Question: Determine whether the coefficient \( m_{a,b,c} \) of the monomial \( x_1^{a+c+1}x_2^{a+b+1}x_3^{b+c+1} \) in the expansion of the polynomial \( f_{a,b,c}(x_1,x_2,x_3) = (x_1-x_2)^{2a+1}(x_2-x_3)^{2b+1}(x_3-x_1)^{2c+1} \) is zero when \( a, b, c \) are pairwise unequal nonnegative integers.

Answer: 0


Question: Find the highest power of 2 that divides \(3^{2^k} - 1\). Prove that it is \(2^{k+2}\) and show that \(2^{k+3}\) does not divide \(3^{2^k} - 1\).

Answer: 2^{k+2}


Question: Let $a$ be an element in $GF(2^9)$ such that $a^9 + a^8 = -1$. Determine the order of the subgroup $H = \langle a \rangle$ in $GF(2^9)$. Provide your answer as a single integer.

Answer: 73


Question: For which primes \( p \) does the series \( \sum_{k = 0}^{\infty} \left(\frac{15}{7}\right)^k \) converge in the \( p \)-adic numbers \( \mathbb{Z}_p \)? Find the sum of the series in each case.

Answer: -\dfrac{7}{8}


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{1-e^{\sin x \ln \cos x}}{x^3}. \]

Answer: \dfrac{1}{2}


Question:
Find the smallest positive real number \( C \) such that for any positive integer \( k \), the following inequality holds:
\[ S = \sum_{i=k-\lfloor \frac{k-1}{2} \rfloor+1}^k \frac{1}{i} + \sum_{i=k-\lceil \frac{k-1}{2} \rceil+1}^k \frac{1}{i} < C. \]

Answer: 2\ln 2


Question: Evaluate the limit: \[ \lim_{x\to0} {\frac{\exp\left(-\frac{1}{x^4}\right)}{x}} \]

Answer: 0


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

Answer: \dfrac{3}{4}


Question: Determine the values of \(a > 0\) for which the expected value of the random variable \(X_1\), with distribution \(\mathbb{P}(X_k \geq x) = \frac{1}{x^a}\) for \(x \in \mathbb{N}\), is finite.

Answer: a > 1


Question: Let $M \in \mathbb{R}^{m\times m}$ be an invertible matrix, and let $c\in\mathbb{R}^{m}$ be a vector. Define $\Omega \triangleq \{x\in\mathbb{R}^{m}: M(x-c) \in \Delta\}$, where $\Delta$ is some region in $\mathbb{R}^{m}$. Let $a$ be the centroid of $\Omega$, and $b$ be the centroid of $\Delta$. Determine if it is true that $M(a-c)=b$. Provide a justification for your answer.

Answer: Yes


Question: Let $k$ be a field and $\Omega$ an algebraically closed extension of $k$. Let $X$ and $Y$ be schemes of finite type over $k$. Denote $\text{Hom}_k(\text{Spec}(\Omega), X)$ by $X(\Omega)$. For $\alpha, \beta \in X(\Omega)$, define $\alpha \sim \beta$ if $\alpha(\eta) = \beta(\eta)$, where $\eta$ is the unique point of $\text{Spec}(\Omega)$. Let $f: X \rightarrow Y$ be a morphism of schemes of finite type over $k$, inducing a map $\bar{f}: X(\Omega) \rightarrow Y(\Omega)$. Suppose $\bar{f}(\alpha) = \beta$ and $\beta \sim \beta'$. Does there exist $\alpha'$ such that $\alpha \sim \alpha'$ and $\bar{f}(\alpha') = \beta'$?

Answer: Yes


Question: In the ring $\mathbb{Z}_5[x]$, consider the ideal $I = \langle x^2 + x + 2 \rangle$. Find the inverse of the element $2x + 3 + I$ in the quotient ring $\mathbb{Z}_5[x]/I$. Provide the polynomial that represents this inverse.

Answer: 3x + 1


Question: Given a finite group $G$, is there an ordering of its elements $G = \{a_1, a_2, \ldots, a_n\}$ such that the product $a_1 \cdot a_2 \cdot \ldots \cdot a_n = a_1$? Provide a justification for your answer.

Answer: Yes


Question: Let $A$ and $B$ be differential modules with a differentiation homomorphism $d$ such that $d^2=0$. Suppose $g: A \to B$ is an epimorphism. Is it possible for the induced homomorphism $H(g): H(A) \to H(B)$, where $H(C) = Z(C)/B(C)$, to not be an epimorphism? Provide a justification for your answer.

Answer: Yes


Question: Determine whether there exists an infinite cardinal \( \kappa \) and a collection \( \mathcal{S} \subseteq \mathcal{P}(\kappa) \setminus \{\emptyset\} \) such that for every element \( x \in \kappa \), the set \( \{s \in \mathcal{S} : x \in s\} \) has cardinality \( \kappa \), each member of \( \mathcal{S} \) has cardinality \( \kappa \), and there exists a choice set \( C \subseteq \kappa \) for \( \mathcal{S} \).

Answer: Yes


Question: Let $a$, $b$, and $c$ be positive real numbers with $a < b < c$ such that $a + b + c = 12$, $a^2 + b^2 + c^2 = 50$, and $a^3 + b^3 + c^3 = 216$. Find the value of $a + 2b + 3c$.

Answer: 26


Question:
Find the limit of the function as \((x,y)\) approaches \((-1,0)\):
\[ \lim_{(x,y)\rightarrow (-1,0)} \frac{(xy - 2)(x-y+1)}{(x - y)(x+y) + 2y - 1} \]

Answer: 1


Question: Consider a formal power series $a(z)=\sum\limits_{n\geq 1} a_n z^n$ that satisfies an algebraic equation with polynomial coefficients, meaning there exists a nonzero polynomial $F(z,y)$ such that $F(z,a(z))=0$. Assume that a finite number of terms of $a(z)$ uniquely determine the entire series as a solution to this equation. Also, assume the sequence $\{a_n\}$ is increasing and grows polynomially, i.e., there exist constants $C$ and $d$ such that $a_n < Cn^d$ for all sufficiently large $n$. Is it true that the infimum of exponents $d_1$ such that $a_n < C_1 n^{d_1}$ for some constant $C_1$ and all sufficiently large $n$ coincides with the supremum of exponents $d_2$ such that $a_n > C_2 n^{d_2}$ for some constant $C_2$ and all sufficiently large $n$?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\to\infty}\left(\frac{\sqrt[n]{a}+\sqrt[n]{b}+\sqrt[n]{c}}{3}\right)^n \).

Answer: \sqrt[3]{abc}


Question: Find the degree of the splitting field of the polynomial \( p(x) = x^5 - 3x^3 + x^2 - 3 \) over \( \mathbb{Q} \).

Answer: 4


Question: Find the norm of the bounded linear operator defined by \( Ax(t) = \int_0^1 e^{ts} x(s) \, ds \), where \( x \in C[0,1] \) and \( t \in [0,1] \).

Answer: e - 1


Question: Evaluate the limit: \[ \lim_{x \to \infty}(1+\frac{4}{x})^{\sqrt{x^2+1}} \]

Answer: e^4


Question: Determine the number of elliptic fibrations on the K3 surface $X$, which is the double cover of $\mathbb{P}^2$ branched along the sextic $X^6+Y^6+Z^6-10(X^3Y^3+Y^3Z^3+X^3Z^3)=0$, with $M_9=Q_8\ltimes C_3^2$ acting symplectically on it.

Answer: 9


Question: Determine whether the inequality \(|e^{ix} - e^{iy}| \leq |x - y|\) holds for all real numbers \(x\) and \(y\).

Answer: Yes


Question: Find the minimum value of \(a\) such that \(x^a \geq \ln(x)\) for all \(x > 0\).

Answer: \dfrac{1}{e}


Question: Find the maximum value of \(|z|\) given that \(\left|z-\frac{4}{z}\right| = 8\).

Answer: 4 + 2\sqrt{5}


Question: Let $X_1$ and $X_2$ be standard normal random variables, and let $U$ be a continuous uniform random variable on the interval $[0,1]$. Define $Z = UX_1 + (1-U)X_2$. Find the variance of $Z$, $Var(Z)$. Express your answer as a single number.

Answer: \dfrac{2}{3}


Question: Calculate the expected order of an element chosen uniformly at random from the alternating group \(A_5\).

Answer: \dfrac{211}{60}


Question: An empty tank has a hole in it. Water enters the tank at a rate of 1 gallon per second. Water leaves the tank through the hole at a rate of 1 gallon per second for every 100 gallons in the tank. How long, in seconds, will it take to fill the tank with 50 gallons of water?

Answer: 100 \ln 2


Question:
Find the exact number of non-negative integer solutions to the following system of equations:
\[ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 6 \]
\[ 2x_1 + x_2 + x_3 = 4 \]
\[ x_2 + 2x_4 + x_5 = 4 \]
\[ x_3 + x_5 + 2x_6 = 4 \]

Answer: 15


Question: In a cyclic group of order 15, there is an element \(x\) such that the set \(\{x^3, x^5, x^9 \}\) contains exactly two distinct elements. Determine the number of distinct elements in the set \(\{x^{13n} : n \text{ is a positive integer} \}.\)

Answer: 3


Question: Let $\phi: \mathscr{F} \to \mathscr{G}$ be a morphism in the category of sheaves of rings on $X$, denoted as ${\bf Sh}(X,{\bf Ring})$. If $\phi$ is an epimorphism, is it true that $\mathscr{G}$ is the sheafification of the image presheaf of $\phi$?

Answer: Yes


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

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


Question: Find the remainder when \(2^{837259739} + 3^{23456}\) is divided by 31.

Answer: 3


Question: Find the smallest value of \(|20^m - 9^n|\) for \(m, n \in \mathbb{N}\).

Answer: 11


Question: Calculate the limit: \[ \lim_{n \to \infty}\sum_{k=1}^n \frac {(n+1)^k} {n^{k+1}} \]

Answer: e - 1


Question: Evaluate the double integral \( \int_{0}^{1}\int_{\arcsin(y)}^{\frac{\pi}{2}} e^{\cos(x)} \, dx \, dy \) by changing the order of integration using Fubini's theorem.

Answer: e - 1


Question: Find the limit: \[ \lim_{n\to\infty}(2-\sqrt[n]{2})^n \]

Answer: \dfrac{1}{2}


Question:
Solve the system of equations for real numbers $x$ and $y$:
\[ \begin{align*}
(\sqrt{x^2 + 1} + x)(\sqrt{y^2 + 1} - y) &= 1, \\
3\sqrt{x + 2y - 2} + x\sqrt{x - 2y + 6} &= 10.
\end{align*} \]

Answer: (2, 2)


Question:
Given the vectors \(A\), \(B\), and \(C\) such that:
\[ A \cdot A = 4, \quad B \cdot B = 4, \quad A \cdot B = 0 \]
\[ (A \times B) \times C = 0, \quad (A \times B) \cdot C = 8 \]
Find \(A \cdot C\). (Hint: Picture the vectors, and think geometrically. Use base, coordinate-free definitions. Avoid long calculations.)

Answer: 0


Question: Evaluate the line integral \( \int_{\Gamma} xy^2 \, dx + xy \, dy \) over the curve \( \Gamma = \{ (x, y) \in \mathbb{R}^2 : y = x^2, x \in [-1, 1] \} \) with a clockwise orientation using Green's Theorem. Close the curve by adding the line segment from \((1, 1)\) to \((-1, 1)\).

Answer: -\dfrac{4}{5}


Question: Evaluate the integral \( \int_{S_2} z^2 \, dS(x) \), where \( S_2 = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\} \) and the surface element is given by \( dS(x) = r^2 \sin \psi \, dr \, d\phi \, d\psi \). Use the parametrization \((0, \psi) \in (0, \pi) \times (0, 2\pi) \mapsto (\sin \phi \cos\psi, \sin\phi \sin \psi, \cos \phi)\).

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


Question: Determine whether the following statement is true: A topological space $X$ is KC if and only if every compact subset of $X$ is KC. Specifically, is it true that if every compact subset of $X$ is KC, then $X$ is KC?

Answer: True


Question: Evaluate the expression \(\delta_{ii}\delta_{jk}\delta_{jk}\) where \(\delta_{ii}\) is defined in \(\mathbb{R}^3\).

Answer: 9


Question: Let $V$ be a finite-dimensional inner product space and $T \in \mathcal{L}(V)$. Define $|T|=\sqrt{T^*T}$. Determine whether the equality $||T||=|T|$ holds, where $||T||$ is the composition of two $|\cdot|$ operations.

Answer: ||T|| = |T|


Question: Calculate the integral \( \int_{F} \! \text{rot} \ a \cdot \ dn \) for the vector field \( a = [-2x-z, -y+z, -2x+z]^T \), where the surface \( F \) is the sphere defined by \( x^2 + y^2 + z^2 = 9 \) and the ground plane \( z = 0 \).

Answer: 0


Question: Calculate the work done by the force field \( F(x,y,z) = \langle 3\sqrt{z}, -x, 4\sqrt{y} \rangle \) along the path composed of the line segments \( C_3 \) from the origin to \( (1,1,0) \) and \( C_4 \) from \( (1,1,0) \) to \( (1,1,1) \). Use the parametrizations \( r_3(t) = \langle t, t, 0 \rangle \) for \( C_3 \) and \( r_4(t) = \langle 1, 1, t \rangle \) for \( C_4 \). Compute the line integrals and find the total work done.

Answer: \dfrac{7}{2}


Question: Three customers, A, B, and C, arrive simultaneously at a bank with two tellers on duty. The tellers were idle when the customers arrived. Customer A goes to one teller, B goes to the other, and C waits until either A or B finishes before receiving service. If the service times are exponentially distributed with a mean of 4 minutes, what is the probability that customer A is still in the bank after customers B and C have left?

Answer: \dfrac{1}{4}


Question: Let $B$ be a finite Blaschke product of degree $n$. Suppose there exists an open subset $U$ of the unit disk $\mathbb{D}$ such that $B$ is univalent on $U$. Is it true that $n=1$?

Answer: No


Question: Determine the dimension of the vector space \( \mathbb{C}[[x,y]]/(2xy-y^2,x^2-2xy) \).

Answer: 4


Question:
Consider the function \( f(z) = \sum_{n \geq 0} nz^n \) defined on the open disk \(|z| < 1\). Choose the correct option regarding the properties of this function:

(a) \( f \) is not injective but attains every complex value at least once.
(b) \( f \) is injective but does not attain every complex value.
(c) \( f \) is injective and attains every complex value.
(d) None of the above.

Answer: B


Question:
Determine the polynomial $p(z)$ in the Hadamard product factorization of the function $e^z - 1$, given that the zeros are $\{2 \pi i n ,n\in \mathbb{Z}\}$, they are all simple, and the function is of order 1. The factorization is given by:
\[ e^z - 1 = e^{p(z)} z \prod_{n=1}^\infty \left(1 + \frac{z^2}{4n^2\pi^2}\right). \]

Answer: \dfrac{z}{2}


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that \( f(\sin(1/n)) = \cos(1/n) \) for \( n \in \mathbb{N} \). Given that the derivative \( f' \) exists at \( 0 \), find \( f'(0) \).

Answer: 0


Question: Determine whether the complement of a sequential-open set is sequential-closed.

Answer: Yes


Question: Find the volume of the region bounded by the paraboloid \(z = x^2 + y^2\) and the plane \(z = 4y\).

Answer: 8\pi


Question:
Determine the set of limit points of the sequence \(1, \frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8}, \frac{1}{16}, \frac{3}{16}, \ldots\). Choose from the following options:

(a) \([0,1]\)
(b) \((0,1]\)
(c) the set of all rational numbers in \([0,1]\)
(d) the set of all rational numbers in \([0,1]\) and of the form \(m/2^n\) where \(m\) and \(n\) are integers.

Answer: a


Question: Let $p$ be an odd prime number and $P(x) = a_0 + a_1 x + \dots + a_{p-1} x^{p-1}$ be a polynomial in $(\mathbb{Z}/p\mathbb{Z})[x]$ with $\deg(P) < p$. Determine whether it is true that if $a_{p-1} \neq 0$, then the polynomial function associated with $P$ is not a permutation of $\mathbb{Z}/p\mathbb{Z}$.

Answer: True


Question: Evaluate the sum: $$\sum_{n=0}^k \frac{(\frac{1}{2})_n}{n!} \frac{(\frac{1}{2})_{k-n}}{(k-n)!}.$$ Does it equal 1?

Answer: 1


Question:
Given the vector fields \( \vec{v} \) and \( \vec{c} \), simplify the volume integral:
\[ \iiint_V{\vec{v} \nabla \cdot \vec{c} \, \mathrm{d}V} \]
Assume that for every fixed control volume \( V \), the relation \( \iiint_V{\nabla \cdot \vec{c}} \, \mathrm{d}V \equiv 0 \) holds. Determine if the integral simplifies to zero.

Answer: 0


Question:
Given the recurrence relations:

\[ a_{2n} = a_{n} - 1 \]
\[ a_{2n+1} = 2a_{n} + 1 \]

and the condition \( a_{20} = 1 \), find the value of \( \sum_{n=1}^{63} a_n \).

Answer: 728


Question: Calculate the voltage difference \( V \) by evaluating the line integral \( V = \int_A^B E \cdot dL \) where the electric field \( E = y \hat{x} + x \hat{y} \), and the path is given by \( x = (1+y)^2 \). The coordinates of points \( A \) and \( B \) are \( A = (4,1,-2) \) and \( B = (9,2,-2) \). Express the integral in terms of a single variable and compute the voltage difference.

Answer: 14


Question: Find all pairs of natural numbers \((x, y)\) such that \(x^3 - 3^y = 2015\).

Answer: (14, 6)


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

Answer: \sqrt{e}


Question: Compute the asymptotic density of the set of prime powers less than or equal to \( x \), defined as \( \Pi(x) = \#\{p^k \leq x : p \text{ is prime}, k \in \mathbb{N}\} \). Find \( \lim_{x \to \infty} \frac{\Pi(x)}{x} \).

Answer: 0


Question: Given the equations \( \phi(f(x) - 1) = 2x + 5 \) and \( \phi(x) = 2f(x+1) + 1 \), find the value of \( f(4) \).

Answer: 5


Question: Evaluate the double integral \( \iint_D \left|x^2 + y^2 - 2y\right| \, d\sigma \), where \( D = \{(x, y) \mid x^2 + y^2 \le 4\} \).

Answer: 9\pi


Question: In \( \mathbb{R}^n \), is weak convergence equivalent to strong convergence? Provide a justification for your answer.

Answer: Yes


Question: Given a continuous random variable U that follows a uniform distribution on the interval (0,1), and a continuous random variable Y that, given U = u, follows an exponential distribution with parameter \( \mu \), find the survival function \( G_Y(y) = P(Y > y) \).

Answer: e^{-\mu y}


Question:
In a metric space, which of the following properties of a subset $A$ implies the other three?\
\(a)\) $A$ is closed\
\(b)\) $A$ is bounded\
\(c)\) The closure of $B$ is compact for every $B \subseteq A$\
\(d)\) $A$ is compact\
Determine the correct option.

Answer: D


Question: Let \(X_1, X_2, \ldots, X_n\) be i.i.d. random variables with \(X_k \sim \text{Bernoulli}(\theta)\) for \(\theta \in (0,1)\). Define \(Y = \sum_{k=1}^n X_k\). Determine \(E(X_k | Y)\).

Answer: \dfrac{Y}{n}


Question: Find the extremal of the functional \( \int_0^1 [(y')^2 + 12 xy] \, dx \) subject to the boundary conditions \( y(0) = 0 \) and \( y(1) = 1 \).

Answer: x^3


Question: Evaluate the limit: $$\lim_{x \to +\infty} \frac{\int_{0}^{x} t|\sin t| \, dt}{x^{2}}.$$

Answer: \dfrac{1}{\pi}


Question: Find the modulus of the complex number $z$ that lies in the region $|z-1| \leq |z-i|$ and $|z-(2+2i)| \leq 1$ for which $\arg(z)$ is least.

Answer: \sqrt{7}


Question: Determine if there exists a constant \( C > 0 \) such that the number of solutions to the congruence \( x^2 \equiv a \pmod{2^L} \), where \( a \) is an odd integer and \( L \geq 1 \), is at most \( C \) for any \( L \).

Answer: 4


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

Answer: \dfrac{n}{m}


Question: Let $H$ be the line at infinity in $\mathbb{CP}^{2}$, and let $P$ and $Q$ be distinct points on $H$. Let $X$ be the blow-up of $\mathbb{CP}^{2}$ at $P$ and $Q$; let $E_{1}$ and $E_{2}$ be the exceptional divisors over $P$ and $Q$, respectively, and let $L$ be the proper transform of $H$. What is the self-intersection number of $L$?

Answer: -1


Question: Determine the number of ordered triples of positive integers \((a, b, c)\) such that \(a \leq b \leq c\) and \(abc = 2014^3\).

Answer: 171


Question: Calculate the residue of the function \( f(z) = \frac{(z^2+1)^5(z^{10}+1)}{64z^{11}i} \) at \( z_0 = 0 \).

Answer: -\dfrac{i}{32}


Question: Let \(x_n\) be a sequence of real numbers such that \( \lim_{n \to \infty}(x_n - x_{n+2})=0 \). Prove that \( \lim_{n \to \infty}\frac{x_n - x_{n+1}}{n}=0 \).

Answer: 0


Question: Let $X$ be the disjoint union of a circle $C$ and an interval $I$. Determine whether it is true that for every homeomorphism $f: X \to X$, $f(C) = C$ and $f(I) = I$. Provide a justification for your answer.

Answer: True


Question: Let $X$ and $Y$ be random variables with cumulative distribution functions $F_X(t)$ and $F_Y(t)$, respectively. Suppose that for all $t \in \mathbb{R}$, $F_X(t) \geq F_Y(t)$. Does this imply that $E(X) \leq E(Y)$? Justify your answer.

Answer: Yes


Question: Find the number of solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 20\) where each variable satisfies \(1 \leq x_i \leq 4\) for \(i = 1, 2, \ldots, 6\). Use the inclusion-exclusion principle to determine the number of solutions.

Answer: 120


Question:
Given a function of two variables \( f = f(x_0,x) \) and an unknown function of one variable \( w = w(x) \), they are related by the equation:
\[
L(x) = \int_{a}^{b} w(x_0) f(x_0,x) \, dx_0
\]
where \( a \) and \( b \) are constants. Find the formula for \( \frac{\partial L(x)}{\partial w(x)} \).

Answer: f(x, x)


Question: Compute the limit: \[ \lim_{x\to\infty}\left(\sqrt{x+\sqrt{x+\sqrt{x + \sqrt x} }}-\sqrt x\right) \]

Answer: \dfrac{1}{2}


Question: Calculate the line integral \( \int (10x^4 - 2xy^3) \, dx - 3x^2y^2 \, dy \) along the path defined by \( x^4 - 6xy^3 = 4y^2 \) from the point \( O(0,0) \) to \( A(2,1) \).

Answer: 60


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}e^{\ln(1-\sin{\frac{1}{n}})^n} \]

Answer: 1


Question: Evaluate the line integral \( \int_{C} \frac{x \, dy - y \, dx}{x^{2}+y^{2}} \) along the oriented broken line \( C \) with vertices \((2,-2)\), \((4,4)\), \((-5,5)\) oriented counterclockwise.

Answer: \pi


Question:
Find the coefficient $a_{-2}$ in the Laurent series expansion \( \sum_{n=-\infty}^{\infty}a_{n}(z+1)^n \) of the function \( f(z)=\sin\left(\frac{z}{z+1}\right) \). Choose the correct answer from the following options:

(a) 1
(b) 0
(c) \( \cos(1) \)
(d) \( -\frac{1}{2}\sin(1) \)

Answer: D


Question: Let $X$ and $Y$ be two independent and identically distributed random variables, each following a uniform distribution on the interval $(0,1)$. Define $T = XY$. Calculate the probability distribution function of $T$. Provide your answer in terms of a function of $t$, where $0 \leq t \leq 1$.

Answer: -\ln t


Question: Let $p: \mathbb{R}^n \rightarrow \mathbb{R}$ be a polynomial with a non-empty zero set $S$. Determine whether it is true that for any $x, y$ in the same connected component $C$ of $S$, there exists a piecewise smooth path $\gamma: [0,1] \rightarrow C$ such that $\gamma(0) = x$ and $\gamma(1) = y$. Provide a justification for your answer.

Answer: True


Question: Does there exist a noncommutative abelian ring with nontrivial idempotents?

Answer: Yes


Question: Calculate \( \lim_{n \to \infty} n^\alpha \int^{\frac{1}{n^\beta}}_0 f(x)dx \) where \( \beta > 0 \) and \( \beta > \alpha \), for a function \( f \) integrable on \([0,1]\).

Answer: 0


Question: Given a graph $G = (W \cup U, E)$ where $W = \{w_1, w_2, ..., w_n\}$, $U = \{u_1, u_2, ..., u_n\}$, and $E = \{\{w_i, u_j\}: 1 \leq i, j \leq n\}$, calculate the number of Hamiltonian cycles in the graph $G$. The graph is a complete bipartite graph $K_{n,n}$. Determine the number of Hamiltonian cycles in $K_{n,n}$.

Answer: \dfrac{(n-1)! \cdot n!}{2}


Question: Is it always possible to color a finite set of grid points with two colors, red and blue, such that in each vertical and horizontal line, the number of red and blue points differ by at most one?

Answer: Yes


Question: Let \( X_1, X_2, \ldots, X_n \sim \mathcal{N}(1,1) \) be independent and identically distributed normal random variables. Determine the limit of \( \frac{\text{Var}(X_1^2 + X_2^2 + \cdots + X_n^2)}{n} \) as \( n \to \infty \).

Answer: 6


Question: Solve the congruence equation \(x^3 + x + 57 \equiv 0 \pmod{125}\).

Answer: 4


Question: Determine the radius of convergence \( R \) for the power series \( \sum_{n=1}^{\infty}(n^{\frac{1}{n}}-1)x^n \).

Answer: 1


Question: Let \((a,b) \in \mathbb{Z}^2\) such that \(a+b\) is a solution of the equation \(x^2+ax+b=0\). Find the smallest value of the product \(ab\).

Answer: -54


Question: Evaluate the integral: $$ \int^{\pi}_{0} \frac{\sin(nx)\cos\left ( \frac{x}{2} \right )}{\sin \left ( \frac{x}{2} \right ) } \, dx $$

Answer: \pi


Question: Find the singular solution of the differential equation given by \( 8ap^3 = 27y \), where \( p = \frac{dy}{dx} \).

Answer: y = 0


Question: Determine the positive integer that cannot be expressed in the form \(x^2 + 2y^2 + 5z^2 + 5w^2\).

Answer: 15


Question: Find the residue and the Laurent series expansion of the function \( w = \sin(z) \cdot \sin\left(\frac{1}{z}\right) \) at the point \( z_0 = 0 \). Determine the coefficient of the \( z^{-1} \) term in the expansion, which represents the residue at this point.

Answer: 0


Question: Determine whether the following statement is true or false: If $G$ is an infinite group, then $G$ must have an element of infinite order.

Answer: B


Question: Determine the largest set \( D \) such that the series \( \sum_{n=1}^{\infty} \frac{1}{1+nx^4} \) converges pointwise for all \( x \in D \).

Answer: \emptyset


Question: Determine whether the following statement is true or false: Let $X$ be a compact Hausdorff space and an infinite set. If $X$ has a topology strictly weaker than the discrete topology, then the compact convergence topology is strictly finer than the pointwise convergence topology.

Answer: A


Question: Determine the value of \( \alpha \) for which Rolle's Theorem is applicable to the function \( f(x) = x^{\alpha} \cdot \ln(x) \) on the interval \([0, 1]\), given that \( f(0) = 0 \).

Answer: \alpha > 0


Question: Given the function \( y = e^x - 2e^{-x} \), find the value of \( (f^{-1})'(1) \).

Answer: \dfrac{1}{3}


Question: For an $n \times n$ square matrix $A$, express the determinant of its cofactor matrix in terms of $\det(A)$ and $n$. What is the determinant of the cofactor matrix of $A$?

Answer: (\det(A))^{n-1}


Question: Evaluate the integral \( \int_{C}\frac{\cos(z)}{\sin(z)}dz \), where \( C \) is a circle centered at the origin and there is no zero of \( \sin(z) \) inside \( C \) other than the point \((0,0)\).

Answer: 2\pi i


Question: Given a sample $X_1, ..., X_n$ from an exponential distribution $\mathcal{E}(\lambda)$ with $\lambda = 3$ and $n = 50$, find the value of $\alpha$ such that $P(\bar{X} \le 1) = \alpha$, where $\bar{X}$ is the sample mean.

Answer: 1


Question:
Determine the nature of the roots of the equation \(x^2-4qx+2q^2-r=0\). Given that \(\alpha\) and \(\beta\) are roots of \(x^2+px+q=0\) and \(\alpha^4\) and \(\beta^4\) are the roots of the equation \(x^2-rx+s=0\), the equation \(x^2-4qx+2q^2-r=0\) always has:

(A) two real roots
(B) two positive roots
(C) two negative roots
(D) one positive and one negative root

Answer: A


Question: Calculate the number of different spanning trees in the graph $G_{k,n}=([n],E)$, where $ij\in E\Leftrightarrow 0 <|i-j|\leq k$ for $i\neq j\in [n]$. Use Kirchoff's Matrix Tree theorem to find the number of spanning trees for $k=3$ and $n=5$. What is the number of spanning trees?

Answer: 75


Question:
Given the function \( f(x) = \lim\limits_{n \to \infty} n(x^{1/n} - 1) \), determine the value of \( f(xy) \). Choose the correct option:

A) \( f(x) + f(y) \)
B) \( \frac{f(x)}{f(y)} \)
C) \( f(x) - f(y) \)
D) \( f(x)f(y) \)

Answer: A


Question: The equation of the line joining the complex numbers \(-5 + 4i\) and \(7 + 2i\) can be expressed in the form \(az + b \overline{z} = 38\) for some complex numbers \(a\) and \(b\). Find the product \(ab\).

Answer: 37


Question: Let $n > 2$ and $\sigma^2 > 0$. In a math exam with $n$ participants, the scores $X_1, X_2, \ldots, X_n$ follow a normal distribution with mean $\mu_X$ and variance $\sigma^2$. The average score is given by \( \overline{X} = \frac{1}{n} \sum_{i=1}^n X_i \). For each $i = 1, \ldots, n$, find the covariance between $X_i - \overline{X}$ and $\overline{X}$. \[ \text{Cov}(X_i - \overline{X}, \overline{X}) = ? \]

Answer: 0


Question: Compute the limit: \[ \lim_{x\to 0} \frac{\sqrt{\cos x} - \sqrt[3]{\cos x}}{\sin^2x}. \] Do not use l'Hospital's rule.

Answer: -\dfrac{1}{12}


Question: Ben chooses random numbers from the set \( \{0, 1, \ldots, 9\} \) independently and with uniform probability, arranging them in a sequence in the order they are picked. What is the expected number of distinct numbers that appear between the first two occurrences of the number 0?

Answer: \dfrac{9}{2}


Question: Let $\omega$ be a $k$-form on an open set $A \subseteq \mathbb{R}^n$. If for any $k$-chain $c$ in $A$, the integral $\int_c \omega = 0$, is it true that $\omega = 0$?

Answer: Yes


Question: Find the limit of the expression \[ c = \frac{\sqrt{n-1}\Gamma\left(\frac{n-1}{2}\right)}{\sqrt{2}\Gamma\left(\frac{n}{2}\right)} \] as \( n \to \infty \).

Answer: 1


Question: Find \( \lim_{n\to\infty}\int_0^1 nf(x)e^{-nx} \, dx \) for a continuous function \( f: \mathbb{R} \to \mathbb{R} \).

Answer: f(0)


Question: Given the sets \(A = \{1, 2, 3, 4\}\) and \(B = \{1, 2, 3, 4, 5, 6, 7\}\), determine the number of one-to-one functions \(f: A \to B\) such that \(f(i) \neq i\) for all \(i \in A\).

Answer: 465


Question: Evaluate the limit \( \lim_{n \to \infty} \{(2+\sqrt{3})^n\} \), where \( \{\cdot\} \) denotes the fractional part of a number.

Answer: 1


Question: Find the minimum value of the expression \(x^{\ln(y)-\ln(z)}+y^{\ln(z)-\ln(x)}+z^{\ln(x)-\ln(y)}\) given that \(x, y, z > 0\).

Answer: 3


Question:
Determine which of the following expressions satisfies the equation \( \phi(x) = f(x) + \int_{0}^{x} \sin(x-t) \phi(t) \, dt \):

(A) \( \phi(x) = f(x) + \int_{0}^{x} (x-t) f(t) \, dt \)

(B) \( \phi(x) = f(x) - \int_{0}^{x} (x-t) f(t) \, dt \)

(C) \( \phi(x) = f(x) - \int_{0}^{x} \cos(x-t) f(t) \, dt \)

(D) \( \phi(x) = f(x) - \int_{0}^{x} \sin(x-t) f(t) \, dt \)

Answer: A


Question: Find the number of digits in the smallest positive integer \( s \) such that both the digit sum of \( s \) and the digit sum of \( s + 1 \) are divisible by 19.

Answer: 20


Question: Find the volume of the region bounded by the surfaces given by the equations \( x = 2y - y^2 - z^2 \) and \( 2y - x - 1 = 0 \). Use polar coordinates to express the limits of integration, where \( 0 \le r \le 1 \) and \( 0 \le \phi \le 2\pi \).

Answer: \dfrac{\pi}{2}


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

Answer: \sqrt{2\pi}


Question: Find an integer $x$ such that $x^{31} \equiv 2 \pmod{81}$. Note that $81$ is not a prime number, so Fermat's Little Theorem cannot be directly applied.

Answer: 47


Question: Evaluate the limit: \[ \lim_{R\rightarrow +\infty}\int_{-1}^{1}\frac{\sin(2\pi R t)}{t}dt. \]

Answer: \pi


Question: Compute the area between the curves \( y = \cos(\sin(x)) \) and \( y = \sin(\cos(x)) \) on the interval \( 0 \leq x \leq 2\pi \).

Answer: 2\pi J_0(1)


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

Answer: -\dfrac{1}{6}


Question:
Given the function \( f(x) = (x^2 + x + 1)(ax + b) \), find \( f'(3) \) given the following conditions:

1. \( \lim\limits_{x\to1}\frac{f(x)-f(1)}{x-1}=3 \)
2. \( \lim\limits_{x\to2}\frac{x^3-8}{f(x)-f(2)}=1 \)

Answer: 27


Question: Find the sum of the values of \(x\), \(y\), and \(z\) that maximize the expression \((1-x)(2-y)(3-z)(x+y/2+z/3)\), given that \(x < 1\), \(y < 2\), \(z < 3\), and \(x + y/2 + z/3 > 0\).

Answer: \dfrac{3}{2}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{n}{2^n}\sum_{k=1}^{n}\binom{n-1}{k-1}\{\sqrt{k^2+2k+2}\} \] where \( \{x\} \) denotes the fractional part of \( x \).

Answer: \dfrac{1}{2}


Question: Show that the extension \(\mathbb{Q}(\zeta_n)/\mathbb{Q}(\zeta_n+\zeta_n^{-1})\) has degree 2, where \(\zeta_n\) is a primitive \(n\)-th root of unity with \(n > 2\).

Answer: 2


Question: The equation \(3x^2+2xy+3y^2-18x-22y+50=0\) is transformed to \(4x^2+2y^2=1\) with respect to a rectangular axes through the point \((2,3)\) inclined to the original axes at an angle \(\theta\). Determine the value of \(\theta\).

Answer: \dfrac{\pi}{4}


Question: Calculate the surface area of the sphere defined by \( x^2 + y^2 + z^2 = R^2 \) using a surface integral. Verify that the surface area is \( 4\pi R^2 \).

Answer: 4\pi R^2


Question: Does there exist a bijective function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that is nowhere-continuous, assuming that both the domain and range have the standard topology generated by open intervals \((a, b) \subset \mathbb{R}\)?

Answer: Yes


Question: Given a parameterized curve \( \gamma : I \rightarrow \mathbb{R}^3 \) by arc length with curvature \( \kappa = 2 \) and torsion \( \tau = 3 \), find the magnitude of the derivative of the normal vector, \( \| N' \| \).

Answer: \sqrt{13}


Question: Let $y_1(t)$ and $y_2(t)$ be solutions of Bessel's equation given by $$t^2y'' + ty' + (t^2-n^2)y=0$$ on the interval $(0,\infty)$, with initial conditions $y_1(1)=1$, $y_1'(1)=0$, $y_2(1)=0$, and $y_2'(1)=1$. Compute the Wronskian $W[y_1,y_2](t)$.

Answer: \dfrac{1}{t}


Question:
Determine the limit function \(f(x)\) as \(n\) approaches infinity for the sequence of functions \(f_n(x)\) defined on the interval \([0, 3]\) by:
\[
f_n(x) = \begin{cases}
  n^2x & \text{if } 0 \leq x \leq \frac{1}{n} \\
  n(2-nx) & \text{if } \frac{1}{n} < x \leq \frac{2}{n} \\
  0 & \text{if } \frac{2}{n} < x \leq 3
\end{cases}
\]
Find \(f(x) = \lim_{n \to +\infty} f_n(x)\).

Answer: 0


Question:
Given the function \( A(x) = B(2x-1) - B(2x) \), where \( B(x) \) is a function such that \( \lim_{x \to \infty} B(x) = 0 \), evaluate the expression:
\[ \int_{b}^\infty A(x) \, dx - \frac{1}{2} \int_{2b-1}^{2b} B(x) \, dx \]
for \( b > 1 \). Show that the result is 0.

Answer: 0


Question: Let \(f\) be a holomorphic function from the upper half-plane (excluding the real line) to the unit disc, such that \(f(i) = 0\). Determine the maximum possible value of \(|f(2i)|\).

Answer: \dfrac{1}{3}


Question: Evaluate the line integral \( \int_{(1,0)}^{(2, 2)}{\frac{-y}{x^2 + y^2}dx + \frac{x}{x^2 + y^2}dy} \) along any path that does not pass through the point \((0,0)\).

Answer: \dfrac{\pi}{4}


Question: Find the limit: \[ \lim_{n \to \infty} \left(\frac{n^2+3}{n^2+5n-4}\right)^{2n} \]

Answer: e^{-10}


Question: If $g$ is uniformly continuous on the interval $(0,1)$, does $g(0)$ necessarily exist?

Answer: Yes


Question: Evaluate the limit of the sequence given by \( \lim_{n\to+\infty}\sqrt[n]{n}\cdot\sqrt[n+1]{n+1}\cdots\sqrt[2n]{2n} \).

Answer: \infty


Question: Determine whether the following statement is true or false: If $u \in H^1(\mathbb{R})$, then the difference quotient $(u(x+h)-u(x))/h$ converges to $u'(x)$ in $L^2(\mathbb{R})$ as $h \to 0$. Provide a justification for your answer.

Answer: True


Question: Evaluate the volume integral \( \iiint_ E (x+y) \, dV \) where \( E \) is the pyramid with vertices \((1,1,1), (0,0,0), (0,1,1), (0,1,0), (0,0,1)\).

Answer: \dfrac{7}{24}


Question: Find the area of an isosceles triangle with an inradius of \(\sqrt{3}\) and a largest angle of \(120^\circ\).

Answer: 12 + 7\sqrt{3}


Question: Compute \(\lim_{x\rightarrow \infty} x(\sqrt{x^2+1}-\sqrt[3]{x^3+1})\).

Answer: \dfrac{1}{2}


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

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


Question:
The function \( w(z) = -\left(\frac{1}{z} + bz\right), -1 < b < 1, \) maps \(|z| < 1\) onto which of the following regions?

(A) A half plane
(B) Exterior of the circle
(C) Exterior of an ellipse
(D) Interior of an ellipse

Select the correct option.

Answer: C


Question: Let $k$ be a field, and $A$ a local $k$-algebra with maximal ideal $\mathfrak{m}$. Suppose the residue field $A/\mathfrak{m}$ is isomorphic to $k$ as a ring. Is the map $k \to A \to A/\mathfrak{m}$ an isomorphism, implying that $A/\mathfrak{m}$ has dimension 1 over $k$ with respect to the algebra structure inherited from $A$?

Answer: Yes


Question: Determine whether the following statement is true or false: If $r$ is a primitive root modulo $p$, then $r+p$ is also a primitive root modulo $p$. Justify your answer.

Answer: True


Question: Determine the number of natural numbers \( n \) such that \( 0 \leq n \leq 10,000,000 \) and the sum of the digits of \( n \) is at most 34.

Answer: 6505246


Question: Determine whether the stalks of the presheaf defined by \( U \mapsto \prod_{z \in U} I_z \) are exactly the abelian group \( I_x \) at each point \( x \in X \).

Answer: No


Question: Let $u$ and $v$ be vectors in $\mathbb{R}^n$ such that $u^Tv=0$. Define the matrix $A=uv^T+vu^T \in \mathbb{R}^{n\times n}$. Find an expression for $||A||_2$, the 2-norm of the matrix $A$.

Answer: \|u\| \|v\|


Question: Minimize \(x + 2y\) subject to the constraints \(x^2 + y^2 \leq 1\) and \(3x + 4y \leq -5\).

Answer: -\dfrac{11}{5}


Question: Compute the boundary map $d_0$ for the Cech cohomology with twisted coefficients on the circle, given the open cover $U_0, U_1, U_2$ and restriction morphisms $\rho^i_{ij}:U_i \to U_{ij}$ with $\rho^2_{02} = -1$ and $\rho^i_{ij} = 1$ for all other $i,j$. Express the map in terms of $(a,b,c) \mapsto (a-b,c-b,a-c)$ and determine its determinant.

Answer: 0


Question: Evaluate the limit \( \lim_{x\rightarrow 0} \frac{\sin x^2}{ \ln ( \cos x^2 \cos x + \sin x^2 \sin x)} \).

Answer: -2


Question: Given the sum of $n=14$ iid random variables $X_1, \dots, X_{14}$ with PDF \( f_{X_i}(x) = \frac{1}{\theta} e^{-\frac{x-\delta}{\theta}}, \quad x > \delta \) and 0 otherwise, where \( \theta = 2 \) and \( \delta = 0 \), calculate the probability \( P(Y < 2.467) \) for \( Y = \frac{\sqrt{28}Z}{\sqrt{S_{14}}} \) and \( Z \sim \mathcal{N}(0,1) \).

Answer: 0.99


Question: Let \( f: \mathcal{P}(A) \to \mathcal{P}(A) \) be a function such that for every \( U, V \in \mathcal{P}(A) \), if \( U \subseteq V \) then \( f(U) \subseteq f(V) \). Prove that there exists a set \( W \in \mathcal{P}(A) \) such that \( f(W) = W \).

Answer: W


Question: Evaluate the triple integral \( \int \int \int_{\Omega} xy^{2}z^{2} \, dx \, dy \, dz \) where \( \Omega \) is the region bounded by the surfaces \( z=xy \), \( y=x \), \( x=1 \), and \( z=0 \).

Answer: \dfrac{1}{198}


Question: Let $d$ be an integer. Suppose $A$ and $B$ are subsets of $\mathbb{R}^d$, each homeomorphic to an open $d$-ball, and their intersection $A \cap B$ is also homeomorphic to an open $d$-ball. Is the union $A \cup B$ homeomorphic to an open $d$-ball?

Answer: Yes


Question: Five computers are connected in a network. For each pair of computers, a coin is flipped. If it is heads, a direct link is established between the two computers; if it is tails, no link is established. A message received by a computer is automatically forwarded to all computers directly linked to it. Calculate the probability that every computer can receive messages from every other computer.

Answer: \dfrac{91}{128}


Question: Given a set \( \Omega_3 = \{1, 2, 3\} \), a collection of subsets \( A_1, A_2, \ldots, A_r \) is called a covering of \( \Omega_3 \) if \( \bigcup_{i=1}^r A_i = \Omega_3 \), \( \emptyset \neq A_i \subset \Omega_3 \), and \( A_i \neq A_j \) for all \( i \neq j \). Denote by \( C_3 \) the count of different coverings of \( \Omega_3 \). Given that \( C_1 = 1 \) and \( C_2 = 5 \), determine the value of \( C_3 \).

Answer: 109


Question: Evaluate the integral \( \int_{C}\frac{2z^{2}-5z+2}{\sin(z)}~dz \), where \( C \) is the unit circle oriented counterclockwise.

Answer: 4\pi i


Question: Determine the number of common tangents to the curves \( y^2 = 2012x \) and \( xy = (2013)^2 \).

Answer: 1


Question: Determine the number of $p$-element subsets $A$ of the set $\{1, 2, 3, \ldots, p\}$ such that the sum of the elements in $A$ is divisible by $p$, where $p$ is an odd prime number.

Answer: 1


Question: For any compact surface in \(\mathbb{R}^{3}\) that is not homeomorphic to a sphere, does there always exist a point with zero mean curvature, i.e., \(H=0\)?

Answer: No


Question: There are 755 cards, each numbered uniquely from 1 to 755. If 20 cards are randomly selected, what is the variance of the sum of the numbers on these 20 cards?

Answer: 926100


Question: Evaluate the integral \( \int _\gamma (e^{z^2} + \overline{z}) \, dz \) where \( \gamma \) is the positively oriented unit circle.

Answer: 2\pi i


Question: Let $X$ be a metric space and let $U, V \subset X$ be open and bounded subsets. Is the Hausdorff metric \( h(U,V) = \max(\sup_{x\in U}\inf_{y\in V}d(x,y),\sup_{x\in V}\inf_{y\in U}d(x,y)) \) defined for all possible choices of $U$ and $V$?

Answer: Yes


Question: Evaluate the integral \( \int_{-\pi}^{\pi} f(x) \, dx \) where \( f(t) = 1 + \sum_{n=1}^{\infty} \frac{2}{1+n^{2}}\cos (nt) \). Determine if this integral equals \( 2\pi \).

Answer: 2\pi


Question: Let $M$ be a symplectic manifold with symplectic form $\omega$. Suppose $f$ is a smooth function on $M$. Is there a unique vector field $X_f$ on $M$ such that for all vector fields $Y$, the equation $df(Y) = \omega(X_f, Y)$ holds?

Answer: Yes


Question: Let $X, Y, Z$ be independent and identically distributed random variables, each following a uniform distribution on the interval $(0, 1)$. Calculate the probability $P(X > Y + Z)$. Express your answer as a single value.

Answer: \dfrac{1}{6}


Question: Determine whether the equation $\sin(2x) = x$ has exact solutions for $x \in \mathbb{R}$. If exact solutions exist, find them; otherwise, provide a proof that finding exact solutions is impossible.

Answer: 0


Question:
Let $f, g : \mathbb{R}^n \to \mathbb{R}$ be differentiable at $a \in \mathbb{R}^n$. Evaluate the limit:
\[ \lim_{x \to a} \frac{|(f(x) - f(a))(g(x) - g(a))|}{||x - a||} \] where $||x||$ denotes the modulus of the vector $x$. Assume the value of the limit is zero.

Answer: 0


Question: Suppose $R$ is a Noetherian ring, $P$ is a prime ideal in $R$, and $Q$ is a $P$-primary ideal that contains $P^n$. Does $Q$ necessarily contain $P^{(n)}$, the $n$th symbolic power of $P$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the contour integral \( \int_{\gamma} \frac{1}{z^2+i} \, dz \), where \( \gamma \) is the circle of radius 2 centered at the origin.

Answer: 0


Question: Evaluate the Riemann-Stieltjes integral: \[ \int_0^6 (x^2 + [x]) \, d(|3-x|) \] where \([x]\) denotes the floor function. The solution is expected to be 63.

Answer: 63


Question: Find the geometric locus of the complex number \(z\) that satisfies the equation \(|z+1+3i|=|z-5-7i|\).

Answer: 3x + 5y = 16


Question: Let $L$ be a regular language over the alphabet $A=\{0\}$. Determine whether the language of binary representations of $n$, such that $0^n \in L$, is regular.

Answer: Yes


Question:
Calculate the volume of the region \(E\) defined by:
\[
E = \{(w,x,y,z) \mid 0 < w < 1 - (x^2 + y^2 + z^2)^3\}.
\]

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


Question: Let $f_n: [0, \infty) \to \mathbb{R}$ be a sequence of functions such that $|f_n(x)| \leq |g(x)|$ for every $x$ and $n$, where $g: [0, \infty) \to \mathbb{R}$ is an integrable function. Suppose $\int_0^\infty f_n(x) \, dx$ and $\int_0^\infty g(x) \, dx$ exist. If $f_n \to 0$ pointwise, does it follow that $\int_0^\infty f_n(x) \, dx \to 0$? Justify your answer.

Answer: Yes


Question: Let $X$ and $Y$ be spaces homeomorphic to the Cantor set, and let $x_0 \in X$. Suppose $f: X \to Y$ is a continuous function such that $f$ restricted to $X \setminus \{x_0\}$ is injective. Must $f$ be injective?

Answer: Yes


Question: Find a primitive solution to the differential equation: \[ ydx + xdy + 2zdz = 0 \]

Answer: xy + z^2 = C


Question: Let \( f: S_3 \to S^1 \) be a non-trivial non-injective group homomorphism, where \( S_3 \) is the symmetric group on 3 symbols and \( S^1 = \{z \in \mathbb{C} \mid |z|=1\} \). Determine the number of elements in the kernel of \( f \).

Answer: 3


Question: Evaluate the limit: $$\lim_{(x,y)\to(0,0)}\;\frac{x^5+y^5}{x^3+y^3}$$

Answer: 0


Question:
Determine the value of $R$ that minimizes the residuals in the following model:
\[
\tan(\Delta RA)=\frac{\left(\frac{R}{2115}\right) \cos(14.174550 - 0.003488 t) \sin(0.814907 - 0.000468 t) - \sin(15.04 t)}{\left(\frac{R}{2115}\right) \cos(14.174550 - 0.003488 t) \cos(0.814907 - 0.000468 t) - \cos(15.04 t)}
\]
where $t$ is a variable of time, and all other variables are constants. Find the value of $R$ that minimizes the values of $\Delta RA$ across all times $t$. Assume $R_E$ and $\lambda$ are constants such that $R_E \cos \lambda = 2115$. Provide the value of $R$ that achieves this minimum.

Answer: 2115


Question: Evaluate the triple integral \( \iiint_T \sqrt{x^2+y^2+z^2} \, dx \, dy \, dz \) where the region \( T \) is defined by the inequality \( x^2+y^2+z^2 \leq y \). Determine the limits of integration for this region.

Answer: \dfrac{\pi}{10}


Question: Evaluate the limit \( \lim_{n\to\infty}\int_0^{\pi/4}{\tan^n x \, dx} \).

Answer: 0


Question: Find the integer \( n \) such that \( \mathbb{Z}[i]/(-2+2i) \cong \mathbb{Z}/n\mathbb{Z} \).

Answer: 8


Question: Find the differentiable function \( f: [0,1] \to [0,1] \) such that \( f(f(x)) = x \) for all \( x \in [0,1] \) and \( f(0) = 1 \).

Answer: 1 - x


Question: Evaluate the expression \( \sin(5^{\circ})\sin(55^{\circ})\sin(65^{\circ}) = \frac{A\sqrt{B}+C\sqrt{D}}{E} \), where \( B \) and \( D \) are natural numbers not divisible by the square of any prime, \( A \) and \( C \) are integers, and \( E \) is a natural number. Find the value of \( A + B + C + D + E. \)

Answer: 24


Question: Determine whether the limit \( \lim\limits_{(x,y)\to (0,0)}\dfrac{1-\cos(x^2+y^2)}{(x^2+y^2)^2} \) exists.

Answer: \dfrac{1}{2}


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

Answer: \dfrac{1}{9}


Question: Determine the radical of the ideal \((x^3-y^6, xy-y^3)\) in \(\mathbb{C}[x,y]\).

Answer: (x - y^2)


Question: An object weighing 16 pounds is placed beneath the surface of a lake. The buoyancy of the object is 30 pounds, causing it to rise. The resistance of the water in pounds is numerically equal to the square of the velocity in feet per second. If the object surfaces in 5 seconds, what is the velocity of the object at the instant it reaches the surface?

Answer: \sqrt{14}


Question: Is there a linear transformation from the vector space of all real-valued functions defined on the integers to itself that is not continuous?

Answer: Yes


Question: What is the expected number of local maxima in a circular permutation of $n$ numbers? Arrange $n$ numbers uniformly at random on a circle. A local maximum is defined as a number that is greater than both of its neighbors. Determine if the expected number of local maxima is given by $(n + 1)/3$. Provide your answer as a single expected value.

Answer: \dfrac{n}{3}


Question: Calculate the limit of the series given by \( \lim_{x\to +\infty}\sum_{n=1}^{\infty}\frac{a_n}{n!}x^n \), where \( \{a_n\} \) is the Fibonacci sequence defined by \( a_0 = a_1 = 1 \) and \( a_{n+2} = a_n + a_{n+1} \) for every \( n \geq 0 \).

Answer: \infty


Question: Consider an infinite dimensional unital $\mathrm{C}^*$-algebra $A\subset B(H)$ and a subspace $U\subset H$. Suppose that for all $f\in A$, the restriction of $f$ to $U$ is scalar, meaning there exists $\lambda\in\mathbb{C}$ such that $f(u)=\lambda u$ for all $u\in U$. Can we conclude that $U$ is either zero or one-dimensional?

Answer: No


Question: Evaluate the integral \( \int_{|z|=2} \frac{1}{\sqrt{z^4+4z+1}} \, dz \).

Answer: 0


Question:
Given two sets $X$ and $Y$ with a joint probability distribution $p(x,y)$, and marginal distributions $p(x)$ and $p(y)$, the mutual information $I(X; Y)$ is defined as:
\[ I(X; Y) = \sum_{x,y}p(x,y)\cdot\log\left(\frac{p(x,y)}{p(x)p(y)}\right) \]
Suppose for all $x,y$, the pointwise mutual information satisfies:
\[ -k \leq \log\left(\frac{p(x,y)}{p(x)p(y)}\right) \leq k \]
Find the tightest possible upper bound on $I(X; Y)$ given these constraints.

Answer: k


Question: Eight singers participate in an art festival where m songs are performed. Each song is performed by 4 singers, and each pair of singers performs together in the same number of songs. Find the smallest value of m for which this is possible.

Answer: 14


Question:
Let \( f(x,y) \) be defined as follows:
\[
f(x,y) =
\begin{cases}
\frac{xy(x^2-y^2)}{x^2+y^2}, & \text{if } (x,y) \ne (0,0) \\
0, & \text{if } (x,y) = (0,0)
\end{cases}
\]
Find the value of \( \frac{\partial^2 f}{\partial x \partial y} \) at the point \((0,0)\).

Answer: 1


Question: Given $g(1) = g'(1) = 4$ and the function defined by the integral \( f(x) = \int_0^x (x - t^2)g(t) \, dt \), find the value of \( f''(1) \).

Answer: 0


Question:
Let $k$ be a fixed natural number and consider the matrix
\[A=\begin{bmatrix}a_1&*&*&...&*\\0&a_2&*&...&*\\0&0&a_3&...&*\\ \vdots & \vdots & \vdots & \ddots & \vdots\\0&0&0&...&a_n\end{bmatrix}\]
where $a_i^k=1$ for every natural $i$, and the $a_i$ are pairwise distinct. The $*$ denotes a random complex number. Determine if $A^k=E$, where $E$ is the identity matrix.

Answer: A^k = E


Question: Let $f$, $g$, and $h$ be three functions from $\mathbb{N}$ to $\mathbb{N}$. Assume that $f \in o(g)$ and $g \in O(h)$. If the limit of $\frac{f(n)}{h(n)}$ exists as $n \to \infty$, what are its possible values?

Answer: 0


Question: Given the Orlicz function $M$ with properties $M(0)=0$, $M(x)>0$ for $x>0$, and $M(x)\rightarrow \infty$ as $x\rightarrow \infty$, and the Orlicz-Luxemborg norm defined by $$\|x\|=\inf\left\{r>0:\sum_{n=1}^{\infty}M\left(\frac{x_n}{r}\right)\leq 1\right\},$$ determine if $\sum_{n=1}^{\infty}M\left(\frac{x_n}{r}\right)\leq 1$ implies that the sequence $x_n$ is bounded.

Answer: Yes


Question:
Given two real rectangular matrices $A$ and $B$ such that $AB = I$, where $A$ is an $m \times n$ matrix and $B$ is an $n \times m$ matrix, determine which of the following statements is true:

1. $n > m$
2. $m \ge n$
3. If $BA = I$, then $m > n$
4. Either $BA = I$ or $n > m$

Select the correct statement.

Answer: 4


Question: Let \( f:[0,1] \to \mathbb{R} \) be a continuously differentiable function. Given that \[ \left| \frac{f(0)+f(1)}{2} - \int_0^1 f(x)\, dx \right| \le \alpha \max_{x \in [0,1]} |f'(x)|, \] determine the possible values of \( \alpha \).

Answer: \dfrac{1}{4}


Question: Find the unit normal field of the paraboloid \( P = \{(x, y, z) \in \mathbb{R}^3: z = 4 - x^2 - y^2, x^2 + y^2 < 4 \} \). Then, integrate the vector field \( F(x, y, z) = (x, y, xy + z)^T \) over \( (P, \nu) \), where \( \nu \) is the unit normal field.

Answer: 24\pi


Question: Consider the following scenario: Let $P_u$ and $P_v$ be $p$-subgroups of a finite group $G$ such that $P_u \leq x^{-1}P_v x$ for some $x \in G$. Suppose $X$ is a common indecomposable stable summand of $BP_u$ and $BP_v$. If $\iota: X \rightarrow BP_u$ is an inclusion map of $X$ as a summand of $BP_u$, determine whether the composite map \[ X \xrightarrow{\iota} BP_u \xrightarrow{Bi \circ Bc_x} BP_v \] is an inclusion map of $X$ as a summand of $BP_v$. Assume all objects are $p$-completed spectra, and $X$ is also a summand of $BG$. Is this statement true?

Answer: True


Question: Let $V$ be an infinite-dimensional vector space over the field $\mathbb{F}$, where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$. If the dimension of $V$ is $\alpha$, a cardinal number, determine the dimension of $V^*$, the algebraic dual of $V$. It is known that $\dim V < \dim V^*$ when $V$ is infinite-dimensional.

Answer: 2^\alpha


Question: Consider two distinct irreducible real algebraic sets \(M_1\) and \(M_2\), each of dimension \(n-1\) in \(\mathbb{R}^n\). Determine if their intersection necessarily has dimension at most \(n-2\).

Answer: Yes


Question: Find the residue at $z=0$ of the function \( f(z) = \log\left(\frac{1-az}{1-bz}\right) \), where \( a \) and \( b \) are complex constants.

Answer: 0


Question: Given the function \( u = x^2y^2 + yz - z^3 \) and the constraint \( x^2 + y^2 + z^2 = 11 \), find the value of \( \left( \frac{\partial u}{\partial y} \right)_{x} \) at the point \((u, x, y, z) = (5, 1, -3, 1)\).

Answer: -23


Question: Determine whether the function \( u(x,y) = \frac{x}{x^2+y^2} \) is harmonic.

Answer: Yes


Question: Find the positive value of \(x\) for which the integral \(\int_0^{\sqrt{x}} \sin \left(\frac{2\pi t}{t+2}\right) dt\) is maximized.

Answer: 4


Question: Maximize the variance of a random variable $X$ with support on $[0,1]$ given that $\mathbb{E}[X]=\mu$, where $\mu \in [0,1]$. Determine the distribution of $X$ that achieves this maximum variance.

Answer: \mu(1 - \mu)


Question: Find a function \( f(x) \) such that the sequence \( f(n) = a_n \) converges, but \( \lim_{x \to \infty} f(x) \) does not exist.

Answer: \sin(\pi x)


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{(1+nx)^m-(1+mx)^n}{\sqrt[m]{1+mx}-\sqrt[n]{1+nx}} \] where \(n\) and \(m\) are distinct positive integers.

Answer: -mn


Question: Find the degree of the field extension \( \mathbb{Q}(\sqrt[4]{2}):\mathbb{Q}(\sqrt{2}) \).

Answer: 2


Question: Determine whether the statement "To be concave on $[0,1]$, $f(t)\leq f(0)+tf'(0)$ is enough" is true or false. Provide a justification for your answer.

Answer: B


Question: Determine the values of \( \alpha \) for which the integral \( \int_{0}^{1} (- \ln x)^\alpha \, \mathrm{d}x \) is convergent.

Answer: \alpha > -1


Question: Evaluate \( \lim _{x\to \infty }\int _{\frac{1}{x}}^x\:f\left(t\right)dt \) where \( f\left(t\right)=\frac{1}{\left(1+t^2\right)\left(1+t^3\right)} \).

Answer: \dfrac{\pi}{4}


Question: Find a 1-error-correcting code in \( \mathbf{Z}_{5}^{6} = \mathbf{Z_5}\, \times \, \mathbf{Z_5}\, \times \, \mathbf{Z_5}\, \times \, \mathbf{Z_5}\, \times \, \mathbf{Z_5}\, \times \, \mathbf{Z_5} \) with the maximum number of code words.

Answer: 625


Question: Evaluate the limit: \[ \lim_{x \to +\infty} \frac{2x \cdot \arccos\left(\frac{2x}{1+x^2}\right) - \pi x}{2} \]

Answer: -2


Question: Let S be a compact convex subset of the Euclidean plane E with a non-empty interior. If p is a point in E such that every straight line passing through p bisects the area of S, is S necessarily centro-symmetric with respect to p? Answer yes or no.

Answer: Yes


Question: Compute the integral \( \int_{0}^{2} x \, d\alpha(x) \), where \( \alpha(x) = \begin{cases} x & ,0 < x \leq 1 \\ 2 + x & , 1 < x \leq 2 \end{cases} \).

Answer: 4


Question: Find the value of \( x \) that satisfies the equation \( 4^{217} \equiv x \pmod{391} \).

Answer: 55


Question: Find a closed form expression for the series \( \sum_{k=0}^{\lfloor n/2 \rfloor} (2k+1) {n \choose 2k+1} \).

Answer: n \cdot 2^{n-2}


Question: From a point A, the bearing to the base of a 60m high tower is 53°, and the angle of elevation to the top of the tower is 16°. From a point B, the bearing to the base of the tower is 300°, and the angle of elevation to the top is 20°. Calculate the distance between points A and B.

Answer: 313


Question: Determine the order of the group $G$ generated by the elements $x$ and $y$ with the relations $x^3 = y^2 = (xy)^2 = 1$. What is the order of $G$?

Answer: 6


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

Answer: \dfrac{1}{e}


Question: Evaluate the improper integral \( \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^3} \, dx \) using the residue theorem.

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


Question: Evaluate the expression \( \arcsin\left(\frac{5\sqrt{3}}{14}\right) - \arctan\left(\frac{1}{4\sqrt{3}}\right) \) without using a calculator, assuming you know the standard angles \( \pi/3, \pi/4, \pi/6 \).

Answer: \dfrac{\pi}{6}


Question: Let $f: \mathbb{R} \to \mathbb{C}$ be a Borel function such that $|f|=1$. Is it true that there exist $\alpha < \beta$ such that $\int_{\alpha}^{\beta} f(x) \, dx \neq 0$?

Answer: Yes


Question: Find the arc length of the cycloid described by the parametric equations \( \gamma(t)=(\cos(\frac{3}{2}\pi-t)+t,\sin(\frac{3}{2}\pi-t)+1) \) for \( t \in [0, 2\pi] \).

Answer: 8


Question: Compute the limit: $$\lim_{\varepsilon \to 0}\left|\int_{\partial B(0,\varepsilon)} \frac{e^{-|x|}}{4 \pi |x|}dx\right|$$ where $x \in \mathbb{R}^3$. Determine if this limit is equal to 0.

Answer: 0


Question: Let \( f(x) \) be a polynomial of degree 8 such that \( f(r) = \frac{1}{r} \) for \( r = 1, 2, 3, \ldots, 9 \). Find \( \frac{1}{f(10)} \).

Answer: 5


Question: Find the length of the longest path in the complete graph $K_7$, where vertices may be repeated but edges cannot be repeated.

Answer: 21


Question:
Let \( G \) be an open set in \( \mathbb{R}^n \). Two points \( x, y \in G \) are said to be equivalent if they can be joined by a continuous path completely lying inside \( G \). Determine the number of equivalence classes. Choose from the following options:

- Only one.
- At most finite.
- At most countable.
- Can be finite, countable, or uncountable.

Answer: C


Question: Find the smallest positive integer \(x\) such that \(7^x \equiv 23 \pmod{29}\).

Answer: 4


Question: Determine if there exists a deficient number $N$ such that $\sqrt{D(N)} \mid A(N)$, where $D(N) = 2N - \sigma(N)$ is the deficiency and $A(N) = \sigma(N) - N$ is the sum of the aliquot parts of $N$. If such a number exists, provide an example.

Answer: 14


Question: Evaluate the integral \( \int_0^{\infty}\int_0^{\infty}e^{-(x+y)^2}\,dx\,dy \).

Answer: \dfrac{1}{2}


Question: Find an equation of the plane that passes through the point \((1, -2, 1)\) and contains the line of intersection of the planes \(x + y - z = 2\) and \(2x - y + 3z = 1\).

Answer: 7x + y + 3z = 8


Question: Find the value of \( m \) such that the equation \(|x^2+4x+3|-mx+2m=0\) has exactly three solutions.

Answer: -8 + 2\sqrt{15}


Question: Does there exist a smooth transitive action of a finite-dimensional Lie group on the Hantzsche-Wendt manifold? Specifically, is there a Lie group $G$ and a closed subgroup $H$ such that $G/H$ is diffeomorphic to the Hantzsche-Wendt manifold?

Answer: No


Question: In the complex vector space $\Bbb C^2$, determine the number of real unit vectors whose projection onto the vector $|1\rangle$ has a length of $\sqrt{3}/2$. Provide your answer as a non-negative integer.

Answer: 4


Question: Let $A$ and $B$ be sets such that $|A| = 8$ and $|B| = 5$. Find the number of functions $f: A \rightarrow B$ such that no element in $B$ has exactly 3 pre-images from $A$. In other words, for all $b \in B$, $|\{ a \in A \mid f(a) = b \} | \neq 3$. Determine the total number of such functions.

Answer: 154305


Question: Can each finite rank projection in $B(H)$ be expressed as a finite sum of rank one projections? Provide a concise proof.

Answer: Yes


Question: Let $C$ be a smooth curve over a field $k$, with $ar{C}$ as its smooth compactification and $S = \bar{C} \setminus C$. Consider $S$ as a reduced divisor defined over $k$. For the sheaf of logarithmic differentials $\Omega^1(\log S)$, take a non-zero rational section $\eta$. Let $s$ be a point in $S$. Is the residue $\mathrm{Res}_s(\eta)$ an element of the field $k$ or the residue field $k(s)$?

Answer: k(s)


Question: Solve for $x$ in the equation: $$\sqrt{\log_x\left(\sqrt{3x}\right)} \cdot \log_3 x = -1$$

Answer: \dfrac{1}{9}


Question: Evaluate the line integral \( \int_{\kappa} F \, dx \) for the vector field \( F(x,y,z) = \begin{pmatrix} x^{2}+5y+3yz \\ 5x +3xz -2 \\ 3xy -4z \end{pmatrix} \) along the curve \( \kappa: [0, 2\pi] \to \mathbb{R}^{3}, \, t \mapsto \begin{pmatrix}\sin t\\ \cos t \\ t \end{pmatrix} \). Use the fact that the vector field is conservative to simplify the calculation.

Answer: -8\pi^2


Question: Determine whether the function \( f(x) = \sum_{n=0}^{\infty} 2^{-n}\chi_{[n,n+1)}(x) \) is Lebesgue integrable on \( \mathbb{R} \). Justify your answer.

Answer: Yes


Question: Determine if there exists a function \( f(x) \neq x \) such that \( f(f'(x)) = f'(f(x)) \).

Answer: Yes


Question:
Given a strictly positive sequence \((a_n)_{n \ge 1}\) such that \(\lim_{n \to \infty} \frac{a_{n+1}}{n a_n} = a > 0\), find the value of:
\[ \lim_{n \to \infty} \left( \frac{(n+1)^2}{\sqrt[n+1]{a_{n+1}}} - \frac{n^2}{\sqrt[n]{a_n}} \right). \]

Answer: \dfrac{e}{a}


Question: Calculate the sum \( S(z) \) of the power series \( \sum_{n=1}^{\infty}\frac{(-1)^n}{n}(z-i)^n \) for \( z \) within the convergence disk.

Answer: -\ln(1 + z - i)


Question: Given that \((1+x+x^2)^n = \sum_{r=0}^{2n}a_{r}\cdot x^{r}\) and \(\sum_{r=0}^{n}(-1)^r\cdot a_{r}\cdot \binom{n}{r} = k\cdot \binom{n}{\frac{n}{3}}\), find the value of \(k\) when \(n\) is a multiple of 3.

Answer: 1


Question: Let \( A, B \in \mathcal{M}_2(\mathbb{C}) \) be two diagonalizable matrices, and let \( D := P^{-1}BP \) be a diagonal matrix such that \( AD = DA \). If \( P^{-1}BPA = AP^{-1}BP \), does it follow that \( AB = BA \)?

Answer: Yes


Question: Evaluate the integral \( I = \int\int_{D} |\cos(x+y)| \, dx \, dy \), where \( D = \{ (x, y) \mid |x| + |y| \leq 2\pi \} \).

Answer: 16\pi


Question: Given the group presentation \(\langle x, y \mid x^n = y^2 = 1, xy = yx^2 \rangle\), determine the order of the element \(x\) when \(n = 3k\). Can we conclude that the order of \(x\) is equal to 3 under this condition?

Answer: 3


Question:
Given a collection of sets, determine the number of unique sequences that can be formed such that each sequence contains all elements from the sets. For example, if the sets are:

\[ S = \{\{0\}, \{1, 2, 3\}, \{1, 2, 3\}, \{3\}\} \]

The number of unique sequences is 2, which are:

\[ s_0 = (0, 1, 2, 3) \]
\[ s_1 = (0, 2, 1, 3) \]

Calculate the number of such unique sequences for the given sets.

Answer: 2


Question: Calculate the expected value of the product of the order statistics, $E[X_{(1)}X_{(2)}]$, where $X_1$ and $X_2$ are independent and identically distributed as $N(0, \sigma^2)$. Provide your answer in terms of $\sigma$.

Answer: 0


Question:
Let the pressure $p$ and temperature $T$ at a point $(x,y,z)$ be given by:
\[ P(x,y,z)=\frac{x^2+2y^2}{1+z^2}, \quad T(x,y,z)=5+xy-z^2 \]
If the position of an airplane at time $t$ is \((x(t),y(t),z(t))=\left(2t,t^2-1,\cos t\right)\), find \(\frac{\mathrm{d}}{\mathrm{dt}}(PT)^2\) at time \(t=0\) as observed from the airplane.

Answer: -16


Question: Grandma has 8 grandchildren and 4 different types of popsicles: 6 Vanilla, 6 Strawberry, 5 Banana, and 3 Chocolate. Each grandchild requests one popsicle of a particular flavor. What is the total number of different sets of requests that Grandma can fulfill?

Answer: 57750


Question: Consider the functions \(f(x)= [x]^2 - [x+6]\) and \(g(x)= 3kx^2+ 2x + 4(1-3k)\), where \([x]\) denotes the floor function. Let \(A= \{x ~|~ f(x)= 0\}\) and \(k \in [a,b]\) such that every element of set \(A\) satisfies the inequality \(g(x)\ge 0\). If the set \(A\) is given by \([-2, -1) \cup [3,4)\), find the value of \(6b - 3a\).

Answer: 2


Question: Given the hyperbola \( \frac{(3x-4y-1)^2}{100} - \frac{(4x+3y-1)^2}{225} = 1 \), find the length of the latus rectum.

Answer: 9


Question: On a hyperbola $f(x)$, two points $(t,f(t))$ and $(t,-f(t))$ lie on $f(x)$. The coordinates of the two foci are $F(0,5)$ and $F^{'}(0,-5)$. The difference in distance from the two foci is 6. Determine the value of \( \lim_{t\to\infty}{\frac{f(t)}{t}} \) given that $f(t) > 0$. \( \)

Answer: \dfrac{3}{4}


Question: Find the minimum value of \( \left(\frac{1}{x}-1\right)\left(\frac{1}{y}-1\right)\left(\frac{1}{z}-1\right) \) given that \( x+y+z=1 \) and \( x, y, z \) are three distinct positive real numbers. Choose the correct answer from the following options: 1, 4, 8, or 16.

Answer: 8


Question: Determine whether the integral \( \int_{-\infty}^\infty e^{-\pi (a+z)^2} \, da \) converges for every \( z \in \mathbb{C} \).

Answer: Yes


Question:
Determine whether the following statement is true:

Let $G$ be a finite group. For any irreducible $\mathbb{R}G$-module $U$, its character has one of the following forms:

- $\chi_U$, where $U$ is an irreducible $\mathbb{C}G$-module,
- $2\operatorname{Re}\chi_W$, for some irreducible $\mathbb{C}G$-module $W$.

Moreover, for any irreducible $\mathbb{C}G$ module $V$, one of the following is true:

- If $\iota(V)=1$, then $V$ is an irreducible $\mathbb{R}G$-module,
- If $\iota(V)\neq 1$, then $2\operatorname{Re}\chi_V$ is a character of some irreducible $\mathbb{R}G$-module.

Answer: True


Question: Determine the number of subgroups of index 2 in the group \((C_2)^3 \times C_3\).

Answer: 7


Question: If a function $f$ is of bounded variation on the interval $[a,b]$, is $f$ Riemann integrable on $[a,b]$?

Answer: Yes


Question: Evaluate the limit: \[ I = \lim_{n \to \infty} \int_0^{2\pi} \frac{\cos nx}{x+1} \, \text{d}x \]

Answer: 0


Question: If an analytic function \( f : \mathbb{R} \to \mathbb{R} \) satisfies \( f(\mathbb{Q}) \subseteq \mathbb{Q} \), must \( f \) be a polynomial? Provide a justification for your answer.

Answer: Yes


Question: Find the mass of a wire shaped as the curve of intersection between the sphere \(x^2 + y^2 + z^2 = 1\) and the plane \(x + y + z = 0\), given that the density of the wire is \(x^2\).

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


Question: Find the smallest positive integer greater than 1 that has both its leading and ending digits equal to 1 in both its base-8 and base-9 representations. Provide your answer in decimal form.

Answer: 793


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

Answer: -\dfrac{1}{8}


Question: Let $f, g: \mathbb{R} \rightarrow \mathbb{R}$ be functions such that the product $fg$ has the Intermediate Value Property, $g$ also has this property, and $f$ is injective. Is it true that $f$ must also have the Intermediate Value Property?

Answer: No


Question: Prove that a group of order 160 is not a simple group.

Answer: G


Question: Simplify the expression \( \sum_{k=0}^n\binom{n}{k}(-1)^k\left(\sum_{j=0}^k\binom{k}{j}3^j\right) \).

Answer: (-3)^n


Question: Find the minimum value of the function \( f(x) = 2^{x^2} - 1 + \frac{2}{2^{x^2} + 1} \).

Answer: 1


Question: Evaluate the integral \( \int_{C_r} \frac{100z^{100}+99z^{99}}{z^{100}+z^{99}+1} \, dz \) using the Cauchy integral theorem, where \(C_r\) is a contour that encloses all the roots of the denominator.

Answer: -2\pi i


Question: How many elements in the set \( S = \{0, 1, \ldots, 5399\} \) are quadratic residues modulo 5400?

Answer: 363


Question: Let $N = \lceil\delta^{-\gamma}\rceil$, with $0 < \gamma < 1$. Determine whether \( \lim_{\delta \to 0} f(t,\delta,\lceil \delta^{-\gamma}\rceil)=0 \) for all \( t > 0 \), where \( f(t,\delta, N)= e^{-t/\delta} \sum_{k=0}^{N-1}\frac{(t/\delta)^k}{k!} \).

Answer: 0


Question:
Determine the pointwise limit \( f(x) = \lim_{n \to \infty} f_n(x) \) for the function sequence \( f_n : [0, 2] \to \mathbb{R} \) given by:
\[
f_n(x) = \begin{cases}
 n^3 x^2, & 0 < x < 1/n; \\
n^3\left(x - \frac{2}{n}\right)^2, & 1/n \le x < 2/n; \\
0, & \text{otherwise.}
\end{cases}
\]

Answer: 0


Question: Determine whether the following statement is true or false: For functions $f, g \in L^1(\mathbb{R})$, if their Fourier transforms are equal, i.e., $\hat{f}(y) = \hat{g}(y)$ for all $y \in \mathbb{R}$, then $f(x) = g(x)$ for almost every $x \in \mathbb{R}$.

Answer: True


Question: Evaluate the asymptotic behavior of the probability that a randomly chosen monic polynomial of large degree $n$ in $\mathbb{F}_p[x]$ is irreducible. This probability can be expressed as \(\lim_{n\to\infty}\frac{N_p(n)}{p^n},\) where \(N_p(n)=\frac{1}{n}\sum_{d|n}p^d\mu\left(\frac{n}{d}\right)\) is the number of monic irreducibles of degree $n$ in $\mathbb{F}_p[x]$. Determine this limit.

Answer: 0


Question: Evaluate the complex integral \( \int_{|z|=2} \frac{e^z}{z-1} \, dz \) using the residue theorem.

Answer: 2\pi i e


Question: Determine the number of sequences \( (a_0, a_1, \ldots, a_{18}) \) such that \( a_i \ge 0 \) for all \( i \), \( a_0 = a_{18} = 0 \), and \( a_i = a_{i-1} \pm 1 \) for all \( 1 \le i \le 18 \).

Answer: 4862


Question: Calculate the gradient \( \nabla_x v^\top A x \) for a fixed matrix \( A \in \mathbb{R}^{m \times n} \) and a fixed vector \( v \in \mathbb{R}^m \), where \( x \in \mathbb{R}^n \) is the variable. Express your answer in terms of \( A \) and \( v \).

Answer: A^\top v


Question: Let \(a, b, c\) be positive real numbers satisfying the equations \(a+b+c=1\) and \(a^2+b^2+c^2=\frac{3}{8}\). Determine the maximum value of \(a^3+b^3+c^3\).

Answer: \dfrac{5}{32}


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

Answer: 3


Question: Given a positive real number \( n \), find the positive real number \( m \) that minimizes the function \( m \log_m{n} \).

Answer: e


Question: Determine the maximum value of \(a^2 + b^2 + c^2\), where \(a, b, c\) are the sides of a triangle inscribed in a unit circle.

Answer: 9


Question: In a permutation search game, the natural numbers $1$ through $n$ are arranged in a random order, which is unknown and has a uniform distribution. You make a sequence of guesses, one number and one slot at a time, and are told after each guess whether it is correct. The game ends when the order of the $n$ numbers is determined. Is there a strategy that takes fewer guesses than the trivial elimination in both the worst case and the average case?

Answer: Yes


Question:
Given the joint probability density function of two random variables $X$ and $Y$:
\[f(x, y) = \begin{cases}
\frac{3}{5}x(y + y^2)  & \text{if }0<x<2\text{ and } 0 < y < 1\\
0 & \text{otherwise}
\end{cases}\]
find $\Pr(Y > X)$.

Answer: \dfrac{27}{200}


Question: Find the remainder when the polynomial $f(x^{12})$, where $f(x) = 1 + x + x^2 + x^3 + x^4 + x^5$, is divided by $f(x)$. Express your answer as a polynomial in terms of $x$. \( \boxed{\text{remainder}} \)

Answer: 6


Question: Find the positive integer \( n \) such that the equation \( 2\pi\sin{nx} = 2\pi - x \) has exactly 2018 different solutions for \( x \) in the interval \( 0 \le x < 2\pi \).

Answer: 1009


Question: Let \( E = \mathbb{R}^n \). Let \( \mathcal{V} \) be a linear subset of \( E \) such that for every non-zero vector \( X = \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} \in \mathcal{V} \), each component \( x_i \neq 0 \) for all \( i \in \{1, 2, \ldots, n\} \). Determine the dimension of \( \mathcal{V} \).

Answer: 1


Question: Let $P$ be an $s \times s$ matrix and $Q$ be an $s \times r$ matrix. If $\text{rank}(P|Q) = s$, is it possible to find an $r \times s$ matrix $R$ such that $\text{rank}(P + QR) = s$?

Answer: Yes


Question: Given that $y_1 = x^2 \ln(x)$ is a solution to the differential equation $x^2 y'' - 3x y' + 4y = 0$, use the method of reduction of order to find a second linearly independent solution.

Answer: x^2


Question: Evaluate the limit: \[ \lim_{x \to \pi/6} \frac{(2\sin x + \cos(6x))^2}{(6x - \pi)\sin(6x)} \]

Answer: -\dfrac{1}{12}


Question: An insurance company offers coverage for ship breakdowns, where breakdowns follow a Poisson process. On average, a breakdown occurs once every 1000 days. The damage from a breakdown is distributed as Binomial($2 \times 10^7, 0.5$), independent of other breakdowns and the Poisson process. Calculate the expected total damage over one year (365 days).

Answer: 3650000


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

Answer: 2\pi i


Question: Find the Maclaurin series representation of the function \( f(x) = \ln\left(\pi\sqrt{\frac{2+x}{2-x}}\right) + \arctan{\frac{1-x}{1+x}} \) and determine the radius of convergence of this series.

Answer: 1


Question: Let \( \mu \) be a positive Radon measure on a bounded domain \( \Omega \) such that \( \mu \in H^{-1}(\Omega) \). The measure \( \mu \) has support in a subdomain \( Z \subset \Omega \). Suppose \( z \in H^1_0(\Omega) \) is a function such that \( z = 0 \) almost everywhere in \( Z \). Is it true that \( \langle \mu, z \rangle_{H^{-1}, H^1_0} = 0 \)?

Answer: 0


Question: Calculate the expected value of an $n$-digit number $P = \sum\limits_{i=0}^{n-1}10^ic_i$, where each digit $c_i \in \{0, \dots, 9\}$ is removed independently with probability $p$. If all digits are removed, the resulting number is considered $0$. What is the expected value of the number obtained?

Answer: (1 - p) \sum_{i=0}^{n-1} c_i (10 - 9p)^i


Question: Let $g(x)$ be a strictly increasing and twice-differentiable function over the interval $[a,b]$. Suppose $f(x)$ is another twice-differentiable function such that for all $x \in (a,b)$, the inequality $g'(x) f'(x) \geq 0$ holds. Given that $g'(x) \geq 0$ for all $x$ and $g'(x) > 0$ except possibly at finitely many points, does this imply that $f(x)$ is weakly increasing over $(a,b)$?

Answer: Yes


Question: If $X$ and $Y$ are two discrete random variables taking values in the set of non-negative integers, and if $E(t^X) = E(t^Y)$ for all $t \in (0,1)$, do $X$ and $Y$ have the same distribution?

Answer: Yes


Question: Consider a sequence of functions \(f_n\) that are continuously differentiable on \((0,1)\), such that \(f_n \to f\) pointwise on \((0,1)\). If the derivatives \(f_n'\) are uniformly bounded and equicontinuous, does the limit function \(f\) belong to \(C^1((0,1))\)? In other words, is \(f\) also continuously differentiable on \((0,1)\)?

Answer: Yes


Question: Suppose you have a function \( g : [0, \infty) \to [0, \infty) \) such that \( \lim_{x \to +\infty} (2g(2x) - g(x)) = 0 \). Additionally, for every \( M > 0 \), the restriction \( g|_{[0,M]} \) is bounded. Does it follow that \( \lim_{x \to +\infty} g(x) = 0 \)?

Answer: 0


Question: Determine whether the set $S$ in $\mathbb{R}^3$ given by the equation $x^3 + 2y^3 + 3z^3 = 0$ is a surface. Answer true or false.

Answer: B


Question: Given the power series \( P(x) = \sum_{n=0}^\infty (-1)^n 2^n (n+1) x^n \), determine the power series for \( \frac{1}{P(x)} \).

Answer: 1 + 4x + 4x^2


Question: Let $f$ be a continuous function on the interval $(0,1)$. Determine whether the following statement is true or false: If $(x_n)$ is a Cauchy sequence such that $0 < x_n < 1$ for all $n$, then $f(x_n)$ is a Cauchy sequence.

Answer: B


Question: Determine whether the function \( f(x) = \begin{cases} x\left\lfloor \frac{1}{x}\right\rfloor, & x\neq 0 \\ 0, & x=0 \end{cases} \) has a removable discontinuity at \( x = 0 \).

Answer: Yes


Question:
Find the number of quadratic polynomials \( ax^2 + bx + c \) that satisfy the following conditions:

1. \( a, b, c \) are distinct integers.
2. \( a, b, c \in \{1, 2, 3, \ldots, 999\} \).
3. \( (x+1) \) divides \( ax^2 + bx + c \).

Answer: 498002


Question: Let $e = e^2 \in R$ such that $ReR = R$. Suppose $\mathfrak{B}$ and $\mathfrak{C}$ are ring direct summands of the ring $eRe$ with $eRe = \mathfrak{B} \oplus \mathfrak{C}$. Prove or disprove that $R\mathfrak{B}R \cap R\mathfrak{C}R = 0$. Assume that any ring direct summand is generated by a central idempotent, and use Theorem 21.11(2) from "A First Course in Noncommutative Rings" by T. Y. Lam, which states that $e(R\mathfrak{A}R)e = \mathfrak{A}$ for any ideal $\mathfrak{A}$ of $eRe$, and that the map $\mathfrak{A} \rightarrow R\mathfrak{A}R$ defines a one-to-one correspondence between the ideals of $eRe$ and those of $R$.

Answer: R\mathfrak{B}R \cap R\mathfrak{C}R = 0


Question: Let $S\subset\mathbb{R}^3$ be an orientable surface with a local parametrization $X:(u_1,u_2)\in U\subset \mathbb{R}^2\longrightarrow X(U)\subset S$. Define the normal vector $N=\dfrac{X_{u_1}\times X_{u_2}}{\lVert X_{u_1}\times X_{u_2} \rVert}$ and take the positive orientation. Suppose $R\subset X(U)\subset S$ is a simple region with boundary $\partial R$, a closed, piece-wise smooth curve. Let $A=X^{-1}(R)\subset U\subset\mathbb{R}^2$ and compute $\partial A$. If $\tilde{\alpha}\equiv A$ has the positive orientation on the plane, and $\alpha(t)=X(\tilde{\alpha}(t))$ is the boundary of $\partial R$, is $\alpha(t)$ positively oriented?

Answer: Yes


Question: Is the \(\omega\)-limit set \(\omega(x)\) always \(f\)-invariant for a continuous function \(f: X \to X\) on a metric space \(X\), regardless of whether \(X\) is complete?

Answer: Yes


Question: Determine the value of the initial bet \( k \) such that the expected winnings in the following game are zero: A fair coin is tossed until the first tails appears. If the first tails appears on the \( n \)-th toss, you receive a total of \( 2^n \) dollars. What should \( k \) be for the game to be fair?

Answer: \infty


Question: Evaluate the definite integral \( \int_{-\pi/2}^{\pi/2} \frac{9x^4 \cdot \sin(9x)}{1+x^8} \, dx \) using the method of u-substitution.

Answer: 0


Question: Calculate the limit: \[ \lim_{n \to \infty} \left(4n^2\left(\frac{n^3+3n^2+3}{2n^3+n-1}\right)^n\right) \]

Answer: 0


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

Answer: 2


Question: Prove that for all positive real numbers $a$, $b$, and $c$, the following inequality holds: \[ \sum_{cyc} \frac{a^2}{b^2 + c^2 + bc} \geq 1. \]

Answer: 1


Question: Calculate the value of the expression: \[ \cos\frac{\pi}{13} + \cos\frac{3\pi}{13} + \cos\frac{5\pi}{13} + \cos\frac{7\pi}{13} + \cos\frac{9\pi}{13} + \cos\frac{11\pi}{13}. \]

Answer: \dfrac{1}{2}


Question: Determine the number of ring homomorphisms from the ring of real numbers to itself.

Answer: 1


Question: In a perfect graph, does there exist a clique that intersects all maximal independent sets?

Answer: Yes


Question: Let $(X, \tau)$ be a topological space with a topological base $\mathcal{E}$. Denote Bor$(\mathcal{E})$ as the smallest $\sigma$-algebra containing $\mathcal{E}$. Assume that Bor$(\mathcal{E}) = $ Bor$(\tau)$. If $O$ is an open set in $X$, does there exist a sequence of open sets $\{O_n\} \subseteq \mathcal{E}$ such that $O = \bigcup O_n$?

Answer: Yes


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

Answer: 1


Question:
Determine the number of solutions to the equation:
\[ \frac{(x-b)(x-c)}{(a-b)(a-c)} + \frac{(x-a)(x-c)}{(b-a)(b-c)} + \frac{(x-a)(x-b)}{(c-a)(c-b)} = 1 \]
where \(a\), \(b\), and \(c\) are parameters.

Answer: \infty


Question: Find the Taylor series expansion for \( f(z) = \frac{1}{(z-i)(z-2)} \) and determine its radius of convergence.

Answer: 1


Question: Is the wedge sum operation associative? Specifically, is the wedge sum $X \vee Y \vee Z$ homeomorphic to $(X \vee Y) \vee Z$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the triple integral \( \int_{0}^{6} \int_{0}^{12-2y} \int_{0}^{4-\frac{2y}{3}-\frac{x}{3}} x \, dz \, dx \, dy \).

Answer: 144


Question: Evaluate the limit: \[ \lim_{x\rightarrow 1}\left(\frac{\sqrt[3]{7+x^3}-\sqrt[2]{3+x^2}}{x-1}\right) \] without using L'Hôpital's rule.

Answer: -\dfrac{1}{4}


Question:
Consider continuous functions from the real line \(\mathbb{R}\) to a discrete space \(Y\). Determine which of the following statements is true:

(a) Any two continuous functions are never homotopic.
(b) Two continuous functions are homotopic if one of them is a constant function.
(c) Two continuous functions are homotopic if both are constant functions.
(d) Any two continuous functions are always homotopic.

Select the correct option.

Answer: C


Question: Determine the number of transpositions required to sort the sequence $(1, 3, 4, 2)$ into ascending order. Express your answer as the number of moves needed to transform the permutation matrix corresponding to this sequence into the identity matrix.

Answer: 2


Question: Evaluate the double integral \( \int_0^a \int_0^a \delta(x-y) \, dx \, dy \).

Answer: a


Question: Identify the group \((D, \circ)\) of the symmetries of the circle.

Answer: O(2)


Question: Given the function \( f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3) \) for all \( x \in \mathbb{R} \), find the value of \( f(1) - f(0) \).

Answer: -2


Question: Evaluate the integral \( \frac{1}{2 \pi} \int_{0}^{2 \pi} \ln (|r e^{it} - z|) \, dt \) for \( r > |z| \).

Answer: \ln r


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

Answer: \dfrac{1}{6}


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

Answer: 2


Question: Find \( \lim \limits_{(x,y) \rightarrow (0,0)} \frac{x^3 y^3 }{x^2+y^2} \).

Answer: 0


Question: Find the point on the curve \( \langle t^3, 3t, t^4 \rangle \) where the plane perpendicular to the curve is parallel to the plane \( 24x + 6y - 64z = 0 \).

Answer: (-8, -6, 16)


Question: Given a circle with radius \(a\) and a diameter \(AB\), chords are drawn perpendicular to \(AB\), intercepting equal arcs along the circumference of the circle. Find the limit of the average length of these chords as the number of chords tends to infinity.

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


Question: Evaluate the limit: \[ \lim_{x \to 0, y \to 0} \left(1 + x^2y^2\right)^{-1/(x^2+y^2)}. \]

Answer: 1


Question: Let \( f:[1,\infty)\to \mathbb{R} \) be a monotonic and differentiable function with \( f(1)=1 \). Determine the number of solutions \( N \) to the equation \( f(f(x)) = \frac{1}{x^2-2x+2} \). Find \( N \).

Answer: 1


Question: Find the absolute maximum and minimum values of the function \(-8x^3-6y^2+12xy=0\) in the region where \(x \geq 0\) and \(y \geq -1\).

Answer: \frac{1}{2}


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

Answer: 0


Question: Determine if the function \(f(x) = \frac{x^2 + x}{2} \mod 2^n\) produces all values from 0 to \(2^n - 1\) for integer inputs \(x\) in the range \([0, 2^n - 1]\).

Answer: Yes


Question: Let $S$ be the set of $8 \times 8$ symmetric positive definite matrices with trace $8$ and determinant $1$. Determine the cardinality of the set $S$. \( \text{card}(S) \).

Answer: 1


Question: Determine the fundamental period of the discrete sequence \( x[n] \), which is the sum of two product terms. The first product term has a fundamental period of \( \frac{16 \times 15}{\text{gcd}(16,15)} = 240 \), and the second product term has a fundamental period of \( \frac{8 \times 6}{\text{gcd}(8,6)} = 24 \). Calculate the fundamental period of \( x[n] \).

Answer: 240


Question: Let $z = 1 + i$ and $c = 1 - i$. If $z^c = \exp(w)$, find the value of $Re(w) + Im(w)$. \( \text{(where } Re(w) \text{ and } Im(w) \text{ are the real and imaginary parts of } w \text{)} \)

Answer: \dfrac{\pi}{2}


Question: Find the constant $c$ such that \( \lim_{x\to11}\frac{|x^2-6x|-|x^2+cx|}{x-11} = 22 \).

Answer: -16


Question:
Determine the number of different ordered quadruples \((a, b, c, d)\) of complex numbers such that:
\[a^2 = 1\]
\[b^3 = 1\]
\[c^4 = 1\]
\[d^6 = 1\]
\[a + b + c + d = 0\]

Answer: 7


Question: Determine whether the group $S_3 \oplus S_3$ contains an element of order 4, 6, 9, or 18. Justify your answer.

Answer: 6


Question:
Find a continuous function \( f: [0,1] \rightarrow \mathbb{R} \) that can be expressed as two distinct infinite polynomial series:
\[ f(x) = \sum_{n=0}^\infty c_n x^n \]
where the sequence \((c_n)_n\) is not unique.

Answer: 0


Question: Simplify the expression: \[ \frac{1}{\sqrt{12-2\sqrt{35}}} - \frac{2}{\sqrt{10+2\sqrt{21}}} - \frac{1}{\sqrt{8+2\sqrt{15}}} \]

Answer: \sqrt{3}


Question:
Determine the values of $n$ that satisfy the inequality:

$$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$

where $p_i$ denotes the $i$th prime number, and $A$ is defined by the relationship $p_n = 6A + r$ with $0 \leq r < 6$. What are the possible values of $n$?

Answer: n \geq 8


Question: Determine the number of solutions to the equation \( x \tan x = 1 \) for \( -2\pi \le x \le 2\pi \).

Answer: 4


Question: Define a sequence \(a_n\) such that \(a_n = n^n \mod 101\) for each positive integer \(n\). Determine the smallest positive integer \(d\) such that \(a_n = a_{n+d}\) for all \(n\).

Answer: 10100


Question: Is the set \( H = \{ h:[0,1] \rightarrow \mathbb{R} : h(x)= \sum_{j=1}^n a_j e^{b_jx} , a_j,b_j \in \mathbb{R}, n \in \mathbb{N} \} \) dense in \( C([0,1], \mathbb{R}) \) with respect to the uniform norm?

Answer: Yes


Question: For which \(\alpha\) is the integral \(\int_Q \frac{1}{(x^2+2y^2)^{\alpha}}\,dx\,dy\) finite, where \(Q=[0,1]\times[0,1]\)?

Answer: \alpha < 1


Question: Find the minimum value of the function \( f(x) = \frac{(1+x)^{0.8}}{1+x^{0.8}} \) for \( x \geq 0 \) without using derivatives.

Answer: 2^{-1/5}


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

Answer: \dfrac{4}{e}


Question: Find the largest positive integer $N$ such that \( \sqrt{64 + 32^{403} + 4^{N+3}} \) is an integer.

Answer: 2008


Question: Is the following statement true? For any prime number \( p \) and any pair of integers \( r_1 \) and \( r_2 \) between 1 and \( p-1 \), the ratio \( \frac{U(p, r_1, n)}{U(p, r_2, n)} \) approaches 1 as \( n \) approaches infinity, where \( U(p, r, n) \) denotes the number of primes less than \( n \) that are congruent to \( r \) modulo \( p \).

Answer: True


Question: Given that \(4a^2+9b^2+16c^2+\frac{1}{9a^2}+\frac{1}{16b^2}+\frac{1}{25c^2}=\frac{133}{30}\), find the value of \(1440a^2b^2c^2\), where \(a\), \(b\), and \(c\) are non-zero real numbers.

Answer: 1


Question: Provide an example of an ideal $A \subseteq \Bbb{C}[X]$ such that $A \subsetneq I(Z(A))$, where $Z(A)$ is the set of zeroes of all polynomials in $A$ and $I(Z(A))$ is the ideal of polynomials that vanish on $Z(A)$. Verify that $A$ is strictly contained in $I(Z(A))$.

Answer: (X^2)


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{a^{\tan x} - a^{\sin x}}{\tan x - \sin x} \]

Answer: \ln a


Question: Evaluate the limit: \[ \lim _{n\to \infty}\frac{e^n\times n!}{n^n} \]

Answer: \infty


Question: Evaluate the integral \( I = \int_{0}^{2\pi} e^{\cos x} \cos (\sin x)\,dx \). Verify if the solution process leads to \( I = 0 \).

Answer: 2\pi


Question: Find the coefficient of \(x^{24}\) in the expansion of \((1 + x + x^2 + x^3 + x^4 + x^5)^8\).

Answer: 98813


Question: Show that \( \int_0^{2\pi} \log|1-ae^{i\theta}| \, \mathrm{d}\theta = 0 \) when \(|a| = 1\).

Answer: 0


Question: Determine the value of \( \lim\sup_{n\to\infty} \frac{E_n}{n} \), where \( E_n \) is the expected value of the maximum period length of functions \( f:[n]\to[n] \).

Answer: 0


Question: Determine the order of convergence for the fixed point iteration of the function \( g(x) = \cos(x^{50}) - 1 \) at the fixed point \( p = 0 \).

Answer: 100


Question: Compute the integral \( \oint_{\partial D(3i,5)} \frac{z}{1-e^z} \, dz \) using the Cauchy Integral Formula.

Answer: 4\pi^2


Question: Let $H$ be a Hilbert space, and let $A$ and $B$ be positive linear operators from $H$ to $H$. If $A \ge B$, meaning $A - B$ is a positive operator, does it follow that $\|A\| \ge \|B\|$? Provide a proof or counterexample.

Answer: \|A\| \ge \|B\|


Question: Determine the closure of the set \( W = \{ f \in C[0,1] \mid f(0) = 0 \} \) with respect to the \( \|\cdot\|_1 \) norm.

Answer: L^1[0,1]


Question: Given $n = p \cdot q$, where $p$ and $q$ are large primes, and $m$ is an arbitrary value in $Z_n$, let $r_1$ be a value chosen uniformly at random from $Z^*_n$. Define $r_2 = r_1 \cdot m$. Is $r_2$ a uniformly random value in $Z_n$? Assume that factorizing $n$ into $p$ and $q$ is computationally hard.

Answer: No


Question: Find the variance of the random variable $x$ given that $x + y + z = 3$, where $x$, $y$, and $z$ are independent standard normal random variables.

Answer: \dfrac{2}{3}


Question:
Given the function \( f(x,y) \) satisfying the conditions:
\[ f(x,0) = 1, \quad x \in \mathbb{R} \]
and
\[ f[f(x,y),z] = f(z,xy) + z, \quad x,y,z \in \mathbb{R} \]
find the value of:
\[ f(2014,6) + f(2016,6) \]

Answer: 24182


Question: Determine the minimum number of times a biased die must be rolled so that the probability of rolling a 6 can be estimated with an uncertainty of 0.01 and a confidence level of at least 0.9. Use the Central Limit Theorem to approximate this number.

Answer: 6766


Question: Diagonalize the quadratic form \( q(x_1, x_2, x_3, x_4) = x_1x_2 + x_2x_3 + x_3x_4 \) and determine its signature.

Answer: (2, 2)


Question: Given the probability density function \( \frac{1}{\pi a_0^3}e^{-2r/a_0} \), find the most probable value of \( r \) for the wavefunction of the lowest energy state of hydrogen, \( \psi(r) = \sqrt{\frac{1}{\pi a_0^3}}e^{-r/a_0} \), where \( r \) is the radial distance between the electron and nucleus and \( a_0 \) is the Bohr radius.

Answer: a_0


Question: Let $X$ and $Y$ be topological spaces where $X$ is locally contractible and has no dense finite subset, and $Y$ is path-connected. Given $n$ pairs of points $(x_i, y_i)$ with $x_i \in X$ and $y_i \in Y$ for $1 \leq i \leq n$, and a continuous map $f: X \to Y$, determine if there exists a continuous map $g: X \to Y$ homotopic to $f$ such that $g(x_i) = y_i$ for each $i$. Provide a justification for your answer.

Answer: Yes


Question: A person rolls a 6-sided die until they get six consecutive 6's. If each roll takes 5 seconds, how long is the expected time to achieve this sequence?

Answer: 279930


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

Answer: 0


Question: Determine the value of \( \lambda \) that maximizes or minimizes the expression \( \psi(\lambda) = \int_{-\infty}^{\infty}\{\Phi(\lambda x)\}^{K}\phi(x)dx \), where \( \Phi() \) and \( \phi() \) are the cumulative distribution function and probability density function of the standard normal distribution, respectively.

Answer: 0


Question: Let \( \alpha \) be a root of \( x^7 - 1 \) with \( \alpha \neq 1 \). Determine the degree of the field extension \( [\mathbb{Q}(\alpha + \alpha^{-1}):\mathbb{Q}] \).

Answer: 3


Question: Given a prime $p$ and a generator $g$ of the multiplicative group $\mathbf{Z}_p^*$, find the smallest positive integer $k$ such that for all integers $x$, the congruence $g^x \equiv g^{x \mod k} \pmod{p}$ holds.

Answer: p - 1


Question: Determine whether the following statement is true for the Cantor function \( c : [0,1] \to [0,1] \): \[ c( \inf \{ x : c(x) \ge y \}) = y \] for all \( y \in [0,1] \).

Answer: True


Question: Prove that \( \lim_{n \to \infty} \int_{\|x \|^2 \le n} \frac{1}{(2\pi)^{n/2}} e^{-\|x\|^2/2} \, dx = \frac{1}{2} \). Here, \( B_n = \{x \in \mathbb{R}^n : \|x \|^2 \le n \} \) and the Gaussian measure is the probability measure on \( \mathbb{R}^n \) with density \( \frac{1}{(2\pi)^{n/2}} e^{-\|x\|^2/2} \).

Answer: \dfrac{1}{2}


Question:
Compute the following limit involving the hypergeometric function \(_2F_1(1,b;b+1;z)\) around the branch point at \(z=1\):
\[ \lim_{\epsilon\rightarrow 0^+} \ _2F_1(1,b;b+1;1+\epsilon)-\ _2F_1(1,b;b+1;1-\epsilon) \]

Answer: -i \pi b


Question:
Determine the number of ordered triples \((A_1, A_2, A_3)\) of sets such that:

1. \(A_1 \cup A_2 \cup A_3 = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\)
2. \(A_1 \cap A_2 \cap A_3 = \emptyset\)

Express the answer in the form \(2^{a}3^{b}5^{c}7^{d}\), where \(a, b, c,\) and \(d\) are nonnegative integers.

Answer: 2^{10}3^{10}


Question: Evaluate the limit \( \lim_{\epsilon\to 0} \int_{[-M, M]\setminus(-\epsilon, \epsilon)} \frac{1}{x} \,dx \).

Answer: 0


Question:
Find the characteristic curves for the partial differential equation \( \left(\frac{\partial u}{\partial x}\right)\left(\frac{\partial u}{\partial y}\right) - u = 0 \) using the Charpit method, given that the curves pass through the initial curve \( x = 0, y = \tau, u = \tau^2, \tau \in \mathbb{R} \). Choose the correct set of characteristic curves from the following options:

1. \(x = 3\tau(e^s-1), y = \dfrac{\tau}{2}(e^{-s}+1), u = \tau^2e^{-2s}\)
2. \(x = 2\tau(e^s-1), y = {\tau}(2e^{2s}-1), u = \dfrac{\tau^2}{2}(1+e^{-2s})\)
3. \(x = 2\tau(e^s-1), y = \dfrac{\tau}{2}(e^{s}+1), u = \tau^2e^{2s}\)
4. \(x = \tau(e^{-s}-1), y = {-2\tau}\left(e^{-s}-\dfrac{3}{2}\right), u = \tau^2(2e^{-2s}-1)\)

Answer: 3


Question: Let $X$ be an infinite set and ${\cal P}(X)$ its powerset. A subset ${\cal F} \subseteq {\cal P}(X)$ is said to have the splitting property (SP) if there exists $A \in {\cal P}(X)$ such that for all $F \in {\cal F}$, we have $F \cap A \neq \emptyset \neq F \cap (X \setminus A)$. Let $\text{SP}(X)$ denote the collection of all subsets of ${\cal P}(X)$ with the splitting property, ordered by $\subseteq$. If ${\cal F} \in \text{SP}(X)$, does there exist ${\cal M} \in \text{SP}(X)$ such that ${\cal M}$ is maximal in $(\text{SP}(X), \subseteq)$ and ${\cal F} \subseteq {\cal M}$?

Answer: Yes


Question: Find the variance of the AR(2) process defined by \(X_i = 0.3X_{i-2} + u_i\) for \(i=1,...,n\), where \(X_0 = X_1 = 0\) and \(u_i\) are independent and identically distributed as \(N(0, 3^2)\).

Answer: \dfrac{900}{91}


Question: Find the term with the highest power of \(x\) in the expression \((x^3-2)^{16}-(x^4 + 3)^{12}\).

Answer: -32x^{45}


Question: Construct an uncountable set $X \subset \mathbb{R}_{\geq 0}$ such that for all distinct $a, b \in X$, $a + b \not\in X$. Provide an example of such a set.

Answer: [1, 2)


Question: Given the polynomial \(x^4 + px^3 + qx^2 + rx + s = 0\) with rational coefficients and four complex roots, two of the roots add up to \(3 + 4i\) and the other two have a product of \(13-i\). Compute the value of \(q\).

Answer: 51


Question: True or false: There exists a field $K$ and a positive integer $n$ such that every polynomial over $K$ of degree $n$ is reducible, but there exists an irreducible polynomial of degree $n+1$.

Answer: True


Question: Calculate the mass of the surface \( S = \{(x,y,z) \mid x^2+y^2\le 2x, z=\sqrt{x^2+y^2} \} \) with mass density \( f(x,y,z) = x^2+y^2+z^2 \).

Answer: 3\sqrt{2}\pi


Question:
Consider a simple Lie algebra \( L \) over \( \mathbb{C} \) with a maximal toral subalgebra \( H \) and a root system \( \Phi \) relative to \( H \). The Lie algebra \( L \) has a Cartan decomposition:
\[ L = H \oplus \bigoplus_{\alpha \in \Phi} L_{\alpha}. \]
Let \( \sigma \) be an automorphism in the Weyl group of \( \Phi \). This automorphism induces an automorphism of \( H \) and can be extended to an automorphism of \( L \). If \( \sigma \) maps a root \( \alpha \) to another root \( \beta \), does the extension of \( \sigma \) necessarily map the root space \( L_{\alpha} \) to \( L_{\beta} \)?

Answer: Yes


Question: Is it true that if a function \( f: D \subseteq \mathbb{C}^n \to \mathbb{C} \) is holomorphic, where \( D \) is a domain in \( \mathbb{C}^n \) and \( n > 1 \), then \( f \) is \( C^{\infty} \)?

Answer: Yes


Question:
Determine whether there exists a function \( f: (-\delta, \delta) \to \mathbb{R} \) that satisfies the following conditions for a real number \( \delta > 0 \):

1. \( f \) is differentiable on \((-\delta, \delta)\).
2. The second derivative \( f''(0) \) exists.
3. There exists a sequence \( \{x_n\} \) with \(-\delta < x_n < \delta\) and \( \lim_{n \to \infty} x_n = 0 \) such that \( f' \) is not continuous at any \( x_n \).

Answer: Yes


Question:
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be independent and identically distributed random variables with \(P(X_{i} = 1) = P(X_{i} = -1) = p\) and \(P(X_{i} = 0) = 1-2p\) for all \(i = 1, 2, \ldots, n\). Define
\[a_{n} = P\left( \prod_{i=1}^{n} X_{i} = 1 \right), \quad b_{n} = P\left( \prod_{i=1}^{n} X_{i} = -1 \right), \quad c_{n} = P\left( \prod_{i=1}^{n} X_{i} = 0 \right).\]
Which of the following is true as \(n\) tends to infinity?

(A) \(a_n \rightarrow \frac{1}{3}, \; b_n \rightarrow \frac{1}{3}, \; c_n \rightarrow \frac{1}{3}\)

(B) \(a_n \rightarrow p, \; b_n \rightarrow p, \; c_n \rightarrow 1-2p\)

(C) \(a_n \rightarrow \frac{1}{2}, \; b_n \rightarrow \frac{1}{2}, \; c_n \rightarrow 0\)

(D) \(a_n \rightarrow 0, \; b_n \rightarrow 0, \; c_n \rightarrow 1\)

Answer: D


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

Answer: \infty


Question: Find the positive integer \( n \) such that \( \int_{0}^{\frac{\pi}{2}} x^n \sin x \, dx = \frac{3}{4}(\pi^2 - 8) \).

Answer: 3


Question: Evaluate the improper integral: \[ \int_{1}^{\infty} \frac{dx}{x \sqrt{x^2 - 1}}. \]

Answer: \dfrac{\pi}{2}


Question: Maximize \( \int_{0}^{1} f(x)^5 \, dx \) over all functions \( f: [0,1] \to [-1,1] \) such that \( \int_{0}^{1} f(x)^3 \, dx = 0 \) and \( \int_{0}^{1} f(x) \, dx = 0 \).

Answer: 0


Question: Determine the genus $g$ of a closed Riemann surface if its tangent bundle $T$ is trivial, given that the degree of the tangent bundle is given by the formula $\deg(T) = 2 - 2g$. What is the value of $g$?

Answer: 1


Question: Determine which function is asymptotically larger: \(\lg(\lg^* n)\) or \(\lg^*(\lg n)\).

Answer: \lg^*(\lg n)


Question: Evaluate the limit using l'Hopital's Rule: $$ \lim _{x\to 6}\left(\frac{x-5}{x-6}- \frac{1}{\ln(x-5)}\right)$$

Answer: \dfrac{1}{2}


Question: Evaluate \( \lim_{n\to\infty} n \int_0^1 x^n e^x \, dx \).

Answer: e


Question: Determine the set of values of $x$ for which the Maclaurin series of $\ln(1+x)$ converges to the function itself.

Answer: (-1, 1]


Question: Let $A$ and $B$ be positive operators on a finite-dimensional Hilbert space $H$ such that $A \leq B$. Is it true that $||Ax|| \leq ||Bx||$ for all $x \in H$?

Answer: No


Question: Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(x) = f(\sin x) \) for all \( x \in \mathbb{R} \).

Answer: f(x) = c


Question: Evaluate the integral: $$\int_0^\infty \int_{-x}^0 \frac{e^{-x^2/2}}{\sqrt {2\pi}} \frac{e^{-y^2/2}}{\sqrt {2\pi}}\, dy\, dx.$$

Answer: \dfrac{1}{8}


Question: Evaluate the contour integral \( \int_C \frac{(z+a)(z+b)}{(z-a)(z-b)} \mathrm{d}z \), where the modulus of \(a\) and \(b\) are less than 1, and the contour \(C\) is the anticlockwise unit circle \(|z|= 1\).

Answer: 4\pi i(a + b)


Question: Calculate the integral of the Gaussian distribution's probability density function (PDF) from a constant \( a > 0 \) to infinity. Express your answer in terms of the cumulative distribution function (CDF) of the normal distribution.

Answer: 1 - \Phi(a)


Question: Evaluate the integral: \[ \int_{-\infty}^{\infty} \frac{\sin^2(\pi N x)}{(\pi x)^2} \, dx. \]

Answer: N


Question: Determine the number of ways to distribute 80 balls into 5 distinguishable bins such that no bin contains more than 24 balls.

Answer: 116371


Question: Find \( \partial_{\overline{z}}\left(\frac{z}{z-w}\right) \) using the fact that \( \partial_{\overline{z}}\left(\frac{1}{z-w}\right) = \delta(z-w) \).

Answer: w \delta(z - w)


Question: Find the number of ways to collect $50 from 15 distinct people, where the first person can give either 0, 1, or 8 dollars, and each of the other 14 people can give either 1 or 5 dollars. Use generating functions to determine the coefficient of $x^{50}$ in the expansion of \((1 + x + x^8)(x + x^5)^{14}\).

Answer: 5434


Question: Find the value of \( x \) such that \( 2^{12^7 + 3} \equiv x \pmod{36} \).

Answer: 8


Question: Find \( F''(\pi) \) if \[ F(x) = \int_x^{\sin x}\left( \int_0^{\sin t}\sqrt{1+u^4} \, du \right) dt. \] Use the fundamental theorem of calculus to solve this problem.

Answer: 2


Question: Evaluate the integral: \[ \int_0^{\infty} \frac{\ln x}{1+x^2} \, dx \]

Answer: 0


Question: Given an arithmetic function $f$ such that \( \sum_{d\mid n}f(d)=\begin{cases}0 & \text{ if } n \text{ is divisible by a square of a prime} \\ n & \text{ otherwise}\end{cases} \), calculate $f(6300)$. Use the Möbius inversion formula to find your answer.

Answer: -180


Question: Suppose $W(t)$ is a standard Brownian motion. Calculate the probability $P(-1 < W(3) - W(1) < 1)$. Provide your answer as a percentage.

Answer: 52.05\%


Question: Evaluate the integral \( \int^{1}_{0}\int^{\sqrt{1-x^{2}}}_{0}\int^{\sqrt{1-x^{2}-y^{2}}}_{0} \dfrac{1}{x^{2}+y^{2}+z^{2}}dzdxdy \) using spherical coordinates.

Answer: \dfrac{\pi}{2}


Question: If $z^5 - 32$ can be factorized into linear and quadratic factors over real coefficients as $(z^5 - 32) = (z - 2)(z^2 - pz + 4)(z^2 - qz + 4)$, find the value of $p^2 + 2p$. Use the identity $a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$ to assist in your solution.

Answer: 4