Question: Given the quartic polynomial \(x^4 - 3x^3 + ... - 10 = 0\) with real coefficients, and two complex roots \(a + 2i\) and \(1 + bi\), where \(a\) and \(b\) are real numbers, determine the sum of the real roots of the polynomial.

Answer: 1


Question: Let \(a, b,\) and \(c\) be the roots of the equation \(x^3 - 9x^2 + 11x - 1 = 0\). Define \(s = \sqrt{a} + \sqrt{b} + \sqrt{c}\). Find the value of \(s^4 - 18s^2 - 8s\).

Answer: -37


Question: Is there a finite distributive lattice $L$ such that there is no injective lattice homomorphism $f: L \to \omega^\omega$, where $\omega^\omega$ is considered as a lattice with component-wise join and meet?

Answer: No


Question: Given that $m$ and $n$ are positive real numbers satisfying the equation \(m+4\sqrt{mn}-2\sqrt{m}-4\sqrt{n}+4n=3\), find the value of \(\frac{\sqrt{m}+2\sqrt{n}+2014}{4-\sqrt{m}-2\sqrt{n}}\).

Answer: 2017


Question: Find an integer \( x \) such that \( x^7 \equiv 23 \mod 143 \).

Answer: 23


Question: Evaluate the integral \( \int_0^\pi \arctan(\cos x)\,\mathrm{d}x \).

Answer: 0


Question: Find all linear functionals \( \varphi: X \to \mathbb{R} \) such that \( |\varphi(x)| \leq \|x\|^2 \) for all \( x \in X \), where \( X \) is a normed space.

Answer: 0


Question: Calculate the flux of the vector field \( \mathbf{F} = xi + yj + zk \) outward across the sphere defined by \( x^2 + y^2 + z^2 = a^2 \).

Answer: 4\pi a^3


Question: Is the product topology on \( \mathbb{R}^d \), induced by the standard topology on \( \mathbb{R} \), equivalent to the standard topology on \( \mathbb{R}^d \)? Justify your answer.

Answer: Yes


Question: Find the residue of the function \( f(z) = \frac{\cot(z)\coth(z)}{z^3} \) at the point \( z = 0 \).

Answer: -\dfrac{7}{45}


Question: If a real $n \times n$ matrix $U$ satisfies $U + U^T \geq 0$, does its unitarily similar counterpart $V = WUW^T$, where $WW^T = W^TW = I$, also satisfy $V + V^T \geq 0$?

Answer: Yes


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

Answer: 4


Question: Compute the derivative of \( f(x) = e^{-1/x^2} \) at \( x = 0 \) using the definition of the derivative. Assume that \( \lim\limits_{x\to\infty}\dfrac{\ln x}{x^b} = 0, \) where \( b > 0. \)

Answer: 0


Question: Let $D$ be a division algebra over a field $k$, and let $R = M_n(D)$ be the $n \times n$ matrix ring over $D$. Consider a simple $R$-module $M$ that is the quotient of $R$. Given that $R = \bigoplus_j I_j$, where $I_j$ is the subring of $R$ with all columns except the $j$-th being zero, determine if the induced map $R \to M$ results in an isomorphism between $M$ and some $I_j$. Is this statement true?

Answer: True


Question: Calculate the limit \( \lim_{x \to \infty, n \to \infty} f^n(x) \), where \( f(x) = \ln(1+x) \) and \( f^n(x) = f(f^{n-1}(x)) \) for all integers \( n \geq 2 \) with \( f^1(x) = f(x) \).

Answer: 0


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

Let $Y$ be a random variable with finite mean and $X$ a random vector. $Y$ can always be decomposed as follows:
\[ Y = \mathbb{E}[Y|X] + \epsilon, \]
where $\epsilon$ is a scalar random variable that is uncorrelated with any function of $X$. Provide a justification for your answer.

Answer: False


Question: Determine the number of integers $x$ from 1 to 1000 such that $x^{100} \equiv 1 \pmod{1000}$. How many such integers exist?

Answer: 400


Question: Let $(f_n)$ be a sequence in $L^2(\mathbb{R})$ that converges to $f$ in the $L^2$-topology. If $\int_{\mathbb{R}} f_n(x) \, dx = 0$ for all $n$, does it follow that $\int_{\mathbb{R}} f(x) \, dx = 0$?

Answer: No


Question: Consider the n-cube $H_n = (V, E)$, where $V = \{0,1\}^n$ and edges are directed such that $(x, y) \in E$ if and only if $x \subseteq y$ and the Hamming distance $\text{dist}_{Hamm}(x, y) = 1$. Let $0^n = (0, 0, \ldots, 0)$ and $1^n = (1, 1, \ldots, 1)$. Suppose $S \subseteq V \setminus \{0^n, 1^n\}$ with $|S| = n$, and in $H_n \setminus S$, there is no directed path from $0^n$ to $1^n$. Is it true that $H_n \setminus S$, when viewed as an undirected graph, is disconnected?

Answer: Yes


Question: Find the value of \( a \) such that \( \lim_{x\rightarrow 0}(1+a\sin x)^{\csc x} = 4 \).

Answer: \ln 4


Question: Let $D$ be an integral domain. If the polynomial ring $D[x]$ is a unique factorization domain (UFD), does it imply that $D$ is also a unique factorization domain? In other words, does the equivalence $D$ is a UFD if and only if $D[x]$ is a UFD hold?

Answer: Yes


Question: Let \( a < b \), \( n \in \mathbb{N} \), and \( f : [a, b] \to \mathbb{R} \) be a continuous function such that \( \int_a^b t^k f(t) \, dt = 0 \) for all \( k = 0, 1, \ldots, n \). Determine the minimum number of zeros of \( f \) in the interval \( (a, b) \).

Answer: n + 1


Question: Find the limit as \( n \to \infty \) of the expression \( \sum\limits_{k=1}^n \tan\left(\frac{2k}{n^2}\right) \).

Answer: 1


Question: Given that the function \( f(x) = \frac{\sin 3x + A \sin 2x + B \sin x}{x^5} \) is continuous at \(x=0\), find the value of \(A + B + f(0)\).

Answer: 2


Question: Let $M$ be a module over the polynomial ring $\mathbb{C}[x]$ such that $x+n$ is invertible in $M$ for every integer $n$. Define $N = \bigoplus_{n>0} \mathbb{C}_n$, where $\mathbb{C}_n := \mathbb{C}[x]/(x+n)$. Determine whether $\operatorname{Ext}^1(M, N) = 0$.

Answer: 0


Question: Let \( f \) be a function that is continuous on \( \mathbb{R} \) and differentiable for all \( x \neq 0 \). Given that \( \lim_{x \rightarrow 0} f'(x) = 1 \), prove that \( f'(0) \) exists and is equal to 1.

Answer: 1


Question: Suppose the function $f(z)$ has an isolated singularity at the point $a \in \mathbb{C}$ and is even in $(z-a)$, i.e., $f(z-a) = f(-(z-a))$. Prove that the residue of $f(z)$ at $a$ is zero.

Answer: 0


Question:
Consider the recursive sequence defined by \( x_n = \frac{x_{n-1}}{2} + \frac{3}{x_{n-1}} \) for \( n \geq 1 \), with an initial point \( x_0 > 0 \). Determine which of the following statements is true about the convergence of the sequence \( \{x_n\} \):

(a) The sequence converges only if \( x_0 < 3 \).
(b) The sequence converges for any \( x_0 \).
(c) The sequence does not converge for any \( x_0 \).
(d) The sequence converges only if \( x_0 > 1 \).

Select the correct option.

Answer: B


Question:
Consider the surface $S = \{(x,y,z) \in \mathbb{R}^3 : z = x^2 + y^2, x^2 + y^2 \leq 1\}$, which is oriented by the function $\alpha: B \to S$ defined by $\alpha = (x,y,x^2 + y^2)$, where $B = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1\}$. Identify the boundary of $S$, denoted as $\partial S$, and compute the line integral:
\[ \int_{\partial S} (\cos(z))\,dx + (xz + \tan(y))\,dy + y^2z^3\,dz. \]
Apply Stokes' Theorem to evaluate this integral.

Answer: \pi


Question:
Determine the order of consistency \( p \) for the initial value problem (IVP) \( \dot{y}(t)=f(t, y(t)), y(0)=y_0 \) using the following Butcher Tableau:
\[
\begin{array}{c|ccc}
0 & 0 & 0 & 0\\
\frac{1}{3} & \frac{1}{3} & 0 & 0\\
\frac{2}{3} & \frac{2}{3} & 0 & 0 \\
\hline
& 0 & \frac{1}{2} & \frac{1}{2}  \\
\end{array}
\]

Answer: 2


Question: Let $f: [0,1] \to \mathbb{R}$ be a continuous function such that $f(0) = f(1)$. For every $r \in [0,1]$, is it possible to find two points $a, b \in [0,1]$ at a distance $r$, satisfying $f(a) = f(b)$, given that $r < 1/2$? Justify your answer.

Answer: Yes


Question: Determine the number of solutions for the equation \( C \equiv M^e \; (\text{mod}\; n) \), where \( n, e \in \mathbb{Z} \) are fixed, and \( C \in \mathbb{Z}_n \). Specifically, find the number of solutions \( M \in \mathbb{Z}_n \) for the equation \( 11 \equiv M^7 \; (\text{mod}\; 187) \) with \( M < 187 \).

Answer: 1


Question:
Given a car with a speedometer error of a constant value \( c \), you are provided with \( n \) drives. For each drive, you have the distance traveled \( d_i \) and the speedometer reading \( s_i \). The total time driven over all drives is \( t \). Assume the speed was constant throughout each drive. Find the speedometer error \( c \) given the equation:

\[ t = \sum_{i=1}^{n} \frac{d_i}{s_i + c} \]

where \( d_i \) is the distance traveled during drive \( i \) and \( s_i \) is the speedometer reading for drive \( i \).

Answer: c


Question: Let $X = \mathbb{R}^{\mathbb{R}}$ with the product topology. Consider the set $A$ of functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x) \in \mathbb{Q}$ for all $x \in \mathbb{R}$ and $f(x) = 0$ for all but finitely many $x$. Also, consider the set $B$ of functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x) = 0$ when $x \in \mathbb{Q}$. Determine whether $A$ or $B$ is dense in $X$. Provide a justification for your answer.

Answer: A


Question: Let $G$ be a finite multiplicative abelian group contained in a field $K$, with $-1 \in G$. Denote $\mu(G)$ as the product of all elements of $G$. Prove that $\mu(G) = -1$.

Answer: -1


Question: Determine whether the operator \( L: \mathbb{R}[X] \rightarrow \mathbb{R}[X] \) defined by \( L\left(\sum\limits_n a_nX^n\right)=\sum\limits_n a_{2n}X^{2n} \) is bounded with respect to the norm \( \|p\|=\int\limits_{-1}^{1}|p(t)|dt \). If it is bounded, find its norm.

Answer: 1


Question: Let $X \subset \mathbb{R}$ and let $f, g: X \to X$ be continuous functions such that $f(X) \cap g(X) = \emptyset$ and $f(X) \cup g(X) = X$. Which one of the following sets cannot be equal to $X$? \[ \text{(a) } [0,1] \quad \text{(b) } (0,1) \quad \text{(c) } [0,1) \quad \text{(d) } \mathbb{R} \]

Answer: A


Question: Simplify the expression \((1+i)^{2017} + (i-1)^{2017}\), where \(i = \sqrt{-1}\).

Answer: 2^{1009} i


Question: Evaluate the surface integral of the function \( G(x,y,z) = x + y + z \) over the surface cut from the first octant by the plane \( 2x + 2y + z = 2 \).

Answer: 2


Question: Evaluate the limit of \( \frac{x^x}{x!} \) as \( x \to \infty \).

Answer: \infty


Question: Find the expected value of the sample median for a random sample of three observations from the standard normal distribution.

Answer: 0


Question: Determine the conditions under which the linear operator $T$ on $\mathbb{R}^4$, represented by the matrix \( Q = \begin{bmatrix} 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & b & 0 & 0\\0 & 0 & c & 0 \end{bmatrix} \) relative to the standard basis, is diagonalizable.

Answer: a = b = c = 0


Question:
Given that \((3,0,0,\beta)\) is an optimal solution to the linear programming problem:

Minimize \(x+y+z-\alpha t\)

subject to:

\[ 2x-y+z=6 \]
\[ -x+y+t=3 \]
\[ x,y,z,t \ge 0 \]

Find the maximum value of \(\beta - \alpha\).

Answer: 7


Question: Evaluate the integral \( \int_0^{2\pi}\dfrac{4ie^{it}}{4e^{it}-3}\,\mathrm{d}t \).

Answer: 2\pi i


Question: Given that \( f(x) \) and \( g(x) \) are functions satisfying \( f(g(x)) = x^2 \) and \( g(f(x)) = x^3 \) for all \( x \ge 1 \), and \( g(16) = 16 \), compute \( \log_2 g(4) \). Assume \( f(x) \ge 1 \) and \( g(x) \ge 1 \) for all \( x \ge 1 \).

Answer: \dfrac{4}{3}


Question: Determine the correct representation of the real hyperbolic plane \(\mathbb{RH}^2\) as a homogeneous space. Is it \(SO(2,1)/SO(2)\) or \(SL(2)/SO(2)\)?

Answer: SO(2,1)/SO(2)


Question: Given 365 unique days in a year, how many friends would you need on average to ensure that at least one friend's birthday falls on each day? Assume that birthdays are equally distributed across all days of the year.

Answer: 2365


Question: Calculate \( \lim_{n \to \infty} \int_{[n,n+1)} f(x) \, dx \) given that \( f \in L^1(\mathbb{R}) \).

Answer: 0


Question: Let \( x_0 \) be an arbitrary value in \( (0, \pi) \). Define the sequence \( x_{n+1} = \sin(x_n) \). Determine the limit of \( \sqrt{n} \sin(x_n) \) as \( n \to \infty \).

Answer: \sqrt{3}


Question:
Given the power spectral density function $S_X(w)$ at two frequencies:
\[ S_X\left(\frac{\pi}{4}\right)=10+3\sqrt{2}, \quad S_X\left(\frac{\pi}{6}\right)=11+3\sqrt{3}. \]
The autocorrelation function $R_X$ has the properties $R_X(0) = 10$ and $R_X(m) = 0$ for $|m|\geq 3$. Determine the value of \( \frac{R_X(1)}{R_X(2)} \).

Answer: 3


Question: Simplify the expression: $$\frac{1}{a(a-b)(a-c)} + \frac{1}{b(b-a)(b-c)} + \frac{1}{c(c-a)(c-b)}.$$

Answer: \dfrac{1}{abc}


Question: Determine the range of values for \( k \) such that the equation \( x^4 - 14x^2 + 24x - k = 0 \) has four real and unequal roots.

Answer: (8, 11)


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

Answer: 0


Question: Compute the limit: $$\lim_{x\to0}\frac{(x^2+1) \ln(x^2+1)-\sin(x^2)}{x\sin(x^2)}.$$

Answer: 0


Question: Find the limit of the expression \( \cos(kt + kux) \) as \( k \to 0 \), where \( k = 2\sqrt{1-a^2} \) and \( u = b(6-k^2) \), with \( a \) and \( b \) being real numbers.

Answer: 1


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

Answer: 2


Question: Find the greatest common divisor (GCD) of all the numbers in the set \(\{(n+2014)^{n+2014}+n^n \mid n \in \mathbb{N}, n > 2014^{2014}\}\).

Answer: 4


Question: Evaluate the integral \( \int_{0}^{2\pi} \lfloor \cot^{-1} x \rfloor \, dx \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function.

Answer: \cot 1


Question: Let \( \gamma \) be the ellipse defined by \( x^2 + 4y^2 = 4 \), oriented anticlockwise. Compute the line integral \( \int_{\gamma} (4y - 3x) \, dx + (x - 4y) \, dy \).

Answer: -6\pi


Question: Evaluate \( \lim_{n\rightarrow \infty} \sum_{r= 0}^{n} \frac{r}{n^2+ r} \).

Answer: \dfrac{1}{2}


Question: Is there a base in which all prime numbers end with 5 different symbols? If such a base exists, identify it.

Answer: 8


Question: Suppose $A$ and $B$ are non-zero linear transformations on a finite-dimensional vector space $V$ such that $A^2 = B^2 = 0$. If the dimensions of the ranges of $A$ and $B$ are equal, can $A$ and $B$ be similar?

Answer: Yes


Question: Determine whether the statistic \( \max\{-X_{(1)},X_{(n)}\} \) is one-dimensional or two-dimensional.

Answer: one-dimensional


Question: Find the unique positive constant \( a \) such that the inequality \( a^x \geq \frac{x}{2} + 1 \) holds for all real numbers \( x \). Calculate \( \lfloor 100a \rfloor \).

Answer: 164


Question:
Evaluate the following limit for an irrational number $x$:
\[ \lim_{m \to \infty} \lim_{n \to \infty} \left[ \cos(n!\pi x) \right]^{2m}, \qquad (m,n \in \mathbb{N}) \]

Answer: 0


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that for some constant \( k \) and \( \alpha > 1 \), the inequality \( |f(x) - f(y)| \leq k|x-y|^{\alpha} \) holds for all \( x, y \in \mathbb{R} \). Prove that \( f(x) = f(0) \) for all \( x \in \mathbb{R} \).

Answer: f(x) = f(0)


Question: Consider the function \( w(x,y) = x^3 \), where \( (x,y) \) is defined implicitly by the equation \( x^2 + y^2 = 5 \). Compute the gradient of \( w(x,y) \) when \( x = 2 \) and \( \frac{\partial y}{\partial x} = 0 \).

Answer: (12, 0)


Question: Given the system of equations: \(x + \frac{1}{y} = 10\), \(y + \frac{1}{z} = 10\), and \(z + \frac{1}{x} = 10\), find the maximum possible value of \(z\).

Answer: 5 + 2\sqrt{6}


Question: Consider a bounded, convex set $S \subset \mathbb{R}^n$. Let $d = \mathbb{E}_{a\sim S}[\mathbb{E}_{b\sim S}[d(a, b)]]$ be the average distance between two randomly chosen points from $S$. Is it true that for every fixed point $a \in S$, the expected distance $\mathbb{E}_{b\in S}[d(a, b)]$ equals $d$?

Answer: No


Question: Evaluate the expression \( \sum_{cyc} \frac{a^4}{(a-b)(a-c)} \) where \( a = -\sqrt{3} + \sqrt{5} + \sqrt{7} \), \( b = \sqrt{3} - \sqrt{5} + \sqrt{7} \), and \( c = \sqrt{3} + \sqrt{5} - \sqrt{7} \).

Answer: 30


Question:
Find the limit of the sequence of partial sums of the harmonic series from \(k+1\) to \(2k\):
\[ \lim_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}} \]

Answer: \ln 2


Question: Compute the contour integral \( \int_{\mathbb{R} - 10i} dz \frac{e^{iz}}{1 + z^2} \). The contour \( \mathbb{R} - 10i \) is a horizontal line in the complex plane, parallel to the real axis, and shifted down by 10 units. The integral is oriented from left to right. Express your answer in terms of \( \pi \) and \( e \).

Answer: \pi \left( \dfrac{1}{e} - e \right)


Question: Evaluate the series \( \sum_{r=0}^{n} (-1)^r \frac{\binom{n}{r}}{r+1} \) for a positive integer \( n \).

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


Question: Find the largest natural number \( n \leq 100 \) such that the polynomial \( Q(x) = x^{624} + x^{524} + x^{424} + x^{324} + x^{224} + x^{124} + x^{n} \) is divisible by \( P(x) = x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1 \) without a remainder.

Answer: 94


Question: Evaluate the line integral \( \int \mathbf{A} \cdot d\mathbf{r} \) from \((0,0,0)\) to \((1,1,1)\) along the path \(x=t, y=t^2, z=t^3\), where \( \mathbf{A} = (3x^2-6yz)\mathbf{i} + 2(y+3xz)\mathbf{j} + (1-4xyz^2)\mathbf{k} \).

Answer: 3


Question: Given integers \(0 \leq a_1 \leq a_2 \leq \dots \leq a_n \leq b\) such that \(4^b \leq 4^{a_1} + \dots + 4^{a_n}\), determine if \(4^b\) can always be expressed as a sum of some subset of \(4^{a_1}, \dots, 4^{a_n}\).

Answer: Yes


Question:
Find the limit of the sequence as \( n \to \infty \):
\[ \lim_{n \rightarrow\infty}  \frac{1}{n}\left(\sqrt{\frac{1}{n}}+\sqrt{\frac{2}{n}}+\sqrt{\frac{3}{n}}+\ldots+1\right) \]

Answer: \dfrac{2}{3}


Question: How many ordered triples \((a, b, c)\) of distinct integers satisfy \(-47 \leq a, b, c \leq 47\) and \(a + b + c > 0\)?

Answer: 411930


Question:
Find the least degree polynomial \( f \in \mathbb{R}[x] \) such that for two finite intervals \([a,b],[c,d]\subsetneq\mathbb{R}\) where \(a<b<c-1<c<d\) and \(b-a=d-c=s<1\), the following conditions hold:

1. \( \forall x\in[a,b],\ f(x) \in\left[\frac{a+b}{2} - r, \frac{a+b}{2} + r\right] \)
2. \( \forall x\in[c,d],\ f(x) \in\left[\frac{c+d}{2} - r, \frac{c+d}{2} + r\right] \)

where \( r\in(0,\frac{s}{2}) \). Consider whether Chebyshev polynomials can be used to construct such a polynomial. What is the degree of the resulting polynomial?

Answer: 4


Question: Determine the maximum number of positive real zeroes that the function \( f(x) = x^a - b(c^x) \) can have, where \( a, b, c \) are positive real numbers.

Answer: 2


Question: Consider a set $X = \{ a, b, \ldots \}$ of symbols and define $X' = \{ a, a^{-1}, b, b^{-1}, \ldots \}$ as a set of symbols. Let $R$ be the set of words of the form $xx^{-1}$ or $x^{-1}x$. Let $M$ be the monoid having $X'$ as generators and $R$ as relations. Then $M$ is a group since the word $x_{n}^{-\varepsilon_{n}}\cdots x_{1}^{-\varepsilon_{1}}$ serves as the inverse of the word $x_{1}^{\varepsilon_{1}}\cdots x_{n}^{\varepsilon_{n}}$ (with $\varepsilon_{i} \in \{ 1, -1 \}$ and $x^{1} := x$). Can it be shown that this group is free over the set $X$?

Answer: Yes


Question: Maximize the function \(f(x, y) = xy\) subject to the constraint \(x^2 + y^2 \leq 1\).

Answer: \dfrac{1}{2}


Question: Find the limiting distribution of \( \frac{T_n}{S_n} \) as \( n \to \infty \), where \( T_n = \sum_{i=1}^{n^2} X_i \) and \( S_n = \sum_{i=1}^{n^2} Y_i \), with \( X_i \sim \text{Bernoulli}\left(\frac{\lambda}{n}\right) \) and \( Y_i \sim \text{Poisson}\left(\frac{\lambda}{n}\right) \), and both sequences are independent.

Answer: 1


Question: Evaluate the infinite series: \[ \sum_{n=0}^{\infty}{\frac{2^n}{(2n+1){2n\choose n}}} \] using the identity \[ \int_0^{\pi/2}{\sin^{2k+1}x\;dx}=\frac{2^{2k}k!^2}{(2k+1)!}. \]

Answer: \dfrac{\pi}{2}


Question: Evaluate the integral \( \int_0^{2\pi} \frac{\sin(t) + 4}{\cos(t) + \frac{5}{3}} \, dt \).

Answer: 6\pi


Question: Find the number of ordered quadruples \((a,b,c,d)\) of nonnegative real numbers such that \(a^2 + b^2 + c^2 + d^2 = 4\) and \((a + b + c + d)(a^3 + b^3 + c^3 + d^3) = 16\).

Answer: 15


Question: Is 510511 the smallest Euclid number \( E_n \) that is divisible by the next prime \( p_{n+1} \)? Specifically, verify if \( 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 + 1 = 19 \times 97 \times 277 \).

Answer: 510511


Question: Evaluate the complex integral \( \int_{|z-1|=1}\frac{1}{z^2-1}dz \) over the circle centered at 1 with radius 1 in the complex plane.

Answer: \pi i


Question: Evaluate the sum \( S(n) = \sum_{i=0}^n (-1)^i \binom{n+i}{i} \binom{n}{i} \) and show that \( S(n) = (-1)^n \).

Answer: (-1)^n


Question: Given two independent random variables, where \( X \sim \mathcal{U}[-1,5] \) and \( Y \sim \mathcal{U}[-5,5] \), calculate the conditional expectation \( \mathbb{E}\{Y - X \mid X \le 1, Y > X, Y \in [-1,1] \} \).

Answer: \dfrac{2}{3}


Question: Let \( y = f(x) \) be a function that is discontinuous for exactly 3 values of \( x \) but is defined for all \( x \in \mathbb{R} \). Let \( y = g(x) \) be another differentiable function such that \( y = f(x)g(x) \) is continuous for all \( x \in \mathbb{R} \). Find the minimum number of distinct real roots of the equation \( g(x)g'(x) = 0 \).

Answer: 5


Question: Given an infinite set \(P\), is there an ordering relation \(\leq\) such that the poset \((P, \leq)\) is surjectively rigid, meaning the only order-preserving surjective map \(f: P \to P\) is the identity \(\text{id}_P\)?

Answer: Yes


Question: Given a positive real number $x$, a sequence $\{a_n(x)\}$ is defined as follows: \(a_1(x)=x\) and \(a_n(x)=x^{a_{n-1}(x)}\) recursively for all \(n\geq 2\). Determine the largest value of $x$ for which \(\lim_{n\to\infty}a_n(x)\) exists.

Answer: e^{1/e}


Question: Determine whether there exists a bijection \( f: \mathbb{R} \setminus \mathbb{R}^{-} \rightarrow [0,1) \) that is continuously differentiable on its entire domain.

Answer: Yes


Question: Evaluate the Riemann-Stieltjes integral \( \int_{0}^{1}{x\,\mathrm{d}\alpha(x)} \), where \( \alpha(x) = \begin{cases} 0 & x=0 \\ \frac{1}{2^n} & \frac{1}{3^n} < x \leq \frac{1}{3^{n-1}}, \quad n=1,2,... \end{cases} \).

Answer: \dfrac{1}{10}


Question: Determine whether the set $S'$, constructed from $S \cup (-S)$ by closing it under the operation of taking divisors in $\mathbb{Z}[X]$, is maximal for the property that if $A$ is a ring such that for every $a \in A$ there is an $f \in S'$ such that $f(a) = 0$, then $A$ is commutative. Is $S'$ maximal for this property?

Answer: Yes


Question: Evaluate the limit: $$\lim_{|(x,y,z)|\to+\infty}(x^4+y^2+z^2-x+3y-z).$$

Answer: +\infty


Question: Evaluate the limit: \[ \lim_{x\to0}(\cosh 3x)^{\csc^2x} \]

Answer: e^{\frac{9}{2}}


Question: Given that $a > b > 1$ and \( \frac{1}{\log_a(b)} + \frac{1}{\log_b(a)} = \sqrt{1229} \), find the value of \( \frac{1}{\log_{ab}(b)} - \frac{1}{\log_{ab}(a)} \).

Answer: 35


Question: Evaluate \( \lim_{x\to 1}\int_{x}^{x^2}\frac{1}{\ln {t}}\mathrm dt \).

Answer: \ln 2


Question: Calculate \((2^{2^{19}} + 1) \mod 19\).

Answer: 5


Question:
Evaluate the expression \( \sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt{2(1+\cos\theta)}}}} \) for \( n \) iterations, where \( 0<\theta <180 \). Choose the correct option:

A) \( 2\cos\left(\frac{\theta}{2^n}\right) \)
B) \( 2\cos\left(\frac{\theta}{2^{n-1}}\right) \)
C) \( 2\cos\left(\frac{\theta}{2^{n+1}}\right) \)
D) None of these

Answer: A


Question: For an integrable function $f$ over a set $E$, does there always exist a simple function $\phi$ such that the integral $\int_E |f - \phi|$ can be made arbitrarily small?

Answer: Yes


Question: Determine the degree of the splitting field of the polynomial $x^6+1$ over the rational numbers.

Answer: 4


Question: Determine the period of a function that satisfies the condition \( f(a-x) = f(a+x) \), where \( a \) is any positive integer.

Answer: 2


Question: Let $a$ and $b$ be the roots of the polynomial $x^2 - 10cx - 11d$, and let $c$ and $d$ be the roots of the polynomial $x^2 - 10ax - 11b$. Given that $a, b, c, d$ are distinct real numbers, find the value of $a + b + c + d$. \[ a + b + c + d = ? \]

Answer: 1210


Question: Given a real sequence \((x_n)\) in \(l^p\) for a fixed \(p>1\) but not in \(l^1\), determine if there exists a subsequence \((x_{\phi(n)})\) that belongs to \(l^{1+\varepsilon}\) for every \(\varepsilon > 0\) and is still not in \(l^1\).

Answer: Yes


Question: Assume that \(x_1, \dots, x_n\) are non-negative real numbers such that \(x_1 + \dots + x_n \in \mathbb{Q}\) and \(x_1 + 2x_2 + \dots + nx_n \in \mathbb{Q}\). Does this imply that the numbers \(x_1, \dots, x_n\) are rational?

Answer: No


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: Determine whether the limit \( \lim_{(x,y)\rightarrow (0,0)} \frac{e^{-x^{2}} - y}{x^{2}+y^{2}} \) exists. If it does, find its value.

Answer: \infty


Question: Does there exist a function \( f(x) \) such that \( f(x) = a \) if \( x \) is rational and \( f(x) = b \) if \( x \) is irrational, where \( a \neq b \)?

Answer: Yes


Question: Solve the Diophantine equation in positive integers: \(7^x = 3^y - 2\).

Answer: (x, y) = (1, 2)


Question: Given the permutation \(3, 5, 1, 2, 4, 6\), determine the number of ascending subsequences of length 3.

Answer: 6


Question:
Prove that for all positive integers \( l \), the following identity holds:
\[ 0 = \sum_{k=0}^{3l-1} (-3)^{k} \binom{6l}{2k+1} \]

Answer: 0


Question: Determine the number of automorphisms of the group $\Bbb Z \oplus \Bbb Z_2$. Provide a convincing proof for your answer.

Answer: 4


Question: Find the smallest number of vertices in a tree that has exactly two vertices of degree 3, one vertex of degree 4, and two vertices of degree 6. Provide an example of such a tree.

Answer: 19


Question: Determine the number of terms needed in the series \( \sum_{n=0}^\infty \frac{(-1)^n(7)^{2n+1}}{(2n+1)(2n+1)!} \) to approximate the integral \( \int_0^1 \frac{\sin(7x)}{x} \, dx \) to four decimal places.

Answer: 11


Question: Find a number \(a\) such that \(0 \leq a \leq 50\) and \(a \equiv (105^{70} + 1)^{15} \pmod{51}\).

Answer: 16


Question: Let $M$ be a submanifold of $\mathbb{R}^n$. Is there an open set $\Omega$ in $\mathbb{R}^n$ such that the distance function $d(x, M)$ is smooth on $\Omega$?

Answer: Yes


Question: Compute \( \lim\limits_{n \to \infty} \int_0^{\frac{\pi}{2}} \frac{(\sin(x))^{n}}{1-\sin{(x)}}\,\mathrm{d}x \).

Answer: \infty


Question: Find the infimum of the expression \(\frac{(n+1)^2}{2^n}\) for natural numbers \(n\). Use Bernoulli's inequality to assist in your solution.

Answer: 0


Question: Let $f$ be a uniformly continuous function on the real line that is integrable with respect to the Lebesgue measure. Prove that $\lim_{x \to \infty} f(x) = 0$.

Answer: 0


Question: Let \( f = \sum_{i \ge m} a_i t^i \) be a formal Laurent series, where \( m \in \mathbb{Z} \). Consider the formal differential \( df \). What is the coefficient of degree \(-1\) in the differential form \( \frac{df}{f} \)? Provide an explicit expression for this coefficient.

Answer: m


Question: Evaluate the limit: \[ \lim _{n\to \infty }\left(\frac{n3^n\left(\sqrt{n+1}-\sqrt{n}\right)}{3^n\sqrt{n+1}+2^n}\right) \]

Answer: \dfrac{1}{2}


Question: Let $a, b, c \in \mathbb{Z}[\omega]$ where $\omega$ is a cube root of unity, and none of $a, b, c$ are zero. If $a^3 + b^3 + c^3 = 0$, prove that at least one of $a, b, c$ is divisible by $1-\omega$.

Answer: 1 - \omega


Question: Given the elliptic curve $y^2 = x^3 + 2x + 3 \mod 97$ and the point (3, 6), compute the y-coordinate of the point obtained by doubling (3, 6) on the curve, given that the x-coordinate of the doubled point is 80.

Answer: 10


Question: Evaluate the limit of the integral: \[ \lim_{n \to \infty} \int_{0}^{\infty} \frac{\sin(x/n)}{x^2+x} \, dx. \]

Answer: 0


Question: Given a matrix \( A \in M_{2\times 2}(\mathbb{R}) \) such that \( A + A^{-1} = I_2 \), calculate the sum \( I_2 + A + A^2 + \cdots + A^{2019} \).

Answer: 2A - I_2


Question: Consider two posets $(P, \preccurlyeq_P)$ and $(Q, \preccurlyeq_Q)$, each generated by their respective covering relations $\lessdot_P$ and $\lessdot_Q$. Is it true that $P$ and $Q$ are isomorphic if there exists a bijective map $f: P \to Q$ such that for all $a, b \in P$, $a \lessdot_P b \Leftrightarrow f(a) \lessdot_Q f(b)$?

Answer: Yes


Question: Verify the three-dimensional delta function property: \[ \int_{a}^{b} \int_{c}^{d} \int_{e}^{f} \delta(x-X)\delta(y-Y)\delta(z-Z) f(x)f(y)f(z) \, dx \, dy \, dz = f(X)f(Y)f(Z) \] where \( a<X<b, c<Y<d, e<Z<f \) define the volume where the delta function is located.

Answer: f(X)f(Y)f(Z)


Question: Let $G$ be a group and $N$ be a minimal normal subgroup of $G$. Let $G_p \in \text{Syl}_p(G)$, where $p$ is an odd prime divisor of $|G|$. Suppose that $M/N$ is a maximal subgroup of $G_pN/N$. Prove that $M = PN$ for some maximal subgroup $P$ of $G_p$.

Answer: M = PN


Question: Apply the linear multistep method given by the equation \( x_{n+2} - 2x_{n+1} + x_{n} = h(f_{n+1} - f_{n}) \) to solve the initial value problem \( x'(t) = -x(t) \) with the initial condition \( x(0) = 1 \) and starting values \( x_0 = x_1 = 1 \). Express the solution in exact form.

Answer: 1


Question: Let $T$ be a topology on a set $X$. Suppose $S$ is a collection of subsets of $X$ such that $\emptyset$ and $X$ are in $S$, and $S$ is closed under finite unions and arbitrary intersections. Prove that $T = \{X-A : A \in S\}$ is a topology on $X$.

Answer: T


Question: Evaluate the integral: \[ \int_{0}^{4\pi}\frac{x|\sin x|}{1 + |\cos x|} \, dx \]

Answer: 16\pi \ln 2


Question: Let $X_1, X_2, X_3$ be independent random variables each uniformly distributed over the interval $(0, 1)$. Let $X_{(1)}, X_{(2)}, X_{(3)}$ be the order statistics of $X_1, X_2, X_3$. Compute the probability $P(X_{(1)} + X_{(3)} \leq 1)$. Provide your answer as a single probability value.

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{C} (x,y,z) \cdot d\overline{r} \) around the circle centered at \((0,0,1)\) passing through \((0,0,2)\) and \((0,1,1)\). Assume the circle lies in the plane described by \(x=0\) and \(y^2+(z-1)^2=1\).

Answer: 0


Question:
Consider the estimator \( W = \dfrac{1}{2n^2}\sum_{i}\sum_{j}(X_i-X_j)^2 \) for \( \sigma^2 \), where \( X_1, X_2, \ldots, X_n \) are i.i.d. random variables with a normal distribution \( N(\mu, \sigma^2) \). Determine the nature of \( W \) as an estimator of \( \sigma^2 \):

\(\text{(A)}\) biased and consistent

\(\text{(B)}\) unbiased and consistent

\(\text{(C)}\) biased and inconsistent

\(\text{(D)}\) unbiased and inconsistent

Choose the correct option.

Answer: A


Question:
Given a Brownian motion $W$, let $J_{k,n} = ((k-1)2^{-n}, k2^{-n}]$ for $n \in \mathbb{N}_0$ and $k = 1, \dots, 2^n$. Assume $n \ge m$ and $J_{k,n} \subset J_{l,m}$, with $J_{k,n}$ lying in the left half of $J_{l,m}$. Define $\Delta W([a,b]) = W_b - W_a$. Show that the following expectation is true:
\[ E[(\Delta W (J_{2k-1,n+1}) - \Delta W(J_{2k,n+1})) \Delta W(J_{2l-1,m+1})] = 0. \]

Answer: 0


Question: Determine the norm \( \|\phi\| \) for the linear functional \( \phi \in l_p^* \) defined by \( \phi(x) = \sum_{k=1}^N \frac{x_k}{N^{\frac{1}{q}}} \), where \( x = (x_1, x_2, \ldots) \in l_p \), \( N \in \mathbb{N} \), \( 1 < p, q < \infty \), and \( \frac{1}{p} + \frac{1}{q} = 1 \). Use the definition of the \( l_p \) space norm \( \|x\|_{l_p} = \left(\sum_{k=1}^{\infty} |x_k|^p\right)^{\frac{1}{p}} \) and the definition of the norm of a functional \( \|\phi\| = \sup \{ |\phi(x)| : \|x\|_{l_p} = 1 \} \).

Answer: 1


Question: Find the maximum volume of a rectangular box that can be inscribed in a tetrahedron bounded by the coordinate planes and the plane \( \frac{2x}{5} + y + z = 1 \). Do not use Lagrange multipliers in your solution.

Answer: \dfrac{5}{54}


Question: Determine the CW complex that is homotopy equivalent to the set defined by the equation $x^2 + y^2 - 1$.

Answer: S^1


Question: Calculate the rank of the coherent sheaf \( \widetilde{k[t]/(t)} \) on \( \text{Spec}(k[t]) \).

Answer: 0


Question: Calculate the square of the distance from the point \((1,1,1,1)\) to the subspace \(W\), where \(W = \text{span}\left(\frac{1}{\sqrt{2}}(0,0,1,1),\frac{1}{\sqrt{2}}(1,-1,0,0)\right)\) in \(\mathbb{R}^4\).

Answer: 2


Question:
Consider a function $f:\mathbb{R}^n\longrightarrow\mathbb{R}^m$ with partial derivatives everywhere so that the Jacobian matrix is well-defined. Let $L>0$ be a real number. Determine if the following statement is true:
\[|f(x)-f(y)|\leq L|x-y|,\forall x,y \Longleftrightarrow |J_f(x)|_2\leq L,\forall x\]
where $|\cdot|$ denotes the Euclidean vector norm and $|\cdot|_2$ the spectral matrix norm.

Answer: True


Question: Given the equations \( \frac{1}{a} + \frac{1}{b} + \frac{1}{a+x} = 0 \), \( \frac{1}{a} + \frac{1}{c} + \frac{1}{a+y} = 0 \), and \( \frac{1}{a} + \frac{1}{x} + \frac{1}{y} = 0 \), where \( a \neq 0 \), \( b \neq 0 \), and \( c \neq 0 \), find the value of \( a + b + c \).

Answer: 0


Question: Assume that \( f(x) \) is an even function in \( L_1 \), meaning \( \int_{\mathbb{R}} |f(x)| \, dx < \infty \). If \( \int_{\mathbb{R}} \cos(rx) f(x) \, dx = 0 \) for all \( r \geq 0 \), does it follow that \( f(x) = 0 \) almost everywhere? Justify your answer.

Answer: Yes


Question: Given the matrix \( A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix} \), where \( A = B + C \), \( B \) is symmetric \( (B = B^T) \), and \( C \) is skew-symmetric \( (C = -C^T) \). If \( (CB)^T = D \) and \( D = [ d_{ij} ]_{3 \times 3} \), find the value of \( d_{12} + d_{21} + d_{13} + d_{31} \).

Answer: 9


Question: Determine the order of the group $SL_2(F_3)$, where $F_3$ is the finite field with 3 elements.

Answer: 24


Question: Provide an example of a field that has infinite dimension over \(\mathbb{C}\).

Answer: \mathbb{C}(t)


Question: Determine the number of twin prime pairs of the form \((2^n - 1, 2^n + 1)\). The first pair is \((3, 5)\) for \(n = 2\). Are there any other pairs besides this one?

Answer: 1


Question: Identify all closed sets in \(\mathbb{R}\) that are subgroups of \((\mathbb{R}, +)\) and have at least one limit point.

Answer: \mathbb{R}


Question: Given that \(a \leq b \leq c\) are prime numbers, prove that if \(abc \mid (ab + ac + bc - 1)\), then \(a = 2\).

Answer: 2


Question: Use Green's Theorem to evaluate the line integral \( \int_C 2xy^{3} \, dx + 4x^{2}y^{2} \, dy \), where \( C \) is the boundary of the triangular region in the first quadrant, enclosed by the x-axis, the line \( x=1 \), and the curve \( y=x^{3} \).

Answer: \dfrac{2}{33}


Question: Minimize the function \( f(x,y,z) = \ln(x^2+1) + \ln(y^2+1) + \ln(z^2+1) \) subject to the constraint \( x^2 + y^2 + z^2 = 12 \) using Lagrange multipliers.

Answer: \ln 13


Question: Calculate the volume of the region defined by the inequality \((4x+4y+2z)^2+(4x+2y+4z)^2+(2x+4y+4z)^2 \leq 1\) using the change of variables \(u=4x+4y+2z\), \(v=4x+2y+4z\), \(w=2x+4y+4z\). The Jacobian determinant of the transformation is 40. Express the volume as an integral in the new variables \(u, v, w\) and find the value of the integral.

Answer: \dfrac{\pi}{30}


Question: Determine whether the following statement is true or false: Every $2 \times 2$ matrix over the complex numbers $\Bbb C$ is a square of some matrix.

Answer: B


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{1}{x}\int_1^{1+x}\frac{\cos t}{t} \, dt \]

Answer: \cos 1


Question: Evaluate the integral: \[ \int_0^\infty \frac{1}{(1+x^{2015})(1+x^2)} \, dx \]

Answer: \dfrac{\pi}{4}


Question:
Determine the nature of the singularities of the function \( f(z) = \frac{\sin(1/z)}{z^{2}+11z+13} \). Choose the correct option:
(a) No singularity.
(b) Only poles.
(c) Only an essential singularity.
(d) Both an essential singularity and poles.

Answer: D


Question: Given that the equation \(Ax = 0\) has only one solution, determine the number of solutions to the equation \((A^T)Ax = 0\).

Answer: 1


Question: Evaluate the integral \( \int_\gamma \frac{dz}{z} \) where \( \gamma = a \cos(t) + i b \sin(t) \) for \( t \in [0, 2\pi] \), using the complex logarithm function as the primitive of \( \frac{dz}{z} \).

Answer: 2\pi i


Question: Suppose $z_n$ is a sequence of complex numbers that converges to $0$. Does there exist a sequence {$\epsilon_n$} where each $\epsilon_n = \pm 1$ such that the series $\sum \epsilon_n z_n$ converges?

Answer: Yes


Question: Calculate the limit: $$\lim\limits_{n \to \infty} \cos^{n^2} \left (\frac{2x}{n} \right)$$

Answer: e^{-2x^2}


Question: Determine the points at which the complex function \( f(x,y) = x^2 - ixy \) is differentiable.

Answer: 0


Question: Evaluate the integral \( \int_{\mathcal{C}} \frac{z^9+e^{iz}+(7654)^{\sin(z)}z}{z-1} \,\mathrm{d}z \), where \( \mathcal{C} \) is the contour given by \( C(\theta) = \frac{e^{i\theta}}{2} \) for \( \theta \in [0, 2\pi] \).

Answer: 0


Question: Determine whether there exists a $C^\infty$ function $f$ defined on $(-\epsilon, \epsilon)$, where $\epsilon > 0$, such that the Maclaurin series $\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$ has a radius of convergence $R = 0$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the value of the summation \( \sum_{k=0}^{20}(-1)^k\binom{k+2}{2} \) without calculating each term separately.

Answer: 121


Question: Determine whether there exists a sum of two squares, expressed as \(a^2 + b^2\), that can be written in more than four distinct ways using positive integers \(a\) and \(b\).

Answer: Yes


Question: Consider a word of length $n$ consisting entirely of the letter 'A': $$AA\ldots A$$. You are allowed to perform $n$ actions, where each action consists of changing any letter to the next letter in the alphabet. How many distinct outcomes are possible, considering outcomes that are permutations of each other as the same?

Answer: p(n)


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{1}{\tan(x)}\int_{x}^{\sin2x}(1+\sin(t))^{1/t}dt \]

Answer: e


Question: Determine the number of 2-Sylow subgroups in the group $S_4 \times S_3$. The number of 2-Sylow subgroups is given by $n_2 = 1 + 2k$, where $n_2$ divides the order of the group $|S_4 \times S_3| = 144$. What is the value of $n_2$?

Answer: 9


Question: Consider a primitive matrix $A$. If a nonnegative matrix $\epsilon E$ is added to $A$, where $\epsilon > 0$, does the limit \( \lim_{\epsilon \to \infty} \rho(A+\epsilon E) \) exist, or does the spectral radius grow without bound?

Answer: \infty


Question: Calculate the flux of the vector field \( v = (3y_1 - 2y_2^2 + y_3^2, 3y_1^2 - 2y_2 + y_3^2, 3y_1^2 - 2y_2^2 + y_3) \) through the surface \( \tilde{Y} \), where \( \tilde{Y} \) is the image of the parameterization \( \varphi: K \to \mathbb{R}^3 \) with \( K = [0, 2\pi] \times [0, 1] \) and \( \varphi(x) = ((\cos x_1)(x_2 - x_2^3), (\sin x_1)(x_2 - x_2^3), x_2^2 + \frac{1}{3} x_2^6) \).

Answer: \dfrac{\pi}{5}


Question: Evaluate the infinite series \( \sum_{n=0}^\infty \frac{S_n}{(n+1)!} \), where \( S_n \) is the sum of the products of the first \( n \) natural numbers taken two at a time. Express your answer in terms of \( e \).

Answer: \dfrac{11e}{24}


Question:
Let $A$ be a $3000 \times 3000$ matrix with real entries such that $A^{2016} = I$. If $\mathrm{rank}(I - A^{1008}) = 1000$, determine $\mathrm{rank}(I + A^{1008})$. Choose the correct answer from the following options:

(a) 504
(b) 2000
(c) 1992
(d) 2016

Answer: B


Question: Is every Banach space a complete topological vector space? Consider a net $(v_\alpha)_{\alpha\in I}$ in a topological vector space $(V, \mathcal{T})$. The net is called Cauchy if for every neighborhood $U$ of $0$, there exists an index $\alpha_0$ such that for all $\alpha, \beta \geq \alpha_0$, the difference $v_\alpha - v_\beta$ is in $U$. A topological vector space is complete if every Cauchy net converges. Given a Banach space $(V,\Vert \cdot \Vert)$, is $V$ a complete topological vector space?

Answer: Yes


Question:
Calculate the volume of the solid bounded by the planes $z = 0$, $x = 1$, $x = 3$, $y = 1$, $y = 2$, and the surface $z = x^2y^2$ using triple integration in Cartesian coordinates. Evaluate the integral:
\[ \int_1^3 \int_1^2 \int_0^{x^2y^2} dz \, dy \, dx \]

Answer: \dfrac{182}{9}


Question: Evaluate the double integral: \[ \frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^y e^{-\frac{1}{2}(x^2+y^2)} \, dx \, dy. \]

Answer: \dfrac{1}{2}


Question: Compute \( \lim_{n \rightarrow \infty} \lim_{R \rightarrow \infty} \int_0^R \sin{(x/n)} \sin{(e^x)} \, dx \).

Answer: 0


Question: Let $X$ be a random variable with a standard normal distribution $X \sim N(0,1)$, and let $Y$ be a random variable with a uniform distribution $Y \sim \text{Uniform}(0,1)$. Compute the probability $P(Y > X)$. \( \text{Express your answer as a single probability value.} \)

Answer: 0.6844


Question: Calculate the line integral \( \oint_C (x^2-2xy)dx+(x^2y+3)dy \) where \( C \) is the closed curve of the region bounded by \( y^2=8x \) and \( x=2 \).

Answer: \dfrac{128}{5}


Question: Let \( p \geq 2 \) and \( s \in (1, 2) \). Prove that the inequality \[ T = \frac{s^{p-1} + s^{p-2} + \cdots + 2}{4^{p-1}} \leq 1 \] holds.

Answer: T \leq 1


Question: Given the function \(F(x) = \int_0^x (x-t)^n u(t) \, dt\), find \(F^{(n+1)}(x)\).

Answer: n! \, u(x)


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

Answer: 0


Question: Evaluate the limit: \[ \lim_{t \to \frac{\pi}{2}} \frac{ \int_{\sin t}^{1} e^{x^2 \sin t} \, dx}{\int_{\cos t}^{0} e^{x^2 \cos t} \, dx}. \]

Answer: 0


Question: Consider a graph $G$ where each vertex represents a 3-element subset of the set $\{1, 2, 3, 4, 5, 6\}$. Two vertices $A$ and $B$ are connected by an edge if the subsets have exactly one element in common, and the cost of the edge is the value of this common element. Find the cheapest spanning tree of the graph $G$.

Answer: 34


Question: Evaluate the limit: \[ \lim _{x\to 0+}\left(\frac{\left(1-\cos \left(2x\right)\right)^{14}\left(1-\cos \left(7x\right)\right)^2\sin ^{14}\left(9x\right)}{\tan ^{14}\left(x\right)\left(\ln \left(8x+1\right)\right)^{30}}\right) \]

Answer: 0


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

Answer: 0


Question:
Find the limit of the sequence \( \{u_n\} \) defined by the recurrence relation:

- \( u_0 > 0 \)
- \( u_{n+1} = \sqrt{u_n + \sqrt{u_{n-1} + \ldots + \sqrt{u_0}}} \)

Answer: 2


Question: Evaluate the iterated integral by changing to polar coordinates: \[ \int_0^{1/2}\int_0^{\sqrt{1-x^2}}xy\sqrt{x^2+y^2}\,dy\,dx \]

Answer: \dfrac{19}{480}


Question: Determine the cardinality of the set of all finite and infinite binary words that do not contain the sequence "01".

Answer: \aleph_0


Question: Calculate the surface integral \( \iint_{S} F \cdot dS \) where \( F(x, y, z) = xy\mathbf{i} + yz\mathbf{j} + zx\mathbf{k} \) and \( S \) is the surface of the cylinder \( x^2 + y^2 \le 1, 0 \le z \le 1 \) oriented by the outwards normal.

Answer: \dfrac{\pi}{2}


Question: Twenty people are sitting around a circular table. In how many ways can you choose three people such that no two of them are neighbors?

Answer: 800


Question: Compute the limit: \[ \lim_{x\rightarrow 0} \frac{x^{\sin x} -1}{x} \]

Answer: -\infty


Question: Evaluate the limit using Taylor series: \[ \lim_{x \to +\infty} \frac{x - \sin(x) \log(1+x)}{x^7} \]

Answer: 0


Question: Find the minimal value of $c$ such that $n^{cn}$ eventually outgrows $n!$ as $n$ increases.

Answer: 1


Question: Determine the values of \(a\) for which the series \(\sum_{n=0}^{\infty}\frac{n!2^n}{e^{an}n^n}\) is convergent.

Answer: a > \ln 2 - 1


Question: Let $g$ be a primitive root modulo a prime $p$. Suppose $m$ is a divisor of $p-1$ with $1 < m < p-1$. Determine the number of integral solutions to the congruence $x^m \equiv g \pmod{p}$. Provide your answer as a single integer.

Answer: 0


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

Answer: -\dfrac{1}{e}


Question: Given that $u_1 = x + 1$ is a solution of the differential equation \( xu'' - (x+1)u' + u = 0 \), find another linearly independent solution using the method of reduction of order.

Answer: e^x


Question: Determine the nature of the singularity at the point \( z = \pi/2 \) for the function \( f(z) = \tan^2 z \). Specifically, identify if it is a regular point, an essential singularity, or a pole, and if it is a pole, state its degree.

Answer: 2


Question: A 2D random walker starts at the origin \((0,0)\) and takes one step randomly per second. There is a square absorbing boundary defined by the points \((2,2)\), \((2,-2)\), \((-2,2)\), and \((-2,-2)\). Determine the average time it takes for the random walker to reach the absorbing boundary.

Answer: \dfrac{9}{2}


Question: Determine the maximum number of real roots of the equation \(e^x + P(x) = 0\), where \(P(x)\) is a polynomial of degree \(n\).

Answer: n + 1


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

Answer: 1


Question: Let $a_1, a_2, \dots$ be a sequence of positive integers such that for all $m < n$, $m+n$ divides $a_m + a_n$. Is it necessary that $n$ divides $a_n$ for all $n$? Justify your answer.

Answer: No


Question: Evaluate \( \frac{1}{2 \pi i} \oint_{|z|=3} \frac{f'(z)}{f(z)} \, dz \), where \( f(z) = \frac{z^2 (z-i )^3 e^z}{3 (z+2 )^4 (3z - 18 )^5} \).

Answer: 1


Question: Let $X$ and $Y$ be random variables such that $X \vert Y=y$ is normally distributed as $N(y,1)$, and $Y$ is a continuous random variable with PDF $f_Y(y)=3y^2$ for $0<y<1$ and $0$ otherwise. Find $\text{Var}(X)$. Provide your answer as a single numerical value.

Answer: \dfrac{83}{80}


Question: Let \(x_n\) be a sequence of real numbers such that \(\lim_{n\to\infty}(x_{n+3}-3x_{n+2}+3x_{n+1}-x_{n})=2\). Find \(\lim_{n\to\infty}\frac{x_n}{n^3}\).

Answer: \dfrac{1}{3}


Question: Evaluate the limit: $$\lim_{n \to \infty}\int_1^n \frac{|\sin x|}{n} \, dx$$

Answer: \dfrac{2}{\pi}


Question: Maximize the probability function \( z = \frac{y}{2x+2y} + \frac{50-y}{200-2x-2y} \) subject to the constraints: \( x+y > 0 \), \( x+y < 100 \), \( x \leq 50 \), \( y \leq 50 \), and \( x, y \geq 0 \). What is the maximum probability?

Answer: \dfrac{3}{4}


Question: Evaluate the limit: \[ \lim_{n\to\infty} 2^{-n}\sum_{k_1 = 1}^2 \sum_{k_2 = 1}^2 \cdots \sum_{k_n = 1}^2 \frac{k_1 + k_2 + \cdots + k_n}{k_1 ^2 + k_2 ^2 + \cdots + k_n ^2} \]

Answer: \dfrac{3}{5}


Question: Determine the minimum sample size required to construct a 90% confidence interval for the proportion of vegetarian recipes in a cookbook, such that the interval length is at most 0.06. Assume the cookbook contains 2311 recipes and use a conservative estimate for the proportion.

Answer: 568


Question: Evaluate the limit: $$\lim_{x \to 0 ,\, y \to 0} \frac{x^3y-xy^3}{(x^2+y^2)^{3/2}}$$

Answer: 0


Question: Given that $z_1, z_2, z_3, z_4$ are the roots of the equation $z^4 + z^3 + z^2 + z + 1 = 0$, find the least value of $\lfloor |z_1 + z_2| \rfloor + 1$. Here, $\lfloor \cdot \rfloor$ denotes the floor function.

Answer: 1


Question: Let $X_1, X_2, \ldots$ be i.i.d. Bernoulli random variables with parameter $\frac{1}{4}$, and let $Y_1, Y_2, \ldots$ be another sequence of i.i.d. Bernoulli random variables with parameter $\frac{3}{4}$. Let $N$ be a geometric random variable with parameter $\frac{1}{2}$, where $\mathrm{P}(N = k) = \frac{1}{2^k}$ for $k = 1, 2, \ldots$. Assume the $X_i$’s, $Y_j$’s, and $N$ are all independent. Compute the covariance $$\mathrm{Cov}\left(\sum_{i=1}^N X_i, \sum_{i=1}^N Y_i\right).$$

Answer: \dfrac{3}{8}


Question:
Find \( \lim\limits_{n\to\infty}x_n \) where
\[x_{n+1}=\begin{cases}\frac{x_n}{2} & \text{if } x_n \text{ is even} \\\frac{1+x_n}{2} & \text{if } x_n \text{ is odd}\end{cases}\]
and \(x_0 > 0\).

Answer: 1


Question:
Given a probability space \((\Omega, \mathcal{F}, \mathbb{P})\) where \(\Omega = (0,1]\), \(\mathcal{F} = \mathcal{B}((0,1])\), and \(\mathbb{P}\) is the Lebesgue measure \(\lambda\), consider the following random variables:

- \(X_1(\omega) = 0\) for all \(\omega \in \Omega\).
- \(X_2(\omega) = \mathbf{1}_{1/2}(\omega)\), the indicator function at \(1/2\).
- \(X_3(\omega) = \mathbf{1}_{\mathbb{Q}}(\omega)\), the indicator function for the set of rational numbers in \((0, 1]\).

It is given that \(\mathbb{P}(X_1 = X_2 = X_3) = 1\). Find the standard deviation \(\sigma(X_i)\) for each \(i = 1, 2, 3\).

Answer: 0


Question:
Let \(X\) be a continuous random variable with the probability density function \(f(x)= \frac{1}{(2+x^2)^{3/2}}\) for \(-\infty<x<\infty\). Determine \(E(X^2)\). Choose the correct option:
(a) \(0\)
(b) \(1\)
(c) \(2\)
(d) does not exist.

Answer: D


Question: Evaluate the double integral \( \int\int_{D} [x\sin(x^4+y)+1] \, dx \, dy \), where \( D \) is the circle centered at the origin with radius 1.

Answer: \pi


Question: Evaluate the triple integral \( \int_{\Omega}(x+y+z)^2xyz \, dxdydz \) over the region \( \Omega=\{ (x,y,z)\in \mathbb{R}^3:0\leq x+y+z\leq 1 , \:1\geq x\geq0, \: \: 1\geq y\geq 0, \: \:1\geq z\geq 0 \} \). Use the change of variables \( \begin{cases}x=u\\y=v\\z=w(1-u-v)\end{cases} \) to find the value of the integral. The result should be \( \frac{1}{960} \).

Answer: \dfrac{1}{960}


Question: Suppose $f_n$, $f:X\to K$ where $K$ is a finite set and $(X,d)$ is a metric space. Given that $f_n(x)\to f(x)$ for all $x\in X$ (pointwise convergence), and $d_H$ is the Hausdorff metric on subsets of $X$, determine if these conditions guarantee Hausdorff convergence of the preimages $f_n^{-1}(y)=\{x\in X : f_n(x)=y\}$, i.e., is it true that \(d_H(f_n^{-1}(y), f^{-1}(y))\to 0\) for each \(y\in K\)?

Answer: No


Question: Let $X$ be a Banach space and $(x_n)$, $(y_n)$, $(f_n)$ be bounded sequences in $X$, $X$, and $X^*$ respectively, such that $f_m(x_n) = \delta_{mn}$ for all $m, n$. Given that $\epsilon = \sum \|x_n - y_n\| < \infty$, prove that there exists an invertible operator $T: X \to X$ such that $Tx_n = y_n$ for all $n$, provided $\epsilon$ is sufficiently small.

Answer: T


Question: Evaluate whether the series \( \sum_{n=1}^{\infty} \left[ \zeta(2n) - \zeta(2n+1) \right] \) equals \( \frac{1}{2} \).

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$\lim_{n \to \infty} \frac{\sum_{k=1}^n\lfloor kx\rfloor}{n^2}$$

Answer: \dfrac{x}{2}


Question: Determine whether the following statement is true or false: If $f_i(X)$ is connected for all $i=1,2,...,n$, where $f_i: \mathbb{R}^n \to \mathbb{R}$ is defined by $f_i(x) = x_i$ for all $x = (x_1, ..., x_n) \in \mathbb{R}^n$, then $X$ is connected.

Answer: B


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

Answer: -\dfrac{1}{2}


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

Answer: \dfrac{\pi}{8}


Question: Determine whether the following statement is true or false: If $f \in L^1 (X, \mu)$, then \( \int_E |f| \, d\mu + \int_{E^c} |f| \, d\mu = \int_X |f| \, d\mu. \)

Answer: True


Question: What is the expected number of fair coin flips needed to obtain the sequence HTT using the conditional expectation technique?

Answer: 8


Question: Determine the value of $0^0$ in terms of ordinals.

Answer: 1


Question: Evaluate the limit: $$\lim_{n\rightarrow\infty}{\int\int}_{(0,\infty)\times (0,\infty)} \frac{\sin^2(nxy)xy}{e^{(x^2+y^2)}}\,dx\,dy.$$ Use convergence theorems such as the Dominated Convergence Theorem to justify your solution.

Answer: \dfrac{1}{8}


Question: Find the natural number \( n \) such that the equation \( F_{n-1} \cdot x^2 - F_n \cdot y^2 = (-1)^n \) has a solution in positive integers \( x \) and \( y \), where \( \{F_n\} \) is the Fibonacci sequence defined by \( F_1 = F_2 = 1 \) and \( F_{n+1} = F_n + F_{n-1} \) for \( n \geq 2 \).

Answer: 3


Question: Show that the ideal \((2,x)\) is maximal in \(\mathbb{Z}[x]\) by using a homomorphism \(\phi: \mathbb{Z}[x] \to \mathbb{Z}_2\). Demonstrate that \(\ker \phi = (2,x)\).

Answer: (2, x)


Question: Determine the number of elements of order 30 in the symmetric group \( S_{10} \).

Answer: 120960


Question: Find the principal value of \( \left(\left(\frac{1+\sqrt{3}i}{2}\right)^{-3}\right)^{1-i} \).

Answer: -e^{\pi}


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

Answer: 6


Question: Given the function \( f(x,y) = \frac{1}{1-xy} \), find the value of \( \frac{\partial ^8 f}{\partial x^4\partial y^4}(0,0) \).

Answer: 576


Question:
Evaluate \( \lim_{x \to -\infty} x \cdot f(x) \), where \( f(x) = \lim_{h \to 0} \frac{1}{h} \int_{x}^{x+h} \frac{dt}{t + \sqrt{1 + t^2}} \). Choose the correct answer from the following options:

(A) 0 \hspace{1cm} (B) \frac{1}{2} \hspace{1cm} (C) 1 \hspace{1cm} (D) non-existent

Answer: D


Question: Evaluate the integral \( I = \int_{|z-a|=r} \frac{e^{2z}}{(z-a)^3} \, dz \) using Cauchy's Integral Formula, given \( r > 0 \) and \( a \in \mathbb{C} \).

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


Question: Let $X$ be a group and $Y$ be a set. Suppose there is a bijection $h: X \longrightarrow Y$. Can $Y$ be endowed with a group structure such that $h$ becomes a group isomorphism? Justify your answer.

Answer: Yes


Question: Use the residue theorem to evaluate the integral \( \int_\gamma \frac{2z^2-5z+2}{\sin z}\,\mathrm{d}z \), where \( \gamma \) is the unit circle.

Answer: 4\pi i


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(\sum_{k=0}^{n-1}(\zeta(2)-H_{k,2})-H_n\right) \] where \( H_{k,2}:=\sum_{j=1}^k\frac1{j^2} \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{\sum_{i=1}^n \frac{1}{i}}{n} \]

Answer: 0


Question: Determine the values of \(a \in \mathbb{R}\) for which the integral \(\int_{0}^{\infty}\frac{x^a}{x+1}dx\) converges.

Answer: (-1, 0)


Question: Determine the values of $p$ for which the series \( \sum_{n=1}^{\infty}\left( \sqrt[n]{n}-1 \right)^p \) converges.

Answer: p > 1


Question: Evaluate the integral: $$\int_0^\infty \left( \frac{x^{10}}{1+x^{14}} \right)^{2} \, dx$$

Answer: \dfrac{\pi}{28}


Question: Determine the number of conjugacy classes in the dihedral group $D_8$. Use the formula for the number of conjugacy classes given by \( \frac{1}{|G|}\sum|C_{G}(g)| \), where \( C_{G}(g) = \{h \in G ; gh=hg\} \).

Answer: 5


Question: Evaluate the integral \( \int_{0}^{2} g(x) \, dx \), where \( g(a) \) is the solution to the equation \( x^5 + x = a \).

Answer: \dfrac{4}{3}


Question: Determine the Löwenheim number of the extended first-order logic $L(Q^c)$, which includes Chang's quantifier $Q^c$. What is this Löwenheim number?

Answer: \aleph_1


Question: Given that \( A^4 = -1.6A^2 - 0.64I \), find \( \lim_{m \to \infty} A^m \).

Answer: 0


Question:
Evaluate the integral \( \int_{1}^{2} \frac{1}{x} \, dx \) using the limit as a sum method. Use the formula:
\[ \int_{a}^{b} f(x) \, dx = \lim_{h \to 0} h \times \sum_{r=1}^{n-1} f(a + rh) \]
where \( nh = b-a \). Here, \( a = 1 \), \( b = 2 \), and \( f(x) = \frac{1}{x} \).

Answer: \ln 2


Question: Four points in order \(A, B, C, D\) lie on a circle. The extension of \(AB\) meets the extension of \(DC\) at \(E\), and the extension of \(AD\) meets the extension of \(BC\) at \(F\). Let \(EP\) and \(FQ\) be tangents to the circle with points of tangency \(P\) and \(Q\) respectively. Given that \(EP = 60\) and \(FQ = 63\), find the length of \(EF\).

Answer: 87


Question:
Determine the intersection set \( E \) of the family of planes given by the equation \( n^2x + (2n-1)y + nz = 3 \) for all natural numbers \( n \). Choose the correct option for \( E \):

- \( E = \varnothing \)
- \( E \) is the plane \( x+y+z=3 \)
- \( E \) is the line \( \begin{cases} x+y+z=3, \\ y=-3 \end{cases} \)
- \( E \) is the point \( (0,-3,6) \)

Answer: D


Question: Find an example of an $S$-composite number from the set $S = \{1, 5, 9, \ldots\}$, which can be expressed as a product of $S$-primes in more than one way. Here, $S$-primes are numbers in the set with only $1$ and themselves as factors, and $S$-composites are the rest.

Answer: 441


Question: Let \( f \) be an endomorphism on \( \mathbb{R}^3 \) such that \(-1\) is the only eigenvalue. Given the basis \( B = \{v, u, w\} \) where \( f(v) = -v \) and \( f(u) = -u \), find the determinant of the matrix associated with \( f \) with respect to the basis \( B \).

Answer: -1


Question: Compute the limit of \((e^x + x)^{1/x}\) as \(x\) approaches zero.

Answer: e^2


Question: Find the smallest positive integer \( n \) such that \( 2^n \equiv 1 \pmod{5 \cdot 7 \cdot 9 \cdot 11 \cdot 13} \).

Answer: 60


Question: Assume that \(y=x^2 \ln (x)\) is a solution to the differential equation \( x^2y'' +axy' +by=0\). What is the value of \(a+2b\)?

Answer: 5


Question: Find the derivative \( y' \) of the function \( y = \frac{(1+2x)^{1/2} \cdot (1+4x)^{1/4} \cdot (1+6x)^{1/6} \cdots (1+100x)^{1/100}}{(1+3x)^{1/3} \cdot (1+5x)^{1/5} \cdot (1+7x)^{1/7} \cdots (1+101x)^{1/101}} \) at \( x = 0 \).

Answer: 0


Question: Let $F:M\rightarrow N$ be a map between orientable compact connected $n$-manifolds such that $F^{-1}(\partial N)=\partial M$. The degree of $F$, $\deg(F)$, is given by the equation $$F_{\#}([M])=\deg(F)[N],$$ where $F_{\#}:H_n(M,\partial M)\rightarrow H_n(N,\partial N)$ is the homomorphism induced by $F$ in the $n$-dimensional relative homology groups, $[M]\in H_n(M,\partial M)$ and $[N]\in H_n(N,\partial N)$ are the chosen fundamental classes of $M$ and $N$. Consider $S$ and $\Sigma$ orientable compact connected hypersurfaces in $M$ and $N$, respectively, such that $\partial S=S\cap \partial M$ and $\partial\Sigma=\Sigma\cap \partial N$. Determine if it is true that, if $\deg(F)=1$, $F(S)=\Sigma$ and $F^{-1}(\partial\Sigma)=\partial S$, then $\deg(f)=1$, where $f=\left.F\right|_S$. Provide a justification for your answer.

Answer: 1


Question:
Determine if the determinant of the transformation matrix from Cartesian to orthogonal curvilinear coordinates is equal to 1. The transformation is given by:
\[
\begin{bmatrix}
\hat e_1\\
\hat e_2
\end{bmatrix}
= \begin{bmatrix}
\frac{1}{h_1} \frac{\partial x_1}{\partial y_1} & \frac{1}{h_1} \frac{\partial x_2}{\partial y_1}\\
\frac{1}{h_2} \frac{\partial x_1}{\partial y_2} & \frac{1}{h_2} \frac{\partial x_2}{\partial y_2}
\end{bmatrix} \times
\begin{bmatrix}
\hat i_1\\
\hat i_2
\end{bmatrix}
\]
where
\[h_1 = \sqrt{ \left ( \frac{\partial x_1}{\partial y_1} \right )^2 + \left ( \frac{\partial x_2}{\partial y_1} \right )^2}, \quad h_2 = \sqrt{ \left ( \frac{\partial x_1}{\partial y_2} \right )^2 + \left ( \frac{\partial x_2}{\partial y_2} \right )^2}\]
Is the determinant of this matrix equal to 1 for orthogonal coordinate systems?

Answer: 1


Question: Determine whether the following statement is true or false: If $0 \leq A \leq B$ for positive definite real matrices $A$ and $B$, does it imply $0 \leq B^\dagger \leq A^\dagger$ where $\dagger$ denotes the Moore-Penrose inverse?

Answer: True


Question: Let \( \mathcal{L} \) be the first-order language of the theory of rings. Let \( A \) and \( B \) be two infinite fields such that \( A \) is existentially closed in \( B \). We know that \( A \) is existentially closed in \( B \) if and only if \( B \) can be embedded in an \( \mathcal{L} \)-structure \( B^* \) which is an elementary extension of \( A \). Since \( A \) is existentially closed in \( B \), \( A \) is also algebraically closed in \( B \). The proposition also tells us that \( A \) is algebraically closed in \( B^* \) since \( B^*/A \) is a regular extension. Is \( B \) necessarily algebraically closed in \( B^* \)?

Answer: No


Question: Let \( f = (x^4 - 1)(x + 2) \) and \( g = (x^2 - 4)(x + 1) \). Find a single polynomial that generates the ideal \( I = (f,g) \) in \( \mathbb{R}[x] \).

Answer: x^2 + 3x + 2


Question: Two players, A and B, take turns rolling a fair six-sided die, with A rolling first. To win, a player must roll a 4 immediately after the other player rolls a 1. What is the probability that player A wins the game?

Answer: \dfrac{36}{73}


Question: Find the limit of the sequence \( a_n = \frac{1}{n^2} \sum_{k=1}^{n} \lfloor kx \rfloor \), where \( \lfloor a \rfloor = \max\{k \in \mathbb{Z} : k \leq a\} \) for \( a \in \mathbb{R} \).

Answer: \dfrac{x}{2}


Question: Consider two second countable Hausdorff topological spaces $A$ and $B$, and let $M = A \times B$. If $M$ is an $n$-manifold with $n \geq 0$ finite, does it follow that $A$ and $B$ are $k$- and $l$-manifolds with $k+l = n$?

Answer: Yes


Question: Find the residue of the function \( \cot(z) \) at its poles \( z = n\pi \), where \( n \) is an integer.

Answer: 1


Question: In triangle ABC, from vertex A, draw perpendiculars AM and AP to the bisectors of the exterior angles at vertices B and C, respectively. If the perimeter of triangle ABC is 10, find the length of segment PM.

Answer: 5


Question: Evaluate the integral: $$\displaystyle \int_{-\infty}^{\infty} \frac{\cos x}{x^2+1}~dx$$

Answer: \dfrac{\pi}{e}


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(0) = 1$ and $f(x) - f\left(\frac{x}{2}\right) = \frac{3x^2}{4} + x$. Find $f(3)$. \( \text{Provide your answer in the form } \boxed{\text{answer}}. \)

Answer: 16


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0} \frac{\exp (\arcsin (x)) - \exp (\sin (x))}{\exp (\arctan (x)) - \exp (\tan (x))} \]

Answer: -\dfrac{1}{2}


Question: Calculate the integral \( \int_D dx \wedge dy \) over the manifold with boundary \( D = \{ x \in \mathbb{R}^2 : \Vert x \Vert \leq 1 \} \) using the definition of the integral for manifolds. Assume the identity map as a chart for \( D \).

Answer: \pi


Question:
Suppose $A$ and $B$ are two square matrices such that the largest eigenvalue of $AB-BA$ is positive. What must be the smallest eigenvalue of $AB-BA$?
(a) must be positive
(b) must be negative
(c) must be $0$
(d) is none of the above

Answer: B


Question: Evaluate the limit: \[ \lim_{s \to 0} \sin(s)\,\Gamma(s) \]

Answer: 1


Question: Find the directional derivative of the function \( f(x,y)=\frac{x}{x^2+y^2} \) at the point \((1,2)\) in the direction of the path \( \textbf{r}(t) = (t^3, 2t) \). Round your answer to the nearest three decimal places. (Hint: The direction of the path \( \textbf{r}(t) \) at each \( t \) is given by \( \textbf{r}'(t) \).)

Answer: 0.011


Question: Evaluate the limit: \[ \lim_{n \to \infty} n \sin(2\pi\sqrt{1+n^2}) \] where \( n \in \mathbb{N} \).

Answer: \pi


Question: Find the value of \( \alpha \) that minimizes the condition number \( k(A) = \Vert A \Vert \Vert A^{-1} \Vert \) for the matrix \( A = \begin{bmatrix} 0.1\alpha & 0.1\alpha \\ 1 & 1.5 \end{bmatrix} \), assuming the \( \infty \)-norm.

Answer: 12.5


Question: Compute the limit \( \lim_{a \to \infty} \operatorname{erf}(a+z) \) for a given \( z \in \overline{\mathbb{C}} \). Can you conclude that this limit is equal to 1 by considering the behavior of \( \operatorname{erf}(a) \) as \( a \to \infty \)?

Answer: 1


Question: Evaluate the definite integral: \[ \int_{0}^{1}\left(2x\sin \frac{1}{x}-\cos \frac{1}{x}\right)dx \]

Answer: \sin 1


Question: Determine the value of \( \lambda \) such that the line given by the equations \( 3\lambda x - 2 = 3y + 1 = \lambda z \) and the plane given by the equation \( \lambda x - 3y + 2z - 3 = 0 \) do not intersect.

Answer: 3


Question: Determine whether the following statement is true or false: If $x_n \to x$ in $L^1(X,H)$, where $H$ is a Hilbert space, then $\langle x_n, h \rangle \to \langle x, h \rangle$ in $L^1(X,\mathbb{R})$ for any fixed $h \in H$. Justify your answer.

Answer: True


Question: Find the smallest positive integer \( n \) such that \( 3^n \equiv 1 \pmod{1000000007} \).

Answer: 500000003


Question: For the function \( f(x) = \ln(x^2 + \alpha x + \alpha) \), where \( \alpha \) is in the interval (0, 4), determine the value of \( \alpha \) for which the global minimum of \( f(x) \) is maximized.

Answer: 2


Question: Evaluate the limit \( \lim _{n\to \infty}\left(1-\left(\frac{y}{n}\right)^n\right)^n \), where \( y \) is a positive constant.

Answer: 1


Question: Given independent random variables $X$ and $Y$ with density functions $f_X(x) = \frac{1}{2\sqrt{x}}$ for $x \in [0,1]$ and $f_Y(y) = \frac{1}{2\sqrt{y}}$ for $y \in [0,1]$, calculate the probability $P(S \leq 1)$ where $S = X + Y$. Use the convolution of probability distributions to find your answer.

Answer: \dfrac{\pi}{4}


Question: Determine the greatest common divisor (gcd) of the set \( \{0\} \) in an integral domain.

Answer: 0


Question: Determine whether the infinite product \( \prod\limits_{n=0}^{\infty} (1+x^{2^{n}}) \) converges.

Answer: |x| < 1


Question: Let $m$ be an odd prime and $t$ an integer. Suppose $a$ is an integer such that $\gcd(a, tm) = 1$ and $a < tm$. For an integer $b$ with $1 \leq b \leq m-1$, determine if $\gcd(a + btm, tm^2) = 1$.

Answer: 1


Question: Calculate the number of permutations of the letters ABCDEFGHI such that none of the sequences ABC, DEF, or GHI appear consecutively in the permutation.

Answer: 348114


Question: Determine whether the equation \(x^2 + y^2 - \sqrt{2}z^2 = 0\) has solutions in the ring \(\mathbb{Z}[\sqrt{2}]\).

Answer: No


Question: The expression \( i^{i^{i^2}} \) can be written as \( e^\alpha \) for some real number \( \alpha \in [0, 2\pi) \). Determine the value of \( \alpha \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the line integral \( \int_C (y-z) \, dx + (z-x) \, dy + (x-y) \, dz \) along the curve \( C \), which is the intersection of the cylinder \( x^2 + y^2 = a^2 \) and the plane \( \frac{x}{a} + \frac{z}{b} = 1 \). Use the parametrization \((a\cos t, a\sin t, b-b\cos t)\) for the curve \( C \).

Answer: -2\pi a(a + b)


Question: Determine whether the following statement is true or false: If $Y = X'\beta + e$, $\mathbb{E}[e|X] = 0$, and $\mathbb{E}[e^{2}|X] = \sigma^{2}$, then $e$ is independent of $X$. Provide a justification for your answer.

Answer: False


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{1}{xe^{ \frac{1}{2x^2}}}.$$

Answer: 0


Question: Determine the number of inequivalent spin structures on the manifold \( M = \#_{k}(S^2 \times S^1) \).

Answer: 2^k


Question: Find all solutions in prime numbers \((p, q, r)\) such that \(p^4 + q^4 + r^4 + 119 = s^2\) for some integer \(s\).

Answer: (2, 3, 5)


Question: Prove that \(\lim_{x \to \infty} \frac{e^{e^{x}}}{x!} = \infty\).

Answer: \infty


Question: Let \( P \in \mathbb{C}[X] \) be a polynomial and \( a \in \mathbb{C} \) such that \( P(a) \neq 0 \). Assume that \( a \) is a root of order \( k \) of \( P - P(a) \). Prove that for \( \rho > 0 \) small enough, there exist \( 2k \) points on a circle centered at \( a \) with radius \( \rho \) such that \( |P(z)| = |P(a)| \).

Answer: 2k


Question: What is the relative density of abundant numbers among the positive integers?

Answer: 0.2476


Question: Determine whether a meromorphic function on the entire complex plane with at most one pole can have infinitely many fixed points.

Answer: Yes


Question: Let $(X, \mathcal{A}, \mu)$ be a measure space and $(A_n)_{n \in \mathbb{N}}$ be a sequence of measurable sets. Show that if $\sum_{n \in \mathbb{N}} \mu(A_n) < \infty$, then \[ \int_X \limsup_{n \to \infty} \mathbb{1}_{A_n} \, d\mu = 0. \]

Answer: 0


Question: Determine whether the following statement is true or false: Let $\phi_t(x)$ be a solution of $x'=f(x)$ where $x\in\mathbb{R}^n$, $ f:\mathbb{R}^n\to\mathbb{R}^n$, and $f\in C^1.$ If $\lim_{t\to\infty}{\phi_t(x)}=x_0$, then $f(x_0)=0.$

Answer: True


Question: Evaluate the line integral \( \int_0^\pi F \cdot dr \) for the vector field \( F(x,y)=\langle 2y^2+y^2\cos(xy^2),4xy+2xy\cos(xy^2)\rangle \) along the path \( r(t)=\langle e^t\sin(t),e^t\cos(t)\rangle \), where \( t \in [0,\pi] \).

Answer: 0


Question: If a real matrix $A$ of size $n \times n$ has each eigenvalue with an algebraic multiplicity of 1, can we conclude that $A$ has exactly $n$ distinct eigenvalues and is diagonalizable?

Answer: Yes


Question: Let $V$ be a $K$-vector space with basis $B = (v_1, v_2, v_3, v_4)$ and let $f \in \text{End}(V)$ be a linear transformation defined by $f(v_i) = v_{i+1}$ for $i = 1, 2, 3$ and $f(v_4) = v_1$. Calculate the characteristic polynomial $P_f(t)$ of the endomorphism $f$. \( \text{Note: } M^B_B(f) \text{ is the transformation matrix of } f \text{ with respect to the basis } B. \)

Answer: t^4 - 1


Question: Find the residues of the function \( f(z) = e^z \csc^2(z) \) at all its poles in the finite plane.

Answer: e^{n\pi}


Question: Find a 6-digit number with distinct digits such that moving the last digit to the front results in a number that is 5 times the original number.

Answer: 142857


Question: Find the value of the nested radical expression: \[ \sqrt{7+\sqrt{7-\sqrt{7+\sqrt{7-\cdots}}}} \]

Answer: 3


Question: Find the smallest positive integer \(a\) such that \(5n^{13} + 13n^5 + a(9n) \equiv 0 \pmod{65}\).

Answer: 63


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{\ln(1+x^{2018})-\ln^{2018}(1+x)}{x^{2019}}.$$

Answer: 1009


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{(\cos x)^{1/2}-(\cos x)^{1/3}}{(\sin x)^2} \]

Answer: -\dfrac{1}{12}


Question: Is the entropy of a probability measure equal to the entropy of its push-forward by a measurable map? Let \( \mu \) be a probability measure on \( \mathbb{R}^d \) with a density, and let \( f: \mathbb{R}^d \to \mathbb{R}^d \) be a measurable map. Define the push-forward measure \( f_{\#}\mu \) and the entropy of \( \mu \) as \( H(\mu) = \int \mu \log \mu \, dx \). Is it true that \( H(\mu) = H(f_{\#}\mu) \)?

Answer: No


Question:
Given that $b = 9$, find the value of $a$ such that:
\[ \frac{2}{3}\log_b{a} + \frac{3}{5}\log_c{b} + \frac{5}{2}\log_a{c} = 3 \] where $a$, $b$, and $c$ are real numbers each greater than 1.

Answer: 27


Question: Evaluate the triple integral \( \iiint_\Omega{\sqrt{x^2+y^2+z^2} \, dx \, dy \, dz} \) where \( \Omega = \{(x, y, z) \mid x^2 + y^2 + z^2 \le x\} \) by converting to spherical coordinates.

Answer: \dfrac{\pi}{10}


Question: Compute the integral of \( \tanh(z) \) over the circle of radius 12 centered at \( z_0 = 0 \). Identify the poles of \( \tanh(z) \) and determine which poles lie inside the circle to find the residues needed for the integral.

Answer: 16\pi i


Question: Evaluate the integral: \[ \int_0^1 \left((1-x^7)^{1/3} - (1-x^3)^{1/7}\right) \, dx \]

Answer: 0


Question: Let $A$ be a Banach algebra, and suppose $e \in A$ such that $e^2 = e$ and $eAe$ is a division algebra. Define the operators $T_e: A \to A$ by $T_e(a) = eae$ and $S_e: Ae \to Ae$ by $S_e(ae) = eae$. If $S_e$ is not a weakly compact operator, is $T_e$ also not a weakly compact operator?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \to \infty}\dfrac{\log (1 - n + n^2)}{\log (1 + n + n^{10})^{1/3}}.$$

Answer: \dfrac{3}{5}


Question: Evaluate the limit \( \lim_{R \to \infty} \int_{[1+iR, 1+2iR]} \frac{e^{z^2}}{z^2} \, dz \). Determine whether this limit is equal to zero.

Answer: 0


Question: Find the limit \( \lim\limits_{x\to 0}\left(\frac{\sin(x)}{x}\right)^{\frac{\sin(x)}{x-\sin(x)}} \).

Answer: \dfrac{1}{e}


Question: Calculate the area bounded by the inverse of the function \( f(x) = x^3 + 3x + 2 \) between \( x = -2 \) and \( x = 6 \).

Answer: \dfrac{9}{2}


Question: Given the curve \( y = 5^{(x-2)} - \frac{1}{25} \), bounded by \( y = 5 \) and \( x = 0 \), find the height \( x \) at which the volume of the solid formed by rotating the curve about the y-axis is half of the total volume. The total volume is given by \( \int_0^5 \pi \left( \log_5(y + \frac{1}{25}) + 2 \right)^2 \, dy \approx 95.976 \). Determine the height \( x \) such that \( \int_0^x \pi \left( \log_5(y + \frac{1}{25}) + 2 \right)^2 \, dy = \frac{95.976}{2} \).

Answer: 2.72


Question: Evaluate the limit: \[ \lim_{x\rightarrow \infty  } \lim_{n\rightarrow \infty} \sum_{r=1}^{n} \frac{\left [ r^2(\sin x)^x \right ]}{n^3} \]

Answer: 0


Question:
Determine all points \( a \) at which the function \( h(x) \) is differentiable, where:
\[ h(x) = \begin{cases} x^3 - x^2, & \text{if } x \in \mathbb{Q} \\ 2x - 2x^2, & \text{otherwise} \end{cases} \]

Answer: \emptyset


Question: Evaluate the integral \( \int_C \omega \), where \( C \) is the meridian determined by the plane \( x+y+z = 0 \) on the sphere \( \mathbb{S} \subset \mathbb{R}^3 \) of radius \( \rho \) centered at the origin, and \( \omega = (y+z)\mathrm{d}x + (x+z)\mathrm{d}y + (x+y)\mathrm{d}z \). Determine if the integral vanishes.

Answer: 0


Question: Calculate the determinant of the matrix \( D = \begin{pmatrix}\cos a & \sin a & \cos a & \sin a \\ \cos 2a & \sin 2a & 2\cos 2a & 2\sin 2a\\\cos 3a & \sin 3a & 3\cos 3a & 3\sin 3a\\ \cos 4a & \sin 4a & 4\cos 4a & 4\sin 4a\end{pmatrix} \).

Answer: 0


Question: Let $G=(L\cup R,E)$ be a bipartite graph with $n$ nodes on each side. Determine the maximum value of $n$ such that if any two vertices in $L$ have at least two neighbors in $R$, then $G$ has a perfect matching.

Answer: 2


Question: Calculate the double integral \( \iint\limits_H \, (x+y) \, \mathrm{d} A \), where \( H \) is the area of the cardioid given by \( r=1+\cos(\theta) \).

Answer: \dfrac{5\pi}{4}


Question: Find the closure of the set \( \{ z \in \mathbb{C} : |z| \in \mathbb{Q} \} \) in the complex plane.

Answer: \mathbb{C}


Question: Calculate the limit: \[ \lim_{x \to 0^+} \frac{3e^{\log^3x}\log^2x}{x} \]

Answer: 0


Question: Evaluate the triple integral \( \int_{-2}^{1} \int_{-2}^{1} \int_{-2}^{1} \frac{x^2}{x^2+y^2+z^2} \,dx \,dy \,dz \).

Answer: 9


Question: Find the limit as \( n \to \infty \) of \( \left(1 + \frac{2}{n}\right)^n \).

Answer: e^2


Question:
Determine the cardinality of the set \( S = \{ T: \mathbb{R}^3 \to \mathbb{R}^3 \mid T \text{ is a linear transformation with } T(1,0,1)=(1,2,3), T(1,2,3)=(1,0,1) \} \). Choose the correct option:

A. a singleton set
B. a finite set containing more than one element
C. a countably infinite set
D. an uncountable set

Answer: D


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

(a) $1$
(b) $-1$
(c) $i$
(d) $-i$

Answer: B


Question:
Given that \( \det
\begin{pmatrix}
  a & x & x & x \\
  x & b & x & x \\
  x & x & c & x \\
  x & x & x & d
\end{pmatrix}
=f(x)-xf'(x) \), where \( f'(x) \) denotes the derivative of \( f(x) \) with respect to \( x \), determine the function \( f(x) \). Choose the correct option:

(a) \( (x-a)(x-b)(x-c)(x-d) \)
(b) \( (x+a)(x+b)(x+c)(x+d) \)
(c) \( 2(x-a)(x-b)(x-c)(x-d) \)
(d) \( 2(x+a)(x+b)(x+c)(x+d) \)

Answer: A


Question: Consider a graph $G = (V, X)$ with $n$ vertices, where exactly $n-1$ vertices have odd degree. Determine the number of vertices of odd degree in the complement graph $\overline{G}$.

Answer: n - 1


Question: Find the limit of the function \( f(x,y) = (x^2+y^2)\log|x+y| \) as \((x,y) \rightarrow (0,0)\).

Answer: 0


Question:
Let X be a real normed space with finitely many non-zero terms, equipped with the supremum norm. Consider the linear operator \( T: X \to X \) defined by \( T(x_1, x_2, x_3, \ldots) = (x_1, \frac{x_2}{4}, \frac{x_3}{9}, \ldots) \). Determine which of the following statements is true:

1. T is bounded but \( T^{-1} \) is not bounded.
2. T is not bounded but \( T^{-1} \) is bounded.
3. Both T and \( T^{-1} \) are bounded.
4. Neither of the two is bounded.

Select the correct option.

Answer: 1


Question: Determine whether the statement \( \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n}=L \implies \lim_{n \rightarrow \infty} \sqrt[n]{a_n}=L \) is true. If it is not true, provide a counterexample.

Answer: True


Question: Determine whether the following statement is true or false: If any rearrangement of the series \( \sum_{n} a_n \) converges, then the series \( \sum_{n} a_n \) converges absolutely.

Answer: True


Question: Consider a hypergraph $H = (V, E)$ where each edge $e \in E$ has more than one element and finitely many elements. Define a weak coloring $c: V \to \kappa$ such that for all $e \in E$, the restriction $c|_e$ is not constant. Define a strong coloring $c: V \to \kappa$ such that for all $e \in E$, the restriction $c|_e$ is injective. Let $\chi_w(H)$ be the smallest cardinal for a weak coloring and $\chi_s(H)$ for a strong coloring. Is it possible that $\chi_w(H) < \aleph_0$ but $\chi_s(H) \geq \aleph_0$?

Answer: Yes


Question: Determine whether \(\mathrm{SO}(2)\) is a subgroup of \(\mathrm{SU}(2)\).

Answer: Yes


Question: Calculate the remainder when $73^{1567}$ is divided by 990 using the Chinese Remainder Theorem.

Answer: 127


Question:
A rod AB of length $l$ slides along two lines intersecting at point P and inclined at an angle $\theta$. Determine the nature and area of the locus of the circumcenter of triangle PAB. Choose the correct option:

(A) The locus of the circumcenter of triangle PAB is a circle.
(B) The area of the locus of the circumcenter of triangle PAB is $\frac{1}{8}l^2\theta\ \csc^2\theta$.
(C) The locus of the circumcenter of triangle PAB is the major segment of a circle.
(D) The area of the locus of the circumcenter of triangle PAB is $\frac{1}{2}l^2\theta\ \csc^2\theta$.

Select the correct option.

Answer: A


Question: If \(U\subset\mathbb{R}^n\) is an open and contractible subset with a continuous function \(f: U \to \mathbb{R}\) that has only one minimum and whose level curves are connected by paths, is \(U\) homeomorphic to \(\mathbb{R}^n\)?

Answer: Yes


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

Answer: \ln 3


Question: Evaluate the limit: \[ \lim\limits_{x \to 0}\frac{\int_0^{x^2}\cos t^2\,dt}{(\sin x)^2}. \]

Answer: 1


Question: Find the residue of the function \( z^3 \cosh \left( \frac{1}{z} \right) \) at \( z = 0 \).

Answer: \dfrac{1}{24}


Question: Calculate the limit using the Maclaurin series: \[ \lim_{x \to 0}\frac{1-(\cos{x})^{\sin{x}}}{x^3} \]

Answer: \dfrac{1}{2}


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

Answer: \dfrac{1}{3}


Question: Approximate \( \text{erf}(1) \) using the series expansion \( \frac{2}{\sqrt{\pi}} \sum_{k=0}^{\infty} \frac{(-1)^{k}1^{2k+1}}{(2k+1)k!} \) to within \( 10^{-7} \).

Answer: 0.8427008


Question: Calculate the maximum value of \[ \frac{bc}{(b + c)^3(a^2 + 1)} + \frac{ca}{(c + a)^3(b^2 + 1)} + \frac{ab}{(a + b)^3(c^2 + 1)} \] where \(a, b, c\) are positive real numbers satisfying \(abc = 1\).

Answer: \dfrac{3}{16}


Question: Determine the topology of the quotient space \((S^1 \times S^1)/C_n\), where a cyclic group \(C_n\) of order \(n\) acts on each \(S^1\) by rotation. The actions may be distinct and may not be faithful.

Answer: S^1 \times S^1


Question: Evaluate the limit: $$ \lim_{x\to 1}\dfrac{\sqrt{x+1}+\sqrt{x^2-1}-\sqrt{x^3+1}}{\sqrt{x-1}+\sqrt{x^2+1}-\sqrt{x^4+1}} . $$

Answer: \sqrt{2}


Question:
Given the constraints $0 < x \le 1 \le y$, calculate the minimum value of the expression:
\[ \frac{x}{y + 1} + \frac{y}{x + 1} + \frac{x^2 + 3y^2 + 2}{6xy(xy + 1)}. \]

Answer: \dfrac{3}{2}


Question: Find an integer value of $s$, other than square $s$ and $s=3$, such that the equation \( p^2-3(108s^6t^6-1)q^2=s \) has a non-trivial solution in integers $p$, $q$, and $t$.

Answer: 7


Question: Evaluate the limit: \[ \lim_{n\to \infty}{n\left[\left(1+\frac{c}{n}\right)^{n}-\left(1-\frac{c}{n}\right)^{-n}\right]} \] without using L'Hopital's Rule.

Answer: -c^2 e^c


Question: Find the value of \((65!)^2\) modulo 131.

Answer: 1


Question:
Find the distance between the two skew lines given by the parametric equations:

Line 1: \(x = 2 + t, \; y = 1 + 6t, \; z = 2t\)

Line 2: \(x = 1 + 2s, \; y = 6 + 14s, \; z = -1 + 5s\)

Answer: 3


Question: Find the gradient of the function \( f(u) = u^{T}Au - 2ub \) with respect to the vector \( u \), where \( u \) is an \( n \times 1 \) vector, \( A \) is an \( n \times n \) symmetric positive definite matrix, and \( b \) is an \( n \times 1 \) vector.

Answer: 2Au - 2b


Question: Determine whether the statement is true: If integers \(x\) and \(y\) satisfy the equation \(4x^3 = y^2 + 27\), then \(y\) must be a multiple of 3.

Answer: True


Question: Consider the sequence defined by $a_0 = 1$, $a_1 = 1$, and $a_{n+1} = a_n a_{n-1} + 1$ for $n \geq 1$. Determine whether $a_{2007}$ is divisible by 4.

Answer: No


Question: Determine whether there exists a linear mapping \( \psi: \mathbb{R}[x] \rightarrow \mathbb{R}[x] \) such that \( \psi(x) = x^2 \) and \( \psi(x^2) = x \), where \( \mathbb{R}[x] \) is the vector space of real polynomials. Answer true or false.

Answer: True


Question: Find all complex numbers \( z \) that satisfy the equation \( z^2 + z|z| + |2z|^2 = 0 \).

Answer: 0


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that for all $x \in \mathbb{R}$, the limit \[ \lim_{h \to 0} \frac{f(x+h) - f(x-h)}{h} \] exists. Does this imply that $f$ is differentiable on $\mathbb{R}$?

Answer: No


Question: Find the value of \( \lfloor (\sqrt{21}+\sqrt{29})^{1984}\rfloor \mod 100 \).

Answer: 71


Question: Let $P=\langle 7, 5 +\sqrt{-10}\rangle$ be an ideal in $\mathbb{Z}[\sqrt{-10}]$. Determine if $P \cap \mathbb{Z} = 7\mathbb{Z}$.

Answer: P \cap \mathbb{Z} = 7\mathbb{Z}


Question: Evaluate the line integral \( \int_{C}{\frac{-x}{x^2+y^2}dx+\frac{y}{x^2+y^2}dy} \) where the curve \( C \) is parameterized by \( x = \cos t \) and \( y = \sin t \) for \( 0 \le t \le \frac{\pi}{2} \).

Answer: 1


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

Answer: 3


Question: Find all pairs of positive integers \((a, b)\) such that \(13^a + 3 = b^2\).

Answer: (1, 4)


Question: Find the maximum possible product of two positive integers whose digits form a permutation of \( \{1, 2, 3, 4, 5, 6, 7, 8\} \), using each digit exactly once.

Answer: 65193902


Question: Let $M$ be a smooth manifold homeomorphic to a torus. Suppose that $M$ admits a smooth transitive action by a finite dimensional Lie group $G$. Is $M$ necessarily diffeomorphic to the standard torus $T^n \cong \mathbb{R}^n/\mathbb{Z}^n$?

Answer: Yes


Question: Find the point of intersection of the functions \( y = e^{-x} \) and \( y = x \).

Answer: (0.567, 0.567)


Question: Find the point of intersection $P_3$ of the plane tangent to the sphere $x^2+y^2+z^2 = 6$ at the point $P_1(1, 1, 2)$ and the normal line to the ellipsoid $2x^2 + 3y^2 + 4z^2 = 5$ at the point $P_2(1, -1, 0)$. Determine the coordinates of $P_3$.

Answer: (-11, 17, 0)


Question: Find the best approximation of the function \(x^3 + x^2 + e^{-x}\) in the subspace \(G\) of \(C[-1,1]\), where \(G = \{ g \in C[-1,1] \mid g(x) = g(-x) \text{ for all } x \}\). Assume \(C[-1,1]\) is equipped with the standard \(L^2[-1,1]\) inner product.

Answer: x^2 + \cosh(x)


Question:
Consider the operator norm of the following $n \times n$ matrix:
$$
\|(I + 11^* + X^*X)^{-1} 11^* (I + 11^* + X^*X)^{-1}\|,
$$
where $X$ is an $n \times n$ matrix. Is this norm bounded by $\frac{1}{n}$?

Answer: \dfrac{1}{n}


Question: A particle is randomly positioned with a uniform distribution inside a sphere centered at the origin with a radius of 7. Calculate the expected value of the particle's distance from the origin.

Answer: \dfrac{21}{4}


Question: Determine whether the series \( \sum_{n\ge0}\frac{(2k\pi i)^n}{(n+1)!} \) can be zero for any integer \( k \neq 0 \). If the series is zero, the limit \( \lim_{z\to2k\pi i}\frac{z}{e^z-1} \) is infinite; otherwise, it is a finite complex number.

Answer: \infty


Question: Let $A$ and $B$ be commutative rings, and let $f: A \to B$ be a surjective ring homomorphism. Consider the induced group homomorphism $f^{*}: A^{*} \to B^{*}$, where $A^{*}$ and $B^{*}$ are the groups of units of $A$ and $B$, respectively. Is $f^{*}$ necessarily surjective?

Answer: No


Question: Evaluate the integral \( \int_{0}^{\infty} t^2 e^{-t} \sin(t) \, dt \).

Answer: \dfrac{1}{2}


Question: Find the norm of the linear functional \( T(f) = \int_0^2 f(x) \, dx - \int_3^{10} f(x) \, dx \) on the space \( L^2[0,10] \), where \( L^2[0,10] = \{ f : [0,10] \rightarrow \mathbb{R} \mid f \text{ is Lebesgue measurable and } \int_0^{10} f^2(x) \, dx < \infty \} \) equipped with the norm \( || f || = \left( \int_0^{10} f^2(x) \, dx \right)^{1/2} \). Determine \( || T || \).

Answer: 3


Question: If a linear operator $X$ on a Hilbert space satisfies $X^* = -X$, is $X$ necessarily equal to $0$?

Answer: No


Question: Determine the smallest integer $n$ such that for all $x \in \left(-\frac{1}{2}, \frac{1}{2}\right)$, the inequality \[ \left|e^x - \sum_{k=0}^n \frac{x^k}{k!}\right| \leq \frac{|e^x|}{10^{16}} \] holds.

Answer: 14


Question: Find the recursive formula with starting conditions for $D_n$, the number of valid binary trees with $n$ inner vertices, where a binary tree is valid if for each inner vertex $v$, the left sub-tree connected to $v$ contains an even number of inner vertices. Additionally, calculate $D_6$. Provide the recursive formula and the value of $D_6$.

Answer: 12


Question: Prove the binomial equality: \[ \sum_{k=0}^{n} \binom{2n}{2k} = 2^{2n-1} \]

Answer: 2^{2n-1}


Question: Determine the degree of the splitting field of the polynomial \( f(x) = x^3 - 5 \) over the finite field \( \mathbb{F}_7 \).

Answer: 3


Question: Determine whether there exists a finite non-abelian 2-group $G$ of order $2^9$ without a non-trivial elementary abelian direct factor, such that $$Z(G) \cap Z(\Phi(G))= \langle \prod_{i=1}^{2^d} a^{x_i} \;|\; a\in Z(\Phi(G)) \rangle,$$ where $\{x_1,\dots,x_{2^d} \}$ is a right transversal of $\Phi(G)$ in $G$. Here, $\Phi(G)$ is the Frattini subgroup of $G$, and $Z(H)$ denotes the center of a group $H$. It is known that such groups of order $2^8$ exist, with exactly 10 groups of order $2^8$ having this property.

Answer: Yes


Question: Evaluate the line integral \( \int_A (\cos x)y\,dx+(x+\sin x)\,dy \) over the ellipse \( A=\{(x,y)\in \mathbb{R}^2\mid \frac{x^2}{a^2}+\frac {y^2}{b^2}=1\} \).

Answer: \pi a b


Question:
Determine whether there exists an ordinal \( \alpha \) such that \( \alpha = \omega_{\alpha} \), where \( \omega_{\alpha} \) is defined recursively as follows:
1. \( \omega_{0} = \omega \)
2. \( \omega_{\alpha+1} = \gamma(\omega_{\alpha}) \)
3. \( \omega_{\lambda} = \sup\{\omega_{\alpha} \mid \alpha < \lambda\} \) for a non-zero limit ordinal \( \lambda \)
Here, \( \gamma(\alpha) \) is the least ordinal that cannot be injected into \( \alpha \).

Answer: Yes


Question:
Evaluate the limit of the following definite integral as \( n \to \infty \):
\[
\lim _{ n\rightarrow \infty  }{ \int _{ 0 }^{ 1 }{ \int _{ 0 }^{ 1 }{ \underbrace{\dots}_{n-3 \, \text{times}} \int _{ 0 }^{ 1 }{ \cos ^{2} { \left\{ \frac { \pi  }{ 2n } \left({x}_{1}+{ x }_{ 2 }+\dots +{ x }_{ n }\right)\right\}  \quad { dx }_{ 1 }{ dx }_{ 2 }\cdots { dx }_{ n } }  }  }  } = \dfrac{1}{2}
\]

Answer: \dfrac{1}{2}


Question: Let \( a_{10} = 10 \), and for each integer \( n > 10 \), let \( a_n = 100a_{n - 1} + n \). Find the smallest integer \( n > 10 \) such that \( a_n \equiv 0 \pmod{99} \).

Answer: 45


Question: Determine the number of semi-direct products $H \rtimes Q$ that can be constructed for $H = C_{42}$ and $Q = C_{3}$, where $C_{n}$ denotes the cyclic group of order $n$. Specifically, find the number of homomorphisms $\theta: C_{3} \rightarrow \text{Aut}(C_{42})$. Given that $\text{Aut}(C_{42}) = C_{12}$, reduce this to finding homomorphisms $\theta: C_{3} \rightarrow C_{12}$. How many such homomorphisms exist?

Answer: 3


Question: Calculate \(\lim\limits_{(x,y) \to (0,0)} \frac{\sin(2x)}{x(y^{2} - 1)} \).

Answer: -2


Question: Determine the minimum number of consecutive integers needed such that any coloring with $m$ colors contains a 3-term arithmetic progression with the first two elements being the same color.

Answer: 2m + 1


Question: Find the limit: \[ \lim_{x \to \frac{\pi}{4}} \frac{4\sqrt{2} - (\cos x + \sin x)^5}{1 - \sin 2x} \]

Answer: 5\sqrt{2}


Question: Let \(f(x) = x^4 + ax^3 + bx^2 + cx + d\) be a polynomial with real coefficients and real roots. Given that \(|f(\iota)| = 1\), find the value of \(a + b + c + d\).

Answer: 0


Question: For a connected n-manifold M, the Lie algebra of all smooth vector fields is denoted by \( \chi^{\infty}(M) \). For a point \( p \in M \), we define \( L_{p} = \{ X \in \chi^{\infty}(M) \mid X(p) = 0 \} \). \( L_{p} \) is a Lie subalgebra of \( \chi^{\infty}(M) \) with codimension equal to \( n \). Is it true that every codimension-\( n \) Lie subalgebra of \( \chi^{\infty}(M) \) is necessarily in the form of (or isomorphic to) \( L_{p} \) for some \( p \in M \)?

Answer: Yes


Question: Evaluate the integral \( I = \int_{-\infty}^{+\infty} \left(a+\frac{x}{2}\right)^2 \delta\left(a^2-(a+x)^2\right) \, dx \), where \( \delta(x) \) is the Dirac delta function.

Answer: \dfrac{|a|}{2}


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

Answer: e^{-2}


Question: In a group of $n$ pairs of girlfriends (totaling $2n$ girls) and another group of $2n$ boys, each girl must be matched with a boy such that she can defeat him at poker. For any given pair of girlfriends, the $i$th pair, both girls can defeat at least $2i - 1$ boys. Additionally, if one girl in the pair cannot defeat a boy, her partner can. Determine if it is possible to find a matching where each girl defeats her matched boy.

Answer: Yes


Question: Find the expected area of a triangle formed by three random points inside a triangle with unit area.

Answer: \dfrac{1}{12}


Question: Given non-negative numbers \(a, b, c\) such that \(a + b + c = 1\), find the maximum value of the expression \(B = ab + bc + ca - 3abc\).

Answer: \dfrac{1}{4}


Question: Let \( A, B \subseteq [n] \) be chosen uniformly at random. Compute the variance of \(|A \cup B|\).

Answer: \dfrac{3n}{16}


Question: Evaluate the limit \( \lim_{k \to \infty} \int_0^\infty e^{-kx^2} \, dx. \)

Answer: 0


Question: Find the area enclosed by the inverse function of \( f(x) = x^3 + 3x + 1 \), between \( x = -3 \) and \( x = 5 \), and the x-axis. Use techniques of finding area by definite integrals.

Answer: \dfrac{9}{2}


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

Answer: \dfrac{14}{3}


Question: Determine the largest integer $k$ such that $\Bbb{Z}^{k}$ can be embedded in $\mathcal{GL}(n,\Bbb{Z})$. What is this largest value of $k$?

Answer: n - 1


Question: Let $E$ be a finite degree extension field of $F_q$, where $q = p^n$, $p$ is a prime, and $n$ is an integer. Let $r \in E$ and $g(x)$ be the minimal polynomial of $r$ over the prime field $F_p$ with degree $d$. Determine the number of elements in the field $F_p(r)$. Express your answer as a power of $p$. \( \boxed{} \)

Answer: p^d


Question: Compute \( \lim_{n\to \infty}\int_0^{1/n}\frac{n}{1+n^2x^2+n^6x^8}dx \).

Answer: \dfrac{\pi}{4}


Question: Consider a smooth curve \( \gamma:[0,1]\to \mathbb S^2 \) on the unit sphere. Determine whether the set \( \bigcup_{t\in [0,1]}\mathrm{span}(\gamma(t))\subset \mathbb R^3 \) has a Hausdorff dimension of at most 2, and consequently, whether it has zero 3-dimensional Lebesgue measure.

Answer: Yes


Question: Determine the number of orbits of the symmetric group Sym(6) acting on itself by conjugation.

Answer: 11


Question: Find the expected value of the maximum distance of three jumps, where each jump has a probability of \( \frac{1}{3} \) of being disqualified (resulting in a distance of 0) and a probability of \( \frac{2}{3} \) of having a distance uniformly distributed over \([2, 3]\).

Answer: \dfrac{23}{9}


Question: Evaluate the integral: \[ \int_{0}^{\infty}\frac{e^{-t} - e^{-4t}}{t}\,dt. \]

Answer: \ln 4


Question: Can the sine of a complex number be greater than 1? Justify your answer.

Answer: Yes


Question: In the figure, let $P$, $Q$, and $I$ be the incenters of triangles $\triangle AHB$, $\triangle BHC$, and $\triangle ABC$, respectively. If $MN = a$, calculate the area of the shaded region $BPIQ$. Express your answer in terms of $a$.

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


Question: Let \( P(z) = z^n + a_{n-1}z^{n-1} + \ldots + a_0 \) be a polynomial with complex coefficients. Prove that if \( r > 1 + |a_{n-1}| + \ldots + 2|a_0| \), then \( |P(r)| > |P(0)| \).

Answer: |P(r)| > |P(0)|


Question:
Solve for \( n \) in the equation:
\[ \frac{\sin\left(\frac{90^\circ}{2^n}\right)}{\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\ldots\sqrt{2}}}}}_{n \text{ radicals}}} = \frac{-1+\sqrt{2}}{2} \]

Answer: 2


Question: Determine the number of distinct roots of the polynomial \( f(x) = x^{81} - 1 \) in the algebraic closure of the field \( F_2 \).

Answer: 81


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

Answer: -\ln 2


Question: Find all automorphisms of the dihedral group $D_3$ and determine the group to which $\text{Aut}(D_3)$ is isomorphic.

Answer: S_3


Question: Is the solvable radical of the commutator subgroup $G'$ of a group $G$ normal in $G$?

Answer: Yes


Question:
If $f$ and $g$ are integrable and real-valued functions on $(X, M, \mu)$, which of the following assertions is correct?

1. $fg \in L^1(\mu)$
2. $fg \in L^2(\mu)$
3. None of the above.

Answer: C


Question: Find the radius of convergence of the power series \( \sum\limits_{n = 0}^\infty \frac{(1+\frac{1}{n})^{n^{2}}}{n^{3}}z^{n} \).

Answer: \dfrac{1}{e}


Question: Let $G$ be a connected compact Lie group and $H$ be a connected Lie subgroup of $G$ such that $G$ and $H$ have the same rank. Is the dimension of the quotient space $G/H$ even?

Answer: Yes


Question: Let $a$, $b$, and $c$ be positive real numbers satisfying the following system of equations: \[ a + \frac{1}{b} = 3, \] \[ b + \frac{1}{c} = 4, \] \[ c + \frac{1}{a} = \frac{9}{11}. \] Find the value of $a \times b \times c$. \( \boxed{?} \)

Answer: 1


Question: Determine the radius of convergence of the power series \( \sum_{n\geq 0}a_{n}x^{n} \), where \( a_{0}=\frac{2}{3} \) and \( a_{n}=2-\frac{1}{2}a_{n-1} \) for all \( n \geq 1 \). It is given that \( \frac{2}{3} \leq a_{n} \leq \frac{5}{3} \) for all \( n \geq 1 \). Use the ratio test to find the radius of convergence.

Answer: 1


Question:
Find the limit of the sequence as \( n \to \infty \):
\[ \lim_{n \to +\infty} \frac{1}{\sqrt{n}}\left(1+\frac{2}{1+\sqrt{2}} + \frac{3}{1+\sqrt{2}+\sqrt{3}} + \ldots + \frac{n}{1+\sqrt{2}+\sqrt{3}+\ldots+\sqrt{n}}\right), \ n \in \mathbb{N} \]

Answer: 3


Question: Find all harmonic functions on \(\mathbb{R}^2\) of the form \(f(x^2+y^2)\), where \(f\) is a \(C^2\) class function.

Answer: C


Question: Two kings are placed on the same square of a 3x3 chessboard. Each king moves independently and randomly to any adjacent square (including diagonally) with equal probability. Determine the probability that, after a large number of moves, both kings will be on the same square.

Answer: \dfrac{1}{8}


Question: Evaluate the integral \( \oint_\Gamma \frac{\cos z}{(z-\pi)^2}dz \), where \( \Gamma \) is the circle \( |z|=1 \).

Answer: 0


Question: Evaluate the integral \( \int_C \frac{5z^4 - 3z^3 + 2i}{(z-1)^3} \, dz \), where \( C \) is any simple closed curve in a counterclockwise direction with \( z = 1 \) inside \( C \).

Answer: 42\pi i


Question: In the acute triangle $ABC$, the distance between the feet of the altitudes to sides $AB$ and $BC$ is $24$. Given that $\angle B = 37^\circ$, calculate the measure of the circumradius of triangle $ABC$.

Answer: 25


Question: Suppose $f$ is an analytic function in the region $|z|<2$ and satisfies $|f(z)|\le 2$ for all $z$ in this region, with the condition $f(1)=0$. Determine the best possible bound for $f(1/4)$. Provide your answer as a single value or expression.

Answer: \dfrac{4}{5}


Question: Determine the limit of the sequence defined by the recurrence relation \(x_{n+1} = \frac{1}{2} x_n^2 - 1\) with the initial condition \(x_1 = \frac{1}{3}\).

Answer: 1 - \sqrt{3}


Question: Calculate the Riemann-Stieltjes integral \( \int_0^4 x^2 \, d\alpha(x) \), where \( \alpha(t) = n^2 \) for \( t \in [n, n+1) \).

Answer: 170


Question: Find an integer \( n > 1 \) such that \( k^n + 1 \equiv 0 \pmod{n^2} \), where \( k \) is a prime number.

Answer: 3


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

Answer: 2


Question: Determine whether the limit \( \lim_{n \to \infty}\frac{n-H_{n}-\sum_{j\le n}\frac{\Omega\left(j\right)}{j}+\sum_{j\le n}\frac{\omega\left(j\right)}{j}}{n} \) exists, where \( n \) is a positive integer, \( H_n \) is the \( n \)th harmonic number \( (H_n = \sum_{j\le n}\frac{1}{j}) \), \( \Omega(j) \) denotes the sum of the distinct prime factors of \( j \), and \( \omega(j) \) denotes the number of distinct prime factors of \( j \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{1}{x^3}\int_0^x\frac{t^2}{t^4+1}dt \]

Answer: \dfrac{1}{3}


Question: Find the integrating factor for the differential equation: \((3x+\frac{6}{y})\mathrm{d}x+(\frac{x^2}{y}+\frac{3y}{x})\mathrm{d}y=0\). Determine if it is possible to find a common integrating factor for the entire equation.

Answer: xy


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

Answer: \dfrac{\pi}{4}


Question: In a self-complementary graph G with exactly m vertices of degree k, determine the number of vertices that must have a degree of n - k - 1.

Answer: m


Question: If $R(x)$ is a formal power series such that $[R(x)]^2 = x + 1$, must $R(x)$ be the formal power series of $\sqrt{x+1}$ or $-\sqrt{x+1}$?

Answer: Yes


Question:
Compare the values of the following two expressions:

\[ A = \frac{1.0000004}{(1.0000006)^2} \]

and

\[ B = \frac{(0.9999995)^2}{0.9999998} \]

Determine which of the two expressions is larger, or if they are equal.

Answer: A


Question: Determine whether the following statement is true or false: All critical points of a function must be either extrema or saddle points. In other words, if \( f'(x) = 0 \) or \( f'(x) = D.N.E \), there are only two possibilities: \( f(x) \) is an extrema or \( f(x) \) is a saddle point. There are no other possibilities.

Answer: B


Question: Find the product of the x-coordinates of the points of intersection of the circle \(x^2 + y^2 = 4\) and the hyperbola \(xy = 4\).

Answer: 16


Question: Evaluate \(\lim\limits_{n\rightarrow\infty}\int_0^n x^i\left(1-\frac{x}{n}\right)^n dx\) for \(i \geq 0\), and show that it equals \(i!\).

Answer: i!


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

Answer: 1