Question: Is the index of $G_{pq}$ in $G_p \cap G_q$ always finite, where $G$ is a group, $G_n = \langle\{g^n| g \in G\}\rangle$, and $p$ and $q$ are coprime integers?

Answer: No


Question: In triangle \(ABC\), where \(AB = AC\), let \(I\) be the incenter. Given that \(BC = AB + AI\), find the measure of \(\angle BAC\).

Answer: 90^\circ


Question: Let $F = \{E_1, E_2, \ldots, E_n\}$ be a collection of $n$ subsets of a set $X$, where $n$ is a positive integer. Determine the number of distinct sets in the sigma-algebra $\sigma(F)$ generated by $F$. Provide your answer as a single integer.

Answer: 2^{2^n}


Question: Determine the set of real numbers \( x \) for which the series \( \sum^{\infty}_{n=1} \frac{n!x^{2n}}{n^n(1+x^{2n})} \) converges. Choose from the following options: \( \{0\} \), \( \{-1 \leq x \leq 1\} \), \( \{-1 < x < 1\} \), \( \{-\sqrt{e} \leq x \leq \sqrt{e}\} \), or \( \mathbb{R} \).

Answer: \mathbb{R}


Question: Determine the number of integer pairs \((a, b)\) such that \(1 \le a, b \le 42\) and \(a^9 \equiv b^7 \pmod{43}\).

Answer: 42


Question: Evaluate the integral \( \int_{0}^{2\pi} e^{-it} e^{e^{it}} \, dt \).

Answer: 2\pi


Question: Determine the residues at all isolated singularities of the function \( f(z) = \tan^3(z) \).

Answer: 1


Question: Determine whether there exists a connected topological space $X$ such that $X$ is homeomorphic to $X \times X$. Provide a justification for your answer.

Answer: Yes


Question: Is Faa di Bruno's formula applicable to the composition \((f \circ g)^{(n)}(x)\) where \(f: \mathbb{R} \rightarrow X\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\), with \(X\) being a real Banach space?

Answer: Yes


Question: A line through the point \(P(l,3)\) intersects the ellipse \(\frac{x^2}{16} + \frac{y^2}{9} = 1\) at points \(A\) and \(D\), and intersects the \(x\)-axis and \(y\)-axis at points \(B\) and \(C\) respectively. Given that \(PA \cdot PD = PB \cdot PC\), find the minimum value of \(|l|\).

Answer: 8


Question: A large playlist consists of songs with lengths that have a mean of 2 minutes and 10 seconds and a standard deviation of 15 seconds. What is the probability that more than 36 randomly chosen songs are required to fill a program that is 76 minutes long?

Answer: 0.0912


Question: Evaluate the limit: \[ \lim_{a \rightarrow 0^+}a \int_{b}^{-\log a}\frac{e^y}{y} \, dy \]

Answer: 0


Question: Determine the maximum possible value of the expression \[ T = \frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3} + \frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3} + \frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3} \] where $x, y, z \in \mathbb{R}_+$. Provide your answer as a single value.

Answer: \dfrac{3}{16}


Question: Consider a Noetherian normal local domain $(A,m)$ with $K = \text{Quot}(A) \subset E, F$ as Galois extensions of $K$. Let $B=\overline{A}^{E}$, $C=\overline{A}^F$, and $D=\overline{A}^{EF}$. Choose primes $m_B, m_C$, and $m_D$ in the corresponding rings over $m$. Is it true that the separable part of the residue field of $m_D$ is generated by the separable parts of the residue fields of $m_B$ and $m_C$ over the residue field of $m$? The 'separable part' refers to the maximal separable extension of the residue field of $m$ contained in the residue field of $q$, where $q$ is one of $m_B, m_C$, or $m_D$. Provide a justification for your answer.

Answer: Yes


Question: Let \((G, \cdot)\) be a group with a group action \(s : G \to S_4\) such that \(s(f) = (12)\) and \(s(g) = (34)\) for certain elements \(f, g \in G\). Can \((G, \cdot)\) be a cyclic group? Justify your answer.

Answer: No


Question: Evaluate the line integral \( \int_C F \cdot ds \) for the vector field \( F(x,y,z) = \left( 2xyz + \sin(x)\right)\mathbf{i} + x^2z \mathbf{j} + x^2 y \mathbf{k} \) along the parametrized curve \( c(t) = \left(\cos^5(t),\sin^3(t),t^4\right) \) for \( 0 \le t \le \pi \).

Answer: 0


Question: Let $M$ be an $n \times m$ matrix over $\mathbb{F}_2$ with no repeated columns, and suppose that $m \leq 2^{n-1}$. Is it always possible to find a smaller matrix with distinct columns by taking a linear combination of the rows of $M$?

Answer: Yes


Question: Find the residue of the function \( f(z) = (z+1)^2 e^{3/z^2} \) at \( z = 0 \).

Answer: 6


Question: Maximize $z$ given the constraints $x + y + z = 3$ and $x^2 + y^2 + z^2 = 6$, where $x$, $y$, and $z$ are real numbers. What is the largest possible value of $z$?

Answer: 1 + \sqrt{2}


Question: Let $(U_n)_n$ be an arbitrary sequence of open convex subsets of the unit disk $D(0,1) \subseteq \mathbb{R}^2$ such that $\sum_{n=0}^\infty \lambda(U_n) = \infty$, where $\lambda$ is the Lebesgue measure. Does there exist a sequence $(q_n)_n$ in $\mathbb{R}^2$ such that $D(0,1) \subseteq \bigcup_{n=0}^\infty (q_n + U_n)$? Here, $q_n + U_n$ is defined as $\{x \in \mathbb{R}^2 \mid x - q_n \in U_n\}$.

Answer: Yes


Question: Let $T_j$ be a sequence of compact operators on a Hilbert space $H$ which converges strongly to the identity, i.e., for each $v \in H$, the sequence $\| T_j v - v \|$ tends to zero. Determine if there must exist an index $j$ such that the spectrum of $T_j$ contains a non-zero number.

Answer: Yes


Question: Calculate the flux integral of the vector field \( \vec{F} = (x,y,1+z) \) out of the surface defined by \( z = 1-x^2-y^2 \) for \( z \geq 0 \).

Answer: \dfrac{5\pi}{2}


Question: Let $V$ be a $K$-vector space of dimension $n$. Consider a hyperplane $H$ given by the kernel of the linear map $l_H: V \to K$ defined by \( l_H(x) = a_1x_1 + \cdots + a_nx_n \), where $x_1, \ldots, x_n$ are the coordinates of $x$ in a chosen basis of $V$. Let $P(V)$ be the projective space associated with $V$. Define the map $l_{H'}: P(V) \to K$ by \( l_{H'}([x_1: \cdots : x_n]) = a_1x_1 + \cdots + a_nx_n \). Does the kernel of $l_{H'}$ define a projective hyperplane in $P(V)$?

Answer: Yes


Question: Consider a sequence of tosses of a balanced die. Let \(X_k\) be the result of the \(k\)-th toss, and define \(Y_n = \max(X_1, X_2, \ldots, X_n)\). Determine the almost sure convergence of the sequence \((Y_n)_{n \geq 1}\).

Answer: 6


Question: Is the natural map from the purely algebraic pushout $A \circledast_C B$ to the amalgamated free product $A *_C B$ injective for all choices of unital C*-algebras $A$, $B$, and $C$?

Answer: Yes


Question: Let $(X_n)_{n\in \mathbb{N}}$ be independent continuous random variables with cumulative distribution function given by \( f_n(x) = \frac{n+1}{2}|x|^n \mathbb{1}_{[-1,1]}(x) \). Define \( S_n = \sum_{k=1}^n X_k \). Calculate the weak limit of \( \frac{S_n}{\sqrt{n}} \).

Answer: \mathcal{N}(0,1)


Question: For which natural numbers \(x > 0\) is the expression \(\sum_{n=0}^{4}{x^n}\) a perfect square?

Answer: 3


Question: What is the probability that a positive divisor of 8748 million is the product of exactly 20 non-distinct primes?

Answer: \dfrac{1}{168}


Question: Evaluate the surface integral \( \int_{S} \int (\nabla \times \vec{F}) \cdot d\vec{s} \) where \( S \) is the ellipsoid defined by \( x^2 + y^2 + 2z^2 = 16 \) and \( \vec{F} = \sin(xy)\vec{i} + e^x\vec{j} - yz\vec{k} \).

Answer: 0


Question: Let $R$ be a ring with identity such that each (right) ideal of $R/J(R)$ is idempotent, where $J(R)$ is the Jacobson radical of $R$. Is $R/J(R)$ necessarily von-Neumann regular?

Answer: Yes


Question: In a finite measure space $(X, \mu)$, does almost uniform convergence imply $L^1$ convergence?

Answer: No


Question: If $\alpha, \beta, \gamma$ are real roots of the equation \( \frac{1}{x} + \frac{1}{x-1} + \frac{1}{x-2} = \frac{11}{54}x^2 \), find the value of \( \lfloor \alpha \rfloor + \lfloor \beta \rfloor + \lfloor \gamma \rfloor \).

Answer: 4


Question: If \( \mathbb{E}[f \mathbb{1}_{[0,\theta]}]=0 \) for all \( \theta \geq 0 \), does it imply that \( f=0 \) almost surely for a function \( f \) on the nonnegative real line?

Answer: Yes


Question: Determine the maximum possible number of solutions for the equation \( f(z) = az + b \), where \( a \) and \( b \) are complex numbers, given that \( f : \mathbb{C} \to \mathbb{C} \) is a holomorphic function with the property that the real part of \( f''(z) \) is strictly positive for all \( z \in \mathbb{C} \).

Answer: 2


Question: Determine the minimal dimension of an affine subspace in \(\mathbb{R}^n\) that can divide an open, contractible set \(U \subseteq \mathbb{R}^n\) into disconnected components, where \(n \ge 3\).

Answer: n - 1


Question: Find the last two digits of \(3^{3^{2014}}\).

Answer: 83


Question: Evaluate the improper integral \( \int_{-8}^1 \frac{dx}{x^{1/3}} \).

Answer: -\dfrac{9}{2}


Question: Does there exist a $C^{\infty}$-function $f:\mathbb{R}\to\mathbb{R}$ whose Taylor series has zero radius of convergence at every point in $\mathbb{R}$?

Answer: Yes


Question:
Evaluate the following limit to study the differentiability of the function at \( x = 3 \):

\[ g(x) = \frac{x - 3}{\sqrt[3]{x + 5} - 2} \quad \text{for} \quad x \neq 3, \quad g(3) = 12. \]

Calculate the limit:

\[ \lim_{x \to 3} \left( \frac{\frac{x - 3}{\sqrt[3]{x + 5} - 2} - 12}{x - 3} \right) \]

Answer: \dfrac{1}{2}


Question: Minimize the function \(f(n,k) = (n-1) - \sqrt{(n-1)^2 - 4(k-1)(n-k-1)}\) over the integers \(n\) and \(k\), subject to the constraints \(k \in \mathbb{N}\), \(n-2 \ge k \ge 2\), and \(n \in \mathbb{N}\), \(n \ge 4\).

Answer: 3 - \sqrt{5}


Question: In the polynomial quotient ring \(B = \mathbb{Z}_2[x]/(x^3+x^2+x+1)\), find a polynomial \([g(x)]\) such that \([x^2+1] \cdot [g(x)] = [0]\).

Answer: [x + 1]


Question: Find the value of \( \sin^{-1}\frac{12}{13} + \cos^{-1}\frac{4}{5} + \tan^{-1}\frac{63}{16} \).

Answer: \pi


Question: Find the Lanczos derivative of the function \(f(x) = |x|\) at \(x = 0\).

Answer: 0


Question: Let $P$ be a $p \times p$ projection matrix, and $I$ be the identity matrix of the same size. If $(I-P)A = 0$ for a non-zero matrix $A$, prove that $A = PC$ for some $p \times p$ constant matrix $C$, assuming $P \neq I$. Provide a justification for your answer.

Answer: A = PC


Question:
Compute the following limit:
\[ \lim_{n\to\infty}\max_{x\in[0,\pi]}\sum_{k=1}^n\frac{\sin(kx)}{k} \]
Choose the correct answer from the options below:

1. \(\displaystyle\int_0^\pi\frac{\cos^2(x)}{x^2}~\mathrm{d}x\)
2. \(\displaystyle\int_0^\pi\frac{\sin^2(x)}{x^2}~\mathrm{d}x\)
3. \(\displaystyle\int_0^\pi\frac{\sin(x)}{x}~\mathrm{d}x\)
4. \(\displaystyle\int_0^\pi\frac{\cos(x)}{x}~\mathrm{d}x\)

Answer: 3


Question: Determine whether the quartic polynomial \( P(x) = x^4 - 6x^3 + 9x^2 - 3x \) can take on rational square values for infinitely many rational values of \( x \).

Answer: Yes


Question: Evaluate the integral \( \int_C \frac{f(z)}{(z-1)^{2020}}dz \), where \( f(z) = u(x,y) + iv(x,y) \) is an entire function satisfying \( au + bv \ge \ln(ab) \) for \( a > 1, b > 1 \). The contour \( C \) is an equilateral triangle of side 1 with centroid at \( z = 1 \).

Answer: 0


Question: Can a small part of a 2-Torus be locally isometrically embedded in an affine \( \mathbb{R}^3 \) space?

Answer: Yes


Question: Evaluate the triple integral \( \iiint\limits_R (xy + z^2) \, dV \), where the region \( R \) is defined by \( \{(x,y,z): 0 \le z \le 1-|x|-|y|\} \).

Answer: \dfrac{1}{15}


Question: Let $A$ be an $8 \times 8$ matrix with real entries satisfying the equation $A^2 - 6A + 9I_8 = 0_8$. Determine the eigenvalues of the matrix $A$.

Answer: 3


Question: A board game is played on a hexagonal grid of 19 tiles. A 'traveler' token starts on the center tile. Each turn, a die is rolled to determine which neighboring tile the traveler moves to, with all six directions being equally likely. The game ends on the turn that the traveler leaves the board. What is the expected number of turns until the game ends?

Answer: \dfrac{55}{9}


Question: Evaluate the integral \( \int_0^\infty e^{-x^{1/3}} \, dx \).

Answer: 6


Question: Evaluate the integral \( \int_{-\infty}^0 \delta(t) \, dt \), where \( \delta(t) \) is the Dirac delta function.

Answer: 1


Question: Let A be the sum of the digits of \(16^{16}\). Let B be the sum of the digits of A. Find the sum of the digits of B without calculating A.

Answer: 7


Question: Find the norm of the operator \( T: \mathbb{R}^2 \to \mathbb{R}^2 \), defined by \( T(x,y) = (x+2y, 2x+y) \). Use the Lagrange multiplier theorem to determine the operator norm \( \|T\| = \sup_{\|x\|=1} \frac{\|T(x)\|}{\|x\|} \).

Answer: 3


Question: Compute the Galois group of the splitting field of the polynomial \( t^4 - 3t^2 + 4 \) over \( \mathbb{Q} \).

Answer: V_4


Question: Determine the number of elements in the quotient ring \( \mathbb{Z}[i]/(2-i) \).

Answer: 5


Question: Find all functions \( f : ( 0 , + \infty ) \to ( 0 , + \infty ) \) such that \[ f \big( f ( x y ) + 2 x y \big) = 3 x f ( y ) + 3 y f ( x ) \] for all \( x , y \in ( 0 , + \infty ) \).

Answer: f(x) = 4x


Question: Let $S$ be a smooth projective surface over $\mathbb{C}$ with Picard number $\rho(S) = 1$, and let $L$ be an ample line bundle on $S$. For a positive integer $d$ such that $dL$ is very ample, consider a very general member $D$ of the linear system $|dL|$. Let $T_D \rightarrow S$ be the degree $d$ cyclic cover branched along $D$. Is the Picard number $\rho(T_D)$ equal to 1 for a sufficiently large integer $d$?

Answer: 1


Question: Calculate the limit: \[ \lim_{n \to \infty} \sqrt[n]{|\sin n|} \]

Answer: 1


Question: Given an integer $N$, consider the set $S = \{0, 1, \ldots, 2^N - 1\}$. Determine the number of subsets $A \subseteq S$ such that the XOR of all elements in $A$ is zero. Note that the XOR sum of an empty set is zero, and XOR here is a bitwise operation.

Answer: 2^{2^N - N}


Question: Determine the positive values of $P$ for which the series \( \sum_{n=1}^{\infty}\frac{(n!)^2}{(Pn)!} \) converges.

Answer: P \geq 2


Question: Let $A$ be a $3 \times 3$ real matrix such that $A^4 = I$ and $A \neq \pm I$. Does it imply that $A^2 + I = O$?

Answer: No


Question: Determine whether the following statement is true: For the function \( \rho(x, x') \) defined as \( \rho(x, x'):=\inf_{\lambda\in\Lambda}\Big\{\max\Big(||x\circ \lambda-x'||, ||\lambda-I||\Big)\Big\} \) for all \( x, x'\in\Omega \), where \( \Omega:=D([0,1],\mathbb{R}) \) is the space of cadlag functions and \( \Lambda \) is the collection of non-decreasing continuous onto functions \( \lambda: [0,1]\to [0,1] \), does \( \rho(x,x')=0 \) imply \( x=x' \)?

Answer: True


Question: Compute the limit of the series \( \sum_{n=3}^\infty \frac{n^2}{2^n} \).

Answer: \dfrac{9}{2}


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

Answer: \dfrac{\pi}{4}


Question: Calculate the work done by the force field \( F(x,y,z) = e^{yz}i + e^{y}j - e^{x}k \) along the curve \( C \), which is the intersection of the surfaces \( z = x^2 + y^2 \) and \( x = 1 \), from the point \((1,2,5)\) to \((1,0,1)\). Use the parameterization \( x(t) = 1 \), \( y(t) = -t \), and \( z(t) = 1 + t^2 \) for \(-2 \leq t \leq 0\). Verify if the work integral \( \int_{-2}^{0} -e^{-t} - e \, dt \) is correctly calculated as \( 1 - e^{2} + 2e \).

Answer: 1 - e^{2} + 4e


Question: Is it possible that \(\liminf_{n\to \infty } nU_n > 0\) where \(U_n \sim \mathcal{U}[0,1]\) are independent and identically distributed random variables? Justify your answer.

Answer: No


Question: Does there exist a subring of \( \mathbb{Q} \) that is not Noetherian?

Answer: No


Question: Suppose \( x \) is a positive real number such that \( \{x\}, [x] \), and \( x \) are in a geometric progression. Find the least positive integer \( n \) such that \( x^n > 100 \). Here, \([x]\) denotes the integer part of \( x \) and \( \{x\} = x - [x] \).

Answer: 10


Question: Given the total differential \( dF = 2xy e^{xy^2} dy + y^2e^{xy^2} dx \), find the function \( F(x,y) \).

Answer: e^{xy^2}


Question: Find the integer solution for $x$ in the congruence $28^x \equiv 2 \pmod{43}$. Provide the smallest non-negative integer $x$ that satisfies this equation.

Answer: 39


Question: Determine the number of ring homomorphisms from \( \mathbb{Z}[x,y] \) to \( \mathbb{F}_2[x]/(x^3+x^2+x+1) \).

Answer: 64


Question: Evaluate the limit: \[ \lim_{n \to \infty}\frac{2^{\sqrt{\log (\log(n))}}}{(\log(\log(n)))^2\sqrt n} \]

Answer: 0


Question: Consider a sequence of equicontinuous functions \( f_k : \mathbb{R} \to \mathbb{R} \) such that for every \( k \), \( f_k(0) = 0 \). Does the sequence \( \langle f_k \rangle \) have a pointwise convergent subsequence?

Answer: Yes


Question: For what values of $k$ do the quadratics $x^2 + kx + 1$ and $kx^2 + x + 1$ share exactly one root?

Answer: -2


Question: Let $A$ be a set and $G$ a function. Suppose $f_1$ and $f_2$ are functions mapping the natural numbers $\omega$ into $A$. Assume that for each $n$ in $\omega$, both $f_1 \restriction n$ and $f_2 \restriction n$ belong to the domain of $G$, and that $f_1(n) = G(f_1 \restriction n)$ and $f_2(n) = G(f_2 \restriction n)$. Prove that $f_1 = f_2$.

Answer: f_1 = f_2


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

Answer: \ln 2


Question:
Let \( f \) be a continuous function on the interval \([0,1]\). Evaluate the limit:
\[
\lim_{n\to \infty} (n+1) \int_0^1 x^n f(x) \, \mathrm{d}x
\]

Answer: f(1)


Question: Evaluate the triple integral \( \iiint_D \frac{y}{1+\sqrt{z}}\,dx\,dy\,dz \) over the region \( D=\{(x,y,z)\in\mathbb{R^3}: x^2+y^2\le z\le 1\} \). Can the integral be concluded to be zero due to the symmetry of the integrand?

Answer: 0


Question: Is every convex subset of a product of two trees with the $l_1$-metric a median space? Specifically, for every three points A, B, and C in the subset, does there exist a point D in the subset such that D lies on some geodesics connecting A and B, B and C, and A and C?

Answer: Yes


Question: Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be defined by $f(x) = cx$ for some constant $c \neq 0$, and let $S$ be a regular surface. Express the Gaussian curvature of $f(S)$ in terms of the Gaussian curvature of $S$. Provide your answer in terms of $c$ and the Gaussian curvature of $S$.

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


Question: Let $\pi:Y\rightarrow \mathbb{P}^3$ be the blow-up of two points $p,q\in\mathbb{P}^3$, and then of the strict transform of the line $L$ spanned by them. Let $E_p,E_q, E_{p,q}$ be the exceptional divisors over $p,q$, and $L$, respectively. $E_p$ and $E_q$ are isomorphic to the blow-up of a point in $\mathbb{P}^2$, and $E_{p,q}\cong \mathbb{P}^1\times\mathbb{P}^1$. Let $\sigma_{p,q}\subset E_{p,q}$ be the class of the $\mathbb{P}^1$ which is not contracted by $\pi$. Given that $E_p\cdot \sigma_{p,q} = E_q\cdot\sigma_{p,q} = 1$, find the intersection number $E_{p,q}\cdot \sigma_{p,q}$. Provide your answer as a single number.

Answer: -1


Question:
Determine whether the following product of $q$-binomial coefficients is a polynomial in $q$:
\[
\frac{\binom{n}{j}_q\binom{n+1}{j}_q \cdots\binom{n+k-1}{j}_q}{\binom{j}{j}_q\binom{j+1}{j}_q\cdots\binom{j+k-1}{j}_q}
\]
where $n$, $j$, and $k$ are non-negative integers.

Answer: Yes


Question: Evaluate the nested summation: \[ \sum_{m_1=0}^{9}\sum_{m_2=0}^{m_1-1}\sum_{m_3=0}^{m_2-1}\sum_{m_4=0}^{m_3-1}m_4 \]

Answer: 252


Question: Let $f$ be an entire function on $\mathbb{C}$ such that $|f(z)| \le 100\log|z|$ for each $z$ with $|z| \ge 2$. If $f(i) = 2i$, find the value of $f(1)$. \( \text{(Note: An entire function is a function that is holomorphic over the entire complex plane.)} \)

Answer: 2i


Question: Let $X_1, X_2, \ldots$ be a sequence of independent exponential random variables, each with mean 1. Given a positive real number $k$, define $N=\min\left\{ n: \sum_{i=1}^n X_i >k \right\}$. Compute the expected value $E[N]$. Use Wald's equation if applicable.

Answer: k + 1


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}n^{-n^2}\left\{(n+1)\left(n+\frac{1}{2017}\right)\left(n+\frac{1}{2017^2}\right)\cdots\left(n+\frac{1}{2017^{n-1}}\right)\right\} \]

Answer: 0


Question: Evaluate the line integral \( \int_\ell \frac{-1}{1+(y-x)^2}\,dx + \frac{1}{1+(y-x)^2}\,dy \) along the path from the point \((0,0)\) to \((1,2)\) following the ellipse \((x-1)^2 +(y/2)^2 =1\).

Answer: \dfrac{\pi}{4}


Question: Compute \( \lim_{n\rightarrow\infty}\frac{n^n}{(n!)^2} \).

Answer: 0


Question:
Determine for which of the following functions the series \(\sum_{x \in S(f)} \frac{1}{x}\) converges, where \(S(f) = \{x : x > 0, f(x) = x\}\):

1. \(f(x) = \tan x\)
2. \(f(x) = \tan^2 x\)
3. \(f(x) = \tan \sqrt{x}\)
4. \(f(x) = \sqrt{\tan x}\)
5. \(f(x) = \tan 2x\)

Provide your answer with reasoning for each function.

Answer: 3


Question: Find the limit of $E[X^p]$ as $p$ approaches infinity, where $0 \leq X \leq 1$.

Answer: P(X=1)


Question: Solve for \(x\) in the equation: \[ 7000 = \sum_{n=1}^{5} \frac{1000}{(1 + x)^n} + \sum_{n=6}^{10} \frac{2000}{(1 + x)^n}. \]

Answer: 0.14


Question: Let $1 < p < 2$. Consider a normalized weakly null sequence $(f_n)_{n}$ in $L_{p}$ that contains no subsequence equivalent to the unit vector basis of $l_{p}$. Determine whether there exists a subsequence $(f_{k_n})_{n}$ such that for all $m \in \mathbb{N}$ and all scalars $a_1, a_2, \ldots, a_m$, the inequality \[ \left\| \sum_{n=1}^{m} a_n f_{k_n} \right\|_{p} \leq C_{p} \left( \sum_{n=1}^{m} |a_n|^{2} \right)^{\frac{1}{2}} \] holds, where the constant $C_{p}$ depends only on $p$. Justify your answer.

Answer: Yes


Question: Can the real numbers on the interval $[0,1]$ be well-ordered?

Answer: Yes


Question: In the sequence defined by $x_1 = \lfloor \frac{2^n}{n} \rfloor$ and $x_{i+1} = x_i - \lfloor \frac{2^{n-i}}{n-i} \rfloor \text{sgn}(x_i)$, determine if $x_{n-1} \in \{-1, 0, 1\}$ for a given positive integer $n$.

Answer: Yes


Question: Find the sum of all roots of the equation \((\sin x + \cos x)^{(1 + \sin 2x)} = 2\) for \(x \in [-2\pi, 4\pi]\).

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


Question: Let \( X \) be a positive random variable in \([0, \infty)\) such that \( \mathbb{E}(X) < \infty \). Prove or disprove that \( \lim_{N \to \infty} \mathbb{E}(X \mathbb{I}_{\{X > N\}}) = 0 \).

Answer: 0


Question: Evaluate the integral \( \int^{\infty}_{0} e^{-x}x^{100} \, dx \).

Answer: 100!


Question: Draw all connected graphs of order 5 in which the distance between every two distinct vertices is odd. Explain why you have drawn all such graphs.

Answer: K_5


Question: Determine if there exists a perfect number other than 28 that can be expressed in the form \( n^3 + 1 \), where \( n \) is an integer.

Answer: No


Question: Determine whether the following statement is true or false: If $X_n \to X$ in distribution and $\{X_n^2\}$ are uniformly integrable, then $X_n \to X$ in $L^2$. Provide a justification for your answer.

Answer: False


Question: Evaluate the line integral \( \int_{L}{(2x-y)dx+(x-y)dy} \) without using Green's theorem, where the curve \( L \) is defined as the union of the semicircle \( \{(x,y): x^{2}+y^{2}=2y, x\geq0\} \) and the quarter-circle \( \{(x,y):x^{2}+y^{2}=4, x\leq0,y\geq0\} \), oriented counter-clockwise starting from the point \((0,0)\).

Answer: 3\pi + 4


Question: Let \((X, \mathcal{T})\) be a compact Hausdorff topological space, and let \(f: X \to X\) be a one-to-one and continuous function. Is \(f\) surjective?

Answer: No


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{ \sqrt{x+\sqrt{x+\sqrt{x}}} }{ 2\sqrt{x}+4\sqrt[8]{x}} \]

Answer: \dfrac{1}{4}


Question: Determine whether the Lebesgue measure of the interval (0,1) is equal to the Lebesgue measure of the interval [0,1]. Justify your answer.

Answer: Yes


Question: Compute \( sd(sd(sd(1976^{1976}))) \), where \( sd(n) \) represents the sum of the digits of \( n \).

Answer: 7


Question: Determine whether the following statement is true or false: Every refinement of a cover of a topological space is a star-refinement.

Answer: B


Question: Find all positive integers $z$ such that $5^z - 1$ can be expressed as the product of an even number of consecutive positive integers.

Answer: 2


Question: Find the smallest integer value of \( n \) such that \( 1000n^3 < 2^n \).

Answer: 24


Question:
Calculate the expected value of the following expression, where $a_i$, $b_i$, and $c_i$ for $1 \leq i \leq 10$ are independent standard normal random variables:
\[ E\left[\sum_{i=1}^{10} \left(\frac{a_i^2 c_i^2}{a_i^2 + b_i^2}\right)\right] \]

Answer: 5


Question: Let $Y$ be a uniform random variable on the interval $[0,1]$, and let $X$ be a uniform random variable on the interval $[1,e^Y]$. Find the expected value $E[X]$.

Answer: \dfrac{e}{2}


Question: What is the probability that \(n\) randomly chosen vectors in \(\mathbb{R}^n\) are linearly independent?

Answer: 1


Question: Let $1, w_1, w_2, \ldots, w_9$ be the distinct complex $10^{th}$ roots of unity. Calculate the value of \((1 - w_1)(1 - w_2) \cdots (1 - w_9) \sum_{j=1}^9 \frac{1}{1-w_j}\).

Answer: 45


Question: In triangle $ABC$, the median $AM$, the angle bisector $CN$, and the cevian $BD$ are concurrent. A line $NT$ is drawn perpendicular to $AC$. If $CD = 6$ and $S$ is the intersection of $AM$ and $ND$, find the length of segment $TS$.

Answer: 3


Question: If $f: \mathbb{N} \to \mathbb{N}$ is a strictly increasing function such that $f(f(x)) = 2x+1$ for all natural numbers $x$, find $f(13)$.

Answer: 19


Question: Find all positive integer pairs \((a, b)\) such that \((1 + a)(8 + b)(a + b) = 27ab\).

Answer: (2, 4)


Question: Calculate the line integral of the vector field \( F = (2x\sin(\pi y)-e^z, \pi x^2\cos(\pi y)-3e^z, -xe^z) \) along the curve \( C \) from \((0,0,0)\) to \((1,1,\ln(2))\), where the curve is defined by the conditions \( z=\ln(1+x) \) and \( y=x \).

Answer: -\dfrac{13}{2}


Question: Evaluate the integral \( \int_0^\pi \cos^4\theta \sin^3\theta \,d\theta \) using de Moivre's theorem.

Answer: \dfrac{4}{35}


Question: If \( A = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{1000}} \), find \( \lfloor A \rfloor \).

Answer: 60


Question: Compute the limit: \[ \lim_{n \rightarrow\infty}\ln\left(\frac{n-1}{n+1}\right)^n \]

Answer: -2


Question: Find the smallest integer \( n \geq 1000 \) such that the sum \( 1 + 11 + 111 + \cdots + \underbrace{11\cdots1}_{n \text{ digits}} \) is divisible by 101.

Answer: 1121


Question: Calculate the number of non-isomorphic, simple, connected, labeled graphs with 4 vertices and 3 edges.

Answer: 2


Question: Let $X \subset \mathbb{P}^3_\mathbb{C}$ be a smooth algebraic surface of degree $d$, and let $L \subset X$ be a line on this surface. Compute the self-intersection number $L^2$ of the line $L$ on the surface $X$.

Answer: 2 - d


Question: Consider $k$ positive integers $a_1, a_2, \ldots, a_k$ such that $a_r \not\equiv 0 \pmod{k}$ for all $1 \le r \le k$. Determine whether there exist indices $m$ and $n$ with $1 \le m \le n \le k$ such that $\sum_{i=m}^n a_i$ is divisible by $k$. Is this generalization true?

Answer: Yes


Question: Determine the length of the non-periodic part of the infinite decimal expansion of \( \frac{1}{n} \) for any natural number \( n > 1 \).

Answer: \max(a, b)


Question: Evaluate the limit: \[ \lim_{x \to 0}\frac{1-\cos (1- \cos x)}{x^4} \]

Answer: \dfrac{1}{8}


Question: Consider an onto linear isometry \( T: (\mathbf{R}^2, \|\cdot\|_{\infty}) \rightarrow (\mathbf{R}^2, \|\cdot\|_{\infty}) \). The norm \( \|\cdot\|_{\infty} \) is defined as \( \|(a,b)\|_{\infty} = \max\{ |a|,|b|\} \). A unit ball with this norm is a square centered at the origin with unit length. For every \((a,b) \in \mathbf{R}^2\), denote \( T(a,b) = (c,d) \). If \( |a| = |b| = 1 \), must \( |c| = |d| = 1 \) hold true? In other words, does an onto isometry map a 'corner' point of a square to another 'corner' point of a square?

Answer: Yes


Question: Evaluate the expectation \( E\left[\left(\int_0^t B_s \, ds\right)^2\right] \) for a Brownian motion \( B_s \).

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


Question: Find the expected value of \( Z = 2Y_1 + Y_2 \), where \( Y_1 = \min(X_1, X_2) \) and \( Y_2 = \max(X_1, X_2) \). Each \( X_i \) is an independent exponential random variable with parameter \( \theta = 2 \).

Answer: \dfrac{5}{4}


Question: Evaluate the sum \( S = \sum_{r \in \mathbb{N}} (n-2r+1)^2 \binom{n}{2r-1} \) and find its closed form expression. The final closed form is \( n(n+1)2^{n-3} \).

Answer: n(n+1)2^{n-3}


Question: Find the 6-digit number such that when it is multiplied by 2, 3, 4, 5, and 6, the resulting numbers are permutations of the original number's digits.

Answer: 142857


Question: Find a complex function that is holomorphic at $z = -i$ but not conformal at this point.

Answer: (z + i)^2


Question: Calculate the value of \(18^{19^{20}} \mod 21\) using properties of modular arithmetic.

Answer: 18


Question: Evaluate the definite integral \( \int_{1}^{2} \int_{1}^{2} \int_{1}^{2} \int_{1}^{2} \frac{x_{1}+x_{2}+x_{3}-x_{4}}{x_{1}+x_{2}+x_{3}+x_{4}} \, d x_{1} \, d x_{2} \, d x_{3} \, d x_{4} \).

Answer: \dfrac{1}{2}


Question: Let \( L: \mathbb{R}^n \rightarrow \mathbb{R}^n \) be an orthogonal projection such that the image \( L(\mathbb{R}^n) \) has rank 1. Prove that there exists a unit vector \( v \) such that \( L \) is defined by the matrix \( vv^T \).

Answer: vv^T


Question: Solve the differential equation \(( D^4 + 6 D^3 + 9 D^2) y=0\), where \(D = \frac{d}{dx}\), with the initial conditions \(x=0,  y=0,  y'=0,  y''= 6\), and the asymptotic condition that as \(x\) tends to \(+\infty\), \(y'\) tends to \(1\). Find the value of \(y\) when \(x = 1\).

Answer: 1 - e^{-3}


Question: For a function \( f: \mathbb{R}^{n} \to \mathbb{R} \), determine the values of \( \alpha > 0 \) for which the condition \( |f(x)| \leq |x|^{\alpha} \) implies that \( f \) is differentiable at 0.

Answer: \alpha > 1


Question: Calculate the line integral \( \int_K \underline{u} \bullet \underline{t} \, ds \) for the vector field \( \underline{u}(x,y,z) = (x^3+2y+z,\,y^3+2x+z,\,1/2(x^2+y^2)+x+y) \), where \( K \) is the curve of intersection of the surfaces \( V: x+y-z=1 \) and \( S: z = 1-x^2-y^2, \ z \geq 0 \), along the path from \((1,0,0)\) to \((0,1,0)\). Use Stokes' theorem to find the value of the integral.

Answer: 0


Question:
Given a bounded sequence \(X = (x_n)\) of real numbers, and a real number \(x^*\), prove the following implication:

If for every \(\epsilon > 0\), there are at most a finite number of \(n \in \mathbb{N}\) such that \(x^* + \epsilon < x_n\), but an infinite number of \(n \in \mathbb{N}\) such that \(x^* - \epsilon < x_n\), then \(x^* = \inf\{u_m : m \in \mathbb{N}\} = \lim(u_m)\), where \(u_m = \sup\{x_n : n \geq m\}\).

Answer: x^*


Question: Determine the bound given by the Hamming bound for the largest possible size of a $t$-error-correcting code of length $2t + 1$ over the alphabet $\mathbb{A} = \{0, 1\}$, where $t$ is a positive integer.

Answer: 2


Question: In a directed bipartite graph $G=(V,E)$, is it always possible to select a subset of vertices $V' \subseteq V$ such that no two vertices in $V'$ are connected by an edge, and every vertex in $V \setminus V'$ has an edge pointing to a vertex in $V'$?

Answer: Yes


Question:
Verify the following computation in general relativity for a Lorentzian manifold $(M,g)$ with metric components $g_{ij}$ and inverse metric components $(g^{-1})^{ij}$. Given that $u$ and $\underline{u}$ satisfy the eikonal equations:
\[(g^{-1})^{\mu\nu}\partial_{\mu}u\partial_{\nu}u=0,\quad(g^{-1})^{\mu\nu}\partial_{\mu}\underline{u}\partial_{\nu}\underline{u}=0,\]
define the vector fields $L'$ and $\underline{L'}$ by:
\[ L'^{\mu}=-2(g^{-1})^{\mu\nu}\partial_{\nu}u,\quad \underline{L}'^{\mu}=-2(g^{-1})^{\mu\nu}\partial_{\nu}\underline{u}.\]
Show that in arbitrary coordinates:
\[g_{\lambda\mu}L'^{\nu}\nabla_{\nu}L'^{\mu}=-2L'^{\nu}\nabla_{\nu}\partial_{\lambda}u=-2L'^{\nu}\nabla_{\lambda}\partial_{\nu}u=4g^{\nu k}\partial_{k}u\nabla_{\lambda}\partial_{\nu}u=2\partial_{\lambda}(g^{\nu k}\partial_{\nu}u\partial_{k}u)=0.\]

Answer: 0


Question:
Let $S_1, S_2 \subseteq \mathbb{R}^2$ be two finite disjoint sets of points in the plane with $\text{diam}(S_1) \leq 1$ and $\text{diam}(S_2) \leq 1$. Determine whether there always exists a transformation $f: S_1 \cup S_2 \rightarrow \mathbb{R}^2$ such that:

- $\text{diam}(f(S_1)) \leq 1$ and $\text{diam}(f(S_2)) \leq 1$; and
- for any pair of points $s_1 \in S_1$ and $s_2 \in S_2$, $d(s_1,s_2) \leq 1$ if and only if $d(f(s_1), f(s_2)) > 1$, where $d(x,y)$ is the Euclidean distance between $x$ and $y$. Provide a justification for your answer.

Answer: Yes


Question:
Consider the polynomial \( p(z) = a_0 + a_1 z + a_2 z^2 + \cdots + a_n z^n \) of degree \( n \geq 1 \), where \( a_0 \) and \( a_n \) are both non-zero. Define the function \( f(z) = \frac{1}{p(1/z)} \), which is meromorphic on \( \mathbb{C} - \{0\} \). Determine the nature of the singularity of \( f(z) \) at \( z = 0 \). Choose the correct option:

1. \( f(z) \) has a removable singularity at \( z = 0 \) and is non-vanishing there.
2. \( f(z) \) has a removable singularity at \( z = 0 \) and has a zero of order \( n \) at \( z = 0 \).
3. \( f(z) \) has a pole of order \( n \) at \( z = 0 \).
4. \( f(z) \) has an essential singularity at \( z = 0 \).

Answer: 2


Question: Consider an analytic complex function $f(z)$ in the upper half-plane, $z=x+iy$, such that $\int_{-\infty}^{\infty}|f(x)|^{2}dx < \infty$, and $f$ is continuous on the real axis. Determine whether $f$ is bounded on the real axis, i.e., whether there exists $M > 0$ such that $|f(x)| \leq M$ for all $x \in \mathbb{R}$. Provide a justification for your answer.

Answer: Yes


Question: Determine the smallest natural number \( k \) such that \((y-x^2)^k\) lies in the ideal \((x-1,y-1)\).

Answer: 1


Question: Find the last three digits of the product of the terms in the arithmetic progression: \(7, 23, 39, \ldots, 2071\), where the common difference is 16.

Answer: 625


Question: Determine the number of prime numbers that can be expressed in the form \(x^6 + y^6\), where \(x\) and \(y\) are integers.

Answer: 1


Question:
Consider the function \(f:[2, \infty) \rightarrow \mathbb{R}\), defined by \(f(x)=x^{4}+2(a-4) x^{2}-8 a x+1\), where \(a \in \mathbb{R}\). Determine the maximal set of values of \(a\) for which the inverse function \(f^{-1}\) exists. Choose from the following options:

A. \((-9, \infty)\)
B. \((-
\infty, 1)\)
C. \([-9,1]\)
D. \([-8, \infty)\)
E. \((-
\infty,-8]\)

Answer: D


Question: A firm will employ $N$ workers next year according to this year’s profit or loss $X$. What is the probability that exactly two workers will be employed? Assume that $\Lambda := \exp(X) \sim \text{Gamma}(2, 2)$ and, given $\Lambda = \lambda$, $N \sim \text{Poisson}(\lambda)$. Calculate the probability that exactly two workers will be employed.

Answer: \dfrac{4}{27}


Question: Determine the values of \( \alpha \in \mathbb{R} \) for which the integral \( \int_0^1 \int_0^1 \frac{1}{|y-x|^\alpha} \, dx \, dy \) is convergent.

Answer: \alpha < 1


Question: Evaluate the integral \( \int_{0}^{2\pi} \log|e^{i \theta}-1| \, d\theta \) and determine if it is equal to zero.

Answer: 0


Question:
Given a random variable \( X \) with the density function:
\[
f(x; \theta) = \frac{\theta + x}{\theta + 1} \cdot e^{-x}, \quad \text{if } x \geq 0
\]
\[
f(x; \theta) = 0, \quad \text{if } x < 0
\]
and the constraint \( 0 \leq \theta \leq 4 \), determine the maximum likelihood estimate of the parameter \( \theta \) based on the sample realization \( x_1 = \frac{1}{2} \).

Answer: 4


Question: Evaluate the integral \( \int_{|z|=1} e^{\overline{z}} \, dz \), where \( z(t) = e^{it} \) and \( t \) ranges from 0 to \( 2\pi \).

Answer: 2\pi i


Question: Evaluate \( \frac{1}{zx+y-1}+\frac{1}{zy+x-1}+\frac{1}{xy+z-1} \) given that \( x+y+z=2 \), \( x^2+y^2+z^2=3 \), and \( xyz=4 \).

Answer: -\dfrac{2}{9}


Question: In parallelogram ABCD, points P, Q, R, and S are located on sides AB, BC, CD, and DA, respectively, such that AP = BQ = CR = DS = \( \frac{1}{3} \) of the length of AB. Determine the ratio of the area of quadrilateral PQRS to the area of parallelogram ABCD.

Answer: \dfrac{5}{9}


Question: Calculate the flux of the vector field \( \vec{F} = x^2\widehat{i} + y^2\widehat{j} + z^2\widehat{k} \) through the lateral surface of the cylinder defined by \( x^2 + y^2 = 4 \), bounded by the planes \( z = 0 \) and \( z = 8 \). Exclude the flux through the top and bottom planes of the cylinder.

Answer: 0


Question: Let $f \in \mathcal{C}^\infty(\mathbb{R}^n)$ be a smooth function, and let $(x_k)_k$ be a sequence in $\mathbb{R}^n$ such that $d(x_k, \text{supp}(f)^C) \leq \frac{1}{k}$. For any fixed $N > 0$, does the sequence $k^N f(x_k)$ converge to $0$?

Answer: Yes


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \( f(x^2+y) = f(x) + f(y^2) \) for all real numbers \( x \) and \( y \).

Answer: 0


Question: Let $A$ be a $55 \times 55$ diagonal matrix with characteristic polynomial $(x-c_1)(x-c_2)^2(x-c_3)^3 \ldots (x-c_{10})^{10}$, where $c_1, c_2, \ldots, c_{10}$ are distinct. Let $V$ be the vector space of all $55 \times 55$ matrices $B$ such that $AB = BA$. Determine the dimension of $V$.

Answer: 385


Question: Determine the order of the function \( f(n,k) = \sum_{i=1}^{n-1}(n-i)(k-i)\log(k-i) \) in terms of Big O notation, given that \( n < k \).

Answer: O(n^2 k \log k)


Question: Find the function \( f(x) \) that satisfies the equation \( f(x)f(y) + 2 = f(x) + f(y) + f(xy) \) for all \( x, y \), given that \( f'(0) = 0 \) and \( f'(1) = 2 \).

Answer: x^2 + 1


Question: If the incidence matrix of a graph $G$ with $n$ vertices has rank $n-1$, does this imply that the graph $G$ is connected?

Answer: Yes


Question: Find the initial value $a_1$ such that the sequence defined by $a_{n+1} = \frac{1}{4 - 3a_n}$ for $n \ge 1$ is convergent. If the sequence converges, determine its limit.

Answer: \dfrac{1}{3}


Question:
Find the rank of the skew-symmetric matrix $M$, where $M$ is a $(2n+1) \times (2n+1)$ matrix with entries given by:
\[ a_{ij} = \begin{cases} \,\,\,\, 1 & \text{if } i-j\in \{-2n, -2n+1, \dots, -n-1\} \cup \{1, 2, \dots, n\}\\ -1 & \text{if } i-j\in \{-n, -n+1, \dots, -1\}\cup \{n+1,n+2, \dots, 2n\} \end{cases} \]
Determine $\text{rank}(M)$. Provide a justification for your answer.

Answer: 2n


Question: Evaluate the limit: \[ \lim _{x\to 0^+}\frac{\sin 3x}{\sqrt{1-\cos ^3x}} \]

Answer: \sqrt{6}


Question: Let \(A\) be a positive definite matrix and \(x\) be a vector on the unit sphere (i.e., \(||x|| = 1\) with the Euclidean norm). Determine whether there exists a constant \(c > 0\), dependent on \(A\), such that \(c < x^T A x\) for all vectors \(x\) with \(||x|| = 1\).

Answer: Yes


Question: Determine if there is exactly one root of the polynomial \(x^4 - 7x^3 + 9 = 0\) in the interval \((1, 2)\). Use the Intermediate Value Theorem and Rolle's Theorem to justify your answer.

Answer: Yes


Question: Determine whether there exist any integers \( n \) such that Euler's totient function \( \phi(n) < \frac{n}{5} \). Provide a proof or a counterexample.

Answer: Yes


Question: Suppose \( f(x) \) is a polynomial of degree 5 with a leading coefficient of 2009. Given that \( f(1) = 1 \), \( f(2) = 3 \), \( f(3) = 5 \), \( f(4) = 7 \), and \( f(5) = 9 \), find the value of \( f(6) \).

Answer: 241091


Question: Calculate the residue of the function \( \frac{1}{z^{2017}} \) at the pole \( z = 0 \).

Answer: 0


Question: Consider the action of the general linear group $GL(E)$ on a vector space $E$, defined by $g \cdot v = g(v)$ for all $g \in GL(E)$ and $v \in E$. Determine the number of orbits for this action.

Answer: 2


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

Answer: \infty


Question: Calculate the limit: \[ \lim_{n \to \infty} \frac{(2n)!}{(2^n (n!)^2)} \]

Answer: \infty


Question: Determine the number of different ring homomorphisms from \(\mathbb{Z}[i]\) to \(\mathbb{Z}/(85)\).

Answer: 4


Question: Consider the (strong) Bruhat order, \(\leq_B\), on the symmetric group \(S_n\). Suppose there are permutations \(\pi,\sigma \in S_n\) such that \(\pi \geq_B \sigma\). Additionally, if \(i\) precedes \(j\) in \(\sigma\) written in single line notation, then \(i\) precedes \(j\) in \(\pi\) written in single line notation. Does there exist a maximal chain in the Hasse diagram of the Bruhat order from \(\sigma\) to \(\pi\), say \(\sigma \leq_B \sigma_2 \leq_B \cdots \leq_B \sigma_k = \pi\), such that each \(\sigma_i\) satisfies the same property with respect to \(\sigma\) for all \(1 \leq i \leq k\)?

Answer: Yes


Question: Let $V$ be an infinite dimensional vector space over $\mathbb{C}$ or $\mathbb{R}$. Suppose there are two norms on $V$ such that \( \| \cdot\|_1 \leq \| \cdot \|_2 \). Is there always a continuous linear surjection from the completion of $V$ with respect to \( \| \cdot \|_2 \) to the completion of $V$ with respect to \( \| \cdot \|_1 \)?

Answer: No


Question: Evaluate the infinite product: $$\prod_{n=1}^\infty\left(\frac{3}{2^n}\right)^{1/2^n}$$

Answer: \dfrac{3}{4}


Question: Let $T_A, T_B, T_C$ be independent and identically distributed random variables following an Exponential distribution with parameter $\lambda = 1$. Determine the probability $P(T_A < \min(T_B, T_C))$. Provide your answer as a single probability value.

Answer: \dfrac{1}{3}


Question: Evaluate the contour integral \( \oint \frac{1}{1+z^5} \, dz \) where the contour is a circle with radius 3 centered at the origin in the complex plane. Use the residue theorem to find the value of the integral.

Answer: 0


Question: Find a closed form expression for the sum \( \sum_{k=n/2+1}^n k \binom{n}{k} \).

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


Question: Find the last two digits of \(77777^{77777}\).

Answer: 97


Question: Prove that the ring $k[x,y]/(y-x^2)$ is isomorphic to the ring $k[x]$.

Answer: k[x]


Question: Evaluate the limit: \[ \lim_{x\to 0}\ (\sqrt {2x+1}\ -\ \sqrt[3]{1-3x})^{x} \] without using L'Hospital's Rule.

Answer: 1


Question: Suppose 10 people consisting of 5 married couples are randomly seated around a round table. Let \( X \) be the number of wives who are seated next to their husbands. Calculate the variance \( \text{Var}(X) \).

Answer: \dfrac{80}{81}


Question: Given that \(x^3 - \frac{1}{x^3} = 108 + 76\sqrt{2}\), find the value of \(x - \frac{1}{x}\).

Answer: 3 + 2\sqrt{2}


Question: Evaluate the following expression: \[ \frac{{ \int_{0}^{\infty} e^{-x^2}\, dx}}{{\int_{0}^{\infty} e^{-x^2} \cos (2x) \, dx}} \]

Answer: e


Question: Let \( f(x) \) and \( g(x) \) be complex polynomials. If \( f(x) \mid g(x) \) and \( g(x) \mid f(x) \), prove that there exists a non-zero complex number \( a \) such that \( f(x) = a g(x) \).

Answer: f(x) = a g(x)


Question: Determine the dimension of the cut space for an undirected graph, given that the cut space is a vector space over \( \mathbb{Z}_2 \).

Answer: n - c


Question: Given the change of basis matrix \( S = \begin{pmatrix} p & q \\ 1 & 1 \end{pmatrix} \), the vector \( x = (8, 3) \) transforms to \( x' = (1, 2) \) and the vector \( y = (5, 2) \) transforms to \( y' = (1, 1) \). Determine the new coordinates of the vector \( z = (-1, 0) \) after the change of basis.

Answer: (1, -1)


Question: Evaluate the integral \( \int_{-1}^1 \left( \sum_{j=1}^5 \sqrt{j(2j+1)} P_j(x) \right)^2 dx \), where \( P_j(x) \) are Legendre polynomials. Use the orthogonality property of Legendre polynomials to find the value.

Answer: 30


Question: Calculate the volume of the solid body $D$ bounded by the $xy$-plane, the surface $z=2y$, and the surface $y=4-x^2$. Express your answer as a fraction.

Answer: \dfrac{512}{15}


Question: Find the minimal distance between the origin and the intersection of the surfaces given by the equations \(x^2 = 2yz\) and \(x^2 + 3y^2 + 2z^2 = 30\).

Answer: \sqrt{10}


Question: Calculate the limit: \[ \lim_{(x,y) \to (0,0)} \frac{xy\cos(xy)-\sin(xy)}{x^2} \]

Answer: 0


Question: Given the function \( f(x) = x^3 - 12x + 3 \), determine the number of real solutions to the equation \( f(f(x)) = 0 \).

Answer: 9


Question: Find the minimum value of the function \( f(x) = (x + a + b)(x + a - b)(x - a + b)(x - a - b) \), where \( a, b \in \mathbb{R} \).

Answer: -4a^2b^2


Question: Consider a self-adjoint operator \( T: D \to D \) where \( D \subset L^2 \) is a finite-dimensional subspace. Determine whether there exists a self-adjoint operator \( S: L^2 \to L^2 \) such that \( S = T \) on \( D \).

Answer: Yes


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

Answer: e^2 + 1


Question: Let $S_n$ be a simple random walk over $\mathbb{Z}$, starting at $0$, where $S_n = X_1 + X_2 + \ldots + X_n$ and each $X_i$ is an independent random variable taking the value $1$ with probability $\frac{1}{2}$ and $-1$ with probability $\frac{1}{2}$. Calculate $\mathbb{E}[S_n^4]$. Provide your answer in terms of $n$.

Answer: 3n^2 - 2n


Question: Find the number of multisets of size five from the set \( \{A, B, C, D, E\} \) such that at least one letter occurs at least three times. Use generating functions to solve the problem.

Answer: 75


Question: Determine which of the following functions is entire: \(\sin(\sqrt{z})\) and \(\cos(\sqrt{z})\).

Answer: \cos(\sqrt{z})


Question: Suppose \( f:\mathbb{R}\rightarrow\mathbb{R} \) is a continuous, strictly monotonic, and odd function. Is \( f \) necessarily a polynomial function? Answer with 'Yes' or 'No'.

Answer: No


Question:
Find all functions \( f: \mathbb{R}^{*} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R}^{*} \), the following equation holds:
\[
f(y^2f(x)+x^2+y)=x(f(y))^2+f(y)+x^2.
\]

Answer: f(x) = x


Question: Find the values of constants \(a\) and \(b\) such that \( \lim _{x\rightarrow -\infty }\sqrt {x^{2}+6x+3}+ax+b=1 \). Then, calculate \(a+b\).

Answer: 5


Question: In the metric space \(C([0,1])\) with the metric \(d(f,g) = \sup_{x \in [0,1]} |f(x) - g(x)|\), is it true that a sequence \(f_n\) converges to \(f\) if and only if \(f_n\) converges to \(f\) uniformly?

Answer: Yes


Question:
Which of the following approximations of \( \pi \) best delimits the propagation of uncertainty?
(a) \( \pi = 4(1-1/3+1/5-1/7+1/9...) \)
(b) \( \pi = 6 \left(0.5 + \frac{0.5^3}{2 \times 3} + \frac{3 \times 0.5^5}{2 \times 4 \times 5} + \frac{3 \times 5 \times 0.5^7}{2 \times 4 \times 6 \times 7} + ... \right) \)

Answer: b


Question: Evaluate the limit: $$ \lim_{m\rightarrow\infty}\sum_{n=0}^{\infty}\frac{1}{m^n} \sum_{k=0}^{\infty}  {2k \choose k }  {2n \choose k}\left( \frac{-1}{2} \right)^k $$ and determine if the result is zero.

Answer: 1


Question: Provide an example of a nonabelian group in which all proper subgroups are normal.

Answer: Q_8


Question: Given two bounded closed regions $A, B \subset \mathbb{R}^n$, let $A \bigtriangleup B$ denote the symmetric difference of $A$ and $B$, defined as $(A - B) \cup (B - A)$. If $m(S)$ represents the Lebesgue measure of a set $S$, does $m(A \bigtriangleup B) = 0$ imply that $A = B$? In other words, can $m(X \bigtriangleup Y)$ define a metric on the set of all closed regions in $\mathbb{R}^n$?

Answer: No


Question: Let $X$ and $Y$ be two Banach spaces with respective norms $\|\cdot\|_X$ and $\|\cdot\|_Y$. Suppose that $X$ and $Y$ are subsets of a vector space $Z$. Define $K(t,x)$ for $t\in (0,\infty)$ and $x\in X+Y$ by \[K(t,x)=\inf_{x=a+b,\ a\in X,\ b\in Y}\{\|a\|_X+t\|b\|_Y\}\] Assume that for a fixed $x$ and $p\in [1,\infty)$, \[\int_0^\infty \frac{|K(t,x)|^p}{t}dt<\infty.\] Can we conclude that $x=0$?

Answer: Yes


Question: Does there exist a twice differentiable function that approximates a threshold such that \( f(x) = 0 \) for \( x \leq a \), \( f(x) = L \) for \( x \geq a + \epsilon \), and \( f(x) \) takes values between these two for \( x \in (a, a+\epsilon) \), where \( \epsilon \) is a small constant?

Answer: Yes


Question: Consider the sequence given by \( a_n = \frac{3n^{3} + 7n^{2} + 1}{4n^{3} - 8n + 64} \). Using the epsilon-N method, determine if this sequence converges to \( \frac{3}{4} \). Specifically, verify if the treatment of the denominator in the expression is correct when bounding the terms to show convergence.

Answer: \frac{3}{4}


Question: Is the subgroup of homotopically trivial isometries a closed subgroup of the isometry group? Let $(M,g)$ be a connected Riemannian manifold. According to the Steenrod-Myers-Theorem, the isometry group $\text{Isom}(M,g)$ of $(M,g)$ is a compact Lie group with the compact-open topology. Is the subgroup $G$ of isometries, which are homotopically trivial (i.e., homotopic to the identity), a closed subgroup of $\text{Isom}(M,g)$?

Answer: Yes


Question:
Consider a family of topological spaces \((X_\alpha)_{\alpha \in A}\) with open subsets \(X_{\alpha\beta} \subseteq X_\alpha\) and homeomorphisms \(\psi_{\alpha\beta} \colon X_{\alpha\beta} \to X_{\beta\alpha}\) satisfying \(\psi_{\beta\alpha}^{-1} = \psi_{\alpha\beta}\) and \(\psi_{\alpha\gamma} = \psi_{\beta\gamma} \circ \psi_{\alpha\beta}\) for all \(\alpha,\beta,\gamma\) on appropriate domains. Define the space \(X\) as the quotient:
\[ X := \bigl(\bigsqcup_{\alpha \in A} X_\alpha\bigr)\big/\sim \]
where \((x,\alpha) \sim (y,\beta)\) if \(x \in X_{\alpha\beta}\), \(y \in X_{\beta\alpha}\), and \(\psi_{\alpha\beta}(x) = y\). Is the natural projection \(\pi \colon \bigsqcup_{\alpha \in A} X_\alpha \to X\) an open map?

Answer: Yes


Question: In triangle $\triangle ABC$, point $A$ is a focus of an ellipse with a major axis of length $10$ and a minor axis of length $8$. Point $B$ is an endpoint of the minor axis, and point $C$ is on the ellipse such that the other focus lies on line $BC$. Calculate the inradius of $\triangle ABC$.

Answer: \dfrac{30}{17}


Question: If the inverse of the function \(y = \sqrt{x + \sqrt{2x - 1}}\) is given by \(y = ax^2 + bx + c\), find the value of \(a^2 + b^2 + c^2\).

Answer: 4


Question: Calculate the divergence of the unit vector \( \hat{a} \) that is perpendicular to the surface of a sphere with radius \( R \).

Answer: \dfrac{2}{R}


Question: An urn contains 100 balls, some of which are white and some are black. Let $p$ be the probability of drawing a white ball. You draw one ball at a time, replacing it after each draw. After 100 draws, you observe that all 100 balls drawn are white. What is the value of $p$?

Answer: \dfrac{99}{100}


Question: Determine whether there exists an elliptic curve $E/\mathbb{Q}(T)$ such that any morphism $X_1(N) \to E$ for any $N > 0$ is constant.

Answer: Yes


Question: Evaluate the integral \( \int_{y=0}^{3}\int_{x=y}^{\sqrt{18-y^2}} (7x + 3y) \, dx \, dy \).

Answer: 72 + 54\sqrt{2}


Question: Let $a, b, c$ be non-negative real numbers such that \(abc[(a − b)(b − c)(c − a)]^2 = 1\). Find the minimum value of $a + b + c$.

Answer: \sqrt[3]{36}


Question: Evaluate the limit using Landau's symbols: $$\lim_{x\to0^+} \left(\frac{x^{\sin(x)}-1}{x}\right)$$

Answer: -\infty


Question: Given a random variable $X$ from a normal distribution with parameters $\mu=0$ and $\sigma=\theta$, find the critical region of the Neyman-Pearson test with $\alpha=0.1$ for the hypotheses $H_0: \theta=4$ and $H_1: \theta=1$. Why is the critical region $\{x^2 < c\}$ with $c=0.25$?

Answer: x^2 < 0.25


Question: Determine the values of $p > 0$ for which the series \( \sum_{n=2}^\infty \frac{1}{n \log_2^p n} \) is convergent.

Answer: p > 1


Question: Calculate the triple integral of the vector field \( F = (z^2, x^2, y^2) \) over the region \( D \) defined by \( 0 \leq x, y, z \leq 1 \) using Gauss's theorem. Determine the value of the integral.

Answer: \left( \dfrac{1}{3}, \dfrac{1}{3}, \dfrac{1}{3} \right)


Question: Evaluate the set \( \bigcup_{s \in \mathbb{R}} \bigcap_{t \geq s} A_t \), where \( A_t = (\cos t - 2, \cos t + 2) \) for \( t \in \mathbb{R} \).

Answer: (-1, 1)


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{e^{\tan x} - e^x + \ln(\sec x + \tan x) - x }{\tan x - x} \]

Answer: \dfrac{3}{2}


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

Answer: 8


Question: Compute the integral \( \int_B g \, d\lambda^3 \) where \( g(x,y,z) = xyz \) and \( B = \{(x,y,z) \in \mathbb{R}^3 \mid x, y, z \geq 0, x^2 + y^2 + z^2 \leq R^2\} \), with \( R > 0 \) being arbitrary.

Answer: \dfrac{R^6}{48}


Question: Find a polynomial $f \in \mathbb{Q}[x]$ such that the sum of the ideals $I = \langle x^2 + 2x + 3 \rangle$ and $J = \langle x^3 + x + 1 \rangle$ is equal to the ideal $\langle f \rangle$. What is $f$?

Answer: 1


Question:
Calculate the expected value of the following series:
\[ \sum_{i=4}^\infty 500 \frac{e^{-4} 4^i}{i!} \]

Answer: 283.26


Question: A busy railway station has two taxi stands at separate exits, A and B. At stand A, taxis arrive according to a Poisson Process with a rate of 2 per minute. Passengers choose stand A with probability 0.7 and stand B with probability 0.3. At stand B, taxis arrive according to a Poisson Process with a rate of 1 per minute. If a passenger has been waiting for a taxi for 3 minutes, what is the probability that they are waiting at taxi stand A?

Answer: \dfrac{7}{7 + 3e^3}


Question: A borrower takes out a loan of $2000 for two years. Construct a sinking fund schedule if the lender receives 10% effective annual interest on the loan, and the borrower makes semiannual deposits into a sinking fund earning 8% interest convertible quarterly. Calculate the amount of each deposit needed to replace the loan amount.

Answer: 470.70


Question: True or False: Every finite dimensional vector space can be made into an inner product space with the same dimension.

Answer: A


Question: Calculate the hyperbolic distance between the points $2$ and $5+i$ in the upper half-plane $H = \{ z : \text{Im}(z) > 0 \}$. Use the metric \( d_H(z, w) = 2\tanh^{-1}\left(\left|\frac{z-w}{z-\bar{w}}\right|\right) \).

Answer: \infty


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2} \cos^2\theta ~ \delta(\sin\theta) ~d\theta. \)

Answer: 1


Question: Evaluate the infinite power tower \( \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdots}}} \).

Answer: 2


Question: Suppose $H$ is an infinite-dimensional Hilbert space. Can we find $n$ isometries $s_1, s_2, \ldots, s_n \in B(H)$ such that $1 = s_1s_1^* + s_2s_2^* + \cdots + s_ns_n^*$, where $1$ is the identity operator on $H$?

Answer: Yes


Question: An object occupies the solid region in the first octant bounded by the coordinate planes and the cylinders \(x^2 + y^2 = 4\) and \(y^2 + z^2 = 4\). If the charge density at any point is given by \(x\), calculate the total charge within this region.

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


Question: Let \( c:[0,1]\to\mathbb{R}^2\backslash\{\mathbf{0}\} \) be a closed path with winding number \( k \). Define \( \tilde{c} = \rho(t)c(t) \), where \( \rho:[0,1]\to(0,\infty) \) is a function satisfying \( \rho(0)=\rho(1) \). Determine the winding number of \( \tilde{c} \).

Answer: k


Question: In a cube ABCD.EFGH with a side length of 12 cm, point P is the intersection of lines BG and FC. Point Q is the intersection of segment EP with the plane DBFH. Determine the length of segment EQ.

Answer: 4\sqrt{6}


Question: Evaluate the line integral \( \oint_C \frac{dz}{z-2} \) where \( C \) is the circle \( |z-1|=5 \).

Answer: 2\pi i


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

Answer: 0


Question: What is \( \Theta(f(n)) - \Theta(f(n)) \)?

Answer: O(f(n))


Question: Consider \(A \subseteq B\) where both \(A\) and \(B\) are \(\sigma\)-algebras, and \(B\) is countably generated. Is it possible for \(A\) to not be countably generated?

Answer: Yes


Question: Compute the intersection form for a smooth subvariety $\iota:X=V_+(f)\subset\mathbb{P}^2\times\mathbb{P}^1$, where $f\in H^0(\mathcal{O}(1,2))$. Determine the value of the intersection $\iota^*\mathcal{O}(2,3).\iota^*\mathcal{O}(2,3)$. Provide your answer as a single value or expression.

Answer: 20


Question:
Consider a holomorphic function $W(t_1,\dotsc,t_n)$ defined on a connected open set $U$ of $\mathbb{C}^n$. Let $\mathbf{t}^{(0)}$ be a point in $U$. Suppose there exists a cycle $\gamma$ in $\mathbb{C}^m$ and a rational function $F(\mathbf{t}, x_1,\dotsc, x_m)$ such that for all $\mathbf{t}$ in a neighborhood of $\mathbf{t}^{(0)}$:

1. The map $\mathbf{x} \mapsto F(\mathbf{t}, \mathbf{x})$ is continuous on $\gamma$.
2. $W(\mathbf{t}) = \oint_\gamma F(\mathbf{t}, \mathbf{x})\mathrm{d} \mathbf{x}$.

Is it true that for every point $\mathbf{t}^{(1)}$ in $U$, there exists another cycle $\gamma_1$ such that these properties hold in a neighborhood of $\mathbf{t}^{(1)}$?

Answer: Yes


Question: Let \( A = [a_{ij}] \in \mathcal{M}(n,n; K) \) with \( \det A = a \), and let \( c \in K \) with \( c \neq 0 \). Define the matrix \( B = [b_{ij}] \in \mathcal{M}(n,n; K) \) where \( b_{ij} = c^{i-j}a_{ij} \) for \( i,j \in \{ 1,2,\dots,n \} \). Prove that \( \det B = a \).

Answer: a


Question: If $A_n = \frac{a_1 + a_2 + \ldots + a_n}{n}$ is convergent, is the sequence $a_n$ necessarily bounded?

Answer: No


Question: Find the coordinates of the vector \( \vec{u} \) that has a magnitude of \( 3\sqrt{3} \), is orthogonal to both \( \vec{v} = (2, 3, -1) \) and \( \vec{w} = (2, -4, 6) \), and forms an acute angle with the vector \( (1, 0, 0) \).

Answer: (3, -3, -3)


Question: Determine the values of \(m\) for which the integral \(\int_{B_1(0)} \frac{1}{|z|^{2m}} \, \mathrm{d}\operatorname{vol}\) is finite in \(\mathbb{C}^n\), where \(B_1(0)\) is the unit ball centered at the origin.

Answer: m < n


Question: Determine the number of group homomorphisms from the dihedral group $D_m$ to the cyclic group $\mathbb{Z}_n$, where $m$ is odd and $n \in \mathbb{N}$. Assume $D_m = \{r, f : r^m = f^2 = (rf)^2 = e\}$. Verify if the number of homomorphisms is given by $2(m,n)$, where $(m,n)$ is the greatest common divisor of $m$ and $n$. Is this calculation correct?

Answer: \gcd(2, n)


Question: Let $R = \mathbb{C}[T]$ be the ring of polynomials with coefficients in the field of complex numbers $\mathbb{C}$. Consider the $R$-module $M$ generated by elements $x$ and $y$, with the relations $Tx = y$ and $T^2x = y$. Determine the dimension of $M$ as a vector space over $\mathbb{C}$.

Answer: 2


Question: An urn contains 18 black balls and 2 white balls. What is the expected number of draws (without replacement) needed to obtain two balls of different colors?

Answer: \dfrac{135}{19}


Question:
Evaluate the series:

\[ 1 + \frac{2}{3}\cdot\frac{1}{2} + \frac{2\cdot5}{3\cdot6}\cdot\frac{1}{2^2} + \frac{2\cdot5\cdot8}{3\cdot6\cdot9}\cdot\frac{1}{2^3} + \ldots \]

Choose the correct value from the following options:

(A) \(2^{1/3}\)
(B) \(2^{2/3}\)
(C) \(3^{1/2}\)
(D) \(3^{3/2}\)

Answer: B


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

Answer: \dfrac{4}{e}


Question:
In triangle $ABC$, points $B'$ and $C'$ are located on sides $AB$ and $AC$ such that $BB' = CC'$. Let $O$ and $O'$ be the circumcenters of triangles $ABC$ and $AB'C'$, respectively. The line $OO'$ intersects lines $AB'$ and $AC'$ at points $B''$ and $C''$. Given that $AB = \frac{1}{2} AC$, determine the relationship between $AB''$ and $AC''$:

A) $AB'' < \frac{1}{2} AC''$
B) $AB'' = \frac{1}{2} AC''$
C) $\frac{1}{2} AC'' < AB'' < AC''$
D) $AB'' = AC''$
E) $AB'' > AC''$

Answer: D


Question: Given that $x^2 + yx + y^2 = 4$ and $x^4 + x^2y^2 + y^4 = 8$ for $x, y \in \mathbb{R}$, calculate the value of $x^6 + y^3x^3 + y^6$. \( \text{Provide your answer in a simplified form.} \)

Answer: 19


Question: In a sequence of $n$ integers $x_1, x_2, \ldots, x_n$, is there always a contiguous subsequence whose sum is a multiple of $n$? Provide a proof or counterexample.

Answer: Yes


Question: Find the value of \( n \) for which the quadratic equation \( \sum_{k=1}^{n}(x+k-1)(x+k) =10n \) has solutions \( \alpha \) and \( \alpha+1 \) for some \( \alpha \).

Answer: 11


Question: Evaluate the limit: \[ \lim_{(x,y)\rightarrow(+\infty, +\infty)}\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)} \]

Answer: 0


Question: Calculate the limit of the sequence: \[ \lim_{n \to \infty} n^{\frac{1}{n}} (n+1)^{\frac{1}{n+1}} \cdots (2n)^{\frac{1}{2n}} \]

Answer: \infty


Question:
Suppose that $y_1$ solves $2y''+y'+3x^2y=0$ and $y_2$ solves $2y''+y'+3x^2y=e^x$. Which of the following is a solution of $2y''+y'+3x^2y=-2e^x$?

(A) $3y_1-2y_2$

(B) $y_1+2y_2$

(C) $2y_1-y_2$

(D) $y_1+2y_2-2e^x$

(E) None of the above.

Answer: A


Question: Evaluate the surface integral \( \int \vec{F} \cdot \vec{n} \, dS \) over the closed surface \( S \) of the solid bounded by \( x^2 + y^2 = a^2 \), \( z = x + 1 \), and \( z = 0 \), where \( \vec{F} = \langle 2x, -3y, z \rangle \) and \( \vec{n} \) is the outward unit normal vector. Use the divergence theorem to find the value of the integral.

Answer: 0


Question: Evaluate the limit \( \lim_{n\rightarrow\infty}\int_{2n\pi}^{2(n+1)\pi}x\ln x\cos x\,dx \).

Answer: 0


Question: Let $F$ be a Banach space with the closed unit ball $B$. Let $E \subset F^*$ be a total subspace such that $B$ is complete with respect to the norm $|||f|||=\sup \limits_{e\in E,~e\ne 0} \frac{|\left<f,e\right>|}{\|e\|}$. Does it follow that $E$ is norming, i.e., there exists $r>0$ such that $|||f||| \ge r\|f\|$ for all $f\in F$?

Answer: Yes


Question: Calculate \( \lim_{x\rightarrow0}\frac{\int_0^x\cos\frac{1}{t^{2}}dt}{x} \).

Answer: 0


Question: Determine the minimum number \( k \in \mathbb{N} \cup \{\infty\} \) such that every topological partition into connected sets of \( \mathbb{R}^2 \) with the usual topology is \( k \)-colorable. If the Four Color Theorem is applicable, explain why its conditions are satisfied in this context.

Answer: 4


Question: Evaluate the limit: $$\lim \limits_{x \to +\infty}\sin(x)(\ln{(\sqrt{x}+1})-\ln(\sqrt{x+1}))$$

Answer: 0


Question: Evaluate the limit of the infinite product: \[ P = \frac{7}{9} \cdot \frac{26}{28} \cdot \frac{63}{65} \cdots \frac{n^3-1}{n^3+1} \cdots. \]

Answer: \dfrac{2}{3}


Question: Evaluate the limit: \[ \lim_{x\to \infty}\frac{x^{2011}+2010^x}{-x^{2010}+2011^x} \]

Answer: 0


Question: How many pairs of positive integers \((x, y)\) satisfy the equation \(x^2 - 10! = y^2\)?

Answer: 105


Question: Calculate the residue of \( \frac{e^{2z}}{(e^{iz} +1)} \) at \( z = \pi \).

Answer: i e^{2\pi}


Question: Determine the number of solutions to the equation \(X^6 - 1 = 0\) in the ring \(\mathbb{Z}/(504)\).

Answer: 144


Question: Let \((a_n)_{n\geq 1}\) be a sequence of real numbers satisfying \(|a_n|\leq 1\) for all \(n\). Define \(A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n)\) for \(n\geq 1\). Find \(\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)\).

Answer: 0


Question: Given that $0 < p \leq |a_n| \leq q$ for all $n \in \mathbb{N}$, find the radius of convergence of the power series $\sum a_n x^n$. Provide your answer as a single number.

Answer: 1


Question: Evaluate the limit: \[ \lim\limits_{x \to 0^{-}} (1-8^x)^{\sin 3x} \]

Answer: 1


Question: The polynomial $x^3 - 3x^2 + 4x - 1$ is a factor of $x^9 + px^6 + qx^3 + r$. Determine the value of $p + q + r$. \( \boxed{} \)

Answer: 36


Question: In how many ways can you select $k$ (not necessarily distinct) numbers from the set $\{1, 2, \ldots, p\}$ such that their sum is congruent to $0$ modulo $p$?

Answer: p^{k-1}


Question: Evaluate the integral \( \int \frac{e^{z^2}}{z^3} \, dz \) using Cauchy's Integral Formula and determine if it equals \( \pi i \).

Answer: 2\pi i


Question: Determine whether the function $f \equiv 1$ is the unique normalized solution to the equation \( f(x) = \frac{1}{x} \int_0^x f(t) \, dt \) for \( f \in L^1(0,1) \), where \( f \) is non-negative almost everywhere and satisfies \( \int_0^1 f(x) \, dx = 1 \).

Answer: f(x) = 1


Question: How many integers between 30,000 and 100,000 have a sum of digits that is 15 or less?

Answer: 5852


Question: Evaluate the integral \( \int_{|z+2i|=1} \frac{\sin(z)}{z^3+16z} \, dz \).

Answer: 0


Question:
Determine whether there exists a positive integer \( x \) such that the probability mass functions of two Poisson distributions with different rate parameters \( \lambda_0 \) and \( \lambda_1 \) are equal. Specifically, solve the equation:
\[
\frac{e^{-\lambda_0} \lambda_0^x}{x!} = \frac{e^{-\lambda_1} \lambda_1^x}{x!}
\]
where \( x \in \mathbb{Z}^+ \) and \( \lambda_0 \neq \lambda_1 \).

Answer: Yes


Question: Evaluate the line integral \( \int_C P\,dx + Q\,dy \) where \( P(x,y) = \frac{y+x}{x^2+y^2} \) and \( Q(x,y) = \frac{y-x}{x^2+y^2} \) around the circle \( C: x^2+y^2=a \) oriented clockwise.

Answer: 2\pi


Question: Given the function \( f(x) = \frac{1}{2}\int_{0}^{x}(x-t)^2g(t) \, dt \) where \( g(x) \) is a function such that \( g(1) = 6 \), find \( f'''(1) \).

Answer: 6


Question: Solve the discrete logarithm problem \( 6 \equiv 2^x \pmod{101} \) using the Baby Step, Giant Step method. Find the value of \( x \).

Answer: 70


Question: Find the remainder when the sum \( S = \sum_{n=0}^{2015} (n! + n^3 - n^2 + n - 1) \) is divided by 1000.

Answer: 578


Question: Find the remainder when \(2^{{2014}^{2013}}\) is divided by 41.

Answer: 16


Question: Solve the recurrence relation \( T(n) = 2T(\sqrt{n}) + n \).

Answer: \Theta(n)


Question: Find a maximal ideal in \( \mathbb{Z}[x] \) that properly contains the ideal \((x-1)\).

Answer: (2, x - 1)


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

Answer: \dfrac{\pi}{8}


Question: Determine whether the following statement is true or false: If \( U \) and \( T \) are normal operators that commute with each other on an inner product space, then \( U + T \) is normal. Provide an explanation for your answer.

Answer: True


Question: Let \( \nu \) be a complex measure on a measurable space \((X, M)\) such that \( \nu(X) = |\nu|(X) \). Prove that \( \nu = |\nu| \).

Answer: \nu = |\nu|


Question: Simplify the trigonometric expression: \[ \prod_{k=1}^{59} \left(1 - \frac{\cos(60+k)^{\circ}}{\cos(k)^{\circ}}\right) \]

Answer: 1


Question:
Let $V$ be a vector space of dimension $m \geq 2$. Let $T: V \to V$ be a linear transformation such that $T^{n+1} = 0$ and $T^n \neq 0$ for some $n \geq 1$. Which of the following statements is necessarily true?

(a) $\text{Rank}(T^n) \leq \text{Nullity}(T^{n+1})$

(b) $\text{Trace}(T) \neq 0$

(c) $T$ is diagonalizable

(d) $n = m$

Answer: a


Question:
Determine the radius of convergence of the power series:
\[ \sum_{n=0}^{\infty} (-1)^n \frac{n^{2n}}{(4n+1)^n}(x+2)^{n^2} \]

Answer: 1


Question: Evaluate the integral \( \iiint_E \frac{ \,dx \,dy \,dz}{\sqrt{x^2+ y^2 +(z-2)^2}} \) where \( E \) is the region defined by \( x^2+y^2+z^2 \le 1 \).

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


Question: Is the subset of \( \mathbb{Q} \) defined by \( \mathbb{Z}_p=\{ \frac{a}{b} \,\,|a,b\in\mathbb{Z}, p\nmid b\} \) isomorphic to the ring of rational \( p \)-adic integers as considered in Huppert's book, "Character Theory of Finite Groups"?

Answer: Yes


Question:
Consider the \((p,q)\)-tensor given by
\[T_{i_1,\dots,i_p}^{j_1,\dots,j_q}e_{i_1}\otimes\dots\otimes e_{i_p}\otimes\varepsilon^{j_1}\otimes\dots\otimes\varepsilon^{j_q}.\]
If you contract the last indices, do you obtain the tensor
\[T_{i_1,\dots,i_{p-1},k}^{j_1,\dots,j_{q-1},k}e_{i_1}\otimes\dots\otimes e_{i_{p-1}}\otimes\varepsilon^{j_1}\otimes\dots\otimes\varepsilon^{j_{q-1}}?\]

Answer: Yes


Question: Find a non-constant polynomial in \( \mathbb{Z}/8\mathbb{Z}[x] \) that is a unit.

Answer: 1 + 4x


Question: Determine the number of ways to arrange the letters \(z, z, y, y, x, x, w, w, v, v\) such that no two identical letters are adjacent.

Answer: 39480


Question: Let \(A\subset\mathbb{R}^n\) be a set of positive Lebesgue measure. Determine whether the set \(\{\|x\|:x\in A\}\subset [0,\infty)\) has positive Lebesgue measure in \([0,\infty)\).

Answer: Yes


Question: Let $X$ be a locally convex space over $\mathbb{R}$, and let $D \subset X$ be dense. Consider a Banach space $B$ over $\mathbb{R}$ with a Schauder basis $\{b_i\}_{i=1}^{\infty}$. Is the set \( D^+ \triangleq \left\{\sum_{j=1}^n \beta_j d_j\otimes b_j: \, d_j \in D, \, \beta_j \in \mathbb{R} \right\} \) dense in $X \otimes B$ with respect to the projective tensor product topology?

Answer: Yes


Question: Consider the infinite series \(\mathbb{S}=\sum_{n=0}^{\infty}\frac{a_n}{10^{2n}}\), where the sequence \(\{a_n\}\) is defined by \(a_0=a_1=1\), and the recurrence relation \(a_n=20a_{n-1}+12a_{n-2}\) for all positive integers \(n \geq 2\). If \(\sqrt{\mathbb{S}}\) can be expressed in the form \(\frac{a}{\sqrt{b}}\) where \(a\) and \(b\) are relatively prime positive integers, determine the ordered pair \((a, b)\).

Answer: (45, 1997)


Question: Given a square $ABCD$ with center $O$, points $E, F, G, H$ are located on sides $AB, BC, CD, DA$ respectively, such that $AE = BF = CG = DH$. The diagonals $OA, OB, OC, OD$ intersect $HE, EF, FG, GH$ at points $X, Y, Z, W$ respectively. If the area of quadrilateral $EFGH$ is 1, calculate the product of the areas of quadrilaterals $ABCD$ and $XYZW$. Assume $E, F, G, H$ are midpoints of the sides of $ABCD$. What is the value of $(\text{Area } ABCD) \times (\text{Area } XYZW)$?

Answer: 1


Question: Let \( \{f_n\}_{n=1}^\infty \) and \( f \) be integrable functions on \([0,1]\) such that \( \lim_{n \rightarrow \infty} \int_{0}^1 |f_n(x) - f(x)| \, dx = 0 \). Suppose \( \{g_n\} \) is a uniformly bounded sequence of continuous functions converging pointwise to a function \( g \). Determine whether the following statement is true: \( \int_0^1 |f_n(x) g_n(x) - f(x) g(x) | \, dx \rightarrow 0 \) as \( n \rightarrow \infty \).

Answer: True


Question: Find the value of the Lagrange multiplier \( \lambda \) for the function \( f(x_1, x_2, x_3) = 9x_1 \cdot x_2 \cdot x_3 \) subject to the constraint \( 2x_1 + x_2 + x_3 = m \), where \( m > 0 \) and \( x_1, x_2, x_3 > 0 \).

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


Question: Calculate the value of the expression \((0.5)!(-0.5)!\) using the Gamma function, where \(n! = \Gamma(n+1)\) for non-integer values.

Answer: \dfrac{\pi}{2}


Question: Given the characteristic polynomial \(z^4(z-1)^3(z-2)^2\), determine the dimension of \(\text{null}(T-2I)^9\).

Answer: 2


Question: Is it true that an abelian group is uniquely $p$-divisible if and only if it has a natural structure of a $\mathbb{Z}[\frac{1}{p}]$-module?

Answer: True


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{\log (1 + a_n)}{\sin (a_n + \frac{\pi}{2})} \), where \( \{a_n\} \) is a sequence of real numbers such that \( \lim_{n \to \infty} {a_n}^{\frac{1}{n}} = \frac{1}{4} \).

Answer: 0


Question: Find all natural numbers $n > 0$ such that the sum of the squares of their four smallest divisors equals $n$. Specifically, if $1$, $a$, $b$, and $c$ are the smallest four divisors of $n$ with $1 < a < b < c$, then $1^2 + a^2 + b^2 + c^2 = n$. Determine all such $n$.

Answer: 130


Question: Maximize the function $f(x) = x_1 + x_2$ subject to the constraint $g(x) = x_1^2 + 3x_1x_2 + 3x_2^2 - 3 = 0$. Find the point $x = (x_1, x_2)$ that satisfies these conditions.

Answer: (3, -1)


Question: Given the recursive sequence $a_{n+1} = \log(1 + a_n)$ with $a_1 > 0$, find $\lim_{n \to \infty} n \cdot a_n$. Assume that $\log$ denotes the natural logarithm.

Answer: 2


Question: Compute \(\lim_{n\to\infty}\int_{0}^{1}\frac{nx}{1+n^2x^4}dx\).

Answer: \dfrac{\pi}{4}


Question: Given a non-trivial homomorphism \( \chi \) from \( \mathbb{Z}/p \mathbb{Z}^* \) to \( \mathbb{C}^* \), evaluate the sum \( \sum_{n \in \mathbb{Z}/p \mathbb{Z}^* } \chi(n) \).

Answer: 0


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

Answer: 1


Question: Evaluate the integral \( \int _C ze^{\frac{3}{z}}dz \), where \( C \) is the contour \( |z|=2 \) oriented in the anti-clockwise direction.

Answer: 9\pi i


Question: Determine the order of the quotient ring \( \mathbb{Z}[x]/\langle(x^2+x+1)(x^3+x+1)\rangle \).

Answer: \infty


Question: Consider a continuously differentiable function \( v \). If the determinant of the Jacobian of \( v \) is non-zero and \( v \) is bijective, does the inverse of \( v \) exist?

Answer: Yes


Question: Find the function $f(x)$ if $f: \mathbb{R} \to \mathbb{R}$ is continuous and satisfies $f(x+y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$. What is $f(x)$?

Answer: f(x) = kx


Question: Determine the values of the damping coefficient \( b \) for which the solution to the differential equation approaches the equilibrium position most rapidly, given the characteristic equation \( \lambda^2 + b\lambda + 3 = 0 \).

Answer: 2\sqrt{3}


Question: Evaluate the limit: \[ \lim_{t \to 1} \frac{\sqrt{2t^2-1}\sqrt[3]{4t^3-3t}-1}{t^2-1} \]

Answer: \dfrac{5}{2}


Question: Provide an example of an element \( a \in K((T)) \) that is algebraic over \( K(T) \) but not an element of \( K(T) \), for some field \( K \).

Answer: \sqrt{1 + T}


Question: Does the absolute value function \( u(x) = |x| \) have a second weak derivative?

Answer: No


Question: Evaluate the surface integral \( \iint_S (z + x^2 y) \, dS \), where \( S \) is the part of the cylinder \( y^2 + z^2 = 1 \) that lies between the planes \( x = 0 \) and \( x = 3 \) in the first octant \( x, y, z \ge 0 \).

Answer: 12


Question: How many ways are there to divide 30 objects, consisting of 5 pears, 5 apples, 5 doughnuts, 5 lollipops, 5 chocolate cats, and 5 candy rocks, into two unordered piles of 15 objects each?

Answer: 2166


Question: Determine whether the statement "A matrix $A$ is Hermitian if and only if $A$ has real eigenvalues" is true or false.

Answer: False


Question: Evaluate the path integral \( \int_{0}^{i} \sin(z) \, dz \).

Answer: 1 - \cosh(1)


Question: Let $z$ be a complex number with $|z| \le \frac{4}{5}$. If $S \subset \mathbb{N}^+$ is a finite set, prove that \( \sum_{n \in S} z^n \neq -\frac{20}{9} \).

Answer: -\dfrac{20}{9}


Question: Evaluate the limit: $$ \lim_{x\to1} \frac{x^{\frac{1}{13}}-x^{\frac{1}{7}}}{x^{\frac{1}{5}}-x^{\frac{1}{3}}} $$

Answer: \dfrac{45}{91}


Question: Find the limit of the sequence \( a_n = \frac{(n!)^2 \cdot (2n)!}{(4n)!} \) as \( n \to \infty \).

Answer: 0


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

Answer: 1


Question: Compute the limit: $$\lim_{n\to\infty} \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)^{\frac{1}{2}}\cdots\left(1+\frac{n}{n}\right)^{\frac{1}{n}}$$

Answer: e^{\frac{\pi^2}{12}}


Question:
Find the flux of the vector field \( \vec{G}=\operatorname{curl} \vec{F} \), where
\[ \vec{F}(x, y, z)=y^{3} \vec{i}+x^{3} \vec{j}+z^{3} \vec{k} \]
through the upward oriented part of the paraboloid
\[ z=1-x^{2}-y^{2} \]
lying over the plane \( z=0 \).

Answer: 0


Question: Determine the largest possible number of points $n$ ($n \ge 3$) in the plane such that any three of them form a right-angled triangle.

Answer: 4


Question: Let $u$ be a positive element in a $C^*$-algebra $A$ with $\|u\| \le 1$. Prove or disprove that $u^2 \leq u$.

Answer: u^2 \leq u


Question: Find the minimum value of \( P = \left(a + b + c + d\right)\left(\frac{1}{a} + \frac{1}{b} + \frac{4}{c} + \frac{16}{d}\right) \) where \(a, b, c,\) and \(d\) are positive integers. Also, determine the values of \(a, b, c,\) and \(d\) when this minimum is achieved.

Answer: 64


Question: Given the differential equation \( y'' + q(t)y = 0 \) where \( q(t) \) is a continuous periodic function with period \( T \), and \( y_1 \) and \( y_2 \) are linearly independent solutions, calculate the value of \( y_1(T) + y_2'(T) \) if there exists a non-trivial solution \( y_* \) that is periodic with period \( T \).

Answer: 2


Question: A rabbit runs in a perfect circle of radius $r$ with a constant speed $v$. A fox chases the rabbit, starting from the center of the circle and also moves with a constant speed $v$, always positioning itself between the center of the circle and the rabbit. Determine the time it will take for the fox to catch the rabbit.

Answer: \dfrac{\pi r}{2v}


Question: Let \(a < b\) be real numbers and \(m\) be a positive integer such that \(3^{-m} < \frac{b-a}{6}\). Determine if there exists a positive integer \(k\) such that the open interval \(\left(\frac{3k+1}{3^m}, \frac{3k+2}{3^m}\right)\) is contained within the open interval \((a, b)\).

Answer: Yes


Question: Determine the group homomorphisms from the multiplicative group of non-zero complex numbers, \( \mathbb{C}^\ast \), to the additive group of integers, \( \mathbb{Z} \).

Answer: 0


Question: Determine the number of real roots of the function \( f(x) = x^4 - x^3 \sin x - x^2 \cos x \).

Answer: 3


Question: Find the limit: \[ \lim_{\alpha\rightarrow 0}\alpha^{-3}e^{-\frac{1}{2\alpha^2}}. \]

Answer: 0


Question: Let $H$ be a closed hypersurface in $\mathbb{A}^{n}$, where $n$ is sufficiently large, over $\mathbb{C}$. Let $U$ be the open subset that is the complement of $H$. Given a point $x \in H$, determine if there exists a curve $C \subset \mathbb{A}^{n}$ such that $C \cap H = \{x\}$. Provide a justification for your answer.

Answer: Yes


Question: Find the dimension of the algebraic set $X = V(y-x^2, y^2-y+z^2)$ over an algebraically closed field $F$. The dimension of an algebraic set is defined as the transcendence degree of its rational function field over $F$ if the set is irreducible, or the maximum dimension of its irreducible components if the set is reducible.

Answer: 1


Question: Find the limit of \( G(x) = \int_x^{x^2} \frac{1}{\ln t} \, dt \) as \( x \rightarrow 0 \).

Answer: 0


Question: Can every function in the space of continuous functions vanishing at infinity, $C_{0} (\mathbb{R}^{n})$, be uniformly approximated by a sequence of functions from the Schwartz class, $\mathcal{S}(\mathbb{R}^{n})$?

Answer: Yes


Question: Compute \( \lim \limits_{x\to\infty} \left(\frac{x-2}{x+2}\right)^x \).

Answer: e^{-4}


Question: Find the number of divisors of \( N = 3^5 \times 5^7 \times 7^9 \) that are of the form \( 4n+1 \), where \( n \geq 0 \).

Answer: 240


Question: Evaluate the limit: \[ \lim_{x\to 1} \frac{\sin(x^2-1)}{x-1} \] using the known limit \( \alpha = \lim_{x\to 0} \frac{\sin(x)}{x} \). Do not use L'Hôpital's rule.

Answer: 2


Question:
Calculate the limit of the multivariable function as \( x \to 0 \):
\[
\lim_{x\to 0} \frac{(\ln(1+x_2)-x_2)(1-\cos(x_3))\tan(x_1)}{\|x\|^4}
\]
where \( \|x\| \) indicates the Euclidean norm and \( x \in \mathbb{R}^3 \).

Answer: 0


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

Answer: \pi


Question:
Let \( a > 1 \) and \( f, g: [-a,a] \rightarrow \mathbb{R} \) be twice differentiable functions such that:
- \( f(x) = 0 \) only for \( x = -a, 0, a \)
- \( f'(x) = 0 = g(x) \) only for \( x = -1, 0, 1 \)
- \( g'(x) = 0 \) only for \( x = -c, c \) where \( 0 < c < 1 < a \)

Which of the following relations between \( f \) and \( g \) is possible?
\[ f = g' \quad \text{or} \quad f' = g \quad \text{or} \quad f = -g' \]

Answer: f' = g


Question: Let \(G\) be a group and \(x, y \in G\) such that \(x\) and \(y\) commute and \(\text{gcd}(|x|, |y|) = 1\). Does it follow that \(|xy| = |x||y|\)? Justify your answer.

Answer: |xy| = |x||y|


Question: Simplify the expression \(\left|\frac{x-y-(x+y)e^{2iyL}}{x+y-(x-y)e^{2iyL}}\right|^2\) and verify that it equals 1.

Answer: 1


Question: Find the shortest distance from the triangle with vertices \((1,1,0), (3,3,1), (6,1,0)\) to the point \((9,5,0)\).

Answer: 5


Question:
Given a linear transformation \( L: \mathbb{R}^3 \to \mathbb{R}^3 \) such that:
\[ L(1,1,0) = (0,0,1), \quad L(1,2,1) = (2,1,2), \quad L(0,1,2) = (2,0,0), \quad L(3,5,2) = (4,2,5) \]
Determine \( L(7,12,100) \).

Answer: (10, -90, -83)


Question: Find the probability density function of the random variable $U = Y_1 + Y_2$, where $(Y_1, Y_2)$ is uniformly distributed over the region defined by $0 \leq y_1 \leq 1$, $0 \leq y_2 \leq 1$, and $0 \leq y_1 + y_2 \leq 1$. Express your answer in terms of $u$. \[ \int_0^1\int_0^{u-y_1}\,dy_2\,dy_1 \]

Answer: 2u


Question: Determine the limit of \( \frac{1}{n} \sum_{i=1}^n i^{\frac{1}{i}} \) as \( n \to \infty \).

Answer: 1


Question: Find the limit: \[ \lim_{x\to\infty} \frac{\displaystyle \sum\limits_{i = 0}^\infty \frac{x^{in}}{(in)!}}{\displaystyle\sum\limits_{j = 0}^\infty \frac{x^{jm}}{(jm)!}} \] for natural numbers \(n\) and \(m\).

Answer: \dfrac{m}{n}


Question: Determine the number of ways to express 100 as a sum of positive even integers.

Answer: 204226


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(\mathbb{Q}) \subset \mathbb{N}$. Determine whether $f$ is (1) unbounded, (2) constant, or (3) non-constant bounded.

Answer: B


Question: Given the equation \(a + b - 1 = 1 + \frac{\ln(2^a - 1)}{\ln 4} + \frac{\ln(2^b - 1)}{\ln 4}\), where \(a, b > 0\) are real numbers and \(\ln\) denotes the natural logarithm, determine if \(a = b\).

Answer: a = b


Question: Evaluate the triple integral \( \int \int \int z(x^2+y^2+z^2)^{-\frac{3}{2}} \, dx \, dy \, dz \) over the region defined by \( x^2+y^2+z^2 \leq 36 \) and \( z \geq 3 \).

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


Question: Calculate the vector result of the operation \( \nabla \times (\mathbf{k} \times \mathbf{r}) \), where \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \).

Answer: 2\mathbf{k}


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

Answer: 584


Question: Find the limit as \( n \to \infty \) of the integral \( \int_n^{n+2} e^{-x^3} \, dx \).

Answer: 0


Question: Given the equation \( a^2 + \frac{5}{2}b^2 + c^2 = 2ab + bc + ca \), find the value of \( a + 2b + 2c \).

Answer: 0


Question: Given non-negative real numbers $a, b, c, d, e, f$ such that $a+b+c+d+e+f=1$, find the maximum value of the expression $ab+bc+cd+de+ef$. Provide your answer as a single number.

Answer: \dfrac{1}{4}


Question:
Calculate the area of the region defined by the set:
\[ \{(x,y) \in \mathbb{R}^2 : 0 \leq x \text{ and } 0 \leq y \leq 2^{-\lfloor x \rfloor} \} \]

Answer: 2


Question: Given the equation \( f\left(\sin x - \frac{\sqrt{3}}{2}\right) = f(3x - \pi) + 3x - \pi \) for \( x \in [-\pi/2, \pi/2] \), find the value of \( f'(0) \).

Answer: -\dfrac{6}{5}


Question: Find the function $f(n)$ that satisfies the equation \( f(m + n) + f(m - n) = 2f(m) + 2f(n) \) for all integers $m$ and $n$, given that $f(4) = 16$.

Answer: n^2


Question: Let $E$ be an idempotent matrix different from the identity matrix. Prove or disprove that $\det(E) = 0$. Provide a counterexample if the statement is false.

Answer: 0


Question: Find the degree of the field extension \(\mathbb{Q}(\sqrt[5]{7} + \sqrt[5]{49})\) over \(\mathbb{Q}\).

Answer: 5


Question: Evaluate the double integral \( \iint \left(\frac{x-y}{x+y}\right)^4 \, dx \, dy \) over the triangular region bounded by the lines \( x+y=1 \), the \( x \)-axis, and the \( y \)-axis.

Answer: \dfrac{1}{10}


Question: Are the $L^p$ norms ordered by $p$ in the same way that power means are ordered?

Answer: Yes


Question: Let \( \Omega \subset \mathbb{R}^N \) be a bounded regular domain and \( p \in (1,\infty) \). Suppose that \( u \in W_0^{1,p}(\Omega) \) and \( u \) is locally essentially bounded. Does this imply that \( u \) is globally essentially bounded in \( \Omega \)?

Answer: Yes


Question: Determine the possible values of the integral \( \frac{1}{2\pi i}\int_{\gamma}\frac{2i}{z^2+1}dz \), where \( \gamma \) is a closed continuously differentiable path in the upper half-plane that does not pass through \( i \).

Answer: \mathbb{Z}


Question: Determine the degree of the splitting field of the polynomial $X^4 + X^3 + X^2 + X + 1$ over the field $\mathbb{F}_3$.

Answer: 4


Question: Calculate the area under the curve defined by the function \( \frac{x^3+2x^2+x+1}{x^2+2x+1} \) from \( x=0 \) to \( x=\infty \).

Answer: \infty


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

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


Question: Evaluate the limit: \[ \lim_{x \to 1} \frac{(1-x^{1/2})(1-x^{1/3})\cdots(1-x^{1/n})} {(1-x)^{n-1}} \]

Answer: \dfrac{1}{n!}


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

Answer: 0


Question: Determine whether the existence of a unique solution for the equation \(A\mathbf{x} = \mathbf{b}\) for some vector \(\mathbf{b}\) implies that the matrix \(A\) has a left inverse. Assume \(A\) is an \(m \times n\) matrix.

Answer: Yes


Question: Given the joint density function \(f(x,y)=\frac{1}{18}e^{-(x+y)/6}\) for \(0<x<y\), calculate the conditional variance \(\operatorname{Var}(Y |X = 2)\).

Answer: 36


Question: Determine the correct value of the limit: \[ \lim_{x\rightarrow 0}\frac{x}{2}\sqrt{\frac{1+\cos(x)}{1-\cos(x)}} \]

Answer: 1


Question: Evaluate the triple integral \( \iiint\limits_E \frac{yz}{x^2+y^2+z^2} \, dx \, dy \, dz \) using spherical coordinates, where the region \( E \) is bounded by the equation \( x^2+y^2+z^2-x=0 \).

Answer: 0


Question: Evaluate the triple integral \( I = \int_{0}^{2} \int_{0}^{2+3x} \int_{0}^{2-y+3x} x \, dz \, dy \, dx \).

Answer: 38


Question: In a complete metric space $E$ with the property that for any bounded sequence $x_n$ and for every $\epsilon > 0$, there exist indices $i \neq j$ such that $d(x_i, x_j) < \epsilon$, determine whether every bounded sequence in $E$ has a convergent subsequence.

Answer: Yes


Question: Let $X$ be a compact metric space and $\mathcal{M}(X)$ be the set of probability measures on $X$ equipped with the topology of weak convergence. Consider $A\subseteq X$ to be any measurable set in $X$. Is the map $P \mapsto P(A)$ from $\mathcal{M}(X)$ to $[0,1]$ measurable?

Answer: Yes


Question: Given the smooth function $f(x,y)$ with partial derivatives $f_{x}(1,1)=2$ and $f_{y}(1,1)=-3$, and the parameterization $\vec{r}(t)=(t,t^{2})$, determine if the derivative of the composite function $g(t)=(f\circ \vec{r})(t)=f(\vec{r}(t))$ at $t=1$ is equal to $1$. State whether this is true or false.

Answer: B


Question: Suppose $\xi$ is a measurable partition on a compact metric space $X$, generated by countable measurable sets $E_i$, such that $\xi = \vee_{i=1}^{\infty} \{ E_i , X \backslash E_i \}$. Let $\mathcal{B}_n$ be the finite sigma-algebra generated by sets $E_1,...,E_n$. For a sequence of uniformly bounded measurable functions $\phi_m$ that converges pointwise to a function $\phi$, does $E(\phi_m | \mathcal{B}(\xi))$ converge pointwise to $E(\phi | \mathcal{B}(\xi))$?

Answer: Yes


Question: Show that if \( f(x) = \frac{\cosh x}{\sinh x}f'(x) \), then \( f(x) = \cosh x \).

Answer: \cosh x


Question:
Evaluate the limit given that the derivative \( f'(1) = 3 \):
\[
 \lim_{h \to 0} \frac{f(1-5h^2)-f(1+3h^2)}{h^2(h+1)}
\]

Answer: -24


Question: Evaluate the contour integral \( \oint_{C} z \sin\left(\frac{1}{z-1}\right) \, dz \) where the curve \( C \) is defined as \( C = \{z : |z| = 2\} \). Determine if the solution to this integral is \( 2\pi i \).

Answer: 2\pi i


Question: Find the pointwise limit of the sequence of functions \( f_n(x) \), where \( f_n(x) = 1 \) if \( x \in [n, n+1] \) and \( f_n(x) = 0 \) otherwise, as \( n \to \infty \).

Answer: 0


Question: Let $G$ be a group and $\phi: G \rightarrow G$ be an isomorphism such that $\phi(g) = ga$ for some $a \in G$ and for all $g \in G$. If $\phi$ can be expressed as a single cycle including all elements of $G$, is $G$ necessarily a cyclic group? Justify your answer.

Answer: Yes


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

Answer: \ln x


Question: If $P(A^c \text{ i.o.}) = 0$, does it imply that $P(A \text{ i.o.})=1$? Here, "i.o." stands for "infinitely often."

Answer: Yes


Question: Find the exact value of the sum: \( \sin(1^\circ) + \sin(3^\circ) + \sin(5^\circ) + \ldots + \sin(179^\circ) \).

Answer: \dfrac{1}{\sin(1^\circ)}


Question: Evaluate the limit \( \lim_{x\to 0} \frac{x-\sin x - \cos^{-1} (e^{-\frac x2})}{x^2} \).

Answer: -\infty


Question:
Given the function \( u(x,t) = \sum_{k=-\infty}^{\infty} a_k e^{-\lambda_k^2 t} \cos(\lambda_k x) \) with \( \lambda_k = \frac{\pi}{2} + \pi k \) and the condition \( \lambda_{-k-1} = -\lambda_k \) for \( k \in \mathbb{N} \), express \( u(x,t) \) as \( u(x,t) = \sum_{k=0}^{\infty} b_k e^{-\lambda_k^2 t} \cos(\lambda_k x) \). Determine the correct expression for the coefficients \( b_k \):

1. \( b_k = 2 \int_0^1 u(x,0) \cos(\lambda_k x) \, \mathrm{d}x \)
2. \( b_k = \int_0^1 u(x,0) \cos(\lambda_k x) \, \mathrm{d}x \)

Which expression correctly represents \( b_k \)?

Answer: 1


Question: Consider the curve given by \( y = x^3 \). A tangent at a point \( C_1 \) (where \( C_1 \neq (0,0) \)) on this curve meets the curve again at \( C_2 \). The tangent at \( C_2 \) meets the curve at \( C_3 \), and this process continues. The x-coordinates of the points \( C_1, C_2, C_3, \ldots, C_n \) form a geometric progression. Determine the ratio of the area of the triangle \( C_1C_2C_3 \) to the area of the triangle \( C_2C_3C_4 \).

Answer: \dfrac{1}{16}


Question: Determine \(\lim_{n\to\infty} x_n\) if \(\left(1+\frac{1}{n}\right)^{n+x_n}=e\) for all \(n \in \mathbb{N}\).

Answer: \dfrac{1}{2}


Question: Calculate \( \lim\limits_{n \rightarrow +\infty} \int_{0}^{1} f_n(x) e^{-x^2} \, dx \), where \( f_n(x) = nxe^{-\sqrt{n}x} \) for \( x \in [0,+\infty) \). The sequence of functions \( f_n(x) \) converges pointwise to the null function on \( [0,+\infty) \) but not uniformly. However, there is uniform convergence on sub-intervals \( [a,+\infty) \) with \( a > 0 \).

Answer: 1


Question: Compute the infinite sum \( \sum_{x=0}^\infty P(X>x) \) for a discrete random variable \( X \) defined on the points \( x = 0, 1, 2, 3, \ldots \).

Answer: E[X]


Question: Let \(X\) be a projective scheme over a Noetherian ring, \(\mathcal{F}\) an invertible sheaf on \(X\), and \(U\) an arbitrary open subset of \(X\). Is \(\Gamma(U,\mathcal{F})\) a \(\Gamma(U,\mathcal{O}_X)\)-module of finite type?

Answer: Yes


Question: Find the smallest integer \( n \) such that \( 2^{2014} \mid 17^n - 1 \).

Answer: 2^{2010}


Question: Let \( g: \mathbb{R} \to \mathbb{R} \) be a continuous function such that \( \lim_{x \to \infty} (g(x) - x) = \infty \) and \( g \) has finitely many fixed points. Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function satisfying \( f \circ g = f \). Is it true that \( f \) is constant?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\rightarrow \infty} \frac{x \int_{0}^{x} e^{x^2} \, dx} {e^{x^2}} \]

Answer: \dfrac{1}{2}


Question: Let $G$ be a finitely presented group, and let $\mathbb{Z}G$ be the corresponding group ring. Consider $I$, the kernel of the augmentation morphism $\mathbb{Z}G \rightarrow \mathbb{Z}$. Is $I$ always a finitely generated $\mathbb{Z}G$-module as a right module?

Answer: Yes


Question: Consider a compact subset $K \subset \mathbb{R}^n$ and the set $C(K)$ of real-valued continuous functions on $K$, equipped with the sup norm. Let $g \in C(K)$ and $f: K \to K$. Define the operator $T_\beta: C(K) \to C(K)$ by \[(T_\beta F)(x) = \max_{a \in [0,1]} \{ a g(x) + (1-a) \beta F(f(x)) \}.\] Suppose for any $\beta \in [0,1]$, there is a unique fixed point $F_\beta$ of the equation $F = T_\beta F$. Additionally, for $\beta \in [0,1)$, $T_\beta$ satisfies Blackwell's conditions, ensuring a unique fixed point $G_\beta$. For a sequence $(\beta_n)_{n \geq 1}$ with $\beta_n \in [0,1)$ and $\beta_n \uparrow 1$, does $G_{\beta_n}$ converge to $F_1$?

Answer: Yes


Question: Evaluate the double integral \( I = \int_{-1}^{1}\int_{0}^{x^2} \sqrt{x^2-y}\,dy\,dx \).

Answer: \dfrac{1}{3}


Question:
Consider the Cauchy problem given by:
\[ (PC_{a,b}) \begin{cases} 3x^2y'' - 10xy' + 4y = 0 \\ y(2) = a \\ y'(2) = b \end{cases} \text{ with } a, b \in \mathbb{R} \]
Determine under which condition(s) the solution can be uniquely extended to all of \( \mathbb{R} \). Choose the correct option:

(A) We can uniquely extend the solution to all \( \mathbb{R} \) if and only if \( b = 2a \).
(B) We can extend the solution to \( \mathbb{R} \) and there is an infinite number of prolongations of the solution of \( (PC_{a,b}) \) if and only if \( b = 2a \).
(C) We can extend the solution to \( \mathbb{R} \) if and only if \( b = a = 0 \).
(D) The solution is always extendible to \( \mathbb{R} \) without imposing any condition on \( a \) and \( b \).

Answer: A


Question: Compute the limit: $$\lim_{n\to\infty}\left(n-(\arccos(1/n)+\cdots+\arccos(n/n))\right).$$

Answer: \dfrac{\pi}{4}


Question: Determine whether there exist pseudo-regular polygons with interior angles \(\theta\) such that \(\theta \neq (1-2/n)\pi\) for all \(n \ge 3\).

Answer: Yes


Question: Evaluate the limit: $$\lim _{x\to0}\dfrac{\displaystyle \int _0^{x^2}(\sin t)^{\frac{3}{2}} dt}{\displaystyle\int ^x_0t\left(t-\sin t\right)dt}$$

Answer: 12


Question: Find the limit of the sequence \( \frac{\sin{1}+\sin{2}+\cdots+\sin{n}}{n} \) as \( n \to \infty \).

Answer: 0


Question: Consider the multiplicative group $S = \{z : |z| = 1\} \subset \Bbb C$. Let $G$ and $H$ be subgroups of $S$ with orders $8$ and $10$, respectively. Determine the order of the intersection $G \cap H$. Note that the order of $G \cap H$ must divide both $o(G)$ and $o(H)$.

Answer: 2


Question: Find all positive numbers $T$ such that \( \int_T^\infty x^{-\log{x}} \, dx = \int_0^T x^{-\log{x}} \, dx \).

Answer: \sqrt{e}


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{\int_x^{2x} \frac{2t+3}{t+2} \, dt}{x} \]

Answer: 2


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

Answer: 0


Question: Calculate the surface integral \( \int_M \langle f, v \rangle \, dS \) over the ellipsoid \( M = \{(x,y,z) \mid x^2 + \frac{y^2}{4} + \frac{z^2}{9} = 1\} \), where \( f(x,y,z) = \begin{pmatrix} x + \sin(y)\cos(z) \\ y + e^{x^2+y^2} \\ -z + \log(1+x^2+y^2) \end{pmatrix} \) and \( v \) is the outward unit normal vector field. Use the divergence theorem to assist in your calculation.

Answer: 8\pi


Question: Let $U$ be an ultrafilter on $\mathcal{P}(\omega)$ and $\langle \sigma _\alpha \mid \alpha < \omega_1 \rangle$ be a sequence of elements of $U$. Is there an element $\sigma \in U$ such that for every finite $F \subset \sigma$, the set $\{\alpha\in \omega_1 \mid F \subset \sigma _\alpha \}$ is uncountable?

Answer: Yes


Question: What is the official name for an additive abelian group equipped with a multiplication that is distributive over addition but not necessarily associative?

Answer: non-associative ring


Question: For which values of \( p > 0 \) does the integral \( \int_{0}^{1}\frac{\sin(x^p)}{x}dx \) converge?

Answer: p > 0


Question:
Use Green's theorem to compute the area of the region bounded by the $x$-axis and the arch of the cycloid given by the parametric equations:

$$ x = t - \sin(t), \quad y = 1 - \cos(t), \quad 0 \leq t \leq 2\pi $$

Answer: 3\pi


Question: Suppose there are \(N\) realizations of a Gaussian process denoted as the vectors \(z_{j} \in \mathbb{R}^{n}\) for \(j = 1, \ldots, N\). Let \(y\) be a random variable such that \(y = \sum_{j=1}^{N}(Bz_{j})[i]\), where \(B\) is a unitary matrix satisfying \(B^* B = BB^* = I\). Determine the variance of \(y^{2}\).

Answer: 2N^2


Question: Find the maximum value of \( \int_0^1 (f(x))^3 \, dx \), given that \(|f(x)| \leq 1\) and \(\int_0^1 f(x) \, dx = 0\), where \(f\) is a real-valued function.

Answer: \dfrac{1}{4}


Question: Determine whether the statement "For any nontrivial ideal $I$ of a ring $R$, the quotient module $R/I$ is not a free $R$-module" is true or false. Provide a justification for your answer.

Answer: True


Question: Find the limit: $$\lim_{n \to \infty} \frac{((n+1)(n+2)\cdots(n+n))^{\frac{1}{n}}}{n}.$$

Answer: \dfrac{4}{e}


Question: Calculate the residues at the singular points of the function \( f(z) = z^{100} \cos\left(\frac{1}{z}\right) \).

Answer: 0


Question: From points on the line \(3x-4y+12=0\), tangents are drawn to the circle \(x^2+y^2=4\). The chord of contact of these tangents passes through a fixed point. Determine the slope of the chord of the circle that has this fixed point as its midpoint.

Answer: \dfrac{3}{4}


Question: Determine whether a proper subfield of the real numbers \( \mathbb{R} \), considered as a topological field, must be disconnected.

Answer: Yes


Question: Evaluate the summation \( S = \sum_{n=1}^{\infty} n^2 e^{in\theta} \) where \( \theta = \frac{\pi}{3} \).

Answer: -i\sqrt{3}


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{(2-1)\times (3-1) \times (5-1) \times \dots \times (p_n-1)}{2\times 3 \times 5 \times \dots \times p_n} \] where \(p_n\) is the \(n\)-th prime number.

Answer: 0


Question: Let G be a cyclic group of order n, where n is odd. Determine the infimum of the average of the sum of the element orders in G.

Answer: 1


Question: Is the conditional variance Var(X|\(\mathcal{P}(\Omega)\)) equal to 0 for any random variable X?

Answer: 0


Question: Evaluate the integral \( R(0) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{dk}{ik+1} \) using contour integration and the residue theorem. Verify that \( R(0) = \frac{1}{2} \).

Answer: \dfrac{1}{2}


Question: For a primitive 7th root of unity \( \omega \), calculate the magnitude of the expression \( |1+2\omega + 2\omega^2 + 2\omega^4| \).

Answer: \sqrt{7}


Question: Let $V$ be a convex region in $\mathbb{R}^3$ with boundary $S$, a closed surface, and let $\vec{n}$ be the unit outer normal to $S$. Suppose $F$ and $G$ are two continuously differentiable vector fields such that $\mathop{\rm curl} F = \mathop{\rm curl} G$, $\mathop{\rm div} F = \mathop{\rm div} G$ everywhere in $V$, and $G \cdot \vec{n} = F \cdot \vec{n}$ on $S$. Determine if $F = G$ everywhere in $V$.

Answer: F = G


Question: Find the integer closest to the expression: \[ 10^{2019} - \sqrt{10^{2019}(10^{2019} - 1)} \]

Answer: 1


Question: Find the number of ordered pairs \((m, n)\) of positive integers that satisfy the equation \(4^m - 3^n = 1\).

Answer: 1


Question: Suppose $X$ is a random variable such that $E[2^X] = 4$. Determine an upper bound for $P(X \ge 3)$ and justify your answer.

Answer: \dfrac{1}{2}


Question: Assume $g,f\colon A\subset\mathbb{R}^M\rightarrow\mathbb{S}^2$ are two bijective functions defined on the set $A$. Consider the constraint $C$: $\forall x,y\in A, \exists R\in SO(3)\colon Rf(x)=f(y)\iff Rg(x)=g(y)$. Does this constraint imply that $f=\pm g$?

Answer: f = \pm g