1)/x^2} $$

Answer: 0


Question: Evaluate the limit: $$ \lim_{n\to\infty} \frac{n!\times(2n)!}{(3n)!} $$

Answer: 0


Question: Find the exact sum of the series \( \sum_{n=1}^{\infty}\frac{(-1)^n n^2}{3^n} \).

Answer: -\dfrac{3}{32}


Question: Evaluate the integral \( \int_{-1}^{1} \int_{0}^{\sqrt{1-x^2}} \int_{0}^{2} \frac{1}{1+x^2+y^2} \, dz \, dy \, dx \) using cylindrical coordinates.

Answer: \pi \ln 2


Question: Evaluate the integral \( \int_0^{\frac{ \pi }{2}} \frac{\sin^{1395}x}{\sin^{1395}x + \cos^{1395}x}\ dx \).

Answer: \dfrac{\pi}{4}


Question: Determine the number of roots of the polynomial \( P(z) = z^5 + 2z^3 + 3 \) that lie within the closed unit disk \( \{z \mid |z| \leq 1\} \).

Answer: 1


Question: Calculate the limit: \( \lim_{x \to 0} \sin(x)\sin(1/x) \).

Answer: 0


Question: Given that X is a continuous random variable with density function \( f(x) = \frac{2}{x^2} \) for \( x \geq 2 \) and 0 otherwise, find the density function of \( Y = \frac{X}{X-1} \) for \( 1 < y \leq 2 \).

Answer: \dfrac{2}{y^2}


Question: Suppose $A$ and $B$ are diagonalizable matrices over the real numbers with the same characteristic polynomial. Are $A$ and $B$ necessarily similar?

Answer: Yes


Question: Find the minimum value of the function \( f(x, y) = 3x^2 - 2xy \) subject to the constraint \( \frac{x^2}{4} - y^2 = 1 \), where \( x, y \in \mathbb{R} \).

Answer: 6 + 4\sqrt{2}


Question: Determine whether there exist conformal maps that are not analytic and are defined on any open subset of the complex plane.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{n!\cdot (2n)!}{(3n)!}$$

Answer: 0


Question: Evaluate the limit: $$\lim_{n\to\infty} \frac{n!e^n}{n^n}.$$

Answer: \infty


Question: Determine the structure of a group $G$ with order 51.

Answer: C_{51}


Question: Assume $u$ is a harmonic function on $\mathbb{R}^3$, and $u(x,y,z)=1+x$ on the surface of the unit sphere $x^2+y^2+z^2=1$. Determine the value of $u(0,0,0)$. \( \boxed{?} \)

Answer: 1


Question: Can two distinct formulas or series of formulas have the same Gödel number?

Answer: No


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{n}\sqrt[n]{(n+1)^5+(n+2)^5+\cdots+(3n)^5}. \]

Answer: 0


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

Answer: 1


Question: Calculate \(2^{31} \mod 2925\) using the Chinese Remainder Theorem.

Answer: 1298


Question: Determine the coefficient of \(x^{50}\) in the generating function \(g(x) = (1 + x + x^2 + \ldots)(1 + x^2 + x^4 + \ldots)\).

Answer: 26


Question: Solve for $x$ in the equation \( \frac{(x!)^3}{x} - 1 = 3455 \), where \( x! \) denotes the factorial of \( x \).

Answer: 4


Question: Evaluate the limit: $$\lim_{x \to \infty}\frac{1}{x^3}\int^x_1\frac{t\ln(1+t)}{t^4 + 4} \, dt$$

Answer: 0


Question: Find the limit of the sequence \( X_n = \cos\left(\left(\frac{3^n+\pi^n}{3^n+\left(\pi-\frac{1}{4} \right)^n} \right)^{1/n}\right) \) as \( n \to \infty \), where \( n \in \mathbb{N} \).

Answer: \dfrac{1}{2}


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

Answer: a > 0


Question:
Find the fixed (equilibrium) points of the following system of differential equations:

$$\dot{x} = -x + xy$$

$$\dot{y} = x - y - x^2 - y^3$$

Set \(\dot{x} = 0\) and \(\dot{y} = 0\) and solve for \(x\) and \(y\).

Answer: (0, 0)


Question: Determine the number of ways to paint 10 houses on a street using 4 different colors, such that each house is painted in only one color, each color is used at least once, and no two colors are used on the same number of houses.

Answer: 302400


Question: Let $v:[0,1]\rightarrow\mathbb{R}$ be an absolutely continuous function such that $v(1)=0$ and $\int_0^1|v'(x)|^2dx<\infty$. Find the value of $\lim_{x\rightarrow 1^-}\dfrac{v(x)}{(1-x)^{1/2}}$. \(\text{Express your answer as a number or a specific expression.}\)

Answer: 0


Question: Let $a_n$ be a sequence defined by $a_n=\frac{b_{n+1}}{b_n}$, where $b_1=2$, $b_2=3$, and $b_{n+2}=2b_n+b_{n+1}$ for $n\geq 1$. Determine the limit of $a_n$ as $n$ approaches infinity.

Answer: 2


Question:
Given $a, b, c \in \mathbb{R}$ with $0 \leq a, b, c \leq 1$, determine the maximum value of the expression:
\[ \frac{a+b+c}{3} + \sqrt{a(1-a) + b(1-b) + c(1-c)}. \]

Answer: \dfrac{3}{2}


Question: Determine the degree \([K_p : \mathbb{Q}_p]\) for the number field \(K = \mathbb{Q}(\theta)\) with \([K : \mathbb{Q}] = n\), where \(\mathbb{Q}_p\) denotes the field of \(p\)-adic numbers and \(K_p = \mathbb{Q}_p(\theta)\).

Answer: n


Question: Given two independent random variables $X$ and $Y$ with moment-generating functions (mgfs) $M_{X}(t) = (1-t)^{-1}$ for $t<1$ and $M_{Y}(t) = (1-2t)^{-1}$ for $t<0.5$, respectively. Let $U = \frac{1}{2}(X+Y)$ and $V = \frac{1}{2}(X-Y)$. Determine the covariance, $\text{Cov}(U,V)$. Use the relationships $U+V = X$ and $U-V = Y$ if they simplify the calculation.

Answer: -\dfrac{3}{4}


Question: Given the quadratic equation \(x^2 - 14x + m = 0\), where \(m\) is a positive integer, and the roots \(p_1\) and \(p_2\) are distinct, positive, prime integers, find the value of \(K = (p_1 + p_2)^2 + 2p_1p_2\).

Answer: 262


Question: Find the largest possible value of $f(4)$ if the continuously differentiable function $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfies $f(0) = 0$ and $f(x)f'(x) \leq 2$ for all $x \in \mathbb{R}$. The function is defined on the entire real line.

Answer: 4


Question: Consider a sequence of positive real numbers \((b_n)_{n\in\mathbb{N}}\) that is strictly decreasing and converges to 0. If \(\displaystyle\lim_{n\to\infty}\frac{b_{n}}{b_{n+1}}=1,\) is the set \(Y = \left\{ \frac{b_i}{b_j}: i,j\in\mathbb{N} \right\}\) dense in \([1,\infty)?\) Answer yes or no.

Answer: Yes


Question: Determine the support of the Dirichlet function, defined as follows: if $x$ is rational, $f(x)=0$; if $x$ is irrational, $f(x)=1$. What is the support of this function?

Answer: \mathbb{R}


Question: Determine whether the following statement is true or false: If a group $G$ has order divisible by two distinct primes $p$ and $q$, then $G$ must have an element of order $pq$. Provide a justification for your answer.

Answer: False


Question: Calculate the volume of the subset \( \left\{ \begin{pmatrix} 2 & 33 & 444\\ 0 & 3 & 4 \\ 0 & 0 & 4 \end{pmatrix}\begin{pmatrix}x \\y \\ z \end{pmatrix}:x^2+y^2+z^2 \leq 1\right\} \) of \( \mathbb{R}^3 \).

Answer: 32\pi


Question: Suppose that a real function $f$ has a limit at every point in a set $K \subset \mathbb{R}$, and we define $g(x) = \lim_{t \to x} f(t)$. Is $g$ necessarily continuous on $K$? Provide a justification for your answer.

Answer: Yes


Question: Compute the flux \( \iint_{\partial E} \vec F \cdot d\vec S \) for the vector field \( \vec F = (x y^2 + \cos(yz))\hat i - (x^2 + \sin(zx))\hat j + (z + \cos(x y))\hat k \), where \( \partial E \) is the outward-oriented boundary of the solid \( E \), which is the intersection of the cylinders \( x^2 + y^2 \leq 1 \) and \( y^2 + z^2 \leq 1 \).

Answer: \dfrac{32}{5}


Question: Find the remainder when \(34^{45^{56}}\) is divided by 13.

Answer: 8


Question: Determine the value of $t > 0$ that minimizes the arc length of the curve $y = x^3 + x^2 - \frac{29x}{2} + 1$ over the interval $x \in [t, t+1]$. Use the arc length formula $L = \int_{t}^{t+1} \sqrt{1 + [f'(x)]^2} \, dx$.

Answer: \dfrac{4}{3}


Question: Evaluate the integral \( \int_{0}^{2\pi} \sin\left(\frac{\sqrt{2} \sin(x)}{3}\right) \cos\left(\frac{\cos(x)}{3}\right)\,\mathrm{d}x \) using Bessel functions.

Answer: 0


Question: In a soccer match between Team A and Team B, the final score is a 9-9 tie. During the match, Team A had the lead at one point, and later, Team B also had the lead. How many different sequences of 18 goals can represent the course of the match, considering the order in which the goals were scored?

Answer: 38896


Question: Determine the values of $t$ for which the set $A = \{ (x_1, x_2, x_3, x_4) \mid 3x_1 - 2x_2 + x_3 + x_4 - t(x_1^2 + x_4^2) = 0 \}$ forms a subspace of $\mathbb{R}^4$, given that $x_1^2 + x_4^2$ is always non-negative.

Answer: 0


Question: Evaluate the expression \( \frac{1}{1+x^{b-a}+x^{c-a}} + \frac{1}{1+x^{a-b}+x^{c-b}} + \frac{1}{1+x^{b-c}+x^{a-c}} \) for positive real numbers \(x\) and rational numbers \(a, b, c\).

Answer: 1


Question: Given a real function $f$ that is continuously differentiable with $|f'(x)| \leq \frac{4}{5}$ for all $x$, determine whether there exists an $x$ such that $f(x) = x$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether there exists a point in the bar where the temperature remains constant at 5°C as time approaches infinity for the solution to the heat equation given by \[ u(x,t)=2+x+\sum_{n=1}^{\infty} c_{n}e^{-n^{2}\pi^{2}kt}\sin(n\pi x). \]

Answer: No


Question:
Calculate the Fourier coefficient $c_1$ for the function $f(x) = x \cos(x)$ on the interval $(-\pi, \pi)$. The Fourier coefficients are given by the formula:
\[ c_n = \frac{1}{2 \pi} \int_{-\pi}^{\pi} x \cos(x) e^{-inx} \, dx \]

Answer: \dfrac{i}{4}


Question: Calculate the limit: \( \lim _{ x\to0^+}{x\displaystyle\int _{x}^{1}\frac{\cos(t)}{t^2}}dt \).

Answer: 1


Question: If $f(x)$ is a Borel function, is $f(x+a)$ also a Borel function for any real number $a$? Answer yes or no.

Answer: Yes


Question: Determine if the Gaussian prime $\mathfrak{q} = 3 - 2i$ can be expressed as the sum of two squares in the ring of Gaussian integers $\mathbb{Z}[i]$. Provide your answer as either "Yes" or "No".

Answer: Yes


Question: Determine the values of $b$ for which the improper integral \( \int_{-r}^r \frac{\cos (\pi x)}{|x|^b+1} \, dx \) exists.

Answer: (-\infty, \infty)


Question: Determine the measure of the simplex \(\{(x_1, x_2, \ldots, x_d) \in \mathbb{R}^d : 0 < x_1 < x_2 < \ldots < x_d < 1\}\).

Answer: \dfrac{1}{d!}


Question: Determine whether the sequence $(Z_n)_{n\ge1}$, where $Z_n$ is the minimum of the first $n$ results of tosses of a balanced die, converges almost surely.

Answer: 1


Question: Evaluate the limit: $$\lim_{n \to \infty}\frac{\ln(1+n^4)-\ln(n^2)}{\sin^2(\frac{1}{n})}$$

Answer: \infty


Question: Given the group $G = \mathbb{Z}_6 \times \mathbb{Z}_3$, an element $a = (4, 1)$ in $G$, the subgroup $H = \langle a \rangle$, and the quotient group $G_1 = G/H$. Let $k$ denote the order of the coset of $(3, 1)$ in $G_1$, and let $\ell$ represent the order of the coset of $(2, 2)$ in $G_1$. Calculate $(k + \ell) \mod{5}$.

Answer: 2


Question: Evaluate \( \lim\limits_{x\to0}\ \left(\frac {\tan 2x} {x}\right)^\frac {1} {x^2}\ .\)

Answer: \infty


Question: Evaluate the limit: $$\lim_{h \to 0} \dfrac{\displaystyle\int_{1-h}^{1+h} f(x) \,dx}{h}$$ where $f$ is a continuous function defined on $\mathbb{R}$. What is the value of this limit?

Answer: 2f(1)


Question: Evaluate the integral \( \int_{-1}^{1} \frac{x^3\,e^{\sin \left( \cos \left( x \right) \right)}+1}{x^2+1}\mathrm{d}x \).

Answer: \dfrac{\pi}{2}


Question: Determine the prime ideals of the ring of integers \( \mathcal{O}_K \) of the number field \( K = \mathbb{Q}(\sqrt{5}) \) that lie over the prime ideal \( 7\mathbb{Z} \).

Answer: 7\mathcal{O}_K


Question: Given 3 cities, each pair of cities is connected by a road with an exponentially distributed cost with parameter 1. Calculate the expected value of the minimum cost to connect all cities, denoted as \( E(c_3) \).

Answer: \dfrac{7}{6}


Question: Express the empty set $∅$ using only the set $A$, the operation $*$, and parentheses if necessary.

Answer: A * A


Question: Find all natural number triplets \((a, b, c)\) such that \(a \leq b \leq c\) and \(\frac{3}{7} = \frac{1}{a} + \frac{1}{ab} + \frac{1}{abc}\).

Answer: (3, 4, 7)


Question: Determine the number of distinct ways to distribute 12 identical action figures among 5 children, ensuring that each child receives no more than 3 action figures, using generating functions.

Answer: 35


Question: Compute the conditional expectation \( E[X_{n+2}|F_n] \).

Answer: X_n


Question: Consider the sequence \(\{x_n\}_{n=1}^{\infty}\) recursively defined as follows: \(x_1 = x_2 = 1\), and for \(n > 2\), \[x_{n+1} = x_n - \frac{1}{n}x_{n-1}.\] Determine the sum of the infinite series: \[\sum_{i=1}^{\infty}x_i.\]

Answer: e


Question: Determine whether the following statement is true or false: If $f : [-\pi ,\pi] \rightarrow \mathbb{C}$ is a continuous function such that $\int_{-\pi}^{\pi}f(-t)dt=- \int_{-\pi}^{\pi}f(t)dt$, then $\int_{-\pi}^{\pi}f(t)dt=0$. Provide a justification for your answer.

Answer: True


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}(a_n)^{-1/n^2} \] where \( a_n = \sum_{k=1}^n \left(1 + \frac{k}{n^2} \right)^k \).

Answer: 1


Question: If $f(x)$ is a continuous function such that $f(x) + f\left(\frac{1}{2} + x\right) = 1$ for all $x \in \left[0, \frac{1}{2}\right]$, find the value of $\int_0^1 f(x) \, dx$. \( \text{Express your answer as a single number.} \)

Answer: \dfrac{1}{2}


Question: Determine the values of \(x\) for which the power series \(\sum \frac{(x-b)^n}{na^n}\), where \(a, b > 0\), is conditionally convergent. Express your answer in terms of \(a\) and \(b\).

Answer: b - a


Question: If $\tau_{1}$ and $\tau_{2}$ are two topologies on a set $\Omega$ such that $\tau_{1}$ is weaker than $\tau_{2}$ (i.e. $\tau_{1}\subset\tau_{2}$) and $\tau_{1}$ is metrizable, is $\tau_{2}$ also necessarily metrizable? Answer with 'Yes' or 'No'.

Answer: No


Question: Determine whether it is possible for an infinite group to have a finite number of generators, each of which has infinite order.

Answer: Yes


Question: Determine whether the function \( g(x) = \begin{cases} \arccos(1-x^2) & x \ge 0 \\ -\arccos(1-x^2) & x < 0 \end{cases} \) is analytic at \( x=0 \).

Answer: Yes


Question: In a group $G$, if $f$ is a class function on $G$, does the condition $f(g) = f(h)$ necessarily imply that $g$ and $h$ are in the same conjugacy class in $G$? Provide a justification for your answer.

Answer: No


Question: Calculate the limit: $$\lim_{x\to 0}\frac{\cos(\ln(x+1))-\ln(\cos(x)+1)}{\cos^4(\frac{x}{4})}$$

Answer: 1 - \ln 2


Question:
Let $\Omega = D(0,2)/\{\frac{1}{2}\}$, where $D(0,2)$ is a disc, and let $f$ be holomorphic in $\Omega$. Given that $\frac{1}{2}$ is a simple pole for $f$ with residue 1, calculate the limit:
\[ \lim_{n \to \infty} \frac{f^{(n)}(0)}{2^{n}n!} \]

Answer: -2


Question: Find the covariance of two random variables X and Z, where Z = X^3, given that X follows a standard normal distribution.

Answer: 3


Question: Find the maximum and minimum values of the function \( f(x,y,z) = x^4 + y^4 + z^4 \) subject to the constraints \( x^2 + y^2 + z^2 = 1 \) and \( x + y + z = 0 \) using the method of Lagrange multipliers.

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true or false: A Borel measurable function $f:\mathbb{R} \rightarrow \mathbb{R}$, for which $\int_{a}^{b}f(x)dx=0$ holds for every $a<b$ in $\mathbb{R}$, must satisfy $f(x)=0$ for every $x\in \mathbb{R}$. Provide a justification for your answer.

Answer: B


Question: For any set of $n \geq 4$ consecutive positive integers, determine if there are always three distinct elements in the set that are not divisible by an odd prime $p < n$. Answer yes or no.

Answer: No


Question: Does the requirement for a map and its inverse to be continuous in a homeomorphism imply that the map is bijective?

Answer: Yes


Question: Determine the number of $2 \times 2$ matrices $A$ with entries in the finite field $\mathbb{Z}/3\mathbb{Z}$ such that $A^{50} = I$, where $I$ is the identity matrix.

Answer: 14


Question: Calculate the work done by the vector field \( f(x,y,z) = (x,z,2y) \) along the curve formed by the intersection of the surfaces \( x^2 + y^2 = 1 \) and \( z = x^2 - y^2 \), traversed in the anti-clockwise direction as viewed from the point (0,0,100).

Answer: 0


Question: Given $m > n$, determine which is larger: $(m!)^n$ or $(n!)^m$?

Answer: (m!)^n


Question:
Determine whether the function \( g(x) \) defined on \([0,1]\) by
\[
g(x) = \begin{cases} \hfill a_n \hfill & \text{if $x$ is rational and $x = \frac{p}{q}$ with $p$ and $q$ relatively prime and $q = 2^n$} \\
\hfill 0 \hfill & \text{otherwise} \end{cases}
\]
is Riemann integrable. If it is, compute \( \int_0^1 g(x) \, dx \).

Answer: 0


Question: Find the restriction on the constant $a$ such that the function \( f(x)=\dfrac{x^2+2x+a}{x^2+4x+3a} \) is surjective from its domain to the set of real numbers \( \mathbb{R} \).

Answer: (0, 1)


Question: Suppose $V$ is a Hilbert space and $T: V \rightarrow V$ is a linear map such that there exists a constant $C$ satisfying \(|\langle T(v), w \rangle| \leq C\|v\| \|w\|\) for all $v, w \in V$. Is $T$ necessarily bounded?

Answer: Yes


Question:
Suppose $f$ is analytic on $\mathbb{D}$ with power series

$$f(z) = \sum_{n = 0}^\infty a_n z^n.$$

Let

$$f_n(z) = a_0 + a_1z + \ldots + a_nz^n.$$

If $f_n \to f$ uniformly on the closed unit disk, is the radius of convergence of the series strictly greater than $1$?

Answer: No


Question: Evaluate the limit: $$\lim_{n \to \infty} \int_{0}^{1} (n + 1)x^n f(x) \, dx$$ where $f$ is continuous on $[0,1]$ and it is given that $\int_{0}^{1} (n + 1)x^n \, dx = 1$. Determine the value of the limit.

Answer: f(1)


Question: Evaluate the expression \( I(a) = \lim_{\Omega \to \mathbb{R}^3} \, \int_\Omega \nabla \left( \frac{1}{|x-a|} - \frac{1}{|x|} \right) dV \), where \(x\) and \(a\) are vectors in \(\mathbb{R}^3\). Determine the value of \(I(a)\) or explain why the evaluation fails.

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


Question:
Given a continuous random variable $Z$ with a density function defined as follows:
$$ f_{Z}(z)=\left\{\begin{array}{ll}
2z & \text{if } 0 \leq z \leq 1 \\
0 & \text{otherwise}
\end{array}\right. $$

Let $X$ and $Y$ be two independent random variables, conditioned on $Z=z$, such that:
$$ \begin{array}{l}
\mathbf{P}(X=1 \mid Z=z) = z = 1 - \mathbf{P}(X=0 \mid Z=z) \\
\mathbf{P}(Y=1 \mid Z=z) = z = 1 - \mathbf{P}(Y=0 \mid Z=z)
\end{array} $$

Calculate the conditional probability:
$$ \mathbf{P}(Z \leq \frac{1}{2} \mid X=1 \text{ and } Y=1) $$

Answer: \dfrac{1}{16}


Question: Determine the number of integer values of $N$ such that $N(N-50)$ is a perfect square, given that the greatest common divisor of $N$ and $N-50$ is either $1$ or $50$. Provide your answer as a single integer.

Answer: 2


Question: Evaluate the limit: $$\lim_{t\to 0}\left(\frac{1}{\ln(1 + t)}+\frac{1}{\ln(1-t)}\right).$$

Answer: 1


Question: Evaluate the series \( \sum^{\infty}_{n=0}\frac{1}{(n+3)\cdot n!} \).

Answer: e - 2


Question: Calculate the directional derivative of the function \( f(x,y) = (x^2-2y^2)e^{x^2+y^2} \) at every point on the unit circle in the direction of the positively oriented tangent vector. Use the gradient and the tangent vector \( \vec{t} = (-\sin(t), \cos(t)) \) where \( x = \cos(t) \) and \( y = \sin(t) \). Find \( \nabla f \cdot \vec{t} \).

Answer: -3e \sin(2t)


Question:
Calculate the limit of the sequence as \( n \to \infty \):
\[ \lim_{n\rightarrow\infty}\left( \frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1) }{ 2\cdot4\cdot6\cdot\ldots\cdot(2n)}\right)^3 \]

Answer: 0


Question: For independent Gamma random variables $G_1, G_2 \sim \Gamma(n,1)$, if $\frac{G_1}{G_1+G_2}$ is independent of $G_1+G_2$, does this imply that $G_1+G_2$ is independent of $G_1-G_2$? Answer with 'Yes' or 'No'.

Answer: No


Question: Determine the number of points of discontinuity of the function \( f(x) = \int_0^x{t\sin{\frac1t}}dt \) for \( x \in (0,\pi) \).

Answer: 0


Question: Find the minimum value of the product $PA \cdot PB$, where $P(2,2)$ is a point inside the ellipse $\frac{x^2}{25}+\frac{y^2}{16} = 1$, and $A$ and $B$ are the points where a chord passing through $P$ intersects the ellipse.

Answer: \dfrac{236}{25}


Question:
Given the joint probability density function of random variables $X$ and $Y$:
\[ f_{X,Y}(x,y) = \begin{cases} \frac{1}{y}, & 0 < x < y, \ 0< y< 1 \\ 0, & \text{otherwise} \end{cases} \]
find the variance of $X$. \( \text{Var}(X) \)

Answer: \dfrac{7}{144}


Question: Calculate \( \lim_{n \rightarrow \infty} E[X_n] \) for the sequence defined as follows: \( X_1 \sim U[0,1] \), and if \( X_1 = a \), then \( X_2 \sim U[0,a] \). If \( X_2 = b \), then \( X_3 \sim U[0,b] \), and so on. Use properties of supermartingales to find the limit.

Answer: 0


Question: Evaluate the integral \( A = \int_{0}^{2\pi} \left| \sin\theta + \cos\theta \right| \mathrm{d}\theta \).

Answer: 4\sqrt{2}


Question: Determine whether a noncommutative ring can have a proper non-zero ideal consisting entirely of nilpotent elements.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to \infty} \frac{1}{n^2} \sum_{k=1}^n \frac{f(b+\frac{2k}{n^2}) -f(b)}{\frac{2k}{n^2}}$$ where $f$ is a differentiable function.

Answer: 0


Question: Evaluate the definite integral \( I = \int_2^{343} (x - \lfloor x \rfloor)^2 \, dx \), where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \).

Answer: \dfrac{341}{3}


Question: Determine the number of $k$-dimensional subspaces in an $n$-dimensional vector space over the finite field $\mathbb{F}_q$. Express your answer using the Gaussian binomial coefficient.

Answer: \dbinom{n}{k}_q


Question: Suppose we have a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $\lim_{x \to \infty}f(x)=0$ and $\lim_{x \to \infty}f(f(x))= \infty$. Determine whether $\lim_{x \to 0^+}f(x)=\infty$.

Answer: \infty


Question: Evaluate the limit: $$\lim\limits_{x\to 0}\frac{1}{2x}\int_{0}^{x}\left[1-\cos\left(\frac{2}{u}\right)\right]du$$

Answer: \dfrac{1}{2}


Question: Find the covariance function $C_X(s, t)$ for the random process $X(t) = \varepsilon_1 \cos(t) + \varepsilon_2 \sin(t)$, where $\varepsilon_1$ and $\varepsilon_2$ are independent and identically distributed standard normal random variables ($N(0, 1)$).

Answer: \cos(s - t)


Question: Evaluate the limit: $$ \lim_{n \to \infty} \left[1- p(n) \right]^{n^2} $$ given that $p(n) = o \left( \frac{\log n}{n} \right)$.

Answer: 0


Question: Evaluate the definite integral: $$\int_{\sqrt[3]{4}}^{\sqrt[3]{3+e}}x^2 \ln(x^3-3)\,dx.$$

Answer: \dfrac{1}{3}


Question: Suppose $G$ is a group and $M$ is a normal subgroup of $G$. Let $q: G \rightarrow G/M$ be the natural quotient map. If $\langle B^G \rangle$ is a non-abelian conjugate closure of a subset $B$ of $G$, is it possible for $\langle (q(B))^{G/M} \rangle$ to be abelian?

Answer: Yes


Question: Let $X_1$ and $X_2$ be independent random variables with distributions $P_1(x)$ and $P_2(x)$, respectively. Suppose the distribution of $X_1$ is replaced with a new distribution $\tilde{P}_1(x)$ such that $\mathbb{E}[\tilde{X}_1] = \mathbb{E}[X_1] + \epsilon$ for some $\epsilon > 0$. Does it follow that $\mathbb{E}[\max(X_1,X_2)]$ will also increase? Answer yes or no.

Answer: No


Question: Consider a zero vector of length $2n$. Randomly choose $k$ entries of this vector to be $1$, with the choice taken uniformly from all $\binom{2n}{k}$ possibilities. Let $S_1$ be the sum of the first half of the vector, and $S_2$ be the sum of the second half. Compute the probability that $\frac{|S_1-S_2|}{n} > \epsilon$ as $n$ and $k$ become large.

Answer: 0


Question: Find the exact value of the operator-norm of the Laplace transform, considered as a linear operator from $\mathcal{L}^{2}(\mathbb{R}^{+},\lambda)$ to $\mathcal{L}^{2}(\mathbb{R}^{+},\lambda)$. Provide your answer as a single value.

Answer: \sqrt{\pi}


Question: Find the limit of the sequence \(b_n=\sqrt[n!]{\frac{1}{3^{n!}}-\frac{1}{4^{n!}}}\) as \(n\) approaches infinity.

Answer: \dfrac{1}{3}


Question: Find the value of $z$ such that $\log(iz^2) = \frac{i\pi}{2} + 2\log(z)$. \(z\) is a complex number.

Answer: z \neq 0


Question: Determine the fundamental group of the space \(X = \left\{\left(\sqrt{x^2+y^2}-2\right)^2 + z^2 = 1\right\}\cup \left\{(x,y,0)\;\; :\;\; x^2 + y^2 \leq 9\right\}\subset\mathbb R^3\,\).

Answer: \mathbb{Z}


Question: Determine whether the function \( g(x) = b_1|x+2| + b_2|x-3| + \ldots + b_m|x-m+5| \) is a zero function, given that \( g(i) = 0 \) for all \( i \in [-2, m+3] \).

Answer: Yes


Question: Given a sequence of identically distributed continuous random variables $\xi_1, \xi_2, \ldots, \xi_n, \ldots$, where each variable has the same distribution, let $\nu$ denote the minimum index $k$ such that $\xi_{k-1} > \xi_k$. Find the expected value of $\nu$, denoted as $\mathbb{E}(\nu)$. \( \text{Express your answer in terms of known constants or functions.} \)

Answer: e


Question:
Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be an analytic function such that $f(0) = 0$ and $f(1) = 2$. Determine which of the following statements is correct:

(a) There exists a sequence $\{z_n\}$ such that $|z_n| > n$ and $|f(z_n)| > n$.
(b) There exists a sequence $\{z_n\}$ such that $|z_n| > n$ and $|f(z_n)| < n$.
(c) There exists a bounded sequence $\{z_n\}$ such that $|f(z_n)| > n$.
(d) There exists a sequence $\{z_n\}$ such that $z_n \rightarrow 0$ and $f(z_n) \rightarrow 2$.

Choose the correct option.

Answer: a


Question: A machine produces a weighted coin that lands on heads with an unknown probability $p$, where $P(p \le x) = x^4$. You flip the coin 5 times, and it lands on heads each time. What is the probability that the next flip will also land on heads?

Answer: \dfrac{9}{10}


Question: Calculate the expected value of the length $n$ for a sequence of positive integers $(a_1, a_2, \ldots, a_n)$ such that the sum of the sequence is $2023$. Given that the number of solutions to the equation $a_1 + a_2 + \ldots + a_n = 2023$ with positive integers is $\binom{2022}{n-1}$, and the total number of possible sequences with any length is $2^{2022}$, find the expected value of $n$. Provide your answer as a single number.

Answer: 1012


Question: Provide an example of a function $f(x)$ whose derivative $\frac{df}{dx}$ is periodic, but the function $f(x)$ itself is not periodic.

Answer: x + \sin(x)


Question: Find the value of $b+d$ given that \[\int\sqrt{\frac{x}{a^3-x^3}}\,dx = \frac{d}{b}\sin^{-1}\left(\frac{x^{3/2}}{a^{3/2}}\right) + C,\] where $b$ and $d$ are coprime integers.

Answer: 5


Question: Evaluate the limit: $$\lim_{n\to \infty } \frac {2^3+5^3+8^3+...+(3n+2)^3}{[2+5+8+...+(3n+2)]^2}$$

Answer: 3


Question: Evaluate the integral \( \int_{|z| = 4} \frac{\cos z}{z^2 - \pi^2} \, dz \) using the Cauchy integral formula.

Answer: 0


Question: Derive the expression for $e^{-4y}$ as a function of $w$ from the following system of equations: \[ e^{-2y} \cos(2x)= \frac{1-w^2}{1+w^2} \quad \text{and} \quad e^{-2y} \sin(2x)= \frac{2w}{1+w^2}. \]

Answer: 1


Question: Evaluate the surface integral \( \iint_{S} \text{curl} \vec{F} \cdot d\vec{S} \), where \( \vec{F} = \begin{pmatrix} xyz \\ x \\ e^{xy}\cos (z)\end{pmatrix} \) and \( S \) is the upper half of the sphere \( x^2+y^2+z^2 =1 \) with \( z \geq 0 \), and the normal vector is oriented outwards.

Answer: \pi


Question: For $n \geq 3$, determine how the degree $m$ map $\delta_m^{(n)} \colon S^n \to S^n$ acts on the generator $\eta_n$ of the homotopy group $\pi_{n+1}S^n$. Provide your answer in terms of $m$ and $\eta_n$.

Answer: m \eta_n


Question: Calculate the present value of a continuously varying, continuously paid annuity with payments $(8k + tk)$ made to an account, where $0 \leq t \leq 10$. The interest is credited at a force of interest $\delta_t = \frac{1}{8+t}$. After 10 years, the account's value is $20,000$. Find the value of $k$.

Answer: \dfrac{1000}{9}


Question: Consider a measure space $(\Omega,\Sigma,\mu)$ where $\Omega$ is an unbounded subset of $\mathbb{R}^n$. Determine whether there exists a non-zero linear operator $P$ on $L^1(\mu)$ that preserves positivity, is norm-decreasing, but does not satisfy the inequality $\alpha \|f\| \leq \|Pf\|$ for any $\alpha \in (0,1]$. Provide a justification for your answer.

Answer: Yes


Question:
Let $A$ be a skew-symmetric $n \times n$ matrix with coefficients in $\mathbb{R}$ or $\mathbb{C}$. Consider the $2n \times 2n$ matrix:

$$M = \begin{pmatrix}0&A\\-A&0 \end{pmatrix}.$$

Determine if the signature of $M$ is zero, meaning the number of positive eigenvalues of $M$ is equal to the number of negative eigenvalues.

Answer: 0


Question: Let $B$ be a nilpotent $n \times n$ matrix with complex entries, such that $B^m = 0$ for some integer $m$. Define $A = B - I$, where $I$ is the identity matrix. Find the determinant of $A$. Justify your steps in the transformation of the determinant.

Answer: (-1)^n


Question: Evaluate the integral \( I = \frac{1}{2\pi i}\oint_{|z+1|=2}\frac{\zeta'(z)+1}{\zeta(z)+z}dz-1 \), where \( \zeta \) is the Riemann Zeta function, and determine if \( I = 1 \).

Answer: 1


Question: Calculate the value of the sum \(S=\sum_{i=1}^{4}\frac{1}{\alpha_i^4-15\alpha_i^2-10\alpha_i+24}\), where \(\alpha_1,\alpha_2,\alpha_3,\alpha_4\) are the roots of the polynomial \(p(x)=x^4+4x^3+x^2-6x-1=0\), expressed in the form \(\frac{m}{n}\) with \(\text{GCD}(m,n)=1\) and \(m,n\in\mathbb{N}\). What is the value of \(m+n\)?

Answer: 715


Question: Determine the conditions on the complex numbers $a$, $b$, and $c$ such that the matrix \(\begin{bmatrix}2&0&0\\a&2&0\\b&c&-1\end{bmatrix}\) has a minimal polynomial of \((x-2)(x+1)\).

Answer: a = 0


Question: Determine how the prime number 2 splits in the ring of integers of the splitting field of the polynomial $f(X) = X^3 - 3X + 1$ over $\mathbb{Q}$.

Answer: (2)


Question: Let $\mathcal{H}$ be a Hilbert space. Suppose that $T$ and $S$ are closed range operators such that the subspaces $T\mathcal{H}$ and $S\mathcal{H}$ are orthogonal. Is the sum operator $T+S$ necessarily a closed range operator? Provide a yes or no answer.

Answer: Yes


Question: Find the sum of the squares of the real solutions to the equation \( \sqrt[3]{5x - 2} = \frac{1}{5}(x^3 + 2) \).

Answer: 10


Question: Simplify the integral \( \lim_{n\to\infty}\int_{\left(0,\frac{n}{3}\right)}x^3e^{2x}\left(1-\frac{3x}{n}\right)^nd\lambda(x) \).

Answer: 6


Question: Determine the number of zeros of the function \( f(z) = 10 \sin(z^{2019}) - z \) within the unit disk \( D = \{ z \in \mathbb{C} : |z| < 1 \} \).

Answer: 2019


Question: Calculate the limit: $$\lim_{x \to 0^+} \left( e^\frac{1}{\sin(x)} - e^\frac{1}{x} \right).$$

Answer: \infty


Question: Determine whether every group of even order has a normal subgroup of index 2.

Answer: No


Question: Evaluate the limit \( \lim_{n \to \infty} n^2 \int_0^1 \frac{dx}{(1+x^2)^n} \).

Answer: \infty


Question: Evaluate the integral \( \int_D{\omega} \) where \( \omega=\frac{dx \wedge dy}{(x-y)^2} \) and \( D(u,v)=(u+v,u-v) : I_2 \rightarrow \mathbb{R}^2 \).

Answer: -\infty


Question: In a non-compact topological space X, is it possible for the intersection of a decreasing sequence of closed and connected subsets to be disconnected?

Answer: Yes


Question: Given that the quadratic equations $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$ have a common root and $a \neq 0$, determine the value of \( \frac{a^3 + b^3 + c^3}{abc} \).

Answer: 3


Question: Evaluate the limit: $$\lim_{x \rightarrow\infty}\tan\left(\frac{\pi x}{2x+1}\right)^{\frac{1}{x}} $$

Answer: 1


Question: Given a polynomial $P(x)$ with real coefficients that satisfies the equation $P(P(x) - 2) = 4P(x) + 3$, find the value of $P(-1)$. Provide your answer as a single number.

Answer: 7


Question: Evaluate the limit \( \lim_{n\rightarrow \infty} \int\limits_B \sin(nxy) \, d\lambda_2 \) for any Borel set \( B \subset \mathbb{R}^2 \) with finite Lebesgue measure.

Answer: 0


Question: Determine whether the congruence $x^3 \equiv 5 \pmod{23}$ has a solution.

Answer: 19


Question: Determine whether the expression \( \frac{f(x_{0} + h ) - 2 f(x_{0} + h/2 ) +f(x_{0} )}{h^2 / 4} - f^{(2)}(x_{0}) \) is \( O(h^2) \).

Answer: O(h)


Question: Evaluate the series \( \sum_{n\geq 1} \frac{2^n \cdot n^2}{n!} \) when \( x=2 \).

Answer: 6e^2


Question: Evaluate \( \lim\limits_{n\to\infty} \left(\frac{n^n}{n!}\right)^{1/n} \) using the root test.

Answer: e


Question: Is $\mathbb{R}^m$ an abstract $m$-dimensional $C^k$ manifold?

Answer: Yes


Question: Triangle \(\Delta ABC\) is inscribed in a circle of radius one unit. The internal angle bisectors of angles \(\angle A, \angle B, \angle C\) meet the circle at the points \(A_1, B_1, C_1\) respectively. Find the value of \[ S = \frac{\sum AA_1 \cos\left(\frac{A}{2}\right)}{\sum \sin A} \]

Answer: 2


Question: Given a matrix $A$ with $m$ rows and $n$ columns, where $m \leq n$, each row of $A$ has a norm of $1$, and any two rows of $A$ are orthogonal. Prove or disprove that the sum of the squares of the minors of order $m$ of matrix $A$ is equal to $1$.

Answer: 1


Question: Calculate the flux of the vector field $\mathbf{F} = x\hat{i} + y\hat{j} + z\hat{k}$ through a sphere of radius $a$ centered at the origin, with the outward normal vector $\mathbf{n}$. Assume $\mathbf{n} = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{a}$. What is the flux?

Answer: 4\pi a^3


Question: Evaluate the double integral: $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_0^{a\sin\theta} r^2 \, dr \, d\theta$$

Answer: \dfrac{2a^3}{9}


Question: Find the coefficient of \(x^{12}\) in the expansion of the function \( f(x) = \frac{x^{2}+x+2}{(1-x^{2})^{3}} \).

Answer: 77


Question: Find the remainder when $70!$ is divided by $73$ using a method other than Wilson's Theorem.

Answer: 36


Question: Solve the differential equation \( xy'' + y' = -1 \) with the boundary conditions \( y(0) = 1 \) and \( y(1) = 0 \).

Answer: 1 - x


Question: Determine whether it is possible to construct an injective mapping from \(\mathbb{Z}^4\) to \(\mathbb{Z}\) that satisfies similar efficiency constraints as a known mapping for \(\mathbb{Z}^3\).

Answer: Yes


Question: Suppose \((X,Y) \sim BN(\mu_x=0, \mu_y=0, \sigma_x^2=1, \sigma_y^2=1, \rho=0.6)\). Find the value of \(c\) such that the random variables \(2X + 3Y\) and \(6X + cY\) are independent.

Answer: -\dfrac{38}{7}


Question: Given that $|r(0)| = 0$ and $(r \cdot \dot{r})(t) = 6t^2$ for all $t$, find the value of $|r(1)|$.

Answer: 2


Question: Calculate the gradient with respect to the matrix $\pmb{A}$ of the function $\log(\det(\pmb{A}\pmb{X}))$, where both $\pmb{X}$ and $\pmb{A}$ are positive definite matrices.

Answer: \pmb{A}^{-1}


Question: Determine whether there exists a function \( g : [-2, 2] \to \mathbb{R} \) that is continuous on \([-2, 2]\), differentiable on \((-2, 2)\), but whose derivative \( g' \) is not continuous on \([-2, 2]\).

Answer: Yes


Question: Find the value of \( \lim \limits_{n \to \infty} \int_{0}^{\pi} \frac{\sin 2nx}{2nx} dx \) given that \( \lim \limits_{n \to \infty} \int_{0}^{n\pi} {\sin x\over x}dx = \frac{\pi}{2} \).

Answer: 0


Question: Suppose you have two infinite cubic planar graphs, $G$ and $G'$. Both graphs are asymmetric and have the property that every possible combination of faces exists somewhere in the graph. Are $G$ and $G'$ necessarily isomorphic?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\rightarrow\infty} \frac{1}{n}\Big[\log\Big( \frac{\sqrt{2\pi n} (x-q) - q(1-q) e^{-\frac{n}{2}(\frac{x-q}{q(1-q)})^2}}{\sqrt{2\pi n} (x-q)} \Big)\Big]$$ assuming $x\ne q$ and $q(q-1)\ne 0$.

Answer: 0


Question: Evaluate the contour integral \( W = \int_{\frac{\pi}{2} + i \log(2)}^{\frac{\pi}{2} + i \log(5)} \frac{\mathrm{d}z}{g(z)} \) where \( g(z) = \cos(z) + i \sin(z) \).

Answer: 3


Question: Let $G$ be an abelian group and $M$ be a sequence of $m$ elements from $G$. If $S$ and $F$ are subsets of $G$ with $|S|=n$ and $|F|=n-m$, determine if it is possible to find a sequence of elements from $S$ such that the sum of each element in $M$ with the corresponding element in the sequence is distinct and not in $F$. Answer with 'yes' or 'no'.

Answer: yes


Question: Given that $T \sim t_7$, find the value of $t$ such that $$P(-t < T) + P(0 < T) + P(t < T) + P(2t < T) = 1.75.$$

Answer: 0.356


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

Answer: 2


Question: Evaluate the triple integral \( \iiint_{K}\left(x^{2}-z^{2}\right) d x \,d y \,d z \), where the region \( K \) is defined by \( x \geq 0 \), \( y \geq 0 \), \( z \geq 0 \), and \( x+y+z \leq 1 \).

Answer: 0


Question: Calculate \(5^{123} \pmod{1021}\) without using a calculator.

Answer: 657


Question: Determine whether it is possible to partition an uncountable totally ordered set into two uncountable subsets, without assuming completeness.

Answer: Yes


Question: Is the quotient space of any topological space by the equivalence relation that identifies points in the same connected component always totally disconnected? Provide a justification for your answer.

Answer: No


Question: Determine the smallest set of points in a finite projective geometry $PG(d,q)$ that intersects all lines.

Answer: \frac{q^d - 1}{q - 1}


Question:
Determine the continuity of the function \( F(z) \) on the closed unit disk \( \bar{D}(0,1) \), where \( F(z) \) is defined as follows:

\[
F(z) = \begin{cases}
f(z) & \text{if } |z|=1 \\
\frac{1}{2\pi i} \oint_\gamma \frac{f(\zeta)}{\zeta -z} d\zeta & \text{if } |z| <1
\end{cases}
\]

with \( \gamma \) representing the unit circle traced counterclockwise. Specifically, evaluate the limit:

\[
\lim_{z\rightarrow 1} \frac{1}{2\pi i} \oint_{\gamma} \frac{\bar{\zeta}}{\zeta(\zeta-z)} d \zeta
\]

Does this limit vanish as \( z \) approaches 1?

Answer: 0


Question: Calculate the limit: $$\lim_{n \to \infty} \left( \frac{3^{3n}(n!)^3}{(3n)!}\right)^{1/n}$$

Answer: 1


Question: Calculate the covariance of the random walks \( S_t \) and \( S_u \), where \( S_t = Y_1 + \ldots + Y_t \) and \( S_u = Y_1 + \ldots + Y_u \). The variables \( Y_1, \ldots, Y_u \) are independent and identically distributed with \( E(Y_k) = \nu \) and \( \text{Var}(Y_k) = \gamma^2 \). Assume \( t < u \).

Answer: t \gamma^2


Question: Compute the degree of the field extension \([\mathbb{Q}(\sqrt{5},\sqrt[3]{2}):\mathbb{Q}]\). Using the Tower Law, express this degree as \([\mathbb{Q}(\sqrt{5},\sqrt[3]{2}):\mathbb{Q}(\sqrt{5})][\mathbb{Q}(\sqrt{5}):\mathbb{Q}]\). Determine the value of \([\mathbb{Q}(\sqrt{5},\sqrt[3]{2}):\mathbb{Q}(\sqrt{5})]\).

Answer: 3


Question: Determine whether there exist functions that are differentiable at a point but not continuous in any neighborhood of that point, even in higher dimensions (i.e., $\mathbb{R}^n$ with $n > 1$).

Answer: Yes


Question: Determine whether the following statement is true or false: Let $x^n - a$ be a polynomial over $\mathbb{Q}$, where $a$ is a positive rational number. If the constant term of any factor of $x^n - a$ in $\mathbb{Q}[x]$ has an irrational absolute value, does it imply that $x^n - a$ is irreducible over $\mathbb{Q}$?

Answer: True


Question: Let \( \{v_n\} \) be a sequence defined by \( v_0 = b \in [1,3) \) and \( v_n = \frac{v_{n-1}^3 - 1}{n^2} \) for all \( n \in \mathbb{N}^* \). Find \( \lim_{n \to +\infty} (v_n \sqrt[3]{n}) \).

Answer: 0


Question: Determine whether $M_a$, the set of continuous functions $f : [0,1] \to \mathbb{R}$ with $f(a) = 0$, is a maximal ideal in the ring of continuous functions $R$ on the interval $[0,1]$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: Let $G$ be a finite group with a normal subgroup $N$. If $G$ embeds into $G/G'\times Q_8$, $G/N=Q_8$, and $N$ is contained in the center of $G$, then $G$ is isomorphic to $N\times Q_8$. Provide a proof or counterexample.

Answer: True


Question: Is it possible to find a non-negative function $f(x)$ defined on the interval $[a, b]$ such that $f(x)$ is not identically zero on $[a, b]$ and $\int_{a}^{b} f(x) \, dx = 0$?

Answer: Yes


Question: Determine if the function $f(x) = x$ is Riemann-Stieltjes integrable with respect to $\alpha(x)$ on the interval $[-1, 1]$, where $\alpha(x) = \begin{cases} 0 & (x = -1) \\ 1 & (-1 < x \leq 0) \\ 3 & (0 < x < 1) \\ 5 & (x = 1) \end{cases}$. If it is integrable, calculate the integral $\int_{-1}^{1} f \, d\alpha$.

Answer: 1


Question: Consider the sequence defined by \( a_1 = e^i \) and \( a_{n+1} = a_n + e^{a_n} \) for all \( n \geq 1 \). Determine if the imaginary part of the sequence \( \Im(\lim_{n \to \infty} a_n) \) converges, and if so, find the value of this limit.

Answer: 3\pi


Question: Find all pairs of prime numbers \((p, q)\) such that \(p^2 - 2q^2 = 17\).

Answer: (5, 2)


Question: A mathematician is lost on the complex plane and wants to return to the main road, a strip of width 1 around the real axis (from \( \mathbb{R} + \frac{i}{2} \) to \( \mathbb{R} - \frac{i}{2} \)). He has a device that shows the absolute value \(|z|\) of his position \(z\), rounded to the nearest integer. The device displays the number 1000. Can he reach the main road in a finite number of steps, where each step consists of consulting the device, turning an angle \(\phi\), and walking a distance \(d\)? If yes, what is the minimum number of steps required?

Answer: 2


Question: Given a short exact sequence of $R$-modules, $0 \to M_1 \to M_2 \to M_3 \to 0$, if $M_2$ is finitely generated, does it imply that $M_1$ is also finitely generated? Answer yes or no.

Answer: No


Question: Let $X=(X, \|\cdot\|_X)$ and $Y=(Y, \|\cdot\|_Y)$ be Banach spaces such that $X$ is compactly embedded in $Y$. Let $A \subset X$ be a closed subspace of $X$. Is $A$ compactly embedded in $Y$?

Answer: Yes


Question: Find the volume bounded by the surface defined by the parametric equations: $$(a(t-\sin(t)),a(1-\cos(t))\cos(\theta),a(1-\cos(t))\sin(\theta))$$ where both $\theta$ and $t$ vary between $[0,2\pi]$. Express your answer in terms of $a$.

Answer: 5\pi^2 a^3


Question: In an imaginary quadratic field $K$, is a prime element of the ring of integers of $K$ necessarily an integer?

Answer: No


Question: Let $g(x)$ be a polynomial function such that $g(x)g(y) = g(x) + g(y) + g(xy) - 2$ for all real numbers $x$ and $y$. Given that $g(2) = 5$, determine the value of $g(3)$. Express your answer as a single number.

Answer: 10


Question: Determine the coefficient of $z$ in the Laurent series expansion of \( \frac{e^z}{z-1} \) for the region \( \{|z|>1\} \).

Answer: e - 2


Question: Construct a linear transformation from \( \mathbb{R} \) to \( \mathbb{R} \), viewed as vector spaces over \( \mathbb{Q} \), that is not the identity map.

Answer: T(x) = 2x


Question: Given a closed topological disk \( D \) in the plane, determine if it is always possible to find a straight line segment \( L \subseteq D \) with endpoints on \( \partial D \), and whose interior lies inside the interior of \( D \), such that \( L \) divides \( D \) into two parts of equal area. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Evaluate the integral: $$\int_{-1}^{1}\left[\,x^{2012}\sin\left(\,x\,\right) + \frac{1}{x^{2} + 1}\,\right]\mathrm{d}x$$.

Answer: \dfrac{\pi}{2}