Question: Determine the number of functions \( f: \mathbb{R} \setminus \{0, 1\} \to \mathbb{R} \) that satisfy the functional equation: \[ f\left(\frac{x-1}{x}\right) + f\left(\frac{1}{1-x}\right) = 2 - 2x. \]

Answer: 1


Question: Given that $xyz = 32$ and $x, y, z > 0$, find the minimum value of the function $f(x, y, z) = x^2 + 4xy + 4y^2 + 2z^2$.

Answer: 96


Question:
Determine which of the following expressions is equal to the sum \( \sum_{r=0}^n(n-r)^2r^2(n-2r) \):

1. \( \sum_{k=0}^{n-1}e^{-\frac{2k\pi i}{n}} \)
2. \( \sum_{k=0}^{n}e^{\frac{2k\pi i}{n}} \)

Answer: 1


Question: Let \( \mathbb{Z}^n = \{(a_1,...,a_n) \mid a_i \in \mathbb{Z} \text{ for every } i=1,..., n\} \) be the set of lattice points. If \( L \) is an algebraic set in \( \mathbb{C}^n \) such that \( \mathbb{Z}^n \subseteq L \), prove that \( L = \mathbb{C}^n \).

Answer: \mathbb{C}^n


Question: Let $R$ be a $K$-algebra and $ar{\delta}: R[x] \to N^{-1}R[x]$ be an extension of a $K$-derivation $\delta: R \to R$ such that $\bar{\delta}(x) = \alpha$ for some $\alpha \in N^{-1}R[x]$. Show that there is precisely one derivation $d: N^{-1}R[x] \to N^{-1}R[x]$ extending $\bar{\delta}$ such that $d(\frac{f_1}{f_2}) = (f_2\bar{\delta}(f_1) - f_1\bar{\delta}(f_2))(f_2^2)^{-1}$ for $f_1, f_2 \in R[x]$ and $f_2 \neq 0$. Prove that $d$ is uniquely determined by this condition.

Answer: d\left(\frac{f_1}{f_2}\right) = \frac{f_2 \bar{\delta}(f_1) - f_1 \bar{\delta}(f_2)}{f_2^2}


Question: Minimize the expression \( \sqrt{a^{2} + 4} + \sqrt{(3-a)^{2} + (b-2)^{2}} + \sqrt{25 + (6-b)^{2}} \) where \( a \) and \( b \) are positive real numbers.

Answer: 10


Question:
Let \(f(x)\) be a continuous and twice differentiable function such that \(f(x), f'(x), f''(x)\) are all positive for all \(x \in [0,7]\). If \(f^{-1}(x)\) exists, determine the nature of the expression \(f^{-1}(5) + 4f^{-1}(6) - 5f^{-1}(\frac{29}{5})\). Choose the correct option:

A) The expression is always positive.

B) The expression is always negative.

C) The expression is zero for at least one \(x \in (0,7)\).

D) The expression is zero for all \(x \in (0,7)\).

Answer: B


Question: Let \( f \) be a holomorphic and bounded function in the unit disk \( \{z\in \mathbb{C} : |z|\leq 1 \} \). If \( f \) converges uniformly to zero in the sector \( \frac{\pi}{4}<\arg z<\frac{\pi}{3} \) as \(|z|\) tends to 1, prove that \( f \) is identically zero in the closed unit disk.

Answer: 0


Question: Let $G$ be a group and $H$ a subgroup. Assume that $G$ acts on $X$ and that $H$ acts transitively on $X$. Determine whether the statement $G=G_\omega H$ for all $\omega\in X$, where $G_\omega$ denotes the stabilizer of $\omega$ in $G$, is true or false.

Answer: True


Question: Determine whether a number composed of one hundred 0's, one hundred 1's, and one hundred 2's can be a perfect square.

Answer: No


Question: For a given positive integer \( k \geq 2 \), determine whether there exist infinitely many primes \( p_1, p_2, \ldots \) such that the sum of any \( k \) of them is composite, particularly when \( k \) is odd.

Answer: Yes


Question: Calculate the average distance between two randomly chosen numbers from the continuous interval [0, 10].

Answer: \dfrac{10}{3}


Question: A pack of 20 cards is shuffled using the "interlacing" method, where the original order 1, 2, 3, ..., 20 becomes 1, 11, 2, 12, ..., 10, 20 after one shuffle. Determine how many times this shuffle must be repeated for the cards to return to their original order.

Answer: 18


Question: Determine the number of $S^1$-bundles over $\mathbb{RP}^2$. Is it true that there are only two such bundles, one trivial and one non-trivial?

Answer: 2


Question: Determine whether there exists a finite simple group $G$ such that there are distinct orders $d_1, d_2 \in \tau_G$ with $d_1 < d_2$, where $d_1$ does not divide $d_2$, and for elements $x, y \in G$ with $o(x) = d_1$ and $o(y) = d_2$, the elements $x$ and $y$ commute, i.e., $xy = yx$. Provide a justification for your answer.

Answer: Yes


Question: Consider a differentiable function \( f: \mathbb{R}_+ \rightarrow \mathbb{R}_+ \) such that \( f'(x) > 0 \), \( f''(x) < 0 \), and \( f(0) = 0 \). Does this imply that \( f'(x) \) converges to 0 as \( x \) tends to infinity?

Answer: No


Question: Find the value of $f(1)$ given that $f(x) = 17 + \int_0^x \frac{f(t)}{(t+2)(t+3)} \, dt$ for $x > 0$. Differentiate the equation to solve for $f(x)$.

Answer: \dfrac{153}{8}


Question: Given the integral equation \(\int \frac{\cos (6x) +\cos(9x)}{1-2 \cos(5x)}dx=- \frac{\sin(4x)}{k}-\sin(x)+C\), find the value of \(k\).

Answer: 4


Question: Compute \( \mathbb{E}\left(\int_{0}^{T} |B_{t}|^{2} dt\right) \) where \((B_t)_{t\geq0}\) is a standard Brownian motion and \(T\) is the first time the Brownian motion hits \(1\) or \(-1\).

Answer: \dfrac{1}{6}


Question: Find the maximum positive integer $k$ such that for all positive real numbers $a$, $b$, and $c$, the inequality \( a^2 + b^3 + c^4 + 2019 \geq k(a + b + c) \) holds.

Answer: 79


Question: If $f, g: \mathbb{R} \rightarrow \mathbb{R}$ satisfy $\left|f(x) + g(x)\right| = \left|f(x)\right| + \left|g(x)\right|$ for all $x \in \mathbb{R}$, must $f = cg$ for some $c > 0$? Justify your answer.

Answer: No


Question: A function \(f\) is defined by \(f(z) = (4+i)z^2 + az + b\) for all complex numbers \(z\), where \(a\) and \(b\) are complex numbers. If \(f(1)\) and \(f(i)\) are both purely real, find the minimum value of \(|a| + |b|\).

Answer: \sqrt{2}


Question: Let $X$ be a Poisson random variable with mean $1$. Find $E(|X-1|)$.

Answer: \dfrac{2}{e}


Question: In a commutative ring \(R\), let \(A_1\) and \(A_2\) be two ideals, and \(P_1\) and \(P_2\) be two prime ideals. Given that \(A_1 \cap A_2 \subseteq P_1 \cap P_2\), determine if there exists at least one pair of indices \(i\) and \(j\) such that \(A_i \subseteq P_j\).

Answer: Yes


Question: Calculate the circulation of the magnetic field intensity vector \( H(x,y,z) = 9y\hat{x} \) [A/m] around a circle of radius \( r = 7 \) meters in the \( x-y \) plane, traversed in the \( +\hat{\phi} \)-direction. Express your answer in units of [A] and provide three significant figures.

Answer: -1390


Question:
Compute the determinant of the following tridiagonal matrix:
\[
\begin{vmatrix}
3 & 2 & 0 & 0 & 0 & \cdots \\
1 & 3 & 1 & 0 & 0 & \cdots \\
0 & 2 & 3 & 2 & 0 & \cdots \\
0 & 0 & 1 & 3 & 1 & \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\
\end{vmatrix}
\]

Answer: 2^{n+1} - 1


Question: Find a prime number \( p \) such that the number \((2)(3)(5)(7)\cdots(p) + 1\) is not a prime number.

Answer: 13


Question: Consider two Hermitian matrices $A$ and $B$. If $A \geq B \geq -A$, is it true that $|A| \geq |B|$? Justify your answer.

Answer: Yes


Question: Evaluate the integral \( \int^{\pi/2}_{-\pi/2} \frac{1}{1+e^{\sin x}} \, dx \).

Answer: \dfrac{\pi}{2}


Question: Determine whether there exists a function $f \in L^{p}(\mathbb{R}^{n})$ such that $\|If\|_{L^{p}(\mathbb{R}^{n})} = \infty$, where $If(x) = \int_{\mathbb{R}^{n}}\frac{f(y)}{|x - y|^{n}}\, dy$. Justify your answer.

Answer: Yes


Question: Let $U$ be a non-empty open subset of $\mathbb{C}$. Consider a holomorphic function $f$ defined by the series \( f(z) = \sum_{n=0}^\infty a_n e^{2\pi i nz} \), where \( a_n \in \mathbb{C} \) for every \( n \) and \( z \in U \). If \( f(z) = 0 \) for every \( z \in U \), must it be true that \( a_n = 0 \) for every \( n \)?

Answer: Yes


Question: The sum of fifty positive numbers is 1. What is the maximum value of the sum of their inverses?

Answer: \infty


Question: Let \( v \) be a permutation of \( \{1, 2, \ldots, 2n+1\} \) where \( n \) is odd. The permutation \( v \) must satisfy the condition that the middle element \( v_{n+1} \) has an equal number of elements less than \( v_{n+1} \) to its right as there are elements greater than \( v_{n+1} \) to its left. Formally, this means \(|\{i: i > n, v_i < v_{n+1}\}| = |\{i: i < n, v_i > v_{n+1}\}|\). Determine the number of such permutations, denoted as \(|V_n|\).

Answer: (2n)!


Question: Determine the center of the $k$-algebra $A(k)$ with generators $x$ and $y$ and the defining relation $yx - xy = 1$, given that the characteristic of $k$ is greater than 0.

Answer: k[x^p, y^p]


Question: If $B$ is a subring of an Artinian ring $A$, is $B$ necessarily an Artinian ring?

Answer: No


Question: Suppose that the error, in grams, of a balance has the density function \(f(x)=\frac{1}{4}e^{\frac{-|x|}{2}}\) for \(-\infty<x<\infty\), and that 100 items are weighed independently. Use the Central Limit Theorem to approximate the probability that the absolute difference between the true total weight and the measured total weight is more than 50 grams.

Answer: 0.077


Question: Determine whether the congruence $x^2 + x + 1 \equiv 0 \pmod{997}$ has any solutions. Justify your answer.

Answer: Yes


Question: Find the value of the infinite series: \[ \sum_{n=1}^\infty \frac{n^2}{(n+1)(n+2)(n+3)(n+4)} \]

Answer: \dfrac{5}{36}


Question: Determine whether the following statement is true or false: If a monotonic real sequence \(x_n\) converges to 0, then \(n(x_{n+1} - x_n) \to 0\).

Answer: True


Question:
Find the minimal polynomial of the operator $T$ on $\mathbb{C}^5$, where $T$ is represented by the matrix:
\[
\begin{bmatrix}
0 & 0 & 0 & 0 & -3\\
1 & 0 & 0 & 0 & 6\\
0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 1 & 0
\end{bmatrix}
\] with respect to the standard basis. The minimal polynomial is known to be $z^5 - 6z + 3$. Verify this result by finding the minimal polynomial.

Answer: z^5 - 6z + 3


Question: Find the coefficient of \(x^{15}\) in the expansion of \((1-x^2)^{-10}(1-2x^9)^{-1}\).

Answer: 440


Question: Find the maximum value of \(x_5\) for \((x_1, x_2, x_3, x_4, x_5)\) in \(\mathbb{R}^5\) such that \(\sum_{i=1}^5 x_i = 6\) and \(\sum_{i=1}^5 x_i^2 = 8\).

Answer: 2


Question:
Given a random sample \( X_1, \dots, X_n \) from a uniform distribution \( U(\theta, \theta + 1) \), consider the following three estimators for \( \theta \), where \( \theta \in \mathbb{R} \):

1. \( T_1 = X_{(n)} \)
2. \( T_2 = \frac{X_{(1)} + X_{(n)}}{2} \)
3. \( T_3 = \frac{X_{(1)} + X_{(n)}}{2} - \frac{1}{2} \)

Which of these estimators is the maximum likelihood estimator (MLE) for \( \theta \)?

Answer: T_3


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

Answer: [-1, 1]


Question: Let $f(x)$ be a differentiable function satisfying $f\left(\frac{x+y}{2}\right) = \frac{f(x) + f(y)}{2}$ for all $x, y \in \mathbb{R}$ with $f(0) = 0$. If $\int_{0}^{2\pi} (f(x) - \sin x)^2 \,{\rm d}x$ is minimized, find the value of $f(-4\pi^2)$. \( \)

Answer: 3


Question: Given a linear map \( T: \mathbb{R}^2 \longrightarrow \mathbb{R}^2 \), the measure \( \mu(E) = \lambda_2 (T(E)) \) is defined, where \( \lambda_2 \) is the Lebesgue measure in \( \mathbb{R}^2 \). If \( \mu([0,1]^2)=3 \), find the value of \( \mu(B(0,1)) \), where \( B(0,1)=\{ (x,y) \mid x^2 + y^2 \leq 1 \} \).

Answer: 3\pi


Question: Consider a simple undirected graph \( G = (V, E) \) with \( V = \{1, \ldots, n\} \) where \( n > 1 \). For each vertex \( v \in V \), let \( N(v) = \{ w \in V : \{v, w\} \in E \} \) be the neighborhood of \( v \). An edge-coloring \( c : E \to \{1, \ldots, n\} \) is called "good" if for every edge \( \{x, y\} \in E \), the color \( c(\{x, y\}) \) is in \( N(x) \cup N(y) \). Determine whether every graph \( G = (\{1, \ldots, n\}, E) \) has a good edge-coloring.

Answer: Yes


Question: Let \( S = \{1, 2, 3, \ldots, 1337\} \). Find the smallest positive integer \( n \) such that the product of any \( n \) distinct elements in \( S \) is divisible by 1337. What are the last three digits of \( n \)?

Answer: 331


Question: Evaluate the integral \( \int_{V}\vec{r}.\left ( \vec{d}-\vec{r} \right )\delta^{3}\left ( \vec{e}-\vec{r} \right )dV \), where \( \vec{d}=\left \langle 1,2,3 \right \rangle \), \( \vec{e}=\left \langle 3,2,1 \right \rangle \), and \( V \) is a sphere of radius 1.5 centered at \( \left ( 2,2,2 \right ) \).

Answer: -4


Question: Determine the number of bidders Jane should pay David to find in order to maximize the difference between the expected value of the second-highest bid from a uniform distribution of $n$ bidders and the cost $10n$. Each bidder values the item uniformly between $[500, 1000)$. The highest bidder wins and pays the second-highest bid price. Calculate the optimal number of bidders, $n$, to maximize this difference.

Answer: 9


Question: Calculate the area of the region defined by the inequalities: \(0 \leq x \leq \sqrt{y}\), \(0 \leq y \leq \sqrt{x}\), and \(x+y \leq \frac{3}{4}\).

Answer: \dfrac{13}{96}


Question: Find a polynomial \( p(t) = a_0 + a_1 t + \dots + a_n t^n \) such that the inequality \( |a_0| + |a_1| + \dots + |a_n| \le C \sup_{t \in [0,1]} |p(t)| \) does not hold for a given constant \( C > 1 \). Provide a counter-example.

Answer: (1 - t)^n


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

Answer: \dfrac{1}{e}


Question: Find the remainder when the polynomial \(P(x) = x^{2008} + x^{2007} + 1\) is divided by the binomial \(x^2 + 1\).

Answer: -x + 2


Question: Find all \(3\times3\) square matrices \(A\) such that \(AX = XA\) for any \(3\times3\) upper triangular matrix \(X\).

Answer: A = \lambda I


Question: A cube is sectioned by 33 planes parallel to its faces, forming multiple parallelepipeds. If the number of parallelepipeds with faces coinciding with the faces of the cube is half of the total number of parallelepipeds, find the total number of parallelepipeds formed.

Answer: 1344


Question: Given that $a + b + c = 0$, find the value of the expression \[ \left(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c}\right)\left(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}\right). \]

Answer: 9


Question: Let the function \(f(x)= \sqrt{(x^2-2x+2)(-x^2+3x+10)}-\sqrt{(-x^{2}+x-1)(x^{2}+x-6)}\) have a domain \(D=[a,b+c]\). Find the value of \(\frac{a^3+b^3+c^3}{abc}\).

Answer: 3


Question:
Consider the ordinary differential equation \( y'(t) = -y^3 + y^2 + 2y \) with the initial condition \( y(0) = y_0 \) where \( y_0 \in (0, 2) \). Determine the set to which \( \lim_{t \to \infty} y(t) \) belongs. Choose from the following options:

- \{-1, 0\}
- \{-1, 2\}
- \{0, 2\}
- \{0, +\infty\}

Answer: \{-1, 2\}


Question: Determine the interval of initial values \(x_0 \in \mathbb{R}\) for which the sequence defined by \(x_{n+1} = 2^{\frac{x_n}{2}}\) converges to 2 as \(n\) approaches infinity.

Answer: (-\infty, 4)


Question: Evaluate the line integral \( \int_{C} F \cdot \, dr \) where \( C \) is the curve consisting of line segments from \((4,1)\) to \((4,3)\) to \((1,3)\) to \((1,1)\), and \( F(x,y) = (x+y)i + (y-1)^3 e^{\sin(y)}j \).

Answer: -\dfrac{33}{2}


Question: Let $A$ and $B$ be abelian groups such that $\mathrm{Ext}(A, B) = 0$. If $C \cong A$ and $D \cong B$, does it follow that $\mathrm{Ext}(C, D) = 0$? Provide a justification for your answer.

Answer: Yes


Question: In an equilateral triangle with side length 1, 8 points are placed either on the boundary or inside the triangle. Determine the maximum possible value for the shortest distance between any two of these points.

Answer: \dfrac{1}{3}


Question: Determine the set of real numbers \(x\) for which the series \(\sum_{n=0}^{\infty} n! x^{n!}\) converges absolutely.

Answer: (-1, 1)


Question: Given a natural number $n$, find the sum of the binomial coefficients $\binom{n}{r}$ for $r$ ranging from $0$ to $n$. Express your answer in terms of $n$.

Answer: 2^n


Question: Let $A$ and $B$ be matrices in $M_n(F)$, where $F$ is a field. Define the centralizer of $A$ as $C(A) = \{X \in M_n(F) \mid XA = AX\}$. Similarly, define $C(B)$. If $C(A) \subseteq C(B)$, prove that $B \in F[A]$, where $F[A]$ is the field generated by $A$.

Answer: B \in F[A]


Question: Show that the sequence of functions \( f_n(x) = \frac{x}{1+n^2x^2} \) converges uniformly to 0 on \( \mathbb{R} \). Use an \( \epsilon-n_0 \) argument to demonstrate this convergence.

Answer: 0


Question: Evaluate the line integral \( \oint_C (x-y^3) \, dx + x^3 \, dy \) where \( C \) is the unit circle \( x^2 + y^2 = 1 \), oriented counterclockwise.

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


Question: Given that $g(n) = \Theta(n^2)$ and $f(n) = g(n) + g(n-1) + \ldots + g(2) + f(1)$, determine if it is correct to conclude that $f(n) = \Theta(ng(n))$. Assume $g(n)$ and $f(n)$ are asymptotically positive functions.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty} \left(\sqrt[3]{n^{48}+n} - \sqrt[3]{n^{48}+n^2}\right) \left((n^3 +3)^{12} - (n^4+4n)^9\right) \]

Answer: -6


Question: Evaluate the limit \(\lim_{x\rightarrow 1} \frac{\sin(\pi x^a)}{\sin(\pi x^b)}\) where \(a, b \in \mathbb{R}\) and \(b \neq 0\), without using l'Hôpital's rule or any derivative-based methods.

Answer: \dfrac{a}{b}


Question: Suppose $x^*$ is the optimal solution for the convex optimization problem $ \min_x \{f(x) : Cx \leq d\}$, where $f(x)$ is a strictly convex quadratic function. Given that $x^*$ satisfies the additional constraint $Ax = b$, can we conclude that $x^*$ is also the optimal solution for the problem $\min_x \{f(x) : Ax = b, Cx \leq d\}$?

Answer: Yes


Question: Find the value of \(a\) such that \(\int_0^{\infty}\frac{dx}{1+x^{2021}}=\int_0^1\frac{dx}{(1-x^{2021})^{a}}\).

Answer: \dfrac{1}{2021}


Question: Find the minimum value of \(x\) such that \(\sqrt{\left(\frac{x+y}{2}\right)^3} + \sqrt{\left(\frac{x-y}{2}\right)^3} = 27\), where \(x, y \in \mathbb{R}\).

Answer: 9


Question: Let $\theta$ be the angle between the vectors $A = (1,1,...,1)$ and $B = (1,2,...,n)$ in $\mathbb{R}^n$. Find the limiting value of $\theta$ as $n \to \infty$. Use the equation for the angle between two vectors: $\theta = \arccos\frac{A \cdot B}{\|A\|\|B\|}$.

Answer: \dfrac{\pi}{6}


Question: For which positive integers \( n \) does the determinant \( \mathcal{D}_n \) of an \( n \times n \) matrix, filled row-by-row with the first \( n^2 \) elements of the Thue-Morse sequence, not equal zero?

Answer: 2


Question:
Consider a finite group $G$ and the complex group algebra $\mathbb{C}(G)$, consisting of formal sums \( \sum_{g \in G} a_g g \) where \( a_g \in \mathbb{C} \). The algebra structure is defined by:
\[ \sum_g a_g g + \sum_g b_g g = \sum_g (a_g + b_g) g, \]
\[ \left( \sum_g a_g g \right) \left( \sum_g b_g g \right) = \sum_{g,h} a_g b_h (gh). \]
Is there a group $G$ and a non-zero element $x \in \mathbb{C}(G)$ such that $x^2 = 0$?

Answer: Yes


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

Answer: 1


Question: Determine whether the topological space $S^1 \times S^2$ has a nontrivial covering map to itself.

Answer: Yes


Question:
Find the expected value, \( E(Y) \), of the random variable \( Y \) with the probability density function:
\[ f(y) = \frac{y^{\frac{3}{2}}}{\frac{3}{4}\sqrt{\pi}}e^{-y}, \quad 0 < y < \infty \]

Answer: \dfrac{5}{2}


Question: Evaluate the triple integral: \[ \int_{-1}^1 \int_{0}^{2} \int_{-2}^2 \left[1+x^2 z^2 + x y \cos(x^4 + y^4 + z^4) \right]\;\operatorname d\!z \;\operatorname d\!y \operatorname d\!x \]

Answer: \dfrac{208}{9}


Question: Determine whether the function \( \frac{1}{|x|^\alpha} \) is integrable at infinity in dimensions \( d \geq 2 \) for any \( \alpha \).

Answer: \alpha > d


Question: Determine whether the sequence $\{i^{1/n}\}_{n=1}^\infty$ converges to 1. Justify your answer.

Answer: 1


Question: Consider a one-dimensional real-valued stochastic differential equation (SDE) given by \(dX_t = b(t, X_t)dt + \sigma(t, X_t)dB_t\) on a probability space \((\Omega, \mathcal{F}, \mathcal{F}_t, \mathbb{P})\) with a Brownian motion \(B_t\). Assume path-wise uniqueness holds and there exists a strong solution for every initial condition \(X_0\), independent of \(B_t\). Suppose there are two constant solutions \(X^1_t = K_1\) and \(X^2_t = K_2\) with \(K_1 < K_2\) and \(K_1, K_2 \in \mathbb{R}\). If a solution starts in \([K_1, K_2]\), i.e., \(X_0 \in [K_1, K_2]\) almost surely, does it follow that \(\mathbb{P}(X_t \in [K_1, K_2] \text{ for all } t \geq 0) = 1\) when \(b(t, X_t) = b(X_t)\) and \(\sigma(t, X_t) = \sigma(X_t)\) (time-independent coefficients)?

Answer: Yes


Question: Find the maximum order of a permutation in the symmetric group $S_7$. Provide a specific permutation in $S_7$ that achieves this maximum order.

Answer: 12


Question: Evaluate the limit: \[ \lim_{m \to \infty} \int_{-\infty}^{\infty} \sin(e^t) e^{-(t - m)^2} \, dt. \] Determine if this limit equals zero.

Answer: 0


Question: Let $X_1, \dots, X_n$ be independent random variables, each following a normal distribution with mean $1$ and variance $2$. Find the moment-generating function for the random variable $n^{-1/2}(S_n-n)$, where $S_n = \sum_{i=1}^n X_i$. Provide your answer in terms of $n$.

Answer: e^{t^2}


Question: Let $A$ and $B$ be $n \times n$ matrices such that $A^2 = A$, $B^2 = B$, and $(A + B)^2 = A + B$. Prove that $AB = 0$.

Answer: AB = 0


Question: Find the remainder when \( 2016^{{2017}^{2018}} \) is divided by 77.

Answer: 70


Question: Evaluate the integral \( \int_{0}^{1} \frac{1}{x+\sqrt{1-x^2}} \, dx \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the tensor product \( \mathbb{C} \otimes_{\mathbb{R}} \mathbb{R} \).

Answer: \mathbb{C}


Question: Let \( z \) be a complex number such that \( \arg\left(\frac{z^3-1}{z^3+1}\right) = \frac{\pi}{2} \) on the Argand plane. Determine the length of the arc of the locus of \( z \) for which \( \text{Re}(z) > 0 \) and \( \text{Im}(z) < 0 \).

Answer: \dfrac{\pi}{6}


Question: Evaluate the double integral \( \iint xy\ dx\ dy \) over the region defined by the inequalities \( x^2 + y^2 \leq 25 \) and \( 3x + y \geq 5 \).

Answer: \dfrac{135}{4}


Question: Find the next least positive integer after 38 such that its square ends with three fours.

Answer: 462


Question: Find the minimal scalar \( m \in \mathbb{R} \) such that for the quadratic form \( q(x,y,z) = 2zx + 4yz - 2xy \), the inequality \( q(x,y,z) \leq m(x^2 + y^2 + z^2) \) holds for all \( x, y, z \in \mathbb{R} \).

Answer: 2


Question: Given that \( y = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \dots \) and \( z = -y - \frac{y^2}{2} - \frac{y^3}{3} - \dots \), find the value of \( \ln \left( \frac{1}{1-e^z} \right) \) for a nonzero number \( x \).

Answer: x


Question: Determine whether the sequence \( x_n = \frac{1}{n} \) is a Cauchy sequence in the metric space \((\mathbb{R}, d)\), where the metric is defined as \( d(x, y) = |\arctan x - \arctan y| \).

Answer: Yes


Question: If $A$ is an $m \times n$ matrix and $v$ is an $n \times 1$ vector that belongs to a subspace of dimension $d \le n$, must the vectors of the form $Av$, which are $m \times 1$, be elements of a subspace of dimension $e \le d$? Answer with 'Yes' or 'No' and justify your answer.

Answer: Yes


Question: Let $f: (X, \mu) \to [0, \infty]$ be a measurable function. Define $S$ as the set of all real numbers $\alpha$ such that $\mu(f^{-1}((\alpha, \infty])) = 0$. If $S = \emptyset$, let $\beta = \infty$. If $S \neq \emptyset$, let $\beta = \inf S$. Determine whether $\beta \leq f$ or $f \leq \beta$. Provide a justification for your answer.

Answer: f \leq \beta


Question:
Let $N \geq 2$ be an even natural number and define the set $A_N=\{k=(k_1,k_2) \in \mathbb{N} \times \mathbb{N}:1<k_2<k_1\leq N\}$. Consider the set $U_N=\{(k,l): k \in A_N, l \in A_N, \angle(k,l)>0\}$, where $\angle(k,l)$ is the angle between two vectors $k$ and $l$ in the plane. Determine the limit:
\[ \lim_{N \to \infty} N \inf_{(k,l)\in U_N}\angle(k,l). \]

Answer: 0


Question: Evaluate the limit of the integral \( \int_0^\pi \left| i\epsilon e^{i\theta}\cdot\frac{e^{i\alpha}e^{i\alpha \epsilon e^{i\theta}}}{(1+\epsilon e^{i\theta})^4-1}\right|\,d\theta \) as \( \epsilon \to 0 \).

Answer: \dfrac{\pi}{4}


Question: Given the polynomial \( P(x) = 4x^3 + 4x - 5 = 0 \) with roots \( a, b, \) and \( c \), evaluate the expression \((b+c-3a)(a+b-3c)(c+a-3b)\).

Answer: -80


Question: Let \( f \) be a convex differentiable function. Prove that if \( \lim_{x \to +\infty} f(x) = l \), then \( \lim_{x \to +\infty} f'(x) = 0 \).

Answer: 0


Question: Given that \( X \) and \( Y \) are continuous random variables with the joint probability density function \( f(x,y) = 3y \) for \( 0 \le x \le y \le 1 \), find the probability density function of the random variable \( T = XY \).

Answer: 3(1 - \sqrt{t})


Question:
Let \(\{X_i\}_{i=1}^n\) be a sequence of i.i.d random variables with common pdf:
\[ f(x;a,\theta) =\theta a^\theta x^{-(\theta+1)} \boldsymbol 1_{(a,\infty)}(x) \quad \text{where } \theta, a > 0\]
Find the asymptotic relative efficiency of the maximum likelihood estimator (MLE) of \(a\) with respect to the uniformly minimum variance unbiased estimator (UMVUE) of \(a\).

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{(1+x)^{1/x} - e + \frac{1}{2}ex}{x^2} \]

Answer: \dfrac{11e}{24}


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2} \frac{\sin^{2012}{x}}{\left(1+ \alpha^x\right)\left(\sin^{2012} {x}+\cos^{2012}{x}\right)}\;{dx} \) for \( \alpha \in \mathbb{R^+} \).

Answer: \dfrac{\pi}{4}


Question: Find the Maclaurin polynomial of order 6 for the function \( f(x) = x \ln(1+x^3) \ln(1-x^2) \).

Answer: -x^6


Question: Evaluate the integral of the complex conjugate squared over the path $C = [1-i, 1+i]$, where $C$ is the straight line segment from $1-i$ to $1+i$. Parametrize the path by $x(t) = 1-i + t((1+i) - (1-i))$ for $t \in [0,1]$. Compute the integral \( \int_C \overline{x(t)}^2 \, dt \).

Answer: \dfrac{2}{3}


Question: Determine the number of homomorphisms from a finite extension of \(\mathbb{Q}\), denoted as \(K\), to \(\mathbb{R}\).

Answer: r_1


Question: What is the relationship between the dual space of the dual space, \(V^{**}\), and the original vector space \(V\)?

Answer: V \cong V^{**}


Question: In a scenario where 8 letters are placed into 8 addressed envelopes, calculate the probability that exactly 3 letters are placed in their correct envelopes, and none of the remaining letters are placed in their correct envelopes.

Answer: \dfrac{11}{180}


Question: Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function satisfying the equation \( f(x) = 5 \int_{0}^{x} f(t) \, dt + 1 \) for all \( x \in \mathbb{R} \). Find the value of \( f(1) \).

Answer: e^5


Question: Determine whether there exists a simple group that is non co-hopfian. A simple group is defined as a group where all normal subgroups are either the group itself or the trivial group, and a group is non co-hopfian if there exists an injective group endomorphism that is not surjective.

Answer: Yes


Question:
Find the minimum value of the expression:
\[ P = \frac{x^3}{(y+2z)(2z+3x)} + \frac{y^3}{(z+2x)(2x+3y)} + \frac{z^3}{(x+2y)(2y+3z)} \]
subject to the condition:
\[ x^2y^2z^2 + (1+x)(1+y)(1+z) \ge x+y+z+xy+yz+zx+3 \]
where \(x, y, z > 0\).

Answer: \dfrac{1}{5}


Question: Determine whether the following statement is true or false: Let $F$ be a field extension of a field $k$, and let $R$ be an algebra over $k$. Is it true that the center of the tensor product $R \otimes_k F$ is equal to $Z(R) \otimes_k F$, where $Z(R)$ denotes the center of $R$?

Answer: True


Question: Let $\Phi$ be a homogeneous Poisson point process (PPP) of intensity $\lambda$ on $\mathbb{R}$. Consider the ordered points $\{x_0, x_1, \dotsc\} = \Phi \cap \mathbb{R}^+$, such that $0 < x_0 < x_1 < \dotsc$. Find the complementary cumulative distribution function (CCDF) of the ratio $x_0/x_k$, where $x_k$ is Erlang distributed with probability density function $f_{x_k}(x) = \frac{\lambda^{k+1}x^k}{k!}e^{-\lambda x}$. Provide your answer in terms of $\lambda$ and $k$.

Answer: (1 - r)^k


Question:
Determine the number of real roots of the polynomial equation \( P(x) = 2013x^{2012} - 2012x^{2011} - 16x + 8 \) within the interval \( x \in \left[0, 8^{\frac{1}{2011}}\right] \). Choose the correct option:

(a) exactly one real root.
(b) no real root.
(c) at least one and at most two real roots.
(d) at least two real roots.

Answer: D


Question: Given a group $G$ and an element $a \in G$ such that $a^8 = a^t$ and $a^{88} = a^{74}$, find the value of $t$ in the set $\{0, 1, 2, 3, 4\}$, given that $a^2 \neq a^0$. What is the value of $t$?

Answer: 1


Question: If \(\nu\) is a finite signed measure, does it imply that its total variation \(|\nu|\) is also finite? Justify your answer.

Answer: Yes


Question: Find the density of the $xy$ coordinates in a 3-dimensional unit ball, given that $Z = 0$. Express your answer as a function of $x$ and $y$.

Answer: \dfrac{1}{\pi}


Question:
Consider the set \( S = \{ f \in C[0,1] : \lim_{n \to \infty} f(1/n) = 0 \} \) with the supremum norm. Determine which of the following statements is correct:

(a) \( S \) is a vector space but not closed in \( C[0,1] \).
(b) \( S \) is closed but does not form a vector space.
(c) \( S \) is a closed vector space but not an algebra.
(d) \( S \) is a closed algebra.

Select the correct option.

Answer: D


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{\sqrt{2(2-x)}(1-\sqrt{1-x^2})}{\sqrt{1-x}(2-\sqrt{4-x^2})} \]

Answer: 4


Question: Find a non-zero real polynomial \( F(x, y) \) with degree \( \leq 2n-2 \) such that \( F(p(t), q(t)) = 0 \) for all real \( t \), where \( p(x) \) and \( q(x) \) are real polynomials of degree \( \leq n \) and are both non-constant. Can such an \( F \) be found?

Answer: Yes


Question: Determine whether every Baire function $f: \mathbb{R} \to \mathbb{N}$ must be constant. A function $f: X \to Y$ is a Baire function for metrizable spaces $X$ and $Y$ if it belongs to $F(X,Y)$, where $F$ is the smallest class of functions from $X$ to $Y$ that contains all continuous functions and is closed under pointwise limits.

Answer: Yes


Question: Find the coefficient \( k \) in the joint probability density function \( f(x,y) = k e^{-\frac{1}{2}(x^2 - 2xy + 5y^2)} \) of a multivariate normal distribution.

Answer: \dfrac{1}{\pi}


Question: Find the center of the circle described by the complex function \( z = \frac{3i-t}{2+it} \) where \( t \in \mathbb{R} \).

Answer: \dfrac{5}{4}i


Question: Let $R$ be a Noetherian ring and $I$ be a proper ideal of $R$. Suppose $b \notin I$ is such that $I + Rb$ and $(I:b) = \{x \in R : xb \in I\}$ are projective as $R$-modules. Is the ideal $I$ projective as an $R$-module?

Answer: Yes


Question:
Verify whether the following inequality holds for all complex numbers \( z \):
\[ |e^{z}| \geq 2 - e^{|z|} \]

Answer: Yes


Question: Evaluate the infinite series: $$\sum_{n=1}^{\infty} \frac{n}{n^4+n^2+1}$$

Answer: \dfrac{1}{2}


Question: Let the number $2^n + n^2$ be prime, where $n \geq 2$. Prove that the number $(n-3)$ is divisible by 6.

Answer: 6


Question: Evaluate the integral \( \int_0^1 e^{\frac{1}{\log(\theta)}} \,d\theta \) using polar coordinates.

Answer: 2 K_1(2)


Question: Evaluate the integral \( \int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x \).

Answer: \dfrac{\pi}{4}


Question: Determine whether there exists a natural number $x$ such that $x^3 = \sum_{i=1}^{2000} a_i^3$ for some natural numbers $a_i$. The $a_i$'s need not be distinct.

Answer: Yes


Question: Calculate the average distance in maximum norm from the origin to the nearer of two random points uniformly distributed in a 1x1 unit square.

Answer: \dfrac{8}{15}


Question: Let \( \{f_n\} \) be a sequence of differentiable functions on \( \mathbb{R} \) such that \( f_n' \) is bounded for each \( n \). If \( \{f_n'\} \) converges uniformly to \( f \) on \( \mathbb{R} \), is it true that \( f \) is bounded and the derivative of some function?

Answer: Yes


Question:
Find the norm of the point evaluation functional given by \( g(t_0)=\int_0^1g(t)\delta(t-t_0) \text{dt} \), where \( \delta(t) \) is the Dirac delta function defined as:
\[ \delta(t)= \begin{cases}
0, & t \in [0,1] \setminus \{t_0\} \\
\infty, & t=t_0 \in [0,1]
\end{cases} \]
Use the formula for the norm of an operator:
\[ \Vert A\Vert = \sup_{g\neq 0}\frac{|Ag|}{\Vert g \Vert} \]

Answer: 1


Question: Solve the inequality \( \sin x > \ln x \). Determine the solution set where the sine function is greater than the natural logarithm function.

Answer: (0, 2.219)


Question: For any integers \(m, n, r > 1\), does there exist an infinite group \(G\) with elements \(a, b \in G\) such that \(o(a) = m\), \(o(b) = n\), and \(o(ab) = r\)? Provide a justification for your answer.

Answer: Yes


Question: Suppose $f, u, u_n \in L^p(\Omega)$ where $\Omega \subset \mathbb{R}^N$ is a bounded domain and $f, u, u_n \geq 0$. Given that \( \|u+u_n\|_p \rightarrow \|u+f\|_p \), \( u_n \rightarrow f \) almost everywhere in \( \Omega \), and \( u_n > f \), does it follow that \( \|u_n-f\|_p \rightarrow 0 \)?

Answer: Yes


Question: Find a tight upper bound for the sum \( \sum_{m=2}^n \frac{\log m}{m^2} \) and express it using Big O notation.

Answer: O(1)


Question: Determine whether there exists a constant \(c > 0\) such that the series \(\sum_{k=1}^\infty \frac{1}{k^\alpha} \exp(-c k^{1-\alpha})\) converges for \(0 < \alpha < 1\).

Answer: Yes


Question: Given a non-negative valued sequence $x_n$ such that $\lim\limits_{n\to\infty} x_n = 0$, and a polynomial function $p$ of degree at least 1 satisfying $\|x_n\| \leq p(n^{-1})$, with $x_n \neq 0$ for any $n > 0$, determine if there exists a rational function $q$ and a positive integer $N$ such that $q(0) = 0$ and $\|x_n\| \leq q(n^{-1})$ for all $n \geq N$. Provide a justification for your answer.

Answer: Yes


Question: The expression \( \frac{(\text{cis}\ 75^\circ)(\text{cis}\ 80^\circ−1)}{\text{cis}\ 8^\circ−1} \) can be written in the form \( r\ \text{cis}\ \theta \), where \( \text{cis}(\theta)=\cos \theta+i\sin \theta \). Find \( \theta \) in degrees, ensuring that \( 0 \le \theta < 360^\circ \).

Answer: 111


Question: Solve the initial value problem given by the differential equation \( z' = z(z-5)(z-20) \) with the initial condition \( z(3) = 10 \). Determine \( \lim_{x \rightarrow \infty} z(x) \).

Answer: 5


Question:
Find the largest integer K such that for integers a > 0, b > 0, and c > 0 satisfying the inequality:
\[ 126 > 2^{a+1}(2^b(2^c-1) - 3) > 0 \]
the sum a + b + c is less than K.

Answer: 9


Question:
Let $V \subseteq \mathbb{C}^9$ be a vector subspace of dimension $6$. Suppose there exist matrices $A, B \in M_{3 \times 6}(\mathbb{C})$ such that:
\[ V = \{(x_1, \ldots, x_9) \in \mathbb{C}^9 : (x_3, x_6, x_9)^T = A(x_1, x_2, x_4, x_5, x_7, x_8)^T \} = \{(x_1, \ldots, x_9) \in \mathbb{C}^9 : (x_3, x_6, x_9)^T = B(x_1, x_2, x_4, x_5, x_7, x_8)^T \} \]
Is it true that $A = B$?

Answer: A = B


Question:
Given two real polynomials $f(x)$ and $g(x)$ of degrees $m(\geq 2)$ and $n(\geq 1)$ respectively, satisfying the equation \( f(x^2+1) = f(x)g(x) \) for every \( x \in \mathbb{R} \), determine which of the following statements is true about the roots of $f(x)$:

(A) $f$ has exactly one real root $x_{0}$ such that $f^{\prime}(x_{0}) \neq 0$.
(B) $f$ has exactly one real root $x_{0}$ such that $f^{\prime}(x_{0})=0$.
(C) $f$ has $m$ distinct real roots.
(D) $f$ has no real root.

Select the correct option.

Answer: D


Question: Let \((W_t)_t\) be a Brownian motion with respect to \((F_t)_t\). Compute the expectation \(E[(W_t^2 - t)(W_s^2 - s)]\) for \(0 < s < t\).

Answer: 2s^2


Question: Given an AR(1) model with 144 observations, where \( y_t = \phi y_{t-1} + \epsilon_t \) and \( \epsilon_t \) is white noise with mean zero and variance \( \sigma^2 \), find an estimate for \( \phi \). The initial and final observations are \( y_1 = -1.7 \) and \( y_{144} = -2.1 \). Additionally, \( \sum_{t=2}^{144} y_t y_{t-1} = -128.6 \) and \( \sum_{t=1}^{144} y_t^2 = 246.4 \). Use the formula for the first sample autocorrelation: \[ \hat{\phi} = \frac{\sum_{t=2}^{144} (y_t - \overline{y})(y_{t-1} - \overline{y})}{\sum_{t=1}^{144} (y_{t-1} - \overline{y})^2} \] to find \( \hat{\phi} \).

Answer: -0.53


Question:
Solve the partial differential equation in polar coordinates:
\[ U_{rr} + \frac{1}{r^2}U_{\theta\theta} + \frac{1}{r}U_r = 0 \]
subject to the initial conditions:
\[ U(1,\theta) = 1, \quad U_r(1,\theta) = 1, \quad U_\theta(1,\theta) = 0. \]
Find the solution \( U(r,\theta) \).

Answer: 1 + \ln r


Question: Find the minimum value of \( r \) such that the limiting value of the Markov chain \( X_n \), defined by \( X_0 > 0 \) and \( X_n = \begin{cases} 1.5X_{n-1} & \text{with probability } 0.5 \\ rX_{n-1} & \text{with probability } 0.5 \end{cases} \) for every \( n \geq 1 \), is greater than or equal to \( X_0 \).

Answer: \dfrac{2}{3}


Question: Determine the number of real roots of the equation \(1+\frac{x}{1}+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\cdots+\frac{x^{7}}{7}=0\).

Answer: 1


Question: Find the non-zero singular value(s) of the matrix \( M = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{bmatrix} \) without using singular value decomposition (SVD).

Answer: 2\sqrt{5}


Question: Is it possible to choose an origin in $\mathbb{C}^n$ such that for a discrete sequence $\{p_j\}$ in $\mathbb{C}^n$ (without repetition), the condition $0 < |p_1| < |p_2| < \dots$ holds? Assume $n > 1$. Provide a justification for your answer.

Answer: Yes


Question: Find all positive integer values of \( A \) such that the fraction \( \frac{A}{41} \) ends with the repeating decimal \( \overline{24390} \).

Answer: 10


Question: Consider 9 distinct apples and 7 distinct bananas to be placed into 6 identical bins. Determine the number of ways to configure the bins such that exactly one bin contains 5 apples, exactly one bin contains 3 bananas, and the remaining 4 bins each contain 1 apple and 1 banana. Provide your answer as a single number.

Answer: 105840


Question: Evaluate the limit \( \lim_{n\rightarrow\infty}\int_0^\infty \dfrac{1+nx^2+n^2x^4}{(1+x^2)^n}d\mu. \)

Answer: 0


Question: Determine the derived set of the subset \( A = \{a + \pi b : a, b \in \mathbb{Z}\} \) of the real line.

Answer: \mathbb{R}


Question: Determine the radius of the circle onto which the bilinear transformation \( f(z) = \frac{i z + 2}{4z + i} \) maps the real and imaginary axes.

Answer: \dfrac{9}{8}


Question: Solve for \(x\) in the equation: \[ \left(\sqrt{5+2\sqrt{6}}\right)^x - \left(\sqrt{5-2\sqrt{6}}\right)^x = 96 \]

Answer: 4


Question: Given the homographic function \( y = \frac{ax+b}{cx+d} \), derive the expression \( ad-bc \) using an alternative method to the one provided. Show your steps clearly to arrive at the condition \( \boxed{ad-bc} \).

Answer: ad - bc


Question: Calculate the work done by the force field \( \mathbf{F} = \frac{2x}{y}\mathbf{i} - \frac{x^2}{y^2}\mathbf{j} \) in moving a particle from the point \((-1, 1)\) to the point \((3, 2)\).

Answer: \dfrac{7}{2}


Question: Compute the line integral \( \int_{C(0,4)^+} (x^2+y^3)dx + (y^4+x^3)dy \), where \( C(0,4) \) is the positively oriented circle with center at the origin and radius 4, using Stokes' Theorem.

Answer: 0


Question: Let $f(x):[0, 1] \rightarrow \mathbb{R}$ be a continuous function that attains its maximum at exactly three points. Determine whether the function $h(x) = f^3(x) - f^2(x) + 5f(x) - 14$ also attains its maximum at exactly three points.

Answer: Yes


Question: Let \( \xi_{19} \) be a primitive 19th root of unity. Calculate the real part of the sum \( \sum_{k=1}^9 \xi_{19}^{k^2} \).

Answer: -\dfrac{1}{2}


Question: Determine if there are any positive integer solutions $(a, b)$ to the equation $5^a + 1 = 2 \cdot 3^b$ other than the solution $(a, b) = (1, 1)$. If so, find such a solution.

Answer: (1, 1)


Question:
Let $N$ be a positive integer not equal to $1$. Consider the number $N! - 1$. Which of the following statements is true?

(A) $N! - 1$ is a prime number.
(B) At least one of the numbers $N + 1, N + 2, \ldots, N! - 2$ is a divisor of $N! - 1$.
(C) The smallest number between $N$ and $N!$ which is a divisor of $N! - 1$, is a prime number.

Answer: C


Question: Suppose $f$ is a real function defined on $\mathbb{R}$ which satisfies \(\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0.\) Does this imply that $f$ is continuous?

Answer: No


Question: Consider an irreducible polynomial $f(x) \in \mathbb{Q}[x]$ with degree at least 2. Let $K$ be the splitting field of $f(x)$, and let $G$ be the Galois group of $K$ over $\mathbb{Q}$. If $\alpha_1, \dots, \alpha_n \in K$ are all the roots of $f(x)$, does there exist such a polynomial $f(x)$ such that for each $g \in G$, there exists a root $\alpha_i$ with $g(\alpha_i) = \alpha_i$?

Answer: No


Question: Determine the number of distinct ways to label the vertices of a cube with the integers 1 through 8, without repetition, such that the sum of the numbers on the four vertices of each face is the same. Arrangements that can be obtained through rotations of the cube are considered identical. How many different arrangements are possible?

Answer: 6


Question: Suppose $(F_{n})$ is an increasing sequence of closed sets in $\mathbb{R}^{d}$, and define \( F := \bigcup_{k\geq1}\bigcap_{n\geq k}\partial F_n . \) Given that $F$ is a closed set with no isolated points, determine if there can be a point $x_{0}\in F$ such that every open neighborhood $V$ of $x_{0}$ contains the closure of an open set $\omega$ where $\omega$ contains $x_{0}$ and the boundary of $\omega$ is in $F$. Can such a point exist?

Answer: Yes


Question: Differentiate the expression $x^t Ax$ with respect to the vector $x$, where $x$ is a column vector and $A$ is a symmetric square matrix. Express your answer in terms of $A$ and $x$.

Answer: 2Ax


Question: Find the degree of the field extension \([\mathbb{Q}(2^{1/4})(\sqrt{5}) : \mathbb{Q}(\sqrt{5})]\).

Answer: 4


Question: Let $M$ be a free module over a principal ideal domain (PID) $R$ with basis $X$. Consider a proper subset $Y$ of $X$ and a submodule $N$ of $M$. Define $N_Y = \langle Y \rangle \cap N$. If $N_Y$ is free with a basis $B$, prove or disprove that $|B| \leq |Y|$.

Answer: |B| \leq |Y|


Question: Evaluate the integral \( \iint_D \frac{x}{\sqrt{x^2+y^2}} \,dx\,dy \), where \( D \) is the region in the first quadrant of the unit circle.

Answer: \dfrac{1}{2}


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

If $U$ is any domain, $f$ is any function continuous on $U$, and $\gamma$ is any closed contour such that its image and its interior lie in $U$, then \( \int_\gamma f(z)\,dz = 0 \).

Answer: B


Question: Determine whether the ideal \((I, J)\) is equal to the ideal \((I, J, IJ)\), where \(IJ\) is the product of the ideals \(I\) and \(J\).

Answer: (I, J) = (I, J, IJ)


Question: Find \( \lim_{n \to \infty} \left( \frac{1}{\sqrt{n^2+1}} + \frac{1}{\sqrt{n^2+2}} + \cdots + \frac{1}{\sqrt{n^2+n}} \right) \).

Answer: 1


Question: Let $\phi:U\longrightarrow V$ be a diffeomorphism between the open sets $U, V\subseteq \mathbb R^n$. Provided $J\phi(x)\neq 0$ for all $x\in U$, consider the map $J\phi:U\longrightarrow GL_n(\mathbb R)$ defined by $x\mapsto J\phi(x)$. Is this map continuous?

Answer: Yes


Question: Let $f: X \to X \times X$ be the diagonal inclusion map defined by $f(x) = (x, x)$, where $X$ is a manifold in $\mathbb{R}^m$. Show that the derivative of $f$ at a point $x$, denoted $df_x$, maps a tangent vector $v \in T_x(X)$ to $(v, v) \in T_{(x,x)}(X \times X)$. Verify that $df_x(v) = (v, v)$.

Answer: df_x(v) = (v, v)


Question: Evaluate the integral \( \int_{0}^{\pi}\left\{\int_{y^2}^{\pi}\frac{y\sin x}{x} \, dx\right\}dy \).

Answer: 1


Question: Let $V$ be the vector space of polynomials in the variable $t$ of degree at most $2$ over $\mathbb{R}$. An inner product on $V$ is defined by \(\langle f,g \rangle = \int_0^1 f(t)g(t)\,dt .\) Let \(W=\operatorname{span}\{1+t^2,1-t^2\}\) and $W^{\perp}$ be the orthogonal complement of $W$. Which of the following conditions is satisfied for all $h\in W^{\perp}$? \((a)\) $h$ is an even function. \((b)\) $h$ is an odd function. \((c)\) $h(t)$ has a real solution. \((d)\) $h(0)=0$. \[ \] Determine the correct condition.

Answer: C


Question: Evaluate the limit: \[ \lim_{k \to \infty} \int_0^\infty \frac{1}{1+kx^{10}} \, dx. \]

Answer: 0


Question: What is the probability that no card remains in its initial position after a standard 52-card deck is well shuffled?

Answer: \dfrac{1}{e}


Question:
Solve the system of congruences for \(x\) in \(\mathbb{Z}\):
\[
\begin{align*}
x &\equiv 2 \pmod{5}, \\
x &\equiv 1 \pmod{8}, \\
x &\equiv 7 \pmod{9}, \\
x &\equiv -3 \pmod{11}.
\end{align*}
\]

Answer: 1537


Question: Let $I$ be an ideal in the polynomial ring $R = k[x,y]$. Consider the algebraic closure $\overline{k}$ of $k$ and the ring $S = \overline{k}[x,y]$. Determine whether $IS \cap R = I$.

Answer: IS \cap R = I


Question:
Find a number \( a \) such that:
1. \( a \equiv 7^{3003} \pmod{3750} \)
2. \( 1 \leq a \leq 5000 \)
3. \( a \) is not divisible by 7.

Answer: 4093


Question: Compute the expected value \( E[e^{vS}] \), where \( v \) and \( S \) are two independent random variables, each exponentially distributed with parameters \( \lambda_1 \) and \( \lambda_2 \), respectively.

Answer: \infty


Question: If \( \gcd(f,g)=1 \) in \( R[X] \), is it true that \( \gcd(f,g)=1 \) in \( K[X] \)? Assume \( R \) is an integral domain and \( K \) is its field of fractions. Justify your answer.

Answer: Yes


Question:
Determine the maximum value of \( \alpha \) such that for any continuously differentiable function \( f: [0,1] \rightarrow \mathbb{R} \), the following inequality holds:
\[
\left|\frac{f(0)+f(1)}{2}-\int_{0}^{1} f(x) \, dx\right| \leq \alpha \max_{x \in [0,1]}\left|f^{\prime}(x)\right|.
\]

Answer: \dfrac{1}{4}


Question:
Evaluate the following double integral:
\[
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \exp \left( -\frac{A}{2}(x-y)^2 + B (x-y) \right) dx \, dy
\]
where \(A\) and \(B\) are real and positive constants. Determine the value of this integral.

Answer: \infty


Question: Determine the number of incongruent solutions to the congruence equation \(x^2 \equiv 49 \pmod{10^6}\).

Answer: 8


Question: Given the stochastic matrix \(P = \begin{bmatrix} 0 & 1/2 & 1/2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1/4 & 3/4 & 0 & 0 & 0 \\ 1/3 & 1/3 & 0 & 1/3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1/6 & 2/3 & 1/6 & 0 \\ 0 & 0 & 0 & 2/3 & 0 & 0 & 1/3 \\ 0 & 0 & 0 & 1/6 & 0 & 0 & 5/6 \\ 0 & 1/2 & 0 & 0 & 0 & 0 & 1/2 \end{bmatrix}\), calculate the probability \(\mathbb{P}_4(H^1<H^7)\), which is the probability of reaching state 1 before state 7 when starting from state 4.

Answer: 0


Question: Find the value of the infinite series \( \sum_{n=0}^{\infty} (-1)^n \) using Ramanujan's summation method. Express your answer in a boxed form.

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true or false: If \((na_{n})_{n=1}^{\infty}\) converges to 0, then the series \(\sum_{n=1}^{\infty}a_{n}\) is convergent.

Answer: B


Question: Let $A$ be a $C^*$-algebra, and $R: A \to A$ be its right multiplier. Determine if the existence of $b \in A$ such that for all $a \in A$, $R(a)b = a$ implies that $A$ is unital.

Answer: Yes


Question: Let \(\alpha\) be a non-real fifth root of unity and \(\beta\) be a non-real fourth root of unity. Evaluate the expression \((1+\alpha)(1+\beta)(1+\alpha^2)(1+\beta^2)(1+\alpha^3)(1+\beta^3)(1+\alpha^4)\).

Answer: 0


Question: Given the equation \[ \frac{\left(1^4+\frac{1}{4}\right)\left(3^4+\frac{1}{4}\right)\ldots\left((2n-1)^4+\frac{1}{4}\right)}{\left(2^4+\frac{1}{4}\right)\left(4^4+\frac{1}{4}\right)\ldots\left((2n)^4+\frac{1}{4}\right)}=\frac{1}{k_1n^2+k_2n+k_3} \] find the value of \(k_1 - k_2 + k_3\).

Answer: 5


Question: Let $f: [a,b] \to \mathbb{R}$ be a function, and let $a = t_0 < t_1 < \ldots < t_n = x$ be a partition $P$ of $[a,x] \subseteq [a,b]$. Define the arc length of $f$ over $P$ as \( l(f,P) = \sum_{i=1}^{n} \sqrt{(t_i - t_{i-1})^2 + (f(t_i) - f(t_{i-1}))^2} \). Let \( L(x) = \sup l(f,P) \) where the supremum is taken over all partitions of $[a,x]$. Let $d(x)$ be the Euclidean distance between $(a,f(a))$ and $(x,f(x))$. Suppose $L(b)$ is defined. Prove that \( \lim_{x \to a} \frac{L(x)}{d(x)} = 1 \).

Answer: 1


Question: Given that \(|z+1| \le 1\) and \(|z^2+1| \le 1\) for a complex number \(z\), prove that \(|z| \le 1\).

Answer: |z| \le 1


Question: Find the remainder when $7777^{5555}$ is divided by $191$. Use Fermat's Little Theorem to simplify the calculation.

Answer: 185


Question: A standard deck of 52 cards is distributed equally among 4 players, with each player receiving 13 cards. What is the probability that one of the players receives 13 cards, each with a different rank? Express your answer as a fraction or a decimal rounded to four decimal places.

Answer: 0.0004


Question: Evaluate the infinite series \( \sum_{n=1}^{\infty}\frac{2n-1}{2^n} \).

Answer: 3


Question: Let \( f(x) \) be a function defined on the interval \([0, 2\pi]\) such that \( f(x) \) yields positive real numbers and satisfies the normalization condition \( \int_0^{2\pi} f(x) \, dx = 1 \). Additionally, suppose \( \int_0^{2\pi} f(x) \sin(x) \, dx = r \), where \( r \) is a real number between -1 and 1. Determine the maximum value of \( \int_0^{2\pi} f(x) \cos(x) \, dx \) across all functions \( f(x) \) that satisfy these conditions.

Answer: \sqrt{1 - r^2}


Question: Determine the smallest positive integer \( k \) such that \( k^{2018} + 2018 \) is a prime number.

Answer: 3


Question: Determine the conditions that the matrix \( A \in \mathbb{R}^{n \times n} \) must satisfy such that \( \lim_{k \to \infty} ||e_k|| = 0 \), where the sequence is defined by \( e_{k+1} = A e_k \) and \( e_0 \) is given.

Answer: \rho(A) < 1


Question: Find \( \lim_{t \to \infty} P (X>t+ \frac{x}{t} | X > t) \) where \( X \sim N(0,1) \).

Answer: e^{-x}


Question: Let $S$ be the vector space of real $n \times n$ skew-symmetric matrices. For a nonsingular matrix $A$, compute the determinant of the linear map $T_A : S \rightarrow S$ where $T_A(X) = AXA^{T}$. What is the determinant of this map?

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


Question:
Determine all continuous real-valued functions \( f \) defined on the set of real numbers that satisfy the following conditions:

1. \( f \) is continuous at \( x = 0 \).
2. \( f(x) = f\left(\frac{x}{1-x}\right) \) for all \( x \neq 1 \).

Answer: f(x) = c


Question: Consider an urn containing balls labeled from 1 to n. You draw all the balls one by one without replacement. What is the probability that the ball labeled k is drawn exactly on the k-th draw?

Answer: \dfrac{1}{n}


Question: Evaluate the limit: \[ \lim_{x\to1}(3 \sqrt[3]{x}-2\sqrt{x})^{\frac{1}{\ln x}} \]

Answer: 1


Question: Is every ring homomorphism from a non-zero commutative unital ring to the integers surjective? Justify your answer.

Answer: Yes


Question: Prove that \( \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n = e \), where \( e \) is defined as the unique positive number such that \( \int_1^e \frac{1}{x} \, dx = 1 \).

Answer: e


Question: Driver A has been leading archrival B by a steady 3 miles. Two miles from the finish, driver A runs out of gas and decelerates at a rate proportional to the square of his remaining speed. One mile later, driver A's speed is exactly halved. If driver B's speed remains constant, determine who won the race.

Answer: A


Question: Evaluate the limit \( \lim_{n \to \infty} n^2 \sum_{k=1}^n \frac{1}{(n^2+k^2)^2} \) using the concept of Riemann sums.

Answer: 0


Question: Determine whether the following statement is true or false: If $R$ is a Euclidean domain, then any subring $S$ of $R$ is a unique factorization domain.

Answer: B


Question: Find the splitting field $K$ of the polynomial $x^{12} - 9$ over the field of rational numbers $\mathbb{Q}$, and determine the degree of the extension $[K: \mathbb{Q}]$. Provide your answer as a single number.

Answer: 12


Question: A casino produces coins which, when flipped, will land on heads with probability $p$. The coins are weighted, and for some $x \in [0, 1],$ $P[p \leq x] = x^{3}.$ At a game in the casino, a coin is flipped 4 times and results in 4 heads. What is the probability that the next flip will be heads?

Answer: \dfrac{7}{8}


Question:
Evaluate the limit of the series as \( n \to \infty \):
\[
\lim_{n\to\infty} \left[\frac{n^2}{{(n^2+1)}^{3/2}} + \frac{n^2}{{(n^2+2)}^{3/2}} + \dots + \frac{n^2}{{(n^2+(n+1)^2)}^{3/2}}\right]
\]

Answer: \infty


Question: Find the value of $i^i$, where $i$ is the imaginary unit.

Answer: e^{-\pi/2}


Question: Determine if the limit of the sequence \( a_n = \left(1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}\right)\log \left(\frac{n+1}{n}\right) \) exists. If it does, calculate the limit.

Answer: 1


Question: Find the remainder when \(x^{1001}\) is divided by \(x^4 + x^3 + 2x^2 + x + 1.\)

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


Question: Suppose \( f_i \) is a sequence uniformly bounded in \( W^{1,p} \) for some \( +\infty > p > 1 \). By passing to a subsequence, assume \( f_i \) converges weakly to \( f \) in \( W^{1,p} \). If \( 2p > n \), can we conclude that \( f_i \) converges to \( f \) in \( L^{2p} \)?

Answer: Yes


Question: Determine the number of distinct roots of the equation \(3x^2-12x+11+\frac{1}{5}(x^3-6x^2+11x-6)\) within the interval \((1,2)\).

Answer: 1


Question: Determine whether the matrix \( A = \begin{pmatrix} -2 & 0 \\ 0 & -1 \end{pmatrix} \) can be expressed as \( A = \exp(B) \) for some matrix \( B \). Justify your answer.

Answer: Yes


Question: Let \( \mathbf{v}:(a,b)\to\mathbb{R}^2 \) be a continuous function such that \( ||\mathbf{v}(t)||=1 \) for all \( t \in (a,b) \). Determine if there exists a continuous function \( \mathbf{r}:(a,b)\to\mathbb{R}^2 \) such that \( \mathbf{v}(t_0)=\lim\limits_{t\searrow t_0} \frac{\mathbf{r}(t)-\mathbf{r}(t_0)}{||\mathbf{r}(t)-\mathbf{r}(t_0)||} \) for all \( t_0\in (a,b) \).

Answer: Yes


Question: Evaluate the integral \( \int_0^1 f(x) \, dx \), where the function \( f \) is defined as follows: \[ f(x) = \begin{cases} 1, & \text{if } x \in \mathbb{R} \setminus \mathbb{Q} \\ 2, & \text{if } x \in \mathbb{Q} \end{cases}. \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{x\sin(nx)}{1-\frac{\sin x}{x}} \]

Answer: 6n


Question: Given a random vector \((X,Y)\) with density \(f(x,y)=\frac{6}{7}(x^2+\frac{xy}{2})\) for \(0<x<1\) and \(0<y<2\), find the conditional probability \(\mathbb{P}(Y>\frac{1}{2}|X<\frac{1}{2})\). Use the conditional probability formula \(\mathbb{P}(Y>\frac{1}{2}|X<\frac{1}{2})=\frac{\mathbb{P}((Y>\frac{1}{2})\cap (X<\frac{1}{2}))}{\mathbb{P}(X<\frac{1}{2})}\) and set up the appropriate integrals to solve for this probability.

Answer: \dfrac{69}{80}


Question: Consider the space $X = \mathbb{R} \times (0, +\infty)$ with the topology induced from $\mathbb{R}^2$. Define an equivalence relation $R$ on $X$ by $$(x_1, y_1)R(x_2, y_2) \iff x_1 = x_2.$$ Is the quotient space $X/R$ homeomorphic to $\mathbb{R}$?

Answer: Yes


Question: Provide an example of a transcendental element \( \alpha \) such that a field monomorphism \( \sigma: F \rightarrow \bar{F} \) cannot be extended to a field monomorphism \( \tau: F(\alpha) \rightarrow \bar{F} \), where \( E/F \) is a field extension, \( \alpha \in E \) is transcendental over \( F \), and \( \bar{F} \) is the algebraic closure of \( F \).

Answer: \pi


Question: In a Cartesian coordinate system, you can move from point \((a,b)\) to \((a+1,b)\), \((a,b+1)\), or \((a+1,b+1)\). However, you must ensure that no right angles are formed when drawing lines during the move. Determine the number of ways to move from \((0,0)\) to \((5,5)\) under these conditions.

Answer: 83


Question: Find a subspace \( Y \subseteq l^2 \) such that \( Y \) is not closed. Define \( Y = \{ x = (x_n) \in l^2 \mid x_n = 0 \text{ except for a finite number of indices} \} \). Prove that \( Y \) is not closed in \( l^2 \).

Answer: Y


Question: Let $H = H_1 \times H_2$ be a closed subgroup of a second-countable locally compact Hausdorff group $G = G_1 \times G_2$, with $H_i \leq G_i$. Let $\chi = \chi_1 \otimes \chi_2$ be a unitary character of $H$, where $\chi_i$ is a character of $H_i$. Consider the induced representation $I(\chi) = \operatorname{Ind}_H^G\chi$, which is the Hilbert space of measurable functions $f: G \rightarrow \mathbb{C}$ satisfying $f(hg) = \chi(h)f(g)$ and the norm condition $\int_{H \backslash G} |f(g)|^2 \, dg < \infty$. Let $V \subset I(\chi)$ be the linear space spanned by functions of the form $(g_1,g_2) \mapsto f_1(g_1)f_2(g_2)$, for continuous $f_i \in I(\chi_i)$. Is $V$ norm-dense in $I(\chi)$?

Answer: Yes


Question: Determine whether every group action can be represented as a conjugation by unitaries. Specifically, let $G$ be a countable discrete group, and define $l^\infty(G) := \{f \colon G \to \mathbb{C} \mid \|f\|_\infty < \infty\}$. Consider the group action of $s \in G$ on $f \in l^\infty(G)$ defined by $(s \mathbin{.} f)(g) = f(s^{-1}g)$ for all $g \in G$. Is it true that for every $s \in G$, there exists a $u_s \in B(l^2(G))$ such that $s \mathbin{.} f = u_s f u_s^*$, given that $l^\infty(G)\subseteq B(l^2(G))$?

Answer: Yes


Question: Given a distribution $P$ and a set of distributions $Q$ such that each $Q$ is at most $d$ distance from $P$ in terms of Total Variation Distance, determine the maximum distance between $P$ and the convex combination $S$ of all such distributions $Q$. Assume the Total Variation Distance is used as the metric.

Answer: d


Question: Determine the number of pairs of subsets \((A, B)\) of the set \(X = \{1, 2, 3, \ldots, 100\}\) such that the symmetric difference \(A \Delta B = \{2, 4, 6, \ldots, 100\}\).

Answer: 2^{100}


Question: Determine the interval of values for $p$ such that the integral \( \int_1^{\infty} \frac{\log{x}}{x^p} \,dx \) converges.

Answer: (1, \infty)


Question: Find all functions \( f: \mathbb{N}_{+} \to \mathbb{N}_{+} \) such that for all positive integers \( m \) and \( n \), the divisibility condition \( m^2 + f(n) \mid mf(m) + n \) holds.

Answer: f(n) = n


Question: Find the coefficient of \(x^{15}\) in the expression \(\frac{(x-x^7)^4}{(1-x)^5}\).

Answer: 861


Question: Evaluate the limit: $$\lim_{n \to \infty} \int_0^{\pi/2} \frac{x \sin^n x}{\sqrt{1+\sin^2 x}}\, dx.$$

Answer: 0


Question:
Let $V$ be a $
abla{C}$-vector space of dimension $N \geq 2$, and let $d$ and $l$ be positive integers with $l < N$. Consider a linear subspace $U \subseteq S^d(V)$ of codimension $k=\binom{l+d-1}{d}$. Suppose there exists a basis of $U$ of the form $\{u_1^{\vee d},\dotsc, u_r^{\vee d}\}$ such that $\operatorname{span}\{u_1^{\vee d-1},\dotsc, u_r^{\vee d-1}\}=S^{d-1}(V)$, where $r=\binom{N+d-1}{d}-k$. Does there exist a linear subspace $W \subseteq V$ of dimension $l$ for which $S^d(V)=U \oplus S^d(W)$? Here, $\vee$ denotes the symmetric product.

Answer: Yes


Question: Given four points $O, A, B, C$ in the plane such that \( \overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC} = \overrightarrow{0} \) and \( 2|\overrightarrow{OA}| = 2|\overrightarrow{OB}| + |\overrightarrow{OC}| \), and the angle \( \theta \) between \( \overrightarrow{AB} \) and \( \overrightarrow{OC} \) satisfies \( \tan{\theta} = \frac{3}{2} \), find the value of \( \frac{S_{\Delta ABC}}{\overrightarrow{OA} \cdot \overrightarrow{OB}} \).

Answer: -2


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

Answer: -\infty


Question: A sample from a normal population has a variance of 4.0. Determine the size of the sample needed so that the sample mean deviates from the population mean by no more than 2.0 with a probability of at least 0.95.

Answer: 4


Question: Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a differentiable function with $f(0, -2) = 3$ and $\nabla f(0, -2) = (-6, 2)$. Define $g(t) = \|(t, t^2, f(3t, t-2))\|$. Find $g'(0)$. Provide your answer as a single value.

Answer: -16


Question: Determine which is larger: $100^{300}$ or $300!$? Provide a method to compare these two values without using a calculator.

Answer: 300!


Question: Compute the expected value \( \mathrm{E} [ B_s B_t^2 ] \) for a Brownian motion \( B \) where \( s \leq t \).

Answer: 0


Question: Calculate the Hamming distance of a linear block code with parameters (7,2), where the code is 7 bits long, has 2 data bits, and 5 error correction bits.

Answer: 4


Question: Determine the set $A = \{x > 0 \mid \sum_{n=1}^{\infty} (\sqrt[n]{x} - 1) \text{ converges}\}$. What is the set $A$?

Answer: \{1\}


Question: Evaluate the integral \( I = \int_V (\nabla \cdot F) \, dv \) where \( F = rr = xi + yj + zk \) and \( V \) is the volume of a sphere with radius \( R \) centered at the origin. Use both direct computation and the divergence theorem to find the value of the integral.

Answer: 4\pi R^3


Question: Consider a disk with two holes, resulting in three boundary circles. Demonstrate that there are only two distinct ways to identify these three boundary circles.

Answer: 2


Question: Show that \( \sqrt[3]{5} \) is not contained in any cyclotomic extension of \( \mathbb{Q} \). Determine the Galois group of the polynomial \( x^3 - 5 \) over \( \mathbb{Q} \) and use this information to justify your conclusion.

Answer: S_3


Question: Determine whether the topological semigroup \( T = [0, \frac{1}{2}] \cup \{1\} \) with the usual topology and multiplication operation satisfies the SFC-condition. The SFC-condition states that for every finite set \( H \subseteq T \) and every \( \epsilon > 0 \), there exists a finite set \( K \subseteq T \) such that \( |K \setminus tK| < \epsilon |K| \) for all \( t \in H \).

Answer: Yes


Question: Compute the limit: \[ \lim\limits_{N \to +\infty} \sqrt{N+1} \log \left(1+\frac{x}{N+1}\right) \] where \( x \ge 0 \) is fixed.

Answer: 0


Question: Use Stokes's Theorem to evaluate the integral \( \iint (\nabla \times V) \cdot n \, d\sigma \) over the part of the surface \( z = 9 - x^2 - 9y^2 \) above the xy-plane, where \( V = 2xy\ \mathbf{i} + (x^2-2x)\ \mathbf{j} -x^2z^2\ \mathbf{k} \).

Answer: -6\pi


Question: Find the degree of the field extension \( \mathbb{Q}(i, \sqrt{-1 + \sqrt{-3}}) : \mathbb{Q} \).

Answer: 8


Question: Calculate the fundamental group of the space \(X = \mathbb{R}^3 \setminus \{\text{union of } n \text{ lines passing through the origin}\}\).

Answer: F_{2n-1}


Question: Let $(M, \omega)$ be a symplectic manifold and $N$ an isotropic submanifold. For a point $p \in N$, determine if it is always possible to find coordinates $(x_1, \ldots, x_n, y_1, \ldots, y_n)$ in a neighborhood $U$ of $p$ such that both $\omega = \sum dx_i \wedge dy_i$ (Darboux coordinates) and \[ N \cap U = \{(x_1, \ldots, x_n, y_1, \ldots, y_n) : x_{k+1} = \cdots = x_n = y_1 = \cdots = y_n = 0\} \] where $k = \dim N$.

Answer: Yes


Question:
Suppose the limit \(L=\lim_{n\to\infty}\sqrt n\int_0^1\frac{\mathrm dx}{(1+x^2)^n}\) exists and is larger than \(\frac{1}{2}\). Determine the range in which \(L\) falls:
A. \(\frac{1}{2}<L<2\)
B. \(2<L<3\)
C. \(3<L<4\)
D. \(L\ge4\)

Answer: A


Question: Evaluate the contour integral \( \int_{\mathcal{C}} \frac{1}{z^3(z+4)} \, dz \), where \( \mathcal{C} \) is the circle \( |z + 2| = 3 \) described in the anti-clockwise direction in the complex plane.

Answer: 0


Question: Let $E$ be the set of all nonnegative continuous functions $f:[0,1]\to \mathbb{R}$ such that \(f(x)\,f(y)\ge |x-y|\) for all \(\{x,y\}\subset [0,1]\). Find the minimum value of \(\int_0^1f(x) \,dx\) for \(f \in E\).

Answer: \dfrac{3}{4}


Question: For which values of $c \in \mathbb{R}$ can the level set $M_c = \{(x, y) \mid x^2 + 2xy + 3y^2 = c\}$ be described locally as a $C^1$ graph at every point?

Answer: c > 0


Question: Is it possible to construct an open set with measure less than a given positive real number that contains the Cantor set?

Answer: Yes


Question: Consider the matrix $M \in \mathbb{R}^{n \times n}$ defined by $M_{ij} = 1$ if $|i-j| \leq k$, and $0$ otherwise. Is it true that the spectral norm $\|M\|_2 \leq O(k)$?

Answer: Yes


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

Answer: \ln 2


Question:
Consider a boolean function \( f: \{0,1\}^n \to \{0,1\} \) computed by a logical circuit comprising only binary AND and OR gates, without any feedback. Determine which of the following scenarios is NOT possible for such a function:

1. \( f(0,0,...,0) = f(1,1,...,1) = 0 \)
2. \( f(0,0,...,0) = f(1,1,...,1) = 1 \)
3. \( f \) is a MAJORITY function.
4. \( f \) is a PARITY function.
5. \( f \) outputs 1 at exactly one assignment of the input bits.

Identify the scenario that cannot occur.

Answer: 4


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{x}{\ln (x^2+2x+4) - \ln(x+4)} \] without using L'Hôpital's Rule.

Answer: 4


Question: In \(\mathbb{R}^n\), a path-connected subset \(A\) is also arc-connected. Given any finite subset \(B\) of \(A\), is there an arc that passes through all the points of \(B\)?

Answer: Yes


Question: Evaluate the series: \[ \sum_{i=0}^{\infty} \left(1-(-1)^{\frac{i(i+1)}{2}}\right)\cdot \left(\frac{1}{3}\right)^i \]

Answer: \dfrac{9}{10}


Question: Find the sum of the power series \( \sum\limits_{n=0}^\infty \frac{x^n}{(n!)^2} \).

Answer: I_0(2\sqrt{x})


Question: Evaluate the integral \( \int_{\Gamma} \frac{f'(z)}{f(z)} \, dz \) where \( f(z) = z^3 e^z \) and \( \Gamma \) is the circle \( z = e^{i\theta} \) with \( \theta \) varying from \( 0 \) to \( 4\pi \).

Answer: 12\pi i


Question:
Find all integer solutions \((a, b, c, d, e)\) to the following system of equations:
\[
\begin{align*}
  a^2 &= a + b - 2c + 2d + e - 8, \\
  b^2 &= -a - 2b - c + 2d + 2e - 6, \\
  c^2 &= 3a + 2b + c + 2d + 2e - 31, \\
  d^2 &= 2a + b + c + 2d + 2e - 2, \\
  e^2 &= a + 2b + 3c + 2d + e - 8.
\end{align*}
\]

Answer: (3, 2, 1, 5, 4)


Question: Consider an ellipse $E_1$ with semi-major axis $r_x$ and semi-minor axis $r_y$. Define another ellipse $E_2$ with semi-major axis $r_x + D$ and semi-minor axis $r_y + D$. If a point $P_a$ lies within $E_2$, does this imply that the minimum distance from $P_a$ to $E_1$ is less than $D$?

Answer: Yes


Question:
Determine which of the following integrals correctly represents the convolution of the functions \( x(\alpha t) \) and \( y(t) \):

1. \( \int_{-\infty}^{\infty} y(t) x(\alpha(t-\tau)) \, d\tau \)
2. \( \int_{-\infty}^{\infty} y(t) x(\alpha t - \tau) \, d\tau \)

Answer: 1


Question: Let \( f:[0,1]\rightarrow \mathbb{R} \) be a differentiable function such that \( f(0)=f(1)=0 \) and satisfies the inequality \( f''(x)-4f'(x)+4f(x)\ge e^{2x} \) for \( x\in [0,1] \). Find the minimum value of \( \frac{f'(x)}{f(x)} \) for \( x \in \left( 0, \frac{1}{2} \right) \).

Answer: 2


Question: Let \( F/K \) be a finite extension field with \( \text{Aut}_K(F) = \{f_1, \ldots, f_n\} \). Suppose there exists an element \( a \in F \) such that \( \{f_1(a), \ldots, f_n(a)\} \) forms a basis for \( F \) over \( K \). Is the extension \( F/K \) a Galois extension?

Answer: Yes


Question: Determine the largest possible size of a deck for which the following card-guessing puzzle is solvable: Alice is dealt five random cards from the deck, looks at them, and shows four of them to Bob in a chosen order. Using only these four cards and their order, Bob must determine the identity of the fifth card. Given that for a deck of size \( n \), the condition \( 4!\binom{n}{4} \geq \binom{n}{5} \) must hold, find the maximum value of \( n \).

Answer: 124


Question:
Determine the form of the set containing the real solutions to the inequality:
\[ \frac{\log_{2^{x^2+2x+1}-1}(\log_{2x^2 + 2x + 3}(x^2 - 2x))}{\log_{2^{x^2+2x+1}-1}(x^2 + 6x + 10)} \geq 0 \]
The set of all real solutions to this inequality is of the form:
\[(a) \ (a,b) \cup (b,c) \\
(b) \ (-\infty,a) \cup (c,\infty) \\
(c) \ (a,b) \]
for some real numbers \(a,b,c\) such that \(-\infty<a<b<c<+\infty\). Determine which form correctly represents the solution set.

Answer: A


Question: Let $x$ be an infinite set and let $\text{Part}(x)$ be the collection of all partitions of $x$. For a partition $P \in \text{Part}(x)$ and a subset $t \subseteq x$, define $P_{[t]} = \{p \in P : p \cap t \neq \emptyset\}$. The tiling relation on $\text{Part}(x)$ is defined by $P \triangleleft Q$ if and only if for all $S \subseteq P$, we have $\mathsf{card}(S) \leq \mathsf{card}(Q_{[\bigcup S]})$. Let $\mathcal{A} \subseteq \text{Part}(x)$ be a non-empty collection of partitions such that there is a set $M$ of subsets of $x$ where every member of $\mathcal{A}$ refines $M$. Is there a partition $Z \in \text{Part}(x)$ such that $Z$ refines $M$ and $Z \triangleleft A$ for all $A \in \mathcal{A}$?

Answer: Yes


Question: Find the radius of convergence for the Taylor expansion of the hyperbolic secant function, \( \operatorname{sech}(x) \), at \( x = 0 \).

Answer: \dfrac{\pi}{2}


Question: Suppose $A^T = -A$ and $AA^T$ is a positive definite symmetric matrix. Given that $AA^T = O\Lambda O^T$ with positive eigenvalues, define $\sqrt{AA^T} = O\sqrt{\Lambda}O^T$, where $\sqrt{\Lambda} = \text{diag}(\sqrt{\lambda_1}, \ldots, \sqrt{\lambda_n})$. Does $A$ commute with $\sqrt{AA^T}$, i.e., is $A\sqrt{AA^T} = \sqrt{AA^T}A$? Provide a justification for your answer.

Answer: Yes


Question: Let $G \subset \mathbb{C}^{n}$ be a domain and let $f$ be an injective holomorphic map from $G$ into a complex Hilbert space $H$. Is $f$ necessarily a homeomorphism?

Answer: Yes


Question: Let $A$ and $B$ be topological spaces and $f : A \rightarrow B$ be a function. A function $f$ is called locally continuous if every $a \in A$ has an open neighborhood $U \subseteq A$ such that $f : U \rightarrow B$ is continuous. Determine whether every locally continuous function is also continuous.

Answer: Yes


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

Answer: n


Question: Three numbers are chosen at random between 0 and 2. What is the probability that the difference between the largest number and the smallest number is less than \( \frac{1}{4} \)?

Answer: \dfrac{11}{256}


Question: Let \( f \) be a real-valued function with domain \( \mathbb{R} \) satisfying the equation \( f(x+k) = 1 + (2 - 5f(x) + 10f(x)^2 - 10f(x)^3 + 5f(x)^4 - f(x)^5)^{\frac{1}{5}} \) for all real \( x \) and some positive constant \( k \). Find the period of \( f(x) \) if it is periodic.

Answer: 2k


Question:
Let \( f(x) \) be a polynomial function of degree \( n \) with \( n \) real and distinct roots. Define \( g(x) = f'(x) + 100f(x) \). What is the minimum number of roots that \( g(x) \) must have?

(a) \( n \)
(b) \( n+1 \)
(c) \( n-1 \)
(d) None of these

Answer: A


Question: Evaluate the line integral \( \int_{\gamma} |z-1| |dz| \) where \( \gamma(t) = e^{it} \) for \( 0 \leq t \leq 2\pi \).

Answer: 8


Question: Let \( f: \mathbb{N}^{*} \to \mathbb{N}^{*} \) be a function satisfying the equation: \[ \frac{f(x+1)f(x)-2x}{f(x)} = \frac{2f^2(x)}{x+f(x)} - 1. \] Determine all such functions \( f \).

Answer: f(x) = x


Question: Evaluate the limit: \[ \lim _{n\rightarrow \infty }\sin\biggl(\frac{1}{n}\biggr)\cdot \sum_{k=1}^{n}\sec^2\biggl(\frac{3k-2}{3n}\biggr) \]

Answer: \tan 1


Question: Find the integer \(x\) such that \(x^{11} \equiv 10 \pmod{42}\).

Answer: 40


Question: Simplify the expression \( \frac{( b \times (a \times b)) \cdot (b \times (a\times b))}{(a\times b) \cdot (a\times b)} \), where \(a\) and \(b\) are 3-dimensional vectors, and express it without a denominator.

Answer: b \cdot b


Question: Determine the number of Sylow 3-subgroups of the special linear group $G = SL(2,\mathbb{F_{3}})$, where $G$ is the group of $2 \times 2$ matrices with determinant $1$ over the field of order $3$, $\mathbb{F_{3}}$. Given that $|G|=24$, use Sylow's theorems to find the possible values for the number of Sylow 3-subgroups.

Answer: 4


Question:
Can the value of the Riemann zeta function at 0, \(\zeta(0)=-1/2\), be deduced from the identity \(E(z)=E(1-z)\), where
\[E(z)=\pi^{-z/2}\Gamma(z/2)\zeta(z)?\]

Answer: -\dfrac{1}{2}


Question: Solve the integral equation: \[ \int_0^x e^{-x} u(y)\, dy = e^{-x} + x - 1. \]

Answer: u(x) = x e^x


Question: Find the infimum \( \alpha \) for which the series \( \sum_{n=1}^\infty \frac{(2n)!}{n!(\alpha n)^n} \) converges.

Answer: \dfrac{4}{e}


Question: Evaluate the definite integral \( I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1+\sin x} \, dx \). Verify if the result is indeed 1, considering the steps involve terms that appear to cancel out due to division by zero.

Answer: 1


Question: Determine if the operator \(I + M\), where \(M(f)(x) = x^2f(x)\) for \(f \in L^2([0,1])\), is invertible. If it is, find the norm \(|| (I+M)^{-1} ||\).

Answer: 1


Question:
Given a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are positive integers, determine if it is possible to paint a point \( C \) on a segment \( AB \) such that \( \frac{AC}{CB} = \frac{p}{q} \) using the following operations:
1) Choose two painted points \( X, Y \) and paint the midpoint of \( XY \).
2) Choose two painted points \( X, Y \), a positive integer \( n \), and paint the point \( Z \) on the segment \( XY \) such that \( \frac{XZ}{ZY} = \frac{n}{n+1} \).

Is it true that for any fraction \( \frac{p}{q} \), Matías can paint the point \( C \) such that \( \frac{AC}{CB} = \frac{p}{q} \)?

Answer: Yes


Question: Let \( A, B \in M_2(\mathbb{R}) \) be real \( 2 \times 2 \) matrices such that \( A^2 + B^2 + 2AB = 0 \) and \( \det A = \det B \). Compute \( \det(A^2 - B^2) \).

Answer: 0


Question: Calculate the limit: \[ \lim _{n\rightarrow \infty}((4^n+3)^{1/n}-(3^n+4)^{1/n})^{n3^n} \]

Answer: e^{-12}


Question: Let \( f \) be a homeomorphism between metric spaces. Is the preimage of an open ball under \( f \) necessarily an open ball?

Answer: No


Question:
Determine the value of the constant \( A \) such that the limit \( \lim_{(x,y) \to (0,0)} f(x,y) \) exists, where the function \( f(x,y) \) is defined as follows:

\[
f(x,y) = \begin{cases}
\frac{x^4+y^4}{y(x^2+y^2)}, & \text{if } y \neq 0 \\
A, & \text{if } y = 0
\end{cases}
\]

Answer: 0


Question: Determine the number of zeros of the polynomial \( f(z) = z^{13} - 2z^7 + 10z^3 - z + 4 \) within the annular region \( P = \{ z : 1 \leq |z| < 2 \} \).

Answer: 10


Question: For which values of \( a \) are the flows of \( \overrightarrow{x}' = A \overrightarrow{x} \) and \( \overrightarrow{y}' = B \overrightarrow{y} \) topologically conjugate, given that \( A \) is a hyperbolic matrix such that all solutions of \( \overrightarrow{x}' = A \overrightarrow{x} \) tend to the origin as \( t \to \infty \), and \( B = \begin{bmatrix}a-3 & 5 \\ -2 & a+3\end{bmatrix} \)?

Answer: a < 0


Question: Let $\mathcal O$ be a normalized discrete valuation ring with a surjective valuation $v: \text{Frac}(\mathcal O) \rightarrow \mathbb{Z} \cup \{\infty\}$. Consider a proper subring $R \subset \mathcal O$ with Krull dimension greater than 1. Determine whether $v(R^\times) = 0$. Justify your answer.

Answer: Yes


Question: Let \( f : \mathbb{C} \longrightarrow \mathbb{C} \) be a holomorphic function such that for all \((n,m) \in \mathbb{Z}^2\), \( f(n+im) = 0\). Is \( f \) constant?

Answer: No


Question: Determine the sign of the integral \( \int{xyz\space dV} \) over the region \( W \), where \( W \) is bounded by the cone \( z=\sqrt{x^2+y^2} \) and \( z=2 \).

Answer: 0


Question: In triangle $ABC$, draw the altitude $AH$. Let $HM \perp AB$ and $HN \perp AC$. If the perimeter of the pedal triangle $DEH$ of triangle $ABC$ is 26, calculate the length of segment $MN$.

Answer: 13


Question:
Consider the natural representation \(\pi\) of the group \(D_4\) on the vector space with orthonormal basis \(\{e_1, e_2, e_3, e_4\}\). This representation gives rise to a \(C^*\) algebra \(A = \tilde{\pi}(\mathbb{C}(D_4))\), where \(\tilde{\pi}: \mathbb{C}(D_4) \rightarrow B(H)\) is a *-homomorphism. The matrices for rotation and reflection are given by:

\[ \mathrm{Rotation} = \rho = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \]

\[ \mathrm{Reflection} = \sigma = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix} \]

Determine the dimension of the commutant \(A'\) of \(A\).

Answer: 3


Question: Determine whether the function \( f(x) = \frac{x}{2} + \frac{1}{x} \) is a contraction map on the set \( X = \{ x \in \mathbb{R} : x \ge 1 \} \) with respect to the usual metric on \( \mathbb{R} \). If it is a contraction map, find the smallest contraction constant \( \lambda \).

Answer: \dfrac{1}{2}


Question:
Consider the damped wave equation on \(\mathbb{R}^{2}/2\pi\mathbb{Z}^{2}\):
\[
u_{tt} - \Delta_{x}\nu + a\nu_{t} = 0\]
with initial conditions \(\nu(x,0) = 0\) and \(\nu_{t}(x, 0) = f(x)\), where \(t \geq 0\), \(x \in \mathbb{R}^{2}/2\pi\mathbb{Z}^{2}\), \(a \in (0, 1)\), and \(f \in C^{\infty}(\mathbb{R}^{2}/2\pi\mathbb{Z}^{2})\). Define the energy function:
\[E(t) := \frac{1}{2}\int_{\mathbb{R}^{2}/2\pi\mathbb{Z}^{2}}|\nabla \nu|^{2} + |\nu_{t}|^{2}\, dx.\]
Given that:
\[E'(t) = -a\int_{\mathbb{R}^{2}/2\pi\mathbb{Z}^{2}}\nu_{t}^{2}\, dx,\]
determine whether \(E(t)\) must decrease exponentially over time.

Answer: Yes


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

Answer: 0


Question: Determine the number of homomorphisms from the symmetric group $S_5$ to the alternating group $A_6$. Consider the possible kernels of these homomorphisms and use the first isomorphism theorem to aid in your calculation.

Answer: 46


Question: Find the argument of the complex expression \( \left( \frac{i+e^{i\Theta x} }{i-e^{i\Theta x }}\right)^{2} \), where \( \Theta \geq 0 \).

Answer: \pi


Question: Determine whether the congruence \(x^2 - 6x - 13 \equiv 0 \pmod{127}\) has solutions.

Answer: Yes


Question: Solve the congruence equation: \(x^3 + 2x + 2 \equiv 0 \pmod{25}\).

Answer: 13


Question: Determine the total number of 7-digit natural numbers whose digits sum to 20.

Answer: 139068


Question: In a Galois Field of order 8, determine the maximum order of an element in the multiplicative group of the field.

Answer: 7


Question: Find all natural number solutions to the system of equations: \[ \frac{1}{a} = \frac{1}{b} + \frac{1}{c} - \frac{1}{abc} \] and \[ a^2 + b^2 = c^2. \]

Answer: (8, 15, 17)


Question: Evaluate the line integral \( \int_C (y+\sin x)dx + (z^2+\cos y)dy +x^3dz \) where \( C \) is the curve given parametrically by \( r(t) = \langle \sin t, \cos t, \sin 2t \rangle \), for \( 0\leq t\leq 2\pi \).

Answer: \pi


Question: Given that $\alpha, \beta, \gamma, \delta$ are distinct roots of the equation $x^4 + x^2 + 1 = 0$, find the value of $\alpha^6 + \beta^6 + \gamma^6 + \delta^6$. Express your answer as a single number.

Answer: 4


Question: Let \( f \) be a differentiable function satisfying \( f(x+y)=(f(x))^{\cos y}\cdot (f(y))^{\cos x} \) for all \( x,y \in \mathbb{R} \). Given that \( f(0)=1 \) and \( f'(0)=\ln 2 \), evaluate the integral \( \int_{\frac{\pi}{2}}^{\frac{17\pi}{2}} \frac{f(x)}{f(x)+f(\pi /2-x)} \, dx \) and find the value of \( k \) such that the integral equals \( k\pi \).

Answer: 4


Question: Evaluate the following expression: \[ G = \lim_{n\to\infty}(n+1)\int_{0}^{1} x^{n} f(x) \, dx \] where $f(x)$ is a real-valued continuous function on the interval $[0, 1]$. Determine the value of $G$. (Hint: $G = f(1)$)

Answer: f(1)


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

Answer: 0


Question: Determine whether there exists a bijective map \( f: \mathbb{Z} \to \mathbb{Q} \) such that for all integers \( n \leq 0 \), \( f(n) \geq 0 \).

Answer: Yes


Question: Let \(T: V \to V\) be a linear transformation such that \(T^2 = T\). Find the expression for \(e^T\), where \(e^T\) is the exponential of the linear operator \(T\).

Answer: I + (e - 1)T


Question: In an acute triangle ABC, the base BC has the equation \(4x - 3y + 3 = 0\). The coordinates of the orthocentre (H) and circumcentre (P) of the triangle are \((1, 2)\) and \((2, 3)\) respectively. The radius of the circle circumscribing the triangle is \(\frac{\sqrt{m}}{n}\), where m and n are relatively prime. Find the value of \(m + n\). (You may use the fact that the distance between the orthocentre and circumcentre of the triangle is given by \(R \sqrt{1 - 8\cos A\cos B\cos C}\).)

Answer: 63


Question: Consider a matrix $A$ with $m$ rows and $n$ columns, where $m \leq n$. Each row of $A$ has norm 1, and any two rows are orthogonal. Is the sum of the squares of the minors of order $m$ of $A$ equal to 1? Justify your answer.

Answer: 1


Question: Let \( F = \langle a, b \rangle \) be a free group. If \( N \) is the normal subgroup of \( F \) generated by the set \( \{ b^2, ab \} \), find the index of \( N \) in \( F \).

Answer: 2


Question: Determine the number of complex numbers \( z \) with \( |z| < 1 \) that satisfy the equation \( az^4 + z^3 + bz^2 + cz + d = 0 \), given that \( a, b, c, \) and \( d \) are positive numbers such that \( a + b + c + d < 1 \).

Answer: 3


Question:
Given the quadratic equation \(x^2 - 4x + 7 = 0\) with roots \(x_1\) and \(x_2\), calculate the value of the expression:
\[
\frac{x_1^4 - 2x_1^3 + 3}{x_1^3 + x_1^2} + \frac{x_2^4 - 2x_2^3 + 3}{x_2^3 + x_2^2}
\]
using Vieta's formulas without manually calculating the roots.

Answer: -\dfrac{29}{49}


Question: Determine the number of ways to color $n$ distinct objects using at most $n$ different colors, where permutations of colors are considered identical. For example, the colorings $1,1,1,1$ and $2,2,2,2$ are considered the same, while $1,1,2,3$ and $2,3,1,1$ are different. Express your answer in terms of $n$.

Answer: B_n


Question: Determine whether there exists a sequence of indices \( \{n_k\} \) such that both subsequences \( \{\sin n_k\} \) and \( \{\cos n_k\} \) converge.

Answer: Yes


Question: Find the Lebesgue measure of the Cantor-type set $C_\alpha$ constructed as follows: For $0 < \alpha < 1$, in the first step, remove from the interval $[0, 1]$ a "middle" open interval of length $(1 - \alpha)3^{-1}$. In the nth step, remove $2^{n-1}$ open intervals of length $(1 - \alpha)3^{-n}$. Calculate the measure of $C_\alpha$.

Answer: \alpha


Question: Let \(z\) and \(w\) be complex numbers such that \(|2z - w| = 25\), \(|z + 2w| = 5\), and \(|z + w| = 2\). Find \(|z|\).

Answer: 9


Question: Find the asymptotic limit \( \mu = \lim_{n \to \infty} \mu_n \), where \( \mu_n \) is the expected value of \( m_\sigma \) over all permutations \( \sigma \) of the set \( \{1, 2, \ldots, n\} \). Here, \( m_\sigma \) is defined as the number of entries in the permutation \( (\sigma(1), \ldots, \sigma(n)) \) that exceed any later entry. Calculate \( \mu \).

Answer: \infty


Question:
Given the systems of linear inequalities:

\[ P = \{ x \in \mathbb{R}^m : Ax = b, Bx \leq d, x \geq 0 \} \]

and

\[ Q = \{ (x, y) \in \mathbb{R}^{m+n} : Ax = b, Bx + y = d, x \geq 0, y \geq 0 \} \]

Assume that $Q$ is an integral polyhedron with integral extreme points. Is $P$, the projection of $Q$ onto $x$-space, also an integral polyhedron?

Answer: Yes


Question: Determine whether the following statement is true or false: Let $f$ be a continuous function on $[a,b]$, and differentiable on $(a,b)$ such that $f^{'}$ is bounded on $(a,b)$. Does it follow that $f^{'}$ is continuous?

Answer: B


Question: Does the real part of the Riemann zeta function, \( \Re\zeta(s) \), ever vanish for \( \frac{1}{2}<\Re(s)\leq 1 \)?

Answer: Yes


Question: In a finite-dimensional Hilbert space, let \( T \in B(H) \). Is it true that there exists a unitary operator \( V \) such that \( T = V|T| \)?

Answer: Yes


Question: Evaluate the integral \( \oint \left(\frac {\arctan(x^3+3)}{3+x^2} +2\cos(x^2+y^2)\right)\,dx +\left(\frac{\sin(\cos^2(1+y^2))}{y^2+100}+2y\cos(x^2+y^2)\,dy\right) \) using Green's Theorem, where \( C \) is the positively oriented ellipse \((x-4)^2 + y^2/4=1\).

Answer: 0


Question: Assume that \(z\) and \(w\) are differentiable functions of \(x\) and \(y\), satisfying the equations \(xw^3+yz^2+z^3=-1\) and \(zw^3-xz^3+y^2w=1.\) Find \(\frac{\partial z}{\partial x}\) at \((x,y,z,w)=(1,-1,-1,1).\)

Answer: -\dfrac{5}{4}


Question: Determine the asymptotic behavior \( \Theta(S_n) \) of the summation \( S_n = \sum_{k=1}^{n}\frac{n-k+1}{k} \), where \( n \in \mathbb{Z}^{+} \).

Answer: \Theta(n \log n)


Question: How many 9-permutations have descent sets that are subsets of \(\{1, 4, 7\}\)?

Answer: 5040


Question: Evaluate the moment generating function $M_{Z^2_1}(t)$ of $Z_1^2$, where $Z_1$ is a standard normal variable, at the point $t=0.14$. Provide your answer to two decimal places.

Answer: 1.18


Question: Consider the metric space $H$ of uniformly continuous homeomorphisms (with uniformly continuous inverses) of some open bounded subset $G$ of $\mathbb{C}$, with the uniform metric \(d(f, g) = \sup_{z \in G} \left|f(z) - g(z)\right|.\) Is the inversion map \(f \mapsto f^{-1}\) a continuous map from $H$ to $H$? In other words, if \((f_n) \to f\) uniformly on $G$, where $f$ and each $f_n$ is a uniformly continuous homeomorphism of $G$ with a uniformly continuous inverse, does it follow that \((f_n^{-1}) \to f^{-1}\) uniformly on $G$?

Answer: Yes


Question: Determine the values of $s$ for which the integral \( \int_{0}^{\infty} \frac{\sin x}{x^s} \, dx \) converges.

Answer: 0 < s < 2


Question: Consider the sequence space of real-valued sequences that eventually become zero, denoted as \(c_{00}(\mathbb{R})\). Define the normed spaces \(X_1=(c_{00}(\mathbb{R}),\lVert\cdot\rVert_1)\) and \(X_2=(c_{00}(\mathbb{R}),\lVert\cdot\rVert_2)\), where \(\lVert\cdot\rVert_1\) and \(\lVert\cdot\rVert_2\) are the usual \(\ell_p\)-norms. Let \(B_1\) denote the closed unit ball in \(X_1\). Is there a point \(p\in X_1\) such that \(p\) lies on the boundary of \(B_1\), but is closer to the zero-sequence \((0,0,0,\ldots)\) with respect to \(\lVert\cdot\rVert_2\)?

Answer: Yes


Question: Consider a sequence of nested intervals defined as follows: \( L_1 = [0,1] \). For each even \( n \), let \( L_n = [a_n, b_n] \) where \( a_n \) and \( b_n \) are irrational numbers such that \( a_n < b_n \) and \( a_n, b_n \in L_{n-1} \). For each odd \( n > 1 \), let \( L_n = [x_n, y_n] \) where \( x_n \) and \( y_n \) are rational numbers such that \( x_n < y_n \) and \( x_n, y_n \in L_{n-1} \). The sequence of intervals is such that \( L_1 \supset L_2 \supset L_3 \supset L_4 \supset L_5 \dots \). Define the set \( \mathcal{S} = \bigcap_{n=1}^{\infty} L_n \). Determine the number of elements in the set \( \mathcal{S} \) and whether these elements are rational, irrational, both, or neither.

Answer: 1


Question: Determine the limit of the sequence \( a_n = \sqrt[n]{b^n + c^n} \) as \( n \to \infty \), where \( b, c \ge 0 \). Prove that the sequence is convergent and find its limit.

Answer: \max\{b, c\}


Question:
Determine which of the following functions are metrics on a non-empty set $X$, given that $d$ and $d'$ are metrics on $X$:

1. $\rho_1(x, y) = d(x, y) d'(x, y)$ for all $x, y \in X$.
2. $\rho_2(x, y) = \max\{5, d(x, y)\}$ for all $x, y \in X$.
3. $\rho_3(x, y) = \max\{d(x, y), d'(x, y)\}$ for all $x, y \in X$.
4. $\rho_4(x, y) = \min\{d(x, y), d'(x, y)\}$ for all $x, y \in X$.

Identify which of these functions satisfy the properties of a metric.

Answer: \rho_3


Question: Evaluate the integral \( \int_0^{\frac{\pi}{2}} \frac{\sin^3(x)}{\cos^3(x)+\sin^3(x)} \, dx \).

Answer: \dfrac{\pi}{4}


Question: Compute the Euler characteristic of the $n$-sphere $S^n$ for $n \ge 2$ using the standard triangulation of the $(n+1)$-simplex.

Answer: 1 + (-1)^n


Question: Find the point at which the probability density function (pdf) of the chi-squared distribution with $r \ge 2$ degrees of freedom attains its maximum.

Answer: r - 2


Question: What is the expected number of coin flips needed to achieve a sequence of alternating heads and tails 10 times, starting with heads?

Answer: 1364


Question:
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that \( f(f(x)) = -x \) for all \( x \in \mathbb{R} \). Determine which of the following statements is correct:

(a) \( f \) is an injective map.
(b) \( f \) is strictly increasing.
(c) \( f \) is strictly decreasing.
(d) \( f \) is continuous.

Answer: a


Question: Show that the differential equation \( \frac{du}{dx} = |u| \) for \( x \in \mathbb{R} \) has solutions of the form \( u = Ae^x \).

Answer: u = Ae^x


Question: Determine the number of zeros of the function \( f(z) = z^{10} + 10ze^{z+1} - 9 \) within the unit disk \( |z| < 1 \).

Answer: 1


Question: Determine all real numbers \( a \) such that the function \( f(x) = x^a \sin\left(\frac{1}{x}\right) \) for \( x \neq 0 \) and \( f(x) = 0 \) for \( x = 0 \) is continuous.

Answer: a > 0


Question: Find the sum of the series \( \sum_{n=1}^{\infty}(-1)^n \frac{16^n \pi^{2n+1}}{(2n)!} \).

Answer: 0


Question: Evaluate the integral \( \int^1_0 e^{-x^2} \, dx \) correct to three decimal places using the power series expansion of \( e^{-x^2} \).

Answer: 0.747


Question: Verify the equality \( e^{-\lambda} = \lim_{n\rightarrow \infty} \left(1-\frac{\lambda}{n}\right)^n \) given that \( \lambda = np \) where \( p \) is a fixed probability in the context of a Poisson distribution.

Answer: e^{-\lambda}


Question: Let $f$ be an increasing function with $f(0) = 3$. The derivative of $f$ is given by $f'(x)=\cos(\pi x)+x^4+6$. Let $g$ be the function defined by $g(x) = f(\sqrt{2x^2+7})$. Find $g'(3)$. \( \boxed{} \)

Answer: 756


Question: Find the supremum of the expression \( \left|\int_E e^{it} \, dt \right| \) over all subsets \( E \subset [-\pi, \pi) \).

Answer: 2


Question: Find the sum of the infinite series \( \sum_{i=1}^{\infty} \left( \frac{1}{2} \right)^i \left( \frac{1}{i}\right) \).

Answer: \ln 2


Question: Let $A$ be the set of all natural numbers less than or equal to $100$. Determine the number of relations $R \subseteq A \times A$ such that $R$ is reflexive, $R$ is symmetric, and $(2,x) \in R$ if and only if $x$ is an even number.

Answer: 2^{4851}


Question: Evaluate the limit: \[ \lim_{x \to -0.5^{-}} \left\lfloor\frac{1}{x} \left\lfloor \frac{-1}{x} \right\rfloor\right\rfloor \]

Answer: -2


Question: In an extension of the Monty Hall problem, there are two types of hosts. Host A is the standard host who gives you a choice to switch doors. Host B only gives you a choice if you initially choose the correct door; otherwise, he reveals that you have lost. Host A appears with probability P(A), and Host B with probability 1-P(A). Determine the value of P(A) such that there is no dominant strategy if you are given the choice to switch.

Answer: \dfrac{1}{2}


Question: Determine whether every simply connected open subset of \(\Bbb R^n\) is contractible.

Answer: No


Question: Find the maximum value of \( \sum_{i \in B} \frac{i}{|B|} \) where \( B = A \cup \{1\} \) and \( A \subseteq \{2, 3, 4, 5, 6\} \).

Answer: 4


Question: Evaluate the line integral \( \int_C \frac{1}{1+\frac{z}{2}} \, dz \) where \( C \) is the curve parameterized by \( \gamma(t) = (-2\sin t, 2\cos t, 2t^2) \) for \( t \in [0, 1] \).

Answer: 2 \ln 2


Question: Solve the equation $x^{x^{x^{...}}}=2$ for $x$. Then, solve the equation $x^{x^{x^{...}}}=4$ for $x$. Explain why both equations yield the same real solutions.

Answer: \sqrt{2}


Question: Use Burnside's Lemma to determine the number of distinct 3-colorings of a regular octagon, where each vertex can be colored red, blue, or green.

Answer: 498


Question: Show that for \( n \in \mathbb{N} \), the integral \( \int_{-\pi}^{\pi}  \frac{1 - \cos (n+1) x}{1- \cos x} \, dx = (n+1) 2 \pi \).

Answer: (n+1) 2\pi


Question: Is there a sequence of functions \( \{f_n(x)\} \) such that \( \int_0^1 |f_n(x)| \, dx \to 0 \) as \( n \to \infty \), but the sequence \( \{f_n(x)\} \) is pointwise divergent for every \( x \in [0,1] \)?

Answer: Yes


Question: Let X and Y be jointly absolutely continuous random variables. Suppose X follows an Exponential distribution with rate 2, i.e., X~Exponential(2), and the conditional probability P(Y > 5 | X = x) is given by \( e^{-3x} \). Compute the unconditional probability P(Y > 5).

Answer: \dfrac{2}{5}


Question: Evaluate the integral \( \int_\mathbb{N} f \, d\nu \) where \( f(n) = 2^{-n} \) and \( \nu = \mu + \delta_2 + \delta_5 \), with \( \mu \) being the counting measure on \( \mathbb{N} \) and \( \delta \) representing the Dirac measure. Express your answer as a single numerical value.

Answer: \dfrac{41}{32}


Question: Let \((X_1, X_2, \ldots)\) be an independent sequence of random variables, where each \(X_n\) is uniformly distributed on \([0, n]\). Calculate the probability \(P(\{w: X_n(w) \rightarrow \infty \text{ as } n \rightarrow \infty\})\).

Answer: 0


Question: Determine whether the sequence \( x_n = \sqrt[n]{\left\lvert \sin(2^n) \right\rvert} \) converges as \( n \to \infty \).

Answer: 1


Question: Determine whether the converse is true: If a commutative ring with an identity satisfies the ascending chain condition on ideals, does it imply that every ideal is finitely generated?

Answer: Yes


Question: Suppose that vectors $x_1, x_2, \dots, x_n$ have the following property: for each $i$, the sum of all vectors except $x_i$ is parallel to $x_i$. If at least two of the vectors $x_1, x_2, \dots, x_n$ are not parallel, what is the value of the sum $x_1 + x_2 + \dots + x_n$?

Answer: 0


Question: Evaluate the infinite sum \( \sum_{n=0}^{\infty} |(-0.8)^n \theta(n)-(-0.8)^{n-1} \theta(n-1)| \), where \( \theta(n) = 1 \) for \( n \geq 0 \) and 0 otherwise. Determine the correct value of this sum.

Answer: 10


Question: Given that $a, b, c, d > 0$ and $a + b + c + d = 1$, find the maximum value of the expression $M = abc + abd + acd + bcd - abcd$. Provide your answer as a single number.

Answer: \dfrac{15}{256}


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

Answer: 0


Question:
Let $X$ and $Y$ be convex compact subsets in $\mathbb{R}^n$, and let $\langle \cdot, \cdot \rangle$ denote the standard inner product. Determine whether the following equality holds:
\[ \max_{y \in Y} \left[ \langle y, z \rangle - \max_{x \in X} \langle y, x \rangle \right] = \max_{y \in Y} \min_{x \in X} \langle y, z-x \rangle. \]

Answer: Yes


Question: Compute the exterior derivative \(d\omega\) of the 1-form \(\omega = \frac{x^1dx^2 - x^2dx^1}{(x^1)^2+(x^2)^2}\) for \((x^1,x^2) \neq 0\).

Answer: 0


Question:
Determine the limit as \(x \to \infty\) of the expression:
\[
x - \frac{x-1}{x}2e^{-1/x}K_1(2/x) = x \left[1- \frac{x-1}{x^2}2e^{-1/x}K_1(2/x) \right],
\]
where \(K_1(\cdot)\) denotes the modified Bessel function of the second kind. Verify if the limit is indeed \(2\) as calculated, or if there is an error in the reasoning.

Answer: 2


Question: Determine the correct value of the exponent in the expression \(|S \cap [1,N]| = O(N^{0.43})\), where the subset \(S \subset \mathbb{N}\) is defined by the following conditions: \(5, 8 \in S\), if \(n \in S\), then \(7n-3\) and \(7n+5\) are also in \(S\), and if \(n \in S\) and \(n = 5m+4\), then \(11m+9\) is also in \(S\).

Answer: 0.43


Question: Find the length of the graph of the function \( y = f(x) \) over the interval \([0, 3]\), where \( f(0) = 0 \) and \( f'(x) = \frac{\ln(x+2)}{x^2+1} \) for \( x > -2 \).

Answer: 3.32


Question:
Let \((X,Y)\) be a bivariate random variable with a Gaussian distribution on \(\mathbb{R}^2\), mean zero, and variance-covariance matrix:
\[C=\begin{pmatrix} 0.42 & -0.42\\-0.42 & 0.42\end{pmatrix}\]
Find the expected value \(\mathbb{E}[|XY|]\).

Answer: 0.42


Question: Let $X_1, X_2, \ldots, X_{100}$ be independent and identically distributed continuous random variables. Calculate the probability that $X_{12}$ is the smallest and $X_{20}$ is the largest among these 100 random variables.

Answer: \dfrac{1}{9900}


Question: Determine the radius of convergence for the power series \( \sum_{n=0}^\infty c_n z^n \), where the coefficients are defined by \( c_0 = 0 \), \( c_1 = 1 \), and \( c_n = \frac{c_{n-1} + c_{n-2}}{2} \) for \( n \geq 2 \). Assume that \( C = \lim_{n\to\infty} \frac{c_{n+1}}{c_n} \) exists and find the value of \( C \) to determine the radius of convergence.

Answer: 1


Question: In \((\mathbb{R}^{2},\|\cdot\|_{p})\) with \(1 \leqslant p < \infty\) and under the standard basis \(\{e_{1},e_{2}\}\), find the largest possible constant \(c_{p} > 0\) such that for all \(a_{1}, a_{2} \in \mathbb{R}\), the inequality \( \|a_{1}e_{1} + a_{2}e_{2} \|_{p} \geqslant c_{p}(|a_{1}| + |a_{2}|)\) holds.

Answer: 2^{\frac{1}{p} - 1}


Question:
Consider the initial value problem \(y'(t)=f(t)y(t)\) with \(y(0)=1\), where \(f:\mathbb{R}\rightarrow \mathbb{R}\) is a continuous function. Determine which of the following statements is true:

1. Infinitely many solutions exist for some \(f\).
2. A unique solution exists on \(\mathbb{R}\).
3. No solution exists on \(\mathbb{R}\) for some \(f\).
4. A solution exists in an interval containing 0, but not on \(\mathbb{R}\) for some \(f\).

Answer: 2


Question: Use Green's Theorem to evaluate the line integral \( \int_C \langle 2y^2 + \sqrt{1+x^5}, 5x-e^y \rangle \cdot d\mathbf{r} \), where \( C \) is the positively oriented circle \( x^2 + y^2 = 4 \).

Answer: 20\pi


Question: Determine the kernel of the ring homomorphism $R[T] \to R[w]$ defined by $T \mapsto w$, where $w = u/v$ with $u, v \in R$ and $R$ is an integrally closed domain.

Answer: (vT - u)


Question: Let $S = \{c_1, \dots, c_n\}$ be a set of vectors in $\mathbb{R}^M$. Determine the maximum cardinality of a subset $S' \subset S$ such that $\dim(\text{span}(S')) < \dim(\text{span}(S))$. What is the largest size of such a subset?

Answer: n - 1


Question:
Given i.i.d. random variables with mean \( \mu \) and variance 1, let \( \bar{X}_n = \frac{1}{n}(X_1 + \cdots + X_n) \). Use the Central Limit Theorem to approximate the probability:

\[ P\left(|\bar{X}_n - \mu| \ge \frac{2}{\sqrt{n}}\right) \]

Answer: 0.0456


Question: Evaluate the integral \( \int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx \) and show that it equals \( \pi(1-e^{-a}) \) for \( a \ge 0 \).

Answer: \pi(1 - e^{-a})


Question: Let $R$ be a commutative Noetherian ring with unity, and let $M, N, P$ be finitely generated projective modules over $R$. Suppose for some $n \ge 1$, we have $M \otimes_R N \cong M \otimes_R P \cong R^n$. Is it true that $N \cong P$?

Answer: Yes