Question:
Determine the big O notation for the depth-first-search scheduling algorithm described below:

Given $n$ tasks and $m$ agents, where each agent can perform one task at a time, assign agents to tasks such that all tasks are completed in the minimum amount of time. The algorithm constructs a tree of every possible assignment and finds the shortest path (minimum time required). For a problem with 2 agents and 3 tasks, calculate the total number of edges in the network as $t$, where:

$t=\sum\limits_{i=1}^{n} \frac{n!}{(n-i)!}m^i$

What is the big O notation of this algorithm?

Answer: O(n! \cdot m^n)


Question: For all $n \geq 3$, does there exist an $n$-vertex 3-edge-colored graph such that (a) there are exactly 6 automorphisms which preserve the set of edge color classes, and (b) each of these 6 automorphisms results in a different permutation of the edge colors?

Answer: Yes


Question: Let $(A, \mathfrak{m}, k)$ be a Noetherian local ring, $M$ a finitely generated $A$-module, and $I$ an $\mathfrak{m}$-primary ideal. If $\operatorname{Ext}^{i}_{A}(A/\mathfrak{m}, M) = 0$, does it follow that $\operatorname{Ext}^{i}_{A}(A/I, M) = 0$?

Answer: Yes


Question: Provide an example of a finite, simple, undirected graph that is not $\mathbb{R}^2$-realizable.

Answer: K_5


Question:
Consider a sequence \( \{a_n\} \) of non-zero integers satisfying the following conditions: \( |a_n| < |a_{n+1}| \) for all \( n \), \( a_n \) divides \( a_{n+1} \) for all \( n \), and every integer is a divisor of some \( a_n \). Determine the nature of the series \( \sum_{n=1}^{\infty} \frac{1}{a_n} \). Is it:

1) Absolutely convergent with a rational sum
2) Absolutely convergent with an irrational sum
3) Absolutely convergent with a positive sum
4) None of the above

Answer: 2


Question: Let $Q \subseteq \mathbb{R}^n$ be a box, and let $f : Q \to \mathbb{R}$ be an integrable function on $Q$. Prove that if $f(x) > 0$ for all $x \in Q$, then \( \int_Q f > 0 \).

Answer: \int_Q f > 0


Question: Find a function $f(x)$ such that \( \lim_{x\to\infty}\frac{f(x)}{x}=1 \) and \( \lim_{x\to0}\frac{f(x)}{x^2}=C \) for a constant \( C \neq 0 \).

Answer: \dfrac{C x^2}{1 + C x}


Question: Compute the limit \( \lim_{n \to \infty} \int_{0}^{1} \frac{n x \log x}{1 + n^2 x^2} \, dx \).

Answer: 0


Question: Find the maximum value of \( \text{Re}(z) + \text{Im}(z) \) for a complex number \( z \) that satisfies the conditions \( |z - i| \leq 1 \) and \( \frac{\pi}{4} \leq \arg(z-i) \leq \frac{3\pi}{4} \).

Answer: 1 + \sqrt{2}


Question: Determine the Galois group of the polynomial \( f(x) = x^3 - 3x - 5 \) over \( \mathbb{Q} \).

Answer: S_3


Question: Consider a set $A \subset \mathbb{R}$ with the property that if $x \in A$, then $2x^2 - 1 \in A$. Can the set $A$ have exactly 100 elements?

Answer: Yes


Question: Solve for \( a \) in the congruence \( 22 \equiv 5^a \pmod{23} \).

Answer: 11


Question: Determine whether there is a natural bijection between $\text{Hom}_{\text{cont}}(G, \mathbb{Q}_p/\mathbb{Z}_p)$ and $\text{Hom}_{\text{cont}}(G, \mathbb{Q}/\mathbb{Z})$ for a pro-p group $G$, where $\mathbb{Q}_p$ and $\mathbb{Z}_p$ denote the p-adic rationals and integers, respectively.

Answer: Yes


Question: Unit circles $O_1$ and $O_2$ are tangent to each other and line $L$. Circle $O_n$ is tangent to $O_1$, $O_{n-1}$, and $L$. Find the radius of $O_{20}$.

Answer: \dfrac{1}{361}


Question: Consider the set of integers \( \mathbb{Z} \) with the usual less-than relation. Determine whether the well-founded part of this set, \( W(\prec) \), is the empty set.

Answer: \emptyset


Question: Consider the function \( f(y) = \int_0^\epsilon \frac{\left(\frac{p-t}{p(1-t)}\right)^y-(1-t)^{y(1-p)/p}}{t\log(1-t)}dt \) for \( p \in (0,1) \) and \( \epsilon > 0 \) a small constant. Is it true that \( f(y) = O(1) \) for large \( y \)?

Answer: O(1)


Question: Evaluate the integral \( \int_{0}^{\pi} \max \{ \sin(x), D(x) \} \, dm(x) \), where the Dirichlet function \( D(x) \) is defined as \( D(x) = 0 \) if \( x \) is rational and \( D(x) = 1 \) if \( x \) is irrational, for \( x \in [0, \pi] \).

Answer: \pi


Question:
For what positive value of \(c\) does the equation \(\log x = cx^4\) have exactly one real solution for \(x\)?
\(\text{(A) } \frac{1}{4e} \quad \text{(B) } \frac{1}{4e^4} \quad \text{(C) } \frac{e^4}{4} \quad \text{(D) } \frac{4}{e^{1/4}} \quad \text{(E) } 4e^{1/4}\)

Answer: A


Question: Evaluate the double integral \( \iint_S \frac{(x+y)^4}{(x-y)^5} \,dA \) where \( S = \{-1 \leq x + y \leq 1, 1 \leq x - y \leq 3\} \) using the change of variables \( u = x + y \) and \( v = x - y \).

Answer: \dfrac{4}{81}


Question: Determine the number of zeroes of the polynomial \( z^5 + 6z^3 - 10 \) within the annulus \( 2 < |z| < 3 \).

Answer: 2


Question: Find the maximum volume of an equilateral triangular prism inscribed in a sphere of radius 2. The volume of an equilateral triangular prism is given by \( \frac{\sqrt{3}}{4}a^2h \), where \( a \) is the side length of the base triangle and \( h \) is the height of the prism. Express this volume in terms of the radius of the sphere and determine the maximum volume.

Answer: 8


Question: Evaluate the limit: \[ \lim_{n\to \infty}\frac{\binom{2n}{n}}{4^n} \]

Answer: 0


Question: Consider the group $G = SL_2(F_3)$, where $F_3$ is the finite field of order 3. The order of $G$ is 24. Determine the number of Sylow-3 subgroups of $G$ and verify whether it is congruent to 1 modulo 3. If not, identify the mistake in the reasoning that leads to a contradiction with the Sylow Conjugation Theorem.

Answer: 4


Question: Let $X_1, X_2, \ldots, X_{25}$ be a sequence of independent and identically distributed random variables, each following a normal distribution $N(0,1)$. Define $Y = X_1 + X_2 + \cdots + X_{10}$ and $Z = X_1 + X_2 + \cdots + X_{25}$. Calculate the conditional variance $\text{Var}(Y|Z)$.

Answer: 6


Question: Compute \( 2^{1212} \mod 2013 \).

Answer: 70


Question: Given the sets $E_n = [n, \infty)$ for $n = 1, 2, 3, \ldots$, consider the set $E = \bigcap_{n=1}^{\infty} E_n$. Determine $\mu(E)$, where $\mu$ is a measure on the real numbers.

Answer: 0


Question:
Find the minimum value of the function \( f(x,y) = 2x + xy + 3y \) subject to the constraints:
\[ \begin{cases} x^2 + y \geq 3 \\ x + 0.5 \geq 0 \\ y \geq 0 \end{cases} \]
Using the Kuhn-Tucker conditions, determine the point \((x, y)\) that minimizes \(f(x, y)\).

Answer: (\sqrt{3}, 0)


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{- \infty}^{\infty} e^{-2|x| \left( 1+ \frac{\arctan(nx)}{\pi} \right)}dx. \]

Answer: \dfrac{4}{3}


Question: Determine the value of \(a \in \mathbb{R}\) for which all nontrivial solutions of the differential equation \(x'' + ax' + 4x = 0\) are oscillatory about zero.

Answer: (-4, 4)


Question: Let $X$ and $Y$ be finite-dimensional real normed spaces, and consider the set of linear operators $L(X,Y)$ between these spaces. Define the set of equivalence classes \( G(X,Y) := \{[T]; T,S \in L(X,Y) \text{ are equivalent if there exists an isometry } P \text{ on } Y \text{ such that } PT = S \} \). Determine if the condition \( \sup_{x \in X} \left| \|Tx\| - \|Sx\| \right| = 0 \) implies that \( [T] = [S] \).

Answer: No


Question: Suppose you roll 1000 1000-sided dice. What is the probability that exactly one die shows the number 1000?

Answer: \left( \dfrac{999}{1000} \right)^{999}


Question: Determine the values of \( \lambda > 0 \) for which the integral \( \int_0^\infty e^{\sin x} \frac{\sin 2x}{x^\lambda} \, dx \) converges.

Answer: (0, 2)


Question: Evaluate the integral \( \int_a^b [x] \, dx + \int_a^b [-x] \, dx \), where \([.]\) denotes the greatest integer function, and \(a\) and \(b\) are real numbers.

Answer: a - b


Question: Calculate the integral \( \int_C e^z\,dz \) where \( C \) is the circle \( |z - j\pi/2| = \pi/2 \) from the point \( z = 0 \) to the point \( z = j\pi \).

Answer: -2


Question: Determine the annular region of convergence for the Laurent series \( \sum_{n=-\infty}^{\infty} \frac{z^n}{3^n + 1} \).

Answer: 1 < |z| < 3


Question: Find the value of \( T \) such that the length of the curve \( C \), parameterized by \( s(t) = (t\cos(t), t\sin(t), \frac{2\sqrt{2}}{3}t^{\frac{3}{2}}) \) for \( 0 \leq t \leq T \), is equal to 1.

Answer: \sqrt{3} - 1


Question: Let $M$ be a $5 \times 5$ matrix with real entries. Suppose $M$ has finite order and $\det(M-I_5) \neq 0$. Find $\det(M)$. \( \boxed{-1} \)

Answer: -1


Question: Find the value of $k$ such that $(x^2 + kx + 1)$ is a factor of the polynomial $(x^4 - 12x^2 + 8x + 3)$. What is $k$?

Answer: 4


Question:
Given that \(f_k = \Omega(g_k)\) where \(f_k\) and \(g_k\) are two strictly positive, non-equal real sequences, and \(\Omega\) is the Big Omega notation in complexity theory, consider the following limit:
\[ \liminf_{k \to \infty} \frac{f_k}{g_k} = a > 0 \]
Assuming that \(\lim_{k \to \infty} f_k = 0\), is it possible to conclude that \(a\) is finite and not equal to infinity?

Answer: No


Question: If a family of congruent convex prisms tiles space, must there exist a tiling of the plane with polygons congruent to the prism's base?

Answer: Yes


Question: Evaluate the line integral \( \oint\limits_{C} xy^2dx +x^2ydy+z^3dz \) using Stokes' theorem, where \( C \) is the curve of intersection of the plane \( x+y+z=1 \) and the cylinder \( x^2 +y^2=4 \), oriented counterclockwise when viewed from above.

Answer: 0


Question: Integrate the vector field \( F(x,y,z) = \frac{1}{\sqrt{x^2+y^2}}(y,-y,1) \) over the surface of the paraboloid defined by \( z = 1 - x^2 - y^2 \) for \( 0 \leq z \leq 1 \), which corresponds to the region \( x^2 + y^2 \leq 1 \).

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


Question: Consider smooth vector bundles $\pi_{1} \colon E_{1} \to M$ and $\pi_{2} \colon E_{2} \to M$ over a smooth manifold $M$. Let $T \colon \Gamma(E_{1}) \to \Gamma(E_{2})$ be a map from the smooth sections of $E_{1}$ to the smooth sections of $E_{2}$. If $T$ is $\mathbb{R}$-linear, does $(T\sigma)_{p}$ depend only on the values of $\sigma$ in an arbitrarily small neighborhood of $p$?

Answer: No


Question: Assume $H \subset G$ is a closed subgroup, and the inclusion of groups is a homotopy equivalence. If $X$ is a CW complex and $E$ is a principal $G$-bundle over $X$, determine if there exists a principal $H$-bundle $E'$ over $X$ such that $E'(G) := (E' \times G)/H$ is isomorphic to $E$, where $H$ acts on $G$ by left multiplication.

Answer: Yes


Question: Given a function \( F: \mathbb{R}^2 \rightarrow \mathbb{R} \) that is \( C^2 \) with \( F_x(1,1)=F_{yy}(1,1)=1 \), \( F_y(1,1)=F_{xx}(1,1)=0 \), and a function \( g: \mathbb{R}^2 \rightarrow \mathbb{R} \) defined by \( g(r,\theta)=F(r\cos\theta,r\sin\theta) \), find the value of \( g_{r\theta}(\sqrt{2},\pi/4) \).

Answer: 0


Question: Determine whether the following assertion is true or false: If $T : X \rightarrow \mathbb{R}$ is a non-bounded linear operator, then for every $\alpha \in \mathbb{R^{+}}$, there exists a sequence $\{x_n\} \subseteq X$ such that $x_n \rightarrow 0$ implies $Tx_n \rightarrow \alpha$. Provide a justification for your answer.

Answer: A


Question: Determine the values of $x$ for which the series $$\sum_{n=1}^\infty\ln\sqrt[n]{1+\frac{x}{n}}$$ converges.

Answer: (-1, \infty)


Question: Find the terminating polynomial series solution for the differential equation \((1-x^2)y''-xy'+4y=0\) using the series form \(y(x) = x^q \sum_{n=0}^\infty a_n x^n\) with \(q = 0, 1\) and the recurrence relation \(a_{n+2} = \frac{(n+q)^2-4}{(n+q+1)(n+q+2)} a_n\).

Answer: 1 - 2x^2


Question: Does any additive, covariant functor preserve direct sums?

Answer: Yes


Question: Jones has been playing daily games for a very long time. If Jones wins a game, then he wins the next one with probability 0.6; if he has lost the last game but won the one preceding it, then he wins the next with probability 0.7; if he has lost the last 2 games, then he wins the next with probability 0.2. Today is Monday and Jones has just won his game. Find the probability that he won yesterday’s game.

Answer: \dfrac{3}{5}


Question: Let \( X_n \sim N(7n, n) \) and define \( Y_n = \frac{X_n}{n} \). Show that \( Y_n \) converges almost surely to a limit \( L \). What is \( L \)?

Answer: 7


Question: Let $A$ and $B$ be two random matrices such that $A=U^T B V^T$, where $U$ and $V$ are deterministic orthogonal matrices. Do the random matrices $A$ and $B$ have the same distribution?

Answer: No


Question: Consider an infinite division ring with prime characteristics. Is this division ring necessarily a field?

Answer: No


Question: Determine the cardinality of the set of chains of \((\mathcal{P}(\mathbb{N}), \subseteq)\).

Answer: 2^{2^{\aleph_0}}


Question: Compute the integral \( \int_{|z|=2}\frac{f'(z)}{f(z)} \, dz \) where \( f(z) = 1 + \frac{1}{z}+\frac{1}{2!z^2}+\dots+\frac{1}{n!z^n} \). Assume that \( f \) has no zeros for \( |z| \geq 2 \). Use the Argument Principle to determine the value of the integral.

Answer: 0


Question: Calculate the work done by the force field \( F = z^2i + 2xyj + 4y^2k \) on a particle that moves along the path from \((0,0,0)\) to \((0,2,1)\) to \((1,2,1)\) to \((1,0,0)\) and back to \((0,0,0)\).

Answer: -3


Question: If \(\lim_{x\to 0} \frac{\ln(1+ax+x^2)}{\sqrt{x+b}-1}=2\), find the value of \(a+b\).

Answer: 2


Question: Compute the total differential of the function \( g(x,y) = f(f(x,y), f(y,x)) \) at the point \((1,1)\), given that \( f(1,1) = 1 \), \( \partial_x f(1,1) = 1 \), and \( \partial_y f(1,1) = 2 \).

Answer: 5\,dx + 4\,dy


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}\frac{1}{\sqrt{n}(n-\sqrt[3]{n^3+2})} \]

Answer: -\infty


Question: Evaluate the limit: $$\lim _{x\to 0} \frac{1}{x}\left(\frac{1}{\arctan x}-\frac{1}{x}\right).$$

Answer: \dfrac{1}{3}


Question: Evaluate the limit: \[ \lim \limits_{x \rightarrow \infty} x \; \sqrt{\frac{3}{2}-\frac{3x}{4\pi}\sin\left(\frac{2\pi}{x}\right)}. \]

Answer: \pi


Question: Calculate the remainder when \((4^{103} + 2 \times 5^{104})^{102}\) is divided by 13.

Answer: 12


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

Answer: p > 0


Question: Given a discrete random variable \(X\) with possible values \(\{1, 2, 3, \dots\}\) and a probability mass function defined as \(c\left(\frac{1}{x} - \frac{1}{x+1}\right)\) where \(c > 0\), find the value of \(c\) and determine whether the expected value \(E[X]\) exists.

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to 0^{+}} \frac{x}{a} \left\lfloor \frac{b}{x} \right\rfloor \] where \( a, b > 0 \) and \( \left\lfloor x \right\rfloor \) denotes the floor function, the greatest integer less than or equal to \( x \).

Answer: \dfrac{b}{a}


Question: Given that \(\frac{p^2}{q}\) and \(\frac{q^2}{p}\) are the roots of the equation \(2x^2 + 7x - 4 = 0\), find the quadratic equation whose roots are \(p\) and \(q\), assuming \(p + q\) is real.

Answer: x^2 - x - 2 = 0


Question: Consider a random walk $X_t = \sum_{s=1}^t D_s$ with i.i.d. increments $D_t \in \mathbb{R}^n$, where $X$ is a martingale with $\mathbb{E}[D_t]=\vec{0} \in \mathbb{R}^n$. The support of $D_t$ is bounded, and $D_{t,i}$ has strictly positive variance for all $i \in \{1,\ldots,n\}$. Is it true that the probability that the random walk eventually pointwise falls below some threshold $k < 0$ equals $1$, i.e., $$ \mathbb{P}[ \exists t \text{ such that } X_{t,i} \leq k \text{ for all } i \in \{1,\ldots,n\} ] = 1 $$ for all $k<0$?

Answer: 1


Question: Determine the value of \( k \) such that the limit \( \lim_{x\to 0} \frac{\sqrt[5]{x+\sqrt[3]{x}}-\sqrt[3]{x+\sqrt[5]{x}}}{x^{k}} = A \) exists, where \( A \) is a non-zero constant.

Answer: \dfrac{11}{15}


Question: Determine the value of \( m \) for which the infinite series \( \sum_{n=1}^{\infty} \left( \sqrt[n^m]{2} - 1 \right) \) converges.

Answer: m > 1


Question: Determine whether there exists an integer $p$ such that for every integer $q$, the expression $4pq + 2p + 1$ is not a prime number.

Answer: 0


Question: In a regular pentagon $ABCDE$, point $M$ is the midpoint of side $AE$, and segments $AC$ and $BM$ intersect at point $Z$. Given that $ZA = 3$, find the length of $AB$. Express your answer in simplest radical form.

Answer: 3\sqrt{5}


Question: Let \( f(x) \) be a differentiable real function defined on the real line such that \( f(0) = 0 \) and \( f'(x) = [f(x)]^2 \) for every \( x \). Prove that \( f(x) = 0 \) for any \( x \).

Answer: 0


Question: In a fair die game, you roll a die for the first time as a preparatory roll, and the result is P. You continue rolling the die, and if you roll a number greater than or equal to P, the game ends. If you roll a number less than P, you add it to your total gain and continue rolling. What is the expected total gain from this game?

Answer: \dfrac{153}{40}


Question: Determine whether the set of all binary sequences $X^\omega$ and the set of countable subsets of $X^\omega$, denoted as $B$, have the same cardinality.

Answer: Yes


Question: Evaluate the integral \( \int_T y \, d(x,y) \), where the region \( T \) is defined as \( T=\{(x,y)\in\mathbb{R}^2:x,y\le 1,\; -x^2\le y\le 1-x,\; y\ge 0,\; -y^2\le x\le 1-y,\; x\ge 0\} \).

Answer: \dfrac{1}{6}


Question: Given a force vector \( \mathbf{F} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k} \) acting on the point \((1,5,2)\), and a line defined by \( \frac{x}{2} = y = \frac{z}{-2} \), find the torque about the given line. Use the formula \( T = \mathbf{n} \cdot (\mathbf{r} \times \mathbf{F}) \), where \( \mathbf{n} \) is a unit vector in the direction of the line and \( \mathbf{r} \) is the position vector.

Answer: 17


Question: Calculate the sum of the infinite series: \[ \sum_{n=1}^{\infty} \frac{1}{16n^2-8n-3} \]

Answer: \dfrac{1}{4}


Question: Find the number of incongruent solutions to the congruence equation \(x^2 \equiv 5 \pmod{1331p^3}\), where \(p = 2^{89} - 1\) is a Mersenne Prime.

Answer: 4


Question: Given \( n \) vectors \( a_1, \ldots, a_n \) in \( \mathbb{R}^{d} \), determine the maximum number of these vectors that can be linearly independent if their first \( k \) entries sum to zero.

Answer: d - 1


Question: Let $X = \mathbb{R}^{\mathbb{N}}$, the Cartesian product of $\mathbb{R}$ with itself $\mathbb{N}$-times. For each integer $n \geq 1$, define the semi-norm \( \rho_{n}(x) = \max\{|x_{i}| : i = 1, \ldots, n\} \) for each \( x = (x_{k})_{k \geq 1} \in X \). Consider $X$ endowed with the topology generated by this family of semi-norms. Assume $K \subset X$ is compact. Is it true that there exists $M > 0$ such that, for each $x \in K$, $|x_{i}| \leq M$ for all $i \geq 1$? Answer with 'True' or 'False'.

Answer: False


Question: Determine whether the expression \( \frac{1+x}{1+2x+x^2} \) can be simplified to \( \frac{1}{1+x} \) in the ring of formal power series.

Answer: \dfrac{1}{1+x}


Question: Solve the Euler-Lagrange equation for the functional \( L(y) = \int_{-1}^7 \sqrt{1 + (y')^2} \, dx \) with the initial conditions \( y(0) = 1 \) and \( y(1) = 2 \).

Answer: y = x + 1


Question:
Given sequences \( \{a_k\} \) and \( \{b_k\} \) with \( a_k, b_k \geq 0 \), and the conditions:

1. \( b_k \to 0 \).
2. \( \sum_{k = 0}^\infty (a_k - b_k) < \infty \).

Determine if it is correct to conclude that \( a_k \to 0 \). Justify your answer.

Answer: Yes


Question: Calculate the area of the surface defined by the parametric equations: \[ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} \exp(u) + \exp(-u) \\ 2u \\ v(\exp(u) - \exp(-u)) \end{bmatrix}, \] where \( 0 \leq u \leq 1 \) and \( 0 \leq v \leq 1 \).

Answer: \cosh(2) - 1


Question:
Consider the following commutative diagrams of sets and functions, where $f$ and $f'$ are invertible:
\[
\begin{array}{ccccccccc}
 A & \overset{f}{\longrightarrow} & B  && &B & \overset{f^{-1}}{\longrightarrow} & A\\
 u\downarrow&  & v\downarrow & &; & v\downarrow &  & u\downarrow\\
 A' & \overset{f'}{\longrightarrow} & B' &  && B' & \overset{f'^{-1}}{\longrightarrow} & A'
\end{array}
\]
Determine whether the first diagram commutes if and only if the second diagram commutes. Specifically, is it true that $v\circ f=f'\circ u$ if and only if $u\circ f^{-1}=f'^{-1}\circ v$?

Answer: Yes


Question: Given that there is a covering map from the closed unit disk \( D^2 \) to a topological space \( X \), determine the space \( X \) up to homeomorphism.

Answer: D^2


Question: Evaluate the integral \( \int_{-\infty}^{\infty} xe^{-2\lambda |x|} \, dx \).

Answer: 0


Question: If \( \lim_{n \to \infty} (2 x_{n + 1} - x_{n}) = x \), is it necessarily true that \( \lim_{n \to \infty} x_{n} = x \)? Justify your answer.

Answer: Yes


Question: Use Stokes' Theorem to evaluate the line integral \( \int_C (xy \,dx + yz\, dy + zx\, dz) \), where \( C \) is the triangle with vertices \((1,0,0), (0,1,0), (0,0,1)\), oriented in a counter-clockwise rotation as viewed from above.

Answer: -\dfrac{1}{2}


Question: Compute the value of the double series: $$ S = \sum\limits_{m = 1}^\infty  {\sum\limits_{n = 1}^\infty  {\frac{{m^2 n}}{{3^m (n \cdot 3^m  + m \cdot 3^n )}}} } $$

Answer: \dfrac{9}{32}


Question: Evaluate the integral \( \int_{|z|=2} \log\frac{z+1}{z-1} \, dz \), assuming that for \( w \in \mathbb{C} - \{z; \text{Re}(z) \leq 0\} \), we have \(-\pi < \text{Im}(\log w) < \pi \).

Answer: 4\pi i


Question: Does there exist an infinitely differentiable function $f: U \to \mathbb{R}$, where $U$ is an open subset of $\mathbb{R}$, such that the Taylor series of $f$ at $x = x_0 \in U$ has a radius of convergence $R > 0$, and $f$ equals its Taylor series only on the subinterval $(x_0 - r, x_0 + r)$, where $0 < r < R$?

Answer: Yes


Question:
Does a nontrivial power-complete finite pasture exist? A pasture is defined as an algebraic structure \(\langle M, 0, +, \times, \wedge \rangle\) where:

- \(\langle M, 0, +, \times \rangle\) is a ring (not necessarily commutative or unital).
- \(\wedge\) distributes over \(\times\) on the left: \((a \times b) \wedge c = (a \wedge c) \times (b \wedge c)\).
- \(\wedge\) distributes \(+\) into \(\times\) on the right: \(a \wedge (b + c) = (a \wedge b) \times (a \wedge c)\).

An element \(x \in M\) is a perfect power if \(x = y \wedge z\) for some \(y, z \in M\). A pasture is power-complete if all its elements are perfect powers. The trivial pasture \(\{0\}\) is power-complete. Determine if there exists a nontrivial power-complete finite pasture.

Answer: Yes


Question: Given that \( \int_0^1 x f(x) \, dx = A \) and \( \int_0^1 x^{2\pi-1} f(x^\pi) \, dx = 1 \), find the value of \( \cos A \).

Answer: -1


Question: Who is credited with the conjecture that all odd squares are solitary numbers?

Answer: Dean Hickerson


Question: Calculate the probability that the range of a random sample of size 3 from a uniform distribution is less than 0.8.

Answer: \dfrac{112}{125}


Question: Find the value of \( k \) such that \( \int^{\infty}_{0}\frac{\ln^2(x)}{(1-x)^2}dx + k\int^{1}_{0}\frac{\ln(1-x)}{x}dx = 0. \)

Answer: 4


Question: Let \( f \) be a continuous function on \([0,1]\). Evaluate the limit: \[ \lim_{x\rightarrow0 ^+} x^2\int_{x}^{1} \frac{f(t)}{t^3} dt \]

Answer: \dfrac{f(0)}{2}


Question: Determine whether there exists a Galois extension over \( \mathbb{Q} \) with degree 11. Provide reasoning or an example to support your answer.

Answer: Yes


Question: Find the limit of the sequence \(a_n = \frac{((n+1)!)((n-1)!)}{(2n)!}\) as \(n\) tends to infinity.

Answer: 0


Question: Find the inverse of \(x-1\) in the ring \(\mathbb{Z}_5[x] / (x^3 + x + 1)\).

Answer: 3x^2 + 3x + 1


Question: Let \( K \) be a field and \( a \in K \) such that \( f(x) = x^n - a \) is irreducible. If \( m \in \mathbb{N} \) divides \( n \) and \( \alpha \) is a root of \( f(x) \) in an extension field of \( K \), find the minimal polynomial of \( \alpha^m \).

Answer: x^{n/m} - a


Question:
Determine whether there exist constants \(\delta \in (0, \pi)\) and \(C > 0\) such that for all \(N \in \mathbb{N}\), the inequality \(\sup_{s \in [-\delta, \delta]} \left|\int_0^s D_N(t) \operatorname{d}t\right| \le C\) holds, where the \(N\)-Dirichlet kernel is defined as:
\[ D_N(t) = \frac{\sin\left(\left(N+\frac{1}{2}\right)t\right)}{\sin\left(\frac{t}{2}\right)}. \]

Answer: Yes


Question: Determine the domain of definition for the function \( f(x) = \int_{0}^{\infty} \frac{t^x}{1+t^4 \sin^2 t} \, dt \).

Answer: (-1, 1)


Question: Determine the order of infinitesimal of the function \( h(x) = \frac{\ln^5(1+x\ln^{1/8}(x))}{x^x-1} \) with respect to \( x \) as \( x \to 0^+ \).

Answer: 4


Question: Determine the index of the image of the subgroup \( \mathbb{Z}^2 \) under the linear transformation given by the matrix \( T = \begin{bmatrix} 1 & -3 \\ 1 & 2 \end{bmatrix} \).

Answer: 5


Question: In an examination consisting of 20 questions, each question can be scored as -1, 0, or 4. Determine the number of distinct total scores a student can achieve.

Answer: 95


Question: Is an element of the group \(\mathrm{SO}(3)\) uniquely determined by its action on the four points of \(S^2/\sigma\) corresponding to \(\frac{1}{\sqrt{3}}(1, 1, 1)\), \(\frac{1}{\sqrt{3}}(1, 1, -1)\), \(\frac{1}{\sqrt{3}}(1, -1, 1)\), and \(\frac{1}{\sqrt{3}}(1, -1, -1)\)?

Answer: Yes


Question: Find a function \( g(s) \) such that \( g(s) \sim \log_{\zeta(s)}{e} \) as \( s \to \infty \), where \( \zeta(s) \) is the Riemann zeta function.

Answer: 2^s


Question: Provide an example of a group that cannot be expressed as the direct product of its Sylow subgroups.

Answer: S_3


Question: In triangle ABC, angle B is 90 degrees. A point D is located inside the triangle such that AD = 20, DC = 15, DB = 10, and AB = 2BC. Calculate the area of triangle ABD.

Answer: 20\sqrt{5}


Question: Determine whether the expression \(W_{-1}(x)=\ln \cfrac{-x}{-\ln \cfrac{-x}{-\ln \cfrac{-x}{\ldots}}}\) is asymptotically bounded by \(\Theta(\ln x)\).

Answer: Yes


Question: Calculate the limit \( \lim_{c\rightarrow 1+}\sum_{j=0}^{\lfloor\frac{\log n}{\log c}\rfloor}(-1)^j\binom{z}{j}c^j \) for constant values of \(n\) and \(z\).

Answer: 0


Question: Let \( X = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 = 1\} \cup ([-1,1] \times \{0\}) \cup (\{0\} \times [-1,1]) \). Determine the maximum number of distinct points \( n_0 \) that can be removed from \( X \) such that the remaining set \( X - \{p_1, p_2, \ldots, p_{n_0}\} \) is still connected. What is the value of \( n_0 \)?

Answer: 4


Question: Solve the system of equations: \[ \begin{aligned} x^2 + y^2 &= 8 \\ \sqrt[2018]{x} - \sqrt[2018]{y} &= (\sqrt[2019]{y} - \sqrt[2019]{x})(xy + x + y + 2020) \end{aligned} \]

Answer: (2, 2)


Question: Compute the integral \( \int_\Omega  \left|\,x\,\right|\,dx\,dy\,dz \) over the volume of a torus. In \(\mathbb{R}^3\), let \(C\) be the circle in the \(xy\)-plane with radius 2 and the origin as the center, i.e., \(C= \Big\{ \big(x,y,z\big) \in \mathbb{R}^3 \mid x^2+y^2=4, \ z=0\Big\}.\) Let \(\Omega\) consist of all points \((x,y,z) \in \mathbb{R}^3\) whose distance to \(C\) is at most 1. Use appropriate coordinate transformations and symmetry considerations to simplify the computation.

Answer: 17\pi


Question: Compute the Riemann-Stieltjes integral \( \int_{-1}^1 \frac{1}{1+x^2} \, dg(x) \), where \( g(x) = \begin{cases} 1 & \text{if } x<0, \\ 0 & \text{if } x=0, \\ -1 & \text{if } x>0. \end{cases} \)

Answer: -2


Question:
Let \( F \in \mathbb{Z}[X_{1},X_{2},X_{3},X_{4},X_{5}] \) be defined as \( F = (X_{1}+X_{2}+X_{3})^{2}X_{4}X_{5} + X_{1}X_{2}X_{3}(X_{4}+X_{5}) \). If \( A = \{ a_1, a_2, a_3, a_4, a_5 \} \subseteq \mathbb{R} \) is a set with \(|A| = 5\), find the maximum number of distinct elements in the set:
\[ \{ F(a_{\sigma(1)}, a_{\sigma(2)}, a_{\sigma(3)}, a_{\sigma(4)}, a_{\sigma(5)}) \mid \sigma \in S_5 \} \]

Answer: 10


Question: Determine the values of \( \alpha \) for which the series \( \sum_{n=3}^{\infty} \frac{1}{n \cdot \ln(n) \cdot (\ln(\ln(n)))^{\alpha}} \) converges.

Answer: \alpha > 1


Question: Find the particular solution to the differential equation \( \frac{d^2y}{dx^2} = \left( \frac{dy}{dx} \right)^2 + 1 \) with the initial conditions \( y(0) = 0 \) and \( \frac{dy}{dx}(0) = 0 \).

Answer: -\ln(\cos x)


Question: Find the volume enclosed between the surfaces \(x+y+2z=2\) and \(2x+y+z=4\) in the first octant.

Answer: \dfrac{10}{3}


Question: Given a matrix $A$, let $B = AA$. Define matrix $C$ as $C = \begin{pmatrix} A & 0 & 0 \\ 0 & A & 0 \\ 0 & 0 & A \end{pmatrix}$ and matrix $D$ as $D = \begin{pmatrix} B & 0 & 0 \\ 0 & B & 0 \\ 0 & 0 & B \end{pmatrix}$. Prove or disprove that $D = CC$. Provide a justification for your answer.

Answer: D = CC


Question:
Given that \( f \) is a function with continuous derivatives, express \( f(x) \) in terms of \( x \) from the equation:
\[
f(x)=\lim_{t\to 0} \frac{1}{2t} \int_{x-t}^{x+t} s f'(s) \, ds
\]

Answer: Cx


Question: Determine whether the recursive sequence defined by \( b_{n+1} = 2b_n - c(b_n)^2 \) is convergent, where \( c > 0 \) is a constant and \( b_0 \in \left(0, \frac{1}{c}\right) \). If the sequence is convergent, find its limit.

Answer: \dfrac{1}{c}


Question: You have twelve boxes, each containing between 1 and 6 marbles. Each marble weighs 1 gram, and the marbles are fixed in place within the boxes. Using a simple balance scale and a reference weight of your choosing, determine the number of marbles in each box. What reference weight should you choose?

Answer: 7


Question: Compute the limit: $$\lim_{x\rightarrow\infty}\left(\cos\sqrt{\frac{2\pi}{x}}\right)^x$$

Answer: e^{-\pi}


Question: A uniform container of height $6m$ is full of water and has three identical drainage pipes $A$, $B$, and $C$. Each pipe can independently empty the container (the part above it) in $T$ minutes. Pipe $A$ is fixed at the base, while pipe $B$ is fixed at a height of $5m$. Pipe $C$ is fixed somewhere between them. If all three pipes together can empty the container in $\frac{2T}{3}$ minutes, determine the height at which pipe $C$ is fixed. Assume no variations due to pressure differences at different heights.

Answer: 4


Question: Find the sum of the infinite series given by \[ \sum_{i=0}^{\infty} \sum_{j=0}^{\infty} \sum_{k=0}^{\infty} \frac{1}{3^i} \cdot \frac{1}{3^j} \cdot \frac{1}{3^k} \] with the condition that \(i \neq j \neq k\).

Answer: \dfrac{81}{208}


Question: Is every open affine subscheme of an algebraic $k$-variety an affine $k$-variety? Specifically, if $U=\text{Spec} A$ is an affine open subset of an algebraic $k$-variety, and $A$ is a $k$-algebra, is $A$ finitely generated?

Answer: Yes


Question:
Let $X_n = \{0,1,2\}^n$ be the $n$-tuple Cartesian product of the ternary set. Define the subset $W_n$ as follows: \[ W_n = \{ y \in X_n : y_1 \leq 1, y_1 + y_2 \leq 2, \ldots, y_1 + \cdots + y_{n-1} \leq n-1, y_1 + \cdots + y_n = n \}. \]

Consider the polynomial \[ P_n(x) = \sum_{y \in W_n} x^{\prod_{j=1}^n \binom{2}{y_j}}. \]

Prove or disprove that the derivative of $P_n(x)$ evaluated at $x=1$ equals the $(n+1)$-th Catalan number $C_{n+1}$, i.e., \[ \frac{dP_n}{dx}(1) = C_{n+1}. \]

Answer: C_{n+1}


Question:
Let $f(z)$ be an entire function such that $|f(z)| < K|z|$ for all $z \in \mathbb{C}$, where $K > 0$. Given that $f(1) = i$, determine the value of $f(i)$. Choose from the following options:

1. $1$
2. $-1$
3. $i$
4. $-i$

Answer: 2


Question: Let \( f(x) = \log_{\cos 3x} (\cos 2ix) \) for \( x \neq 0 \) and \( f(0) = K \). Given that \( f(x) \) is continuous at \( x = 0 \), find the value of \( K \). Note: The use of L'Hôpital's rule and series expansion is not allowed.

Answer: -\dfrac{4}{9}


Question:
For \( \alpha > 0 \), define the function \( g(x) \) as follows:
\[
g(x) = \begin{cases}|x|^\alpha \cos(1/x^2) & x \neq 0 \\ 0 & x = 0\end{cases}.
\]
Determine the values of \( \alpha > 0 \) for which \( g(x) \) is differentiable at \( x = 0 \).

Answer: \alpha > 1


Question: Does there exist a positive irrational number \( \alpha \), such that for any positive integer \( n \), the number \( \lfloor n\alpha \rfloor \) is not a prime?

Answer: Yes


Question: Given independent random samples \((X_{1},X_{2},X_{3},...,X_{8}) \sim N(8,16)\), \((Y_{1},Y_{2},Y_{3}) \sim N(1,9)\), and \((Z_{1},Z_{2},Z_{3}) \sim N(6,10)\), find the probability \(P(2\bar{X}-\bar{Z}>3\bar{Y})\). The sample means are distributed as \(\bar{X} \sim N(8,2)\), \(\bar{Y} \sim N(1,3)\), and \(\bar{Z} \sim N(6,\frac{10}{3})\). Calculate \(P(2\bar{X}-\bar{Z}>3\bar{Y})\).

Answer: 0.8708


Question: Determine whether the sequence \((x_n)\) defined by \(x_1=\sqrt{2}\) and \(x_{n+1} = (\sqrt{2})^{x_n}\) converges.

Answer: 2


Question: Suppose $f:\mathbb{C}\rightarrow\mathbb{C}$ is an entire function and $g:\mathbb{C}\rightarrow\mathbb{C}$ is an entire function that may have zeros. If $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, does it necessarily follow that $f(z) = Cg(z)$ for some constant $C \in \mathbb{C}$? Decide whether this statement is true or false.

Answer: True


Question: Use Fermat's Little Theorem to find the remainder when \(5^{15}\) is divided by 1337.

Answer: 944


Question: Find \( \lim_{n \to \infty} \frac{(n!)^{1/n}}{n} \).

Answer: \dfrac{1}{e}


Question: Let $T$ be a compact operator with spectral radius $\rho(T)<1$. Does it follow that $\|T^n\| \to 0$ as $n \to \infty$?

Answer: Yes


Question: Simplify the expression: \[ \sqrt{2+\sqrt{2}} + \sqrt{2+\sqrt{2+\sqrt{2}}} + \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}} + \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}} \]

Answer: 6


Question: Maximize the expression \( xy + yz \) subject to the constraints \( x^2 + y^2 = 2 \) and \( y + z = 2 \).

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( \frac{\sin(2x) - 2\sin(x)}{x \cdot \arctan^2(x)} \right) \] without using L'Hôpital's rule.

Answer: -1


Question: Evaluate the iterated integral \( \int_{0}^{2}\left (\int_{0}^{\sqrt{2x-x^2}} \sqrt{x^2+y^2} \, dy\right ) \, dx \).

Answer: \dfrac{16}{9}


Question: Evaluate the summation: $$\sum_{j, k=-q}^q 1_{(h+j-k=0)}$$ and verify that the result is $2q+1-h$.

Answer: 2q + 1 - h


Question: Solve the congruence \(29x^{33} \equiv 27\ \text{(mod 11)}\) using the primitive root method, where the primitive root is 2 and \(ord_{11} (2) = 10\).

Answer: 6


Question: Find the value of \(a\) such that \(\lim_{x\to\infty} \frac{(ax)!}{x^x} = 1\).

Answer: e


Question: Given the polynomial $p(x) = x^3 + ax^2 + bx + c$, where $a, b, c$ are distinct non-zero integers, and the conditions $p(a) = a^3$ and $p(b) = b^3$, find the value of $p(13)$. Assume $a$ and $b$ are distinct.

Answer: 1927


Question: Given the function $f : \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0}$ satisfying $f(1)=1$ and the functional equation \[ f\left(f(x+1) + f\left( x + f(x)\right)\right) = x+2, \] determine the value of $f(10)$.

Answer: 7


Question: Given that the logarithm is base 2, verify that \((\log(\log n))^{(\log_{\log n}n^{-1})} = 1\).

Answer: 1


Question: Given that $Z = x - iy$ and $Y = p + iq$ is a cube root of $Z$, find the value of \[ \frac{\left(\frac{x}{p} + \frac{y}{q}\right)}{p^2 + q^2}. \]

Answer: -2


Question: Determine whether the following statement is true or false: If $F_0$ is a countable submodule of a free $\mathbb{Z}$-module $F$, then $F/F_0$ does not contain any uncountable injective submodule.

Answer: A


Question: If a complex function \( f(z) = 3 \) everywhere on the unit circle and is analytic everywhere in \( \mathbb{C} \), what is the value of \( f(1/3) \)?

Answer: 3


Question: Find the value of \( \lim_{n \to \infty}\int_{0}^{1}nx^ne^{x^2}dx \).

Answer: e


Question: Given a real sequence \( \{x_n\} \), if the sequence \( \{2x_{n+1} + \sin x_n\} \) is convergent, is it true that the sequence \( \{x_n\} \) is also convergent? Answer with 'True' or 'False'.

Answer: True


Question: Are all known $k$-multiperfect numbers for $k > 2$ not squarefree? A positive integer $N$ is $k$-multiperfect if $\sigma(N) = kN$, where $\sigma(x)$ is the sum of the divisors of $x$ and $k$ is a positive integer.

Answer: Yes


Question: Evaluate the limit: \[ \lim _{n\to \infty }\left(n-n^2\int _0^{\frac{\pi }{4}}(\cos x-\sin x)^n dx\right) \]

Answer: 2


Question: Evaluate the definite integral \( \int_0^{1/2} x^3 \arctan(x) \, dx \) to four decimal places.

Answer: 0.0059


Question: An assembly line samples 150 finished items per day, counting the number of defective items, F. The probability of an item being defective, p, varies daily and follows a uniform distribution on the interval [0, 1/3]. What is the expected number of defective items per day?

Answer: 25


Question: Let $G$ be a finite group with $n$ elements. Consider the group multiplication table of $G$ with an initial order $g_1, g_2, \ldots, g_n$. In this table, each row contains all elements of $G$ exactly once. Define $s_i$ as the product of the elements in the $i$th row, taken in the order they appear. Determine the number of distinct values of $s_i$ that can be obtained.

Answer: 1


Question:
Given independent random variables \( X \sim N(\mu,\sigma_1^2) \) and \( Y \sim N(\mu,\sigma_2^2) \), find the value of \( \mu \) such that:

\[ P(2X + 4Y \leq 10) + P(3X + Y \leq 9) = 1 \]
\[ P(2X - 4Y \leq 6) + P(-3X + Y \geq 1) = 1 \]

Answer: 2


Question: Given a compact metric space $(X,d_X)$ and another metric space $(Y,d)$, where the metric capacity of $(Y,d)$ is at least that of $(X,d_X)$, i.e., \( \kappa_X(\epsilon) \leq \kappa_Y(\epsilon) \) for all \( \epsilon \in (0,1] \), does there necessarily exist a continuous injection of $(X,d_X)$ into a metric ball in $(Y,d)$ of some finite radius? Assume $X,Y \neq \emptyset$. Note that the inclusion does not need to be Lipschitz, just continuous.

Answer: No


Question: Compute the Lebesgue integral of the function \( f(x) \) over the interval \([0, 5]\), where \( f(x) = 1 \) for \( x = \frac{1}{n} \) with \( n = 1, 2, \ldots \), and \( f(x) = 2 \) otherwise. Determine the value of \( \int_0^5 f(x) \, dm(x) \).

Answer: 10


Question: Identify an irreducible representation of degree 2 for the symmetric group $S_4$.

Answer: (2,2)


Question: Find the last two digits of the expression \(808^2 + 3^{305} \times 11^{151}\) using congruences.

Answer: 37


Question: Let $G$ be a profinite group, and let $K$ and $H$ be closed subgroups of $G$ such that $K \vartriangleleft H$ and $(H:K) < \infty$. If $K \subseteq B \subseteq H$, is the subgroup $B$ closed in $G$?

Answer: Yes


Question: Evaluate the integral: $$\frac{1}{2}\int_{-\infty}^\infty x^2e^{-|x|}\ dx$$

Answer: 2


Question: In a right-angled triangle \(\triangle ABC\), the perpendicular from \(A\) to \(BC\) intersects \(BC\) at point \(D\). A point \(P\) is chosen on the circumcircle of \(\triangle ADC\) such that \(CP \perp BC\) and \(AP = AD\). A square is drawn on the side \(BP\) with an area of 350 square units. What is the area of \(\triangle ABC\)?

Answer: 70


Question:
Given that $a^2 - b^2 = 2$, find the least possible value of the determinant:
\[
\begin{vmatrix}
1+a^2-b^2 & 2ab & -2b \\
2ab & 1-a^2+b^2 & 2a \\
2b & -2a & 1-a^2-b^2
\end{vmatrix}
\]

Answer: 27


Question:
Given n independent variables $x_1$ to $x_n$ and another variable $y$ that can take values 0 and 1 with $P(y=0)=P(y=1)=0.5$, and $P(x_i=1|y=1) = a$, $P(x_i=1|y=0)=b$ for all $i$, find $P(y=1|X)$ where $X = [1,0,1,0,\ldots,1,0]$. Use Bayes' theorem to determine the correct expression for $P(X)$ in the denominator:

1. $P(X) = ((a+b)/2)^{n/2} (1-(a+b)/2)^{n/2}$
2. $P(X) = a^{n/2}(1-a)^{n/2}0.5 + b^{n/2}(1-b)^{n/2}0.5$

Which expression correctly computes $P(X)$?

Answer: 2


Question:
Consider a polynomial function \( f(x) \) of degree \( n \) that satisfies the equation \( f(x) = f'(x) \cdot f''(x) \). Determine the nature of \( f(x) \). Is it:

A) an onto function
B) an into function
C) no such function possible
D) an even function

Answer: A


Question: Determine whether an embedded resolution can always be found when $X$ is a smooth variety and $Y$ is a curve.

Answer: Yes


Question: Determine the number of zeros of the function \( f(z) = z^{2019} + 8z + 7 \) that lie inside the unit disk.

Answer: 1


Question: Let $G$ be a non-abelian 2-group of order greater than or equal to 32 with $|Z(G)|=4$. Determine whether there exists an abelian subgroup $H$ of $G$ such that $16 \leq |H| \leq |G|/2$. Provide a justification for your answer.

Answer: Yes


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

Answer: (-1, 1)


Question: Suppose $U$ and $V$ are two irreducible complex representations of a finite group $G$ such that the tensor product $U \otimes V$ is also irreducible. Must it be the case that either $U$ or $V$ (or both) are one-dimensional?

Answer: Yes


Question: Evaluate the limit: \( \lim_{n \to \infty} \frac{(3n)!(1/27)^n}{(n!)^3} \).

Answer: 0


Question: Consider the set \( A = \{1, 2, 3, 4, 5, 6, 7\} \). Determine the number of onto functions \( f: A \rightarrow A \) such that at least 3 elements of \( A \) are mapped to themselves (i.e., have self-images).

Answer: 407


Question: Let $T: V \rightarrow V$ be a linear operator with the minimal polynomial $m(x) = (x-1)x^8$. Find the minimal polynomial of $T^3$. \( \text{Provide your answer as a polynomial.} \)

Answer: (x - 1)x^3


Question: Determine the number of ring homomorphisms from the polynomial ring \(\mathbb{Z}[X]\) to the integers \(\mathbb{Z}\).

Answer: \infty


Question: If a power series \( \sum_{k=0}^\infty a_k z^k \) converges absolutely for \( |z| = c \) where \( c \in \mathbb{R} \), must its radius of convergence \( R \) be strictly greater than \( c \)?

Answer: No


Question: Consider the sequence of functions \(\{y_{m}(x)\}\) on \([a,b]\). Assume that \(|y_{m}(x)|\leq M_{m}\) for all \(x\in [a,b]\) and all \(m\in \mathbb{Z}^{+}\). Suppose \(\sum_{m=1}^{+\infty}M_{m}<+\infty\). Does \(y_{m}(x)\) uniformly converge to zero on \([a,b]\)?

Answer: Yes


Question: Determine whether the contour integral \( \oint_{|z|=2} \frac{e^{\frac{1}{z}}}{z(z^{2}+1)} \, dz \) is equal to zero.

Answer: 0


Question:
Given the recurrence relation for \(a_n\):
\[
a_{n} = \sum_{t=0}^{n-1} \binom{n}{t} (-1)^{n-t-1} 2^{t(n-t)} a_{t}, \quad a_{0} = 1
\]
Evaluate \(a_3\).

Answer: 25


Question: Determine the maximum number of distinct seating plans for 15 people in a row of 15 chairs, where two seating plans are considered the same if they share the same adjacent quadruples. For example, if the people are seated in the order 1, 2, 3, 4, 5, ..., 15, then the quadruples 1234, 2345, 3456, ..., 12131415 cannot be used in other plans. What is the maximum number of distinct seating plans possible?

Answer: 2730


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

Answer: -\dfrac{1}{2}


Question: Evaluate the integral: \[ \int_{0}^{\pi/2} \frac{\sin x \cos^5 x}{(1-2\sin^2x\cos^2x)^2} \, dx \]

Answer: \dfrac{\pi}{8}


Question: If $F_1$ and $F_2$ are subspaces of a Hilbert space $H$, and they satisfy $F_1^\bot = F_2^\bot$, is it true that $F_1 = F_2$? Justify your answer.

Answer: No


Question: Compute the value of the infinite series \( \sum\limits_{n=0}^\infty \frac{n^2}{n!} \).

Answer: 2e


Question:
Find the value of \( x \) that satisfies the equation \( x = \sqrt{2 + \sqrt{2 - \sqrt{2 + x}}} \). Choose the correct answer from the following options:

(A) \( 2 \cos 10^\circ \)
(B) \( 2 \cos 20^\circ \)
(C) \( 2 \cos 40^\circ \)
(D) \( 2 \cos 80^\circ \)

Answer: C


Question: Is there a covering map from the Klein bottle to the torus?

Answer: No


Question: Consider the symmetric group $S_5$ of permutations of five symbols. Let $S$ be the set of subgroups of $S_5$ that are isomorphic to the non-cyclic group of order 4. Two subgroups $H$ and $K$ in $S$ are considered equivalent if they are conjugate in $S_5$. Determine the number of equivalence classes in $S$.

Answer: 2


Question: Calculate \( \lim _{n \rightarrow +\infty} \int_0^{\pi} \sin \left(x+\frac{x^{2019}}{n}\right) \, dx \).

Answer: 2


Question: Evaluate the contour integral \( \oint_C \frac{3}{z + 1 + i} \, dz \) along the circle \( |z| = 2 \).

Answer: 6\pi i


Question: In an equilateral triangle \( \triangle ABC \), a point \( M \) is located inside the triangle such that \( |MA|^2 = |MB|^2 + |MC|^2 \). Determine the measure of \( \angle BMC \).

Answer: 150^\circ


Question: Evaluate the integral \(\int_{C} \frac{1}{(\zeta - z)^{n + 1}} d\zeta\) where \(C\) is a circle centered at \(z\) with radius \(\epsilon\). Determine if the integral evaluates to zero.

Answer: 0


Question: Evaluate whether the expression \( \left[ \left(\left(12\cos^2t+K^2\sin^2t\right)^{\frac{S+3}{2}}-\left(12\cos^2t\right)^{\frac{S+3}{2}}\right)(-\cot t)\right]^\pi_0 \) vanishes when the limits of integration are applied, where \( K \) and \( S \) are constants, \( K \) is small, and \( S \) can take values between 0 and 1.

Answer: 0


Question: Let $X_1, X_2, \ldots, X_{100}$ be independent and identically distributed random variables, each uniformly distributed on the interval $(0,1)$. Determine the constant $k$ such that \( P\left( \prod_{i=1}^{100} X_i \ge k \right) = 0.05 \).

Answer: e^{-83.55}


Question: Suppose $F(x,y)$ is a function such that $F(0,0)=0$, and all partial derivatives of any order with respect to $x$ and $y$ at $(0,0)$ are zero. If $F(x,y)$ behaves like $xy$ for small $x$ and $y$, can we conclude that $|F(x,y)| < Cxy$ for some constant $C$ and for small $x$ and $y$?

Answer: Yes


Question: Find the Maclaurin series for \((\cos x)^6\) up to the \(x^4\) term using the Maclaurin series for \(\cos x\).

Answer: 1 - 3x^2 + 4x^4


Question: Determine for which values of \(\alpha, \beta \geq 0\) the series \(\sum_{n \geq 1} \frac{(\log(n))^\alpha}{n^\beta}\) converges.

Answer: \beta > 1


Question: Let $T$ be a bounded linear operator on a Hilbert space $H$, such that for all $x, y \in H$, the inner product $\langle Tx, y \rangle = 0$. Prove that $T = 0$.

Answer: T = 0


Question: Consider a sequence of independent and identically distributed random variables \(X_1, X_2, \ldots\) with \(\mathbb{E}[X_1^4]\) finite. Let \(\theta = \mathbb{E}[X_1]\) and \(\overline{X}\) be the sample mean. Determine whether \((\overline{X})^2\) is a consistent estimator of \(\theta^2\) as \(n \to \infty\).

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac {8x^2(e^{6x}-1)}{2x-\sin(2x)} \]

Answer: 36


Question: Evaluate the integral: \[ I = \int_0^{2\pi} \frac{e^{|\sin x|} \cos x}{1+e^{\tan x}} \,dx \]

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( 1^{1/\sin^2 x} + 2^{1/\sin^2 x} + \cdots + n^{1/\sin^2 x} \right)^{\sin^2 x} \]

Answer: n


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a convex function. For a given $x \in \mathbb{R}$, does there exist a real number $c$ such that $f(y) \ge f(x) + c(y - x)$ for all $y \in \mathbb{R}$?

Answer: Yes


Question: Is there a subgroup of the free group \(F_2\) that is isomorphic to \(F_4\) and has a finite index that is not a multiple of 3?

Answer: No


Question:
Determine the limit of the sequence defined by the recursive logarithms:
\[ t_1 = \ln(n) \]
\[ t_2 = \ln(n - t_1) = \ln(n - \ln(n)) \]
\[ t_3 = \ln(n - t_2) = \ln(n - \ln(n - \ln(n))) \]
\[ \ldots \]
\[ t_k = \ln(n - t_{k-1}) \]
Find the value to which this sequence converges.

Answer: n - W(e^n)


Question: Evaluate the limit \( \lim_{n\rightarrow\infty}\frac{\sum \limits_{k=1}^n(k\cdot a_k)}{n^2} \) given that \( \lim_{n\rightarrow\infty}a_n = L \). Express your answer in terms of \( L \).

Answer: \dfrac{L}{2}


Question: Given the joint probability density function of continuous variables X and Y, \( f(x,y) = x + y \) for \( 0 < x, y < 1 \), determine the probability \( P(0.5 < X + Y < 1.5) \).

Answer: \dfrac{3}{4}


Question: Evaluate the infinite series: \[ \sum_{k=0}^{\infty} \left[ \tan^{-1} \left( \frac{1}{k^2+k+1} \right) - \tan^{-1} \left( \frac{2}{k^2+2k+1} \right) \right] \]

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


Question: Let $F:[0,\infty) \to [0,\infty)$ be a $C^2$ strictly convex function with $F''$ everywhere positive. Consider sequences $\lambda_n \in [0,1]$, $a_n \le c_0 < b_n \in [0,\infty)$ such that \( \lambda_n a_n +(1-\lambda_n)b_n=c_n>c_0 \) for some constant $c_0$, and $c_n \to r>c_0$. Define $D_n:=F\big(c_n\big)-\lambda_nF(a_n)-(1-\lambda_n)F(b_n)$. If $\lim_{n \to \infty}D_n=0$, must $b_n$ be bounded? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \( \lim_{x\to {0}}\left(\frac{1}{x\arcsin x} - \frac{1}{x^2}\right) \) without using L'Hôpital's rule.

Answer: -\dfrac{1}{6}


Question: Is it true that if an absolutely continuous function defined on a closed interval in \(\mathbb{R}\) has a non-negative derivative almost everywhere, then the function is monotone increasing? Justify your answer.

Answer: Yes


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

Answer: \infty


Question: Determine whether there exists an integer \(a\) such that \(\text{ord}_{17}(a) = 4\), where \(\text{ord}_{17}(a)\) is the smallest positive integer \(k\) such that \(a^k \equiv 1 \pmod{17}\).

Answer: Yes


Question: Determine the radius of convergence of the series \( \sum_{k = 1}^{\infty} \frac{k!}{k^{k}}x^{k} \).

Answer: e


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

Answer: 1


Question: Given \( y = \int_0^x f(t)\sin[k(x-t)]dt \), calculate \( \frac{d^2y}{dx^2} + k^2y \).

Answer: k f(x)


Question: Determine the values of \( \alpha \) for which the series \( \sum_{n=3}^{\infty}\frac{1}{n\ln n(\ln \ln n)^\alpha} \) converges.

Answer: \alpha > 1


Question: Determine whether the matrix equation $XA = XB$ always has a solution for any given $n \times m$ matrices $A$ and $B$ with entries in $\mathbb{R}$. Can a $m \times n$ matrix $X$ always be found such that $XA = XB$?

Answer: Yes


Question: Let \( p \) be an odd prime and \( i \ge 1 \) be an integer. Prove that there exists an integer \( \nu \) such that \( (1+p)^{p^i} = 1 + \nu p^{i+1} \) with \( \gcd(\nu, p) = 1 \).

Answer: \nu


Question: Determine the number of positive integer solution pairs \((x, y)\) for the equation \(y^2 = \frac{(x^5 - 1)}{(x-1)}\), where \(x \neq 1\).

Answer: 1


Question: Evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \) for the vector field \( \mathbf{F}(x,y,z) = x\mathbf{i} + y\mathbf{j} + 5\mathbf{k} \). The surface \( S \) is the boundary of the region enclosed by the cylinder \( x^2 + z^2 = 1 \) and the planes \( y = 0 \) and \( x + y = 2 \). Use the positive (outward) orientation for the closed surface.

Answer: 4\pi


Question: Evaluate the sum: \[ \sum_{k=0}^{n} (-4)^k \binom{n+k}{2k} \]

Answer: (-1)^n (2n + 1)


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left(\frac{a_n}{1} + \frac{a_{n-1}}{2} + \frac{a_{n-2}}{2^2} + \ldots + \frac{a_1}{2^{n-1}}\right) \] given that \( \lim_{n \to \infty} a_n = a \).

Answer: 2a


Question: Find the minimal polynomial of the number $\sqrt{2} e^{2\pi i / 3}$ over $\mathbb{Q}$. Provide your answer as a polynomial in $\mathbb{Q}[x]$.

Answer: x^4 + 2x^2 + 4


Question: Packages arrive at a distribution center according to a Poisson process with rate \( \lambda \) per hour. Given that exactly 2 packages have arrived by 8pm, find the probability that no package has arrived by 5pm. Assume the distribution center opens at 0pm.

Answer: \dfrac{9}{64}


Question: Is there a linear transformation $L: V \rightarrow V$ such that for a set of $n$ linearly independent $d$-dimensional vectors $\{\vec{a}_i\}_{i=1}^{i=n}$ that span the vector space $V$, the transformation maps them to an orthogonal basis $\{\vec{b}_i\}_{i=1}^{i=n}$ where $L(\vec{a}_i) = \vec{b}_i$ and $\vec{b}_i \cdot \vec{b}_j = c_{ij} \delta_{ij}$ for all $i, j$, with $c_{ij} \in \mathbb{C}$ and $\delta_{ij}$ being the Kronecker delta?

Answer: Yes


Question: Find the cardinality of the group \(G = \langle a, b, c \rangle\), where \(a = (1,2)\), \(b = (2,3)\), and \(c = (4,5)\) are elements of the symmetric group \(S_5\).

Answer: 12


Question: Let \(K\) be a number field and \(\mathcal{O}_K\) its ring of integers. Let \(I\) be an ideal of \(\mathcal{O}_K\). Determine whether there exists an element \(\alpha \in I\) such that \(N(\alpha) = cN(I)\), where \(c\) is an integer coprime to \(N(I)\).

Answer: Yes


Question: A pool table has 6 pockets. There are 10 red balls, one brown ball, and one black ball. How many ways are there to distribute the balls into the pockets if each pocket can contain no more than 9 balls and the black ball should not be in the same pocket as the brown ball?

Answer: 89610


Question: Given two smooth functions \( f, g: [a,b] \to \mathbb{R} \) such that \( f(x) \cdot g(x) = 0 \) for every \( x \in [a,b] \), does it follow that \( f \equiv 0 \) or \( g \equiv 0 \) on some subinterval \((m,n) \subseteq [a,b]\)?

Answer: Yes


Question: Calculate the work done by the vector field \( \mathbf{u} = (xyz, xy^2z^3-z, xy^3z^2) \) around the circle defined by \( D = \{y^2+z^2=1, x=1\} \) using Stokes' theorem. Use the normal vector \( \mathbf{N} = (1, 0, 0) \) and verify the work done is \( \pi \).

Answer: \pi


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left( x^2 \ln \left( \cos \frac{4}{x} \right) \right) \]

Answer: -8


Question: Let $W$ be a subspace of the vector space $V$, and $W'$ be a subspace of the vector space $V'$. Consider the tensor product $W \otimes' W'$ as the quotient of the free vector space over $W \times W'$ by the space generated by terms of the form $\lambda (v,w) - (\lambda v,w)$, $(v+v',w)-(v,w)-(v',w)$, $\lambda(v,w)-(v,\lambda w)$, and $(v,w+w')-(v,w)-(v,w')$, where $v,v' \in W$, $w,w' \in W'$, and $\lambda \in \mathbb{K}$. Define a natural linear map $f: W \otimes' W' \to V \otimes V'$ by $f(w \otimes' w') = w \otimes w'$. Is the map $f$ injective?

Answer: Yes


Question: Evaluate the double summation: \[ \sum^{\infty}_{n=1}\sum^n_{k=1} \frac{1}{k^2} \]

Answer: \infty


Question: Does a rotation map \(A: \mathbb{R}^n \to \mathbb{R}^n\) preserve the volume form on \(S^{n-1}\)?

Answer: Yes


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

Answer: f(x) = ax + b


Question: Find the remainder when \(13^{99} - 19^{93}\) is divided by 162.

Answer: 0


Question:
Evaluate the definite integral to find the arc length of the curve given by the equation \(24xy = y^4 + 48\) from \(y = 2\) to \(y = 4\). The derivative with respect to \(y\) is \(\frac{y^2}{6} - \frac{x}{y}\). Use this derivative in the arc length formula to evaluate:
\[ \int_2^4 \sqrt{1+ \left(\frac{y^2}{6} -\frac{x}{y}\right)^2} \, dy \]

Answer: \dfrac{17}{6}


Question: Given that $A$ and $XAX^{-1}$ are matrices in $\Bbb M_3 (\Bbb Z)$ for some matrix $X \in \Bbb M_3 (\Bbb R)$, determine whether $X$ must be in $\Bbb M_3 (\Bbb Z)$. Provide a justification for your answer.

Answer: No


Question: Evaluate the Riemann-Stieltjes integral \( \int_{0}^2 x \, d\alpha \), where \( \alpha(x) = x \) for \( 0 \le x \le 1 \) and \( \alpha(x) = 2 + x \) for \( 1 < x \le 2 \).

Answer: 4


Question: Find the remainder when \( 45^{17^{17}} \) is divided by 204.

Answer: 45


Question: Find the value of the sum \( \sum_{k=0}^{n} k^2 \binom{n}{k} \).

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


Question: Determine the number of ways to select cells from an \( n \times n \) grid such that the number of cells selected from each row and each column is odd.

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


Question: Find the conditional expectation $E[X|X>Y]$ where $X$ and $Y$ are independent random variables uniformly distributed over the interval $[0,1]$. Express your answer as a single value or expression.

Answer: \dfrac{2}{3}


Question: Determine the value of \( m \) for which the area of the triangle formed by the axes and any tangent to the curve \( x^m y = b^m \) is constant.

Answer: 1


Question: Evaluate the limit \( \lim_{n \to \infty} (1 - n^{-a}x)^n \) for \( 0 < a < 1 \) and \( x > 0 \).

Answer: 0


Question: Determine the constant \( k \) such that the function \( f_{XYZ}(x, y, z) = ke^{-(x+y+z)} \) is a probability density function over the region where \( 0 < x < y < z \).

Answer: 6


Question: Determine the total running time, in Big O notation, of counting from 1 to \( n \) in binary, given that the time needed to add 1 to the current number \( i \) is proportional to the number of bits in the binary expansion of \( i \) that must change in going from \( i \) to \( i+1 \).

Answer: O(n)


Question: Suppose $M$ is a square matrix such that $M^2 \neq 0$. Does this imply that $M$ has a non-zero eigenvalue?

Answer: No


Question: Find the degree of the constant map \( f_0 : S^1 \to S^1 \) defined by \( f_0(z) = 1 \).

Answer: 0


Question: Show that the definite integrals \( V = - \int_{\frac{1}{3}}^{\frac{1}{2}} \frac{\ln v}{1 - v^2} \, dv \) and \( X = \int_{\frac{1}{2} \ln 2}^{\frac{1}{2} \ln 3} \ln (\coth x) \, dx \) are equal without evaluating them.

Answer: V = X


Question: Calculate \(28! \pmod{799}\) using Wilson's theorem.

Answer: 204


Question: Compute the residue of the function \( f(z) = \frac{e^{az}}{1 + e^z} \) at the pole \( z = \pi i \).

Answer: -e^{a \pi i}


Question: Find the point on the paraboloid \( z = \frac{x^2}{4} + \frac{y^2}{25} \) that is closest to the point \((3, 0, 0)\) using the method of Lagrange multipliers.

Answer: (2, 0, 1)


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\sin\left(e^{1-\cos^3x}-e^{1-\cos^4x}\right)}{x\arctan x} \]

Answer: -\dfrac{1}{2}


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

Answer: \infty


Question: Determine the radius of convergence of the Taylor series for the function \( \frac{2\tan x}{1+4x^2} \) around \( x=0 \).

Answer: \dfrac{1}{2}


Question: Let $G = \mathbb{Z} \oplus \mathbb{Z}$ and $H = \langle (2,2) \rangle$. Determine the order of the factor group $G/H$. \( \text{Note: } H = \{ (x,y) \mid x,y \in 2\mathbb{Z} \} \).

Answer: 4


Question: Given a matrix with its LU decomposition, if the last corner entry is $A(5,5) = 11$ and the last pivot of $A$ is $U(5,5) = 4$, determine the value of $A(5,5)$ that would make the matrix singular.

Answer: 7


Question: Let $X$ be a compact Hausdorff space with a uniformity $\mathcal{U}$, and let $\mathcal{V}$ be a finite open cover of $X$. Is there an entourage $D$ in the uniformity $\mathcal{U}$ such that if $(x, y) \in D$, then there exists an open set $V \in \mathcal{V}$ with $\{x, y\} \subseteq V$? Provide a justification for your answer.

Answer: Yes


Question: A gas station receives cars at a rate of 20 vehicles per hour and has only one gas pump. If the pump is empty, it receives one client. If a new client arrives and finds the pump occupied, they wait. Each client takes approximately 6 minutes (1/10th of an hour) to fill their vehicle. If two clients are at the pump, one filling and the other waiting, new clients leave the station directly. Given the stationary distribution of the states as \((u_1,u_2,u_3) = (1/7,2/7,4/7)\), find the average number of clients served by the gas station every hour. Express your answer as a fraction.

Answer: \dfrac{60}{7}


Question:
Given that $f(1)=1$ and $f'(1)=2$, find the value of the limit:
$$\lim_{x\to 1} \frac{\sqrt{f(x)} - 1}{\sqrt{x} - 1}$$

Answer: 2


Question: Evaluate the product \((1-\omega)(1-\omega^2)\ldots(1-\omega^{n-1})\), where \(\omega\) is a primitive \(n\)-th root of unity.

Answer: n


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

Answer: 4


Question: For a 1-dimensional Levy process $X_t$ with generating triplets $(\sigma, \gamma, \nu)$, where $\sigma \neq 0$ and $\nu \neq 0$, determine whether $P(\omega: X_{t}(\omega)=x) = 0$ for a fixed point $x$ in the state space and fixed $t \neq 0$. Consider the cases of Brownian motion and pure jump Levy processes as references.

Answer: 0


Question: Given the function \( f(x) = \frac{1}{2} \) for \( -1 \leq x \leq 1 \) and \( 0 \) elsewhere, define \( g(x) = f(x) * f(x) \), where \( * \) denotes the convolution operator. Calculate \( h(x) = g(x) * f(x) \) and evaluate \( h \left( \frac{1}{3} \right) \).

Answer: \dfrac{13}{36}


Question:
Compute the determinant of the matrix \( A \):
\[
A = \begin{pmatrix}
1 + x^2 - y^2 - z^2 & 2(xy + z) & 2(zx-y)  \\
2(xy - z) & 1 + y^2 - z^2 - x^2 & 2(yz + x)  \\
2(zx + y) & 2(yz - x) & 1 + z^2 - x^2 - y^2
\end{pmatrix}
\]
Choose the correct expression for \( \det A \):
(a) \((1 + xy + yz + zx)^3\)
(b) \((xy + yz + zx)^3\)
(c) \((1 + x^2 + y^2 + z^2)^3\)
(d) \((1 + x^3 + y^3 + z^3)^2\)

Answer: C


Question:
Minimize the function \( f(x_1, x_2) = x_1^2 + x_2 + 4 \) subject to the constraints:

1. \( -x_1^2 - (x_2 + 4)^2 + 16 \ge 0 \)
2. \( x_1 - x_2 - 6 \ge 0 \)

Determine the coordinates \((x_1, x_2)\) that minimize the function within the feasible region defined by the constraints.

Answer: (0, -8)


Question: Given a set \( L = (1, 2, \ldots, n) \) where \( n \) is odd, and two functions defined as \( f(L) = (3, 4, \ldots, n, 1, 2) \) and \( g(L) = (3, 4, \ldots, n, 2, 1) \), is it possible to achieve any permutation of \( L \) through combinations of \( f \) and \( g \)?

Answer: Yes


Question:
Calculate the sum \( \sum_{i=1}^{200} f(i) \) where the function \( f(x) \) is defined as follows:
\[
f(x) = \begin{cases}
\left[\frac{1}{\{\sqrt{x}\}}\right] & \text{if } x \neq k^2 \\
0 & \text{if } x = k^2
\end{cases}
\]
Here, \( x, k \in \mathbb{N} \), \([.]\) denotes the greatest integer function, and \(\{.\}\) denotes the fractional part. Determine the value of the sum.

Answer: 629


Question: Prove that \(\sum_{S \subset N \setminus \{i\}} {|S|!(n-|S|-1)!}=n!\), where \(N=\{1, 2, \dots, n\}\) and \(i \in N\).

Answer: n!


Question: Find the sum of the series: $$\sum_{n = -\infty}^{\infty} \cos \left(8n + \frac{2\pi}{3}\right)$$

Answer: 0


Question: Evaluate the limit \( \lim_{x\to0}\frac{e^x-1-x}{x^2} \) using only the rules of algebra of limits.

Answer: \dfrac{1}{2}


Question: Find the residue of the function \( f(z) = (z^2-1)\cos\frac{1}{z+i} \) at the point \( z_0 = -i \).

Answer: i


Question: For which values of \( p \) does the triangle inequality \( ||x + y||_p \leq ||x||_p + ||y||_p \) hold true for the \( \ell^p \) norm in \( \mathbb{R}^n \)? Here, \( ||x||_p = \left( \sum_{k=1}^n |x_k|^p \right)^{1/p} \) for \( p \in (0, \infty) \) and \( ||x||_\infty = \sup |x_k| \) for \( 1 \leq k \leq n \).

Answer: [1, \infty]


Question: Calculate the limit: \[ \lim_{(x,y) \rightarrow (0,0)} \sqrt{x^2+y^2} \log|y| \] without using L'Hopital's rule.

Answer: 0


Question: Given a nilpotent operator $F: V \rightarrow V$ with index $k$, where $k$ is the smallest integer such that $F^k = 0$, prove that the minimal polynomial of $F$ is $M(t) = t^k$.

Answer: t^k


Question:
Evaluate the definite integral \( \int_0^{2\pi} \frac{1}{13-5\sin\theta}\,\mathrm{d}\theta \). Choose the correct answer from the following options:
(a) \(-\frac{\pi}{6}\)
(b) \(-\frac{\pi}{12}\)
(c) \(\frac{\pi}{12}\)
(d) \(\frac{\pi}{6}\)

Answer: D


Question:
Find the upper asymptotic bound for the recursion:
\[ T(k) = 2T(k-1) + \frac{1}{k} \]

Answer: O(2^k)


Question: If the singular values of an \(n \times n\) matrix \(A\) are all 1, is \(A\) necessarily orthogonal? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \sum_{k=0}^{n} \frac{k(n-k)!+(k+1)}{(k+1)!(n-k)!} \]

Answer: 1


Question: Find the sum of the series: \[ \sum_{n=1}^\infty \frac{(-1)^n(4n)}{4n^2-1} \]

Answer: -1


Question: Verify the assertion: If \( \gcd(n,m)=1 \), then \( \gcd(R_n,R_m)=1 \), where \( R_n=\frac{10^n-1}{9} \) is a repunit.

Answer: 1


Question: Compute the relative homology group $H_1([0,1], \{0,1\})$. Use the long exact sequence of the pair $([0,1], \{0,1\})$ to determine the value of $H_1([0,1], \{0,1\})$. Justify your answer.

Answer: \mathbb{Z}


Question: A standard deck of 52 cards is shuffled and dealt to 4 players (N, S, E, and W), each receiving 13 cards. If players N and S together hold exactly 10 cards of a specified suit, what is the probability that the remaining 3 cards of that suit are all in the hand of either player E or player W?

Answer: \dfrac{11}{50}


Question: Does there exist a continuous, nowhere differentiable function defined on a non-compact domain? Specifically, is there a function \( f: A \to \mathbb{R} \) that is nowhere differentiable, where \( A \) is not compact?

Answer: Yes


Question: Determine the value of $x$ at which the Bessel equation $x^2y'' + xy' + (x^2 - n^2)y = 0$ has a regular singularity.

Answer: 0


Question: Find the variance of the random variable \( Z = Y^2 + Y + 2017 \), where \( X \) is uniformly distributed over \([-1, 3]\) and \( Y \) is the distance from \( X \) to the nearest endpoint of \([-1, 3]\).

Answer: \dfrac{139}{45}


Question: Determine whether every field extension of degree 2018 is primitive. That is, for a field extension $L/K$ with $[L:K]=2018$, does there exist an element $\alpha \in L$ such that $L=K(\alpha)$? Answer true or false.

Answer: True


Question: Let $I$ be the ideal of $\mathbb{Z}[X]$ generated by $(X^2+2)(X+1)$ and $5$. Determine the number of elements $f \in \mathbb{Z}[X]/I$ such that $f^{18} = 1$. Express your answer as an integer.

Answer: 12


Question: Let $q$ be a prime power and $n$ be a positive integer. Consider $U$, an $\mathbb{F}_q$-vector subspace of the finite field $\mathbb{F}_{q^n}$ with dimension $n-1$. Let $V$ be the kernel of the trace map $Tr: \mathbb{F}_{q^n} \rightarrow \mathbb{F}_q$, which is also a vector subspace of dimension $n-1$. Determine if $U = aV$ for some non-zero $a \in \mathbb{F}_{q^n}$. Provide a justification for your answer.

Answer: Yes


Question: Find the integer $k$ such that \( \int_{\mathbb{R}} \frac{1 - \cos(2\pi x)}{x^2} \, dx = k\pi^2 \).

Answer: 2


Question: Decrypt the RSA message given that the public exponent is \( e = 3 \) and the message \( m < N^{1/3} \), where \( c \equiv m^3 \pmod{N} \). Find the plaintext message \( m \).

Answer: \sqrt[3]{c}


Question: Determine whether the following statement is true or false: If $Y$ is a random variable and $X$ is a subGaussian random variable, and if $E(\exp(tY)) \leq E(\exp(tX))$ for all $t$, then $Y$ is subGaussian.

Answer: True


Question:
Find the smallest positive integer \( x \) such that \( 32^x \equiv 1 \pmod{2015} \). Given that \( 2015 = 5 \times 13 \times 31 \), use the Chinese Remainder Theorem to solve the system of congruences:
\[
\begin{align*}
32^x &\equiv 1 \pmod{5},\\
32^x &\equiv 1 \pmod{13},\\
32^x &\equiv 1 \pmod{31}.
\end{align*}
\]

Answer: 12


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

Answer: e^2


Question: Find the value of \( \lim _{n \rightarrow \infty} \mathbb{E}\left\{\left|1-X_{n}\right|\right\} \), given that \( X_{1}, X_{2}, \ldots \) are nonnegative random variables such that \( X_{n} \stackrel{\text{a.s.}}{\rightarrow} 0 \) and \( \lim_{n \rightarrow \infty} \mathbb{E}\{X_{n}\} = 2 \).

Answer: 3


Question: Determine the radius of convergence of the series \( \sum\limits_{n=1}^{+\infty} n^n x^{n!} \).

Answer: 1


Question: How many spread-out partitions of 21 are there into 4 parts, where a partition is defined as 'spread-out' if all the parts are distinct and no two parts are consecutive numbers?

Answer: 6


Question: Find the surface area of the part of the plane \(x + 2y + z = 4\) that lies inside the cylinder \(x^2 + y^2 = 4\).

Answer: 4\sqrt{6}\pi


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\tan^{12}x-x^{12}}{x^{14}} \]

Answer: 4


Question: In triangle $\triangle ABC$, point $M$ is on segment $\overline{AC}$ such that $|AM|:|MC|=2:1$, and point $N$ is on segment $\overline{BC}$ such that $|BN|:|NC|=1:2$. Let $P$ be the intersection of segments $\overline{AN}$ and $\overline{BM}$. Determine the ratio in which point $P$ divides segment $\overline{AN}$. \(\text{Express the ratio as } |AP|:|PN|.\)

Answer: 6:1


Question: In a house, there are 40 light bulbs in a room and 40 switches at a switchboard near the entrance, far from the room. Each switch corresponds to one light bulb, but you cannot see the bulbs from the switchboard. You can only check the status (on or off) of each bulb by walking to the room. What is the minimum number of trips you need to make to the room to determine which switch controls each bulb?

Answer: 6


Question: Determine the supremum of \(|f(1)|\) for functions \(f\colon \Bbb C\setminus\{0\}\to \Bbb C\) that are analytic and satisfy \(|f(z)|\le 5|z|^{−\sqrt2}\).

Answer: 0


Question:
Consider a sequence of nonempty subsets of integers, \( \{A_n\}_{n=1}^\infty \). Determine which of the following sets is uncountably infinite:

A) \( A_1 \)

B) \( \bigcap_{n=1}^\infty A_n \)

C) \( \bigcup_{n=1}^\infty A_n \)

D) The set of all functions \( f: \{0,1\} \to A_1 \)

E) The set of all functions \( f: A_1 \to \{0, 1\} \)

Answer: E


Question: Evaluate the sum \( S = \sum_{k=0}^{n} \frac{(-1)^k}{k+1} {n \choose k} \).

Answer: \dfrac{1}{n + 1}


Question: Suppose we have three random variables $x, y_1$, and $y_2$ such that $p(x|y_1) = p(x)$ and $p(x|y_2) = p(x)$. Does this imply that $p(x|y_1, y_2) = p(x)$? Justify your answer.

Answer: No


Question: Calculate the sum of all unique permutations of the digits of the number 1112334.

Answer: 999999900


Question: Determine the points at which the function \( f(z) = \frac{y+ix}{x^2+y^2} \) is differentiable, and find the formula for \( f'(z) \) at those points.

Answer: -\dfrac{i}{z^2}


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

Answer: 9


Question: Let $a_0 = 5$ and $a_n a_{n+1} = a_n^2 + 1$ for all $n \geq 0$. Determine $\left \lfloor{a_{1000}}\right \rfloor$. This problem is from the Singapore Math Olympiad 2017 Open Section.

Answer: 45


Question: Let $\textbf{Z}_n$ be the set $\{0, 1, \ldots, n - 1\}$ with addition mod $n$. Consider subsets $S_n$ of $\textbf{Z}_n$ such that $(S_n + k) \cap S_n$ is nonempty for every $k$ in $\textbf{Z}_n$. Let $f(n)$ denote the minimal number of elements in such a subset. Determine the value of $$\lim_{n \to \infty} {\ln f(n) \over \ln n}.$$

Answer: \dfrac{1}{2}


Question:
Consider the function \( f(x) = x\cos \frac{1}{x} \) for \( x \ge 1 \). Determine which of the following inequalities is true for the domain \( x \ge 1 \):

(A) \( f(x+2) - f(x) < 2 \)

(B) \( f(x+2) - f(x) > 2 \)

Answer: B


Question:
Determine whether the following proof is correct:

Let $0 \leq a \leq b \leq 1$. Then for all natural numbers $m \geq 2$, the inequality $b^{\frac{m}{2}} - a^{\frac{m}{2}} \leq \frac{m}{2}(b-a)$ holds.

Consider the function $f(x) = x^{\frac{m}{2} - 1}$ on the interval $[0,1]$. Since $m \geq 2$, it follows that $\sup_{x \in [0,1]} f(x) = 1$. By the fundamental theorem of calculus, we have:

\[ b^{\frac{m}{2}} - a^{\frac{m}{2}} = \int_{a}^{b} \frac{m}{2} f(x) \, dx = \frac{m}{2} \int_{a}^{b} x^{\frac{m}{2} - 1} \, dx \leq \frac{m}{2} \sup_{x \in [0,1]} f(x) (b-a) = \frac{m}{2} (b-a). \]

Is this proof correct?

Answer: Correct


Question: Determine the order of the pole at $z = 2\pi n$ for the function $f(z) = \frac{1}{(1-\cos(z))^2}$, where $n \in \mathbb{Z}$. Provide your answer as an integer.

Answer: 4


Question: Determine the condition under which equality holds for the inequality \(|x+y| \leq |x| + |y|\).

Answer: xy \geq 0


Question: Determine whether the integral \( \int_{-1}^1 e^{-x^{4}}\cdot(1+\ln(x+\sqrt{x^2+1})+5x^3-4x^4)\,dx \) can be evaluated.

Answer: \dfrac{2}{e}


Question:
Given that $a + b + c = 0$, find $\frac{dy}{dx}$ for the function:
\[ y = \frac{1}{x^b + x^{-c} + 1} + \frac{1}{x^c + x^{-a} + 1} + \frac{1}{x^a + x^{-b} + 1} \]

Answer: 0


Question: Let $K$ be a separable, compact metric space. Is $C(K, \mathbb{R})$ also separable without any additional assumptions on $K$?

Answer: Yes


Question: If $AB = BA$ and $AB$ is a diagonal matrix, are both $A$ and $B$ necessarily diagonal matrices?

Answer: No


Question: Given that \(\frac{(a-b)(b-c)(a-c)}{(a+b)(b+c)(c+a)}=\frac{1}{11}\), find the value of \(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\).

Answer: \dfrac{17}{11}


Question:
Let a sequence be defined by \(a_1 = 0\) and \(a_{n+1} - a_n = 4n + 3\) for all \(n \ge 1\). Find \(a_k\) in terms of \(k\).

Then, evaluate the limit:
\[ \lim_{n \to \infty} \frac{\sqrt{a_{4^0n}} + \sqrt{a_{4^1n}} + \cdots + \sqrt{a_{4^{10}n}}}{\sqrt{a_{2^0n}} + \sqrt{a_{2^1n}} + \cdots + \sqrt{a_{2^{10}n}}} \]

Answer: 683


Question: Let \( B = (B_t)_{t \geq 0} \) be a standard Brownian motion and define \( M_t = 4B_t^2 + e^{4B_t -8t} -4t \). Find \( \mathbb{E}(M_\sigma) \) for \( \sigma = \inf\{t \geq 0 : |B_t| = 1\} \).

Answer: 1


Question: Let $f$ be a complex function defined on an open ball $U$. If $\sup |f(z)|=|f(z_0)|$ and $f'(z_0)$ exists, is it true that $f'(z_0)=0$?

Answer: 0


Question: Find the smallest integer M such that if a number N is divisible by all integers from 1 to M, then N must also be divisible by M + 1, M + 2, M + 3, and M + 4.

Answer: 32


Question:
Consider the function \( g(x) \) defined as:
\[
g(x) =
  \begin{cases}
1,  & \text{if $x$ belongs to $[-1,1]$} \\
0, & \text{otherwise}
\end{cases}
\]

and the function \( f(x) \) defined as:
\[
f(x)=\lim_{h \to 0} \frac{\int_{x-h}^{x+h} g(y) \,dy}{2h}.
\]

Find the value of \( f(1) \).

Answer: \dfrac{1}{2}


Question: For a given \( n \in \mathbb{N} \), determine the values of \( p > 0 \) for which the function \( \frac{1}{|x|^p} \) is integrable over the region \( \mathbb{R}^n \setminus B_r(0) \).

Answer: p > n


Question: Determine whether the following statement is true or false: If $f: \mathbb{R} \to \mathbb{R}$ is $\mathbb{A}$-measurable, then for all $\delta > 0$, there exists $a \in \mathbb{R}$ such that $(f^{-1}(a,a+\delta))^c$ is countable, where $\mathbb{A}$ is the sigma-algebra defined as $\mathbb{A} = \{ A \in P(X) : A \text{ is countable or } A^c \text{ is countable} \}$.

Answer: True


Question: Evaluate the integral \( \int_0^{\infty} r^4 e^{-r} \, dr \).

Answer: 24


Question:
Calculate the y-coordinate of the centroid of the upper half of a unit semicircle using the formula:
\[ \frac{1}{l(s)}\int y \, ds \]
where \( l(s) = \frac{\pi}{2} \) is the arc length of the semicircle. Verify that the y-coordinate of the centroid is \( \frac{4}{3\pi} \).

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


Question: Given a random variable $X$ with probability density function $f(X)$, let $Y$ be the sum of $N$ independent draws of $X$. Determine the probability density function of $Y$.

Answer: f^{*N}(y)


Question: Consider an infinite partition \(P\) of the interval \([0,1]\)\, where each element \(I_i\) of the partition has a Lebesgue measure \(m(I_i)\). The entropy of the partition \(P\) is given by \(\sum_i -m(I_i)\log m(I_i)\). Determine whether this entropy can be infinite.

Answer: Yes


Question: Determine the values of $a$ for which the integral \( \int_0^1 \frac{x^a}{x - \sin x} \, dx \) converges.

Answer: (2, \infty)


Question: Determine whether the series \( \sum_{k=0}^{\infty} \frac{z^k}{(k!)^2} \) has a closed-form expression.

Answer: I_0(2\sqrt{z})


Question: Compute the limit \( \lim_{x\to 0} \frac{e^{-1/x^2}}{x} \) without using L'Hopital's rule.

Answer: 0


Question: Determine the total number of real solutions to the equation \(f(x) = 9^x - 5^x - 4^x - 2\sqrt{20^x} = 0\).

Answer: 1


Question: How many bridge hands consist of exactly two 5-card suits, a void, and a remaining suit with a run of 3 cards?

Answer: 218640708


Question: Find all eigenvalues and eigenvectors of the linear transformation \( T: \mathcal{P}^3(\mathbb{R}) \to \mathcal{P}^3(\mathbb{R}) \) defined by \( T(p) = p(t+1) \), where \( p \) is a polynomial of degree at most 3.

Answer: 1


Question: Consider a large square from which four smaller squares have been cut out and discarded. The edges of the remaining shape are identified according to a given labeling scheme. Determine the fundamental group of the resulting topological space $X$. You may introduce additional edges to aid in your calculations, but ensure that no unauthorized identifications are made.

Answer: F_5


Question: Evaluate the integral \( \int_{C}\frac{e^{iz}}{z(z-\pi)}dz \) where \( C \) is any circle centered at the origin with radius greater than \( \pi \), using Cauchy's Theorem or Cauchy's Integral Theorem, without using the residue theorem.

Answer: -4i


Question:
Let $X$ be the random variable representing the time allocated to Question 1 in a 1-hour quiz, with the probability density function given by:
\[ f_X(x) = 12x(1-x)^2 \quad \text{for} \quad 0 \leq x \leq 1 \]
The remaining time is allocated to Question 2. Define $Z$ as the ratio of the time given to the longer question to the time given to the shorter question, such that $Z \geq 1$. Calculate the probability $P(Z \leq 2)$. Provide your answer as a single probability value.

Answer: \dfrac{13}{27}


Question: Find the maximum value of the expression \( S = 2(x+y+z) - 3(xy+yz+zx) + 4xyz \) given that \( x, y, z \geq 0 \) and \( x+y+z \leq \frac{1}{2} \).

Answer: 1


Question: What is the cardinality of the set of all relations from \( \mathbb{N} \) to \( \mathbb{N} \times \mathbb{N} \)?

Answer: 2^{\aleph_0}


Question: A small boat moving at a speed of $V$ km/h uses fuel at a rate given by the function \( q = 8 + \frac{V^2}{50} \), where \( q \) is measured in litres per hour. Determine the speed \( V \) at which the total fuel consumption for a given journey is minimized.

Answer: 20


Question: Determine the number of natural numbers \( n \) such that \( 5^n \equiv 1 \pmod{p} \) does not hold for any prime \( p > 3 \).

Answer: 2


Question: Find a polynomial in \(\Bbb{Q}[x]\) that is irreducible over \(\Bbb{Q}\), has at least one linear factor over \(\Bbb{R}\), and at least one irreducible quadratic factor over \(\Bbb{R}\).

Answer: x^3 - 2


Question: Let $F$ be a nonarchimedean local field, and let $k$ be a global field with a place $v$ such that $F = k_{v}$. Let $E$ be a finite extension of the completion $F = k_{v}$. Determine whether there exists a global field $K$ containing $k$, and a place $w$ of $K$ lying over $v$, such that $K_w = E$ with $[K : k] = [E : F]$. Use Krasner's lemma to justify your answer.

Answer: Yes


Question: B and P each deposit $100 in separate bank accounts. B's account has a nominal interest rate convertible semiannually, while P's account has a force of interest \( \delta \). After 7.25 years, each account has grown to $200. Calculate the difference \( i - \delta \), where \( i \) is the nominal interest rate for B's account.

Answer: 0.0023


Question: Let $S_a \subset \mathbb{R}$ be a finite set for each real number $a > 0$, and suppose that if $b > a$, then $S_b \subseteq S_a$. Determine whether the set $S = \bigcup_{a > 0} S_a$ has measure zero.

Answer: Yes


Question: Determine the degree of the field extension $[F:\mathbb{Q}]$, where $F$ is the minimal subfield of $\mathbb{C}$ containing all the roots of the polynomial $x^4 - x^2 + 1$. Provide your answer as an integer.

Answer: 4


Question: Determine whether the infinite product \( \Pi_{i=0}^{\infty}\mathbb{R} \) has the same cardinality as \( \mathbb{R} \).

Answer: Yes


Question: Given matrices $N_{1} \in \mathbb{R}^{m \times (n-m)}$ and $N_{2} \in \mathbb{R}^{m \times m}$ such that $\text{rank}(N_{1}, N_{2}) = m$, determine if there exist matrices $M_{1} \in \mathbb{R}^{(n-m) \times (n-m)}$ and $M_{2} \in \mathbb{R}^{(n-m) \times m}$ such that $\text{rank}(M_{1}) = n-m$ and $\text{rank}\left(\begin{array}{cc} M_{1} & M_{2} \\ N_{1} & N_{2} \end{array}\right) = n$. Provide a justification for your answer.

Answer: Yes


Question:
Given the function \( y = \sin^3\left(\frac{\pi}{3}\left(\cos\left(\frac{\pi}{3\sqrt{2}}\left(-4x^3 + 5x^2 + 1\right)^{3/2}\right)\right)\right) \), determine which of the following equations is satisfied at \( x = 1 \):

1. \( 2y' + \sqrt{3} \pi^2 y = 0 \)
2. \( 2y' + 3 \pi^2 y = 0 \)
3. \( \sqrt{2} y' - 3\pi^2 y = 0 \)
4. \( y' + 3\pi^2 y = 0 \)

Answer: 2


Question:
Consider a graph $G$ on $n$ vertices and $2n-2$ edges, where the edges of $G$ can be partitioned into two edge-disjoint spanning trees. Determine which of the following statements is NOT true for $G$:

1. For every subset of $k$ vertices, the induced subgraph has at most $2k-2$ edges.
2. The minimum cut in $G$ has at least 2 edges.
3. There are at least 2 edge-disjoint paths between every pair of vertices.
4. There are at least 2 vertex-disjoint paths between every pair of vertices.

Identify the statement that is NOT true.

Answer: 4


Question: Consider a square $S_n = [0,n] \times [0,n]$ in the plane. A partition $\mathcal{Q} = R_1 \cup \cdots \cup R_t$ of $S_n$ is called simple if each set $R_1, \ldots, R_t$ is connected, has positive area, and has a diameter at most 1. Determine whether there exists a constant $k$ (independent of $n$) such that for any natural number $n$, there are at most $k$ simple partitions $\mathcal{Q}_1, \ldots, \mathcal{Q}_k$ of $S_n$ where any two points $x, y \in S_n$ with distance at most 1 are contained in at least one set of the partitions, meaning there exists an $i \in \{1, \ldots, k\}$ and a set $R \in \mathcal{Q}_i$ such that $x, y \in R$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the contour integral \( \int_C \frac{2z-1}{z^4-2z^2+1} \, dz \) where \( C \) is the circle \( |z|=10 \) oriented clockwise.

Answer: 0


Question:
Consider the following commutative diagram of fiber bundles:
\[
\begin{CD}
F @>{}>> E @>{}>> B \\
@VVV @VVV @VVV\\
F' @>{}>> E' @>{}>> B'
\end{CD}
\]
where $B'$ is a simply connected space, and all spaces are path-connected. Suppose the maps
\[
H^{\ast }(B'; \mathbb{Q}) \rightarrow H^{\ast }(B; \mathbb{Q})
\]
and
\[
H^{\ast }(E'; \mathbb{Q}) \rightarrow H^{\ast }(E; \mathbb{Q})
\]
are isomorphisms. If
\[
H^{i }(F; \mathbb{Q}) = 0
\]
for all $i \geq n$ (where $n$ is fixed), does it follow that
\[
H^{i }(F'; \mathbb{Q}) = 0
\]
for all $i \geq n$?

Answer: Yes


Question: When the polynomial \( f(x) \) is divided by \( x^3 + x + 1, \) the remainder is \( 3. \) When \( f(x) \) is divided by \( x^3 - x + 1, \) the remainder is \( x + 1. \) Let \( r(x) \) be the remainder when \( f(x) \) is divided by \( x^6 + 2x^3 - x^2 + 1. \) Find \( r(2). \)

Answer: \dfrac{83}{2}


Question: Let $U$ and $V$ be open subsets of $\mathbb{R}^{d}$ such that $U \subseteq V$. Does there exist a closed set $E$ such that $U \subset E \subset V$?

Answer: No


Question: Evaluate the integral: $$\int_{ -\infty}^{\infty} \frac{e^{- \frac{1}{2}x^2}}{\sqrt{2 \pi}}\cos (ax) \space dx$$ where $a \in \mathbb{R}$. Provide the result in terms of $a$.

Answer: e^{-\frac{a^2}{2}}


Question: Let $S$ be a group and $G$ be a finite group that is $S$-free, meaning no quotient group of any subgroup of $G$ is isomorphic to $S$. If $N \lhd S$, is $G$ an $N$-free group? Answer yes or no and justify your answer.

Answer: No


Question: Evaluate the integral \( \int_{|z|=3} \frac{dz}{z^{2017} - 1} \).

Answer: 0


Question: Given a random variable X with probability mass function \( P(X = n) = pq^{n-1} \) where \( p, q > 0 \) and \( p + q = 1 \), find the variance \( \text{Var}(X) \) using the generating function method.

Answer: \dfrac{q}{p^2}


Question: Let X and Y be independent random variables, each uniformly distributed on the interval [0, 1]. Calculate the probability that \( \max(2X, Y) - \min(2X, Y) < \frac{1}{3} \).

Answer: \dfrac{11}{36}


Question: Evaluate the limit: \[ \lim_{x \rightarrow \pi}\left \lfloor \frac{3\sin(x^2)-1}{2\cos(x)+1} \right \rfloor \] where \( \left \lfloor x \right \rfloor \) denotes the floor function.

Answer: 2


Question: Find the coefficient of \(x^{17}\) in the expansion of \((x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + \, ...)^3\).

Answer: 78


Question: Solve the equation $x^{x^5} = 5$ for $x$.

Answer: \sqrt[5]{5}


Question: Determine whether the following statement is true or false: \( \lim\limits_{n\to \infty} \frac{P(n)}{n} = 0 \), where \( P(n) \) is the maximal size of a sparse subset of \( \{1,2,\dotsc,n\} \). A set \( S \) of integers is defined as sparse if for any four integers \( a,b,c,d \in S \) such that \( a<b<c<d \), it holds that \( a+d \ne b+c \).

Answer: True


Question: Assume matrix $X \in \mathbb{R}^{m \times n}$ is row orthogonal with rank $m$. Construct a new matrix $Y \in \mathbb{R}^{m \times t}$ by sampling columns from $X$ with or without replacement according to some probability distribution. What is the smallest value of $t$ such that $Y$ has rank $m$?

Answer: m


Question: Define $f(0)$ such that the function $f(x) = (1+x^2)^{1/\tan(x)}$ is continuous on the interval $(-\pi/2, \pi/2)$. Determine the value of $f(0)$. \( \boxed{} \)

Answer: 1


Question: Compute \( \lim_{n\to \infty} P\left(\left|\sum_{k=1}^n \frac{1}{X_k}\right|>\frac{\pi n}{2}\right) \), where \( \{X_k\}_{k\geq 1} \) is a sequence of independent random variables uniformly distributed on \([-1,1]\).

Answer: \dfrac{1}{2}


Question: Verify that the Grothendieck group $K_0(k[x])$ is isomorphic to $\mathbb{Z}$, where $k$ is a field. Use the Quillen-Suslin theorem to support your argument.

Answer: \mathbb{Z}


Question:
Evaluate the integral \( \int_{\pi/4}^{\pi/2}{(2\csc(x))^{17}dx} \) and determine which of the following expressions it is equal to:

(a) \( \int_0^{\ln(1+\sqrt{2})}{2(e^u+e^{-u})^{16}du} \)

(b) \( \int_0^{\ln(1+\sqrt{2})}{2(e^u+e^{-u})^{17}du} \)

(c) \( \int_0^{\ln(1+\sqrt{2})}{2(e^u-e^{-u})^{16}du} \)

(d) \( \int_0^{\ln(1+\sqrt{2})}{2(e^u-e^{-u})^{17}du} \)

Answer: a


Question: Find the degree of the algebraic extension \(E = \mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})\) over \(\mathbb{Q}\).

Answer: 8


Question: Find the residue of the function \( f(z) = \frac{z^3 + 2z + 1}{(z-1)(z+3)} \) at the simple pole \( z = 1 \).

Answer: 1


Question: Calculate the volume of the solid formed by rotating the region bounded by the curve \( y = e^{-x} \sin x \) and the line \( x \ge 0 \) about the x-axis.

Answer: \dfrac{\pi}{8}


Question: Find the volume of the solid bounded by the cylinder \(4x^2 + y^2 = 4\), the plane \(z = x + 5\), and the plane \(z = 0\).

Answer: 10\pi


Question: Find the minimum integer value of \( k \) such that the equation \( e^x = kx^2 \) has the maximum number of solutions.

Answer: 2


Question:
Given the function \( f(x,y) = Ke^{ -( ax^2+2bxy+cy^2)  } \), where \( a>0, c>0, b^2-ac>0 \), determine the value of \( K \) such that \( f(x,y) \) is a probability density function. Use the property that \( \int _{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f(x,y) \, dx \, dy = 1 \). Choose the correct value of \( K \) from the options below:

A. \( \frac{1}{\sqrt{\pi}} (ac-b^2) \)

B. \( \frac{1}{\pi} \sqrt{ac-b^2} \)

C. \( \frac{1}{\sqrt{\pi}} \sqrt{ac-b^2} \)

D. \( \frac{1}{\pi} (ac-b^2) \)

Answer: B


Question: Consider a square cake with $n$ toppings, where each topping is a pairwise-interior-disjoint axis-parallel rectangle. You want to cut a small piece out of each topping such that the remaining cake is connected. The pieces should be pairwise-interior-disjoint axis-parallel rectangles, with each piece having a size more than 0 and less than half of the size of the corresponding topping. Is it always possible to make these cuts while keeping the remainder of the cake connected?

Answer: Yes


Question: Evaluate the improper integral: \[ \int_{0}^\infty \frac{\sin(x)}{x} \, dx. \]

Answer: \dfrac{\pi}{2}


Question: Determine whether there exists a nontrivial positive integer solution with $x \ne y$ for the equation $x^4 + y^4 = 2z^2$. Provide a justification for your answer.

Answer: No


Question: Evaluate \( \frac{x}{yz} + \frac{y}{xz} + \frac{z}{xy} \) given that \( x + y + z = 4 \), \( xyz = -60 \), and \( xy + xz + yz = -17 \).

Answer: -\dfrac{5}{6}


Question: Compute the Legendre symbol \(\left(\frac{307}{379}\right)\).

Answer: 1


Question:
Determine whether the following limit exists:

Let $B(0,R)$ be the open ball of radius $R>0$ in $\mathbb{R}^{n}$ ($n>1$) centered at the origin, and let $0<r<R$. By the smooth version of Urysohn's lemma, there exists a test function $\phi$ with compact support in $B(0,R)$ such that $0\leq\phi\leq1$ and $\phi=1$ on $B(0,r)$. Evaluate whether
$$\lim_{r\rightarrow R}\int_{B(0,R)\setminus B(0,r)}\Delta\phi(x)dx=0$$
exists, where $\Delta$ is the Laplacian.

Answer: 0


Question: Find \( f'(0) \) for the function \( f(x) = \frac{g(x)}{x^2} \) when \( x \neq 0 \) and \( f(x) = 0 \) for \( x = 0 \), given that \( g(0) = g'(0) = g''(0) = 0 \) and \( g'''(0) = 14 \).

Answer: \dfrac{7}{3}


Question: Determine the power of the polynomial $x^2 + xy + y^2$ by which the polynomial $(x+y)^7 - x^7 - y^7$ is divisible.

Answer: 2


Question: Find the primary decomposition of the ideal \((x, y^2)\) in the polynomial ring \(K[x, y]\), where \(K\) is a field.

Answer: (x, y^2)


Question: Simplify the sum: \( \sum_{i = t}^{k} (i - 1)! (k - i)! \binom{k - t}{i - t} \) and verify that it equals \( \frac{k!}{t} \).

Answer: \dfrac{k!}{t}


Question: Calculate the volume of the solid bounded by the surfaces \( z = 3x^2 + 3y^2 \) and \( x^2 + y^2 + z = 4 \). Use polar coordinates to evaluate the integral.

Answer: 2\pi


Question: Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be a differentiable function such that \( f(1, 1) = 1 \), \( f_x(1, 1) = 2 \), and \( f_y(1, 1) = 5 \). Define \( g(x) = f(x, f(x, x)) \) for all \( x \in \mathbb{R} \). Determine \( g'(1) \).

Answer: 37


Question: Determine whether there exists a Schwartz function $g \in \mathcal{S}(\mathbb{R}^n)$ such that for a given continuous function $f: \mathbb{R}^n \to \mathbb{R}$ with $f \not\equiv 0$, the integral \( \int_{\mathbb{R}^n} g^2 f \, dx \neq 0 \).

Answer: Yes


Question: Let $V$ be a finite-dimensional vector space of dimension $n$ equipped with a symmetrical bilinear form. If for every subspace $W \neq \{0\}$ of $V$, $\dim W^{\perp} > \dim V - \dim W$, then is the bilinear form $\phi$ equal to zero? Determine whether this statement is true or false.

Answer: True


Question: Let \( F \) be a field and \( x, y, z \) be indeterminates. Consider the field extension \( L = F(xz, yz, xy) \subseteq K = F(x, y, z) \). Show that the degree of the extension \([K:L]\) is finite and calculate its value.

Answer: 2


Question: Suppose we have an i.i.d. sequence \((y_i, x_i, z_i)_{i=1}^n\), where \(x_i\) is a vector with \(K\) entries, and \(y_i\) and \(z_i\) are scalars. Assume \(z_i\) is independent of \(y_i\) and \(x_i\), and that \(E(y_i|x_i) = x_i'\beta\) for any \(i\). Also, \(E(z_i) = 0\) and \(E(z_i^2) = 2\). Define \(w_i = z_i x_i\) and \(q_i = z_i y_i\). Let \(X = (x_1, ..., x_n)'\), \(Y = (y_1, ..., y_n)'\), \(W = (w_1, ..., w_n)'\), and \(Q = (q_1, ..., q_n)'\). What is the probability limit of \(\gamma = (W'W)^{-1}W'Q\)?

Answer: \beta


Question:
Given the matrix \( A = \begin{bmatrix} 1 & 1 \\ -1 & -1 \end{bmatrix} \), find which of the following values cannot be the determinant of a 2x2 integer matrix \( X \) that satisfies the equation \( AX - XA = A \):

- 10
- 6
- 2
- 0

Answer: 10


Question: Consider the second order gradient-like system with linear damping given by the equation \( \ddot{x}+\dot{x}+\nabla f(x)=0 \), where \( x(0)=x_0 \) and \( \dot{x}(0)=0 \). Suppose \( f\in C^2(\mathbb{R}^n) \) and \( \inf_{x\in\mathbb{R}^n}f(x)>-\infty \). The solution \( x:[0,\infty)\rightarrow \mathbb{R}^n \) is known to be bounded. If the parameter in front of the gradient term is changed from 1 to a positive constant \( g \), i.e., the equation becomes \( \ddot{x}+\dot{x}+g\nabla f(x)=0 \), is the solution to this new equation still bounded?

Answer: Yes


Question: Suppose $f(x) = \int g(u)e^{2 \pi iux}du$ for almost every $x \in \mathbb{R}$. Can we conclude that $g$ is the Fourier transform of $f$? Justify your answer.

Answer: Yes


Question: Evaluate the limit \( \lim_{h\to 0}\frac{1}{h}\int_{-\infty}^\infty g\left(\frac{x}{h}\right)f(x)\,\mathrm{d}x \), where \( g:\mathbb{R}\rightarrow \mathbb{R} \) is a continuous function such that \( g(x)\geq0 \) for all \( x\in \mathbb{R} \), \( g(x)=0 \) if \( |x|\geq 1 \), and \( \int_{-\infty}^\infty g(t) \,\mathrm{d}t=1 \). The function \( f:\mathbb{R}\rightarrow \mathbb{R} \) is continuous. Determine the value of the limit.

Answer: f(0)


Question: Consider a local artinian ring $(A, \mathfrak{m})$ and a finite subgroup $G$ of $1+\mathfrak{m}$. Is it true that $G = 1 + I$ for some ideal $I \subseteq A$? Justify your answer.

Answer: No


Question:
Farmer Felix has harvested 1 ton of rice and wants to sell it all at the market. Before selling, the rice must pass through inspections. The inspector demands a portion of the rice as payment, with the following conditions:

- The first inspection costs all the rice.
- If Felix gives the inspector \( x \) tons of rice, the next inspection costs \( x \) tons less.

For example, if Felix gives the inspector portions of 0.2, 0.3, and 0.5 tons in that order, the total amount of rice reaching the market is:
\[ 0.2 \times 0 + 0.3 \times 0.2 + 0.5 \times (0.2 + 0.3) = 0.31 \]

What is the maximum amount of rice Felix can get to the market, assuming the inspector allows infinite inspections?

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{x \to 0} \left(\frac{e^2}{(1+ 4x )^{\frac{1}{2x}}}\right)^{\frac{1}{3x}} \]

Answer: e^{\frac{4}{3}}


Question:
Consider the following 50-term sums:
\[ S = \frac{1}{1\cdot2} + \frac{1}{3\cdot4} + \frac{1}{5\cdot6} + \ldots + \frac{1}{99\cdot100} \]
and
\[ T = \frac{1}{51\cdot100} + \frac{1}{52\cdot99} + \frac{1}{53\cdot98} + \ldots + \frac{1}{100\cdot51} \]
Express \( \frac{S}{T} \) as an irreducible fraction.

Answer: \dfrac{151}{2}


Question: Given the probability distribution \( P(X=k, Y=j) = C \frac{ \binom{k}{j}}{2^{j+k}} \) for \( k \in \mathbb{N}_{0}, 0\leq j \leq k \), calculate the constant \( C \) using generating functions.

Answer: \dfrac{1}{4}


Question:
During 200 test demands, zero failures were discovered. A business software system is required to achieve at most 1 failure in 1000 demands, equivalent to a probability on demand: \( pfd = 10^{-3} \). The business manager believes there is a 70% chance the system meets this requirement and a 30% chance it is an order of magnitude worse (\( pfd = 10^{-2} \)).

Given the prior probabilities:
\[ P(pfd = 10^{-3}) = 0.7 \]
\[ P(pfd = 10^{-2}) = 0.3 \]

The likelihood of observing \( f \) failures in \( d \) tests is defined by the binomial distribution:
\[ P(f|d,pfd) = \frac{d!}{f!(d-f)!}(pfd)^{f}(1-pfd)^{d-f} \]

Using Bayes' theorem, the posterior distribution for \( pfd \) is:
\[ P(pfd|d,f) = \frac{P(f|pfd,d)P(pfd)}{P(f)} \]

Calculate the posterior probability that the software meets the business requirement after observing zero failures in 200 test demands.

Answer: 0.934


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

Answer: \sqrt{2}


Question: Evaluate the limit: \[ \lim_{h \to 0, h \ne 0} \sqrt[h]{\frac{3^h+2^h}{2}}. \]

Answer: \sqrt{6}


Question: Find a holomorphic function $f(z)$ such that the real part of $f(z)$ is $u(x, y) = 2x^3 - 6xy^2$. Assume $f(0) = i$. Determine the function $f(z)$.

Answer: 2z^3 + i


Question: Determine the number of elements of order 5 in the symmetric group $S_{12}$. Provide your answer as a single integer.

Answer: 4809024


Question: Does there exist a finitely presented group that does not contain a nonabelian free group and is not residually amenable?

Answer: Yes


Question:
If the roots of the equation \(x^4 + ax^3 + bx^2 + cx + d = 0\) are in geometric progression, which of the following conditions is true?

(a) \(b^2 = ac\)
(b) \(a^2 = b\)
(c) \(c^2 = a^2d\)

Use Vieta's relations to determine the correct condition.

Answer: C


Question: Evaluate the limit: \[ \lim_{x\to 0^+} x^2 \left(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+\dots+\left[\frac{k}{x}\right]\right) \] where \( k \in \mathbb{N} \) and \([x]\) denotes the integer part of \( x \).

Answer: 0


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

Answer: \dfrac{1}{3}


Question: In how many ways can three teachers (X, Y, Z) and six students be seated in a row such that no two teachers are adjacent?

Answer: 151200


Question: Determine whether the inequality \( \log_{\sqrt{a}}(a+1) + \log_{a+1}\sqrt{a} \ge \sqrt{6} \) holds for all \( a > 1 \).

Answer: Yes


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

Answer: e^4


Question: Evaluate the complex contour integral \( J = \oint_{C}\frac{1}{\cos z+1}dz \) where \( C: |z|=5 \). Use the residue theorem to find the value of the integral.

Answer: 0


Question: Solve the equation for \( x \): \[ 3 \cdot x^{\log_5 2} + 2^{\log_5 x} = 64 \]

Answer: 625


Question: Let $B$ be a finite abelian group of exponent dividing $m$. For a given $b \in B$, can you find a homomorphism $h: B \to \mathbb{Z}/m\mathbb{Z}$ such that $h(b) = c$, where $c \in \mathbb{Z}/m\mathbb{Z}$ and $\operatorname{ord}(b) = \operatorname{ord}(c)$?

Answer: Yes


Question: Find the maximum value of \(|f(z)|\) for \(f(z) = \frac{z^2}{z + 2}\) as \(z\) varies over the unit disc.

Answer: 1


Question: Determine the $x^8$ term in the expansion of \((x+\frac{1}{\sqrt[3]{x}})^5(2x - \frac{1}{x^2})^7\).

Answer: 1280


Question:
Given the functional equations:
\[ F(F(n)) = n \]
\[ F(F(n + 2) + 2) = n \]
and the initial condition \( F(0) = 1 \), determine the value of \( F(129) \).

Answer: -128


Question: Find the non-zero constant \( c \) such that the integral \( \int_{-1}^\infty \frac{e^{x/c}}{\sqrt{|x|}(x+2)}dx \) is convergent.

Answer: -2


Question: Compute the value of the infinite series \(\sum_{n=1}^\infty{\frac{n}{(2n+1)!}}\).

Answer: \dfrac{1}{2e}


Question: Find the number of 9-digit numbers that can be formed using each of the digits from 1 to 9 exactly once, such that no two consecutive digits appear in consecutive positions.

Answer: 47622


Question: Compute the conditional expectation \( E(e^{-\Lambda}|X=1) \) given that \( X \) has a probability mass function \( P_{\lambda}(X=x)=\frac{\lambda^{x}e^{-\lambda}}{x!} \) for \( x=0,1,2,\ldots \) and \( \Lambda \) is a random variable with probability density function \( f(\lambda)=e^{-\lambda} \) for \( \lambda > 0 \).

Answer: \dfrac{4}{9}


Question: Given a simple, undirected graph $G=(V,E)$, determine if there exists a vertex-transitive graph $G_v$ such that the chromatic number of $G$ is equal to the chromatic number of $G_v$, and $G$ is isomorphic to an induced subgraph of $G_v$. Provide a justification for your answer.

Answer: Yes


Question: Determine the positive value of \(c\) for which the equation \(\log(x) = cx^4\) has exactly one real root.

Answer: \dfrac{1}{4e}